Theme

Measuring the unreachable

Nothing in this subject can be picked up, weighed or visited. Every number in it was inferred from an angle or a brightness.

Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.

Orbits

The law that links period to size, and weighs everything

Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since.

An orbit at i = 42°, Ω = 35°, ω = 55°. The three orientation elements. The orbit is generated in its own plane and rotated by the standard sequence, so the inclination, the node line and the argument of periapsis are the angles that produced the drawn curve rather than labels applied to it.

Orbits

Six numbers that fix an orbit for all time, and the sixth is the awkward one

Five of the orbital elements describe a curve that never changes. The sixth says where on it the body is, and it is the only one that has to keep being measured.

Ceres from five directions and no distance. Ceres seen five times over 41 days, from an Earth on a circular orbit, reduced in the plane. Each sighting gives a direction and no range, so the object is somewhere on its sight line; the five lines here span 1.37° of geocentric arc altogether, and Earth's own motion supplies the only baseline there is — 0.403 AU of its 0.691 AU of travel lies across the sight lines. A planar orbit is four numbers, so five angles over-determine it and one orbit comes out: a = 2.7658 AU, e = 0.0785. That is the answer and not an input — the sightings were generated at a = 2.7658 AU and e = 0.0785, and the solve, which sees only the directions and the dates, returns them to 3e-12. The two shaded sectors are what closes the determination: between the first and middle sightings the radius vector sweeps 0.2993 AU² in 21.0 days and between the middle and last 0.2853 AU² in 20.0 days, a ratio of 1.04918 against a time ratio of 1.04918. Slide all three crossings out along their sight lines together and that equality fails at once, so it fixes the distance by itself, with no propagation anywhere in the argument — and it gives a = 2.7658 AU over again. The two dashed curves are candidates that thread the same three sight lines at 85% and 108% of the recovered distance: a = 1.86 AU at e = 0.35, sweeping its areas in a ratio 4.5% wrong; and a = 5.61 AU at e = 0.49, sweeping its areas in a ratio 2.9% wrong. A few per cent in the distance is an orbit of another kind, which is the same fact the conditioning panel measures: one arcsecond of angle error moves a by 0.60% on this arc.

Orbits

Five directions and no distance among them

An image of a moving point records an angle and throws the range away, so an orbit has to be assembled out of angles alone. How many angles are needed is not a detail of the method — it is the whole of what a determination is.

One admissible root, 0.01% from the truth. Gauss's reduction of three directions to a distance, drawn as the two relations whose intersection it is. Three observations of Ceres on days 0, 20, 40 of an arc, generated from its elements and used only as sight directions — no range, no radial velocity. The rising curve is geometry: the heliocentric distance a candidate at geocentric distance ρ₂ would have, r₂² = ρ₂² + 2ρ₂(R₂·L̂₂) + R₂², which contains no dynamics at all. The falling curve is dynamics: ρ₂ = A + µB/r₂³, with A and B built from the three sight vectors, the three observer positions and the three times, and containing no orbit. Eliminating ρ₂ between them gives r₂⁸ + a r₂⁶ + b r₂³ + c = 0 — an eighth-degree equation, from a problem with exactly as many equations as unknowns. Here they cross once at a positive ρ₂, at r₂ = 2.5893 AU against the true 2.5890. The other 2 real roots are rejected not by fitting but by sign: the ρ₂ each implies is negative, and an object behind the observer was not the thing observed.

Orbits

Three observations and no orbit at all

Three directions in space give six numbers for the six elements of an orbit, which sounds like a solved problem. The algebra that solves it is of the eighth degree, and for a near-Earth asteroid three perfect observations can be consistent with three different orbits.

Mars to five metres and Neptune to five thousand kilometres, in the same file. Present-day heliocentric position uncertainty for each planet, in kilometres, with the range component marked separately below it. The two differ because a transponder measures a distance along the line of sight and says nothing about the two directions across it, so a planet with an orbiter is known radially some 17 times better than it is known altogether. Neptune is 10⁶ times less well determined than Mars and only 20 times further away, which is the whole point: the accuracy is a property of the observations, not of the geometry. Mars has carried a transponder almost continuously since 1976; Neptune has been visited once, in 1989, and everything else known about it is meridian-circle astrometry covering 1.07 of one orbit. The two ice giants are the only entries here whose ephemerides are still limited by nineteenth-century technology, and the only cure is a spacecraft.

Orbits

The table that is a fit

A planetary ephemeris is not evaluated from Kepler's laws and is not evaluated from a theory. It is a numerical integration whose starting conditions were least-squares fitted to a century and a half of observations, and its accuracy is a property of those observations rather than of the mathematics.

Orbits

A collision dated by a scatter plot

Nothing in the solar system carries a date. A collisional family does — because a force that depends on a body's size has been pushing its fragments apart ever since, so the cloud is a V whose slope is an elapsed time, and one of those dates is confirmed by fossil meteorites in Swedish limestone.

$A clock that is a straight line for three billion years and then is not. The lunar chronology function: craters of a kilometre or more per square kilometre against the age of the surface, with the count logarithmic and time not. The dashed line is the present impact rate extrapolated backwards, and it accounts for the whole curve up to 3.1 billion years — over that entire range, dating a surface is dividing a crater count by a constant. The rate itself, read off the slope of the drawn curve at the present day, is 8.4·10⁻⁴ craters per square kilometre per billion years, which over the whole Moon is about 32 new craters of a kilometre or more per million years. Past three and a half billion years the exponential term takes over and the curve turns almost vertical: ground that is 4.1 billion years old carries 26 times the crater density of ground 3.5 billion years old, for a difference in age of six hundred million years. Most of the craters on the Moon were made in a small fraction of its life, and nothing that happened after them is recorded anything like as densely. The six marked ages are laboratory measurements on returned rock, and they are what makes the curve a chronology rather than a shape — the Moon is the only body whose crater counts and whose radiometric ages have ever been measured on the same square kilometre.$ Orbits

A surface dated by counting holes in it

Every age quoted for a surface in the solar system outside the Earth — a Martian lava flow, a crater on Mercury, the ice of Europa — comes from counting craters and passing the count through one curve. That curve was calibrated on nine square kilometres of the Moon, and it is nearly a straight line for three billion years and then is not.

Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.

Gravitation

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

Mercury's perihelion, term by term. The observed advance of Mercury's perihelion is 5600 arcseconds per century against the equinox. Almost all of it is the equinox: the coordinate frame itself turns, and removing it leaves 574. Subtracting the perturbations of the other planets — computed by Le Verrier in 1859 and refined many times since — leaves 42.98 arcseconds a century that nothing in Newtonian gravitation accounts for. The bars are logarithmic in nothing; they are the real proportions, which is why the residual is barely visible beside the frame term.

Gravitation

Forty-three arcseconds, after everything else

Mercury's perihelion moves through 5,600 arcseconds a century. Nearly all of that is the coordinate system turning, and almost all of the rest is the other planets. What was left over was 43 — under one per cent of the raw number, and the most consequential residual in the history of the subject.

G: 14 determinations in two families. Published determinations of G, each with its quoted one-sigma interval, sorted into two families — torsion balance, in one form or another, against beam balance, pendulum, atom interferometry. The shaded band behind each family is that family's inverse-variance weighted mean: 6.67435 ± 0.00004 across 11 of them, against 6.67343 ± 0.00009 across 3. The difference is 0.00092 ± 0.00010 10⁻¹¹ m³ kg⁻¹ s⁻², which is 9.3 standard deviations, computed here from the quoted errors alone. The arithmetic is the same one the Hubble figure uses and here it should be distrusted, because the scatter inside each family already exceeds what the intervals allow: eleven torsion-balance determinations spread over 500 parts per million with quoted intervals of 12 to 130 cannot all be right, whatever the difference between the families comes to. That is why the recommended value's uncertainty is expanded far beyond any single experiment's rather than being the weighted combination drawn here — the disagreement is between laboratories using the same method, not between methods.

Gravitation

Nothing in the sky is weighed in kilograms

The Sun's gravitational parameter is known to eleven significant figures. The Sun's mass is known to five. The two statements are about the same object and the difference between them is a constant measured in basements, which is the worst-determined fundamental constant in physics.

An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation.

Gravitation

An average that weighs what cannot be watched

A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not.

$GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst.$ Gravitation

A distance with no ladder under it

The frequency sweep of an inspiral fixes the chirp mass with no distance in it, and the amplitude then gives the luminosity distance directly, because one expression fixes both. That is a distance measured with nothing calibrated beneath it — and its error budget is one angle.

Gravitation

A ring weighed by the wave crossing it

Saturn's rings are a few tens of metres thick and spread over an area larger than the Earth, made of pieces nobody can resolve, and nothing has ever landed on them. Their mass per unit area is nevertheless known to a few per cent — from the rate at which the crests of a wave crowd together as it travels outwards.

One measured number, and every pair of masses that produces it. The plane of the two component masses of an inspiralling binary, with three curves of constant chirp mass across it. The middle one is GW150914's value of 28.7 solar masses, and the chirp mass recovered from the coordinates of the drawn curve varies along its whole length by 7.4e-14 per cent — which is the point: every binary on that line radiates the same frequency sweep at leading order, so the early inspiral cannot tell them apart. Two of them are marked. An equal pair of 33.0 and 33.0 solar masses and a lopsided pair of 63.6 and 18.4 sit on the same contour, and their total masses differ by a factor of 1.24. What separates them is the mass ratio, which enters the phasing only at the first post-Newtonian order, suppressed by the square of the orbital speed in units of the speed of light — small through the hundreds of cycles that carry most of the signal, and appreciable only in the last few, where that speed approaches a third of c. So the chirp mass is a measurement and the individual masses are an inference from the end of the signal, which is exactly the part a detector's high-frequency noise eats first.

Gravitation

One number where two masses were

The hundreds of orbits an inspiralling binary completes inside a detector's band depend on its two masses only through one combination of them. Every pair on that contour radiates an identical sweep, so the early signal — which carries nearly all the signal-to-noise — cannot say which pair it was.

The observed sky

Where a star is depends on who is asking

The sky needs two coordinate systems because two different things stay still in it — the observer's horizon and the stars themselves. One rotation converts between them, and the angle of that rotation is the time.

$Twilight at latitude 52°. Solar altitude through the second half of the day at 52°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 34 min, 40 min, 42 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. At this latitude the curve for June solstice never reaches −18° at all: astronomical twilight does not end, and there is no astronomically dark night from about 48.6° upward.$ The observed sky

Three definitions of night

Twilight ends at three different depression angles, and each threshold is a statement about what can no longer be done. Its length is not a duration but a rate — how fast the Sun goes down — and above one latitude the deepest of the three never arrives at all.

The observed sky

Five numbers from one wiggle

A star's path across a plate is a straight line with a one-year ellipse laid on it. Nothing measures either alone — one fit yields five parameters at once, and they are separable only because their time signatures differ.

The solar system to three figures, and not one distance in it. Left, why an inferior planet's wandering is a measurement. At greatest elongation the sight line from the Earth is tangent to the planet's orbit, so the angle at the planet is a right angle and a/a⊕ = sin ε — no distance anywhere in the argument, only the angle between two directions. Venus reaches 45.4°–47.1°, giving 0.7224 AU against the modern 0.72333. Right, every planet Copernicus could see, derived this way and by the synodic route for the outer ones — 1/P = 1/E − 1/S for the year, then the harmonic law for the distance — plotted against the catalogue. Mercury is the interesting failure: its elongation runs from 17.9° to 27.8° rather than sitting still, so the method returns a range, 0.307 to 0.466 AU, and the true 0.3871 lies inside it. That spread is not an error in the method; it is Mercury's eccentricity being measured by a technique that assumed a circle.

The observed sky

The solar system measured from inside one orbit

Venus never appears more than 47 degrees from the Sun. That single angle gives its orbital radius as a fraction of the Earth's, with no distance measured anywhere — and every other planet gives one up as easily.

Every quasar in the sky streaming at 5.23 µas a year towards one point. Above: the apparent proper motion of distant quasars, drawn in Galactic coordinates with the centre of the Galaxy at the origin. Quasars do not move — at their distances a real transverse velocity of a thousand kilometres a second would be a hundredth of a microarcsecond a year — so a pattern in their apparent motions is a statement about the observer. Annual aberration displaces every source by v/c and returns it a year later; the Sun's velocity is not constant, and a changing displacement does not return. The Sun is being accelerated towards the centre of the Galaxy at 2.4·10⁻¹⁰ m s⁻², so the aberration vector rotates at a/c and the whole sky streams towards the same point, at (a/c) sin θ for a source θ from it. Below: that amplitude against angle from the apex, with the fitted dipole and the measurement. A circular speed of 248 km s⁻¹ at 8.28 kiloparsecs predicts 5.23 microarcseconds a year; the measured dipole in the proper motions of 1.6 million quasars is 5.05 ± 0.35, pointing to within a few degrees of the Galactic centre. The picture cannot show what took so long: the effect is a twenty-thousandth of annual aberration, it accumulates over the whole mission rather than over a year, and it is degenerate with any real rotation of the quasar frame — so a measurement of the acceleration of the solar system is also, unavoidably, an assumption that the distant universe does not turn.

The observed sky

The whole sky drifting towards one point

Annual aberration is the Earth's velocity, and it closes every year. The Sun's velocity is not constant, so the same effect leaves a residue that never closes — every quasar in the sky creeping towards the Galactic centre at five microarcseconds a year, which is a direct measurement of the Sun's acceleration.

$Resolution stops improving at 10 cm of aperture. Angular resolution against aperture at 500 nm, both logarithmic. The falling line is diffraction alone, 1.03 λ/D, which is what a telescope in vacuum delivers and has no floor. The curve is the same telescope under an atmosphere of Fried parameter r₀ = 10 cm, combining diffraction and seeing in quadrature: it follows the diffraction line while D < r₀ and then bends onto a plateau at 0.98 λ/r₀ = 1.01″. An amateur's 100 mm at 0.1 m would resolve 1.062″ above the air and delivers 1.47″ through it; a metre at 1 m would resolve 0.106″ above the air and delivers 1.02″ through it; the VLT at 8.2 m would resolve 0.013″ above the air and delivers 1.01″ through it; the ELT at 39 m would resolve 0.003″ above the air and delivers 1.01″ through it. At 39 m the atmosphere is costing a factor of 371: the aperture is 390 coherence lengths across and every one of the 152,100 patches it collects arrives with a phase of its own. What the extra aperture still buys is photons and speckles, and those two are what adaptive optics and speckle interferometry respectively spend to get the falling line back.$ The observed sky

A ten-metre mirror that resolves like a ten-centimetre one

The atmosphere delivers a wavefront in patches about ten centimetres across, and an aperture larger than a patch collects patches rather than detail. Resolution stops improving at that size — and what the extra aperture keeps buying is photons and speckles, which is why there are two entirely different ways out.

$A patch 1.5″ wide in the visible and 8″ at 2.2 µm. The isoplanatic angle against wavelength, for r₀ = 10 cm at 500 nm, a turbulence layer at 5 km and a wind of 20 m/s. This is the angle over which one measurement of the wavefront is still valid, and it is the hardest of adaptive optics' three limits: at 1.5 arcseconds in the visible, the guide star has to be inside a patch a hundredth the size of the full Moon. Everything scales as λ^6/5 because r₀ does — measured off the curve at λ^1.200 — so the patch grows to 8 arcseconds at 2.2 µm, and its area by the square of that. With 0.1 stars per square arcminute bright enough to guide on, the fraction of sky reachable goes from 0.018 per cent to 0.5 — a factor of 28. That single curve is why the first working systems were infrared, why a laser is fired to make a star where there is none, and why the laser still does not solve it: a beam launched from the telescope wanders with the same atmosphere it is meant to measure, so it cannot sense the overall tilt, and a natural star is still needed for that.$ The observed sky

The correction has to be faster than the air

Making a large telescope resolve like a large telescope means measuring the wavefront and undoing it. Three numbers bound how well that can work and none of them is the mirror — a frequency of hundreds of hertz, a patch a second and a half wide, and the chance of a bright enough star inside it.

The observed sky

The edge of a shadow is a wave

An asteroid's shadow has an edge because the asteroid is large. The Moon's does not — at visible wavelengths and lunar distance the edge of a shadow is ten metres wide, so a lunar occultation is a diffraction pattern sweeping past at half a kilometre a second, and how blurred its fringes are is the star's own diameter.

The observed sky

A star that blinked before it should have

In March 1977 three teams watching a star pass behind Uranus recorded it dimming five times before the planet arrived — and then, symmetrically, five times again on the way out. The symmetry is the whole argument — nothing but a set of rings concentric with the planet produces a mirror image about closest approach.

Starlight

The triangle that reaches the stars, and stops

Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it.

The magnitude scale, plotted. The logarithm of the received light against magnitude from -27 to 32, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller. 9 landmarks are marked, from the Sun to the deepest exposures, spanning a factor of 1.2·10²³ in received light.

Starlight

A scale that runs backwards, multiplies, and works

Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are.

Blackbody curves at 3000, 5800, 10000 K. Thermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.

Starlight

Colour is a thermometer, and it reads across the galaxy

A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy.

The same spectrum, at rest and at 900 km/s. A set of absorption lines at rest and shifted by a radial velocity of 900 km/s. The displacement is proportional to wavelength, so the reddest line here moves 2.06 nm and the bluest 1.18 nm — which is why the measured quantity is the ratio Δλ/λ and not a distance.

Starlight

A shift in a line is a speedometer, and it works at any distance

A spectral line has a wavelength fixed by physics. Measuring where it actually arrives gives the source's speed toward or away — and the measurement does not degrade with distance.

An absorption spectrum at 5772 K. A blackbody continuum at 5772 K with absorption lines cut out of it, each line's depth computed from how much of the gas is in a state that can absorb it. 8 of the 8 lines are strong enough to see at this temperature, which is the whole reason the spectral sequence is a temperature sequence.

Starlight

Composition, read from what is missing

The dark lines in a stellar spectrum are wavelengths that never arrived. Which ones are absent names the elements present — and the strength of a line says more about temperature than about abundance.

$The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.$ Starlight

Every distance is measured with the last one

No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred.

Starlight

A brightness is a distance only if something is known

The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system.

Starlight

Dust makes everything look further away

A star behind dust is fainter, so a distance taken from its brightness comes out too large. The same dust also makes it redder, and the reddening is measurable where the dimming is not — which is the only reason the correction can be applied at all.

Starlight

How much of a line is not in its depth

An absorption line stops getting deeper long before it stops getting stronger. What keeps growing is its area, and the way the area depends on the number of absorbers has three distinct regimes over five decades — one of which carries almost no information at all.

Starlight

A magnitude has to say which light

The same star is a different magnitude in every filter, and a colour index is a difference of two conventions rather than a property of the star. Both are integrals of a spectrum against a piece of glass, and the zero point is a choice somebody made in 1953.

Starlight

An angle of five hundredths of an arcsecond

No telescope has ever resolved a star other than the Sun, and stellar diameters are measured anyway — by finding the separation of two apertures at which the star's interference fringes vanish. What that returns is an angle; the radius arrives only when a distance is brought in, and the distance is the worse-known half.

Starlight

The light that is missing from the edge

The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

Starlight

The second thing a spectrum says

Two stars of the same colour can differ in luminosity by ten magnitudes, and the difference shows in the widths of their lines rather than in which lines are present. That width is a pressure, the pressure is a gravity, and the gravity is a distance.

Starlight

A magnitude measured where nothing was measured

Every published brightness is an extrapolation off the end of a graph. A star is observed through one atmosphere at least, a line is fitted against airmass, and the number quoted is its intercept at zero — a place no observation is ever taken from.

$What the error bar is made of, on a 1 m in 60 s. The four contributions to a photometric error, against the brightness of the star, for a 1-metre aperture, a 60-second exposure and a sky of 21 magnitudes per square arcsecond. The star's own photons give a line of slope exactly 0.2 — σ ∝ N^−1/2 and N ∝ 10^−0.4m, so a magnitude of extra faintness costs a fifth of a magnitude of precision, and no instrument changes that. The sky and the read noise are fixed counts, so their lines have slope 0.4, twice as steep, and they overtake the star at V = 18.25 — that crossing is the faint limit of the night, and it moves when the Moon rises rather than when the telescope changes. Scintillation is flat, because the atmosphere modulates a bright star and a faint one by the same fraction: at 4.09e-4 relative it is 0.44 millimagnitudes here and it is what caps the bright end, up to about V = 10.8. Below all of them is the systematic floor at 0.3 millimagnitudes, which is flat-fielding and colour terms and does not integrate down at all.$ Starlight

The error bar that comes from counting

A brightness is a number of photons, so the precision of the measurement is fixed before any instrument is chosen. What follows is a slope of exactly 0.2 magnitudes of error per magnitude of star, a slope of 0.4 once the sky wins, and a floor that neither of them explains.

$A 28.7 km/s correction, and a 12.5 m/s planet underneath it. Two years of radial velocities of a star at ecliptic latitude 12°, orbited by a companion whose reflex semi-amplitude is 12.5 m/s — the Sun's own, from Jupiter. The upper panel is what the spectrograph measures: the Earth's motion about the barycentre of the solar system, amplitude 28.72 km/s, which is V⊕ cos β to a fraction of a per cent. The planet is in that curve and is 2,297 times smaller than it, which is a line thinner than the stroke it is drawn with. The lower panel is the same data after the correction, and the correction is not a fit: it is computed from an ephemeris, the observatory's position on a rotating deformable Earth, and the star's own coordinates and proper motion. To leave a centimetre a second it has to be right to one part in 2.9·10⁶ — the light-travel time across the Earth's orbit, the relativistic terms, and the fact that the star moves are all inside that budget. What remains is the planet, at 4333 days, and a scatter of 1.2 m/s that is the star rather than the instrument.$ Starlight

The metre per second that is not the star

A line shift is a speedometer, and the speed it reads is mostly the observer's. Getting to a metre a second means removing thirty kilometres of the Earth's own motion to a part in three million, and then confronting a floor that is the star's own surface rather than the instrument.

Starlight

A spectrum with no temperature in it

Every spectrum in this collection so far has been a thermometer. A radio lobe's is a power law, and a power law has no scale — so there is nothing for a thermometer to read, and what the shape carries instead is the energy distribution of the particles that made it.

The faint star is measured against a brighter sky

For anything at the edge of detection the dominant source of noise is not the object. It is the sky in the same aperture, which is brighter than the star and is subtracted rather than measured — and once that is true, every rule of thumb about apertures, exposure times and image quality changes.

The light dust removes at 0.56 µm comes back at 124 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 6500 K photosphere; the shaded region under it is the part removed by a magnitude of visual extinction, computed from the same CCM law the other modes here draw, and weighted towards the ultraviolet exactly as that law says. The curve on the right is what the grains do with it: a modified blackbody at 20 K with an emissivity index of 1.8, normalised so that the energy under it equals the energy under the shaded region. Integrating the two curves actually drawn returns a ratio of 1.000. The starlight peaks at 0.56 microns and the re-emission at 124, a factor of 220, so nothing about the two is recognisable as the same photons and everything about them is the same joules. The practical consequence is a rule about arithmetic: a galaxy's ultraviolet luminosity and its far-infrared luminosity are not two independent measurements of how many young stars it has. One is the light that escaped and the other is the light that did not, and adding them without noticing counts part of the population twice.

Starlight

The dust is not lost light, it is moved light

An extinction curve says how much starlight dust removes. It does not say the light is gone. Every photon a grain absorbs heats the grain, which radiates it back out at twenty kelvin — so an extinction measurement and a far-infrared spectrum are two halves of one energy budget, and the area under them is the same area.

The Hertzsprung–Russell diagram. Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.

Stars

The diagram that sorted the stars, by plotting two things against each other

Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy.

Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end.

Stars

Mass decides everything, by a power of three and a half

Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars.

Stars

The main sequence is a place stars sit, not a track they travel

The commonest misreading of the Hertzsprung–Russell diagram is that stars slide down the band as they age. They do not. They sit at one point for ninety percent of their lives and then leave sideways.

Stars

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

Stars

The mass a cold star cannot exceed

A degenerate star's radius falls as its mass rises, and nothing in that relation suggests a limit. Making the electrons relativistic softens the pressure law until pressure and gravity scale the same way with radius — and the radius drops out of the balance, leaving one mass and no room to argue.

Stars

An age read off a bend

A cluster is a population of one age and many masses, so the place where its stars leave the main sequence is a clock. The bend needs a population — which is why a cluster has an age and a single star does not.

Stars

The candle that has to be standardised

A Type Ia's light is the decay of half a solar mass of nickel-56 seen through an expanding envelope. Their peak brightnesses span six-tenths of a magnitude — and it is the width of the light curve, not anything else, that says which.

A distance of 52.0 parsecs with nothing underneath it. Two ways to a distance for the same pair. The orbital parallax needs no iteration and no assumption: a double-lined spectroscopic orbit gives the relative orbit's linear size as (K₁+K₂)P√(1−e²)/2π sin i = 0.2268 AU, an astrometric orbit gives its angular size as 4.36 milliarcseconds, and the ratio is 52.0 parsecs — a length divided by an angle, with no step of the distance scale below it and no property of the stars assumed. The curves show the dynamical parallax, the version available when only one spectrum can be measured: guess the mass sum, take the linear size from the harmonic law, divide by the angular size, convert the apparent magnitude to an absolute one and read a new mass sum off a mass–luminosity relation. Three starting guesses spanning a factor of 10 in mass converge to the same distance in 8 passes and agree to 0.001 per cent. It converges because the distance depends on the assumed mass only as its cube root — the measured exponent here is 0.3333 — so a factor of two in the mass is 26 per cent in the distance, and one pass removes most of that. What it converges to is not the orbital parallax: the iteration settles at 54.2 pc against 52.0, 4.2 per cent away, because the fixed point is set by the mass–luminosity relation and the apparent magnitude rather than by anything measured about this orbit. The same insensitivity that makes it converge is why it is never better than the relation it leans on.

Stars

Two orbits of one pair, and a distance falls out

Measure the same binary spectroscopically and astrometrically and the orbit comes back twice — once as a length in kilometres and once as an angle on the sky. The ratio is a distance that owes nothing to parallax, nothing to a standard candle, and nothing to any assumption about the stars.

Seven sources over twelve decades of flux, and one threshold below the one that matters. The Sun's neutrino spectrum at the Earth, with the continua drawn per unit energy and the two monoenergetic lines as spikes at their own energies. The pp reaction supplies 91 per cent of all of them and its endpoint is at 0.4233 MeV. The vertical lines are experimental thresholds, and they are the figure's argument: only gallium sits below that endpoint. Chlorine, which produced the deficit and held it for twenty years, could not see a single pp neutrino — it counted ⁷Be and ⁸B, which are a rare branch of a rare branch, together under a per cent of the total — and the water detectors that followed were higher still. So the discrepancy that eventually turned out to be a property of the neutrino was measured, for two decades, using the least representative one per cent of the flux available. The total drawn here is 6.54·10¹⁰ cm⁻² s⁻¹, and it is checkable without any stellar model at all: every completed chain turns four protons into helium, releases 26.73 MeV of which 0.59 leaves as neutrinos, and emits two neutrinos — so the solar constant of 1361 W m⁻² fixes the number at 6.5·10¹⁰ cm⁻² s⁻¹, within 0.6 per cent of the sum of the model's own branches. The dominant flux is a consequence of the Sun shining and of nothing else.

Stars

The only thing that leaves the centre

A photon made in the Sun's core takes a hundred thousand years to get out and arrives thermalised past recognition. A neutrino takes 2.3 seconds and arrives unchanged, so its flux is the fusion rate now — which is why a factor of three could not be absorbed by any adjustment to the Sun.

Stars

Two radii, from a light curve alone

The radius of a star is not measured. It is inferred, from a temperature and a luminosity, through a model. There is one exception — a pair of stars that eclipse each other, whose light curve and velocity curves between them give both radii, both masses and the ratio of temperatures with no model of a stellar interior anywhere in the chain.

A spectrum belonging to no temperature at all. The disc's summed emission, with the individual annuli drawn faintly beneath it. Each ring is a blackbody at its own temperature, and each is drawn at the area it actually has — the outer rings are cool and enormous, the inner ones hot and small. The sum has three parts and only the two ends belong to a temperature: a Rayleigh–Jeans rise of slope 2 from the outermost ring, a Wien cutoff at the hottest, and between them a stretch of slope 0.316, against the 1/3 that comes out of integrating ν²T(r)r dr with T ∝ r^−3/4. That middle section is the observational signature of a disc: no single blackbody produces it, no photosphere produces it, and its width rather than its peak is what says how far in the disc goes. What the figure cannot show is that a real disc's innermost rings are neither thin nor blackbodies, which is where the model's clean edges stop.

Stars

A spectrum that is a stack of temperatures

An accretion disc is not hot. Its inner edge is, its outer edge is not, and the temperature runs continuously between them as a power of radius — so what leaves the disc is the sum of a great many blackbodies at different temperatures, which is a spectrum with no temperature in it and a slope no single body can produce.

Stars

A radius that decides what matter can be

Nobody can make matter at four times the density of an atomic nucleus, and no calculation settles what it does there. What can be done is to weigh a neutron star — every candidate description of that matter predicts a heaviest star it could hold up, and a single measured mass above that value deletes the description permanently.

Stars

Two explosions told apart by a missing line

The classification of supernovae is a decision tree on absences — is hydrogen there, is silicon there, is helium there — and it was drawn up decades before anybody knew what any of these events were. It nevertheless separates a detonating white dwarf from a collapsing stellar core almost perfectly, and the reason it does is worth the essay.

Spaceflight

An angle measured against a quasar

A tracking station measures how fast a spacecraft is receding, which is one number where three are wanted. The two missing angles come from the Earth's rotation, slowly, and near a planetary encounter there is no time for slowly — so the position is instead measured directly, as a difference of arrival times between two antennas, referred to a quasar a few degrees away.

Exoplanets

A planet measured by the light it removes

A transit gives a depth, and the depth is a ratio of two radii rather than a size. Everything a transit says about a planet is said in units of a star nobody has visited either.

Exoplanets

The star moves, and the mass is a lower bound

A planet is found by watching its star fall towards it. The wobble gives a mass multiplied by the sine of an angle nobody has measured — and the shortfall is not a rounding error.

Exoplanets

A star magnified by a planet nobody will see again

Microlensing weighs a planet by the way its gravity bends light around it. The measurement lasts a few hours, cannot be repeated, and is the only one that does not require the planet's star to be visible at all.

$What each method can see. Planet mass against orbital distance, both logarithmic, with the detection threshold of each method drawn as the boundary it actually is. Radial velocity at 1 m/s needs mass rising as √a; astrometry at 20 µas needs it falling as 1/a, which is the only method that gets easier further out; a 100 ppm transit is a threshold on radius and so a horizontal line at about 1.4 Earth masses, cut off at 1.21 AU by the need for three transits in 4 years; direct imaging begins outside the diffraction limit, 0.6 AU at 10 parsecs for a 39 m aperture at 10 µm. The solar system is drawn on top: for two decades every one of its planets except Jupiter lay outside every region, which is the whole of what the early census was measuring.$ Exoplanets

Every survey draws a different sky

The first exoplanets found were enormous and impossibly close to their stars. That was not a discovery about planets. It was a measurement of what a 10 m/s spectrograph watching for three years is able to see.

Exoplanets

The planets that were not seen

An occurrence rate is a count divided by a probability, and the probability can be a five-hundredth. Everything difficult about saying how common planets are lives in that denominator.

Exoplanets

A radius that depends on the colour it is measured in

Measure a transit in one colour and then another, and the planet is a different size. The difference is a few atmospheric scale heights, which is a few hundred parts per million of an already tiny signal.

Exoplanets

One number sets the zone

The habitable zone is a band of stellar flux, so it scales as the square root of luminosity and moves inward far faster than mass falls. For most stars it lies inside the radius at which a planet is tidally locked.

Exoplanets

The line beyond which ice counts as rock

A disc of gas around a young star gets colder outwards, and at about a hundred and seventy kelvin water stops being vapour and becomes a building material. Crossing that one line multiplies the solid mass available by roughly three and a half, in a single step, and the architecture of every planetary system is downstream of it.

$The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in.$ Exoplanets

A planet ten times larger in one colour

A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving.

The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not.

Galaxies

A rotation curve that refuses to fall

Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure.

Galaxies

The mass that is not the light

A galaxy's mass-to-light ratio is not a number, it is a curve, and it rises without turning over. What stellar populations can plausibly weigh sets a ceiling; the dynamics sit far above it, and the gap is a shape rather than a discrepancy.

the Milky Way measured from inside it — the tangent-point construction. Left: the disc in plan, with the Sun at 8.2 kpc from the centre and four lines of sight at galactic longitudes 20°, 40°, 60°, 80°. Each one grazes a circle of radius R₀ sin l, and at that tangent point the whole circular velocity lies along the line of sight, so the largest velocity in the spectrum belongs to a radius the geometry fixes and no distance has to be measured. Right: the four radii and speeds that yields, on the model curve they are read against. The construction reaches only radii inside the Sun's, which is why the outer curve — the half of it that carries the argument — needs distances after all.

Galaxies

A galaxy measured from inside it

The Milky Way is the one galaxy nobody can photograph, and the only one whose rotation can be measured without knowing a single distance. A line of sight at a chosen longitude grazes a circle of known radius, and the fastest gas along it is orbiting there.

Mass from a velocity dispersion, across nine decades. The virial estimate M = 5 σ²R_e/G, drawn for half-light radii from 0.2 to 20 kpc, with four systems placed on it: M32 at σ = 75 km/s and R_e = 0.1 kpc, giving 6.5e+8 M☉; the Milky Way's bulge at σ = 105 km/s and R_e = 1 kpc, giving 1.3e+10 M☉; M87 at σ = 320 km/s and R_e = 7 kpc, giving 8.3e+11 M☉; the Coma cluster at σ = 1000 km/s and R_e = 1500 kpc, giving 1.7e+15 M☉. The same expression spans from a dwarf elliptical to a cluster of galaxies — nine decades of mass — because the virial theorem does not care what the moving objects are. In the last case they are whole galaxies, and the estimate exceeds everything visible in the cluster by a factor of about fifty, which is where the problem was first noticed.

Galaxies

A galaxy held up by disorder

An elliptical galaxy barely rotates. What holds it up against its own gravity is the randomness of its stars' motions, and the virial theorem turns that randomness into a mass by exactly the reasoning that turns a rotation speed into one.

Galaxies

A line width that is a distance

The width of a galaxy's hydrogen line depends on how fast it rotates, which depends on its mass, which is tied to its luminosity — so a quantity no distance enters gives an absolute brightness, and the distance follows from the brightness that is seen.

What distance does, and does not, do to a galaxy. Three quantities against distance, each relative to its value at 5 Mpc, on logarithmic axes so that a power law is a straight line and its exponent is the slope. Flux falls with slope −2 and angular size with slope −1, both of which are ordinary. Their ratio has slope zero: a galaxy of surface brightness 23.5 magnitudes per square arcsecond has that surface brightness at every distance, and a sky of 22 is brighter than it at every distance too. The contrast against the sky — the quantity that decides whether the thing is detectable at all — is -1.5 magnitudes wherever it is put.

Galaxies

The brightness distance cannot touch

Flux falls as the inverse square of distance and so does solid angle, so their ratio does not fall at all. A galaxy's surface brightness is the same number wherever it is put, which means whole populations can be undetectable at any distance whatever.

Three mass models, 2.3× apart in the disc, agreeing to 3.1 km/s everywhere. NGC 3198's rotation curve, decomposed three ways. The stellar disc has been scaled by 0.2, 0.65, 1.1 times its photometric mass, and for each scaling the halo's asymptotic speed and core radius have been fitted — not chosen — to reproduce the same total. The heavy curve and the marked points are that total: the three models agree with it to 3.08 km/s at every radius, well inside a measurement error of 4.5. The light curves below are the disc's own contribution, and at 17 kpc they differ by a factor of 2.3 — from 36 to 85 km/s — with the halo taking up exactly the slack, 126 down to 97. The curve is one function and the decomposition asks for two. The free parameter is the mass-to-light ratio of the stars, which the kinematics never measures, and it is why a "maximum disc" fit and a halo-dominated fit are both published for the same galaxy. The degeneracy does have one hard edge: scaling the disc to 1.35 times its photometric mass cannot be fitted by any halo in the family — the best leaves 5.6 km/s rms — because past the maximum-disc solution the stars alone already overshoot the curve and a halo cannot have negative mass. That is the one thing a rotation curve says about M/L on its own, and it is an upper limit. What separates the rest has to come from somewhere else: the vertical velocity dispersion of the disc, which weighs the stars alone; gas-rich dwarfs where there is scarcely a disc to argue about; or the baryonic Tully–Fisher relation, which ties the halo's speed to the baryons and would be a coincidence if the two were independent.

Galaxies

The same curve, two galaxies

A rotation curve is one function of radius. Decomposing it asks for two — how much of the speed is the stars and how much is the halo — and the mass-to-light ratio that trades one against the other is not measured by anything in the kinematics. A maximum disc and a halo-dominated fit run through identical points.

Why an Einstein radius is a mass. The geometry, drawn at an angle some ten thousand times larger than the real one so that anything is visible at all. Light from a source directly behind a lens reaches the observer along every path that passes the lens at the same distance, so the image is a ring rather than a point. The ring's angular radius is θ_E = √(4GM/c² · D_ls/D_l D_s), which for a lens of 1.0×10¹² M☉ at these distances is 2.52 arcseconds and encloses 10 kpc at the lens. Rearranged, it is a mass in terms of an angle and three distances — and the mass so obtained is inside a cylinder rather than a sphere, and assumes nothing whatever about the lens being in equilibrium, which is the assumption every other weighing in this collection makes.

Galaxies

Weighed by the light that bends past it

Every other mass in this collection is measured from something orbiting, which requires the system to have settled down. A gravitational lens weighs whatever is in the way with no such assumption — the light does not care whether the mass is in equilibrium.

Galaxies

A cluster weighed three ways

The speeds of a cluster's galaxies, the temperature of its gas and the bending of light behind it are three measurements with almost nothing in common. They agree within a factor of two, and all three exceed the mass of the stars by about a hundred.

The luminosity function has a knee, and the knee is the point. A Schechter function with a faint-end slope of -1.25, a characteristic magnitude of -20.9 and a normalisation of 0.0093 per cubic megaparsec per magnitude, plotted logarithmically. Fainter than the knee the curve is a straight line — a power law — and brighter than it the count falls off exponentially, which is why there is no such thing as a galaxy ten times brighter than the brightest. Integrated across the range drawn, galaxies brighter than the knee are 1.3 per cent of the number and 26 per cent of the light: almost every galaxy is a dwarf, and almost all the light is not in one.

Galaxies

A count with a knee in it

Count galaxies by luminosity and the answer is a power law at the faint end and an exponential cut-off at the bright one. Almost every galaxy is a dwarf; almost none of the light is in one; and the bend between those two statements is where galaxy formation stops being efficient.

Galaxies

A one-per-cent distortion, and a million galaxies to see it

A galaxy's own shape is unknown and scatters with a dispersion of about 0.3, so a coherent one-per-cent shear is thirty times smaller than the noise on any single measurement. Nothing is ever measured about one object; the estimator is an average, and the whole design of a survey follows from 0.3 over the square root of N.

Galaxies

A size measured from a delay

An active nucleus at redshift two is a point source in every telescope ever built, and its central mass is measured anyway. The continuum varies, the broad lines follow days later, and the lag is a light-travel time — which is a length, recovered from two light curves that share no wavelength.

$Gas outweighs stars 6:1, and the baryons come to 15%. Above: enclosed mass against radius for a 7-keV cluster with a beta-model gas profile, β = 0.65 and a 250-kpc core. The total is from hydrostatic equilibrium — the same equation that holds up a star, with the mass following from the density and temperature gradients and from nothing else — and comes to 9.98·10¹⁴ solar masses inside 2 megaparsecs. The gas, integrated from the same profile, is 1.6·10¹⁴; the stars in all the galaxies are 2.5·10¹³, 6 times less. Most of a cluster's ordinary matter is not in anything anybody would call an object. The two total-mass curves differ by the 15 per cent hydrostatic bias: some of the pressure holding the gas up is turbulence left from the last merger rather than heat, and a mass computed from the thermal pressure alone is low by about that — the assumption that makes the measurement possible is the one that biases it. With the correction, the baryons come to 15 per cent of the total, against the 15.7 per cent the microwave background gives for the universe as a whole. A cluster is large enough to have kept everything it started with, so its own accounting is a cosmological measurement. Below: the same gas seen two ways. X-ray surface brightness falls as (1+z)⁻⁴, so a cluster at redshift one is 16 times fainter per unit sky than the same cluster nearby; the Sunyaev–Zel'dovich distortion of the microwave background does not fall at all, because it is a fraction of a background whose own brightness rises by exactly the same factor. That is why a millimetre survey finds clusters at any distance and an X-ray survey finds the near ones.$ Galaxies

The baryons that are not in the galaxies

The stars in a cluster are about a seventh of its ordinary matter. The rest is ten-million-kelvin gas that is invisible optically and dominant in X-rays — and weighing it with hydrostatic equilibrium biases the answer low by exactly the amount the assumption is wrong by.

A delay of 81 days, and a sheet nobody can see that moves H₀ to 82.4. Above: the arrival-time surface of a lensed source, along the line through the lens. The curve is the Fermat potential in days — the geometric cost of taking a longer path, minus the gravitational cost of climbing out of the potential — and the images sit at its stationary points, at -1.20″ and 2.04″, which for an isothermal sphere is β ± θ_E. Fermat's principle is doing all of the work here: light does not take the shortest path or the quickest one, it takes every stationary one, and the number of images is the number of stationary points. The vertical distance between the two is 81 days, and it is measurable — the source is a quasar, quasars vary, and the same wiggle appears in one image and then the other. That single number carries an absolute distance: the delay is D_Δt/c times a dimensionless function of the lens model, and D_Δt goes as 1/H₀, so a monitoring campaign gives the Hubble constant with no calibrated distance beneath it. Below: the two light curves, shifted by exactly that delay. The dashed curve is the second image with the delay removed, and the agreement is the measurement. What the picture also shows is the reason the answer keeps moving. The second arrival-time curve is the same lens with a uniform sheet of convergence added and the source moved to compensate: every image sits at the same place, every flux ratio is the same, every image shape is the same, and the delay is λ = 0.85 times as long. A lens model fitted to positions alone cannot see the sheet, and inferring H₀ from the same delay under it gives 82.4 instead of 70 — a 15 per cent shift with no observable attached. Breaking it needs a mass measured some other way: the velocity dispersion of the deflector, or a count of everything else along the line of sight.

Galaxies

A distance measured with a stopwatch

The images of a lensed quasar sit at the stationary points of an arrival-time surface, and the height between two of them is a delay in days. That delay is proportional to a distance, and the distance is proportional to one over the Hubble constant — so a flickering quasar gives H₀ with no calibrated distance under it.

Galaxies

Two temperatures, and nothing in between

The gas between the stars is not one gas at one temperature. Thermal balance has three solutions at any ordinary interstellar pressure, the middle one cannot survive being nudged, and the two that can differ by a factor of over a hundred in both density and temperature — which is why a map of the interstellar medium looks like clouds rather than like fog.

Galaxies

The cloud that cannot hold itself up

A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star.

$One star in 380 makes essentially all the ionising light. Three cumulative fractions against stellar mass, for a broken power-law initial mass function with slopes 1.3 and 2.3 breaking at 0.5 solar masses. Each curve says what share of one quantity is produced by stars heavier than the mass on the axis, and the three do not resemble one another. Only 0.41 per cent of the hydrogen-ionising photons come from stars below 15 solar masses, because the ionising output of a star climbs by five orders of magnitude between eight and twenty. One star in 380 is above that mass, and between them those stars hold 14 per cent of the mass. Those two numbers are the leverage in every star-formation rate quoted from an Hα line. What is measured is the light of a handful of very massive stars; what is reported is the mass of a whole population; and the number in between is an integral over a part of the mass function that no extragalactic observation reaches. The medians are marked but should be read with care, and the reason is visible in the curves: the mass-weighted median at 1.27 solar masses is a property of the population, while the light-weighted one at 58 is a property of where the plot stops — halving the upper mass limit moves it to 36. An integrand that rises with mass has its median wherever the axis ends.$ Galaxies

A birth rate measured from light nothing young emitted

A star-formation rate is quoted in solar masses per year, and nothing in the measurement counts a star or weighs anything. What is measured is a luminosity produced by about one star in four hundred, and the conversion to a mass is an integral over a part of the mass function that no extragalactic observation reaches.

Cosmology

The same constant, measured twice, five sigma apart

Cepheids and supernovae give an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved.

Cosmology

Four distances to the same galaxy

Inside the Local Group the word "distance" has one meaning. Past a redshift of about a tenth it has four, they disagree by factors of a hundred by the time the light is old, and one of them stops increasing and starts coming back.

Distance modulus against redshift, for three universes. The distance modulus μ = 5 log₁₀(D_L/10 pc) against redshift for three universes, all with H₀ = 67.36 km/s/Mpc, with 60 model supernovae drawn from the ΛCDM curve with 0.15 magnitudes of scatter. The point of the figure is how little difference there is: across two decades of redshift the three curves stay within a few tenths of a magnitude, and at z = 0.5 the accelerating and decelerating cases differ by 0.387 mag. A cosmology is not read off this plot. It is read off the residual, which is the next figure.

Cosmology

An expansion that was supposed to be slowing

Gravity is attractive, so an expanding universe full of matter must be decelerating, and the only question was by how much. Two teams set out to measure the deceleration and both found a quarter of a magnitude of extra faintness at redshift half — which is the wrong sign.

Cosmology

The most perfect blackbody ever measured

The whole sky glows at 2.7255 kelvin, and its spectrum matches the Planck curve to fifty parts in a million. Nothing else in astronomy is thermal to a part in twenty thousand, and a spectrum that exact is not a description of the radiation — it is a constraint on the history of everything that could have disturbed it.

The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has.

Cosmology

A standing wave frozen at one instant

The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning.

$The same universe, four times, as a fraction of itself. Each bar is the fractional contribution of the four components to the total density at one epoch, computed from the Planck 2018 parameters by scaling each component from today: radiation as (1+z)⁴, both kinds of matter as (1+z)³, and Λ as a constant. The familiar figure — five per cent baryons, twenty-six dark matter, sixty-nine dark energy — is the top bar and only the top bar. At recombination the same universe is three-quarters dark matter and Λ is one part in ten million; before matter–radiation equality it is mostly radiation. A pie chart of the contents of the universe is therefore a statement about a moment, and the moment is the one it happens to be drawn in.$ Cosmology

A budget whose familiar part is five per cent

Five per cent ordinary matter, twenty-six per cent dark matter, sixty-nine per cent dark energy. The figures are quoted everywhere and each one comes from a different measurement, the denominator they are fractions of is itself built out of the expansion rate, and the whole chart is a statement about one instant that was a different chart at every earlier time.

A bump at a hundred megaparsecs. The two-point correlation function of galaxies, multiplied by the square of the separation so that the interesting part is not buried under the power law. The smooth dashed curve is the broad-band clustering — the ordinary fact that galaxies are near other galaxies, which carries no cosmological information and is treated as a nuisance term in the real analysis. The bump on top of it is the whole measurement, and its position here is not fitted: it is the comoving sound horizon at the drag epoch, integrated in this file from ∫c_s da/a²H with c_s = c/√(3(1+R)), which comes out at 146.9 Mpc — 99.0 in the h⁻¹ Mpc a survey works in. The excess says that a galaxy is very slightly more likely to have a companion at that separation than at 90 or 115, by about one part in two hundred, and the reason is that a pressure wave in the photon–baryon fluid ran outward from every overdensity for four hundred thousand years and stopped where it was when the photons let go. The points are drawn with the error a survey of a million galaxies achieves; a single pair of galaxies at 100 Mpc means nothing, which is why the measurement waited for the surveys.

Cosmology

The same ruler measured twice, ten billion years apart

The sound wave that produced the acoustic peaks in the microwave background also left a faint excess in how galaxies are spaced, at a separation of about a hundred megaparsecs. It is the only cosmological distance indicator whose length is set by physics rather than by a chain of calibrations.

Cosmology

A horizon three times larger than the age allows

The universe is 13.8 billion years old and light travels one light year a year, so the observable universe should be 13.8 billion light years across. It is 46.1, nothing has outrun light, and the discrepancy is a piece of arithmetic rather than a paradox.

w = −0.9 is 34 millimagnitudes, and one supernova scatters by 120. Above: how much the distance modulus moves when the dark energy is not a constant. Each curve is a universe with the same Ωₘ = 0.315 and a different equation of state w, drawn as a difference from w = −1 in magnitudes. At redshift a half, w = −0.9 is worth 34 millimagnitudes — the shaded band is the 0.12-magnitude intrinsic scatter of a single standardised type Ia supernova, and the signal is a fifth of it. Nothing about one object can see this; the measurement is the mean of 1500, whose error on the mean is 3.1 millimagnitudes, and even that only works because the shape of the curve in redshift is different from every systematic anybody has thought of. Below: why the supernovae are not enough on their own. Each locus is the set of (Ωₘ, w) that a measurement cannot tell apart from the fiducial model — computed, not sketched: the supernova curve is the ridge of the same sum of squares a fit would minimise over 0.02–1 in redshift, and the acoustic-scale curve is the exact set of models with the same comoving distance to last scattering, which is what fixes the angle the microwave background's first peak subtends. They cross at 36 degrees. Neither is a measurement of w and the pair is, which is why the constraint on the equation of state is a picture of two loci crossing rather than a number read off a curve — and why −1.03 ± 0.03 is a statement about how well they cross rather than about how well anything was measured.

Cosmology

The number that would say whether it is a constant

Whether dark energy is a cosmological constant is the question of whether w is exactly −1, and w = −0.9 changes a distance modulus by thirty-four millimagnitudes at redshift a half — a fifth of the scatter of a single supernova, along a degeneracy only the acoustic scale can cut across.

A sphere reconstructed as a spheroid, and two distortions 0.15 apart. Left: the acoustic scale in the plane of separation across the line of sight against separation along it, one quadrant of it. In the cosmology that actually holds, the sound horizon is a sphere of 99.0 h⁻¹ Mpc and its locus here is a quarter circle. That is the whole content of the Alcock–Paczyński test: nothing about the early universe distinguishes the radial direction from the transverse one, so any departure from a circle is a statement about the observer's arithmetic rather than about the ruler. Converting angles into transverse separations needs the transverse comoving distance and converting redshift intervals into radial ones needs H(z), so assuming distances 1.1 times too large and rates 0.94 times too small returns an ellipse of axis ratio 0.855 — and the ellipticity measures that distance times the expansion rate over c, in which the sound horizon has cancelled. A ruler of unknown length still measures a shape. The third curve is the difficulty: peculiar velocities also distort the same correlation function along the same axis, squashing it by 1/(1+β) = 0.704 for β = 0.42, and a squashing is a squashing. Separating a geometric distortion from a dynamical one is the entire art of the measurement, and it is done by using the fact that they have different dependences on scale — the velocities act on the broad-band shape and the ruler is a feature. Right and below: the two numbers the same feature gives at each redshift. Across the line of sight, the transverse distance divided by the sound horizon; along it, c divided by the expansion rate times the sound horizon. Two functions of the expansion history, from one bump in one correlation function, and their agreement with a single model is one of the sharper consistency tests in the subject.

Cosmology

A ruler measured along and across

The sound horizon is a sphere, and a sphere in a redshift survey is measured twice over — across the line of sight it gives an angle, along it a redshift interval. Two different functions of the cosmology out of one feature, and their ratio is a measurement with no ruler in it at all.

$One cluster moved to z = 1.8: one signal unchanged, the other 57 times fainter. The same cluster of galaxies placed at a series of redshifts, with two ways of detecting it compared. The Sunyaev–Zel'dovich decrement is flat, because it is a fraction of the microwave background and the microwave background has the same surface brightness at every redshift a cluster can sit at — moving the cluster further away shrinks it on the sky but does not make its shadow shallower. The X-ray surface brightness of the identical object falls as (1+z)⁻⁴, the dimming every surface brightness suffers in an expanding universe, and by z = 1.8 it is 57 times below where it started. This is why the cluster surveys that reach the early universe are millimetre surveys: an X-ray telescope's cluster catalogue thins out with distance and a Sunyaev–Zel'dovich catalogue is limited only by how large the cluster looks and by how faint a fractional distortion the instrument can measure. What the effect cannot supply on its own is a distance — the signal that does not know how far away the cluster is also cannot say — so every one of these clusters still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it.$ Cosmology

A shadow that does not get fainter with distance

Every other way of finding a cluster of galaxies gets harder the further away the cluster is. One does not. A cluster's hot gas scatters about one microwave background photon in a hundred to a higher frequency, and because the result is a fraction of a background rather than a flux from a source, the same cluster is exactly as detectable at ten billion light years as at one.

Fingers 3.1 times long, a large scale squashed to 0.92, and a test of gravity. Left: a 360-megaparsec slice of a clustered universe, as the galaxies actually sit. Right: the same galaxies as a redshift survey records them, with the line of sight up the page. Nothing has moved sideways, because an angle is an angle; every displacement is along the line of sight, because that coordinate came from a redshift and a redshift is the expansion plus whatever the galaxy is doing on its own. Two effects, opposite in sign and separated by scale. Inside a cluster the motions are virial and random, 720 kilometres a second of them, which at H₀ = 67.36 is 11 megaparsecs of smearing on an object a few across: the clusters become fingers 3.1 times longer than they are wide, all pointing at the observer, which is the one structure in cosmology that is definitely not real. On the scale of a supercluster the motions are coherent — everything is falling in, so the far side is approaching and the near side receding — and the structure is compressed rather than stretched, to 0.92 of its true extent here, measured on cluster centroids so the fingers have already averaged away. Below: why that compression is worth having. Its amplitude is the rate at which structure is currently growing, and the growth rate is Ωₘ(z) raised to a power that general relativity fixes at about 0.55. A theory of gravity that differs from general relativity on cosmological scales while matching every solar-system test changes that exponent and nothing else, and the two curves drawn — γ = 0.55 and γ = 0.68 — differ by only 4 per cent at redshift a half, against error bars of 12 per cent on the points beside them. The worst systematic in a redshift survey is the measurement — and it is a hard one, because a quarter of a change in the exponent that governs how gravity assembles structure moves the observable by less than the width of the curve it is drawn on. What the picture cannot show is the degeneracy that limits it: what is measured is fσ₈, a product, and separating the growth rate from the amplitude of clustering needs something else entirely.

Cosmology

A map stretched by the thing it measures

A redshift survey plots galaxies at distances derived from their redshifts, and a galaxy's redshift contains its own motion as well as the expansion. So the map is systematically distorted — squashed on large scales, drawn out into radial spikes on small ones — and both distortions are caused by the gravity the survey exists to measure.

Galaxies

A flare that puts a ceiling on a mass

A star torn apart by a black hole lights up for a year. The tidal radius grows as the cube root of the hole's mass and the horizon grows as the mass itself, so above about a hundred million suns the star is swallowed whole and nothing is seen — which makes the existence of a flare a measurement.

A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.

Galaxies

A stream is not the orbit it came from

A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.

Orbits

Where the mass is and where the light is

A population that grinds itself up settles into a size distribution with a fixed slope, and that slope puts almost all the mass in the largest bodies and almost all the cross-section in the smallest. So a debris disc's brightness measures a population whose mass it says nothing about, and the two are connected only by the exponent.

Starlight

A speed read off an edge

A hot star drives material off itself at thousands of kilometres a second, and the profile that wind prints on the star's own spectrum has a sharp blue edge. That edge is the terminal velocity, measured with no model of the star, no distance, and no calibration — one of the few numbers in stellar astrophysics obtained from geometry alone.

Galaxies

Red, gas-poor, and still spiral-shaped

A galaxy falling into a cluster meets a headwind of hot gas at two thousand kilometres a second. Where that ram pressure exceeds the disc's own grip on its gas, the gas goes; the stars, which are not a fluid and feel no pressure, do not. The result is a spiral with no fuel, and it is the commonest kind of galaxy in a cluster.

Two bodies at a mass ratio of 2.6 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 2.6 to 1, so the heavier body's path is 2.6 times smaller.

Stars

The companion survives, and it is moving

When one star of a close pair explodes, the other is left holding an orbital velocity and no orbit. Whether the pair stays bound turns on a single question — did more than half the total mass leave? — and the stars that answer no are running across the Galaxy at a hundred kilometres a second with nothing visible behind them.

Stars

An equation of state is already a star

Write down how a gas's pressure depends on its density, insist that the pressure hold the weight up, and everything else follows — the run of density, the fraction of the mass inside each radius, and, at one particular index, a mass that does not care what the radius is.

Stars

A clock with no fuel in it

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling. The faintest ones in the Galaxy are not faint because they are small — they are as faint as anything has had time to become, and the place where they stop is a date.

$The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives.$ Stars

A fluid that turns as one piece

The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that.

Gravitation

How much a world gives

A body pulled on from one side deforms, and how much it deforms is a single dimensionless number. That number is three halves for a uniform fluid, three hundredths for the Sun, and two thousandths for a moon made of ice — so measuring it is a measurement of what is inside.

Gravitation

Whether the heavy material sank

A moment of inertia is 0.4 of MR² for a uniform sphere and less for everything that has differentiated, and it is measured by watching a body wobble. Mercury's 0.346 says most of the planet is iron core; the Moon's 0.393 says almost none of it is.

The 2.3-hour spin barrier, and the small bodies that are allowed through it. Rotation period against diameter for a synthetic asteroid population, both axes logarithmic, with the horizontal lines marking where a body held together by nothing but its own gravity would fly apart. That limit is P = √(3π/Gρ) and it contains only the density: 3.30 hours at 1 gram per cubic centimetre, 2.33 hours at 2 gram per cubic centimetre, 1.91 hours at 3 gram per cubic centimetre. Size does not appear in it, which is what makes the figure's shape informative rather than obvious — a barrier that depended on size would be drawn as a slope, and a horizontal line crossing five decades of diameter is a much stronger statement. The observed population respects it. Above about two hundred metres nothing rotates faster than the 2.3-hour line, and the crowding just below that line is real: bodies pile up against a limit they cannot cross. Below two hundred metres the wall stops applying and the fast rotators appear, some of them turning in minutes. Nothing about gravity changes at that size. What changes is that a body small enough is a single coherent rock with tensile strength, while a body large enough is a pile of fragments with almost none, and the barrier is a measurement of which is which. The picture is a synthetic population drawn at random from a fixed seed rather than a catalogue, so the individual points are not asteroids; what is real is the barrier, its value, and the fact that only the smallest bodies are found beyond it.

Orbits

A wall with no size in it

Spin a body held together by nothing but its own gravity, and past a certain rate it comes apart. The rate depends on density alone — 2.3 hours for the stuff asteroids are made of — and the observed population respects the limit exactly, for every body larger than a couple of hundred metres.

Galaxies

An argument about the innermost kiloparsec

Simulations of cold dark matter have produced the same halo for thirty years — a density rising inward without limit. The rotation curves of the smallest galaxies say the density flattens off. The disagreement is confined to a region a thousandth of the halo's size, and it has not been settled.

$Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.$ Stars

Every note turns back at its own depth

A sound wave heading into a star is refracted by the rising sound speed and turns around before it reaches the middle. Where it turns depends on which mode it is — so a list of frequencies is not one average of the interior but thousands of differently weighted ones, and that is a solvable problem.

Drift against thermal inertia: a peak at Γ = 93, in the same place for every size. How fast an asteroid's orbit drifts under its own re-radiated heat, against the thermal inertia of its surface, at a rotation period of 4.3 hours and 1.13 astronomical units. Both axes are logarithmic, and the curves are four diameters. The non-monotonic shape is the content. A surface that conducts nothing re-radiates its heat the instant it receives it: the emission is then symmetric about the sub-solar point and the transverse push cancels exactly. A surface that conducts perfectly is isothermal, has no temperature contrast at all, and again pushes nowhere. The force lives between those two nothings, and peaks where the surface's thermal time constant is comparable to the rotation period — here at Γ = 93 in SI units, and at the same place on every curve, because the size scales the drift without moving the optimum. That separation is what makes the effect a measurement. A drift rate on its own is a single number with several unknowns in it; a drift rate together with a size from radar, a spin from a light curve and a density from a flyby leaves the thermal inertia as the only thing not measured, and solving for it says what the surface is made of. Fine dust sits near 50, bare rock in the thousands, and the values measured for the bodies spacecraft have visited — Bennu at 310, Ryugu at 225, Itokawa at 700 — straddle the peak, with the two rubble piles a factor of two or three above it and the Moon's dust well below. Being past the optimum is not a small effect but it is a gentle one: the curve falls as one over the thermal inertia on that side, so a surface three times more conductive than optimal still drifts at a third of the best rate, while one three times more insulating drifts at a third as well. The shape is symmetric in the logarithm, which is why the measurement is a good one for telling dust from pebbles and a poor one for telling pebbles from boulders. The curve is one-dimensional linear theory for a rotating half-space: it has the right limits and the right peak, and it omits the body's shape, which for an irregular asteroid changes the answer by tens of per cent.

Orbits

A drift rate that says what the surface is made of

The thermal recoil that moves an asteroid's orbit depends on how long its surface holds heat, and the dependence is not monotonic — a perfect insulator and a perfect conductor both push nothing. The peak in between means a measured drift is a measurement of thermal inertia, which is a measurement of grain size.

$14 glitches, and the 1.5 per cent of Vela that is not slowing down. Accumulated fractional spin-up against time for a Vela-like pulsar over 40 years, in parts per million. The underlying spin-down has been removed, so a perfectly braking pulsar would be a flat line at zero. What is drawn instead is a staircase: 14 sudden jumps of a few parts per million, each rising in less than a minute and then relaxing partway back over a couple of months, leaving a permanent step behind. Nothing outside the star can deliver angular momentum on a timescale of seconds, so the source is internal, and the only internal component that could have any to give is one that has not been slowing down with the rest. That is a neutron superfluid: it carries its rotation in quantised vortices, the vortices pin to the crustal lattice and cannot migrate outward, and so the superfluid keeps the spin it had while the crust brakes past it. The reservoir grows until the pinning fails somewhere, and a glitch is the unpinning. The straight line through the staircase is the glitch activity, 0.68 parts per million per year, and it converts directly into an interior measurement: the crust cannot on average take more than the superfluid stores, so the decoupled component must hold at least 2τ_c times the activity of the star's moment of inertia, which here is 1.5 per cent. It is a lower bound rather than a value, and it is one of the very few quantitative statements about the inside of a neutron star that needs no equation of state at all. The individual glitch times and sizes here are drawn from a seeded generator rather than a catalogue; what is real is the staircase's shape, the partial healing, and the arithmetic that turns a slope into a fraction.$ Stars

The part of a star that never slowed down

A pulsar's spin decays smoothly for years and then jumps upward inside a minute. Nothing outside can deliver angular momentum that fast, so something inside has been storing it — and the rate at which the jumps accumulate is a lower bound on how much of the star is not braking with the rest.

$A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.$ Starlight

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

Starlight

The depth is not the area

A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

$Where each gravity harmonic gets its signal: J₂ from the bulk, J₁₀ from the outer 12 per cent. Why a spacecraft that never enters a planet can say something about its depth. Each zonal harmonic of the external field is an integral over the interior density weighted by r to the power of the degree plus two, and the curves here are those integrals accumulated outward: the fraction of each coefficient that has been contributed by the time the integration reaches a given fractional radius, for an interior of polytropic index 1. The weighting climbs steeply with degree, so the curves separate. Half of J₂ comes from inside 73 per cent of the radius, and half of J₁₀ from inside 88 per cent — the higher coefficients barely know the deep interior exists. That ordering is the whole basis of gravity science as a probe. A single coefficient is one number and constrains almost nothing; a series of them, each weighted differently, is a coarse depth profile, and it is how Jupiter's core turned out to be smeared over half the planet rather than sitting as a distinct sphere at the middle. Two limits are worth stating with it. The information falls off fast: by degree ten the kernel is concentrated in a shell so thin that measuring the coefficient says little about anything below it. And every curve here assumes north–south symmetry, under which the odd harmonics vanish identically — so a measured J₃ or J₅ is not a deeper probe of the same thing but a measurement of something else entirely, which at a giant planet is how fast and how deep the winds run.$ Gravitation

A core weighed by something that never went in

The external gravity field of a planet is a series, and each term of it is an integral over the interior density weighted by a different power of radius. Measure enough terms and the series becomes a coarse depth profile — which is how Jupiter's core turned out to be smeared over half the planet rather than sitting at the middle.

Dissipation against viscosity for Io: a peak at 10^13.5 Pa s, and two solutions at the observed rate. The imaginary part of the Love number — the part that turns tidal work into heat — against the viscosity of the body's interior, for a Maxwell rheology at Io's size, density and forcing period. Both axes are logarithmic and the curve is not monotonic, which is the whole content of the figure. At high viscosity the body is elastic: it stores the energy the tide puts in and gives it back, and dissipates nothing. At low viscosity it is fluid: it deforms all the way and does so in phase with the forcing, and dissipates nothing again. Everything happens in between, at viscosities for which the Maxwell time — viscosity divided by rigidity — is comparable to the orbital period, and the peak here is 0.735 at 10^13.5 pascal seconds. Two consequences follow, and they pull in opposite directions. The peak is an upper limit: a homogeneous body of this size cannot dissipate more than that however its viscosity is chosen, so a measured heat flow above it would refute the model rather than constrain it. And below the peak the observed value is met twice — at 10^11.5 and at 10^15.5 pascal seconds — so a heat flow alone does not say which side of the peak the interior is on. Breaking that degeneracy needs a second observable, and the usual one is the phase of the response rather than its size. The picture treats the body as one homogeneous Maxwell solid, which is certainly wrong for a moon with a molten layer; a partial melt concentrates the dissipation and shifts the peak.

Gravitation

One heat flow, and two viscosities

Io radiates a hundred thousand gigawatts of tidal heat, and that number is supposed to say something about the rock inside it. It does, and not what one would expect — because dissipation vanishes at both extremes of viscosity, the measured heat is produced by two different interiors and cannot choose between them.

$One mass and one radius, and every composition that gives them. A planet of 5 Earth masses and 1.6 Earth radii, and the compositions consistent with it. The horizontal axis is the fraction of the planet's mass in an iron core and the vertical axis the fraction in a water layer outside the rock; the heavy curve is every pair that reproduces the measurement exactly, and the band around it is what the 0.05 Earth-radius uncertainty allows. The answer is a curve, not a point, and that is not a failure of precision. Two numbers cannot determine three components: a planet can be made denser by adding iron or lighter by adding water, and along this locus the two changes cancel exactly. The ends of it are not small variations on one planet. At the left is a body with no iron at all and 0 per cent of its mass in water; at the right, one with an iron core like Mercury's and 23 per cent water. Those have different formation histories, different interiors, different everything, and the same mass and radius to the precision anybody can measure them. Breaking the degeneracy needs an observation that is neither a mass nor a radius. The usual one is a transmission spectrum, which measures the atmosphere's scale height and so its mean molecular weight — a hydrogen envelope and a steam envelope differ by a factor of nine in that, and the corresponding factor in the size of the spectral features. What the picture assumes is that the planet is differentiated into clean layers, which is the standard assumption and is false in detail: water dissolves into silicate melt at these pressures, and a mixed interior sits at neither end of this curve.$ Exoplanets

One density, and every planet that has it

A mass and a radius are two numbers, and a differentiated planet has at least three components. The set of compositions matching a measurement is therefore a curve rather than a point — and its two ends are a body with no iron and half its mass in water, and a body with a Mercury-like core.

The inflation threshold at 2·10⁵ W m⁻², and the 0.69 R_J above it. Radius against the starlight received, for a population of Jupiter-mass planets. The horizontal line is what a structural model gives for a cold, old, Jupiter-mass ball of hydrogen and helium: 1.06 Jupiter radii, and it hardly depends on mass at all in this range, because degeneracy is beginning to set in and the mass–radius relation is flattening toward its turnover. Planets receiving less than about 2·10⁵ watts per square metre sit on that line, with a median of 1.06, which is the control the rest of the figure depends on: the models are not wrong in general. Above the threshold the radii climb, reaching a median of 1.75 — half again the size a cold planet of the same mass can be — and the onset is sharp enough to be called a threshold rather than a trend. Starlight by itself will not do this. Irradiation is absorbed high in the atmosphere and re-emitted from there; it slows the escape of heat from below, which delays contraction, but it cannot deposit energy beneath the radiative–convective boundary, and it is the interior entropy that sets the radius. So the excess is evidence for a mechanism that carries roughly half a per cent of the incident flux down to pressures of tens of bars — ohmic dissipation of currents driven through a partly ionised atmosphere, breaking gravity waves, and tidally forced turbulence are the candidates, and the threshold is the number each of them has to reproduce. The points are a synthetic population from a seeded generator, not a catalogue; what is real is the threshold, the size of the excess, and the fact that the un-irradiated planets sit exactly where they should.

Exoplanets

A radius no cold planet is allowed

A Jupiter-mass ball of hydrogen has a maximum size, and it is about 1.06 Jupiter radii however old or young it is. Hundreds of hot Jupiters are half again that, and the excess switches on sharply above a threshold in the starlight they receive — which means something is putting energy in deep.

$Mercury's forced libration: 38.5″ against the 16.2″ a solid body would give. The measurement that finds a liquid core from a distance. A body on an eccentric orbit does not feel a steady torque: the pull on its equatorial bulge swings back and forth through the orbit, and the body rocks about its mean rotation by a small angle. How small depends on how much moment of inertia has to be rocked, and on nothing else — every other factor in the problem belongs to the orbit or to the body's own measured gravity field. The horizontal axis is therefore the fraction of the total polar moment that participates, one if the whole body turns rigidly together, and the vertical axis is the resulting amplitude. The curve is a rectangular hyperbola, because the same torque applied to less moment produces proportionally more angle. A Mercury turning in one piece would librate by 16.2 arcseconds. Radar measurements of the actual rocking give 38.5 ± 1.6, which is 2.38 times larger and many standard deviations away, so only 42 per cent of the moment is being rocked at all. The other 58 per cent is not following the mantle on an eighty-eight-day timescale, and the only way for an interior not to follow its own mantle is for the two to be separated by a liquid. That is how a planet nobody has landed on was shown to have a molten core — by watching, from Earth, the tiny irregularity of its turning. The figure treats the librating shell as rigid, which is right for a rocky mantle and wrong for an ice shell floating on an ocean, where the shell's own elasticity enters at the same level as the effect.$ The observed sky

Four numbers that weigh a planet's core

Mercury rocks about its mean rotation by thirty-eight arcseconds, which is more than twice what a planet turning in one piece could manage. The excess says that most of the planet's moment of inertia is not following the mantle on an eighty-eight-day timescale, and the only thing that does not follow a mantle is a liquid.

$Two interiors, 2.59 mm/s apart, against a floor of 0.02. What a radio link measures when a spacecraft flies past a moon. The horizontal axis is time from closest approach in minutes and the vertical axis is the accumulated change in the line-of-sight velocity in millimetres per second, after the pull of the moon as a point mass has been fitted and removed. What is left is the part of the field that is not spherically symmetric, and the two curves are what two interiors predict for it. The upper one is a body whose tidal Love number is 0.616 — a shell floating on a global liquid layer, free to deform almost as a fluid would. The lower one is a body solid throughout, at 0.03. They differ by 2.59 millimetres per second in the accumulated deflection, against a floor of 0.02 for a coherent two-way X-band link integrated over tens of seconds: 130 standard deviations in a single pass. That ratio is the whole reason the measurement is possible, and it is worth stating what is being compared. The point-mass deflection itself is 837 metres per second — five orders of magnitude larger than the signature of interest — so the interior is not read off the Doppler curve but off the residual left after a model of everything larger has been subtracted: the moon's mass, the planet's, the spacecraft's own thrusting and outgassing, the plasma along the path, the station's motion, and relativity. Every one of those has to be right to a part in a hundred thousand before the last curve here means anything, which is why gravity science needs many passes and a global fit rather than one flyby and a subtraction. The published uncertainty on Titan's k₂ is eleven per cent rather than the fraction of a per cent this signal-to-noise would suggest, and the difference is entirely correlations with the other parameters in that fit — a reminder that a formal error on a curve is not the error on the number extracted from it. The straight-line path assumed here is exact only in the limit of a fast flyby; a slow one bends, and the bending is solved for rather than approximated.$ Spaceflight

An ocean found in a Doppler residual

A spacecraft flying past a moon is deflected by hundreds of metres a second, and the part of that deflection which says whether the moon has an ocean is two millimetres a second. Everything larger has to be modelled and removed first, exactly, and what is left over is an interior.

$A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.$ Stars

A wind that takes no mass and all the spin

The Sun loses about a ten-thousandth of itself to its own wind over its whole life. It loses most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor — the gas is forced to keep turning with the star until it is a dozen or more radii out from the surface it left.

Stars

The clock that starts by forgetting

An ordinary star tells nothing about its age. It sits on the main sequence for billions of years at almost fixed brightness and colour, and the one property that changes monotonically is how fast it turns — but only because the braking law destroys the initial condition first, and only until it stops.

$A star 1.24 times wider than it is tall, and 19 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.9257 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.245 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.768, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.329 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.320 at the theoretical exponent 0.25 and 1.232 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.19 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.$ Starlight

A temperature that depends on where the observer stands

A star turning near its break-up rate is half again as wide as it is tall, and its equator is thousands of degrees cooler than its poles. Neither of those is a small correction to a spectrum — the effective temperature and the luminosity such a star appears to have are partly statements about which way its axis happens to point.

An oscillation that does not matter, on a ramp that does. Stored angular momentum in a reaction wheel over 160 days at 550 kilometres, with a capacity of 25 newton metre seconds. The total environmental torque is 1.75e-4 newton metres, of which 35 per cent is taken to survive averaging over an orbit. The fast oscillation is the part that does not survive: it has the 95.6-minute orbital period, reaches 0.10 newton metre seconds, and returns to where it started every revolution, so it consumes capacity and nothing else. The ramp under it is the secular part, and its slope measured between two instants a whole number of orbits apart is 6.117e-5 newton metres, which is the secular torque and is how the figure checks itself. The wheel fills in 4.7 days and has to be emptied 33 times in the span drawn. Every attitude-controlled spacecraft in the collection lives on this sawtooth, and the vertical drops are the only part of it that costs anything: the store can be moved between wheels for nothing, and taken out of the vehicle only by pushing against something outside it.

Spaceflight

The spin that has to be put somewhere

A spacecraft holding an attitude is not resisting a force. It is absorbing a slow, one-directional trickle of angular momentum from the gradient of gravity across its own body, from sunlight, from the last of the atmosphere — and every store it has for that trickle fills up.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.

Galaxies

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

Gravitation

The second number a black hole has

A black hole in equilibrium is described by its mass and its spin, and nothing else. The mass decides how strongly it pulls. The spin decides how much light a kilogram of infalling matter can emit before it disappears — and between the two extremes that figure changes by a factor of seven.

Galaxies

A cloud that cannot become a star

A molecular cloud core turning once every twenty million years sounds like a body at rest. Conserve its angular momentum through a collapse by a factor of ten thousand and it stops at the orbit of Neptune, spinning, having failed to make anything at all.

A shear layer that should have spread

The Sun's convection zone turns differentially — its equator laps its poles about once every three months — and the radiative interior below turns as one rigid piece. Between them is a transition four per cent of the radius thick. Nothing in hydrodynamics keeps a velocity discontinuity that thin for four and a half billion years.

Gravitation

Every pair arrives circular

Gravitational radiation drains a binary's energy and its angular momentum at rates that do not keep step, and the mismatch destroys eccentricity far faster than it shrinks the orbit. A pair that starts at 0.99 and spirals in from a fifth of an astronomical unit is round to better than a part in a hundred million by the time a detector can hear it.

Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway.

Stars

A neutron star born turning too slowly

Collapse an iron core a few thousand kilometres across down to twelve, and conservation of angular momentum multiplies its rotation rate by about eighteen thousand. A model with no transport in it delivers a newborn pulsar at the break-up limit; the ones that are observed turn twenty times slower, which is a measurement of the core before it fell.

The last parsec, priced in stars. The two timescales that shrink a pair of 10⁸-solar-mass black holes at the centre of a merged galaxy, against their separation in parsecs, both axes logarithmic, for a stellar velocity dispersion of 200 km/s and a central density of 500 solar masses per cubic parsec. Ejecting stars hardens the binary at a rate proportional to the separation, so that timescale grows as the orbit shrinks; gravitational radiation goes as the fourth power of the separation, so its timescale collapses. Neither alone finishes the job and the crossing of the two is where the answer is set. With the loss cone kept full, the crossing is at 2.9e-2 parsecs and the total is 4.2·10⁸ years, inside a Hubble time. With the supply of low-angular-momentum orbits emptied by a factor of 100 — which is what happens in a smooth spherical nucleus, because the stars that could interact have already been thrown out and two-body relaxation refills the orbits far too slowly — the crossing moves outward to 7.3e-2 parsecs and the total becomes 1.7·10¹⁰ years. That is the final-parsec problem, and it is not a problem about gravity: it is a problem about supply. Real nuclei are not spherical, and the figure cannot show what a triaxial potential does, which is to keep feeding the binary orbits it has not already used.

Galaxies

The last parsec, and the stars that are not there

Two galaxies merge and their central black holes sink towards each other, and then stop. From about a parsec apart, friction no longer works and gravitational radiation is not yet strong enough — and the only mechanism in between throws away the stars it depends on faster than they can be replaced.

Exoplanets

A misalignment only cool stars forget

A third of hot Jupiters orbit at a large angle to their star's equator, and some go round backwards. Sort the same planets by the temperature of their host and the picture changes — below about 6,250 kelvin almost all are aligned, and above it almost none are. The boundary is not about the planets.

Stars

A surface that slowed because the star grew

A star leaving the main sequence expands by a factor of a hundred. Conserve its angular momentum and its equator slows by four orders of magnitude, which is why every red giant is a slow rotator. One or two per cent are not, and a giant turning at fifty kilometres a second has to have been given the angular momentum by something else.

Stars

A field that would have arrived ten thousand times too strong

A collapsing cloud carries its magnetic field with it, and the arithmetic of that is not negotiable — the field grows as the two-thirds power of the density. Run it from a diffuse cloud to a protostellar core and the answer is four orders of magnitude above anything measured on a young star.

Stars

The darkness a field pays for

A sunspot is not cool because something is missing. It is cool because a three-thousand-gauss field supplies part of the pressure that holds the column up, the gas therefore supplies less, and the level at which that gas becomes opaque sits some hundreds of kilometres deeper than the surface around it.

$A slope, not an angle. Above: the polarisation angle of a background source against the square of the observing wavelength, at the five wavelengths a radio survey actually uses. The plane rotates as it passes through magnetised plasma, by an amount proportional to the integral of the electron density times the field along the path — and to λ². The intercept is the angle the source emitted at, which nobody knows, so a measurement at one wavelength contains no information whatever; the slope is the rotation measure, here 42 radians per square metre, and it needs no knowledge of the source at all. Below: the fractional polarisation that survives. A telescope beam covers many lines of sight with slightly different rotation measures, their angles disagree by more at longer wavelengths, and the vector sum collapses — which is why the useful band has a long-wavelength edge that has nothing to do with sensitivity. Divide the rotation measure by the dispersion measure of the same path, 26.8 in the usual units, and the electron density cancels: the mean line-of-sight field is 1.93 µG, obtained without knowing the distance, the density, or where along the path the field was.$ Starlight

A slope that needs no source

Polarised light passing through magnetised plasma has its plane rotated, by an amount proportional to the square of the wavelength. A single measurement of the angle is worthless, because nobody knows the angle the source emitted at. A measurement of the slope against wavelength squared does not need to know.

A vanishing field that changes the answer completely. The growth rate of the magnetorotational instability against wavenumber, both in units the orbital frequency sets: the vertical axis is the growth rate divided by Ω, the horizontal is the wavenumber multiplied by the Alfvén speed and divided by Ω. The maximum is exactly three quarters of the orbital frequency and it does not depend on the field strength at all — it occurs at kv_A = √15Ω/4, so a weaker field simply moves the fastest-growing wavelength to a longer one. The curve vanishes above kv_A = √3Ω, which is the only thing the field decides: modes shorter than that are stabilised by magnetic tension. An unmagnetised Keplerian disc is Rayleigh-stable and would never accrete; a disc with a field a millionth of the strength needed to matter dynamically grows a mode that doubles in about a fifth of an orbit. The limit is discontinuous, which is the reason this was found late and by algebra rather than early and by observation.

Gravitation

The weakest field changes the answer

A Keplerian disc is stable by every hydrodynamic test there is. Add a magnetic field of any strength whatever — a millionth of what would matter dynamically — and it becomes violently unstable, at a growth rate of three quarters of an orbit per radian that does not depend on the field at all.

A boundary that a sixty-fourfold gust moves by a factor of two. The standoff distance of a dipole magnetosphere against a wind, in planetary radii, against the wind's proton density, for three wind speeds. The boundary sits where magnetic pressure equals ram pressure — the only balance available, since the field exerts no force on neutral gas and the wind touches nothing. Because a dipole falls as the cube of distance its pressure falls as the sixth power, so the standoff goes as the ram pressure to the −1/6: at 450 km/s a density of 6 per cubic centimetre puts the nose at 9.3 radii — against a measured average of ten to eleven — and a sixty-fourfold compression moves it only to 4.7. That exponent is the reason a magnetosphere is a stable thing to have. It is also the reason the boundary's position is a poor measurement of the field: a factor of two in the distance is a factor of sixty-four in what caused it, so the standoff measures the wind badly and the planet's moment hardly at all.

The observed sky

A boundary that hardly moves

The magnetopause sits where the planet's magnetic pressure equals the solar wind's ram pressure. Because a dipole falls as the cube of distance, its pressure falls as the sixth power — so a sixty-fourfold gust in the wind moves the boundary by a factor of two, and a boundary that will not move is what makes a magnetosphere a stable thing to have.

A field strength read off the scatter of directions. The Davis–Chandrasekhar–Fermi estimate: field strength against the dispersion of polarisation angles across a cloud, at a density of 10⁴ molecules per cubic centimetre and a turbulent velocity of 0.9 kilometres a second. The polarisation directions themselves contain no field strength — an aligned grain reports which way the field points and nothing about how hard it is pulling. The strength is in the disorder. Turbulence of a given energy bends a stiff field less than a weak one, so the angular scatter is the ratio of the turbulent velocity to the Alfvén speed, and inverting it gives the field: 220 µG at a 9-degree dispersion. The relation is exactly inverse, so the estimate is most trustworthy where the field is most ordered and worst where it matters most. The factor of 0.5 in front is not theory — it is what simulations of a known field, analysed this way, turn out to need, and the honest statement is that this method is calibrated rather than derived.

Galaxies

A direction measured by something with no strength in it

An interstellar grain spins with its long axis across the magnetic field, so starlight through a cloud is polarised along the field and the cloud's own emission across it. The observable contains no field strength whatever — and the strength is recovered anyway, from how much the directions disagree.

The energy at which the sky begins to point. Gyroradius against energy for a singly charged particle, in three field strengths, with the thickness of the galactic disc and the size of its halo marked. A cosmic ray is not an image of anything: its path is a helix about a field line, and by the time it arrives the direction it came from has been erased. The erasure is quantitative. At 3 µG a proton's gyroradius equals the disc's half-thickness at 4.2·10¹⁷ electronvolts and the halo's radius at 4.2·10¹⁹, so below the first the particle is stored and stirred for tens of millions of years, and only above the second does it travel in something like a straight line. The measured spectrum has a break — the knee — at about 3×10¹⁵ eV, which is within a factor of a few of the first of those; the sky only starts showing structure above 10¹⁹, which is the second. Two features of a spectrum measured on the ground, both located by one straight line on this plot.

Galaxies

The energy at which a sky begins to point

A cosmic ray arrives from a direction that has nothing to do with where it came from. Its path is a helix about a galactic field line, and only above the energy at which that helix is bigger than the galaxy does the arrival direction start to mean anything — which is one particle per square kilometre per century.

Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.

Cosmology

The seed that cannot be remembered

A galactic dynamo affords something like thirty e-foldings over the age of a galaxy, which multiplies any seed field above a ten-thousandth of a billionth of a microgauss up to the microgauss actually observed. The field a galaxy has today therefore says nothing about the field it started with, and the only place a seed survives unamplified is the emptiness between clusters.

The field nobody measured, and the reason the number is quoted anyway. The energy content of a radio source against the field strength assumed for it, both logarithmic. A synchrotron luminosity constrains only the product of the relativistic electron content and the field: a bright source can be many particles in a weak field or few in a strong one, and no observation of the radiation distinguishes them. The two curves are the two costs. Assuming a weak field is expensive in particles, because the particle energy needed rises as the field to the minus three halves; assuming a strong one is expensive in the field itself, which rises as B². Their sum has a minimum at 5.0e-5 gauss, and at that minimum the particle energy is exactly four thirds of the field energy — which is where the word equipartition comes from, and it is a property of where a curve turns rather than a statement about nature. The estimate is quoted because it is robust: the minimum-energy field goes as luminosity to the two sevenths, so an order of magnitude of ignorance about the luminosity is a factor of 1.93 in the answer. It is also, for the same reason, nearly uninformative — a number that hardly moves is a number that hardly measures.

Starlight

A minimum that was mistaken for a principle

A radio source's brightness fixes the product of its particle content and its field, and nothing about the radiation separates them. What is quoted instead is the field that makes the total energy least — and at that field the particles happen to carry four thirds of what the field does, which is where the word equipartition comes from and why it is not a physical assumption at all.

Two effects that are blind in opposite directions. The sensitivity of two magnetic diagnostics against field strength, on a logarithmic axis spanning five decades. The Zeeman effect measures the line-of-sight component and adds it up along the path, so a field tangled into 100 independent cells inside one resolution element averages down by a factor of 10 and reports almost nothing — which is exactly the situation in a chromosphere or a turbulent cloud. The Hanle effect is a different device altogether: a field precesses the atom between absorption and re-emission, so a scattering line's polarisation is rotated and reduced, and the amount depends on how far the precession gets in one radiative lifetime. That makes it sensitive around 1 gauss for a 100-nanosecond level, and — the useful part — it does not care about sign, so a tangled field does not cancel. Above the crossing at 27.7 gauss the Hanle signal has saturated and carries no strength information, and the Zeeman effect is the instrument. Neither is a measurement of the field; each is a measurement of what the field did to something else.

Starlight

Two instruments blind in opposite directions

The Zeeman effect measures the component of a field along the line of sight and adds it up, so a field tangled into a hundred cells reports a tenth of one cell's strength. The Hanle effect measures how far an atom precesses between absorbing and re-emitting, does not care about sign, and saturates just where the Zeeman effect becomes useful.

The observed sky

The flare that arrives from somewhere else

The solar wind blows radially outward and the Sun rotates, so the field frozen into the wind is wound into a spiral making forty-five degrees to the radius at the Earth. Energetic particles follow the field rather than the line of sight, which is why the flares that deliver particle storms are the ones on the western limb rather than the ones facing the planet.

A formula everyone uses, and the number no pulsar has. Above: the braking index measured for the 4 pulsars whose spin-down has been followed long enough to give a second derivative, against the value a magnetic dipole rotating in vacuum requires. That value is exactly 3, and it is what every catalogued field strength and every characteristic age assumes. Not one measurement reaches it: they run from 1.4 to 2.839, and all of them fall short in the same direction, which is the signature of a systematic rather than of noise. Below: what that costs. The age a spin-down history gives is the period divided by (n − 1) times its derivative, so the ratio to the quoted characteristic age is 2/(n − 1) — 5.00 for Vela. The numbers are not thereby useless: an exponent recovered from the data is exactly the kind of correction a measurement can absorb. What has gone is the claim that the field strength printed beside a pulsar is a measurement of a field. It is a measurement of a spin-down rate, read through a model the same pulsar refutes.

Stars

The exponent no pulsar has

Every field strength in the pulsar catalogue comes from one formula, which assumes the star is a magnetic dipole rotating in a vacuum and therefore that its spin-down obeys an exponent of exactly three. Where that exponent has been measured it is 2.51, 2.84, 1.4 — never three, and always short.

Gravitation

The period a star is pulled towards

A magnetised neutron star does not let a disc reach it. The disc stops where the field's stress wins, and if that radius lies inside the corotation radius the star is spun up while outside it the star is spun down — so there is one period at which nothing changes, and an accreting star walks to it.

Galaxies

A disc held open by what cannot be photographed

Solve hydrostatic equilibrium for the galaxy's gas layer using its temperature and the disc's gravity, and the answer is a layer several times thinner than the one that is there. What is missing from the calculation is turbulence, a magnetic field and a population of cosmic rays — three pressures of comparable size, two of which emit nothing.

The shield that does not shield. Above: measured ion escape rates for three planets against their surface magnetic field. Venus and Mars have no dynamo at all and Earth has one, and the three rates lie within a factor of 9 — with the magnetised planet losing the most. The intuition that a magnetosphere protects an atmosphere is not a small correction away from being right; the measurement does not support it. Below: why. A dipole's field lines are not all closed. Those emerging within a polar cap reconnect with the wind's own field and lead straight to space, and the cap's area is set by how far the magnetosphere reaches — a boundary at ten planetary radii still leaves 5.1 per cent of the surface open. So a magnetosphere is both a shield and a funnel: it deflects the wind from most of the planet and collects ions from the whole ionosphere into the polar wind, which is exactly what an instrument above the poles measures leaving. Whether the net is protection depends on quantities nobody can compute from the field strength alone, and the three points above are the state of the evidence.

Exoplanets

The shield that is also a funnel

A magnetic field is supposed to protect an atmosphere from the stellar wind. Venus and Mars have no dynamo and Earth has one, and their measured ion escape rates lie within a factor of a few — with the magnetised planet losing the most, because a dipole's polar field lines are open and lead straight to space.

Two geometries, one bit of data, and the bit decides. Rotation measure against galactic longitude for two field geometries that look identical in every image ever taken. An axisymmetric field runs the same way round the disc at every azimuth, so the component along the line of sight changes sign exactly 2 times as the longitude goes round — once where the field turns toward the observer and once where it turns away. A bisymmetric field reverses its own direction from one side of the galaxy to the other, and that doubles the count to 4. Nothing about the brightness, the arms or the colour distinguishes the two; the sign of a rotation measure does, and a sign is one bit. The measurement is made against several hundred background sources, each contributing one rotation measure through the whole disc — the sources are not the object of study, they are the illumination. The Milky Way's own answer is untidy: broadly axisymmetric with at least one reversal inside the solar circle, which is neither clean case, and is why the field's origin is still argued about.

Galaxies

Two geometries and one bit to choose between them

A galaxy's large-scale magnetic field either runs the same way round the disc everywhere or reverses from one side to the other. No image distinguishes the two. The sign of a rotation measure does — two reversals around the sky for one geometry and four for the other — and a sign is one bit of information.

Stars

A threshold with no free parameter in it

Divide a cloud's mass by the magnetic flux threading it. Gravity and the magnetic force both fall as the inverse square of the radius, so the ratio cannot change during a collapse — and the critical value that separates a cloud which must collapse from one that never can is one over two pi root G, a pure constant.

Orbits

An eccentricity that cannot be zero

Fit an orbit to noisy data and the eccentricity that comes back is never zero, not even when the orbit is a perfect circle. The reason has nothing to do with the data and everything to do with the fact that a length cannot be negative.

Three passes, and the fourth is below the noise. How far the position is still wrong after each pass of the light-time iteration, for five targets, on a logarithmic scale of kilometres. The first pass uses the body's position now and is wrong by exactly the distance the body travels while its light is in transit — 6254 kilometres for Mars near opposition, which is about twenty-five arcseconds and larger than any residual in an ephemeris fit. Each further pass multiplies the error by the body's speed divided by the speed of light, so the lines are straight and their slopes are that ratio and nothing else. Three passes take every one of these below a metre, which is why the loop in an ephemeris code has a fixed trip count rather than a convergence test. The same argument run backwards is why the correction cannot be applied to the observation instead: the direction light arrived from is what the observation is, and the position it corresponds to is not known until the body's orbit is.

Orbits

Where a planet is and where it is seen

An ephemeris is fitted to observations, and no observation is of a position. It is of a direction light arrived from, at a time that is not the time the light left, bent by a Sun that is nowhere near the line of sight. Three corrections stand between the two, and all three are larger than the residuals.

A branch that adds nothing above 1 km and everything below it. Cumulative crater counts on a 3.5-billion-year-old surface, with the population split into the craters made by objects arriving from outside and the craters made by blocks thrown out of larger ones on the same surface. The two are indistinguishable in a photograph and completely different as a statistic. Secondaries stop at about 1 kilometre, because that is the largest crater a block leaving at a few hundred metres a second can excavate, so the upper half of the plot is unaffected. Below it they are steeper — slope -3.2 against the primaries' -2 — and by the smallest diameter drawn they outnumber the primaries 292 to one. A count taken at 100 metres and read through the primary production curve returns an age of 4.34 billion years for ground that is 3.5, and it returns it with a small formal error, because the counting statistics are excellent. The error is not in the counting.

Orbits

The craters that were not primary

Counting craters dates a surface, and the method works because impacts from space arrive at a known rate. Some of the holes were not made from space. They were made by rock thrown out of the larger holes on the same surface, and they are far more numerous than anything that arrived.

Orbits

An acceleration that was the spacecraft's own heat

Two probes leaving the solar system were tracked for thirty years and both drifted from their predicted paths by a tenth of a nanometre per second squared. The residual was real, it was constant, and it was the same on both. It was also the waste heat of the reactors that powered them, radiating slightly more one way than the other.

A moment of inertia that is a shape plus an assumption. The Darwin–Radau relation: the polar moment of inertia a body would have, given the ratio of its rotation parameter to its flattening, if it were a hydrostatic fluid. The curve passes through exactly 0.4 at q/f = 0.8, which is the uniform sphere and the one point that needs no interior model, and falls as the body becomes more centrally condensed. Four of the five worlds drawn sit on it within a few per cent of their independently measured moments, which is what makes the relation usable at all. The world that does not appear on the plot is the Moon, whose q/f is 0.025 — far outside the window in which the relation has a real solution, because its shape is a fossil frozen in when it was much closer to its primary and has nothing to do with its present rotation. That failure is the useful one. A relation that returns a wrong answer quietly is dangerous; this one returns no answer at all.

Gravitation

The part of a shape the spin cannot explain

A rotating planet bulges by a predictable amount. Subtract that amount from the shape actually observed and something is usually left over — a few parts in a hundred thousand for the Earth, and ninety-seven per cent of the whole bulge for the Moon. The residue is not an error. It is the only remote measurement of what a planet's interior is doing that does not average over the whole body.

One amplitude, and every point on this curve fits it. The set of distances and inclinations that produce the same measured amplitude in one detector. A binary seen face-on radiates most strongly along its spin axis, so it can be twice as far away as an edge-on binary and still arrive with the same strain — the curve is (1 + cos²ι)/2 and the factor between its ends is exactly two. Nothing in a single detector's data distinguishes the two ends. What makes this worse rather than merely awkward is that the prior pulls the other way: an isotropically oriented population has half its members beyond 60 degrees, so most binaries really are closer to edge-on, and a posterior that combines a flat likelihood along this curve with that prior returns a distance biased low with an error bar that understates the range. The whole business of standard-siren cosmology is the business of cutting across this curve.

Gravitation

A distance tangled with an angle

A gravitational wave carries its own distance, with no ladder underneath it and nothing to calibrate. What it also carries, inseparably, is the orientation of the orbit that made it — and one detector cannot tell a nearby binary seen edge-on from one twice as far away seen face-on.

Four defensible choices, and a factor of 2.9 between them. The Coulomb logarithm against the ratio of the two impact parameters it is cut off at. The curve is a logarithm, so it is flat — a factor of ten in the ratio buys 2.3 — and that is usually offered as the reason not to worry. The four marked conventions are all in current use and all defensible, and they give ln Λ from 3.4 to 9.9. Since the drag force is proportional to ln Λ and not to its logarithm, that is a factor of 2.9 in every sinking time computed from it. The flatness protects the answer from a wrong guess about the ratio; it does not protect it from there being no correct guess, which is the actual situation.

Gravitation

A drag computed with a logarithm nobody can pin down

Chandrasekhar's drag formula is exact, derived from first principles, and contains a logarithm of a ratio of two lengths that the derivation does not supply. Every sinking time in astronomy is proportional to that logarithm, and the four conventions in current use differ by a factor of three.

$One measurement, one line, and every point on it is an interior. The Love number against the quality factor, both logarithmic. An orbital measurement — a moon observed to be receding, a spin observed to be slowing — determines only the ratio of the two, so it picks out a diagonal band rather than a point, and every interior along that band reproduces the observation exactly. A body that deforms twice as easily and dissipates half as efficiently is indistinguishable from one that does the opposite. What breaks it is a measurement of the deformation on its own: a spacecraft tracking the body's gravity field through a tidal cycle measures k₂ directly, which is a vertical line here, and the intersection gives Q = 5.36·10⁴. The uncertainty on that answer is the two fractional errors added in quadrature, 20 per cent, and it is dominated by whichever of the two was worse — which for every body in the solar system is the orbital rate rather than the Love number.$ Gravitation

A heat flow that depends on a number nobody can compute

Every tidal rate in astronomy — a moon receding, a spin slowing, an orbit circularising, a satellite melting — is proportional to one combination of two quantities that no orbital measurement can separate. One of them describes how much a body deforms and the other how badly it leaks, and only a spacecraft can tell them apart.

The observed sky

A second that depends on where the clock is

A clock in orbit runs fast, a clock at the bottom of a gravity well runs slow, and a clock at rest far from the Sun runs faster than anything on Earth by fifteen parts in a thousand million. Astronomy therefore has four different seconds, two of them longer than the others, and an ephemeris has to declare which one it is tabulated against.

A frame whose precision stops improving because its sources move. The uncertainty in the orientation of a celestial reference frame, in microarcseconds, against the number of extragalactic sources it is built from. The falling line is what averaging alone would give: each source's position is measured to about 200 microarcseconds and combining N of them improves the frame as one over the square root of N. The upper curve adds the part that does not average down in the same way — the wander of a quasar's radio centroid as new components are ejected along its jet, which is a real motion of the thing being used as a fixed point. Three catalogue generations are marked. The frame gained an order of magnitude in a quarter of a century, and it gained it by observing more sources rather than by observing any of them better, which is the signature of a limit that is in the objects rather than in the instrument.

The observed sky

A frame made of things that are not points

Every position in astronomy is measured against a set of objects declared to be fixed. The objects chosen are quasars, because they are the most distant things there are — and each of them is a jet whose radio brightness centre wanders by tens of microarcseconds as new material is ejected from the core.

$196 nanometres of wavefront, and a Strehl that depends entirely on the colour. Above, the error budget of an adaptive-optics system, term by term, in nanometres of residual wavefront. The terms are independent and add in quadrature, so the total is 196 nanometres and is dominated by the two largest; removing the smallest term entirely would improve it by under six per cent, which is why an optimisation programme that does not know which term is largest achieves nothing. Below, what that residual delivers: the Strehl ratio, the fraction of the light in the diffraction core, against wavelength. It is the exponential of minus the square of the residual measured in radians, and a radian is a wavelength, so the same physical error is four times smaller in phase at two microns than at half a micron. The system drawn here delivers 1 per cent of its light into the core at 550 nanometres and 73 per cent at 2200, and nothing about it changed between the two.$ The observed sky

An error budget added in quadrature

Adaptive optics does not deliver a resolution. It delivers a fraction of the light in the diffraction core, and that fraction is the exponential of minus a sum of squares. Five independent failures of the correction add in quadrature, the largest one decides everything, and the same hardware is useless in the visible and excellent in the infrared.

The observed sky

A solar radius measured past a mountain range

The most accurate way to measure the Sun's diameter is to time an eclipse. What is timed is the moment sunlight vanishes behind the Moon's edge — and the Moon's edge is a horizon with mountains on it, so the measurement is a difference between the Sun's limb and a lunar landscape that has to be supplied from somewhere else.

Starlight

A velocity that is an average of lines that disagree

A radial velocity measured to a metre a second is not measured from a line. It is the position of the peak of a cross-correlation against a mask of thousands of lines, and those lines do not agree with each other by hundreds of metres a second — because each one forms at a different depth in an atmosphere that is boiling.

Starlight

A response measured pixel by pixel

Two completely different quantities are called the flat field. One is the detector's pixel-to-pixel response, which everybody removes and which averages away anyway. The other is the illumination pattern of the optics, which is smooth, is different for every method of measuring it, and survives into every magnitude the instrument produces.

Starlight

A continuum that was never observed

An equivalent width is an area measured relative to the continuum, and the continuum is not in the data. It is drawn — a curve through the highest points of the spectrum — and in any spectrum with many weak lines those highest points are already below the true continuum, because the weak lines have eaten the gaps.

Four spectra, and 3.9 magnitudes between them at the same redshift. The K-correction — the magnitude that has to be added to compare a redshifted object with a nearby one through the same filter — against redshift, for four power-law spectra. At zero redshift every correction is zero by construction. Beyond that they diverge, because a filter at a fixed observed wavelength samples a different part of the source's own spectrum at every redshift, and how much flux is there depends on the spectrum. A flat spectrum needs no correction at all at any redshift, and the two extremes drawn differ by 3.9 magnitudes by z = 1.2. The circularity is the point: applying the correction requires the spectrum, and the spectrum is what a magnitude is being used to constrain. The dashed line is the way out — observe in a band chosen so that it lands on the rest-frame band of interest, and the spectral term cancels, leaving only the bandwidth stretch.

Starlight

A magnitude in a band the source never had

A filter passes light at a fixed observed wavelength, and a redshifted source emitted that light at a shorter one. Comparing a distant galaxy with a nearby one through the same filter therefore compares two different parts of two spectra — and the correction between them needs the spectrum, which is what the magnitude was going to be used to find out.

$One over a noisy parallax, at three precisions. The distribution of the distance obtained by inverting a parallax, for a star truly at 100 parsecs measured with fractional errors of 5, 10, 20 per cent. At five per cent the distribution is nearly symmetric and inverting is harmless. At twenty per cent it is strongly skewed: the mean sits at 105 parsecs rather than 100, and the tail runs to distances several times the truth, because a parallax scattered a little towards zero is a distance scattered a long way outward. The asymmetry is a Jacobian and nothing else — the parallax measurement is unbiased and symmetric throughout. Above about twenty per cent the mean of the distribution stops existing at all, because the density falls only as the inverse square of the distance and the integral of d times that diverges.$ Starlight

The distance is not one over the parallax

A parallax is measured with symmetric errors and a distance is one over it. Inverting a noisy positive quantity is not a change of units — it is a change of distribution, and the one that comes out is skewed, biased outward, and above about twenty per cent error has no mean at all.

Starlight

A sample brighter than the population it came from

Every survey stops at some apparent brightness. At any distance it therefore contains only the objects luminous enough to make the cut, so the average object in it is brighter than the average object in the universe — by an amount that grows with distance and that has been shortening every distance in astronomy since 1920.

Starlight

The tallest peak in nothing at all

A periodogram of pure noise has peaks in it, and the tallest is not small. How tall it has to be before it means something depends on how many frequencies were searched and on what the noise actually is — and astronomical noise is almost never the white noise the standard formula assumes.

Two diagnostics, two bands, and a crossing to 49 kelvin. The plane of effective temperature against surface gravity, with the constraints from two spectroscopic diagnostics drawn as bands. The wings of a hydrogen line are broadened by collisions, so they respond steeply to the gravity and weakly to the temperature: a narrow, steep band. An ionisation balance — requiring that the same element give the same abundance from its neutral and its singly ionised lines — responds to both, and its band is much shallower. Neither diagnostic determines either quantity on its own. Where the two cross is the answer, and the size of the crossing region is set by the band widths divided by the difference of the slopes — so two diagnostics that respond similarly give a long, thin, nearly useless error region however precise each one is. Choosing diagnostics that disagree in their sensitivities is the whole of the art.

Starlight

A temperature and a gravity that trade against each other

A stellar spectrum contains the star's temperature, its surface gravity and its composition, and no single feature in it contains only one of the three. Every diagnostic is a band in the parameter plane rather than a point, and the answer is where the bands cross — which makes choosing diagnostics that disagree in their sensitivities the whole of the art.

An instrument 1.4 times as polarised as the sky it is measuring. The Stokes plane, with the two linear polarisation parameters as axes. The tight cluster near the centre is a set of stars known to be unpolarised, observed through the same instrument: they should sit at the origin and do not, and their mean is the instrumental polarisation — 0.77 per cent here, which is 1.4 times the real polarisation of the field. The other cluster is the field stars, whose measured values are the sum of their own polarisation and the same instrumental offset. Subtracting one mean from the other recovers 0.59 per cent at the right angle. Two things make this worth doing carefully. The offset is a vector, so leaving it in rotates the measured position angle as well as changing its magnitude — by 31 degrees here — and a calibration that only fixes the scale does not touch that. And the offset depends on where the telescope was pointing, because the reflection angles do, so the standards have to be observed at the same place in the sky and at the same instrument rotation.

Starlight

An instrument more polarised than the sky

Every oblique reflection polarises. A telescope is a stack of oblique reflections, so it adds a fraction of a per cent of polarisation to everything it looks at — which for most astronomical sources is more than they have themselves, and which is a vector rather than a scale error, so it rotates the answer as well as changing its size.

$A high end that is 2.5 in slope steeper than the one stars were born with. The mass function a star count measures against the one stars were born with, for a population that has been forming stars steadily for 10 billion years. Below about a solar mass nothing has had time to die, so the two coincide exactly. Above it the fraction still alive is the main-sequence lifetime divided by the age, and since the lifetime falls as the two-and-a-half power of the mass, the present-day function is steeper than the initial one by exactly that exponent. A count of massive stars in an old population therefore under-represents them by orders of magnitude, and reading it as an initial mass function gives a slope far too steep. The correction is large, it is calculable, and it depends on the star-formation history — which is usually the thing the mass function was going to be used to constrain.$ Stars

A mass function corrected by an age

Counting stars by mass gives the stars that are alive. What every argument needs is the stars that were born, and the two differ by the fraction of each mass still on the main sequence — which is a lifetime divided by an age, and which for massive stars in an old population is a very small number.

$4 per cent in one observable is 48 per cent in an age. How an error in the calibration of the large frequency separation propagates into the quantities derived from it. The two scaling relations are exact in their exponents, so a fractional error in the separation appears as twice that in the radius, four times in the mass, and — because a main-sequence lifetime falls as roughly the two-and-a-half power of the mass — ten times in an age. At the 4 per cent level, which is about what the theoretical corrections to the relation amount to for a red giant, that is 15 per cent in mass and 48 per cent in age. Nothing about the seismology is uncertain at that level; the frequencies are measured to parts in a thousand. What is uncertain is the constant of proportionality, and it is uncertain because it was calibrated on one star.$ Stars

Two scaling relations calibrated on one star

Asteroseismology gives a star's mass and radius from two numbers read off its oscillation spectrum. The two relations are exact in their exponents and approximate in their constants, and the constants were fixed by requiring that the Sun come out right — so an error of a few per cent in one observable is tens of per cent in a mass and nearly a factor in an age.

A free parameter worth 88 kelvin across its plausible range. The effective temperature a stellar model predicts, against mass, for three values of the mixing-length parameter. The parameter has no derivation: it is the distance a convective blob is supposed to travel before dissolving, in units of the local pressure scale height, and it is fixed by requiring that a model of the Sun reproduce the Sun. The three curves span 88 kelvin, which at fixed luminosity is a radius difference of 1.5 per cent — comparable to the precision with which radii are now measured by interferometry and by eclipsing binaries. Every stellar age, every isochrone and every mass inferred from a position in the temperature–luminosity plane depends on the value chosen, and there is no reason beyond convenience to expect the solar value to apply to a red giant or to a metal-poor dwarf.

Stars

A length nobody derived, fitted to one star

Convection in a star is turbulent, three-dimensional and impossible to compute inside an evolution code. What is used instead is one number — how far a blob of gas travels before dissolving — fixed by requiring that a model of the Sun come out with the Sun's radius, and then applied to every star ever modelled.

Stars

A convective boundary with no theory to fix it

A convective core has an edge where buoyancy vanishes. A blob arriving there still has momentum, so it carries on, mixing fresh hydrogen into the core and extending the star's life. How far it carries on is a fitted parameter, and across its plausible range every stellar age changes by nearly half.

A flux that measures a temperature to 0.12 per cent. Neutrino flux against central temperature, in units of the standard model's, for the three main solar channels. The exponents are not arbitrary: each reflects how far up the Gamow peak the reaction has to reach, so the channel with the largest Coulomb barrier is the steepest. The boron-8 flux goes as roughly the twenty-fourth power, which means a measurement good to 3 per cent constrains the Sun's central temperature to 0.12 per cent — better than any other technique by an order of magnitude. The same steepness is why the flux is useless as a check on anything else: a stellar model whose central temperature is uncertain at the half-per-cent level predicts this flux to within a factor, and the disagreement between two model families is far larger than the measurement.

Stars

A flux that is a thermometer to a tenth of a per cent

The boron-8 neutrino flux from the Sun's core rises as roughly the twenty-fourth power of the central temperature. That makes it the sharpest thermometer in astrophysics and simultaneously the most fragile prediction — a model uncertain in its central temperature by half a per cent predicts the flux to within a factor.

A density that spans a factor of 25 at 400 kilometres. Thermospheric density against altitude, for three levels of solar activity, with the model's own uncertainty band drawn around the middle curve. The extreme ultraviolet output of the Sun heats the upper atmosphere, so the scale height rises with activity and the density at a fixed altitude rises with it — by a factor of 25 at 400 kilometres between solar minimum and maximum. Superposed on that are a diurnal bulge of about a factor of two, semiannual variations, and geomagnetic storms that raise the density by tens of per cent within hours. The best empirical models reproduce past conditions to about 15 per cent, and orbital lifetime is inversely proportional to density, so a re-entry predicted a year ahead carries that error and the far larger one of not knowing what the Sun will do.

Spaceflight

A density model wrong by a factor of two

Everything about a low orbit's future depends on the density of the air at four hundred kilometres, and that density varies by a factor of twenty-five over the solar cycle, by two within a day, and by tens of per cent during a storm nobody predicted. Every model of it is an empirical fit, and re-entry dates are quoted with the honesty that implies.

Four media, and only two of them care about the frequency. The four propagation delays between a tracking station and a spacecraft, in metres of round-trip range, against the angle between the target and the Sun. The troposphere contributes a few metres and depends only on the elevation; the ionosphere is smaller at X band and scales as the inverse square of the frequency; the solar plasma rises steeply towards conjunction and scales the same way; and the Sun's gravitational delay rises as a logarithm and has no frequency dependence at all. That last distinction is the whole of the calibration strategy: transmitting and receiving at two widely separated frequencies measures the two plasma terms and removes them, leaving the troposphere to a weather model and the relativistic term to theory. Near conjunction the plasma exceeds everything else by an order of magnitude, which is why a spacecraft passing behind the Sun is both untrackable and the best available laboratory for measuring the relativistic term.

Spaceflight

Four media between the antenna and the spacecraft

A deep-space range measurement is a round-trip time, and the signal spends that time passing through a troposphere, an ionosphere, the solar wind and a curved region of spacetime. All four delay it, none of them is the orbit, and the whole of the navigation depends on removing them.

Exoplanets

A duration that measures an eccentricity

A transit's length is a measurement of how fast the planet was moving when it crossed, and that speed depends on where it was on its orbit. For a circular orbit the duration gives the star's density; for an eccentric one it gives the density times a factor of up to four — and if the density is known independently, the factor is the eccentricity.

Exoplanets

A planet radius is a stellar radius

A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

$A planet that subtracts 73 per cent of itself at the inner working angle. The fraction of a planet's flux that survives an angular differential imaging subtraction, against its separation from the star in resolution elements, for a sequence covering 25 degrees of field rotation. The reference image is built from the target's own frames, so a planet that has not moved far between them is present in the reference and is removed along with the speckles. How far it moves is the arc length, which is proportional to the separation — so the self-subtraction is severe close in and negligible far out, and the half-throughput point is at 2.1 resolution elements here. The consequence for any contrast curve is that it is a statement about an algorithm as well as about an instrument: the depth reached has to be measured by injecting fake planets into the data and recovering them, because no calculation predicts what fraction of a real one survives.$ Exoplanets

A star subtracted using the star

Imaging a planet means removing a halo of scattered starlight a hundred million times brighter than the planet, and no model of that halo is good enough to subtract. So it is built from the star's own exposures — and since the planet is in those exposures too, it subtracts part of itself.

Three planets that are the same spectrum. A model transmission spectrum, in scale heights of apparent radius, drawn three times: once as it is, once with the reference radius raised by 0.45 scale heights and the abundance reduced to compensate, and once with a cloud deck truncating the features. The three differ by 0.21 scale heights root-mean-square against features of 2.1, which is well inside the error bars of any real observation. The reason is structural rather than observational: a transmission spectrum measures a difference in apparent radius with wavelength and never an absolute radius, so the level is a free parameter, and shifting the level trades against the abundance almost exactly. Adding a cloud deck adds a third parameter that flattens features and trades against both. Three unknowns and one curve is why the quoted abundance uncertainties from transmission spectroscopy are so much larger than the photometric precision suggests.

Exoplanets

A spectrum flattened by cloud, or by nothing

A transmission spectrum measures how a planet's apparent radius changes with wavelength, and never the radius itself. That missing level is a free parameter, it trades almost exactly against the abundance of whatever is absorbing, and a cloud deck adds a third unknown to a curve that constrains two.

Galaxies

A dispersion inflated by orbits nobody resolved

A velocity dispersion measured from single spectra is not the dispersion of the system's centres of mass. Every star in a binary carries its own orbital velocity of a kilometre or so, and for a dwarf galaxy whose real dispersion is smaller than that, the measured value — and the dark-matter content computed from its square — is mostly binaries.

Two bands 2.1 standard deviations apart, and neither is a point. The plane of the matter density against the amplitude of matter fluctuations, with the constraints from a weak-lensing survey and from the microwave background drawn as bands. Lensing measures the shear produced by structure along the line of sight, and that shear depends on how much matter there is and on how clumpy it is in a fixed combination — more matter arranged less clumpily gives the same signal. The locus is a power law of exponent one half, and the combination it fixes is written S₈. The two bands are separated by 2.1 standard deviations, and whatever that separation is, it is a statement about the growth of structure between recombination and now rather than about either measurement's precision. Neither band alone determines either quantity, which is why the disagreement is quoted in the combination rather than in the parameters.

Galaxies

Two parameters that lensing measures as one

A weak-lensing survey measures how much the shapes of distant galaxies are distorted by the matter in front of them, and that distortion depends on how much matter there is and on how clumpily it is arranged. The two enter as a product. What the survey determines is one number, and the disagreement between surveys and the microwave background is stated in that number because neither measures either quantity alone.

Three mass models, one rotation curve. Rotation curves for three decompositions of the same galaxy, from a disc contributing 30 per cent of the outer rotation to one contributing 95. Each is the quadrature sum of a stellar disc, whose shape is fixed by the light distribution and whose amplitude is an unknown mass-to-light ratio, and a dark halo with parameters of its own. All three reproduce the same flat outer rotation, because the halo's amplitude is adjusted to make up whatever the disc does not supply. They differ in the inner few kiloparsecs, by 27 kilometres a second here — which is more than the measurement error and less than the uncertainty in the disc's own contribution, since that depends on a mass-to-light ratio nobody measures directly. The maximum-disc assumption picks the largest disc consistent with the data, and it is a convention rather than a result.

Galaxies

Three mass models that fit the same curve

A rotation curve is one function of radius and it is fitted with three components, only one of which is known. The stellar disc's contribution scales with a mass-to-light ratio nobody measures, and whatever the disc does not supply the dark halo does — so a family of models reproduces the same curve exactly, and choosing among them is a convention rather than a measurement.

A sheet of mass that changes 32 km/s/Mpc and no image. The mass-sheet degeneracy, drawn as what it does and does not change. Adding a uniform sheet of convergence and rescaling the unobservable source position leaves every image position relative to the lens, every image shape and every flux ratio exactly as it was — the transformation is an exact symmetry of the lens equation, not an approximation. What it does change is the time delay between images, in proportion, so a Hubble constant inferred from a measured delay is multiplied by the inverse of the rescaling. Across the range of sheets that a plausible line of sight can supply, the inferred Hubble constant moves by 32 kilometres a second per megaparsec — which is larger than the disagreement between the early and late measurements the technique is meant to arbitrate. Nothing in the lensing data can fix it; the constraint has to come from the lens galaxy's stellar kinematics or from the environment along the line of sight.

Galaxies

A sheet of mass that changes nothing but the answer

A gravitational lens's images are unchanged by adding a uniform sheet of matter and rescaling the source. Every position, every shape and every flux ratio stays exactly as it was; only the time delays change, in proportion. So a Hubble constant measured from a delay is multiplied by a number the lensing itself cannot determine.

Galaxies

One velocity and two distances

Inside the Sun's orbit a line of sight crosses each galactocentric radius twice, and the two crossings have identical radial velocities. So a cloud's velocity gives two candidate distances, near and far, differing by a factor — and nothing about the velocity says which, though the choice decides whether the object is ordinary or extraordinary.

$A dipole 187 times the signal, and its own harmonics under it. The amplitude of each harmonic of the observer's own motion imprinted on the microwave sky, against the anisotropies of the sky itself. Moving at 369.8 kilometres a second through a blackbody field makes it hotter ahead and cooler behind by a fraction β = v/c, giving a dipole of 3.36 millikelvin — 187 times the 18 microkelvin anisotropies. Each further harmonic is smaller by another factor of β, so the kinematic quadrupole is 4.15 microkelvin, which is comparable to the real quadrupole and has to be subtracted separately. The dipole is not a nuisance in one respect: it is the measurement of the solar system's motion with respect to the radiation, and it is the most precisely known velocity in astronomy.$ Cosmology

A dipole a hundred times the signal

The largest structure in the microwave sky is the observer. Moving through a blackbody radiation field makes it hotter ahead and cooler behind by three and a third millikelvin — nearly two hundred times the anisotropies that all of cosmology is read from — and removing it is the first operation on any map.

$A 5 per cent continuum error, and an optical depth wrong by 1.1. The mean transmitted flux of the Lyman-alpha forest against redshift, with a 5 per cent uncertainty in the quasar continuum drawn as a band. The continuum is not observed: at these redshifts every part of the spectrum blueward of the emission line is absorbed, so the level has to be extrapolated from the red side across a region where the quasar's own spectrum has structure. A fractional error in that level is a fractional error in the flux, and since the optical depth is minus the logarithm of the flux, the resulting error in the optical depth is the fractional error divided by the flux — which grows without bound as the forest goes black. At z = 2 it is 0.06; at z = 6.2 it is 1.1. That is why measurements of when reionisation ended are quoted as limits rather than values above about redshift six.$ Cosmology

A forest with no continuum left

Measuring how much neutral hydrogen sits between here and a distant quasar means measuring the fraction of its light that survives, which means knowing how much light there was. At high redshift nothing survives at the wavelengths that would show it, so the level is extrapolated across the region being measured — and the optical depth is the logarithm of a number divided by a guess.

One curve the temperature fixes, and one line the polarisation adds. The plane of the optical depth to reionisation against the amplitude of the primordial fluctuations. The temperature power spectrum of the microwave background measures the product of the amplitude and the exponential of minus twice the optical depth, so it constrains a curve rather than a point: more electrons scattering the photons out is indistinguishable from fewer fluctuations to begin with, and the two trade along the drawn locus across a factor of 1.22 in amplitude. What breaks it is the polarisation at the largest angular scales, where rescattered photons regenerate a signal whose amplitude is proportional to the optical depth itself rather than to its exponential. That constraint is nearly vertical here, it comes from a handful of multipoles at the very largest scales, and it is the single hardest measurement the microwave background has demanded — because at those scales the Galaxy's own polarised emission is larger than the signal.

Cosmology

An amplitude and a depth that arrive multiplied

The microwave background's temperature fluctuations are the primordial ones damped by everything that scattered them since. The damping is uniform, so a smaller starting amplitude and more scattering produce identical maps — and separating them requires a signal from a handful of the largest angular scales, where the Galaxy's own emission is larger than what is being measured.

Take 8 per cent off the horizon and the tension is gone. The Hubble constant against the sound horizon at recombination, along the locus the microwave background's measured angular scale fixes. What is measured is an angle — the angular size of the horizon, to a part in three thousand — and an angle is a length divided by a distance, so extracting an expansion rate requires the length. That length is computed from the physics of the first four hundred thousand years: the baryon density, the radiation density, the number of relativistic species and the recombination history. Change any of those and the locus is unchanged while the point on it moves. The horizontal band is the late-universe measurement from the distance ladder, and it meets the locus at 136 megaparsecs — 8 per cent shorter than the standard model gives. That is the arithmetic behind every proposal to resolve the disagreement by changing the early universe rather than the late one.

Cosmology

A constant that is an angle divided by a length

The microwave background does not measure an expansion rate. It measures one angle — the apparent size of the sound horizon at recombination — to a part in three thousand, and converting that angle into a rate requires the horizon's physical length, which is computed from a model of the first four hundred thousand years rather than observed.

$Residuals of 112 per cent nearby and 2.2 far out. Deviations from a pure Hubble flow, in per cent, against distance. Each galaxy carries a peculiar velocity of a few hundred kilometres a second — part a coherent bulk flow shared with its neighbours and part a random dispersion — and that velocity is added to its recession. Since the recession grows with distance and the peculiar velocity does not, the fractional error falls as one over the distance: it is 112 per cent at 5 megaparsecs and 2.2 at 250. The practical consequence is a lower cut-off on any Hubble-constant measurement: below about 40 megaparsecs the motions dominate, and the coherent part does not average away over a sample because neighbouring galaxies share it. Choosing that cut-off is one of the analysis decisions a local expansion rate depends on.$ Cosmology

A residual that is somebody else's velocity

A redshift is not a distance until the galaxy's own motion has been removed, and galaxies move at a few hundred kilometres a second. Nearby that is comparable to the expansion itself, so the local Hubble diagram's scatter is motions rather than measurement — and the motions are shared between neighbours, so they do not average away.

$The helium abundance as a count of neutrino species. The primordial helium mass fraction against the number of light neutrino species, integrated at a deuterium-bottleneck temperature of 0.0855 MeV and a neutron lifetime of 877.75 s. The curve rises at 0.01348 in Y_p per species near the standard model, and it is not quite a line: freeze-out temperature goes as the sixth root of g_*, so a species added at N_eff = 4.5 buys 25 per cent less helium than one added at 2 — a slope ratio of 0.749 against the 0.769 that scaling requires. The reading below is therefore taken off the drawn curve rather than off a slope. The horizontal band is the measurement — ⁴He (Aver 2015), Y_p = 0.2449 ± 0.004, taken from recombination lines in metal-poor dwarf galaxies and extrapolated to zero metallicity. Where the band crosses the line is the answer: N_eff = 2.84, with the ends of the observed interval giving 2.56 to 3.13. The standard model has three, and 3.046 rather than 3 because the neutrinos are not quite decoupled when the electron–positron pairs annihilate and take a sliver of the heat. The strength of this is not its precision, which is a third of a species and worse than the microwave background's; it is that the two constraints come from utterly different epochs, and that this one is a laboratory result about particle content obtained from an emission line in a galaxy.$ Cosmology

A particle count taken from a dwarf galaxy

Nearly every neutron that survives the first three minutes ends inside a helium nucleus, so the primordial helium abundance is not chemistry — it is the reading of a race between the weak interaction and the expansion. The expansion rate carries the square root of the number of relativistic species, which is why an emission line in a metal-poor galaxy counts neutrinos.

Cosmology

A length in centimetres, measured against an angle

A cluster's hot gas offers two line integrals of the same electrons — one linear in density, one quadratic. Two equations in two unknowns give back the path length in centimetres, and a length divided by the angle it subtends is a distance that leans on no other distance measurement.

Exoplanets

The band moves and the orbit does not

A star brightens as it burns, so the distance at which water can be liquid sweeps outwards by a factor of one and a half across a main sequence. The band a survey quotes is an instant; the band a planet needs is the overlap of every instant, and for the Sun it is a tenth as wide and does not contain the Earth.

Galaxies

The factor that multiplies every quasar mass

A reverberation lag and a line width give a length times a velocity squared, which is a mass multiplied by an unknown number of order one. That number is fixed by making a few dozen nearby active galaxies lie on a correlation measured in quiescent ones — so every black-hole mass at redshift two rests on a fit performed at redshift zero.

Galaxies

A hole that says mass times time

A gap in a stellar stream is the strongest evidence available that dark subhaloes exist, and its depth depends on the perturber's mass, the impact parameter and the elapsed time only through one combination. Two of those three are unobservable, so a gap is a measurement of a product.

$The spin a mismeasured radius reads as. The spin inferred from the disc's inner radius, against the fractional error in that radius, for four true spins. Both spectral methods reduce to one measurement — where the disc stops — and one conversion, the ISCO relation, so this figure is the error propagation both of them share. The behaviour is not uniform and it is the reverse of what the difficulty of the measurement suggests. A ten per cent error reads a hole of true spin 0.3 as anywhere from 0.16 to 0.44, while the same ten per cent moves a hole of true spin 0.99 only between 0.98 and 1.00. The reason is that the ISCO falls from six gravitational radii to one over the whole range of spin and does most of that falling in the last few per cent, so near the extremal limit a small change in radius is a large change in spin and the inversion is stiff. The published spins clustering near 0.9 and above are therefore the ones least sensitive to the systematics, and a published spin of 0.3 carries an error bar the method cannot really support. An overestimated radius always reads slow, so anything that stops the disc outside the last stable orbit biases every measurement the same way.$ Gravitation

A spin that is one length in disguise

Both ways of measuring a black hole's spin from its light measure the same thing — where the accretion disc stops — and convert it through the same relation. That conversion is stiff at high spin and slack at low, so the published spins near one are the trustworthy ones and the published spins near a third are barely measurements.

Nine dates, and a gap from 0.8 to 3.1 billion years. The lunar crater chronology and the samples that fix it: crater density per square kilometre against age, the density axis logarithmic. Every point is a laboratory measurement on returned rock — six mare and highland units dated by crystallisation, four young craters dated by how long their ejecta has been exposed to cosmic rays. The curve is a fit of N(T) = A(e^(λT) − 1) + BT with the exponential rate swept over 6 to 7.9 per billion years and the two amplitudes refitted at each; the best fit lands at λ = 6.95, against the 6.93 the published chronology uses, which is a check on the fitting rather than an input to it. What the figure exists to show is where the calibrating samples fall in time. Six of them lie between 3.15 and 3.92 billion years, four between 0.026 and 0.80, and there is nothing whatever between 0.8 and 3.1 — a gap covering half the age of the solar system. The consequence is drawn as the spread of the acceptable curves: a surface whose crater density says 3.5 billion years is dated to ±0.03 billion by this family, and a surface at 2 billion to ±0.13. The chronology is a measurement where the astronauts landed and an interpolation everywhere else.

Orbits

Nine dates for every surface in the solar system

Every absolute age quoted for a planetary surface — Martian volcanism, Mercury's plains, the resurfacing of Europa — descends from radiometric dates on rocks returned from nine landing sites on one body. Six of them fall between 3.15 and 3.92 billion years, and there is nothing at all between 0.8 and 3.1.

Starlight

A diameter that depends on a model atmosphere

An interferometer measures fringe visibilities and somebody fits a disc. A uniform disc and a limb-darkened one agree to within a per cent across the whole of the first lobe and differ by five in the second — where the visibility is under five per cent and the calibration errors are the same size.

$742 kelvin between a dwarf and a supergiant, at the same ionisation. The fraction of calcium still neutral against temperature, for three surface gravities: a dwarf, a giant and a supergiant. The three curves are the same curve slid sideways. The Saha equation carries the electron pressure in its denominator, and the photospheric pressure follows the surface gravity as its square root — hydrostatic equilibrium gives a gas pressure of order g over the opacity, and the opacity in a cool star is set by the electrons themselves. So the pressure runs from 19 newtons a square metre in the dwarf down to 0.48 in the supergiant, a factor of 40, and the half-ionisation point moves from 4427 kelvin to 3685 — 742 kelvin apart. At fixed temperature the ionisation ratio goes as g^-0.50, exactly the square root the algebra requires. A star's luminosity class is a measurement of the density of its photosphere, read off which stage of an element the lines belong to.$ Starlight

A luminosity class is a density measurement

The Saha equation has an electron pressure in its denominator, and a supergiant's photosphere is forty times less dense than a dwarf's at the same temperature. So the same element sits in different ionisation stages in the two, the spectrum says which, and the second axis of stellar classification is a barometer.

Cosmology

The residue that failed to burn

Deuterium's abundance is not a measure of what the first three minutes made. It is a measure of what escaped being used — a two-body destruction rate losing a race to a one-body expansion, which is why its curve against the baryon density is steep and helium's is flat.

Cosmology

A factor of three, and the flatness that prices every cure

The oldest stars in the Galaxy show a third of the lithium the first three minutes should have left. Three kinds of resolution have been offered, and the datum that rules on all three is not how much lithium is missing but how uniformly it is missing.

$A bound on the baryon density from an abundance nobody can extrapolate. Primordial deuterium against the baryon density, with the upper bounds that the solar system's own deuterium and helium-3 place on it. The inequality is D_p ≤ D_obs + ³He_obs/g₃, and it holds for any star-formation history whatever: a deuteron entering a star becomes ³He, so the pair can only be moved from one member to the other and then destroyed, never increased. With pre-solar values of D/H = 2.0e-5 and ³He/H = 1.5e-5, the bounds are 3.50e-5 at g₃ = 1, 5.00e-5 at g₃ = 0.5, 8.00e-5 at g₃ = 0.25 — and because deuterium falls with density, each upper bound on the abundance is a lower bound on η: η₁₀ > 4.98, η₁₀ > 3.99, η₁₀ > 2.97. The weakest of them, at a survival fraction of 0.25, still excludes 52 per cent of the density range below the answer. That was the state of the measurement for most of the 1980s, and the shape of it is the thing worth carrying: an abundance too poorly understood to be extrapolated at all still constrained the quantity, because its direction of change under processing was known even though its magnitude was not. The microwave background's 6.13 sits above every bound drawn, which is not a coincidence and is not evidence — a bound that excluded the answer would have been an error, and a bound that admits it is only a bound.$ Cosmology

An abundance with no direction to correct in

Every primordial abundance is measured today and extrapolated backwards, and the extrapolation works because processing moves each species one way. Helium-3 is made by small stars and destroyed by large ones, so the sign of its correction is not merely uncertain — it is unknown.

A lumpy universe makes more of both, and one of them was already too much. The deuterium–lithium plane, with the curve a homogeneous universe traces as its baryon density varies and the points a two-zone universe reaches at a fixed mean density of η₁₀ = 6.13. The dense zone occupies 15 per cent of the volume, and the contrast between the zones runs from 1 — which is the homogeneous case — to 100. Every mixture lies up and to the right of its own homogeneous point, and that is not a modelling choice: an abundance is measured per baryon, so what a telescope averages is η times the abundance — and for both deuterium and lithium that product is a convex function of the density, whose average therefore exceeds its value at the average. The excess is large. At a contrast of 100 the mixture gives D/H = 1.13e-4 against 2.51e-5 smooth, and ⁷Li/H = 1.77e-8 against 4.70e-10, a factor of 37.7. The extra deuterium is exactly what the proposal was for: it lets the mean baryon density be raised while the observed D/H is still matched, which in the 1980s was the one way to make the baryons account for all the matter that dynamics required. The extra lithium is what it costs. The measured abundance was already a factor of 2.9 below the homogeneous prediction, and every step toward the lumpy universe that fixes the density makes that discrepancy worse — which is why the answer to "the baryons are lumpy" turned out to be that the extra matter is not baryons.

Cosmology

The universe that was lumpy at one second

If the baryons were unevenly spread when the network fired, each region ran its own nucleosynthesis and what is observed is an average. For a decade that was the one way to make ordinary matter account for all the matter — and the reason it fails is a theorem about convex curves.

Cosmology

The clock on which light travels in straight lines

Cosmic time makes light cones bulge and horizons curve. There is another time coordinate on which a photon's worldline is a forty-five degree line at every epoch, and on it the horizon problem stops being a piece of arithmetic and becomes a question about whether two triangles overlap.

$Whether an event horizon exists at all, against one number. The comoving event horizon today — the distance a signal sent now will ever cover — against the equation of state of the dark energy, for a flat universe with the measured matter density. The curve runs away at w = −1/3 and does not exist above it: that is where the expansion stops accelerating, and in a universe that does not accelerate the integral ∫da/a²E diverges and every galaxy is eventually reachable, however far away. Below −1/3 the horizon is finite and shrinks as w falls, because a more negative equation of state makes the dark energy density grow with time rather than stay constant. At the cosmological constant's w = −1 the horizon is 16.7 billion light years against a particle horizon of 46.1, so 4.7 per cent of the volume now observable is still reachable. The band is the measured −1.03 ± 0.03. What the figure is for is the asymmetry in what the measurement still allows: two sigma toward zero puts the horizon at 17.5 Gly and two sigma the other way at 14.7, and the shape of the curve means that the closer the true value sits to −1/3 the more violently the answer moves. The reachable fraction is not a robust number in the way the particle horizon is.$ Cosmology

Whether there is a horizon at all

An event horizon exists precisely when the expansion accelerates, and its size is not a smooth function of how much. The integral that defines it runs away as the equation of state approaches minus a third, so two sigma either way on a measured number are two very different futures.

Two horizons, one formula, and a factor of two. Horizon temperature against horizon radius, on logarithmic axes, for the two kinds of horizon this collection has. The upper line is a de Sitter horizon at T = ħc/2πk_BR and the lower is a Schwarzschild horizon at T = ħc/4πk_BR — the same expression with the same constants, differing by exactly two, and both falling as one over the radius so that a bigger horizon is a colder one. The cosmological horizon today has a radius of 4,451 megaparsecs and a temperature of 2.65e-30 K, which is thirty orders of magnitude below the microwave background and will never be measured by anything. A solar-mass black hole sits at 6.2e-8 K, and a black hole as cold as the sky would weigh 2.32e+22 solar masses — of the order of the mass inside the observable universe, which is not a coincidence, since both numbers are c³/GH up to factors of order one. The factor of two between the two lines is the one place the analogy is not exact, and it is not a convention: it comes from the periodicity of the Euclidean time coordinate, which is 8πGM/c³ for a black hole and 2π/H for de Sitter space. The entropies, by contrast, agree exactly.

Cosmology

Two horizons that differ only in who is inside

A black hole's horizon and the cosmological one share an entropy formula exactly and differ in temperature by precisely a factor of two. The factor of two is the whole of the difference, and what it encodes is which side of the surface the observer stands on.

$The mass ratio an interstellar probe needs, and the exhaust that decides it. Mass ratio against final speed, for four exhaust speeds given as fractions of c, on a logarithmic vertical axis. The relativistic rocket equation replaces the velocity change with the rapidity artanh(β), which is the quantity that adds when velocities are combined, so the mass ratio is exp(c·artanh β / vₑ) and the dashed curves are the Newtonian exp(βc/vₑ) for comparison. The two agree wherever the speed is small and part company above about a third of c, with the relativistic answer always the dearer of the two. What the figure is really about is which curve a mission sits on: reaching 0.95c needs a mass ratio of 3.29e+26 at vₑ = 0.03c, 9.02e+7 at vₑ = 0.1c, 2.57e+2 at vₑ = 0.33c, 6.24e+0 at vₑ = 1c. A fusion drive with a realistic exhaust speed sits at the left of that list and the numbers are not engineering numbers. Even the photon rocket — vₑ = c, the fastest exhaust physics permits, and requiring the propellant to be converted entirely to directed radiation — needs a mass ratio of 1.73 to reach half of c and 4.4 to reach nine tenths. The equation never forbids a speed. It prices one, and the price is exponential in a quantity that itself runs to infinity.$ Spaceflight

An equation that does not break at the speed of light

Relativity replaces the velocity change in the rocket equation with the rapidity, which adds where velocities do not. The equation therefore never forbids a speed — it prices one, and the price is exponential in a quantity that itself runs to infinity.

A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 90-day event at impact parameter 0.3, computed with the observer's orbital motion included at three values of the microlens parallax. The dashed curve is π_E = 0 and is exactly symmetric in time, because a straight-line track past a point lens has to be. The others are not. The Earth's displacement over the months the event lasts adds a term π_E times its projected orbital motion to the lens–source trajectory, so one wing is pushed closer to the lens and the other further away, and the curve acquires an asymmetry of 16 per cent at π_E = 0.15 and 34 per cent at π_E = 0.35. That asymmetry is the whole measurement. An ordinary event gives one dimensioned number, t_E, which mixes the lens mass with two distances and a proper motion and therefore weighs nothing; the parallax gives a second, and two constraints on the same lens are what a mass requires. It is only available on long events — the Earth has to move appreciably while the magnification is changing — which is why parallaxes are measured for the timescales above about fifty days and not for the short ones that a low-mass lens produces.

Exoplanets

An asymmetry that the Earth's own orbit puts in

A microlensing event delivers one number with dimensions, and one is not enough to weigh anything. The Earth's motion over a long event distorts the light curve, and the distortion is the second constraint — after which the mass follows with no distance in it at all.

Exoplanets

A centroid that moves when the brightness does not

The two images a lens makes are never resolved, but their centre of light is displaced from where the source would be — by an amount that is largest at a separation where the magnification is only 1.34, long after the photometric event is over.

Exoplanets

An event of a few hours and no host

The Einstein crossing time is the square root of the lens mass and nothing else changes, so a free-floating Earth produces exactly the light curve a star does, at exactly the same height, and is over in ninety minutes. What decides whether it is found is cadence rather than sensitivity.

Exoplanets

Two systems that draw the same curve

A binary lens with separation s and one with separation 1/s have central caustics that agree to first order in the mass ratio. The same event is therefore a planet at four astronomical units or one at less than one, and no amount of photometric precision decides between them.

Galaxies

The line a star is swallowed at is not the horizon

A star torn apart by a black hole makes a flare, and a star swallowed whole makes nothing, so the heaviest hole that can produce a flare is a measurement. That ceiling is set not at the horizon but at the closest orbit from which infalling matter can still come back out — twice as far out for a hole that does not spin, and moved by a factor of five by the hole's rotation.

Two galaxies that stopped moving apart, and the mass that did it. The separation of two galaxies on a radial orbit that began together at the big bang, against cosmic time, solved so that after 13.797 Gyr they are 770 kpc apart and approaching at 110 km/s — the present separation and approach speed of the Milky Way and the Andromeda galaxy. The curve is a cycloid, r = A(1 − cos θ) and t = B(θ − sin θ), and only one cycloid passes through that point with that slope. It rose to 1037 kpc, turned round when the universe was 8.5 Gyr old, and on this purely radial orbit the two meet 3.3 Gyr from now. Its period fixes the mass: A³/(GB²) = 4.2·10¹² solar masses. The dashed curve is the same calculation with the cosmological constant's outward push included, integrated rather than solved; to arrive at the same place at the same speed against that push it needs 4.76·10¹² solar masses, 13 per cent more. Far more than the stars of the two galaxies, it is the timing argument's measurement of the Local Group's dark matter.

Cosmology

The age of the universe weighs the Local Group

The Andromeda galaxy is approaching the Milky Way, and in an expanding universe that means the two once moved apart, stopped and turned round. One radial orbit passes through their present separation with their present speed after exactly the age of the universe, and its period fixes the mass that turned them — four trillion suns, twenty-five times what their stars can account for.

Galaxies

The stars a black hole eats come from a narrow band

A star is torn apart only if its orbit happens to point almost exactly at the hole — within a millionth of the possible directions of its angular momentum. Stars on those orbits are gone within one orbit, and the only way new ones arrive is by the slow random walk of encounters with other stars. Close to the hole that walk is too slow; far from it the orbits take too long; and most of the stars a hole destroys come from a narrow band between.

Exoplanets

The stripping runs ahead of the starlight that drives it

If young stars carve the radius valley with their X-ray light, the valley should be finished early — half of it before the star has delivered a third of that light, and nine tenths of it within a few hundred million years. If the planets' own cooling cores carve it instead, planets should still be crossing it billions of years later. The two accounts put the valley in the same place, and they are separated by the one thing a histogram cannot show — when.

The XUV energy that reached one astronomical unit, for three young Suns. The cumulative X-ray and extreme-ultraviolet energy delivered per square metre at the Earth's distance from the Sun, against age, for three histories that differ only in how long the young Sun stayed magnetically saturated: 20 Myr for a slow rotator, 100 Myr for a medium rotator, 300 Myr for a fast rotator. After saturation each declines to a common track by 1 Gyr, as rotation histories are observed to converge. By 4.5 Gyr the totals are 2.06·10¹⁵ J/m² for the slow rotator, 3.66·10¹⁵ J/m² for the medium rotator, 6.47·10¹⁵ J/m² for the fast rotator: the fast rotator delivered 3.1 times as much as the slow rotator, nearly all of it in the first few hundred million years. The Sun's own rotation at that age is not measured; it is inferred from the spread of rotation periods in young clusters, and all three histories are consistent with a Sun that ends up rotating as it does now.

Exoplanets

The young Sun's spin decides what the Earth kept

The calculation that strips sub-Neptunes applies just as well to a planet with a trace of hydrogen. An Earth that captured a few hundredths of a per cent of its mass from the gas it formed in — several times the hydrogen now in its oceans — would have lost all of it, or kept most of it, depending on something nobody has measured — how fast the Sun was spinning in its first few hundred million years.

Cosmology

The Sun's speed counted in quasars comes out twice too large

The microwave background is warmer in one direction by a part in a thousand, and that dipole is read as the Sun's motion through it at 370 kilometres per second. The same motion must crowd the counts of distant galaxies and quasars towards the same direction by an amount that can be calculated exactly. When a million quasars were counted, the direction agreed and the size came out more than twice too large — as though the Sun were moving at 800 kilometres per second relative to the matter.

m²φ²: where the observed scales left, and where inflation ends. The m²φ² potential, drawn as a shape with its height divided out, against the field in reduced Planck masses. The field rolls downhill towards zero and inflation ends where the slow-roll parameter ε reaches one, at φ = 1.41. The shaded band is the stretch of field the scales now seen on the sky left the Hubble radius from: 60 e-folds before the end at φ = 15.56 and 50 before it at φ = 14.21. Nothing about the sky depends on the rest of the curve. Across that band the two numbers the tilt is made of are ε = 9.01e-3 and η = 0.0090, which give a spectral index of 0.9640 and a tensor-to-scalar ratio of 0.1441 at 55 e-folds. The height is not in either: it is fixed separately by the amplitude of the fluctuations, which puts the potential at (2.0 × 10¹⁶ GeV)⁴ there — and multiplying the whole curve by any constant leaves the band, the tilt and the ratio exactly where they are, because every slow-roll quantity is a ratio of the potential to its own derivatives. The field travels 13.49 Planck masses from the middle of the band to the end.

Cosmology

The tilt knows the slope and not the height

The measured spectral index, 0.965, is quoted as the strongest evidence for inflation, and it is a statement about two dimensionless numbers — how steeply the potential fell and how sharply that slope was changing, over the few e-folds the sky can see. The height of the potential is not in it at all, which is why potentials that look nothing alike reproduce it.

6 potentials against the tilt and the tensor bound. Predictions in the plane of spectral index and tensor-to-scalar ratio, each drawn as a short track from 50 e-folds to 60, the range usually allowed for the pivot scale to have left before the end of inflation. The vertical band is the measured index, 0.9649 ± 0.0042 with its two-sigma extent, and the shaded region above 0.036 is excluded at 95 per cent by the B-mode polarisation limit — drawn as two independent limits, which the published constraint is not quite: the real likelihood is a correlated contour, and it is somewhat tighter than this box in the corner where the tilt is high. λφ⁴: nₛ 0.9412, r 0.3137 at 50 e-folds; 0.9508, 0.2623 at 60 — outside; m²φ²: nₛ 0.9604, r 0.1584 at 50 e-folds; 0.9669, 0.1322 at 60 — outside; linear φ: nₛ 0.9701, r 0.0796 at 50 e-folds; 0.9751, 0.0664 at 60 — outside; φ^⅔: nₛ 0.9734, r 0.0532 at 50 e-folds; 0.9778, 0.0443 at 60 — outside; natural, f = 7: nₛ 0.9569, r 0.0906 at 50 e-folds; 0.9628, 0.0670 at 60 — outside; Starobinsky: nₛ 0.9616, r 0.0042 at 50 e-folds; 0.9678, 0.0030 at 60 — inside. The simplest potential of all, a mass term, is excluded over its whole range of e-folds, and not by the tilt, which it matches — by the tensors.

Cosmology

A ratio that is an energy and a distance

The tensor-to-scalar ratio is the one inflationary observable that measures the height of the potential rather than its shape, and the quantity it fixes is an energy — a ratio of 0.01 means inflation happened at 10¹⁶ GeV. It fixes a second thing as well, how far the field travelled, and near the present bound that distance is several Planck masses, which is where the theory stops being able to vouch for itself.

One sky with and without a local non-Gaussianity of fNL·σ = 0.3. The same scale-invariant random field drawn twice, from one seed: on the left as it is, Gaussian, and on the right after the local transformation Φ → Φ + fNL(Φ² − ⟨Φ²⟩) with fNL·σ = 0.3. Solid contours are one and two standard deviations above the mean, dashed ones below, and each map is measured against its own mean and spread. The transformation adds to every value in proportion to its square, so peaks are pushed up and troughs are pulled back towards the mean: the area above +2σ goes from 1.8 to 4.7 per cent of the map and the area below −2σ from 2.4 to 0.0, and the skewness of the values rises from −0.073 to 1.484. This is exaggerated by a factor of about 2200. The primordial potential varies by about 3 × 10⁻⁵, so even fNL = 5 — the size of the current uncertainty — makes fNL·σ about 10⁻⁴.

Cosmology

A test that can only fail one way

Single-field inflation predicts a local non-Gaussianity of 0.015, a skewness in the primordial potential of a few parts in a million. The measurement is −0.9 ± 5.1. A detection at the level of one would eliminate every model with a single clock at once; a null result at any reachable precision confirms nothing, because the prediction lies below anything the sky has enough independent modes to measure.

Exoplanets

One timing curve and five planets that could draw it

A transiting planet whose times wander at a 58-day period, by just over a minute, is being pulled by something — but the period says only how far from some resonance the pull comes, not from which. Perturbers inside and outside the orbit, near four different commensurabilities, each with its own mass, reproduce the same curve to a fraction of a per cent. Timing alone returns a list, and even the detail that shortens it hides a coincidence of its own.

Exoplanets

A companion on the same orbit, seen in the planet's clock

A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve.

The observed sky

An hour that stretched with the season

For most of recorded history an hour was a twelfth of the daylight — seventy-six minutes at a Roman midsummer and forty-six in midwinter — and a sundial read it exactly. In that system there was no equation of time, because nothing uniform was being compared with the Sun. The sixteen-minute correction became real only when the hour was made equal, and measurable only when clocks could keep time more steadily than the Sun by more than it.

A tumble removed with a coil and a compass, at 500 km. The rotation rate of a small spacecraft — principal moments 0.0067, 0.041, 0.043 kg m², the proportions of a three-unit cubesat — tumbling at 8.8° a second after release, against orbits at 500 km, with nothing to control it but magnetic coils driven by the B-dot law: a dipole opposite to the rate of change of the field measured aboard, capped at 0.2 A m². The field is a dipole tilted 9.2° from the Earth's axis and turning with the Earth. In a polar, 97.4°, orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.58 orbits; in a 51.6° orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.32 orbits; in an equatorial orbit the rate settles at 2.44° a second over the last orbit drawn. The law needs no knowledge of the spacecraft's attitude: a tumbling body sees the Earth's field swing round in its own frame, and a dipole opposing that swing produces a torque that removes the part of the spin perpendicular to the field. The polar orbit does not reach zero. It settles at 0.94 of twice the orbital rate, 0.13° a second, and twice the orbital rate is how fast the field direction itself turns round a polar orbit: a body turning with the field sees little change to oppose. An equatorial orbit keeps the field pointing nearly the same way all the way round, so the spin about it is reached only through the dipole's tilt and the Earth's turning, and 28 per cent of the starting rate is still there at the end.

Spaceflight

A tumble stopped by the field it tumbles through

A small satellite leaves its deployer tumbling, and the first thing most of them do is stop, using nothing but a magnetometer and three coils. The law they run needs no idea where the satellite is pointing. What it cannot do, at any instant, is touch the spin about the local field line — so how much tumble survives is decided by how much the field's direction changes along the orbit, and the stillness it reaches is defined by the field rather than by the stars.

Galaxies

The speed a line width stands in for

A galaxy does not have a rotation speed. It has a rotation curve, rising in the smallest galaxies and peaking early in the largest, and the Tully–Fisher relation is fitted to whichever single number is read off it. Build galaxies with the relation placed in their outer speed and read four other speeds from the same curves, and each gives a shallower or looser relation — which is why the choice of speed is a statement about where the relation lives.

How the scatter depends on how much dispersion is added to rotation. The scatter of the baryonic Tully–Fisher relation, in dex of mass, when the same 80 turbulent model discs are measured with S = √(K Vᵣₒₜ² + σ²), against the weight K given to the rotation, for rotation measured at 1 and 2.2 disc scale lengths. Measured at 1 scale length the scatter is smallest, 0.076 dex, at K = 0.55, and is 0.076 dex at K = 0.5; measured at 2.2 scale lengths the scatter is smallest, 0.077 dex, at K = 0.23, and is 0.134 dex at K = 0.5. For an exponential disc with constant dispersion the pressure correction is exactly K = one over twice the radius in scale lengths, so the best weight depends on where the rotation is measured. The widely used K = 0.5 is the value that makes a pure rotator and a pure isothermal sphere of the same mass agree; it is also the pressure correction for rotation measured at one scale length, and not at any other.

Galaxies

A disc that turns slower than its mass requires

The star-forming discs of ten billion years ago were not the thin, cold, orderly discs of today. Their gas moved randomly at tens of kilometres a second, and that motion is a pressure that holds up part of each disc, so it turns more slowly than its mass alone would require. Read those rotation speeds as if rotation did all the work and the Tully–Fisher relation tilts and scatters as if galaxies had evolved, when what differs is how much of their weight is carried by disorder.

Starlight

The pole-on stars a brightness limit prefers

A rapidly rotating star looks brighter and hotter from its pole than from its equator, so a survey that picks stars by how bright they look should pick more of them pole-on. It does — and the surprise is how little. Averaged over random orientations, a gravity-darkened star's apparent luminosity is exactly its true one, because every photon goes somewhere; a brightness limit restores only a fraction of a per cent of bias, while the error in any single star is ten times larger and of either sign.

$Where titanium oxide comes apart, and what the pressure does to it. The fraction of titanium oxide dissociated into Ti + O, against temperature, for a gas of equal parts at 10, 300, 6000 N/m² of total pressure. The equilibrium is Saha's with a free atom in place of the free electron and the pair's reduced mass in place of the electron's — the two atoms' partial pressures multiplied together, divided by the molecule's, equal an equilibrium constant Kₚ(T) — which for equal abundances gives a dissociated fraction of √(Kₚ/(Kₚ+P)) exactly. A molecule is one particle becoming two, so pressure suppresses the reaction exactly as it suppresses ionisation — the half-dissociation point moves from 3145 K at 10 N/m² to 4164 K at 6000. The bond energy is 6.87 eV against hydrogen's ionisation potential of 13.6, which is why this happens at a third of the temperature. Rotational and vibrational partition functions are included, classically and harmonically; the atoms keep their ground-term weights.$ Starlight

A spectrum with no continuum left in it

Below about four thousand kelvin a photosphere stops being a gas of atoms. Titanium oxide, water and carbon monoxide take over, the same equilibrium settles how much of each survives with a bond energy where an ionisation potential used to be — and the bands are so crowded that the level every line depth is measured against is nowhere on the plate.

Starlight

The continuum is made by one atom in ten thousand

The Sun's light leaves through an ion that exists only because a hydrogen atom will hold a second electron by three-quarters of an electronvolt. The electrons it holds come almost entirely from magnesium, silicon and iron — so the level against which every solar line depth is measured is rationed by elements present at one part in ten thousand.

The one number a cloud cannot change by squeezing. The mass-to-flux ratio in units of its critical value, against column density, for clouds threaded by fields of 3, 10, 30, 100 microgauss. λ below one is subcritical, and the field alone holds the cloud up, and no amount of compression changes that, because squeezing raises the magnetic and the gravitational energy at the same rate and leaves their ratio exactly where it was. λ above one is supercritical and the field is irrelevant to whether the cloud collapses. Each line has slope exactly one because λ is proportional to the column density at fixed field, and each crosses the boundary at 3.9·10²⁰ cm⁻² for 3 µG, 1.3·10²¹ cm⁻² for 10 µG, 3.9·10²¹ cm⁻² for 30 µG, 1.3·10²² cm⁻² for 100 µG. The Jeans mass is a threshold a cloud can cross by contracting; this one is a label it is born with, and the only way past it is to let the field leak out.

Galaxies

Support that cannot be squeezed away

A cloud held up by pressure can always be defeated by compressing it, because gravity gains faster than heat does. A cloud held up by a magnetic field cannot — squeezing raises both energies at exactly the same rate, so the ratio a cloud is born with is the one it keeps, and the only way out is to let the field leak.

The ceiling a measured response has to be read against. The fluid Love number of a layered body — the value it would have if its outer shell offered no resistance whatever — against the radius of its core, for cores 1×, 1.6×, 2.4×, 3.6× the density of the material outside. Every curve starts at 3/2, because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, since the tidal forcing is strongest out where there is then very little mass to move. A body 0.6 of the way core at 3.6× density has a fluid ceiling of 0.787. A measurement above the relevant curve is impossible for any interior of that layering; a measurement below it says only that something is resisting, and does not say what.

Gravitation

An ocean is detected and its depth is not

A tidal response thirty-six times too large for any solid body proves a moon has a liquid layer under its crust. It does not say how deep the liquid is, how thick the crust above it is, or what the core beneath it is made of — because the same one number is produced by a whole surface of interiors.

What Io dissipates depends on how fast the tide is applied. The dissipative part of the Love number, −Im k₂, against the period of the forcing, for Io at an interior viscosity of 1e+16 Pa s. Two rheologies are drawn: a Maxwell solid, a spring and a dashpot in series, and an Andrade solid, which adds the anelastic creep every real material shows in the laboratory and a Maxwell body does not, with an exponent of 0.3. They agree for tides slower than the Maxwell time of 1.9 days, where the body has time to flow and the transient is irrelevant. They part completely for faster ones: at the shortest period drawn the Andrade body dissipates 20 times what the Maxwell body does. A quality factor quoted without a period is half a number, and which half is missing depends on a rheology measured in a laboratory rather than derived.

Gravitation

A quality factor quoted without a period is half a number

The tidal response of a solid body is not a constant. It is a function of how fast the tide is applied, and two rheologies that agree perfectly about a slow tide disagree by orders of magnitude about a quick one — so the same moon has one Love number at its orbital period and a different one at the period of its own libration.

$A delay that counts the gas nothing can see. The mean dispersion measure of a radio pulse against the redshift it comes from, for several shares of the baryons residing in diffuse ionised gas between galaxies. A pulse is delayed by free electrons in proportion to the column it crosses, and that column is an integral over the expansion history of a density the baryon budget fixes — so the only unknown in the whole expression is the share itself. At z = 1 the relation gives 1085 pc cm⁻³ if every baryon is out there and 543 at 50 per cent. The measurement runs the other way: a burst with a known host redshift and a measured delay returns the fraction, and the answer came out consistent with nucleosynthesis. A pulse a millisecond long weighs the half of the ordinary matter that no survey could find, and it does so because the thing that delays it is the thing that does not shine.$ Cosmology

Half the ordinary matter was missing, and a millisecond found it

Nucleosynthesis fixes how many baryons there are to better than a per cent. Every survey of where they are came up about half short for two decades — and what closed the gap was the delay a radio pulse picks up crossing gas too thin and too hot for any telescope to have seen.

$A mass measured by the structure it prevents. The fractional suppression of the small-scale matter power spectrum against the sum of the neutrino masses. The relic density follows from the thermal history with no astrophysics in it — the neutrinos’ share of the critical density, times h², is the mass sum divided by 93.14 eV exactly — and the suppression is about eight times the neutrinos' share of the matter, because a particle moving at a large fraction of the speed of light streams out of an overdensity while it is still growing and takes its own gravity with it. The two mass orderings put a floor under the sum at 58 meV and 100 meV; the cosmological upper bound is near 0.12 eV, which is 7.2 per cent of suppression. The floor and the ceiling are within a factor of two of each other, so this is a measurement about to happen or a model about to break, and the quantity it returns is a sum rather than any individual mass.$ Cosmology

A mass measured by what it stopped from forming

Neutrinos were relativistic in the early universe and are not now, so they are the one entry in the cosmic budget that changes category. What cosmology measures is not their density but the hole they leave — they stream out of a growing clump and take their gravity with them, and the missing structure bounds a particle mass more tightly than any laboratory has.

$Structure stops forming above about 14 times the observed Λ. The fraction of matter that ever collapses into a bound object, against the cosmological constant in units of the observed one, holding the primordial fluctuation amplitude fixed. Growth of structure stops once Λ dominates the expansion, so the linear growth factor approaches a finite limit rather than rising for ever, and a larger Λ freezes it earlier and smaller. The asymptotic growth factor at the observed Λ is 1.110; at a hundredth of it, 5.152; at a hundred times, 0.239. The collapsed fraction falls from 0.86 to 0.40 to 0.000 across the same range, and passes a tenth of its present value at 14 times the observed constant. The observed value sits a factor of 14 below the largest one that leaves anything at all — which is the whole of the anthropic argument, and that factor is what has to be compared against the 10¹²⁰ by which the naive theoretical estimate misses.$ Cosmology

A coincidence that is a factor of fourteen

The cosmological constant is famously wrong by a hundred and twenty orders of magnitude, and famously coincidental in sitting near the matter density now. Computing how large it could be and still leave any structure at all turns the second complaint into a number — and the number is fourteen, not a hundred and twenty.

$The solid mass available, as a staircase of 5 fronts. The share of the condensable material that is solid, against distance from a 1 solar-mass star, on a logarithmic radius axis. Each riser is one species freezing out, at the radius where the disc's temperature — falling as the inverse square root of the distance — reaches that species' condensation point. silicates and iron at 1400 K and 0.04 AU; water ice at 170 K and 2.71 AU; carbon dioxide at 70 K and 16.00 AU; methane and ammonia at 30 K and 87.11 AU; carbon monoxide at 20 K and 196.00 AU. Inside every front the solid surface density is 22 per cent of what is available, which is the refractories alone; water alone contributes 53 per cent, more than twice everything else combined, which is why one of these steps is called the snow line and the others are not. A body's composition is decided by which pair of risers it formed between, and the steps are narrow because a vapour pressure is exponential in the inverse temperature.$ Exoplanets

There is not one line, there is a staircase

Water freezing is the biggest step in a protoplanetary disc and it is one of five. Each condensable species has its own temperature and therefore its own radius, and what a body is made of is decided not by which side of a line it formed on but by which pair of risers it formed between.

Exoplanets

A composition that dates a formation rather than placing it

The ice line in a young disc starts six astronomical units out and sweeps inward to under three as the disc drains. A body at four astronomical units therefore formed dry or wet depending only on when — so what it is made of is a clock, and reading it as a map is the mistake the moving line makes easy.

Exoplanets

Two elements in a ratio, and a birthplace read off it

Carbon and oxygen freeze out at different places, so the gas between the fronts is carbon-rich and the solids are oxygen-rich. A giant planet is made mostly of gas it accreted, so measuring the ratio in its atmosphere and inverting the staircase should give the radius it formed at — and the inversion turns out not to be unique.

$A shadow whose centre is brighter than no shadow at all. The flux an observer records against their distance from the centre of the shadow, for a body of half-light radius 1180 km with an isothermal atmosphere of scale height 55 km. Away from the centre the curve is the ordinary occultation light curve — the star fading as refraction spreads its light — and near it the two limbs' contributions both carry a geometric factor of the impact parameter over the shadow position, which grows without bound on the axis. The spherical atmosphere reaches 34.79 of the unocculted flux. The ray that arrives on the axis has impact parameter 1019 km, which is 161 km — 2.9 scale heights — below the half-light level, at a pressure 19 times higher. That is the only part of an occultation that reaches there. The peak is finite only because the star is not a point: the geometric factor is softened at 3 km, which is the star's own size projected to the shadow. The second curve is the same atmosphere flattened by 2.0 per cent, which spreads the focus over 20 km and drops the peak to 19.8. A real flattened body gives a caustic rather than a broad peak — several sharp spikes, spread in two dimensions rather than one — so the width here is right, the structure is not, and the height is an upper bound.$ The observed sky

The brightest instant of an occultation is its middle

A spherical atmosphere is a lens with a focal length of astronomical units, and an observer standing at the exact centre of the shadow is standing at its focus. The star does not disappear there — it brightens, by more than it would have been unocculted, and the ray that arrives has come from far deeper than anything else in the event.

$Below 1.3 km a shadow stops getting smaller. The relative rate at which a star is occulted, against the smallest body a survey can detect, for size distributions with slopes 3.5, 4, 4.5. The cross-section of a body is not its own diameter: diffraction gives every shadow a minimum width of about the Fresnel scale, √(λD/2), which at 40 AU and 550 nm is 1.28 km. Above that the cross-section grows with the body, so lowering the limit gains events as the limit to the power -1.5 for the shallowest distribution drawn; below it the cross-section stops shrinking and only the number of bodies keeps rising, which is a shallower gain by one power. Pushing the limit from 40 km to 0.2 multiplies the rate by 3106196 at the steepest slope and 14131 at the shallowest — so the rate a survey measures is a measurement of the size distribution, which is the quantity a collisional history predicts and nothing else can reach at these sizes.$ The observed sky

A population counted by shadows that never repeat

A body a kilometre across at forty astronomical units is a hundred million times too faint to image and casts a shadow just as dark as a large one. Monitoring enough stars fast enough catches those shadows — each one a single unrepeatable event of a fraction of a second, and the measurement is not any event but the rate.

The astrometry an occultation campaign has to have. How far the shadow lands from where it was predicted, against the angular error in the positions it was predicted from, for a Centaur at 15 AU, a Kuiper belt object at 40 AU, Uranus at 19 AU. The conversion is one line — an angle times a distance — and one milliarcsecond at one astronomical unit is 0.7255 kilometres. The horizontal bands are each body's own shadow width, which is its diameter, and the crossing is the accuracy at which a campaign stops being a lottery: a Centaur needs 23.0 mas, a Kuiper belt object needs 4.1 mas, Uranus needs 3701.0 mas. Before the all-sky astrometric surveys the typical error was tens of milliarcseconds, which is thousands of kilometres at these distances, so events by small bodies were found by accident and not by appointment. The same event then measures the body's position to a few milliarcseconds or better, which improves the ephemeris that predicts the next one.

The observed sky

Each event pays for the prediction of the next

An occultation is predicted from two positions and lands where the arithmetic says. Recording it then measures the occulting body's position to a few milliarcseconds — better than a year of imaging — so the observation that the prediction made possible improves the ephemeris the next prediction comes from.

$The drag coefficient of a sphere runs from 2.03 to 2.79, and 2.2 is a convention. The free-molecular drag coefficient of a sphere against the accommodation coefficient — the fraction of striking molecules that thermalise with the surface and leave in a cosine distribution rather than bouncing — at speed ratios 2, 4, 8, with the surface at 0.3 times the flow's temperature. Specular reflection gives 2.469 at the lowest speed ratio drawn and 2.001 in the hypersonic limit, where every molecule delivers exactly twice its own momentum. Accommodation adds the re-emitted flux, which leaves at the wall temperature in a direction the flow did not choose, and it adds most where the speed ratio is smallest — which is high up, where the light species dominate. The conventional 2.2 lies outside this family at both ends: at s = 8 a sphere reaches only 2.112 even at full accommodation, and at s = 2 it is already 2.469 with none. That is not a defect of the arithmetic — 2.2 is a fitted average for satellite shapes, whose flat panels have a higher coefficient than a sphere of the same projected area, and the sphere is drawn because it is the one geometry with a closed form. What survives the shape is the dependence: a satellite's drag coefficient is an assumption about its surface chemistry and its attitude, and every density inferred from drag carries it in inverse proportion.$ Spaceflight

A coefficient that belongs to the surface, not the satellite

Every density ever inferred from satellite drag was divided by a drag coefficient, and that coefficient is not a property of the spacecraft. It is a property of what happens when an oxygen atom at eight kilometres a second strikes a surface it has already coated — and the conventional 2.2 is a convention.

Spaceflight

A weather forecast made out of orbits

A density model fitted to fifty years of satellite drag is a climatology, and its error does not shrink with more data. Updating it from the drag observed on objects in orbit right now is the manoeuvre a weather forecast makes — and it buys a factor of several for a day and nothing at all for a fortnight.

Cosmology

A background weighed by what it stops

The faintest light in the universe cannot be photographed from inside the Solar System, because the zodiacal foreground is a hundred times brighter. It can be weighed instead, by the bite it takes out of a blazar at a trillion electronvolts.

$What comes back is a ramp, not a threshold. Detection efficiency against signal-to-noise: the fraction of synthetic transits injected into real photometry that the pipeline afterwards finds. The measured curve is a gamma cumulative distribution of shape 4.65 and scale 0.98 beginning at 4.1, which is the form a survey's own injection tests are fitted with; the dashed line is the step at 7.1 that a threshold calculation assumes instead. Half the injections are recovered at 8.33, 1.2 units above the nominal threshold — the ramp is a property of the search and the cut is a separate decision, so the two need not meet anywhere in particular. The rest of the disagreement is the area between the curves. The pipeline does not reach 99 per cent efficiency until 14.9, four units above the threshold, and it recovers 45 per cent one unit above it. Over a population whose signal-to-noise falls as s^-2 — which is what a planet population looks like, because there are far more small planets than large ones — the step function counts 1.21 times as many detections as the ramp does. That factor is not an error bar. It multiplies every occurrence rate computed without it, and it is larger for the small planets than for the large ones, because the small ones live where the ramp is.$ Exoplanets

The threshold that is not a threshold

A survey's detection limit is quoted as a number — seven point one — and a pipeline does not behave that way. Half the injected signals come back at the threshold, and full efficiency arrives four units above it.

Exoplanets

Every method prefers a circle, and not for the same reason

A transit is more likely on an eccentric orbit and shorter when it happens, and the two very nearly cancel. A velocity curve loses amplitude to harmonics no sinusoidal search is looking at, and that one does not cancel at all.

$How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly.$ Exoplanets

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

Galaxies

The count theory predicts, and the inference it costs

A luminosity function is measured. A stellar mass function is inferred, one galaxy at a time, through a ratio that runs by a factor of six from the bluest galaxies to the reddest — so the conversion changes the shape and not merely the units.

Galaxies

Two counts that are not the same shape

Dark matter halos are counted by a calculation that knows nothing about stars. Galaxies are counted by a survey. Laid on the same axes the two curves disagree at both ends and agree nowhere, and the knee is where the two disagreements hand over.

Galaxies

The same census, taken in two places

Fit a Schechter function to a rich cluster and to the field around it and the two come back with different slopes and different knees. Both differences are real, and neither can be read as a cause — a cluster's galaxies are also redder, and the same photometry measures both.

Starlight

The classical law gives every star one colour

The ultraviolet catastrophe is the famous half. The quieter half is that in 2ckT/λ⁴ the temperature is an overall factor, so the ratio of the law at two wavelengths has no temperature in it — and a classical universe has stars of every brightness and one colour.

Two laws that are the peak and the area of one curve. Planck curves at 3,000 K, 5,772 K, 9,600 K on logarithmic axes, with each peak marked. Normalised by its own peak, Planck's law is a universal function of x = hc/λkT, and three exponents follow from that alone and are fitted here off the drawn curves rather than quoted: the peak wavelength goes as T^-1.000, which is Wien's displacement law with a constant of 2.897772 mm K obtained by solving 5(1 − e^(−x)) = x for x = 4.965114; the peak HEIGHT goes as T^5.000; and the area goes as T^4.000. The third is the first two multiplied. A peak five powers high on a curve one power narrow encloses four powers of area, so Stefan–Boltzmann is not an independent fact about radiation — it is Wien's law and the height of the peak, taken together. That is also why the two are worth having at once. A colour gives the temperature and a flux gives the luminosity, and L = 4πR²σT⁴ then gives a radius: for the Sun at 5,772 K receiving 1361 W/m² at 1.000 AU, the arithmetic returns 6.957·10⁸ m against a measured 6.957·10⁸. A thermometer alone cannot do that, because a colour is a ratio and a ratio has no size in it; the radius comes from the one law that is an absolute quantity rather than a shape.

Starlight

Two laws that are one curve read twice

Wien's displacement and Stefan–Boltzmann are the peak and the integral of the same function. The peak is five powers high and one power narrow, so the area is four — and the fourth power that is taught as a separate law is the first two multiplied.

Orbits

A wall measures a ratio, and a ratio is a line

The same circularisation boundary drawn for planets probes the dissipation inside the planet rather than inside the star. But the boundary depends only on age over Q′, so with no cluster to date the system the measurement is a line in a plane and never a point on it.

A cluster moving at +500 km/s, and the frequency where only the motion is left. The two distortions one cluster imprints on the microwave background, against observing frequency, scaled to the largest excursion drawn. The thermal effect is from the random motion of electrons at 8 keV, with a central Compton parameter of 10⁻⁴; the kinematic effect is from the bulk motion of the same gas at 500 km/s along the line of sight, positive meaning receding, through an optical depth of 0.00639, which is the Compton parameter divided by kT/mₑc². A bulk velocity shifts every scattered photon by a common Doppler factor, and a blackbody shifted by a common factor is a blackbody at another temperature — so the kinematic distortion has exactly the shape of a temperature change, ΔT/T = −τv/c, which here is −29.0 µK at every frequency. In intensity that shape is the derivative of the Planck spectrum, and its largest value falls at 217.5 GHz, the same frequency at which the thermal distortion crosses zero, 217.5 GHz: both conditions reduce to x coth(x/2) = 4. At 150 GHz the thermal decrement is −260 µK, so the motion is 11.2 per cent of it there and all of the signal at the null. What the kinematic spectrum cannot be told apart from is the primary microwave background itself, which is also a temperature change with this shape — so the frequency that isolates the velocity from the gas is no help at all against the sky behind it.

Cosmology

A velocity that has the colour of the sky

A cluster moving through the microwave background shifts the light it scatters by a common Doppler factor, which leaves a spectrum shaped exactly like a change of temperature. That shape is loudest precisely where the hot gas falls silent — and it is the one shape the background itself already has.

1020 clusters or 372, and the survey cannot say which parameter moved. The number of clusters per unit redshift a survey finds above an integrated Compton signal of 8·10⁻⁵ arcmin² over 6 per cent of the sky, computed from the Sheth–Tormen halo count grown by the linear growth factor, the comoving volume in each redshift slice, and the calibrated relation between signal and mass. Each curve is one pair of assumptions: the amplitude of structure, σ₈, and the hydrostatic mass bias, 1 − b, which says how far the X-ray masses the relation was calibrated on fall below the true masses. With σ₈ = 0.811 and 1 − b = 0.8 the survey finds 1020; σ₈ = 0.811 with 1 − b = 0.6 gives 372; σ₈ = 0.75 with 1 − b = 0.8 gives 548. The counts fall at low redshift because there is little volume, and at high redshift because massive halos have not yet formed, and the peak sits near z = 0.24. Lowering 1 − b pushes the threshold onto more massive and rarer halos; lowering σ₈ makes every halo rarer. The two lower curves differ in total by 47 per cent and in normalised shape by at most 2 per cent of the peak. A survey that assumed 1 − b = 0.8 would read the 372 clusters of the curve with 1 − b = 0.6 as σ₈ = 0.716, with a redshift distribution that differs from it by at most 12 per cent of the peak — which is the only handle the survey has on the difference, and it is smaller than the counting noise in any redshift bin holding fewer than about 67 clusters. A total count cannot distinguish a universe with less structure from a survey that has misjudged its masses, and it is that degeneracy, not the counting, that has been argued about since the first large catalogue.

Cosmology

Too few clusters, or a scale that reads light

A catalogue selected on the microwave shadow is, past redshift one half, very nearly a catalogue of everything above a fixed mass — so its count by redshift is the growth of structure read almost directly. Almost, because the mass behind the threshold comes from a calibration, and a scale that reads twenty per cent light is indistinguishable from a universe with less in it.

Galaxies

A gradient the old stars have walked away from

The gas in a disc galaxy is richer in metals near the centre than at the edge, by about six-hundredths of a dex per kiloparsec in the Milky Way. Two ingredients of disc growth make that slope, a disc that grows from the inside out makes it flatten with time — and the old stars that should carry the steeper history have moved several kiloparsecs from where they were born.

A right angle short by 0.147°, and a ratio of 389 hanging on it. The ratio of the Sun's distance to the Moon's implied by the angle between them at the moment the Moon is exactly half lit, on a logarithmic scale, for angles from 80° to 89.95°. At that moment the angle at the Moon between the directions to the Sun and the Earth is a right angle, so the ratio is the secant of the observed angle — a construction with no distance in it, the same right triangle that gives an inferior planet's orbit from its greatest elongation. Aristarchus measured 87°, which gives 19.1. The mean distances give 389, which corresponds to 89.853°. The curve's steepness is the whole story: across a tenth of a degree centred on each marked angle the ratio changes by 3.4 per cent at 87°, 10.5 per cent at 89° and 103 per cent at the true angle. The method was exact and it asked for a right angle measured to a hundredth of a degree, at an instant the eye can judge only to within hours, on a terminator that is never quite straight.

The observed sky

A right angle short by a seventh of a degree

When the Moon is exactly half lit, the angle at the Moon between the Sun and the Earth is a right angle, so the angle seen from the Earth gives the Sun's distance in units of the Moon's. Aristarchus measured 87° and concluded the Sun was nineteen times further away. The construction was exact; the angle he needed was 89.85°, and at that angle a tenth of a degree is the whole answer.

Starlight

The best aperture throws away a tenth

Aperture photometry counts every pixel inside a circle equally and every pixel outside it not at all. For a faint star against its sky the best circle is two-thirds of the seeing wide, catches 71.5 per cent of the light, and reaches 90.2 per cent of the signal-to-noise that weighting each pixel by what it is worth achieves — a loss of 23 per cent in exposure time that no algorithm can beat by more.

$A gauge good to 7 per cent with a tenth of the load left, and to 23 per cent with three hundredths. The uncertainty in the propellant remaining in a spacecraft tank, as a percentage of what remains, against the fraction of the 450-kilogram load still in the tank, on a logarithmic uncertainty axis with the tank emptying to the right. Bookkeeping — summing every thruster firing through a flow-rate model — carries an error common to all burns of 2 per cent of the mass used, plus an independent 5 per cent per burn that averages down over 2000 firings; its absolute error grows with the mass used. Gauging by pressure and temperature infers the empty volume of the tank from the gas law applied to a known mass of pressurant, with a combined 0.66 per cent uncertainty in n R T / P and a 0.2 per cent uncertainty in the tank's volume; its absolute error grows as the gas fills the tank. The two methods are independent and are combined by inverse variance. With a tenth of the load left the combined estimate is uncertain by 2.9 kg, 7 per cent of what remains; with three per cent left, by 3.1 kg, 23 per cent. Near empty the absolute error barely changes, so halving what is left doubles the relative error — the gauge is at its worst exactly when the last manoeuvre has to be planned from it.$ Spaceflight

A fuel gauge that is worst when it is needed

A spacecraft's tank has no float and no dial. The propellant left is estimated by adding up every burn or by reading the pressure and temperature of the gas above the liquid, and both methods' errors grow with the propellant used. Relative to what remains, the error doubles every time what remains halves — so a geostationary satellite has to hold back months of station-keeping as a margin against a gauge that cannot see the last few kilograms.

A hole's shadow at 17° to its spin axis. The critical curve — the outline of a black hole's shadow on a distant observer's sky — for spins from 0 to 0.998, seen 17 degrees from the spin axis, in gravitational radii. Every curve is the projection of the unstable spherical photon orbits, and nothing about an accretion flow enters it. The sizes are 5.20, 5.12, 4.92, 4.84: a spread of 6.9 per cent across the entire range of spin, which is what makes an image of one of these a mass measurement with almost no model in it. What spin does instead is push the curve sideways and flatten one edge — the fast hole's outline sits 0.70 gravitational radii off centre, because photons co-rotating with the hole are dragged round and escape from closer in than counter-rotating ones can. That displacement is the spin signal, and it is measured against a centre nothing else marks.

Gravitation

A ring whose size is almost only a mass

The dark patch at the centre of an image of a black hole is the projection of the orbits light cannot escape from, and its size changes by seven per cent across the whole range of spin and inclination. That insensitivity is what makes an image a mass measurement with almost no model in it — and what leaves the spin in the part of the picture that barely moves.

Two assembly histories, and the sign that separates them. The distribution of effective spin produced by two ways of making a black-hole binary, from 40,000 draws each. A pair that evolved together as two stars keeps its spins near the orbital axis — drawn here aligned to within 25 degrees — and cannot produce a negative projection at all: 0.0 per cent of its mergers fall below zero, and its mean is 0.45. A pair assembled by chance encounters in a dense cluster has no memory of any plane, so its tilts are isotropic, its distribution is symmetric, and exactly half of its mergers have an effective spin below zero. The sign of one number distinguishes two histories — and it does so without any of the individual spins being measured, which is the only reason the question is answerable at all from a catalogue in which every event's spins are individually uncertain. The third curve is a mixture with 50 per cent of its mergers drawn isotropically: its mean is 0.22 and 25 per cent of it lies below zero, which is the shape an observed distribution with a small positive mean and a real negative tail requires. Neither channel alone produces it.

Gravitation

A spin that is measured as an angular momentum

Every other method of measuring a black hole's spin reads it from what the hole does to matter. A merging pair's waveform reads it from the orbit instead — and what it recovers is one mass-weighted projection onto the orbital axis, which is a number that two non-rotating holes and two extremal ones can share.

The ratio that is √2 only for a point. Escape speed divided by circular speed, against distance in units of the Sun's, for a point mass and for flat-rotation-curve haloes truncated at four different radii. The point-mass value is √2 at every distance and it is a floor rather than a rule. Mass outside an orbit contributes nothing to the circular speed — a spherical shell exerts no force inside itself — and everything to the depth of the well, so any extended distribution has a ratio above √2, by an amount that measures how much lies beyond. For a flat rotation curve the escape speed is not even defined without an outer edge: the potential grows as the logarithm of radius and diverges, so the ratio is √(2 ln(R/r)) with R the truncation radius, and the answer depends on a boundary nobody can see. The measured value near the Sun is about 2.27, which on this curve puts the edge at 13 times the Sun's distance — and the strength of that inference is exactly the weakness of the method, because a ratio of 2.50 would put it at 23.

Spaceflight

The ratio that is not √2

Escape speed is exactly √2 times circular speed at every distance from a point mass, and the Sun's own escape speed from the Galaxy is 2.3 times its orbital speed. The excess is not an error in either number. It is a measurement of the mass that lies outside the Sun's orbit, which pulls on nothing and holds everything.

Spaceflight

A camera that cannot see how far

The equations that make a close approach tractable are linear and homogeneous, so scaling an entire relative trajectory gives another valid one with every bearing identical. A vehicle navigating on angles alone is therefore not merely imprecise about range — it is exactly blind to it, and the only thing that restores the scale is spending propellant.

Three pictures, one set of measurements. A one-dimensional sky, the transform of it sampled at 10 spatial frequencies out of 128, and three images reconstructed from those samples. Every one of the three reproduces every measured visibility exactly — checked to a part in a million — so the data cannot choose between them. The dirty image is the samples transformed back and nothing else, complete with the negative sidelobes that incomplete sampling produces. The other two are the members of the same family that minimise a penalty: one the total curvature, one the total absolute brightness. They differ from each other by 7 per cent of the peak, and where they differ is exactly where the array did not look. The smooth reconstruction blurs the two compact sources into one another and keeps the broad component; the sparse one splits them cleanly and loses the broad one. Neither is a lie and neither is the sky. A published image is the data plus a criterion, and the criterion is a choice made by whoever ran the reduction.

Starlight

How much of the picture is the prior

An array measures the sky's transform at a few dozen spatial frequencies and at no others, so the data do not determine an image. They determine a family of them, and the reconstruction picks one member by a criterion that is not in the data — smoothness, or sparsity, or entropy. Three reconstructions of one measurement fit it exactly and disagree by a tenth of the peak.

Starlight

The position closure throws away

A closure phase is immune to the atmosphere because it cannot see anything that looks like a per-antenna error — and a displacement of the source on the sky looks exactly like one. Recovering where a source is means giving that immunity up on purpose, measuring the atmosphere on a neighbour instead of cancelling it, and paying for the difference between two directions.

A sight line stops where its own optical depth is one. The source function of a grey atmosphere in radiative equilibrium — a straight line, S = (3/4)(τ + 2/3) in units of the flux — with the depth each sight line reads off it. The emergent intensity at an angle is the source function integrated along the ray with everything in front of it attenuating, and for a source function linear in optical depth that integral is exactly the source function evaluated at τ = μ. Not approximately: the Eddington–Barbier relation is an identity for a linear source function, and the numerical integral agrees with it here to a part in ten thousand, which is the quadrature's error and not the relation's. A vertical ray reads the source function at τ = 1 and a ray at 78 degrees reads it at τ = 0.2, which is higher in the atmosphere and therefore cooler — so the limb is dimmer than the centre by the amount the source function has risen between those two depths. Everything about limb darkening is the slope of this one line, and the famous 2/3 is the intercept: the average ray, over the whole disc, reads the source function at two thirds of a unit of optical depth, which is the depth a stellar "surface" actually means.

Starlight

A sight line stops at two thirds

The whole of limb darkening is one sentence: what leaves a star at an angle is the source function evaluated where the slant optical depth is one. For a source function linear in depth that statement is exact rather than approximate, and it produces the grey atmosphere's (2+3μ)/5 with no fitting anywhere in it.

Cosmology

A blur undone with the field that made it

The acoustic feature is smeared by the same bulk flows whose displacements can be estimated from the density they produced. Moving every galaxy back along that estimate sharpens the peak and removes most of its second-order shift — and what limits the procedure is not the idea but how well a field can be measured from the objects it moved.

Cosmology

A ruler read in absorption

Beyond the reach of any galaxy survey the only tracer left is the hydrogen between here and a quasar, sampled finely along each sightline and coarsely across the sky. That inverts the balance every galaxy measurement is built on — the forest measures an expansion rate better than a distance, which is the harder of the two to get any other way.

Exoplanets

A reference built from other stars

Subtracting a star's halo using the star's own exposures removes part of the planet with it. Building the reference from a library of archival exposures of other stars removes none of the planet and matches the halo worse — so the two techniques fail in opposite places, and which one wins at a given separation depends on how many exposures somebody has kept.

A free libration does not last. How long a free libration survives before tidal dissipation removes it, against the tidal quality factor, for four locked bodies — with the age of the solar system marked. A lightly damped oscillator loses its amplitude in about Q of its own periods, and a free libration's period is years, so the decay time is thousands to hundreds of thousands of years for any plausible Q. Against 4.5 billion years that is instantaneous. A free libration that is observed today was excited within the last geological instant, and one that is not observed bounds how much has happened. The Moon's is detected, at a few arcseconds, and requires something recent — an impact, a core–mantle interaction, or a resonance passage. Mercury's has been searched for and not found, and the upper limit is the constraint.

The observed sky

A rocking nobody started

A locked body is a pendulum, and a pendulum has a period of its own. That free libration is damped by tides in thousands of years, so any body still oscillating at its own frequency was disturbed in the last geological instant — and the Moon is, while Mercury is not.

Stretched towards the mass, sheared across it. Two rings of galaxies around the same concentration of mass. Left, galaxies physically beside it: its tidal field stretches each one along the line to the centre, so their long axes point radially and their mean tangential ellipticity is −0.30. Right, galaxies far behind it: their light is deflected past the mass and each image is sheared tangentially, with mean tangential ellipticity +0.30. The two patterns are perpendicular. A weak-lensing survey correlates the shapes of foreground and background galaxies to measure the shear, and when a foreground galaxy was aligned by the very structure that lenses the background one, their product is negative — −0.090 here — and subtracts from the signal rather than adding noise to it. What the picture exaggerates is the size: real intrinsic alignments are a per cent or two in the mean, buried under a random shape scatter of about 0.3, and are found only statistically.

Cosmology

The shape a galaxy was given before it was lensed

Weak lensing assumes that galaxies point in random directions, so that a coherent stretch can only have been put there by the light's journey. The tidal field that set every galaxy's spin also stretched its shape — towards the very mass that lenses the galaxies behind it — and the resulting error is not noise but a signal of the opposite sign.

A spin that prefers the axis in the middle. Where the spin of a tidally torqued patch points, measured against the three principal axes of the tidal field acting on it: the distribution of the absolute cosine between the spin and each axis, from 20,000 patches with random shapes in one field of principal values −0.75, 0.09, 0.66. Principal values are those of the second derivative of the potential, so a positive one compresses. An isotropic spin would give three flat lines at one. The spin avoids the most compressive and the most stretching axes and prefers the intermediate one, because the torque about each axis is proportional to the difference of the field's principal values about the other two, and the intermediate axis is the one whose two neighbours differ most: squared, those weights are 0.11, 0.66, 0.24 of the total. Fitted as Lee and Pen's alignment parameter, the directions give a = 0.617. The preference is real and it is weak — the densest bin is 3.37 times the isotropic value — which is the whole difficulty of using spins to map a field.

Cosmology

A spin that remembers the field and not its sign

Tidal torque theory says where a galaxy's spin should point, not only how large it should be. It points along the middle axis of the tidal field, it is exactly perpendicular to an ideal filament, and it is an even function of the field — so a map of spins can find the directions of the cosmic web and can never say which of them is collapsing.

Exoplanets

A planet circularised at twice the distance that would destroy it

A giant planet thrown onto an orbit of eccentricity 0.97 by a distant companion loses its orbital energy to tides at every close passage and keeps almost all of its angular momentum. The orbit therefore shrinks to a circle at exactly twice its closest approach — and since a closest approach inside the Roche distance destroys the planet, the hot Jupiters delivered this way should begin at twice the Roche distance, measured in units of each planet's own size.

Exoplanets

One eccentric planet, or two circular ones

The velocity curve of a star with one planet on an eccentric orbit is, to first order in the eccentricity, exactly the curve of two planets on circular orbits with periods in the ratio two to one. The difference between the two readings is a term in the eccentricity squared, and for the modest eccentricities most planets have, it is smaller than the noise.

$An exposure time chosen by the star. The fraction of a star's p-mode velocity signal that survives averaging over an exposure of the length on the horizontal axis, for an envelope of modes centred on νₘₐₓ with the stated width. The dashed curve is a single oscillation at νₘₐₓ = 3.1 mHz, a period of 5.4 minutes: it averages to exactly zero after one period and at every multiple. The solid curve is the real case, a band of modes 1.2 mHz wide, which cannot all be nulled at once: its first minimum is at 5.8 minutes, where 15% of the rms survives, and beyond it the residual falls only as one over the exposure time. Short exposures on bright stars sample the oscillations rather than averaging them, which is why a bright star observed quickly can be noisier than a faint one observed slowly.$ Exoplanets

An exposure time chosen by the star

A Sun-like star's surface rings at five minutes and boils on timescales from ten minutes to a day, and each of those motions shifts its spectral lines by more than the nine centimetres a second an Earth would. The ringing can be averaged away by an exposure tuned to the star; the slowest boiling cannot be averaged away within a night at all, and it sets the number of nights an Earth twin costs.

Galaxies

Where the sky is the measurement

The outskirts of a galaxy at 29 or 30 magnitudes per square arcsecond are a thousand times fainter than the night sky they are seen through. Photon noise averages down with exposure; an error in the sky subtracted from under them does not. A sky wrong by a tenth of a per cent draws a break into a disc that has none, or a halo around a galaxy that has none, and the telescope's own scattered light draws another.

The same disc measured smaller at every redshift. The isophotal radius, at a limit of 26.5 magnitudes per square arcsecond, of three exponential discs that do not change at all — central surface brightnesses 20.5, 21.65, 23, the same scale length throughout — against the redshift they are placed at. Surface brightness falls as (1+z)⁴, three magnitudes at z = 1 and six at z = 3, before anything about the galaxy evolves or any band shift is accounted for, so every isophote moves inward. The discs vanish from an isophotal catalogue entirely at z = 2.98, 2.05, 1.24. The dashed line is the half-light radius, 1.68 scale lengths, which is defined by the galaxy's own light and does not move. Galaxies at high redshift are genuinely smaller than their present-day counterparts; how much smaller depends on which of these two kinds of radius is quoted, and the isophotal kind would report a shrinkage for galaxies that have none.

Galaxies

A radius that is a measurement of the threshold

A galaxy has no edge. Its light fades exponentially for as far as anyone can measure, so every size ever quoted for one is the radius at which the fading crossed some line that somebody drew. Drawn at a fixed surface brightness, the line measures how bright the galaxy's centre is as much as how big the galaxy is — and at high redshift it measures the expansion of the universe as well.

Galaxies

The halo a galaxy drags inward

The cusp that dark-matter simulations predict is a halo with no galaxy in it. When gas cools and settles into a disc at the centre, the dark matter is pulled in after it, and a quantity conserved on slowly changing orbits says by how much. In a large disc galaxy the pull triples the dark matter at the centre and multiplies tenfold the energy needed to dig a core; in a dwarf it barely registers.

$A band and a strip, and the constant where they cross. The plane of the two quantities a planetary ephemeris cannot separate: the fractional rate of change of the gravitational constant, vertically, and of the Sun's mass, horizontally, both in units of 10⁻¹⁴ per year. The orbits respond only to their sum, the rate of change of GM☉, so a measurement of that sum — drawn here as −9.0 ± 3.0 × 10⁻¹⁴ per year, the precision recent ephemerides reach, not any one published value — is a diagonal band. On its own it allows any Ġ/G at all, provided the Sun's mass is changing to compensate. The vertical strip is the Sun's mass loss computed from its luminosity and its wind, −9.9 to −8.8 × 10⁻¹⁴ per year. Where the two overlap, Ġ/G lies between −3.2 and 3.9 × 10⁻¹⁴ per year. The bound on a change in a fundamental constant is therefore set as much by how well the solar wind is measured as by how well the planets are.$ Gravitation

A constant measured through a Sun that is getting lighter

Whether the gravitational constant changes with time is tested by watching the planets, and the planets report only the product of G with the Sun's mass. The Sun is losing mass — four million tonnes a second as light, a million and a half more as wind — at nearly a ten-trillionth of itself a year, and every bound on a changing G is a statement about the solar wind as much as about gravity.

The weakest bodies are a few hundred metres across. The energy per unit mass needed to shatter a body and disperse half of it, against the body's diameter, for basalt and water ice, from fits to impact simulations calibrated on laboratory fragmentation. Small bodies are held together by material strength, which falls with size because a larger body contains larger flaws; large bodies are held together by their own gravity, which rises with size because the fragments have to be thrown apart as well as broken. The two regimes meet in a minimum at 235 m for basalt and 204 m for water ice, where a body needs the least energy per kilogram to destroy of any size — 127 J/kg and 57 J/kg, the kinetic energy per kilogram of motion at only 16 and 11 metres a second. A collisional population is therefore not self-similar: it has a preferred size, and its size distribution records it.

Orbits

The size at which a body is easiest to break

A pebble is held together by the strength of its rock and a moon by its own gravity, and somewhere between them is a size that has too little of either. Bodies a few hundred metres across are the weakest in the solar system — sixteen metres a second of impact energy per kilogram is enough to scatter one — and a population grinding itself down records that weakness as a knee in its size distribution.

$A square field imaged as a squashed one. A square grid of stars 1° across, centred 70° from the zenith, and how refraction moves each one relative to the field's centre after the displacement common to the whole field has been removed. The arrows are magnified 200 times. Refraction lifts every star towards the zenith, but it lifts the lower ones more because refraction grows towards the horizon, so the field is compressed vertically by 2377 parts per million; and every star moves along its own vertical circle, which converge at the zenith, so the field is compressed horizontally by 286 parts per million. The corners move by up to 4.4 arcseconds against the centre — a hundred times the precision of a modern relative astrometric measurement, and a distortion that changes as the field crosses the sky.$ The observed sky

A square field imaged as a squashed one

Relative astrometry measures positions against neighbouring stars so that the atmosphere's refraction, common to them all, cancels. It does not cancel. Refraction changes across the field, so a square patch of sky seventy degrees from the zenith is imaged squashed by a few parts in a thousand — and stars of different colour are moved by different amounts, by more than the parallaxes being measured.

The observed sky

The sky crowded forward by the observer's speed

Annual aberration is the first-order term of a law that has no higher terms Bradley could have seen. Written exactly, it maps the whole sky onto itself, crowding every star towards the direction of motion — by half a milliarcsecond at the Earth's speed, which a modern astrometric catalogue must carry, and at half the speed of light by enough to put three-quarters of the stars in the forward half of the sky.

$How steady each kind of clock is, by how long it is averaged. The Allan deviation — the typical fractional difference in frequency between successive averages of length τ — against τ, for representative noise models of caesium beam, hydrogen maser, caesium fountain, optical lattice clock. Each falls as τ^−½ while random frequency noise dominates, then levels off at a floor set by slower, correlated noise, and a clock with a systematic drift eventually rises again. A caesium beam is still improving at the longest averaging drawn, reaching 10⁻¹⁴; a hydrogen maser reaches its best, 7.3·10⁻¹⁶, after about 27 hours; a caesium fountain is still improving at the longest averaging drawn, reaching 1.2·10⁻¹⁶; an optical lattice clock is still improving at the longest averaging drawn, reaching 10⁻¹⁸. No clock is best at every averaging time — the maser is steadiest over hours and wanders over weeks, the fountain is noisier over minutes and more accurate over months — and a time scale is built by using each where it is best. The curves are typical shapes, not the specification of any particular instrument.$ The observed sky

A time scale that is finished after it is used

The world's reference time is not kept by any clock. It is an average of about four hundred and fifty, weighted by how steady each has been, published a month late, and then recomputed a year later against the handful of clocks that define the second — so the best available time for any moment is known only long after the moment has passed. The best clocks are now so good that averaging them requires knowing each one's height to a centimetre.

The weak spot in the field. Contours of the strength of the Earth's magnetic field at the surface, in microtesla, over longitude and latitude, from an eccentric dipole: a dipole tilted to pass through the geomagnetic poles and displaced about 570 km from the Earth's centre towards the western Pacific. The field is strongest near the poles and weakest near the equator, as any dipole's is, but the displacement makes the side of the Earth furthest from the dipole's centre weaker still: the minimum, 22.9 µT, lies at 8° S, 40° W, over the South Atlantic. The measured anomaly sits further south and west, near 26° S and 55° W, and is slightly deeper, because the real field has higher harmonics that no dipole carries; the eccentric dipole gets the side of the planet and the size right and the exact place approximately. Trapped particles mirror where the field reaches a fixed strength, so wherever the field is weak they mirror closer to the ground — and over this region the inner belt reaches down into the upper atmosphere.

Spaceflight

A belt emptied over one ocean

The Earth's magnetic field is not centred on the Earth. Its dipole sits five or six hundred kilometres towards the western Pacific, so the opposite side of the planet — the South Atlantic — has the weakest field at any altitude, and a trapped particle that mirrors safely above the atmosphere everywhere else dips into it there. The inner edge of the inner radiation belt is drawn by that one region, and it is moving.

$A source that erases its own polarisation. The polarised fraction of a uniform slab that both emits and Faraday-rotates, relative to its intrinsic value, against wavelength, for total Faraday depths of 10, 40, 150 rad m⁻². Light from the back of the slab is rotated by the whole depth and light from the front by none, so the polarisation vectors from different depths fan out and cancel. The fraction falls as |sin(Φλ²)/(Φλ²)| and vanishes altogether at λ² = π/Φ — 56.0 cm for 10, 28.0 cm for 40, 14.5 cm for 150 — then returns weakly and vanishes again. A source measured only at wavelengths beyond its first null looks unpolarised, and a source measured across the null shows a polarised fraction that depends on the wavelength in a way no foreground screen can produce. The shape of the fall is a measurement of how the rotating plasma is arranged along the line of sight.$ Starlight

A source that rotates its own light

The rotation-measure method assumes the plasma that turns the polarisation sits in front of the source. When it is mixed into the source instead, light from the far side is turned more than light from the near side, the contributions cancel, and the straight line in wavelength squared bends, halves its slope and then vanishes. The cure is to stop measuring in wavelength at all.

$The most pressure a core can supply, and what its envelope asks for. The pressure at the boundary of a star's isothermal core, on a logarithmic scale, against the fraction of the star's mass the core holds. The coloured curves are the most pressure a core of that mass can supply at its edge — the maximum of the isothermal sphere, 1.40 c⁸/(G³M²) for a core of mass M — for cores of mean molecular weight 0.6, 1.0, 4/3; the flat grey line is the pressure the envelope, of molecular weight 0.6, exerts on the core at the same temperature. The core's ceiling falls as the square of its mass, and it can hold the envelope up only while it lies above the line. The curves cross it at core fractions of 0.370, 0.133, 0.075: a core of the same molecular weight as its envelope could hold 37 per cent of the star, and a fully ionised helium core, μ = 4/3, under this envelope only 7.5 per cent. The limit exists because the core is heavier per particle than the gas it has to hold up.$ Stars

A core too heavy for the gas around it

When a star's centre runs out of hydrogen, the helium left behind stops making energy, settles to one temperature, and becomes the one member of the polytrope family with no surface of its own. It can hold up the envelope around it only while it is less than about a tenth of the star. Past that there is no balance to be had, and the star crosses to the giant branch in a few per cent of its lifetime — which is why that part of the diagram is empty.

Stars

A survival that depends on the energy

A low-energy neutrino from the Sun reaches the Earth still an electron neutrino about half the time, and a high-energy one about a third of the time. The difference is made by the Sun's own electrons, and the energy at which it switches over — near two million electronvolts — sits in the gap between the neutrinos that never meet a resonance and the ones that all do.

A thousand times the supply, and eight times the light. The luminosity of an accretion disc, in units of its Eddington luminosity, against the rate at which gas is supplied to it, in units of the rate that would make a thin disc shine at exactly that limit — both on logarithmic axes. Below the limit the luminosity is proportional to the supply, the dashed line, with a tenth of the rest-mass energy radiated. Above it the disc cannot radiate the excess: the luminosity grows only as 1 + ln ṁ, so ten times the supply gives 3.3 times the limit, a hundred times gives 5.6, and a thousand times gives 7.9. The limit is a ceiling that bends rather than breaks — and whatever the supply is doing above it, the light does not say, because the same curve follows whether the surplus is blown away in a wind or carried into the hole.

Stars

A ceiling on the light and not on the mass

Pour gas onto a black hole a thousand times faster than the Eddington rate and it shines at only about eight times its limit. The rest of the energy is either blown away or swallowed, and those two fates give the same luminosity while differing by a factor of the supply in how fast the object grows — so the light a super-Eddington source emits cannot say which one it is.

Starlight

A rate the wind has no choice about

A hot star's wind is driven by starlight caught in hundreds of thousands of spectral lines, and the way those lines are distributed in strength fixes everything else. The mass-loss rate turns out not to be a free parameter at all but the one value for which a smooth flow can leave the surface — and multiplying it by the wind's speed and the square root of the star's radius cancels the star's mass, leaving a relation with the luminosity alone.

The power a source loses to its own photons, against how bright it is. The ratio of the power a synchrotron source loses by inverse-Compton scattering of its own photons to the power it radiates as synchrotron emission, against its brightness temperature, both on logarithmic axes. To first order the ratio goes as the fifth power of the brightness temperature over 10¹² K — so at 10¹¹ K it is 10⁻⁵, negligible, and at 10¹² K it is one. The scattered photons are scattered again, and each order multiplies by the same factor, so the second-order curve rises as the tenth power: at twice the limit the total loss is 1056 times the synchrotron power, and the electrons radiate away their energy in hours as X-rays and gamma rays. A steady source cannot sit above the line where the ratio is one. That is the inverse-Compton catastrophe, and it makes 10¹² K a ceiling on the brightness temperature of any incoherent synchrotron source at rest.

Starlight

A temperature that is a speed

No steady synchrotron source can be brighter than about a trillion kelvin, because above that its electrons lose their energy scattering their own photons faster than they can radiate it. So when a radio source is measured brighter — as the cores of quasars routinely are — the excess is not a hotter source. It is a source coming towards the observer, and the brightness temperature becomes a measurement of how fast.

Exoplanets

An escape that stops following the starlight

The standard estimate of how fast a planet loses its atmosphere assumes the loss grows in proportion to the ultraviolet light it absorbs. It does, up to a point. Beyond about ten thousand erg per square centimetre per second the escaping gas is fully ionised, its heat leaves as Lyman-α light instead of lifting anything, and the loss grows only as the square root of the flux — so the one efficiency every calculation needs is really a function of where the planet sits.

$A fringe that is dark on the star and bright on the planet. The fraction of a point source's light an interferometric nuller transmits, against its angle from the star along the baseline, in milliarcseconds, for two telescopes 10.3 m apart observing at 10 μm — the baseline that puts the first bright fringe exactly on a planet 1 AU from a star 10 pc away, 100 milliarcseconds out. The solid curve is the two-telescope null, sin²: it is zero on the star and rises as the square of the angle. The dashed curve is a quartic null, sin⁴, made by combining four apertures: it is darker near the star and narrower at the planet. The shaded strip is the star itself, 0.93 milliarcseconds across. It looks negligible at this scale, but the null is not perfect across it, and what leaks from that strip is the quantity the whole technique has to beat.$ Exoplanets

A null filled in by the star itself

Two telescopes combined out of phase cancel a star on the axis and pass a planet beside it, and the cancellation rises only as the square of the angle away from the axis. A star is not a point, so its own edges sit off the axis — and for the Sun seen with the baseline that finds the Earth, they leak through at a hundred and fifty times the Earth's brightness, at every distance, because the leak depends only on the size of the star compared with the planet's orbit.

The distance a proton keeps its energy, against the energy. The energy-loss length of a cosmic-ray proton — the distance over which it would lose all its energy at its current rate — against its energy, computed from the photon density of the microwave background at 2.725 K and the cross-section for photoproducing a pion, which peaks at the Δ resonance. The dashed line is the loss to cosmic expansion alone, about 4,283 Mpc. Only above about 6.1·10¹⁹ eV does the combined loss length fall below a gigaparsec, because below that few background photons are energetic enough in the proton's frame to cross the 145 MeV threshold. Above it the loss length collapses: 144,327 Mpc at 3.16·10¹⁹ eV, 155 Mpc at 10²⁰ and 19 Mpc at 10²¹, where every proton is above threshold and the loss is set by the photon density alone. A proton seen above 10²⁰ eV has come from within a hundred megaparsecs or so — a horizon drawn by light at three kelvin. Pair production on the same photons, which costs about a gigaparsec near 10¹⁹ eV, is not drawn.

Galaxies

A horizon drawn by light at three kelvin

A proton above about fifty exaelectronvolts cannot cross the universe, because the microwave background, seen from the proton, is a bath of gamma rays that it loses energy to by making pions. The cut-off in the cosmic-ray spectrum sits where that horizon predicts. But a source that simply cannot accelerate particles any harder would put a cut-off in the same place, and what the particles turn out to be made of says the second explanation may be the right one.

A disc galaxy walked down the plane by three dry mergers. Specific angular momentum against stellar mass, both logarithmic. The solid line is the relation disc galaxies follow, j proportional to M to the two-thirds, normalised to 1000 kpc km/s at 10¹¹ solar masses; the dashed line is the spheroids', a fifth as high. The points follow the median galaxy built from discs that start on the line at 2.51·10¹⁰ solar masses by successive equal-mass mergers, each between two galaxies with the same history as each other, adding spin vectors in random directions and keeping none of the orbital angular momentum. Each merger doubles the mass and moves the point right by 0.3 of a decade; the vector sum keeps on average only two thirds of the specific angular momentum while the line rises by the two-thirds power of two, so the point falls below it: to 0.45 of the line after one merger, 0.18 after two and 0.08 after three. Two dry major mergers carry a disc to where spheroids are measured.

Galaxies

Two spins that cancel, and an orbit that leaves

Disc galaxies and ellipticals of the same stellar mass differ by a factor of five in how much angular momentum each gram of their stars carries. Adding spin vectors in random directions explains most of it — two mergers of equals, with the orbit's angular momentum handed to the dark halo, carry a disc down to where ellipticals sit — and whether a merger makes a spheroid or rebuilds a disc turns on one number the vectors do not supply.

The observed sky

A substorm timed by webers and volts

The solar wind opens the Earth's magnetic field at a rate that can be quoted as a voltage — tens of kilovolts, which is webers of flux per second — and the tail closes it again. Half a gigaweber loaded at fifty kilovolts takes three hours, and three hours is how often substorms recur. The aurora's position is the open flux, its surges are the unloading, and above a certain driving the ionosphere starts to limit the circuit that drives it.

One surface, two ages, and a kink between them. The cumulative number of craters per square kilometre larger than a given diameter, both on logarithmic axes, for a surface that formed 3.9 billion years ago and was flooded 3.3 billion years ago by a layer 200 m thick. The dashed lines are the isochrons of the two ages under the lunar chronology. Craters with rims taller than the flood — larger than 5.6 km, since a simple crater's rim stands about 0.036 of its diameter high — survived it, so above the kink the count follows the 3.9-billion-year isochron. Everything smaller was buried, and the craters counted there formed after the flood, so below the kink the count follows the 3.3-billion-year isochron. Fitting the branches separately returns 3.90 and 3.30 billion years. The kink is not noise and not a change in the impactors: its diameter is a measurement of how thick the flood was.

Orbits

A kink that measures how deep the lava was

A surface that was flooded partway through its life carries two crater populations — the old craters too big to bury, and the young ones made since — and its crater count follows one age above a certain diameter and another below it. The diameter of the kink is the thickness of the flood, read through the height of a crater's rim, and a single age fitted across it describes nothing that ever happened.

The spike, and where one passage through the planets sends it. The distribution of the reciprocal semi-major axis 1/a of long-period comets, in units of 10⁻⁴ AU⁻¹ and each curve scaled to its own peak. The narrow shaded peak is the Oort spike: comets arriving from the Oort cloud for the first time, with 1/a of about 4·10⁻⁵ AU⁻¹ and a spread of a few times 10⁻⁵. The broad curve is where those same comets go after one passage through the planetary region, whose perturbations change 1/a by a typical 5·10⁻⁴ AU⁻¹ — ten times the spike's width. 47 per cent end with negative 1/a, on hyperbolic orbits that leave the Solar System; only 4 per cent remain inside the spike's width. A comet in the spike is therefore on its first passage, almost by definition: the planets cannot put a comet back into the spike once it has been through them.

Orbits

A spike only first-time visitors can be in

Long-period comets arriving from the Oort cloud have orbital energies packed into a spike a few hundred-thousandths of an inverse astronomical unit wide. One passage through the planets changes that energy by ten times as much, so no comet can be in the spike twice — and the spike's position, near thirty thousand astronomical units, is where the Galaxy's own tide is just strong enough to lift a comet's perihelion over Jupiter and Saturn in a single orbit.

The sky-averaged 21-centimetre signal, and the trough that was claimed. The brightness temperature of neutral hydrogen's 21-centimetre line relative to the microwave background, averaged over the sky, in millikelvin, against observed frequency (the redshift is 1420.4 MHz over the frequency, minus one). In this model the first stars' ultraviolet couples the hydrogen's spin to the gas near redshift 17, X-rays heat the gas near 11, and reionisation is half complete at 7.7. Because the gas has cooled faster than the background since it decoupled, coupling makes it absorb: the trough reaches −171 mK at 71 MHz (redshift 19.1). Heating turns the signal into a weak emission, and reionisation removes it above about 197 MHz. The dashed curve is the deepest absorption standard physics allows at each frequency — gas that has only cooled, fully coupled — and the point is the one claimed detection, −500 mK at 78 MHz: about 1.9 times deeper than that floor.

Cosmology

A shadow cast on the microwave background

Before the first stars lit it up, the neutral hydrogen between the galaxies was colder than the microwave background behind it, and once starlight tied its 21-centimetre line to that cold, it absorbed. The absorption, redshifted to radio frequencies near 70 megahertz, is a thermometer for gas nobody has seen — and the one claimed detection was twice as deep as cooling alone allows, which is either a discovery about dark matter or a lesson about antennas.

Cosmology

An emptiness that expands faster

A cosmic void is not a hole in the universe but a region with a fifth of the average density, and gravity makes it behave like a small open universe: it expands faster than its surroundings, every galaxy in it streams outward, and in a redshift survey it is stretched along the line of sight — the opposite of what the same gravity does around a cluster. The stretch measures how fast structure grows, and the average void's roundness measures the geometry of the universe.

$A clumped plasma recombines earlier. The fraction of hydrogen still ionised against redshift, decreasing to the right, on a logarithmic axis, from Peebles's three-level atom: for a smooth plasma (solid), and for one whose baryons are clumped into two equal volumes at 1.71 and 0.29 times the mean density, a clumping factor of 0.5 (thick dashed, averaged by mass; thin dashed, each zone alone). Recombination goes as the density squared, so the dense zone recombines first and carries most of the mass; the average ionisation falls through one half at redshift 1292 instead of 1273, and at redshift 1100 it is 0.123 against 0.145. The photons cannot see the individual patches, which are smaller than their mean free path; they see only the average, and the average recombines earlier.$ Cosmology

A field that would have moved the ruler

A magnetic field left over from the early universe, too weak to matter to any galaxy, could still have pushed the baryons into small clumps before the universe became transparent. Clumped hydrogen recombines earlier, the sound waves in the plasma stop sooner, and the ruler that the microwave background is read with comes out shorter — which raises the Hubble constant the background implies, in the direction of the disagreement that has resisted every other explanation.

A chaotic orbit that waits, and then moves. The momentum of one trajectory of the standard map at K = 1.2, unwrapped and counted in whole turns of 2π, against the iteration number, over 39,892 iterations. The motion inside each band is chaotic from the first step — the trajectory's Lyapunov time is 3.7 iterations — but it lingers near one integer resonance for thousands of iterations before reaching the next: the waits were 3,098, 1,464, 9,461, 15,957, 6,058, 3,854 iterations. Each arrival at the next integer resonance, drawn dashed, is marked. Between the integer resonances lie the remnants of the broken invariant curves, which no longer block the trajectory but still hold it, and each passage is quick compared with the wait before it.

Gravitation

A barrier that leaks at a rate that can be computed

When the last invariant curve between two resonances breaks, it does not vanish. It becomes a cantorus — a curve full of gaps — and a chaotic orbit has to find the gaps to get through. The time that takes grows as the cube of how close the system is to the threshold, and it has nothing to do with the Lyapunov time, which is why an orbit can be hopelessly unpredictable and still stay put for the age of the solar system.

Three fits of the dark energy's equation of state, and the line they all cross. The equation of state w of the dark energy against redshift, for the three combined fits published with the second year of acoustic-scale data from the largest spectroscopic galaxy survey — the same galaxy and microwave-background data with three different supernova compilations — each written as w = w₀ + wₐ z/(1 + z). A cosmological constant is the horizontal line at −1. All three fits have w above −1 today (between −0.84 and −0.67) and below it in the past, and they cross it at redshift 0.35, 0.44, 0.41. Below −1 lies the phantom region, where a density grows as the universe expands; a single field with ordinary kinetic energy can be on either side of the line but cannot cross it. The preference for these fits over a constant is quoted at 2.8, 3.8 and 4.2 standard deviations, and the only thing that differs between the three is the supernovae.

Cosmology

A crossing that may belong to the fitting form

Allow the dark energy's equation of state to change and three independent fits say it has: above −1 today, below it before redshift half. The data hold w well at one redshift only, a line crossed from below is one no single field can cross, and a straight line in the scale factor fitted to a curved history crosses it anyway.

$The helium the first three minutes make, against the neutron's lifetime. The primordial helium mass fraction against the free-neutron lifetime, integrated through the first three minutes with the weak conversion rates normalised on that lifetime and free decay at the same rate, at the baryon density the microwave background gives. The line rises by 1.96 parts in ten thousand per second: a longer-lived neutron means slower conversions, an earlier freeze-out and less decay before the neutrons are locked into helium. The two laboratory lifetimes — 877.75 s from storing ultracold neutrons in a bottle, 887.7 s from counting protons in a beam — give 0.2468 and 0.2487, a difference of 19 in the fourth decimal place. The band is helium in metal-poor galaxies (Aver et al. 2015), 0.2449 ± 0.004; read back through the line it says the neutron lives 869 s, anywhere from 848 to 889 — a measurement of a laboratory constant from galaxies, good to about 20 seconds, and twice too coarse to referee a ten-second disagreement. The absolute level of the line is set to a full network's value for the bottle lifetime; the slope is computed.$ Cosmology

The first three minutes use two lifetimes

A free neutron lives 877.75 seconds in a bottle and 887.7 in a beam. Nucleosynthesis uses the lifetime twice — once to set how fast neutrons and protons interconvert, once to set how fast the leftovers decay — and if a neutron can decay into something invisible, those are the two different numbers the laboratories measure.

Exoplanets

A threshold that makes iron matter only to giants

A star with twice the Sun's iron is four times as likely to have a giant planet, and barely more likely to have a small one. The steep law and the flat one look like two mechanisms. A core racing its disc, run over a spread of discs, produces both — steep where winning is rare, flat where it is common.

Exoplanets

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

One stream after three different populations of dark subhaloes. The density of stars along the middle sixteen kiloparsecs of a thin stream, relative to its mean, after four billion years of encounters with dark subhaloes drawn from a mass function falling as the −1.9 power of the mass between 10⁵ and 10⁸ solar masses. Top: the abundance cold dark matter predicts — in this realisation 26 encounters, 4 of them above a million solar masses — which leave 32 per cent of the stream's length more than 30 per cent below the mean density. Middle: 0.3 of that abundance, 8 encounters and 7 per cent of the length in gaps. Bottom: none at all, where the only variation is the counting noise of the 60,000 model stars. No single dip in the top trace is a clean gap: they overlap, and a large one and several small ones are superposed at every point. What differs between the three is the statistics of the whole trace, not any feature in it.

Galaxies

A spectrum that counts what no single gap can

One gap in a stellar stream measures a subhalo's mass multiplied by an unknown time. A stream crossed by dozens of them measures something better — how many subhaloes of each size there are — but only as a statistic of the whole stream's density, where no feature can be pointed to and one stream is one throw of the dice.

Three arrangements of gas, three velocity–delay maps. The response of a broad emission line to a flash of the continuum, as the density of emitting gas in delay (up, days) and line-of-sight velocity (across, km/s; positive is receding), for gas between 10 and 30 light-days from a hundred-million-solar-mass black hole in three kinematic states, 6,000 clouds each. Left: a rotating disc inclined at 30° with orbits tilted up to 15° from its plane — an ellipse symmetric in velocity, with the fastest gas, nearest the hole, responding soonest. Middle: gas falling in at the free-fall speed — the near side is moving away from the observer and responds first, so the red side leads and the blue side lags, by 27 days on average beyond 1,500 km/s. Right: the same gas flowing out — the mirror image, the blue side leading and the red lagging by 27 days. The three maps are as different as maps can be; their projections onto either axis alone are not.

Galaxies

A delay that tells which way the gas is moving

A broad emission line echoes the continuum after a delay, and the delay is a radius. Measure it separately in the line's blue wing and its red wing and it becomes a map — symmetric for gas in orbit, lagging on the blue side for gas falling in, on the red side for gas blown out — and the one route to the geometry factor for an individual black hole.

The delay a companion's gravity puts in a pulsar's pulses, over one orbit. The Shapiro delay of a pulsar's pulses against orbital phase, for a companion of 0.5 solar masses and orbital inclinations of 89.17°, 85°, 70°, from Δ = −2r ln(1 − sin i · sin Φ), where r is the companion's mass times G/c³ = 2.46 µs. Superior conjunction, when the pulsar is behind its companion, is at 90°. For the orbit nearly edge-on (89.17°, the inclination measured for PSR J1614−2230) the delay rises in a narrow spike 48.5 µs high as the pulses graze the companion; at 85° the spike is 30.8 µs and broad, at 70° 17.1 µs and nearly a sinusoid. The height is set by the companion's mass and the sharpness by the inclination, which is how one curve returns two numbers the orbit alone cannot give.

Gravitation

The part of a delay no orbit can imitate

A binary pulsar's pulses are delayed by up to fifty microseconds as they pass its companion, and most of that delay is invisible — a timing fit folds it into the orbit. What survives is the third orbital harmonic and above, whose size goes as the cube of the tangent of half the inclination, and that cube, not the delay, decides which neutron stars can be weighed.

$Perihelia of a young family, run backwards until they meet. The longitude of perihelion of each of 26 members of a synthetic young asteroid family at the Karin family's position, relative to the family's mean at each moment, against time before the present. Today the angles are spread round the whole circle: members only a few thousandths of an astronomical unit apart precess at rates that differ by a fraction of an arcsecond a year, which over millions of years is many full turns. Run backwards — with each member's own drift in semi-major axis included — they converge, and the spread is smallest at 5.75 million years ago, where they were made. The nodes turn the other way, and in the real family, whose rates come from a full theory in which the two frequencies differ, they give a second, independent convergence at the same instant. An age read this way is a measurement of a time in the way a light curve's period is, with an error of a few per cent, rather than an inference from a drift rate.$ Orbits

An age read from angles that meet

A family of asteroids a few million years old has had its members' orbits turned round and round by the planets, each at a slightly different rate, until their orientations look random. Turn them back and they come together at one instant — the collision — with an uncertainty of a few tens of thousands of years, but only if each member's own slow drift in sunlight is put back too.

Stars

A distance that rests on one unmeasured number

A pulsating star's swelling can be measured twice — in kilometres from the Doppler shift of its lines, and in milliarcseconds by an interferometer — and the ratio is its distance, with no calibration against any other star. Except one: the factor that turns a line's velocity into the speed of the surface, which geometry puts at 1.33 to 1.5, measurement puts at 1.2 to 1.35, and every distance carries in proportion.

Two clocks for a radiation-belt electron, at Kp 2. Two times against drift shell L, on a logarithmic scale, for electrons of about 1 MeV. The first is how long radial diffusion takes to move an electron by one Earth radius, the inverse of an empirical diffusion coefficient that grows as the tenth power of L — at Kp 2, 550 years at L = 2, 10 years at L = 3, 21 days at L = 5. The second is how long the electron survives before it is scattered into the loss cone. Inside the plasmapause, here at L = 4.7, that is set by plasmaspheric hiss: years at L = 1.3, 14 days at L = 2.5, 5.2 days at L = 3; outside it a single lifetime of 30 days stands for every other loss. Inside L = 4.8 the electron is lost before diffusion can carry it further in, which is the shaded region; outside, diffusion wins and the belt is fed. The two curves cross at a single place, and that crossing is the inner edge of the outer belt.

Spaceflight

The gap between the belts is a race between two clocks

Electrons reach the inner magnetosphere by diffusing inward from the outer edge, and they are lost there to waves that scatter them into the atmosphere. The diffusion slows as the tenth power of distance and the loss does not, so the two clocks cross at one sharp place. Inside it lies the slot, and at megaelectronvolt energies the inner electron belt is not a steady population at all but what the last great storm left behind.

The evidence for a period, one trial period at a time. The natural logarithm of the Bayes factor for a sinusoid at each trial period against no signal, for 120 radial velocities spread over 359 days with 2.0 m/s white noise per point and a planet of semi-amplitude 1.0 m/s at 11.3 days. The amplitude prior is Gaussian with scale 3.0 m/s in each component. At the planet's period the evidence is 14.4 — odds of about 1.8·10⁶ to one — and almost everywhere else it is near zero or negative, because a sinusoid that fits nothing is penalised for the amplitude range it wasted. The quantity that answers the question "is there a planet at some period" is not the peak but the average of the Bayes factor over the prior on the period, here uniform in the logarithm of the period from 1.5 to 1,000 days, and that average is ln B = 7.7: 6.7 lower than the peak, the Bayesian form of the look-elsewhere penalty, paid because the data had to find the period as well as the amplitude.

Starlight

Odds for a planet that depend on a guess about its size

A false-alarm probability asks how often noise alone would produce a peak this tall. A Bayes factor asks something better — how much more probable the data are with a planet than without one — and pays for it with two numbers nobody measured. The width allowed for the period barely matters. The width allowed for the amplitude can turn strong evidence into none, from the same velocities.

Starlight

An angle too small to measure, read from a colour

A giant star in the Large Magellanic Cloud subtends two microarcseconds, a hundred times below anything an interferometer resolves. Its angle is read instead from its colour, through a relation calibrated on nearby stars that can be resolved — and because dust dims a star and reddens it by amounts that nearly cancel in that relation, the distance it gives hardly depends on the dust at all.

Two lobes, one halo, and two different paths out. A double radio source drawn in the plane containing the line of sight, the observer far to the right. The host galaxy sits at the centre of a hot halo — the rings mark 20, 40, 60, 100 kpc, in a β-model of core radius 40 kpc — and its two lobes lie 60 kpc out along an axis 45° from the line of sight. Light from the near lobe leaves the halo through its outskirts; light from the far lobe has to cross the halo's middle. With a tangled field of 3 µG at the centre in cells of 5 kpc, the rotation measure in front of the near lobe varies across the beam with a dispersion of 9 rad m⁻² and in front of the far lobe 28 rad m⁻². On the sky the two lobes are the same distance from the core, 42 kpc, and nothing in a total-intensity image says which is which.

Starlight

The fainter polarisation belongs to the lobe further away

The two lobes of a radio galaxy lie at the same distance from the core on the sky, and nothing in a picture of them says which is nearer. Their polarisation does. Light from the far lobe crosses the magnetised hot halo of the host on its way out and light from the near lobe mostly does not, so at long wavelengths the far lobe loses its polarisation first — and the one-sided jet, it turned out, points at the lobe that keeps it.

Polarised dust, synchrotron and the signal between them. The amplitude of B-mode polarisation at angular multipole 80, in microkelvin of CMB temperature on a logarithmic scale, against frequency from 20 to 450 GHz, for a patch of sky as clean as the cleanest large field observed. Dust is a modified blackbody with emissivity index 1.53 at 19.6 K, normalised to 4.6 µK² at 353 GHz; synchrotron falls as frequency to the power −3.1 in antenna temperature, normalised to an illustrative 1 µK² at 23 GHz. The horizontal lines are a primordial signal with tensor-to-scalar ratio 0.01, about 0.048 µK, and the lensing B-modes, about 0.071 µK. The foregrounds together are faintest at 81 GHz, at 0.038 µK, 0.8 times the signal. At 150 GHz the dust alone is 0.105 µK, the amplitude a tensor ratio of 0.046 would produce; at 353 GHz it is 2.23 µK, 47 times the signal. No frequency has the signal above the foreground.

Galaxies

A foreground drawn by the field it traces

The grains that polarise starlight along the Galaxy's magnetic field glow, at millimetre wavelengths, with light polarised across it — and that glow is brighter than the primordial signal cosmologists are looking for at every frequency above eighty gigahertz and in every patch of sky. Its pattern on the sky is set by how loosely the interstellar filaments follow the field, and removing it rests on a bet that the clouds along each line of sight share one temperature.

Galaxies

A measurement of dark matter that needs a model of galaxies

Weak lensing weighs all the matter, dark and ordinary, by how it bends light. But the ordinary matter does not sit where gravity alone would put it — galaxies' supernovae and black holes blow a large part of it out of their haloes — and the scales where that happens are the scales that carry most of a lensing survey's statistical power. Ignored, it reads as a less clumpy universe, in the direction of the disagreement lensing has with the microwave background.

The separation of the Milky Way and Andromeda, run forward. The distance between the Milky Way and Andromeda against time from now, in billions of years, integrated from today's separation of 770 kpc and radial approach of 110 km/s, for four values of Andromeda's sideways (tangential) velocity: 0, 17, 57, 100 km/s. The haloes are extended Hernquist spheres of 1.15 and 1.5 trillion solar masses, and dynamical friction drains the orbit at each passage. At 0 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.0 Gyr. At 17 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.8 Gyr. At 57 km/s the first pericentre is 117 kpc at 4.5 Gyr, and the pair merges at 9.2 Gyr. At 100 km/s the first pericentre is 321 kpc at 4.8 Gyr, and the pair has not merged by 14 Gyr. The measured sideways speed runs from about 17 km/s (Hubble) to about 57 km/s (Gaia), and the outcomes across that range differ by billions of years.

Galaxies

A collision decided by a speed across the sky

Andromeda is coming toward the Milky Way at a hundred and ten kilometres a second, and that part is measured to a few per cent. Whether the two galaxies merge in the next ten billion years depends on how fast Andromeda is also moving sideways — a drift of fifteen millionths of an arcsecond a year, measured twice, with answers of seventeen and fifty-seven kilometres a second. One is a near head-on collision in under five billion years, the other a coin toss.

Plasma beta from the corona's base outward. The ratio β of gas pressure to magnetic pressure against distance from the Sun's centre, one to ten solar radii on logarithmic scales, for a static isothermal corona in a dipole field. quiet Sun, 5 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.41 and crosses one again at 3.46 solar radii; quiet Sun, 10 G at the base, with 10⁹ electrons per cubic centimetre at 1.5 million kelvin: β starts at 0.1 and crosses one again at 5.00 solar radii; active region, 30 G at the base, with 3·10⁹ electrons per cubic centimetre at 2.5 million kelvin: β starts at 0.058 and crosses one again at 2.97 solar radii. The gas pressure falls steeply at first and then levels off, because gravity weakens with distance; the field's pressure, falling as the sixth power, never levels. Every corona therefore has a second place where the gas takes over, and it sits at two to four solar radii — where the closed loops of the corona end and the streamers' tips begin.

Stars

A corona held by its field and opened by its gas

A thousand kilometres above the Sun's surface the gas pressure has fallen so far that the magnetic field takes charge, and the corona's loops are the shape the field gives it. But the gas pressure of a hot atmosphere stops falling a few solar radii out, while the field's keeps falling as the sixth power of distance. So there is a second place where the gas takes over — and it is where the loops end, the streamers begin, and the solar wind is let out.

Stars

The same flux in three sizes

A few per cent of hot main-sequence stars carry kilogauss fields that do not change over decades. So do a similar share of white dwarfs, at a million times the strength, and of young neutron stars, at a hundred billion times it. Multiply each field by its star's area and the three classes carry nearly the same magnetic flux — what a field frozen into a star would keep as the star shrank by a factor of a hundred thousand. A collapse makes a field enormous, but it cannot make it important.

A white dwarf cooling track in J−K and absolute V. The track a 0.6 solar-mass hydrogen-atmosphere white dwarf follows as it cools, absolute V magnitude against J−K colour, fainter downward. Dashed, a blackbody of the same radius, which slides steadily redward and fainter. Solid, with collision-induced absorption: the track reddens to J−K = 0.40 at 4250 K and absolute magnitude 16.1, then hooks back to the blue, reaching −0.86 at 3000 K and magnitude 18.1. Temperatures are marked along it. The oldest white dwarfs therefore sit in the same colour range as much younger, hotter ones and are told apart only by being fainter — which is why they were missed by surveys that selected by colour.

Stars

A colour that runs backwards at the end of cooling

A white dwarf reddens as it cools, like every other hot body — until its surface falls below about four thousand kelvin. Then pairs of hydrogen molecules, colliding in an atmosphere dense enough to crush them together, begin to absorb its infrared light, and the star turns blue again while it goes on getting fainter. The oldest white dwarfs in the Galaxy have the colours of young ones, and for years they were looked for in the wrong place.

The extinction curve of a mixture, and of single sizes. Extinction relative to the V band against inverse wavelength, from the infrared at 0.3 inverse microns to the near ultraviolet at 3.3. The thick grey curve is the observed Galactic average, the Cardelli–Clayton–Mathis law at Rᵥ = 3.1. The coloured curves are computed: MRN, 0.005–0.25 µm, Rᵥ = 2.4; all grains 0.05 µm, Rᵥ = 1.8; all grains 0.3 µm, Rᵥ = 18.4. Grains of any single size make a curve with a knee: steep where the wavelength is longer than the grain, flat where it is shorter. Only a spread of sizes, with many more small grains than large ones, makes the nearly straight rise with inverse wavelength that is observed, and the first model follows the observed curve to an r.m.s. of 0.19 in these units. The ultraviolet bump beyond this range needs a separate population of small carbonaceous grains and is not modelled.

Starlight

A size read from the slope of a colour

Interstellar dust dims blue light more than red, and the way the dimming rises toward the blue is nearly a straight line in inverse wavelength. No grain of any single size makes that line. A grain only removes light selectively when it is comparable in size to the wavelength, so the extinction curve read across wavelength is the population of grains read across size — and it says there are vastly more small grains than large ones, in the same proportion that grinding rocks together produces.

$The Hubble constant the microwave background implies, with an early component. The Hubble constant that keeps the microwave background's measured angular sound horizon fixed, against the peak fraction of early dark energy at z = 3500, with the baryon density held at its standard value and the matter fraction at the 0.315 that late-time distances measure, so that the dark-matter density rises with the Hubble constant. With none, it is 67.4 km/s/Mpc and the horizon 144.3 Mpc. At 5 per cent it is 68.7; at 10 per cent 70.2, with the horizon at 138.5 Mpc; the local value of 73 is reached at 19 per cent. The shrinkage of the horizon is what raises the constant: a shorter ruler at the same angle must be nearer, and nearer at a fixed matter fraction means faster expansion. The extra dark matter this requires is the cost the published fits also find: at the value that reaches 73, the physical matter density is 18 per cent higher than in the standard model.$ Cosmology

A ruler shortened by something already gone

The microwave background gives the Hubble constant only through a computed length, the distance sound travelled before the universe became transparent. Add a component of energy that exists for a few thousand years before recombination and then vanishes faster than radiation, and the universe expands a little faster while the sound is travelling, the ruler comes out a few per cent shorter, and the inferred expansion rate rises toward the local value. The price is paid in dark matter and in the finest details of the oldest light.

How far a void field turns the leptons that make each cascade photon. The deflection angle of a cascade lepton over its cooling length, in radians on a logarithmic scale, against the energy of the cascade photon it produces, for fields of 10⁻¹⁸ G, 10⁻¹⁷ G, 10⁻¹⁶ G, 10⁻¹⁵ G, 10⁻¹⁴ G coherent over at least that length. The angle falls as the square of the lepton energy, so as the inverse of the photon energy: the gigaelectronvolt halo is made by the slowest leptons, which are turned most. At 1 GeV the deflections are 1e-3, 0.01, 0.11, 1.06, 10.62 rad. The dashed line is the jet's half-opening angle, 0.1 rad; a lepton turned further than that sends its photons away from the observer, and the halo at that energy disappears.

Cosmology

A lower limit made of three lengths and a missing echo

The only measurement of a magnetic field in the empty space between galaxies is a lower limit, and it comes from light that did not arrive. Teraelectronvolt gamma rays from distant blazars make electron pairs in intergalactic space, the pairs should re-emit the energy at gigaelectronvolts in a halo around each source, and the halos are missing. Whether that absence weighs a field of a ten-thousandth of a billionth of a microgauss depends on three lengths, the field's own tangle, and how long the blazar has been shining.

Exoplanets

Molecules seen only through a change of temperature

The heat a planet gives off, measured colour by colour as it disappears behind its star, carries the fingerprints of the molecules in its atmosphere — but only because the atmosphere's temperature changes with height. Water absorbs where the air cools upward and emits where it warms, and in an atmosphere at one temperature every molecule vanishes from the spectrum, however much of it there is. An emission spectrum is a sounding of a temperature gradient, and the molecules are what it is taken with.

Exoplanets

Dust that does not dim with distance

An Earth-like planet around a nearby star is seen against the glow of the dust in its own system, and the two behave differently with distance. The planet is a point, and it fades as the square of how far away it is. The dust is a surface, and a surface does not fade at all — a more distant system packs more dust into each resolution element in exactly the proportion its light is diluted. So the dust around a star, not the telescope's contrast, sets how far away an Earth can be found, and how many stars a search can reach.

$Debris discs of very different masses, fading to one line. The fractional luminosity of a debris belt at 50 AU — the share of its star's light the dust absorbs and re-radiates — against age, from a million to ten billion years, for belts that start with 1, 10, 100, 1000 Earth masses of planetesimals up to 60 km across. Each stays near its starting brightness until its largest bodies have had time to collide, 16.5 billion years, 1.7 billion years, 165 million years, 17 million years respectively, and then fades as one over the age. A heavier belt starts brighter and grinds itself down faster, and the two effects cancel: after their collisional times all belts lie on the same line, the dashed maximum, whatever their mass. At ten billion years the two heaviest differ by 1 per cent.$ Orbits

A ceiling that forgets how much there was

A belt of planetesimals around a star shows itself only through its dust, and the dust is ground from the belt's largest bodies by collisions. Give the belt more mass and it makes more dust — but it also collides more often, and grinds itself away faster. At late times the two effects cancel exactly, and every belt at a given distance and age lies on the same line of maximum brightness whatever it started with. A disc above that line cannot be a steady cascade, and it is telling of a recent catastrophe.

The nutation lines and the core's resonance. The resonance factor |σ₀ / (σ − σ₀)|, σ₀ being the free core nutation's frequency, on a logarithmic scale, against frequency in the celestial frame in cycles per sidereal day, negative for retrograde motion, with the free core nutation at a period of 430.2 days retrograde. The vertical lines are the largest forced nutations the Sun and Moon drive: the 18.6-year, annual, semiannual and fortnightly terms, each prograde and retrograde, and precession at zero. Any part of the Earth's response that involves the core is multiplied by this factor. The retrograde annual term, 365 days, is 5.6 on this scale; the prograde annual term 0.46; the retrograde 18.6-year term 1.07. The annual retrograde nutation is the one the core's resonance distorts most, because a year is the forced period nearest the core's own.

The observed sky

A core that wobbles at its own period

The Earth's liquid core can turn about an axis slightly tilted from the mantle's, and the flattened boundary between them pushes it back. That gives the planet a second free mode of rotation: in space a slow retrograde nutation of about 430 days, on the ground a wobble just faster than once a day. Its period is a measurement of the shape of a surface under 2,900 kilometres of rock, and it is read twice — in the nodding of the Earth's axis against the quasars, and in the rise and fall of the solid ground under the daily tides.

$How much of a background a survey can resolve, against sources per beam. The fraction of a background resolved into individual sources before confusion stops the count, against X, the number of sources brighter than the break in the counts that fall inside one beam, on a logarithmic axis. The curve is computed for counts falling as S^−2.5 above the break and S^−1.5 below it, with a source counted as recoverable while there is less than one source per 30 beams brighter than it. It is a single curve: telescope, wavelength and integration time enter only through X. Half the background is resolved at X ≈ 0.033, and the fall from nearly all to nearly none takes about four decades. The surveys are placed on it at their own beams and counts: Hubble, optical at X = 8.7·10⁻⁵, 100 per cent; Chandra, soft X-ray at X = 6.4·10⁻⁶, 100 per cent; Spitzer, 24 µm at X = 0.012, 54 per cent; Herschel, 250 µm at X = 0.043, 24 per cent; Herschel, 500 µm at X = 0.18, 11 per cent. The points do not sit exactly on the line because each band has its own slopes.$ Cosmology

A background that stops breaking into sources

A deep enough image turns a glowing sky into a list of galaxies. Past a certain depth it stops doing so, and the depth is set by how many sources share one beam — which a longer exposure cannot change.

Cosmology

The histogram of a sky too crowded to count

When every beam holds more than one faint source, a map stops being a list. It remains a measurement — the histogram of its pixels carries the number of sources nobody can pick out, down to the depth where there are so many that their sum turns Gaussian.

Exoplanets

More small planets round smaller stars

A red dwarf has about twice as many small planets as the Sun's kind of star and a third as many giants. The two populations move in opposite directions along the same axis, and the small planets were built from discs several times poorer.

$The chance that a candidate is a planet, against how strong its signal is. The reliability of a transit candidate — the fraction of candidates at a given detection statistic that are planets rather than false alarms — against the multiple-event statistic, the combined signal-to-noise of all its transits, for orbital periods of 10, 100, 350 days. Below the 7.1 threshold nothing is a candidate. At a statistic of 9 the reliability is 100 per cent, 80 per cent, 23 per cent; at 12 it is 100 per cent, 95 per cent, 44 per cent. The true planets and the false alarms are modelled, not fitted: the planets as a falling power law above the survey's completeness curve, the false alarms as an exponential tail above the threshold that grows longer and taller with period, because a long-period signal is two or three transits and two or three transits average away little of the non-Gaussian noise that makes false alarms. At short periods a candidate near threshold is nearly always real; at a year, a candidate near threshold is usually not.$ Exoplanets

At a year, most candidates are noise

A transit survey's threshold is set so that Gaussian noise yields less than one false planet in the whole catalogue. The noise is not Gaussian, and at year-long periods the candidates just above the threshold are more often artefacts than planets — so the rate of Earth analogues is partly a measurement of the noise.

Exoplanets

A peak that neither survey can see alone

Giant planets become commoner with distance from their star out to a few astronomical units, and then rarer. The rise is measured by radial velocities and the fall by direct imaging, and the peak between them sits where neither method is comfortable.

Three initial mass functions, normalised to the same light. The number of stars per logarithmic interval of mass, on a logarithmic axis, for three initial mass functions — Kroupa, Salpeter, bottom-heavy — each normalised to the same number of stars above one solar mass, so that the stars which make an old population's light are the same in each. They differ only below a solar mass. In a 10 billion year old population the stars below half a solar mass hold 62 per cent, 78 per cent, 91 per cent of the living stellar mass, and they give 15.8, 24.2, 41.7 per cent of the light of the main sequence, which is itself a third of the population's. The difference between them is almost entirely mass that emits almost nothing.

Galaxies

Stars that weigh and do not shine

Most of the mass in an old stellar population is in stars too faint to contribute to its light. How many of them there are is set by the mass function stars are born with, and the mass-to-light ratio every dark-matter measurement divides by can be wrong by a factor of two without a single colour changing.

The observed acceleration against the acceleration the visible matter explains. The acceleration measured from a rotation curve, v²/r, against the acceleration the stars and gas alone would produce at the same radius, both on logarithmic axes in metres per second squared. The solid curve is the fitting function to the measured relation for 153 disc galaxies, g(obs) = g(bar) / (1 − e^(−√(g(bar)/a₀))) with a₀ = 1.2 × 10⁻¹⁰ m s⁻², and the band is its observed scatter of ±0.13 dex, most of which is measurement error. Above a₀ the relation follows the dashed line of equality, where the baryons account for everything; below it, it bends onto the dotted line g(obs) = √(g(bar)·a₀). At ten times a₀ the observed acceleration exceeds the baryonic by a factor of 1.06; at a₀, by 1.58; at a hundredth of a₀, by 10.5. The discrepancy the dark matter is invoked to explain is a function of one variable, and the variable is an acceleration.

Galaxies

One acceleration, and every galaxy's missing mass

Plot the acceleration measured at each radius of a rotation curve against the acceleration its stars and gas would produce there, for every galaxy at once, and the points fall on a single curve. Where the discrepancy begins is set not by a radius or a mass but by an acceleration.

The observed sky

A clock made of neutron stars

The best atomic clocks are rebuilt every decade and averaged every month. A millisecond pulsar has kept its own time for a billion years. Averaged over years, an ensemble of pulsars is as steady as the world's atomic time scale — steady enough to find its errors — but it can never say how fast that time scale runs.

Phase-space density during a storm fed from inside. Electron phase-space density at fixed first and second adiabatic invariants, against the drift-shell parameter L, before a two-day storm (grey) and 0.5, 1, 2 days after it begins, from a model of radial diffusion with a diffusion rate rising as L¹⁰ and a loss lifetime of 4 days. The boundary is held fixed and a source centred at L = 4.5, standing for acceleration by whistler-mode chorus waves, acts for the storm's two days; the density grows a peak at L = 4.5, well inside the boundary, and diffusion then spreads it both inward and outward. A peak in phase-space density inside the outer boundary cannot be produced by diffusion from outside while the boundary is rising — diffusion only smooths gradients — so its presence then is the signature of local acceleration, and its absence is consistent with transport.

Spaceflight

A peak that says where the electrons were made

The outer radiation belt refills with electrons of millions of electronvolts within hours of a storm. Whether they were carried inward from the edge of the magnetosphere or accelerated where they are found shows in one place only — the density of electrons in the coordinates their motion conserves, which a detector does not measure directly.

Starlight

The photons on the sides of the lines

A spectrum measures a star's velocity only through the slopes of its absorption lines. The continuum, however bright, carries no information at all, and a star with many deep, narrow lines gives a velocity twenty times more precise than a hot, fast-rotating one from the same number of photons.

Photometric against true redshift for a simulated sample. The redshift fitted from broad-band magnitudes against the true redshift, for 300 simulated galaxies — half old, half young — measured to a signal-to-noise of 10 in their brightest band in six optical bands, and fitted with the same two templates they were made from. 217 fall within 15 per cent in 1 + z of the diagonal, with a scatter of 0.053 in (the fitted z − z)/(1 + z) among them. 83 are catastrophic failures, more than 15 per cent off, and they are not scattered at random: they sit in clumps far from the diagonal, where a galaxy's break was mistaken for the other kind of break. The scatter can be improved by measuring better; the outliers are a different kind of error, and averaging does not remove them.

Starlight

A redshift that is wrong by three

Most galaxies in the largest surveys will never have a spectrum. Their distances are read from five or six magnitudes, and the reading usually lands close. When it fails it does not fail by a little — a nearby galaxy is placed ten billion years away, because two different breaks in two different spectra fell into the same filter.

Stars

The last fuels burn for the neutrinos

A massive star burns hydrogen for millions of years and silicon for about a day. The collapse in duration is not because the late fuels are poorer; it is because, from carbon burning on, the core loses more energy to neutrinos than the whole star gives out as light, and the neutrino losses rise as the ninth power of the temperature.

Stars

A flow that piles up at the magic numbers

Half the elements heavier than iron were built one neutron at a time, in ageing stars, over thousands of years. The flow of matter up the chart of nuclei settles into a balance in which each nucleus is as abundant as it is reluctant to capture — and it jams at the closed neutron shells, leaving the peaks at strontium, barium and lead that every star's spectrum records.

$A velocity error that grows with distance. The uncertainty in one galaxy's peculiar velocity, v = cz − H₀d, against its distance in Mpc/h, when the distance comes from an indicator with a fractional error of 5 per cent, 20 per cent, 25 per cent (type Ia supernovae, 5%; Tully–Fisher, 20%; fundamental plane, 25%). The redshift is exact to a few kilometres a second, so the whole error is the distance error times H₀, and it grows in proportion to the distance. The horizontal line is the rms velocity of a galaxy along one direction in linear theory for the model drawn, 302 km/s. Each galaxy's own velocity is measured to better than its own size only inside 60 Mpc/h for 5 per cent, 15 Mpc/h for 20 per cent, 12 Mpc/h for 25 per cent; beyond that a single velocity is mostly noise, and a velocity survey is useful only as an average over many galaxies.$ Cosmology

Velocities that see past the edge of the map

A redshift survey infers the motions of galaxies from the distortions of its own map. A peculiar-velocity survey measures them directly, one galaxy at a time, by subtracting a distance from a redshift — and every one of those measurements is dominated by its own error beyond a few tens of megaparsecs. What makes the survey worth doing is that a velocity answers for structure far larger than the volume it is measured in.

Cosmology

A void is not short of dwarf haloes

The centre of a large void holds a fifth of the average density of matter and almost no galaxies, and it is natural to read the two facts as one. They are not. The theory of how haloes form says a void turns its matter into dwarf-sized haloes at least as readily as anywhere else — it is short of big haloes, not small ones — so a void empty of dwarf galaxies is a statement about which haloes were allowed to light up.

Every curvature has a Hubble constant that leaves the peaks where they are. The Hubble constant a universe needs in order to put the acoustic peaks of the microwave background at the measured angle, against its spatial curvature Ωₖ, with the physical densities of baryons, matter and radiation held at their fitted values so that the sound horizon is unchanged. Along this line the angle θ* — the sound horizon over the distance to last scattering — is exactly constant and the peak positions do not move. The flat member has H₀ = 67.4 km/s/Mpc; a closed universe with Ωₖ = −0.044 needs 53.5, with a matter density of 0.50; an open one with Ωₖ = +0.010 reaches the locally measured 73. The peaks alone therefore measure a combination of curvature and expansion rate, not either one, and the statement that the universe is flat to a few parts in a thousand needs a second distance measured somewhere else.

Cosmology

A flatness that needs a second distance

The universe is said to be flat to two parts in a thousand, and the figure is usually credited to the microwave background. On its own the background measures one angle, and one angle fixes one distance — so every curvature has a Hubble constant that reproduces the peaks exactly. Flatness is where that line crosses a second measurement, and it is only as good as the crossing.

One setting Sun drawn through a layer of warm air. The apparent shape of the Sun at five moments of its setting, seen from 20 m above the sea through an atmosphere with an inversion — 0.15 K per metre over 40 m starting 30 m up — computed by tracing rays from the observer out through the layer and mapping each apparent elevation back to the point of the solar disc it came from. The true elevation of the Sun's centre is given under each panel; the dotted outline is the Sun as a standard atmosphere would show it at the same moment, where it is above the sea horizon at all. The layer does not merely flatten the disc: it cuts it into two separate strips at some moments, stacks an inverted image above the upright one, and draws a thin neck between them, because each strip of apparent elevation maps to a strip of the true disc through a curve that folds. The shapes have been recorded for centuries — the Etruscan vase, the omega, the flattened stack — and each is a picture of the transfer curve for the air that night.

The observed sky

A sunset's shape is a sounding of the air

A layer of warm air over a cold sea bends the rays of the setting Sun into shapes — a flattened stack, an inverted copy balanced on the upright one, a vase with a thin neck — and every shape is a drawing of one curve: the map from where light appears to come from to where it really came from. Run backwards, the curve is a temperature profile. It can be read, but only for the lowest air, only from a known height, and only up to a trade between how strong the warm layer was and how thick.

Each kilometre of aim error, turned into a velocity error. The error in the outgoing velocity per kilometre of error in where a spacecraft passes a moon, in metres per second per kilometre, against the flyby altitude on a logarithmic scale, for the four large moons of Jupiter at an encounter speed of 6 km/s. The flyby turns the velocity through an angle δ with tan(δ/2) = aₕ/b, where b is the aim point's distance from the moon's centre and aₕ = μ/v∞²; missing the aim by a kilometre along b changes δ, and missing it by a kilometre across b rotates the plane of the turn, and both move the outgoing velocity by about 2v∞aₕ/b² per kilometre. At 300 km above Ganymede that is 0.32 m/s per km along and 0.32 across. The closer and more massive the flyby, the more it turns — and the more it amplifies whatever error the approach carried.

Spaceflight

A flyby that multiplies its own aiming error

A close flyby of a moon turns a spacecraft's path by ten degrees for nothing, and it turns every error in the aim with it. A kilometre of miss at three hundred kilometres above Ganymede becomes a third of a metre per second of velocity error, and a week later two hundred kilometres of miss at the next moon. A tour of flybys is an amplifier that must be re-zeroed after every pass, and whether a small spacecraft can use one in place of an engine burn depends on how many kilometres it can aim to.

What it costs to follow a turning grapple point. The velocity a chaser must spend each minute to hold station on a grapple point carried round by a flat spin, ω²r × 60 s, against the spin rate in degrees per second, for points 1, 3, 10 m from the spin axis, both axes logarithmic. The dashed line is what holding station 10 m from a target in a 400 km orbit costs in the same units, from the orbital dynamics alone: 2.3 mm/s per minute. At 1° per second, a point 3 m out costs 5 cm/s per minute — 24 times the orbital term; at 5° per second, 1.4 m/s per minute, 595 times. The cost rises as the square of the rate, so a target spinning twice as fast is four times as expensive to follow, and above a few tenths of a degree per second the orbit — the whole subject of the approach up to this point — has become the small correction.

Spaceflight

A grapple point that moves faster than the orbit

Every rendezvous problem up to the last few metres is orbital mechanics: two bodies in nearly the same orbit, drifting apart and together under gravity. A derelict satellite tumbling at a few degrees a second changes the problem entirely. Holding station on a point of its surface costs more each minute than a week of orbital station-keeping, the moment to grab it comes and goes in seconds, and a free tumble never repeats — so the orbit becomes a small correction to a problem in rotation.

Spaceflight

The wall sells inclination at the dearest price

Transfers to Mars that go almost exactly half-way round the Sun are forced into planes tilted steeply out of the ecliptic, which is why they form a wall of expensive dates across every launch chart. A mission that wanted to leave the ecliptic could use the wall on purpose. It would be buying inclination at the Earth's orbital speed — the dearest place to buy it — and the routes that reach the Sun's poles have always gone somewhere slower to make the turn.

A synthetic spectrum fitted with a line list that is missing the weak lines. Four ångströms of a simulated metal-line spectrum near 5000 Å at resolving power 20,000 and a signal-to-noise ratio of 200 (thin, noisy line), and the best-fitting synthesis (heavy line) computed from a line list that omits every line with a central optical depth below 0.15 — 137 of the 160 lines in the ten-ångström window, carrying 32 per cent of its total absorption. The fit adjusts one abundance for all the listed lines, with the continuum held at its true level; it returns an abundance 0.188 dex too high: the missing lines' absorption has nowhere to go but into the parameters the synthesis has. The residuals (below, offset) show the missing lines as dips that no parameter of the synthesis can fill — which is how an incomplete list announces itself at high resolution, and why it does not at low.

Starlight

The lines a synthesis keeps pay for the ones it lacks

Modern abundance work fits a computed spectrum to a whole stretch of the observed one rather than measuring lines one at a time, and it averages away the errors in each line's laboratory data as it does so. It cannot average away a line that is not in its list. The absorption of the missing lines has to be put somewhere, and the fit puts it into the parameters it has — the abundance of the lines it knows, or the level of the continuum — in a direction set by which of the two it was allowed to move.

How much light a fixed mass of dust scatters, as its largest grains grow. The scattering cross-section per unit mass of dust, on a logarithmic scale relative to the largest value drawn, against the size of the largest grains in a power-law distribution n(a) ∝ a^−3.5 from 0.005 µm, at wavelengths of 1.25, 2.2, 3.6 µm. With the largest grains at a quarter of a micron — the size the interstellar extinction curve implies — the dust scatters 3.6 µm light 33 times less efficiently than it does once they reach a micron: grains much smaller than the wavelength scatter as the fourth power of their size over the wavelength and are all but invisible. At 3.6 µm the scattering per mass peaks when the largest grains are 3.5 µm, and falls beyond, as the mass moves into grains too large to scatter efficiently per gram. A dense core that glows at 3.6 µm in scattered starlight is therefore a core whose grains have grown to a micron or more.

Starlight

A glow that weighs the grains reddening cannot see

The interstellar extinction curve reads the sizes of dust grains from how much more they dim blue light than red, and it goes blind at the top end: grains larger than about a micron dim every visible colour alike. Those are the grains that hold most of the mass, and the ones from which planets begin. Dense cloud cores glowing faintly at 3.6 microns in scattered starlight are the first direct sign of them — because only grains grown to a micron or more scatter that light at all.

Amplitudes measured near the noise, against the amplitudes that were there. A simulated survey of 6000 stars, each with one planet whose true radial-velocity semi-amplitude K is drawn from a population rising towards small amplitudes (dN/d ln K ∝ K^−0.6, from 0.2 to 20 m/s), each measured with an amplitude uncertainty of 0.6 m/s; 1500 are drawn. Across: the true amplitude; up: the amplitude fitted at the planet's period; both logarithmic. The dashed diagonal is a perfect measurement; the horizontal line is the detection threshold, 4 times the uncertainty. Points above it are the detections (filled). Well above the threshold they sit on the diagonal; near it they sit above it — the detections within one unit of the threshold have measured amplitudes 1.33 times their true ones on average — and below it the measured amplitudes stop falling at all, because an amplitude is a length and noise alone gives it about 0.75 m/s.

Starlight

A prior the survey measures through its own noise

Whether a weak periodic signal is a planet depends on how large planets are expected to be, and the best source for that expectation is the population the surveys themselves have measured. But the population's small end is built from the marginal detections whose reality depends on the prior. Breaking the circle means measuring the population and the planets together, with the noise and the threshold written into the model — and doing it that way changes the answers by factors of two where they matter most.

Exoplanets

Every planet of a star must agree on its density

A single transiting planet gives the density of its star only if its orbit is known to be circular, and an eccentric orbit can disguise almost any star as almost any other. A star with four transiting planets has four such measurements of one number. They must agree — and the way they disagree measures how eccentric the orbits are, catches the candidate that orbits some other star, and pins down the star better than its own spectrum does.

$One hot spot, seen through more and more gravity. The X-ray flux from a small hot spot at colatitude 60° on a rotating neutron star seen from inclination 60°, over one rotation, each curve divided by its own mean, for no bending, 2GM/Rc² = 0.20, 2GM/Rc² = 0.34, 2GM/Rc² = 0.48 — the last two are a 1.4 M☉ star of 12 km and a 2.0 M☉ star of 12.4 km. Light bending is computed exactly; Doppler effects are left out. With no bending the spot is behind the star for 39 per cent of the turn and the pulsed fraction is 1.00. As the compactness rises the star's gravity bends light from the far side into view: the dark interval shrinks to 27 per cent, then vanishes, and the pulsed fraction falls to 1.00, 0.99, 0.64. The more compact the star, the more of its own back it shows and the smoother its pulse.$ Stars

The pulse that shows the back of the star

A hot spot on a spinning neutron star flashes once a turn, and the flash is weaker than it would be on an ordinary star because the star's gravity bends light from its far side into view. How much weaker measures the ratio of mass to radius. But on its own the flash cannot tell a compact star from a tilted one — the two trade exactly — and what separates them is the lean the spin puts into the pulse, which measures the radius directly.

The polarisation peaks sit in the temperature troughs. A tight-coupling sketch of the two acoustic sources against multipole, each scaled to its own maximum: the temperature, from the compression of the photon–baryon fluid at last scattering plus its Doppler term, and the E-mode polarisation, which is driven by the fluid's velocity gradient. The sound horizon and the distance to last scattering put the acoustic scale at ℓ = 302, the measured phase of the peaks is 0.267 and the damping scale is ℓ = 1400; the baryons' weight, which changes the heights of the temperature peaks and not their places, is left out. The temperature peaks fall at ℓ ≈ 218, 515, 813, 1110, 1408; the polarisation peaks at ℓ ≈ 130, 391, 681, 975, 1272, between them, because a compression is at its extreme when the fluid is momentarily still, and the velocity is largest a quarter of a cycle later. The model has no projection, no decaying potentials and no thickness, so it places the peaks and says nothing about their shapes.

Cosmology

Polarised only where the glow was lopsided

An electron bathed in light from every side scatters it unpolarised, and so does one bathed in light brighter on one side. Only a glow brighter along one axis than across it leaves a polarisation behind, and in the microwave background that glow existed for a short time, in a thin part of a thin shell, because the plasma was moving.

The spectrum ahead is the spectrum behind, shifted. The difference between the local power spectrum of the microwave background measured in the direction of the observer's motion and in the opposite direction, as a percentage of the average, against multipole, at 369.8 km/s. Ahead, the pattern is compressed and every feature moves to a higher multipole by a factor 1 + β; behind, to a lower one by 1 − β, with β = 1.234e-3. The difference is largest where the spectrum is steepest — 2.30 per cent at most in this range — and passes through zero at every peak and trough, where a small shift changes nothing. The spectrum is an interpolation through the measured peaks and troughs, so its steepness between them is known only to that resolution. A per cent of a quantity estimated from millions of independent modes is a measurable number.

Cosmology

A speed read from the finest spots in the map

The dipole in the microwave sky is taken to be the observer's motion and subtracted. The same motion also squeezes the small-scale pattern towards the direction of travel by a part in a thousand, and that squeeze measures the speed a second time without using the dipole at all — which is the only way to ask how much of the dipole is the universe's own.

Exoplanets

A model of the star that can carry the planet

Starspots are modelled as correlated noise with the star's rotation period built in, and the model is good enough to remove signals that would otherwise be announced as planets. It is also good enough to remove planets. Near the rotation period no amount of data in one season separates the two, and away from it a flexible noise model quietly takes most of the evidence a planet would have had.

Exoplanets

A planet has one velocity in every colour

A Doppler shift moves every line in a spectrum by the same velocity, because it is the whole star that moves. A starspot does not move anything; it removes light, and how much light it removes depends on the colour it is seen in. Measuring the velocity in the blue and the red at once separates the two in a single night — until the star's magnetic field, which grows more visible towards the red, takes part of the separation back.

Exoplanets

A slope that is a whole curve of companions

A companion on an orbit longer than the observations shows in a star's velocity only as a steady drift. A drift is one number, an acceleration, and an acceleration is a mass divided by the square of a distance — so a planet at ten astronomical units and a brown dwarf at fifty produce the same straight line. Turning the line into a companion needs the orbit to bend, a picture, or the other two components of the acceleration.

$How often a merged black hole is kicked faster than a given speed. The fraction of black-hole mergers whose remnant recoils faster than the speed on the horizontal axis, for mass ratios spread evenly in logarithm between 0.1 and 1 and spins of 0.9 of the maximum, spins within 10° of the orbit; spins within 30°; spins in random directions, from a fitting formula to numerical relativity over 4000 random mergers each. The median kicks are 113, 234, 534 km/s. Faster than 100 km/s: 62%, 85%, 96%. Faster than 300 km/s: 5%, 40%, 74%. Faster than 1000 km/s: 0%, 2%, 25%. Gas accreted before the merger is expected to drag both spins into line with the orbit, which is why a gas-rich merger keeps its black hole far more often than a gas-poor one.$ Galaxies

A black hole thrown out by its own merger

When two black holes merge, the gravitational waves they radiate carry momentum as well as energy, and unless the two are identical and spinning in line the waves leave more in one direction than the other. The remnant recoils, at anything from a few tens of kilometres a second to several thousand — and whether a galaxy keeps its central black hole comes down to how its spins were pointing in the last orbit before the merger.

Galaxies

The black holes that can be weighed are the heavy ones

A black hole's mass is measured from the motions of stars or gas only inside its sphere of influence, the region where its gravity beats the galaxy's, and that region has to be resolved. At a given galaxy, a heavier hole rules a larger sphere and can be weighed further away, in a much larger volume — so the sample of weighed holes is drawn from the top of the distribution, and the relations built from it describe the heavy end of every galaxy rather than the typical one.

Mass profiles that pinch together near the half-light radius. The enclosed mass of each of the 13 haloes that fit the same flat dispersion of 10 km/s, against radius in units of the stars' three-dimensional half-light radius, 391 parsecs. At a tenth of that radius they span a factor of 17, and at four times it a factor of 2.5; they pinch together near the half-light radius, closest at 1.47 of it, where they agree to 14 per cent. The dot is the estimate 3σ²r/G at the radius where the stars' density falls as r⁻³, 0.94 of the half-light radius; there the fitted masses lie between 0.65 and 1.03 times it. Each model trades a wrong guess about the orbits against a wrong guess about the mass, in opposite senses inside and outside.

Galaxies

The one radius where the orbits stop mattering

A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted.

Gravity per kilometre of topography, by wavelength. The free-air gravity anomaly per kilometre of topography — the admittance — on Earth, against the wavelength of the topography, for elastic plates 5, 25, 100 km thick, with crust of density 2800 kg/m³ over mantle of 3300 and compensation at 35 km. Short wavelengths are uncompensated and give the full attraction of a slab of crust, 117 mGal per km. Long wavelengths are compensated and give nearly nothing, because the mass of the mountain is cancelled by the root beneath it. Where the curve falls between the two is set by the plate's thickness, so a measured admittance spectrum is a measurement of how strong the lithosphere was when the load was applied.

Gravitation

A mountain's gravity says what holds it up

A mountain can stand because the rock beneath it is strong, or because it floats on a deep root of light crust like an iceberg. The two leave different gravity behind them, and the difference depends on the mountain's size — so the ratio of gravity to topography, measured wavelength by wavelength, is a measurement of how stiff a planet's outer shell was when the mountain was built.

One detection, and every density it could mean. The number density of interstellar objects implied by one detection of a given size in 8 years, against the distance from the Sun inside which a survey could have seen it, for objects arriving at 26.3 km/s, if the survey caught 100 per cent, 10 per cent, 1 per cent of those passing inside that distance. Each estimate is the count divided by the rate of passages per unit density, πq²v∞ with gravitational focusing, times the efficiency. If ʻOumuamua-sized bodies could be seen inside 0.5 AU at 10 per cent efficiency, there are about 0.05 per cubic AU; if inside 2 AU with every one caught, 7.9 × 10^−4. The inferred density is a statement about the survey as much as about the visitors.

Orbits

A census of visitors is a census of the survey

Three bodies from other stars have passed through the inner solar system in eight years of looking. Turning that count into a number of such bodies per cubic astronomical unit needs the volume in which they could have been seen, and that volume is set by the Sun's gravity, by each body's size and by how many weeks a survey had to catch it — so the first number the census gives is a measurement of the survey.

The observed sky

Every star in the catalogue is bent by the Sun

The bending of starlight by the Sun is remembered as an eclipse measurement of 1.75 arcseconds at the solar limb. The same bending reaches every direction in the sky, falling only slowly away from the Sun, and at right angles to it is still four milliarcseconds — two hundred times the precision of a modern star position. Every catalogue corrects every star for it, and the correction has become the most precise test of how much space a mass curves.

$The edge of the Sun, computed. The brightness of the Sun at 500 nanometres across its limb, as a fraction of the brightness at the centre of the disc, against height above the level where the photosphere's radial optical depth is one, from the radiative transfer along each line of sight through a grey model atmosphere. Inside the disc the brightness falls gently with limb darkening; at the edge it falls to nothing over about two hundred kilometres. The steepest point of the fall — the inflection point, which is what a solar radius conventionally means — is 335 km above the radial τ = 1 level, because a ray grazing the limb passes through 270 times more atmosphere than a vertical one and becomes opaque that much higher up, at about 336 km. The brightness falls to one per cent 555 km above the τ = 1 level and to a thousandth at 694.$ The observed sky

The Sun is three hundred kilometres larger seen edge-on

The radius of the Sun measured from images and eclipses and the radius measured from its oscillations disagree by about three hundred kilometres, and neither is wrong. A line of sight grazing the limb passes through hundreds of times more atmosphere than a vertical one and becomes opaque higher up — so the edge a telescope sees sits above the surface the waves feel, and every other definition of the edge, from the detector's threshold to the colour of the light, moves it again.

Starlight

A baseline that grows four astronomical units a year

The Earth's orbit gives a parallax a baseline of two astronomical units and never more. The Sun's own motion through the nearby stars gives one that lengthens by four a year without limit — but the stars move too, by as much, so the long baseline yields a distance only for a group whose motions average away, and for a century it was the only way to reach past a few hundred parsecs.

Stars

Brighter for its mass and fainter for its colour

Two stars of the same mass are almost the same star only if they are made of the same material. Take most of the metals out and the opacity falls, the radiation leaks out faster and the star settles brighter and much hotter — so far hotter that, at a given colour, it sits below the ordinary main sequence. The relation between mass and light has composition inside it, and every mass, age and distance read off it inherits that.

Orbital energy spent against the envelope it has to lift. The orbital energy a 0.6 M☉ companion releases by spiralling in from the onset of a common envelope down to a final separation, against that separation (curve), and the binding energy of the 0.85 M☉ envelope of a 1.2 M☉ giant with a 0.35 M☉ core, divided by the efficiency, for αλ of 0.25, 0.5, 1 (horizontal lines). Energies are in units of GM☉²/R☉. The giant fills its lobe at a separation of 118 R☉, and the companion releases nothing net until the orbit has shrunk below 34 R☉, because the envelope's mass leaves with the envelope; from there it falls until the energy released pays for the envelope, so each crossing is a final orbit — 1.29, 2.48, 4.63 R☉. The released energy grows only as one over the separation, so halving the efficiency halves the final orbit: a factor of four in αλ is a factor of four in the outcome, and there is no measurement of αλ that does not run through outcomes like these.

Stars

The least known number in binary evolution

When a giant swallows its companion, the companion spirals inward until the orbital energy it has released is enough to throw the envelope off. The final orbit is set by one product of two unknowns, it moves in proportion to that product, and the merger time of the white dwarfs it leaves moves as its fourth power.

Gravitation

A disc that stores its inflow and spends it in a century

Where only a thin ionised skin of a young disc can transport angular momentum, the gas arriving from the cloud cannot be passed on, and it piles up in the dead midplane until gravity stirs it and the whole column ignites. The star then gains most of its mass in bursts of a century every ten thousand years — which is why almost every protostar ever observed is fainter than its growth requires.

Fitted to one curve, three halo histories agree at the Sun and not at the centre. The dark-matter density from 1 to 15 kpc from the Galactic centre, on logarithmic axes in GeV per cubic centimetre, for three halos fitted to the same flat rotation curve with the same disc of 5·10¹⁰ solar masses: one as dark-matter-only simulations produce it, and two contracted by the disc under different assumptions about the dark matter's orbits. At the Sun's 8.2 kpc they give 0.45, 0.41, 0.40 GeV/cm³, a spread of 12 per cent. At 2 kpc they give 3.2, 4.6, 4.6, a spread of 44 per cent. Inside the region the curve constrains, a fit has no freedom to move dark matter about, and the history drops out; inward of it, where the disc and bulge dominate the curve and the dark matter is a small part of the total, the history decides the answer.

Galaxies

A history the rotation curve absorbs

Whether the Milky Way's dark halo was pulled inward by its disc changes the dark matter at the Sun by half again — if the halo is taken from a simulation. Fit it to the Galaxy's own rotation instead, and the history drops out at the Sun's radius and survives only near the centre. The local density that detection experiments depend on is set by the disc's mass, not by the halo's past.

Galaxies

A cusp a black hole would sharpen

A black hole that grows slowly at the centre of a dark-matter cusp drags the dark matter in after it, turning a density that rises as one over the radius into a spike rising as the radius to the minus seven-thirds. Whatever the cusp was, the spike comes out nearly the same — and the annihilation signal it would produce is set less by the particle than by whether the spike survived the stars around it.

The deuterium in the solar system's water, body by body. The ratio of deuterium to ordinary hydrogen in water, on a logarithmic axis, for the Earth's oceans and for the bodies it has been measured in, with the protosolar hydrogen and the water around young protostars as the two ends of the range. The oceans, 1.6·10⁻⁴, are 7.8 times the protosolar value; the carbonaceous and enstatite chondrites sit within ten per cent of them, as do two comets from the Jupiter family. The comets from the Oort cloud, and 67P from the Jupiter family, are two to three and a half times richer. Water made in a cold molecular cloud is richer still, near 0.001. The ratio is fixed when the water forms or last exchanges with hydrogen gas, and is not changed by anything that happens afterwards except escape, so it is a label that each body's water carries from wherever it was made.

Exoplanets

Water that remembers how cold it was

Every water molecule in the solar system carries a label: the share of its hydrogen that is deuterium, fixed when the water last exchanged with hydrogen gas. Water made in the cold cloud the Sun formed from is rich in it; water that passed through the hot inner disc is poor. The Earth's oceans sit within ten per cent of the carbonaceous meteorites and twice below most comets — a record of where the water was assembled, and of how much of it had ever been hot.

$The mass at which a growing core cuts off the pebbles drifting past it. The pebble-isolation mass — the mass at which a core perturbs the gas around it enough to raise a pressure maximum outside its orbit, where drifting solids stop — against distance from the Sun, for a passively heated disc whose thickness grows as the distance to the 2/7, with α = 10⁻³. The mass goes as the cube of the disc's aspect ratio, because a planet has to push the gas by a fraction of its own pressure, and so it grows outward: 7 Earth masses at 1 AU, 30 at 5 AU where Jupiter's core formed, 138 at 30. A core reaching this mass stops growing from pebbles, starts accreting gas, and becomes a dam: everything drifting inward from outside its orbit is held there.$ Exoplanets

A core that shut the door on the pebbles

Solids in a gas disc drift inward, and a growing planet eats them as they pass until it is heavy enough to bend the gas around it. Then a pressure maximum forms outside its orbit, the drift stops there, and everything beyond is cut off from everything within. Jupiter's core reached that mass early, and the solar system's meteorites still fall into two families that never mixed.

The intergalactic temperature against density, with and without blazar heating. The temperature of the intergalactic gas against its density relative to the cosmic mean, both logarithmic, near a redshift of two. Photoheating by the ultraviolet background alone (dashed) gives a power law with denser gas hotter, T = 12,000 K at mean density rising as the density to the 0.6. Heating by the pair beams of distant blazars, if they lose their energy to the plasma rather than to scattering, is deposited per unit volume, so each particle in a void gets more of it: adding increments of 3,000, 10,000, 30,000 K at mean density turns the relation over below densities of 0.58, 1.23, 2.44 times the mean, and the emptiest gas becomes the hottest. That inversion is what a void heated from the inside looks like, and it would happen whether or not the void holds a magnetic field.

Cosmology

A void heated from the inside

The missing gamma-ray halos around distant blazars are read as a magnetic field in the voids — unless the electron–positron pairs that should make the halos lose their energy to the plasma first. Then the voids are being heated, the heat goes preferentially to the emptiest gas, and the intergalactic medium's temperature–density relation turns over, with the voids hotter than the filaments. The field and the heating predict different skies, and the Lyman-α forest is where they part.

Cosmology

An edge that needs no pulsation

The local measurement of the expansion rate rests on a calibration step, and for decades that step was the Cepheid variables. The tip of the red giant branch replaces it with an edge — the brightness at which old stars ignite helium, nearly the same in every galaxy — found by counting stars rather than modelling them. It gives a slightly lower expansion rate, and whether the difference is a flaw in one calibration or in both is the local half of the Hubble tension.

A spectral age with a ceiling, against the time the lobe took to grow. The spectral age of electrons whose spectrum breaks at 2 GHz, against the magnetic field in microgauss (curve), beside the dynamical age of a 400-kpc lobe whose head has advanced at 0.01c, 0.03c, 0.1c (horizontal lines: 130, 43, 13 million years). The spectral age has a maximum, 109 million years, at a field of 1.9 microgauss — the microwave background's equivalent field divided by √3 — because a weaker field leaves the electrons cooling on the background's photons instead. A lobe that advanced at 0.01c is 1.2 times older than any field can make its spectrum, so the disagreement cannot be removed by choosing a different field. Either the advance speed is overestimated, or the electrons at the break are younger than the lobe — re-accelerated, or freshly mixed in from the hotspot.

Starlight

Two clocks for one lobe

A radio galaxy's lobe can be dated twice: by the break in its spectrum, where the oldest electrons have radiated away their energy, and by its length divided by the speed its head advances. The two clocks disagree, the spectral age usually several times younger, and no choice of magnetic field can make them agree for the slowest lobes. What the disagreement measures is that the electrons are not as old as the lobe.

Starlight

A wind that may be too hot to see

Below a certain luminosity the winds of hot stars are measured at a tenth to a hundredth of what the theory of line-driven winds predicts. Either the theory fails for thin winds, or the winds are there and hot — heated by their own shocks and unable to cool, invisible to the ultraviolet lines that measure them. X-ray line shapes, which weigh the whole wind regardless of its ions, have begun to say which.

$A flat sky, and stars that are wrong. The residual brightness after flat-fielding, in per cent, against distance from the field centre, for the sky (flat line) and for a star (curve), with 3 per cent scattered light at the centre of both the flat and the science exposure. The sky, which fills the field as the flat's illumination did, carries the same scattered component and divides out exactly: the corrected image looks perfectly flat, which is the test an observer applies. The star, a point that scatters a negligible fraction of its light into its own aperture, is divided by the scattered component it never had and comes out 2.9 per cent too faint at the centre. The visible check passes and the measurement fails.$ Starlight

Light that adds, corrected as if it multiplied

A flat field is an exposure of something uniform, and some of the light in it did not come through the optics the way starlight does — it scattered off a baffle, a filter face or a dusty window and landed where the geometry never sent it. Dividing by that flat makes the sky in every science image look perfect and makes every star in it wrong, by an amount the image itself cannot show.

$The fraction of a spin alignment a disc can show. The ratio of a disc population's shape alignment — its mean ellipticity along the projected tidal template — to the Lee–Pen alignment of its spins, against the discs' thickness as a fraction of their diameter, each point from 30,000 discs in a field of shape 0.2. The thinnest disc drawn, 0.05 of its diameter, passes 0.204 per unit of spin alignment; a disc a fifth as thick, typical of a spiral's stellar disc, 0.171; a body as thick as it is wide passes nothing, because a sphere has no shape to align. The ratio is the conversion between what a spin survey measures and what a lensing survey fits, and it is small for two reasons that multiply: a spin alignment is a weak preference among directions, and a disc shows its direction only in proportion to how far it is tipped towards edge-on.$ Cosmology

An alignment a disc shows only when it is tipped

A disc galaxy's outline on the sky is its spin seen in projection, so the torque alignment of spins has to appear in shapes too. It does, multiplied by a factor near a sixth and carried almost entirely by the discs seen close to edge-on — which is how a lensing survey can fit the term and find nothing while a survey of spins finds it.

Cosmology

A cluster weighed against the background behind it

Every cluster of galaxies has the same source behind it — the last-scattering surface, at a distance known to a fraction of a per cent — and bends that source's smooth gradient into a dipole of a few microkelvin. The dipole knows nothing about gas or galaxy shapes, so stacked over hundreds of clusters it is a mass scale that no hydrostatic assumption can bias.

Exoplanets

A spread the projection does not hide

A single projected spin–orbit angle says little about the true one, because the star's tilt is unknown. A population of nearly aligned systems is different: for small tilts the projected angle is one component of a two-component vector, so the rms true obliquity is exactly √2 times the rms projected angle — no star's inclination needed.

Exoplanets

A transit that runs differently backwards

A star spinning near break-up is brighter at its poles than its equator, so a planet crossing it at an angle covers bright surface on the way in and dim surface on the way out. The light curve stops being symmetric in time, and the part that differs forwards and backwards measures the true spin–orbit angle from photometry alone — on exactly the stars where the line-profile method struggles.

The anisotropy each slope of escape predicts. The dipole anisotropy of cosmic rays diffusing out of a halo 4 kpc high, 3D/(cH), against energy, for diffusion slopes of ⅓, 0.5, 0.6, each normalised to the boron-to-carbon grammage at 10 GV — so all three agree at low energy and part above it. The shaded box is the range of dipole amplitudes measured between 1 and 100 TeV, a few parts in ten thousand to about a part in a thousand. At 10 TeV the three predict 0.0043, 0.014, 0.027: the slope that would let shocks inject an index of two overshoots the measurement by 27 times, and even Kolmogorov's ⅓ by 4. The normalisation of the prediction is uncertain by a factor of a few, through the arrangement of the sources and the shape of the halo; its rise with energy is not, because it is the same slope that the spectrum and the secondary ratio require.

Galaxies

One slope read twice, and the reading that disagrees

How fast cosmic rays leak out of the Galaxy depends on their energy through a single power, and that power is measured twice: in the falling ratio of boron to carbon, and in how unevenly the particles arrive across the sky. The first reading says one-third. The second, taken at face value, says the leak must be slower still — and neither leaves room for the spectrum shock acceleration predicts.

How far an electron can have come, by its energy. The distance an electron or positron can diffuse in its radiative lifetime, 2√(∫D dt), against its energy, for the diffusion coefficient fitted to the nuclei with slopes ⅓ and 0.6, and 1 eV/cm³ of field and radiation. At 100 GeV the horizon is 2.7 or 4.7 kpc; at 1 TeV, 1.24 or 2.95; at 10 TeV, 0.57 or 1.86. The marks are the distances to the nearest middle-aged pulsars and to the Galactic centre. Above a teraelectronvolt the electrons and positrons arriving at the Earth were made within about a kiloparsec — inside the Sun's own neighbourhood of the disc — and a feature in their spectrum is a feature of a handful of nearby sources, not of the Galaxy.

Galaxies

A positron that cannot have come far

A proton at a teraelectronvolt can wander the Galaxy for millions of years. An electron or a positron at the same energy radiates its energy away in a few hundred thousand, and in that time it can diffuse about a kiloparsec. So the rising fraction of positrons among high-energy cosmic rays is not a fact about the Galaxy. It is a fact about a few objects within a thousand parsecs of the Sun.

Gravitation

A few thousand clouds outrun a hundred billion stars

Encounters between stars relax the Galactic disc in a time hundreds of thousands of times the age of the universe, so in practice they do nothing. Put a fraction of the same mass into a few thousand molecular clouds and the time falls to the age of the disc — because relaxation counts the mass of each perturber, not their number. The disc's present thickness is the record of that, and of a bound on anything heavier hiding in the halo.

Orbits

A distant planet felt only as a stretch

An undiscovered planet far beyond Neptune pulls the Sun and Saturn almost equally, and an ephemeris, which measures only where Saturn is relative to the Sun, cannot see a pull that moves both together. What it can see is the difference — a tidal stretch falling as the mass over the cube of the distance — and most even of that is absorbed by refitting Saturn's orbit. What is left decides which planets the solar system can be hiding.

$The Moon pushed towards the Sun, once a month. The change in the Earth–Moon distance, in millimetres, if the Earth's gravitational self-energy fell towards the Sun differently from the rest of its mass by a fraction η = 0.001 — the term that would be present at the synodic period, the cycle of lunar phases, in an integration of the Moon's orbit in the Sun's full field. The amplitude is 11.4 mm, which is 11.4 metres times η: the Moon's orbit would be stretched towards the Sun at new moon and squeezed at full, or the reverse, depending on the sign. The shaded band is ±2 mm, roughly the precision of a modern lunar laser ranging normal point. A violation a thousandth of the size of the effect it tests would be twice that precision.$ Orbits

A Moon that would lean towards the Sun

Part of the Earth's mass is its own gravitational binding energy — a few parts in ten billion. If that part fell towards the Sun even slightly differently from the rest, the Earth and the Moon would fall at different rates and the Moon's orbit would be pushed sunward once a month by about eleven metres for every unit of the difference. Fifty years of laser ranging to the Moon have looked for that push at the millimetre level, against a term the Sun's ordinary gravity puts at the same frequency a hundred kilometres high.

A storm's depression at the equator, as an energy. The depression of the horizontal magnetic field at the Earth's surface against the total kinetic energy of the particles trapped in the ring current, on logarithmic axes. The Dessler–Parker–Sckopke theorem makes the two proportional — ΔB/B₀ = −⅔ E/Eₘ, with Eₘ = 8.4·10¹⁷ joules the energy of the Earth's dipole field above the surface — whatever the particles' energies, pitch angles or distribution in radius. The marked storms are read off the line: a moderate storm of 50 nT is 2·10¹⁵ J of trapped particles, an intense one of 100 nT 4·10¹⁵ J, the storm of March 1989 at 589 nT 2.4·10¹⁶ J, and the lowest estimate for the Carrington storm of 1859 3.4·10¹⁶ J — a few hours of the solar wind's power flowing past the magnetosphere.

The observed sky

A storm weighed by a compass needle

During a geomagnetic storm the horizontal field at the equator drops by tens or hundreds of nanotesla, and a theorem says the drop is proportional to the total energy of the particles trapped around the Earth — whatever their energies, their pitch angles or their distances. A ground magnetometer therefore weighs a population tens of thousands of kilometres up, in joules, and the shape of the record says what that population is made of.

Ions picked up from Mars and swept off on loops. The paths of a hydrogen and an oxygen ion created at rest just above the dayside of Mars and picked up by the solar wind, flowing left to right at 420 km/s through an interplanetary field of 3 nT perpendicular to the page. In the planet's frame each moves on a cycloid: it is accelerated by the wind's electric field, turns in the magnetic field, comes momentarily to rest and repeats, drifting downstream at the wind's speed and reaching twice it at the top of each loop — four times the wind's energy per unit mass. The loop height is twice the gyroradius: 2923 km for hydrogen, 46774 km for oxygen, against a planetary radius of 3390 km. The oxygen is thrown clear of the planet in a single plume to one side, which is how a planet without a magnetic field loses its heavy atmosphere to the wind.

The observed sky

A planet that borrows the wind's field

Venus and Mars have no magnetic field of their own, yet the solar wind does not reach their atmospheres. It is stopped by their ionospheres, whose thermal pressure balances the wind's at an altitude that moves logarithmically with every gust, and the interplanetary field piles up and drapes around the obstacle to make a magnetosphere on loan. The arrangement holds only while the ionosphere out-pushes the wind, and it leaks heavy ions on loops wider than the planet.

$The atmosphere's delay, by how high the spacecraft stands. The one-way delay of a radio signal through the troposphere, in metres of range, against the elevation of the line of sight, for an atmosphere whose refractivity falls exponentially with a hydrostatic scale height of 8 km and a wet one of 2 km. At the zenith it is 2.42 m. Integrated along a straight ray through a curved atmosphere (solid), it is 5.0 m at 30°, 13.4 m at 10° and 24.6 m at 5°. The flat-atmosphere rule, zenith delay over the sine of the elevation (dashed), agrees above about twenty degrees and then runs away, giving 46.2 m at 3° where the curved atmosphere gives 35.9: a ray at a low angle leaves the atmosphere through the top of a shell, not through the side of a slab. Ray bending, ignored here, adds centimetres below about five degrees.$ Spaceflight

Two metres overhead and thirty at the horizon

A radio signal crossing the lower atmosphere is delayed by about two and a half metres of range straight up and by more than ten times that near the horizon. Tracking a spacecraft means converting the first number into the second, and the conversion — the mapping function — is where almost all of the troposphere's error lives: under a millimetre above thirty degrees, centimetres to decimetres in the last ten. And the last ten degrees are exactly the observations the geometry cannot do without.

$An absorption line made by scattering alone. Emergent line profiles, as a fraction of the continuum, against distance from line centre in Doppler widths, for a strong line — 10⁵ times the continuum opacity at its centre — in an atmosphere at one temperature throughout, for destruction probabilities ε = 0.01, 10⁻⁴, 10⁻⁶. In local thermodynamic equilibrium this atmosphere makes no line at all: every depth radiates at the same temperature, and the line's extra opacity merely moves the depth from which the light comes. With scattering the line is deep, its central intensity 0.198, 0.021, 0.002 of the continuum, about twice √ε. The darkness is not a low temperature anywhere; it is photons scattered out of the line and back into the gas until they are destroyed or escape through the continuum, never reaching the observer at the line's frequency.$ Starlight

A line darker than any temperature in the gas

An absorption line is usually read as a temperature: the line is dark because its light comes from higher, cooler gas. A strong resonance line breaks that reading. Its photons are scattered thousands of times before a collision can destroy them, so near the surface the line's source function falls to the square root of the destruction probability times the Planck function — and even a gas at one temperature throughout shows a line almost black at its centre.

Starlight

A width that knows how bright the star is

In the dark trough of a cool star's calcium K line sits a small bright emission core made in its chromosphere. In 1957 Olin Wilson and Vainu Bappu found that the core's width predicts the star's absolute magnitude over fifteen magnitudes — from dwarfs to supergiants, whatever the star's temperature and however active its chromosphere — and the simplest physical account of why reproduces the relation's existence and misses its slope by half.

$Where relativity overtakes the margin: 306 Schwarzschild radii for 10⁵ solar masses. Two things against the size of a contracting star, measured in its own Schwarzschild radius, on logarithmic axes. The flat lines are the margin by which the star's pressure-weighted adiabatic exponent exceeds 4/3 — the Newtonian condition for stability — for stars of 1000, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸ solar masses with mean molecular weight 0.61; it does not change as the star shrinks, because it is fixed by the gas fraction of the pressure, and it falls with mass from 0.036 to 1.2·10⁻⁴. The diagonal is what general relativity adds to the requirement, 1.1245 × 2GM/(Rc²), which in these units is the same line for every mass. Contraction moves each star leftward along its flat line, and where the line meets the diagonal the star becomes dynamically unstable: at 31, 97, 306, 966, 3,052, 9,649 Schwarzschild radii respectively. Every one of those crossings lies between 31 and 9,649 Schwarzschild radii out, where the star is not remotely compact — the relativistic correction is tiny, and it decides the matter only because the margin it is compared with is tinier.$ Stars

A star tipped over at a thousand times its horizon

A star of a million suns is held up almost entirely by its own light, which leaves it stable by about a thousandth. General relativity raises the bar by an amount that is tiny until the star is compact, and a thousandth is tiny too — so the star collapses at nearly a thousand times the radius of the black hole it becomes, at a mean density below that of air, and with no fuel yet burning.

The centre goes soft at 19 solar masses; the whole core only at 27. Two criteria for the pair instability, against the mass of the oxygen core, each the deepest value reached as the core contracts through the pair-producing range. The orange curve is local: the lowest Γ₁ − 4/3 at the core's centre. It goes negative at 19.1 solar masses — the centre of every such core passes through gas that, taken by itself, cannot hold itself up. The blue curve is the criterion that actually decides the matter, the pressure-weighted mean over the whole core, and it stays positive until 27.5 solar masses of oxygen. In the shaded band between the two the centre goes soft and the core does not go unstable, because the stiffer gas around the centre outvotes it. The threshold is computed for a core built as an n = 3 polytrope with the pairs ignored in its structure; detailed evolutionary models find the first pair-instability pulses in stars whose helium cores exceed roughly thirty solar masses, so the construction lands in the right neighbourhood rather than on a precise number.

Stars

A soft centre the rest of the core outvotes

Above a billion kelvin, light in a star's core starts turning into electron–positron pairs, and gas that spends its heat on making mass cannot push back hard enough to hold itself up. But stability belongs to the whole core, not to its centre. An oxygen core goes soft in the middle from about nineteen solar masses and holds; it gives way only near twenty-eight, when the soft gas carries two-thirds of the pressure-weighted volume.

$One sight line, two spectra: emission peaks at 46 K while the absorption reaches τ = 1.10. The 21-centimetre line along one sight line through a cold cloud (5·10²⁰ atoms cm⁻² at 60 K) embedded in warm gas (3·10²⁰ cm⁻² at 6000 K, half in front and half behind), computed by radiative transfer. Above: the emission spectrum, brightness temperature against velocity — a broad pedestal from the warm gas (dashed, alone) with the cold cloud's narrow peak on top, capped at 46 K because the cloud is nearly opaque at line centre. Below: the same gas seen in absorption against a bright radio source behind it, the fraction of the source's light transmitted. The absorption shows the cold cloud and almost nothing of the warm gas, whose optical depth is a hundred times smaller per atom. Read in the optically thin approximation, the emission gives a column of 6.26·10²⁰ cm⁻² against a true 8·10²⁰.$ Galaxies

A temperature that belongs to none of the gas

The hydrogen between the stars announces itself at 21 centimetres twice — in emission, which counts atoms whatever their temperature, and in absorption against a radio source behind, which counts atoms divided by their temperature. The ratio of the two is a temperature. Along any real sight line it is the harmonic mean of a cold phase and a warm one, a number no parcel of the gas actually has — and where the cold gas is thick enough, the emission stops counting it at all.

Atomic gas that stops growing: a cloud holds at most 10, 35, 104 M☉ pc⁻² of HI. The surface density of atomic hydrogen in a slab lit on both faces, against the slab's total gas surface density, both logarithmic, for metallicities of solar, 0.3 solar, 0.1 solar. A thin slab is atomic throughout and sits on the one-to-one line (grey). A thick one shields its interior and turns it molecular, so every gram added beyond the transition is a gram of H₂ and the atomic column stops at 10.1 M☉ pc⁻² for solar, 34.7 M☉ pc⁻² for 0.3 solar, 103.7 M☉ pc⁻² for 0.1 solar. The saturation is the same thing the 21-centimetre maps of nearby galaxies show: atomic gas surface densities that rarely exceed about ten solar masses per square parsec in metal-rich discs, however much gas there is, and that climb in metal-poor dwarfs.

Galaxies

An atomic ceiling set by the dust

Pile more gas into a galaxy's disc and the atomic hydrogen stops growing at about ten solar masses per square parsec; everything beyond that is molecular. The ceiling is the column a cloud needs to shield its interior from ultraviolet light, and because dust both makes the molecules and does the final shielding, it scales as the inverse of the metallicity — and hardly at all with the starlight, since the cold gas thickens in step with the field that would break it apart.

Gravitation

A floor under the unseen companion

A star that swings back and forth in velocity with nothing visible beside it is orbiting something. One velocity curve gives one number about the something — the mass function — and that number is not an estimate of its mass but a floor under it, reached only if the visible star weighed nothing and the orbit were seen exactly edge-on. When the floor stands higher than any neutron star can, the companion is a black hole, whatever the inclination turns out to be.

Gravitation

A black hole weighed by a wobble on the sky

A velocity curve weighs an unseen companion only up to the tilt of its orbit. The same orbit followed on the sky, as a loop a few thousandths of an arcsecond across riding on a star's parallax, shows the tilt as the shape of the ellipse — and with the parallax turning the angle into astronomical units, Kepler's third law gives the companion's mass outright. That is how the nearest black holes were found: silent, unfed, and on orbits no X-ray survey could have reached.

The kick speeds read back out of the periods: median 100 m/s against the 100 m/s put in. The distribution of kick speeds recovered from the fragments' periods alone, by p(v) = −2v g′(v) applied to the along-track kicks, for 300 and 3000 fragments (points), against the log-normal distribution actually used, median 100 m/s and width 0.3 dex (line), on a logarithmic speed axis with the density per logarithmic interval. With 3000 fragments the recovery follows the shape of the truth and puts the median at 100 m/s, though differentiating a measured distribution twice smooths its peak and roughens its tails; with 300 it is noisy but already in the right place. The inversion uses one assumption, that the kicks were isotropic, and one measurement per fragment, its period — the quantity tracking networks determine first and best.

Orbits

A kick speed read off the slope of a histogram

Throw fragments at one speed in random directions and the component of each kick along the orbit is spread evenly between plus and minus that speed — Archimedes' theorem about the area of a sphere, applied to debris. Each fragment's period records that component, so the histogram of periods is a stack of flat bands, one per speed, and its slope gives the speeds back: a break-up weighed by its own violence from a catalogue of orbits, if the event was isotropic, and fragment by fragment if it was not.

Two clouds with the same kicks: the explosion's centre at −2.3 m/s, the collision's at 10.9. The along-track kicks read from the periods of 1500 fragments, for an explosion (blue) and for a collision (orange) whose fragments received the same random kicks plus a shared velocity change of 12 m/s along the track — the centre-of-mass motion a projectile's momentum leaves behind. The two histograms have the same width and shape; the collision's is displaced. The number-weighted mean of the explosion's kicks is −2.3 ± 2.6 m/s, consistent with zero, and the collision's 10.9 ± 2.5 m/s. The width of a cloud says how violent the event was; its centre says whether anything arrived from outside.

Orbits

A collision moves the cloud's centre, and an explosion cannot

However violently a satellite blows itself apart, its fragments' momenta add up to the parent's: the centre of the cloud keeps moving exactly as the satellite did. A collision is different, because the projectile brings momentum in, and a collision energetic enough to shatter its target needs a projectile heavy enough to move the centre by at least 80,000 joules per kilogram divided by the impact speed — eight metres a second in low orbit. The width of a debris cloud measures how violent the event was. Its centre measures whether anything arrived from outside.

40 galaxies detected, and a map of all of them: the summed light traces the structure the catalogue misses. A 160 by 160 Mpc/h slice of a simulated universe at redshift 6, its galaxies clustered on a density field and given luminosities from a Schechter function with a faint-end slope of -1.5. Left: the 40 galaxies bright enough to detect individually, above 1.5 times the characteristic luminosity. Right: the same slice as a map of summed line intensity in 32 by 32 cells, shaded linearly between the 5th and 97th percentiles of the cells, with no galaxy detected at all. The catalogue is a sparse scatter in which the structure is barely visible; the map shows it, and its cells correlate with the underlying density at 0.65. The map's light comes overwhelmingly from galaxies below the detection limit.

Cosmology

A map of galaxies none of which was seen

A survey that lists galaxies one by one stops at the faintest it can detect, and at high redshift most of the light comes from galaxies fainter than that. Map one spectral line in summed intensity instead, pixel by pixel and frequency by frequency, and every galaxy contributes whether it is detected or not. The fluctuations of that map have two parts — clustering, which weighs the total light, and graininess, which weighs how unevenly it is shared — so one power spectrum measures two moments of a luminosity function nobody has counted.

[C II] power is round in the plane of k; CO(4–3) mapped at the wrong distance is elongated 5.1 times along k∥. Contours of the clustering power spectrum in the plane of wavenumber across the sky, k⊥, and along the line of sight, k∥, in h/Mpc, at 10², 10²·⁵, 10³, 10³·⁵ (Mpc/h)³ in arbitrary units, for the target line, [C II] from redshift 6.60 (black), and for the CO(4–3) interloper from redshift 0.84 placed at the target's distance (colour). The target's contours are quarter-circles, because its structure has no preferred direction. The interloper's are ellipses, drawn in towards the k∥ axis and elongated along it by the ratio of the two rescalings, α∥/α⊥ = 5.1: a structure that is round where it really is looks magnified across the sky and flattened along the line of sight where the map puts it. That difference of shape, not of amplitude, is what separates the two in a map where both have been summed.

Cosmology

A line from the wrong redshift lands at the wrong distance

A map of ionised carbon from redshift six is built by assuming every photon at a given frequency came from that line. Carbon monoxide from galaxies at redshift one sends photons to the same frequency, and the map places them five billion light-years too far away. Their structure arrives magnified across the sky and squashed along the line of sight — round where it really is, flattened in the map — and that distortion of shape, which no amount of brightness can hide, is how the impostor is found.

Three planes on one sky: the Milky Way crosses the celestial equator at 62.87° and its pole sits at declination 27.13°. The whole sky in equatorial coordinates — right ascension decreasing to the right, declination up — with the celestial equator (grey), the ecliptic (orange) and the galactic equator, b = 0 (blue), each a great circle drawn by rotating its own frame into the equatorial one. The north galactic pole lies at right ascension 192.86° and declination 27.13°, so the galactic plane is inclined at 62.87° to the celestial equator; the zero of galactic longitude, towards the centre, lies at 266.40°, -28.94° in Sagittarius. All three are conventions tied to physical planes — the Earth's equator, its orbit, the Galaxy's disc — and only the first two are defined by measurements that can be repeated to the precision of the coordinates themselves.

The observed sky

A frame fixed before its centre was found

Galactic longitude and latitude look like the Galaxy's own coordinates, and they are a convention agreed in 1958: a pole taken from the radio map of neutral hydrogen, a zero of longitude put on a radio source then catalogued in Sagittarius, an equator drawn through the Sun. The black hole at the centre was found afterwards, three arcminutes away, and nobody moved the origin. The Sun sits twenty parsecs above the disc, so latitude zero is not the Galaxy's plane except far away — and over eighty million years the Sun will swing a hundred parsecs either side of it.

A shadow 77 mm across at 144.3 m: an occulter 1.42 m wide and a 50 mm aperture inside its umbra. Side view of an eclipse made by two spacecraft, distances along the axis in metres and widths across it in metres, the vertical scale magnified about fifty times. Sunlight arrives from the left. The occulting disc, 1.42 m across, casts an umbra — the cone from which none of the Sun's disc is visible — which narrows because the Sun is not a point: its edges converge at a rate set by the Sun's angular radius, 959.6 arcseconds, and would meet 153 m behind the disc. At 144.3 m, where the coronagraph sits, the umbra is 77 mm across and the aperture 50 mm; the penumbra, from which part of the Sun is visible, extends to 2.76 m. From the aperture the occulter's edge hides the Sun out to 1.058 solar radii, so the corona is visible from just beyond the limb.

The observed sky

An eclipse held in place to a millimetre

A total eclipse of the Sun works because the Moon's disc happens to match the Sun's almost exactly, and it lasts a few minutes. Two spacecraft flying 144 metres apart can make one that lasts six hours: a disc 1.4 metres across, with a coronagraph sitting in its shadow. The geometry that sets the shadow's size also sets the price. At that separation the umbra is 77 millimetres across around a 50-millimetre aperture, so the formation has to hold itself in line with the Sun to a centimetre — and to two millimetres in January, when the Sun is at its largest.

$The radius a rocky planet would have, by core fraction: at 5 M⊕ an Earth-like and a Mercury-like planet differ by 8%. Radius against mass for rocky planets of one to eight Earth masses, from Zeng and collaborators' interior models, for core mass fractions of 0, 0.33, 0.5, 0.7, 1 — from a planet with no iron core to one that is all iron. The shaded band is the range of core fractions planets would have if they kept their host stars' iron, magnesium and silicon in the stars' own proportions, for stars whose iron-to-magnesium ratio is within 0.15 dex of the Sun's: 0.26 to 0.41. The Earth, at 0.325, sits in it. At five Earth masses an Earth-like planet and one with a core fraction of 0.7, like Mercury, differ in radius by 8 per cent — a difference that transit and radial-velocity measurements can now resolve.$ Exoplanets

More iron than the star could give

A rocky planet is made of what its star was made of, minus the gases: iron, magnesium and silicon in the star's proportions, which for the Sun predict an iron core of a third of the planet's mass — and the Earth's core is 32.5 per cent. A handful of small exoplanets are so dense that their cores must hold seven-tenths of their mass, more iron than any star known to host planets could have supplied. Either something removed most of their rock after they formed, or the rock and the metal were separated before they did.

Four rocky planets against the bare-rock limit: TRAPPIST-1b at 0.99, TRAPPIST-1c at 0.87, LHS 3844b at 1.01, GJ 367b at 0.99. Published day-side brightness temperatures of four small planets orbiting M dwarfs, each divided by the temperature its day side would have as dark bare rock re-radiating where it absorbs (dot, with its uncertainty), with the value for heat spread evenly over the whole planet marked for comparison (grey tick, at 0.71). TRAPPIST-1b: 503 ± 26 K against a bare-rock 508 K; TRAPPIST-1c: 380 ± 31 K against a bare-rock 434 K; LHS 3844b: 1040 ± 40 K against a bare-rock 1031 K; GJ 367b: 1728 ± 90 K against a bare-rock 1741 K. Three sit at the bare-rock limit within their errors, which leaves room for no thick atmosphere on dark rock. TRAPPIST-1c sits about one in eight below it, which a thin atmosphere could explain, and so could a moderately reflective surface. None comes near the even-redistribution value that a thick atmosphere would give.

Exoplanets

A day side as hot as bare rock

A planet with no air can be no hotter on its day side than bare rock that re-radiates every photon where it lands. An atmosphere carries some of that heat to the night side, so the day side comes out cooler. For a rocky planet too small for its spectrum to be read, that difference is the test: three of the first four small planets around red dwarfs measured this way sit at the bare-rock limit, and the fourth sits below it by an amount that a thin atmosphere, or a brighter surface, could explain equally well.

How many stars it takes to tell a shared axis from chance. The number of stars whose measured sin i, with an error of 0.1, is needed on average before an aligned cluster is preferred over random axes by a likelihood ratio of 370 to one — three standard deviations — against the inclination of the common axis, on a logarithmic scale, for perfectly aligned, κ = 20, κ = 5. Perfect alignment needs a handful of stars at any angle — 4 at 30°, 10 at 85°. A loose alignment, κ = 5, needs 29 about an axis at 30° and 38 about one at 85°, and 985 about one at 55°: a loose cone about an axis near 54.7°, where 3cos² i₀ = 1, has the same mean cos² i as random axes and is the hardest of all to tell from them.

Starlight

A shared axis hides at the angle where it does no harm

If the stars of a cluster were born spinning about a common axis, their inclinations would show it. A tight alignment shows in a handful of stars at any orientation. A loose one about an axis tilted 54.7° from the line of sight has the same average inclination as random axes and takes a thousand stars to find. At that same angle, a shared axis also stops biasing the cluster's brightnesses.

The cancellation that protects the method, and where it fails. The error in distance from 0.1 magnitudes of extinction left uncorrected, against temperature, for the angle read from V and V − K through the blackbody relation's local slope, and for a standard candle in V (dashed, −4.7 per cent). For late-type stars the dimming and the reddening nearly cancel — the empirical relation gives +1.0 per cent. For hotter stars the reddening term grows and nothing offsets it: the error matches the standard candle's near 13,500 K and reaches +7.5 per cent at 20,000 K and +11.9 at 30,000 K, worse than the candle the method was meant to improve on.

Starlight

The cancellation that leaves with the hot stars

The angular size of a cool giant can be read from its colour, and dust barely spoils the answer: the dimming and the reddening dust causes push in opposite directions and nearly cancel. Carried to hot stars the method gets weaker on both counts. Above about 10,000 K the colour V − K stops tracking temperature, a colour error costs more angle, and the reddening outweighs the dimming. Near 13,500 K leaving the dust uncorrected costs as much as it would a standard candle.

$The share of white dwarfs whose centres have begun to freeze. The fraction of a population of carbon–oxygen white dwarfs, with masses spread as a Gaussian of mean 0.62 and width 0.12 M☉, whose centres have begun to crystallise, against cooling age, overall and in three mass ranges, in the toy model of the other figures. If a field appears when freezing begins, this is the shape its incidence must take: it switches on first among the heaviest white dwarfs, half of which have begun freezing by 0.5 Gyr, and last among the typical ones near 0.6 M☉, by 2.5 Gyr. A fossil field, present from the white dwarf's birth, would show no such delay and no such ordering by mass.$ Stars

A field that waits for the core to freeze

A field inherited from the parent star would be present from a white dwarf's birth. Yet magnetic white dwarfs of ordinary mass are rare among the young and common among the old, as if their fields were switched on later. The proposed switch is the freezing of the carbon–oxygen core, which stirs the liquid above it. If that is right, the delay is set by when each mass begins to freeze, and the heaviest should go first.

$Larger and cooler, along a line of constant light. The fractional change in effective temperature against the fractional change in radius for a star whose luminosity is unchanged, and for luminosities −10% and +10%, from L = 4πR²σT⁴. Holding the luminosity, a radius 10 per cent larger needs a temperature 4.7 per cent lower: about half the radius change, with the sign reversed. The shaded box is the range reported for red dwarfs in eclipsing binaries against standard models — radii 5 to 15 per cent too large, temperatures 3 to 5 per cent too low — and it lies along the line of constant luminosity.$ Stars

Too large and too cool by just enough to keep the light

Red dwarfs in eclipsing binaries, whose radii are measured directly, come out several per cent larger than stellar models predict for their masses and a few per cent cooler. They are not brighter or fainter. The light is decided deep inside, by mass and opacity, and whatever disturbs the convecting outer layers can change only the radius and temperature through which that light leaves.

Where solved fields land in the plane the surveys fit. The w₀–wₐ plane, with the cross at a cosmological constant. Each small point is the best two-parameter description, through the same mock survey as the earlier figures, of a scalar field solved from its own equation: four thawing potentials at seven values of w₀ from −0.97 to −0.62, and freezing trackers on inverse-power potentials with exponents from a quarter to two. The thawing fields all land on one line through the constant, wₐ ≈ −1.49(1 + w₀), with no point more than 0.08 from it; the freezing fields land on the other side, with wₐ positive. The three published fits (larger points, with their quoted errors) lie below the thawing line by −0.38, −0.59, −0.49 in wₐ: steeper than any thawing field with their present value would be. The diagonal is w₀ + wₐ = −1. The thawing line is steeper than it, so every thawing field is fitted as having crossed −1 — but early: these fits cross between redshift 1.7 and 3.3, where a line through the constant with this slope must, near z = 1/(1.49 − 1) = 2.0. The published fits cross near redshift 0.4.

Cosmology

Every thawing field falls on one line

A scalar field that is the dark energy sits still while the universe expands quickly, and begins to roll only when the expansion has slowed to the field's own mass. Solve the field instead of fitting a line to it, and four quite different potentials give one history and one line through the plane the surveys fit — a line the published fits do not sit on.

How much each change moves the growth rate. The percentage difference in fσ₈ from growth in relativity inside a cosmological constant's expansion, against redshift, for three alternatives started from the same early amplitude. Changing the expansion to the evolving fit with DES-SN5YR and keeping relativity moves it by −1.2 per cent at redshift 0.5, and by at most 2.1 per cent anywhere below redshift 2. Keeping the constant's expansion and changing gravity to γ = 0.68 moves it by −8.6 per cent; doing both, −10.3. The shaded band is ±8 per cent, about the error of one of the better redshift-space measurements so far. The difference between a constant and the evolving fit is several times smaller than any single measurement; the difference between two laws of gravity is comparable to one. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is gravity.

Cosmology

The growth that tells a substance from a law

In relativity an expansion history leaves no freedom in how fast structure grows inside it, so every fitted history predicts its own growth. The evolving dark energy the surveys prefer changes that growth by one or two per cent; a change in the law of gravity, with the expansion held fixed, changes it by nine. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is there at all.

Exoplanets

The Earth-sized bin is where diluted planets land

A star with an unresolved companion shares its aperture with that companion's light, and every planet found around either star is measured too small. The error runs one way only, so the smallest radius bin receives planets from every larger one and returns none. With a companion beside three stars in ten, more than a quarter of the planets called Earth-sized are something larger.

$The cadence that finds a planet and the cadence that weighs it. The fraction of caustic crossings — seven tracks across the planetary caustic at offsets spanning it, each sampled at 160 phases — in which a survey of a given cadence detects the anomaly (three points deviating from the single-lens curve by more than five times a 1 per cent photometric error, solid) and in which it also measures the source size (at least two of those points where the source's finite size changes the magnification by more than five per cent, which happens only during a fold crossing; dashed), for a Jupiter and an Earth around a third-of-a-solar-mass star. For the Jupiter, detection stays at 100 per cent at a four-hour cadence and 48 per cent at one day, while the source size is measured in 43 per cent of crossings at fifteen minutes, 15 at four hours and 0 at a day. For the Earth what limits the measurement is the track rather than the cadence: the source size is measured in 43 per cent of crossings at fifteen minutes and still 43 hourly — the tracks that cross a fold — while detection falls from 100 to 54 per cent, and at four hours both are 0. A cadence that finds giant planets reliably measures few of them; for small planets the cadence that finds them is already the one that measures them.$ Exoplanets

A cadence that finds a planet is not one that weighs it

A planetary anomaly on a microlensing event is found by a handful of points that deviate, and it is turned into a mass only by points taken while the source crosses the caustic — a few hours for a Jupiter whose anomaly lasts days. Sampled every four hours, every Jupiter crossing in a simulated survey is found and fewer than one in six is weighed; the field layout that finds the most planets is not the one that weighs the most.

Galaxies

A dispersion that weighs the sheet only if the orbits are known

A lens's images cannot tell how much of its mass is in the galaxy and how much is spread through a sheet around it, and the time delay — and so the Hubble constant — scales with the answer. The stars' velocity dispersion can tell, because orbits feel mass in three dimensions where light only integrates along a line. One dispersion through one aperture weighs the sheet only to sixteen per cent, because the shape of the orbits moves it as much; the same dispersion measured in rings weighs it to four.

$How far a sample of lenses can take the Hubble constant. The fractional uncertainty in H₀ from a sample of N lenses, both axes logarithmic: the independent errors of each lens — 2 per cent from its delay, 3 from the rest of its lens model, 2.5 from its line of sight and its own mass sheet as well as its kinematics fix it — fall as 1/√N; the shared errors — 1 per cent from the calibration of the line of sight, and the orbital bias of the previous figure — do not. With one aperture dispersion per lens the independent error is 10.1 per cent per lens and the floor 1.6 per cent, reached at about 42 lenses; with six resolved rings per lens, 5.9 and 1.1 per cent, reached at about 29. Dotted: the two floors. A sample that adds lenses without changing how each is measured stops improving at its floor; the resolved kinematics lower both the slope's starting point and the floor.$ Galaxies

A hundred lenses and one assumption

A time-delay lens gives the Hubble constant to five or ten per cent, and the hope is that a hundred give it to one. The errors that differ from lens to lens — its delay, its own line of sight, its own internal sheet — do average away. The errors every lens shares because every lens was analysed the same way do not, and they set a floor that a sample reaches after a few dozen lenses and never goes beneath.

Orbits

Jupiter keeps only the visitors already nearly bound

An interstellar object passing Jupiter can lose enough speed to stay, but one flyby can only turn the visitor's velocity relative to the planet, never shorten it, and the turn removes at most a few kilometres a second from its speed around the Sun. So capture is common for visitors that arrive with less than a kilometre a second to spare and almost impossible for the speeds the three known visitors had. A census of the slow tail of the interstellar population, and the orbits a capture leaves — more than a third of them backwards — follow from the same arithmetic.

A dropout that partly comes back. The flux of 1 MeV electrons at L = 4.5, relative to before the storm, through a storm whose ring current deepens to Dst = −150 nT over eight hours and decays with a one-day e-folding. Dashed: the adiabatic part alone, with phase-space density unchanged — the flux falls to 18 per cent at the storm's peak and returns completely. Solid: the same with a real loss of 60 per cent of the phase-space density during the main phase, as compression and outward diffusion would cause. The flux falls to 7 per cent and recovers only to 39. Seen in flux at fixed energy, the two causes look the same while the storm is on, and only the recovery — days later — tells how much of the dropout was electrons lost and how much was the field.

Spaceflight

A dropout that is only the field

During a storm the ring current weakens the field the radiation belt sits in, and every electron, keeping its adiabatic invariants, loses energy. A detector counting electrons at a fixed energy and place sees the flux fall by half or more — and not one electron has gone anywhere. When the ring current decays, the electrons regain their energy and the flux comes back. Only in the coordinates the electrons themselves conserve can a storm's real losses be told from the field's sleight of hand, and the bigger the storm, the larger the sleight.

The ratio of two distances, for a universe with dark energy and one with dust. The distance-duality ratio η — the luminosity distance divided by (1 + z)² times the angular-diameter distance — against redshift. In a universe with a cosmological constant, or any universe in which light travels on the geometry and is conserved, it is exactly one (solid). In a universe of matter alone, decelerating, with grey intergalactic dust dimming its supernovae just enough to make their distances match the accelerating one's, it rises with redshift: 1.19 at redshift 0.5, 1.30 at 1, because the dust makes supernovae look farther than the geometry puts them while leaving the acoustic-scale distances untouched. The shaded band is η = 1 ± 0.03z, about what supernova and acoustic-scale distances together allow. The dusty universe's η rises at the start with a slope of about 0.45 per unit redshift, more than ten times the band's, and leaves it at once.

Starlight

Two distances that must agree if light is conserved

A supernova's distance comes from how faint it is; an acoustic ruler's distance comes from how small it looks. In any universe whose light follows the geometry and is neither created nor destroyed, the two are tied by an exact factor of (1 + z)², whatever the universe contains. Dust that dimmed distant supernovae without reddening them could have faked the accelerating expansion — and would have broken that tie by a fifth by redshift a half. The tie holds to a few per cent.

Starlight

A floor the star sets for itself

Above the atmosphere, with comparison stars and a big enough telescope, every noise in a measurement of a star's brightness can be beaten down except one: the star's own surface, a mosaic of convective cells each brighter or fainter than the rest. That flicker does not average away against anything, and it is larger for stars that are bigger and puffier than the Sun. For the brightest stars, an Earth's transit is limited by the star, not the photons — and the stars Kepler watched turned out twice as noisy as the Sun.

Phase curves that agree where they can be measured. Modelled reflected light of a cloud-covered planet — geometric albedo times phase function, on a logarithmic axis — against phase angle, for clouds of asymmetry parameter 0.00 and 0.65, from scattering evenly to throwing most light forward. Each curve's single-scattering albedo — with the planet's radius known, as a transit measures it — has been chosen to match the first curve between 30° and 110° (shaded), the range the measurement is assumed to cover: the fitted albedos are 0.9900, 0.9998, and inside the window the matched curve differs from the first by at most 1%. Beyond it they part: at 165°, a thin crescent, the forward-scattering cloud is 6.0× as bright as the isotropic one. The model is an optically thick cloud with exact single scattering and a similarity-scaled multiple-scattering term.

The observed sky

Two clouds that shine alike until the crescent

A grey cloud that scatters light evenly and a whiter one that throws it forward return almost the same light at every phase angle from thirty degrees to a hundred and ten. They part only at the thin crescent, where a planet round another star is hardest to see — and the measurement that tells them apart in the middle of the orbit is the polarisation, a ratio with no albedo in it.

The observed sky

A clock whose best interval is three years

A pulsar time scale was expected to improve for as long as anyone kept timing it. The background the timing arrays found in 2023 says otherwise — a single pulsar is steadiest over about three years and worse over every longer interval, and how far an array of them can push that floor down depends less on how many pulsars it holds than on how evenly they cover the sky.

$One cluster weighed two ways, and the redshift where the two swap places. The fractional error on the mass of a single cluster of 3·10¹⁴ solar masses, on a logarithmic axis, against the cluster's redshift. Weighed by the shapes of the galaxies behind it (rising curves), the error is 50 per cent at redshift 0.3 and 750 per cent at 1.5 for a survey of today's depth, 10 galaxies per arcmin²; 26 per cent at redshift 0.3 and 187 per cent at 1.5 for a survey of the next depth, 30 galaxies per arcmin²: the galaxies far enough behind the cluster to be lensed by it run out. Weighed by the dipole it draws on the microwave background (nearly flat curves), the error is 461 per cent at 0.3 and 337 per cent at 1.5 for a map with 10 μK-arcmin of noise; 224 per cent at 0.3 and 153 per cent at 1.5 for a map with 3 μK-arcmin of noise — never good for one cluster, and almost the same at every distance, because the source is at redshift 1090. The curves cross at z = 1.22 (10 galaxies per arcmin², 10 μK-arcmin), z = 0.89 (10 galaxies per arcmin², 3 μK-arcmin), z = 1.77 (30 galaxies per arcmin², 10 μK-arcmin), z = 1.41 (30 galaxies per arcmin², 3 μK-arcmin). Below the crossing a galaxy survey weighs each cluster better; above it, only the background weighs it at all.$ Cosmology

A mass scale checked near and carried far

A cluster can be weighed by the galaxies behind it or by the microwave background behind it, and the two share no systematic error. They do not share a redshift range either: the galaxies run out at about one, the background never does, and a comparison can only be made where both still work.

$The number of lit dwarfs at a void's centre, against the date it was reionised. The number of haloes heavier than 3.2·10⁹ M☉/h that had collapsed from the matter of a 5 Mpc/h sphere by the redshift at which that sphere was reionised, on a logarithmic axis, against that redshift — for a sphere at the centre of a void with 20 per cent of the mean density today (solid) and an average sphere of the same size (dashed). If only those haloes could make stars, they are the sphere's dwarf galaxies. Reionised at 5.3: 4.3 in the void sphere, 195 in the average one; reionised at 6: 2.0 in the void sphere, 121 in the average one; reionised at 7.7: 0.22 in the void sphere, 29 in the average one. Each unit of delay in the date multiplies the void's count by 3.2 and the average sphere's by 2.1: the void's dwarfs sit further out in the tail of the distribution, and a tail is steep. A count of a few galaxies at a void centre is a statement about when that void's gas was heated, to within a fraction of a unit of redshift — provided the threshold is known.$ Cosmology

A void dated by the dwarfs that lit in time

Reionisation did not heat the universe all at once, and the centres of the large voids were among the last places it reached. A dwarf halo there could make stars only if it had grown heavy enough before its own gas was heated — so the number that did is a clock for one void, and a clock that runs faster the emptier the void.

The heaviest atom a hydrogen wind can carry, against the strength of the wind. The crossover mass — the heaviest atom an escaping hydrogen flow drags out with it — against the hydrogen flux at the base of the flow, on a logarithmic axis, for the Earth's gravity and gas at 1000 K. Atoms lighter than the crossover leave with the hydrogen; heavier ones stay. The crossover rises in proportion to the flux: lifting neon takes 9.7·10¹² hydrogen atoms per square centimetre per second, argon 1.8·10¹³, krypton 4.3·10¹³ and xenon 6.6·10¹³. The Earth loses about 3·10⁸ today, which lifts nothing heavier than hydrogen. Driven by a young Sun at its saturated X-ray and ultraviolet output, an energy-limited flow would reach 2.2·10¹³ on Mars, a crossover of 114; 2.3·10¹³ on Venus, a crossover of 50; 10¹³ on the Earth, a crossover of 21 — the smaller planet's weaker gravity both raises the flux per unit area and lowers the force holding the heavy atoms back.

Exoplanets

The heaviest atom a hydrogen wind can lift

A planet losing hydrogen fast enough to blow off as a wind drags other gases with it, but only those lighter than a crossover mass set by how strong the wind is. Below it every gas leaves in proportion to its distance from the limit and its isotopes are sorted by the same amount per mass unit; above it nothing goes. The Earth's xenon is sorted and its krypton is not, which no wind of neutral atoms can arrange.

Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find.

Exoplanets

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

Galaxies

Motions across the sky measure the orbits first

A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.

Galaxies

One profile for a galaxy that has two

The worry about the compact galaxies of ten billion years ago was that their growth since might be partly a faint envelope the sky hides. Modelled, the envelope does hide — but not mainly under the sky. A single profile fitted to a core with an envelope reports a compromise at any depth, too large for a tight envelope and too small for a wide one, and the growth it measures is an understatement.

$How long a chaotic orbit stays away, with and without islands. The fraction of recurrence times to a small square in the chaotic sea of the standard map that exceed a given number of iterations, on logarithmic axes, counted along one orbit of 6·10⁷ iterations at K = 1.5 and K = 7 (772,165 recurrences, mean 78; 548,043 recurrences, mean 109). Dashed: exponentials with the same means. At K = 7, where the islands have all but vanished, the recurrences are exponential: the orbit forgets where it was in a few iterations. At K = 1.5 they part from the exponential after a few hundred iterations and fall instead as a power of time, with exponent 1.17 between a thousand and a hundred thousand iterations; the step at the left is the share of visits that simply stay in the square for a second iteration. A recurrence a thousand times longer than the mean is found once in 257,388 recurrences, where an exponential would give one in e to the thousand. Each mean is the area of the chaotic sea divided by the area of the square — 110 when the sea fills the torus, 78 at K = 1.5, where the islands take 29 per cent of it. The islands shorten the average absence and lengthen its tail.$ Gravitation

A chaotic orbit clings to the edge of order

In a chaotic sea with no islands, an orbit forgets where it was within a few steps, and the time it stays away from any region falls off exponentially. Put islands of regular motion in the sea and the long absences multiply by factors of millions: the orbit reaches an island's edge, is caught in the hierarchy of smaller islands round it, and behaves as if it were regular for thousands of steps. The tail becomes a power law, and a rare few absences carry a large share of the orbit's whole history.

Dark companions' masses with and without a gap, as an astrometric survey would see them. The distribution of the masses of dark companions in a sample found by their stars' astrometric orbits, per unit of the logarithm of mass, for two hypotheses: neutron stars near 1.35 solar masses and black holes near 8 in equal numbers with nothing between 2 and 5 (solid), or the same with the gap filled at the density the black holes have at 5 (dashed; 16 per cent of the sample in it). Each is drawn as measured with mass errors of 10 per cent (blue) and 30 per cent (orange). With 10 per cent errors the empty gap stays empty: 0.2 per cent of the sample lands between 2.5 and 4.5 solar masses, against 10 per cent if it is filled. With 30 per cent errors the edges spill in — 4.3 per cent against 12 per cent — and the gap becomes a shallow dip that a modest sample cannot distinguish from a filled one.

Gravitation

A gap that counting can test and impostors can fill

The black holes in X-ray binaries leave a gap between the heaviest neutron stars and about five solar masses, and nobody knows whether it is real or an artefact of how those masses are measured. Astrometric orbits give masses that need no light curve, and with errors of ten per cent a few dozen of them would settle it. The difficulty is not the counting. It is that pairs of white dwarfs and neutron stars, dark and close, can weigh exactly what a gap object would.

The age a handful of craters allows. The probability of each age for a 100 square-kilometre surface on Mars, given a count of 0, 3, 10, 30, 100 craters 50 metres across or larger, per unit of the logarithm of age, with a uniform prior on age. No craters: younger than 45 Myr at 95 per cent, and nothing more; 3: 55 Myr, from 31 Myr to 88 Myr; 10: 158 Myr, from 116 Myr to 212 Myr; 30: 457 Myr, from 382 Myr to 547 Myr; 100: 1,502 Myr, from 1,359 Myr to 1,660 Myr. Each tenfold increase in the count narrows the range by about the square root of ten, from a factor of 2.8 with three craters to 1.22 with a hundred. The counting error is only part of the story: the chronology converting a crater density into an age is uncertain by a factor of about 1.6 on Mars before any crater is counted.

Orbits

A young surface dated by a handful of holes

The surfaces that decide whether a planet is still geologically alive — the freshest lava on Mars, the smoothest ice on Europa — are the ones with fewest craters to count. Their ages are Poisson estimates from a few dozen holes, they cannot be younger than the area and crater size allow, and a fraction of a per cent of older ground in the counting area can double them. What survives best is not when a surface formed but which of two surfaces formed first.

The summer sunshine at Mars's north pole over the last million years. The sunshine reaching Mars's north pole at the summer solstice, relative to its average, over the last 1,000 thousand years (blue), from a forcing built from the main periods of Mars's recent orbit: the tilt varying by about 3.5 degrees in 120,000 years and 1 degree in 99,000, the eccentricity by 0.03 in 95,000 years and 0.02 in 2.4 million, and the perihelion going round the seasons once in 51,000 years. The part due to the tilt alone is dashed. The sunshine swings by a factor of 2.06 between its extremes: the tilt sets a slow swell with a period of 120,000 years, and the precession, whose strength the eccentricity sets, cuts a faster ripple of 51,000 years into it, strongest when the orbit is most eccentric. These two periods, and their ratio of 2.35, are what a record of the sunshine should carry.

The observed sky

Layers that keep the rhythm and lose the dates

Mars's polar caps are stacks of ice and dust laid down while the planet's tilt and orbit changed the summer sunshine, and the question is whether they can be read as a record of that sunshine. They can — for the rhythm. Even when the ice accumulates faster under weak sunshine than under strong, the cycles of the tilt and the precession survive in the depth of the stack in their correct proportion. What does not survive is the date of any one layer.

Starlight

A spot a slow star hides inside its velocity

A dark spot on a rotating star removes light from one part of each spectral line, and the line's centre seems to move: a velocity of metres per second that no planet caused. In principle the spot betrays itself, because it changes the line's shape and a planet does not. In practice the betrayal is a few per cent of the signal on a star that rotates as slowly as the Sun, and seeing it takes a signal-to-noise of hundreds per pixel across thousands of lines.

Spectra that age, and spectra that stop ageing. Synchrotron spectra in a 3-microgauss field, divided by the straight power law the electrons were injected with so that a spectrum with no break would be a flat line at zero, at ages of 10, 30, 100 million years. Dashed: electrons left alone, whose break falls steadily to lower frequency. Solid: the same electrons stirred throughout, gaining energy at a rate of one e-fold every 30 million years against their losses. The stirred spectra's break moves down at first and then stops, at the frequency radiated by electrons of Lorentz factor 41,763, where gain and loss balance. Read as passive spectra, the stirred ones give ages of 8.5, 18.9, 28.8 million years: the last is 3.5 times younger than the electrons are.

Starlight

A lobe that reads the time since it was stirred

If a radio lobe's electrons are re-energised in place by the turbulence around them, the break in their spectrum stops falling and its spectral age stops growing, settling at the time the stirring takes to double their energy a few times over. The spectrum's shape does not betray it: a stirred lobe fits an unstirred one exactly. What betrays it is a map that stops ageing with distance, and a bill for energy that grows exponentially with the time the stirring lasts.

The rise in survival, and what two detectors see of it. The survival probability of ⁸B electron neutrinos against neutrino energy (grey), and what two detectors measure of it against the energy of the electron they record: the ratio of the counted spectrum to the one expected without conversion, for electron scattering in water (blue) and for charged-current capture on deuterium (orange), both smeared by a water detector's energy resolution. Between 3 and 10 MeV the survival probability rises by 0.122. Scattering shows a rise of 0.044 over the same recoil energies, 36 per cent of it; capture on deuterium shows 0.063, 52 per cent.

Stars

A rise a detector shrinks to a tilt

Between 10 and 3 MeV the chance that a solar neutrino arrives as it left should rise by a third of its value, and no detector has clearly seen it. Counting is not the obstacle: a large water detector has many times the exposure the rise needs. What hides it is that scattering shows only a third of the rise, spread over the spectrum as a tilt of a few per cent, and that half of the evidence for the tilt sits where the radioactive background is and the other half where an error in the energy scale looks the same.

Stars

A lone star is dated only as it leaves

A star in a cluster is dated by the bend its neighbours trace. A star alone has one position on the Hertzsprung–Russell diagram, and an age for it means finding which isochrone passes through that point. On the main sequence they almost all do: a Sun-like star measured as well as a parallax and a spectrum allow is anywhere from three and a half to five and a half billion years old. A star the same mass on its way off the main sequence is dated to two per cent.

The echo of a one-day flare, for four void fields. The 1–10 GeV flux of the cascade echo against the time since the start of a one-day teraelectronvolt flare of 10 Crab-days, from a blazar 500 Mpc away, both on logarithmic scales, for void fields of 10⁻²¹ G, 10⁻²⁰ G, 10⁻¹⁹ G, 3 × 10⁻¹⁹ G. The echo carries 1.0 × 10⁻⁴ erg cm⁻² in the band whatever the field — the field decides only when it arrives, and so how thinly it is spread. Half of it has arrived 14 hours after the flare began in the weakest field and 4.1 years in the strongest, and the flux at that moment falls from 1.1 × 10⁻⁹ to 1.3 × 10⁻¹³ erg cm⁻² s⁻¹. The dashed curve is the flux a space telescope needs to detect a source averaged over that long, falling as the square root of the time to a floor of 10⁻¹² erg cm⁻² s⁻¹. Counting only what arrives after the flare has ended, three of the four echoes clear it.

Cosmology

An echo heard only from the weakest fields

A missing gamma-ray halo gives the field in the voids a lower limit and nothing more. A flare's echo could give it a value — the gigaelectronvolt light that arrives late because the void field bent its path. But the echo carries a fixed amount of energy and the field decides only how long it is spread over, so it can be heard only from fields weak enough to deliver it quickly, and only after flares brighter than almost any yet seen from a blazar far enough away to make one.

A conversion read galaxy by galaxy, with or without an alignment. The mean ellipticity of disc galaxies along a tidal template, in bins of each galaxy's own spin estimator — the quantity whose population average is the Lee–Pen alignment — for 60,000 discs a fifth as thick as they are wide, in three samples: spins set by the torque (net alignment 0.36), 85% of them randomised (0.054), and all of them random (0.002). The points trace the same curve through the origin in every sample — not quite straight, flattening at both ends, because a disc's shape saturates when its spin lies in the sky's plane — and a straight line fitted through each sample (dashed) has slope 0.279, 0.290, 0.292. That slope is the projection factor, what a disc's shape shows of its spin, and it does not care whether the population is aligned. A sample with no alignment at all measures the conversion as well as one with a strong one; only the population's mean, where all its galaxies pile up, moves.

Cosmology

A conversion measured without an alignment

The factor that turns a disc galaxy's spin alignment into the shape alignment a lensing survey fits looked like something only a sample with a detected alignment could test. It is not. It is a relation each galaxy obeys on its own, so it can be read off a few hundred discs with velocity maps whether or not their spins point anywhere in particular — and what is left for the hard measurement to test is only whether a disc's shape is anything more than its projected spin.

Exoplanets

A fraction read in each channel's own units

Hot Jupiters arrive by two routes, one that ends at twice the planet's Roche distance and one that ends where the disc stopped. Fitting the mixture in Roche distances alone, as the edge invites, gives a fraction that drifts by almost half if the planets weigh more than assumed. Each route has to be read in the units in which it does not move — and then the answer is limited not by how well the planets are measured but by how their radii have changed since they arrived.

Exoplanets

A helium line that weighs the star first

The metastable helium line at 1083 nanometres made the escape of a planet's atmosphere measurable from the ground. But the atoms it sees exist only in a balance between the star's extreme-ultraviolet light, which fills the level, and its mid-ultraviolet, which empties it — so the same outflow is six times deeper around an active K dwarf than around an F star, one depth allows a fifty-fold range of mass-loss rates across the wind's possible temperatures, and a rate inferred from it inherits a share of every error in a spectrum that cannot be observed.

$The angular momentum a halo has and the angular momentum a disc needs. The fraction of the mass with specific angular momentum below j, against j in units of the mean, for a halo's material — the universal profile fitted to simulated haloes, with shape parameter μ of 1.1, 1.25, 1.5 — and for an exponential disc with a flat rotation curve (black). Both have the same mean, as a disc that kept its halo's angular momentum would. They do not have the same shape. Below a tenth of the mean a disc holds 2% of its mass; a halo with μ = 1.25 holds 11%, and one with μ = 1.1, 15%. The halo also stops at a largest j, 4.0 times the mean for μ = 1.25, beyond which the disc still has a thin tail. A disc built from a halo's gas without any exchange of angular momentum would be too concentrated at its centre and cut off at its edge.$ Galaxies

A halo with too much slow gas for a disc

A disc's size follows from its halo's average spin, and the average is all the simplest account of disc formation uses. The whole distribution does not match. A halo holds five times as much of its material at low angular momentum as an exponential disc can accommodate, so gas that kept its angular momentum exactly would build a dense centre, a thin body and a hard edge — and the bulgeless discs that are common in the nearby universe say that something removed a fifth to a quarter of the slowest gas before it settled.

Galaxies

A stirring supernovae cannot pay for

The gas in a disc forming stars at redshift two moves at fifty kilometres a second in random directions, five times faster than in a spiral today, and turbulence that fast decays in a few tens of millions of years. Supernovae are the obvious source of the energy, and they run out at about thirteen kilometres a second however fast the disc forms stars. What holds the discs up is gravity — gas stirring itself by falling inward — and a disc stirred that way is also a disc pouring its gas into its own centre.

Sixty futures that began a trillionth apart. The momentum of the standard map at K = 3, a kicked rotor whose phase space is almost entirely chaotic, for 60 members started 10⁻¹² apart in angle, against iteration. For 41 iterations they are one trajectory; then the separation, growing by a factor of about 2.0 each step, reaches the size of the system and they become independent random walks. Three of the 60 cross the threshold at |p| = 90 within 400 iterations (dark), the stand-in for an orbit that goes unstable. Which members cross is decided by differences far below anything that could be measured; how many cross is a property of the dynamics, and it is the only thing the ensemble can be asked for.

Gravitation

A probability for a system with one future

The Solar System will do one thing, and an integration of it is wrong about which thing within a hundred million years. What long integrations report instead is a probability — one per cent that Mercury's orbit goes unstable before the Sun dies — and a probability of a single event is a strange object. It is a count over an ensemble of futures that differ by less than anything measurable, which makes sense only because chaos makes those futures independent; and how many of them are needed is a binomial question with an unforgiving answer for rare events.

The flattening a halo has, and the flattening an orbit feels. Three model haloes with flat rotation curves, each built from thin similar-shell layers whose axis ratio changes with radius (dashed), and the axis ratio a nearly circular orbit at each radius responds to (solid) — the a²-weighted mean over the shells inside the orbit, because a shell exerts no force on anything within it. A halo flattened to 0.85 everywhere is felt as 0.85 everywhere. A halo round inside and flattened to 0.8 beyond about 30 kpc is felt as round by every orbit within 20 kpc — 0.988 at 20 — and at 60 kpc as 0.83, where its own shells are 0.80. A halo flattened inside and round outside is felt as flattened well beyond where its shells have become round. An orbit reports a running average of everything inside it, weighted towards its own radius, and nothing at all about what lies outside.

Gravitation

A shell outside the stream turns nothing

A flattened halo turns the plane of every orbit inside it, and a stream of stars several orbits long records how far. But the turning is set only by the halo inside the orbit — a flattened shell enclosing it exerts no force on it at all — and within that, mostly by the shells just inside. So a stream measures a running average of the halo's shape out to its own radius, a halo round inside and flattened outside looks round to every stream in its core, and reading the shape of each layer back out of the streams costs more precision the further out the layer lies.

Which pairs of planets the angular momentum deficit can bring together. For each adjacent pair of planets, the system's angular momentum deficit divided by the critical deficit at which that pair's orbits could touch, on a logarithmic scale; below one the pair cannot collide under secular evolution, whatever happens. Blue: using only the deficit of the pair's own subsystem — the four terrestrial planets (7.8 × 10⁻⁹ in units of a solar mass times √AU) or the four giants (4.3 × 10⁻⁶). Orange: using the whole Solar System's deficit, which the giants hold almost entirely. On their own budget the inner pairs sit at 0.21, 0.12, 0.34 — safe. On the whole budget they sit at 112, 65, 187. The giants are safe either way, at 0.05, 0.18, 0.36. The terrestrial planets are stable because almost none of the giants' deficit reaches them, not because there is too little of it.

Orbits

A deficit the giants hold and Mercury could borrow

The angular momentum the planets' orbits lack of circular motion is conserved by every secular interaction, to every order, so it is a budget no planet can overdraw. Compared with what it would take to bring a pair of orbits into contact, it proves without an integration that the terrestrial planets cannot collide among themselves — and shows that they are safe only because the giants, who hold five hundred times as much, lend them almost none. A loan of half a per cent of Jupiter's would be enough to unsettle Mercury.

Where a spinning sphere of rubble is steepest. The slope of the surface of a spinning sphere of loose material against latitude — the angle between the local vertical and the combined pull of gravity and rotation — for spins of 30%, 60%, 80%, 90%, 97% of the critical rate, measured in its square, at which the equator would lift off: for a density of 1.19 g cm⁻³ the critical period is 3.03 hours and these are periods of 5.5, 3.9, 3.4, 3.2, 3.1 hours. The slope is zero at the pole and at the equator, where the pull and the vertical are aligned by symmetry, and steepest in between: at 30% it peaks at 10° at latitude 40°, at 97% at 70° at 10°. The band marks the angle of repose of loose rock, 30° to 40°. From 80% of the critical rate the steepest band is past it: the surface begins to slide well before the equator has lost its weight.

Orbits

A surface that slides before the body sheds

A rubble pile spun towards its limit does not wait for its equator to lose its weight. Long before that, its mid-latitudes grow steeper than loose rock can stand, and the surface flows — downhill, which on a spinning body means towards the equator. The ridge that so many small asteroids wear is built in that interval, at spins a sixth slower than the one that would fling material off, and whether it is built at all depends on a cohesion of less than a pascal.

$The Moon's Cassini states as it recedes. The angle between the Moon's spin axis and its orbit normal in each Cassini state, signed so that the present state is positive, against the Earth–Moon distance in Earth radii, for the Moon's present shape (β = (C − A)/B fixed) and orbital inclination. Today the Moon sits in Cassini state 2 at 6.68°. Closer in, two more states exist, a low-obliquity state 1 and an unstable state 4; they meet and vanish at 33.8 Earth radii, where the spin-axis precession, which weakens as the cube of the distance, falls to the critical fraction of the orbit's regression. A Moon that formed close to the Earth sat in state 1 at small obliquity, growing to 21° as it receded; at 33.8 Earth radii that state disappeared from under it, and it had to move to state 2 — which at that distance is at 46°. Since then, the obliquity has relaxed along state 2 to its present value.$ The observed sky

A state the Moon had to jump to

The Moon's spin axis sits a degree and a half from the ecliptic pole, locked in a geometry Cassini described in 1693. The lock is an equilibrium between the Earth's torque on the Moon's bulge and the slow regression of its orbit, and both depend on the Moon's distance. When the Moon was closer than about 34 Earth radii there were two more equilibria — and the one it almost certainly occupied vanished as it receded, leaving it to jump to the present one at an obliquity near 45°.

Whether a field helped, depending on what limits the escape. The ratio of the atmospheric loss of a magnetised Earth to that of the same Earth without a field, against the Sun's age, both on logarithmic scales, set at 2 today — the magnetised planet losing more — for two ways an unmagnetised planet's loss can scale. If both planets lose atmosphere in proportion to the solar-wind power they intercept (blue), the magnetosphere's cross-section shrinks as the young Sun's denser wind compresses it while the bare planet's does not, and the ratio falls to 0.46 at 0.7 billion years: the field protected the young Earth, on balance from before 1.9 billion years. If instead the bare planet's loss is set by the ions the Sun's ultraviolet makes above it (orange), the ultraviolet rises more slowly into the past than the wind's pressure does, and the ratio grows to 3.7: the field cost the young Earth more than it saves today. The answer to whether a magnetic field protects an atmosphere is not a property of fields; it is a statement about which process limits the escape, and the young Sun amplifies the difference.

The observed sky

A shield that depends on what limits the leak

Today the Earth, with its magnetic field, loses ions to space about as fast as Venus and Mars without one — a little faster, by most measures. Whether that was also true under the young Sun, whose wind pressed eighty times harder, depends on something no measurement of the present can settle. If escape is limited by the wind's power, the field protected the young Earth; if it is limited by the ions the Sun's ultraviolet makes, the field cost it more than it does now. The young Sun does not decide the question. It amplifies whichever answer is right.

Two arrivals with the same weak thruster. Two spacecraft entering Mars's sphere of influence (outer circle, 577 thousand km) from the right, each thrusting continuously against its velocity at 0.3 mm s⁻² — about what an electric thruster gives a spacecraft of a few tonnes — and aimed to pass 6 planetary radii from the centre. Arriving at 0.40 km/s (blue), the thrust removes the excess while the craft crosses the sphere: it is captured 10 days after entry, having spent 268 m/s, on an orbit whose apoapsis lies inside the sphere, where the planet rather than the Sun keeps it, and the drawn path follows the same thrust winding it inward until it is within one and a half planetary radii. Arriving at 0.80 km/s (orange), it passes through on a bent hyperbola and leaves again: the thrust had too little time, and too little leverage, to remove an excess twice as large.

Spaceflight

A thruster that must arrive slow

A rocket arriving at a planet stops by burning hard at the bottom of the gravity well, where every metre a second it spends removes the most energy. An electric thruster cannot burn hard anywhere. It can capture only arrivals slower than a few hundred metres a second, so it has to arrive with almost no excess at all — and then it pays the full circular speed of its final orbit to spiral down, two and a half times what a burn from the same start would cost, because a spiral spends its thrust at every speed on the way rather than at the highest one.

The last correction: accurate if late, cheap if early. For a spacecraft approaching a moon at 6 km/s with a 30 km aim error left by navigation from the Earth, which locates the moon by camera to 20 microradians and acts an hour after its last image: the error remaining at the flyby after one final correction (blue, km) and that correction's cost (orange, m/s), against how long before the flyby it is made, all on logarithmic scales. The error falls in proportion to the time-to-go, because the moon is seen from closer, until it meets a floor: 3.1 km from how well the limb of a Europa-sized moon can be fitted once it fills the camera's view, with the burn's own two-per-cent error, 0.6 km, beneath it. The cost rises as one over the time-to-go, because the same error has to be removed in less time. The floor is reached 6.2 hours out, and a correction later than that costs more and buys nothing. A correction a day out costs 0.35 m/s and delivers 10.8 km; one six hours out, 1.4 m/s for 3.2 km. With 1 m/s to spend, the best is 4.2 km, at 8.7 hours out.

Spaceflight

An aim bought with the last day of the approach

A spacecraft that photographs the moon it is approaching can aim its flyby far more precisely than the ground can tell it to, because it sees the target from closer every hour. But each hour closer leaves less time for a correction to act, so every kilometre of aim is bought with propellant — and below a few kilometres the camera stops helping at all, because a moon that fills the view can be located only as well as its limb can be fitted. For a small craft with a metre a second to spend, the aim comes out at four kilometres, set by geometry and not by the antenna.

A lobe, the shell it pushed out, and two sight lines through it. A radio lobe of radius 20 kpc drawn in the plane containing the line of sight, the observer to the right, 60 kpc from the host in the same halo as before. Growing at Mach 2, it swept the gas that filled its volume into a shell compressed 2.29 times and 2.9 kpc thick, and the field in that shell was amplified along the boundary. A sight line through the lobe's centre crosses the shell head-on, where the amplified field lies across the line of sight and contributes nothing: the shell there turns the light by about 6 rad m⁻². A sight line at 85 per cent of the radius crosses it obliquely, along 4.8 kpc of shell with the amplified field partly along the path, and the shell turns it by about 18 rad m⁻². The rotation the halo model attributes to the gas in front of the lobe is partly written by the lobe itself, and it is written most strongly around its edge.

Starlight

A rotation measure the lobe writes on its own boundary

The magnetic fields of hot group and cluster gas are mostly measured through the Faraday rotation of the radio galaxies that live in it, on the assumption that a lobe is a passive lamp behind a screen. But a lobe made its own room. The gas that filled it is now a shell around it, compressed if the lobe grew faster than sound, and the shell rotates the light too — so the inferred field comes out too low for a lobe that grew gently and up to three times too high for one that drove a shock. Only sources behind the halo, which carry no shell, can say which.

Starlight

A calibration chosen by the measurement it calibrates

A standard candle is calibrated on stars with good parallaxes, and "good" is decided by the parallax itself — so the calibrators are the stars whose errors happened to push them nearer. Correcting each star's distance afterwards does not repair it, because the correction needs a picture of where the sample's stars really lie, and that picture depends on the luminosity being calibrated. What works is to choose nothing, keep every star with the negative parallaxes included, and fit the candle by predicting the parallaxes rather than inverting them.

$One production rate, many Suns. Global isotope production, as a fraction of today's average, over the modulation potential and the Earth's dipole moment. Each contour is one production rate — 0.85, 1, 1.2, 1.4, 1.7 times today's — and every point on it is a combination of Sun and Earth an isotope record cannot distinguish. Production at today's dipole of 7.8 × 10²² A m² and a modulation potential of 650 MV is reproduced exactly by a dipole of 10 with a potential of 449 MV. Over the last ten thousand years the dipole is thought to have ranged between about 6 and 11 in these units, the band across the middle of the plane. So a record of production becomes a record of the Sun only after a record of the dipole has been divided out, and the solar history inherits every error in the geomagnetic one.$ Stars

A Sun read through a shield that also changed

Carbon-14 in tree rings and beryllium-10 in ice extend the record of solar activity ten thousand years past the first telescope, because the Sun's field keeps cosmic rays out of the atmosphere that makes them. But the Earth's own field keeps them out too, and over those ten thousand years it changed by as much as the Sun does over a cycle. An isotope record measures the two shields together, so the Sun comes out of it only after the Earth has been divided out — and every slow trend in the reconstructed Sun is, until shown otherwise, a trend in that division.

$X-ray activity against Rossby number, with and without a tachocline. The X-ray luminosity of a star as a fraction of its total, against its Rossby number — the rotation period divided by the convective turnover time — for a synthetic sample of 600 stars of 0.4 to 1.2 solar masses, which have a radiative core and a tachocline below their convection zones, and 220 of 0.1 to 0.35 solar masses, which are convective to the centre and have neither. Both are drawn from one relation: saturated at a ratio of 10⁻³·¹³ below a Rossby number of 0.13 and falling as its −2.7th power above, with 0.25 dex of scatter. The saturated stars of the two kinds average −3.09 and −3.16 in the logarithm; 56 of the fully convective stars rotate slowly enough to lie on the unsaturated branch. This is the shape the measured relation has: the stars with no tachocline lie on the same line as those with one, saturated at the same level and falling along the same slope.$ Stars

A dynamo that did not need the shear layer

The Sun's magnetic cycle has long been placed in the thin layer of shear beneath its convection zone, where a rigidly rotating interior meets a differentially rotating envelope. Stars below about a third of a solar mass have no such layer — they are convective to the centre — and they are as magnetically active as the Sun, saturate at the same level, and fade with slowing rotation along the same line. The saturation is the cleaner half of that evidence, because the slope depends on a turnover time that was itself calibrated from activity.

The same ruler, and a wave that arrives early. The acoustic oscillation in the galaxy power spectrum — the ratio to a smooth spectrum, minus one — against wavenumber, for no free-streaming species and for three neutrinos, with the sound horizon held at 99.0 h⁻¹ Mpc in both so that only the phase differs. The two agree on the largest scales and separate as the wavenumber grows: by 0.3 h/Mpc the wave with neutrinos runs 0.21 radians ahead, so its crests sit at smaller wavenumber — near 0.2 h/Mpc by 2.2 thousandths of an h/Mpc. The amplitude 0.05 and the damping of the feature by bulk flows over 7 h⁻¹ Mpc are illustrative; the phase is computed.

Cosmology

A wave pulled ahead by particles that outran it

The neutrinos of the early universe moved at the speed of light while the sound wave that became the acoustic feature moved at little more than half of it, and their gravity dragged the wave forward. The drag survives as a phase in the galaxy power spectrum, a shift that no change in the ruler's length can copy, so galaxy surveys can count the species that streamed freely without asking the microwave background.

The trough a smooth fit leaves behind. A trough 150 mK deep, centred at 75 MHz and 15 MHz wide at half depth (dashed), and what remains of it after the same foreground fit that removes the sky, with 3, 5, 7 terms. The fit cannot tell which part of the spectrum is sky and which is signal, so it absorbs whatever part of the trough looks smooth: what remains is 197, 62, 71 mK deep at its lowest, narrower than the injected trough and flanked by bumps above zero that were never in the sky. Measured jointly with the foreground, the trough is worth, in signal-to-noise, a trough 48, 29, 15 per cent as deep with no foreground to remove. What a global experiment measures is not the trough but its projection onto whatever the foreground fit leaves free.

Cosmology

A depth that belongs to the fit as much as the sky

The trough the first stars should have cast is found by subtracting a smooth function from a sky ten thousand times brighter, and a smooth function flexible enough to remove that sky also removes most of any trough as wide as the models predict. What a global 21-centimetre experiment reports is not the trough but the part of it the fit could not imitate, and that part depends on how many terms the fit was given.

Exoplanets

A shoreline measured in light already spent

In the solar system one line of slope four, in the plane of escape velocity against sunlight, separates every world with air from every world without. Drawn against the light a planet receives today it puts the temperate planets of a small red dwarf safely on the side with air. Drawn against the stripping light their star has already spent, it moves them an order of magnitude toward the airless side — and leaves most of them inside a band that the line's unknown height cannot resolve.

$A second peak, seen through three precisions. The distribution of measured core fractions for a population in which 90 per cent of planets sit at their stars' prediction (0.32, with an intrinsic scatter of 0.03) and 10 per cent at 0.65 (scatter 0.08), each planet compared with its own star, observed with measurement errors of 0.04, 0.08, 0.15 in core fraction. At 0.04 the dense group is a separate hump; at 0.08 it is a shoulder; at 0.15 it has become a long upper tail on a single peak, indistinguishable by eye from a population that is simply broad. The current measurements sit between the second and third curves.$ Exoplanets

A second peak the errors have to let through

Whether the densest small planets are a population of their own or the far edge of the ordinary one is not a question about any single planet. It is a question about the shape of a histogram seen through measurement errors as wide as the gap it is looking for, and the number of planets it needs rises fifteen-fold for every factor of three in lost precision — which puts the answer a few dozen planets beyond the samples that first suggested it.

$A polarisation fraction that should change with frequency. The polarisation fraction of dust emission relative to its value at 353 GHz, against frequency on a logarithmic axis. A single population at a single temperature gives a fraction that does not change at all. If the carbon grains are unaligned and warmer than the silicates, they take a growing share of the emission toward high frequency and dilute the polarisation: at 1200 GHz the fraction falls to 0.90 of its 353 GHz value for a 3 K difference and 0.85 for 5 K. Half-aligned carbon halves the effect. The shaded band is flat to within ten per cent between 100 and 1200 GHz, the precision with which the satellite and balloon measurements across this range agree with no change.$ Galaxies

The grains that glow and the grains that align

Interstellar dust polarises starlight and polarises its own glow, and for decades the standard account gave the two jobs to different grains — aligned silicates, unaligned carbon. Two populations at two temperatures predict a polarisation fraction that falls toward high frequency; what satellites and balloons measured was a fraction that does not change. That flatness, which no grain shape can fake, says the grains that glow are the grains that align.

Galaxies

A disc weighed by how far its stars climb

The rotation curve cannot say how much of a galaxy's speed belongs to its disc, because any split between disc and halo fits it equally well. The stars' motions perpendicular to the disc can, with no halo in the argument at all — and when that was done for thirty face-on spirals the discs came out light. But the vertical weighing needs the disc's thickness, which a face-on galaxy hides, and it needs the dispersion of the stars that hold the disc up, which the light measured may not belong to.

Gravitation

A Love number measured at nuclear density

The number that told Cassini there was water under Titan's ice is defined the same way for a neutron star, and in the last tenth of a second before two of them merge it changes the shape of the gravitational wave they emit. What the wave measures is mostly not the Love number itself but the star's size, raised to the sixth power — which makes a faint tidal signal into one of the few direct measurements of how large matter at nuclear density lets a star be.

Six moons, and how long each takes to move. The time each of Saturn's six largest moons would take to move outward by its own orbital radius, a divided by its rate of change, on a logarithmic scale. Bars: a constant tidal quality factor tuned to the inner moons' measured motion, under which the time grows steeply with distance and falls with mass — 6 billion years for Enceladus, 210 billion for Titan. Dashed: Titan's measured migration, 11 centimetres a year, which is 11 billion years — 19 times faster than the constant-Q prediction and within a factor of two of the inner moons, where the constant-Q picture puts them a factor of 33 apart. Resonance locking predicts exactly that: one time scale for every locked moon, set by the planet rather than by the moon.

Gravitation

A migration timed by the planet and not the moon

Titan is Saturn's most distant large moon, and by the standard account of tides it should be the slowest to move — a two-hundred-billion-year journey across its own orbit. It is moving eleven centimetres a year, twenty times faster, on the same time scale as the small moons close to the planet. A quality factor cannot do that. An oscillation inside Saturn, tuned to each moon in turn and dragged along by the planet's own slow evolution, can.

Orbits

An orbit seen at an angle keeps everything but its focus

No orbit outside the solar system is seen face on. It is seen as a shadow on the sky, and the shadow of an ellipse is an ellipse, swept at equal areas in equal times, indistinguishable in shape and timing from a real orbit. Only one thing gives the tilt away. The star the companion circles projects to where the true focus projects, and that is not where the apparent ellipse has its focus.

Where in the sky satellites still shine. The whole sky above the horizon seen from below, zenith at the centre, north at the top and the Sun 20° below the western horizon. Shaded: directions in which a satellite at 550 km is sunlit — 71 per cent of the sky by solid angle. The lit region is a cap pressed against the western horizon and reaching up past the zenith, because satellites seen low in the west are far away and high above the shadow; the eastern sky is dark. Ignoring the penumbra moves its edge by under a degree.

The observed sky

The shadow that also switches off the satellites

After sunset the sky overhead darkens as the Earth's shadow climbs through the air, and the same shadow, climbing on past the top of the atmosphere, decides which satellites still catch sunlight. For a satellite five hundred and fifty kilometres up the zenith stays lit until the Sun is twenty-three degrees down — five degrees past the end of astronomical twilight — and at the latitudes of most large observatories' summers there is no hour of the night when it is not.

A year that asks for more than a day on some days. The observing time requested per day through the year, with each target's request spread evenly over the days it can be seen, for the same programme filling 90 per cent of the year. The telescope supplies one day per day (dashed). On 104 days the visible targets ask for more than that, peaking at 1.6 days per day when the Galactic centre is in its window; on the rest there is slack. The annual total fits; the seasons do not.

The observed sky

A schedule written by where the Sun is

A space telescope that must keep the Sun at a fixed range of angles sees each part of the sky in windows set by the date, and the sky its users want is not spread evenly. A schedule can fill around a crowded window as long as the crowd fits inside it, by sending everything else to other dates. Once the demand for one region exceeds the share of the year that region is visible, no choice of proposals elsewhere can recover the excess — the limit is each window, not the total.

Four satellites, and which four. Two sets of four navigation satellites seen on the sky from below, zenith at the centre and the horizon at the rim, with circles at 30° and 60° elevation. Left: one overhead and three low and spread around the horizon. Right: four bunched in one part of the sky. Each set fixes a position, and the error it leaves is the range error multiplied by the dilution of precision — 2.0 for the spread set, 62.0 for the bunched one. The same four range measurements, made with the same precision, give positions 31 times more uncertain when the satellites are bunched.

Spaceflight

Four satellites in view, and which four

A satellite navigation receiver needs four satellites to fix its position, and a constellation is designed to keep at least four above every place. Four is the count, not the answer. The same four ranges, measured equally well, give a position thirty times worse when the satellites are bunched in one part of the sky than when they are spread around it, and the constellations carry their extra satellites to pay for the moments when the geometry, not the count, fails.

How fast to switch. The rms false polarisation left by a 1 per cent rms brightness fluctuation, against the modulation frequency, both on logarithmic scales, for fluctuations whose spectrum turns down at 5, 20, 100 Hz. Below the turnover the false signal is the whole fluctuation, about 0.7 per cent; above it, it falls steeply as the pairs are taken faster than the air changes. To reach a part in ten thousand (dashed) — the precision at which stellar and planetary polarisation becomes interesting — the switching must run at 3,311 Hz, 13,183 Hz, 66,069 Hz respectively: switching at hertz is useless, and a single beam needs kilohertz or more.

Starlight

A polarisation measured faster than the air changes

Polarisation is a difference between two brightnesses, and the brightness of a star seen through the air changes by a per cent from one millisecond to the next. Measure the two one after the other and the air's flicker becomes a false polarisation a hundred times larger than the signals worth having. The cure is to switch between them faster than the air can change, or to measure both at once and then swap them — turning a calibration problem into a differential one.

Starlight

A profile that averages a decade of depth

Each point on a star's disc reports the temperature at a different depth, so a limb-darkening profile looks like a depth sounding of the atmosphere waiting to be read. It is one, but a blurred one. Every angle averages a whole decade of optical depth, the averages overlap, and recovering the temperature structure from them is the inversion of a Laplace transform — a problem whose resolution grows only as the logarithm of the data's precision.

Stars

A limit written for free electrons

The Eddington limit says a star cannot shine brighter than the point where its light pushes its own gas away, and the push is calculated for free electrons scattering light. In a magnetic field of a hundred trillion gauss an electron is not free. It is bound to the field, it can barely respond to light polarised across it, and the limit calculated for it is the wrong number by a factor of hundreds — which is one of two ways a neutron star can shine five hundred times brighter than it is supposed to.

$The carbon a star makes, and the carbon it needs. For a 30 solar-mass star born with no carbon, two quantities against its central temperature, on logarithmic scales: the carbon fraction its core needs for the CNO cycle to carry its luminosity (falling steeply, because a hotter core needs less catalyst), and the carbon the triple-alpha process makes in the hundred thousand years the core takes to contract (rising even more steeply). They meet at 108 million kelvin and a carbon fraction of 1.7 × 10⁻¹⁰: 1.7 parts in ten billion by mass, made by the star itself, are enough to switch on the cycle that runs every massive star of the present day. The contraction stops there.$ Stars

A star that has to make its own catalyst

Every massive star now forming burns hydrogen through a cycle of reactions that carbon, nitrogen and oxygen catalyse, and the first stars had none of those elements. A massive star made of hydrogen and helium alone contracts and heats until its core reaches a hundred million kelvin, where helium begins to fuse into carbon; a few atoms of carbon in ten billion are enough to start the cycle, and the star settles there — three times hotter at its centre than any star of the same mass today, and smaller, and bluer.

Arms drawn into a disc that has none. Left: 1400 gas clouds spread uniformly over a disc between 3 and 13 kpc from the Galactic centre (cross), seen from above with the Sun (ring) 8.2 kpc out; the two thin curves mark where a two-armed spiral density wave of pitch 12° has its arms. The clouds have no arms in their positions, only in their motions: each streams by 10 km/s in a pattern locked to the arms. Right: the same clouds placed at their kinematic distances, read from their velocities with a circular rotation curve and on the correct side of every tangent point. The streaming has moved them by up to a few kiloparsecs, along lines of sight, and into bands. Lines of sight within 12° of the Galactic centre and 18° of the anticentre, where the velocities carry no distance, are left out. Counted in square kiloparsecs between 4 and 12 kpc, the densest fifth of the map holds 30 per cent of the clouds in reality and 45 per cent in the kinematic map.

Galaxies

A spiral drawn by velocities alone

A map of the Milky Way's gas is built by turning each cloud's velocity into a distance with a circular rotation curve. Gas in a spiral arm does not move on circles: it streams by ten or fifteen kilometres a second in a pattern locked to the arms, so the distance errors are locked to the arms too. Read that way, a disc with no arms in it at all comes out of the calculation with arms.

Two places one tidal dwarf could sit. The radial acceleration relation — observed acceleration against the acceleration the baryons alone produce, logarithmic, in m/s² — with the fitted relation for ordinary disc galaxies (solid) and the line of equality (dashed). The marks are a tidal dwarf galaxy of 10⁹ solar masses of gas and stars in an exponential disc of scale length 1.5 kpc, sampled every half kiloparsec to 6 kpc. If it carries no dark matter, as debris pulled from a disc should not, its points lie on the dashed line (open). If the relation is a law that every galaxy obeys, they lie on the curve (filled), and at the outermost radius the acceleration is 5.6 times the baryons' — a rotation speed 2.37 times higher. Every point is below a₀, where the two predictions are furthest apart.

Galaxies

A galaxy born without a halo

When two galaxies collide, some of the gas flung out along their tidal tails condenses into new dwarf galaxies. Debris launched from a disc cannot take the dark halo with it, so these dwarfs should rotate at the speed their own gas and stars allow and no faster. The relation that predicts every ordinary galaxy's rotation from its baryons says they should rotate twice as fast. One rotation curve decides between a halo and a law — except that the inclination of a lumpy young disc can move the answer by more than the difference.

Four ways for a galaxy to fade with redshift. The dimming of a galaxy's surface brightness, in magnitudes per square arcsecond, against redshift, for the same galaxy placed at greater and greater distance. In an expanding universe the bolometric surface brightness falls as (1+z)⁴: 3.01 magnitudes by z = 1 and 7.8 by z = 5. Measured per unit frequency, as magnitudes in a filter are, one power is absorbed by the stretching of the band and the dimming is (1+z)³. If photons lost energy on a static journey ("tired light"), only one power would remain — 0.75 magnitudes at z = 1. In a static space with no loss there is no dimming at all, because flux and solid angle both fall as the inverse square of distance. At z = 1 expansion and tired light differ by 2.3 magnitudes, a factor of 8 in brightness.

Cosmology

A dimming short by what young stars repay

In an expanding universe a galaxy's surface brightness should fall as the fourth power of one plus its redshift; if light merely tired on its way through static space, as the first power. Measured on elliptical galaxies at redshift 0.85, it falls as about the two-and-a-half power — neither. The verdict on expansion rests on what the galaxies did in the meantime, and the decisive fact is not the size of the shortfall but its sign: expansion needs the galaxies to have been brighter when their stars were younger, and tired light needs them to have been fainter.

How deep a hole the Hubble tension needs, and how deep holes are. The typical departure of the matter density from the mean, averaged over a sphere — one standard deviation, σ, from the linear power spectrum with σ₈ = 0.81 — against the sphere's radius, both on logarithmic axes. It falls from 0.55 at 20 Mpc to 0.022 at 300 Mpc. A local expansion rate 8.3 per cent above the global one, the size of the Hubble tension, needs the sphere the supernovae sample to be underdense by 0.47 (dashed), with a growth rate of 0.53. Across the radii the supernovae span (shaded, 100–600 Mpc) that is 4 to 67 standard deviations.

Cosmology

A hole too deep for the sky to have made

If the Milky Way sat inside a vast underdense region, everything around it would be expanding faster than the universe as a whole, and the locally measured Hubble constant would come out high — which is what Cepheids and supernovae find. The idea needs no new physics. It needs a void about half empty across the six hundred megaparsecs the supernovae sample, and the same cosmology that predicts the microwave background prices such a void at dozens of standard deviations.

Exoplanets

A zodi measured in heat and needed in light

The dust around nearby stars that will limit the search for Earth-like planets in reflected light has been measured — but in its own glow at ten microns, by an interferometer whose fringes hide the dust nearest the star. Converting that measurement into the visible brightness at the habitable zone needs the dust's radial profile and its grains' albedo and temperature, none of which was measured. The same ten-micron zodi can mean a quarter or ten times the visible dust a coronagraph will see.

$The polarised light HD 189733 b can send. The polarised part of HD 189733 b's reflected light, as a fraction of its star's light in parts per million, against orbital phase from one transit to the next, for an orbit of radius 0.0314 AU, a planet of 1.138 Jupiter radii and an inclination of 85.7°. A planet's reflected light is its geometric albedo times (R/a)² — here 3·10⁻⁴ — times a phase function, and the polarised part is that times the degree of polarisation, which for a scattering atmosphere peaks near quadrature where the planet is half lit. With an albedo of 0.4 and a peak polarisation of 0.5 the signal reaches 24 ppm (solid). The ceiling (dashed) gives the planet the largest albedo a scattering atmosphere can have, 0.75, and perfect polarisation: 91 ppm. An amplitude of 100 ppm (dotted), the order of the signal claimed in 2008, would need the planet to reflect and polarise more light than falls on it.$ The observed sky

A polarisation capped by the light the planet catches

Light scattered by a planet's atmosphere is polarised, most strongly when the planet is half lit, and the polarisation angle turns as the planet orbits — so a planet's polarised light would weigh its atmosphere and draw its orbit on the sky without a transit. In 2008 it was reported for a hot Jupiter at a part in ten thousand of its star's light. That amplitude is more than the planet can send: no atmosphere reflects more than three-quarters of the light that falls on it or polarises more than all of it, and the light that falls on it is set by its size and its distance. The photons for a real detection are cheap; what limits it is a polarimeter stable to a part in a million.

Spaceflight

A height the map already knows

A navigation service for the Moon's south pole needs only a handful of satellites, if they are put in elliptical orbits that hang for hours over the pole at the high point of each lap. But satellites hanging over a pole all stand at much the same height in the sky, on a cone, and four ranges from a cone cannot tell whether the receiver is lower or its clock is late. On the surface the receiver knows its height already, from a map of the terrain — and that one number, used as a measurement, does more for the service than two more satellites.

How steeply the ratio of all the light to one band rises with temperature. The exponent n in (bolometric flux ÷ band flux) ∝ Tⁿ, for a blackbody, against temperature, for bands at V (0.55 µm), J (1.25 µm), H (1.65 µm), K (2.2 µm). The bolometric flux rises as T⁴; the flux in a band rises as T on the long-wavelength side of the Planck peak and much faster on the short side, so n is the difference. For the Sun's temperature it is 2.33 in K, 1.70 in J and −0.56 in V. An error of one per cent in the modelled band flux becomes an error of 1/n per cent in the temperature: 0.43 per cent, 25 K, in K. In the visible the ratio barely depends on temperature and carries no information about it; in the infrared it does, and the temperature is read with its model error divided.

Starlight

A temperature read against the light's long tail

A star's total light depends on its temperature to the fourth power; its light in the infrared depends on it barely more than linearly, because the infrared lies on the long, flat tail of the Planck curve. The ratio of the two cancels the star's size and distance and leaves a steep function of temperature — and the only model it needs is the infrared surface flux, whose errors are divided by two and a half on the way to the answer. That is the infrared flux method, one of the two foundations of the stellar temperature scale, and the other foundation disagrees with it by up to a hundred and fifty kelvin.

One brightness, a mass for every age. Luminosity against age for young objects of 0.01, 0.02, 0.05, 0.08, 0.2, 0.5 solar masses contracting at nearly fixed surface temperature, logarithmic; the dashed line is one observed luminosity, 0.012 of the Sun's. Every object fades as it contracts, as the age to the power −2/3, so the same luminosity belongs to a different mass at every age: 38 Jupiter masses at 1 Myr, 52 Jupiter masses at 2 Myr, 80 Jupiter masses at 5 Myr, 111 Jupiter masses at 10 Myr. A factor of ten in assumed age is a factor of 2.9 in inferred mass. The boundary between stars and brown dwarfs, 0.075 solar masses, is crossed by an error in the age alone.

Stars

A turnover whose position is an age

The mass function of a young cluster is measured by counting its objects by brightness and converting each brightness into a mass. Below the hydrogen-burning limit nothing holds a brown dwarf's brightness steady: it fades as it contracts, so the conversion needs the cluster's age, and an age wrong by a factor of two moves every mass by about forty per cent. The shape of the substellar mass function survives the error; its position on the mass axis and the ratio of brown dwarfs to stars do not — and comparing those between clusters is how the universality of star formation is tested.

$Four clocks for a 1 M☉ dwarf, and the best of them. The fractional age uncertainty each method reaches for an isolated 1 solar-mass main-sequence star, against its age, logarithmic, for isochrones, rotation, oscillations and chemical clock, with representative published precisions. Rotation is best while the star is young, once its spin has forgotten its start and before its braking weakens; isochrones and oscillations improve with age as the star moves further on the diagram. The combination (heavy line) is 14 per cent at 1 Gyr, 9 at 3 and 4 at 8. It is better than ten per cent (dashed) from 2.6 to 9.4 Gyr.$ Stars

Four clocks and the stars none of them reads

An isolated dwarf star can be dated four ways — by its position on the Hertzsprung–Russell diagram, by how fast it spins, by its oscillations, and by the ratio of two elements its galaxy made at different rates. Each clock has an age range where it works and masses it cannot read. Laid side by side for a Sun-like star they cover most of its life to ten per cent or so. For a star of eight-tenths of the Sun's mass only one clock is left, it works only while the star is young, and the commonest stars in the Galaxy's disc are dated one at a time no better than fifteen per cent at any age.

Gravitation

A ring that is thinner from one side

Saturn's A ring is not a sheet of particles but a field of tilted slabs, each a few dozen metres across, pulled together by the ring's own gravity and sheared apart again within an orbit. Starlight passing through it threads the gaps or meets the slabs depending on its direction, so one patch of ring has a different optical depth for every observer — and a uniform model of it hides most of the particles it contains.

A cluster sinking by Chandrasekhar's formula alone. The orbital radius of a 2.0 × 10⁵ solar-mass globular cluster released on a circular orbit at 1 kpc in the two Fornax halo models, decaying under Chandrasekhar's dynamical friction with a velocity dispersion of 11.7 km/s and a Coulomb logarithm of 3. In the cusp it reaches the centre in 6.7 billion years. In the core the formula gives no arrival within 12 billion years, crossing into the core at 2.8. The formula already slows the cluster inside the core, because the circular speed there drops below the stars' velocity dispersion and few stars move slower than the cluster; but it still lets the cluster creep inward without limit. N-body experiments of clusters sinking into cored haloes find something stronger: the decay stops altogether near the core, which the local formula cannot produce at any setting of its logarithm.

Gravitation

A drag that stops where every orbit keeps time

Chandrasekhar's formula reads the density and the velocities where a sinking body is, and nothing about the orbits those stars are on. Inside a constant-density core every orbit has the same frequency, the differential motion that drags a body inward disappears, and the sinking stops — which is why the five globular clusters of the Fornax dwarf galaxy are still where they are.

$What colour removes, and what it can only estimate. The share of a synthetic family's linked membership that is not part of the family, against the clustering cutoff, with the parts a colour test can and cannot see. The true interloper fraction rises with the cutoff, from 0 per cent at 20 m/s to 84 per cent at 110. A colour cut removes the interlopers of the other type — 12 per cent of the list at 60 m/s — and leaves those of the family's own type, 6 per cent, which can only be estimated by scaling the detected ones by the background's mix. That estimate (dashed) tracks the truth on average and scatters about it, because a few dozen interlopers are a few dozen random draws.$ Orbits

A family checked by a colour it was not chosen by

Asteroid families are found by clustering orbits, and the clustering absorbs background asteroids that happen to orbit nearby. Fragments of one body share its composition, and the survey that found them measured their colours for free — so colour is a second membership test the clustering never used. It removes the interlopers that look different, and it can only estimate the ones that look the same.

A reflector made worse so the light lands where it is wanted. The intensity returned by a 3.8 cm fused-silica corner cube against angle from the exact retro-direction, on a logarithmic scale relative to a perfect cube's peak, at 532 nm. A perfect cube returns an Airy pattern whose central lobe is 17 microradians in radius. Offsetting each of its three dihedral angles by 1.25 arcseconds splits the return into six beams, each from a sixth of the face, deflected by (4√6/3)·n·ε = 29 µrad. In the band of offsets a LAGEOS station sees, 32–38 µrad (shaded), the spoiled cube returns 10 times as much light as the perfect one, along the direction of each lobe. Each lobe is drawn here as the Airy pattern of a circular aperture of a sixth of the area, which is an approximation to the true sector shape.

The observed sky

A reflector built wrong so the light comes home

A corner-cube reflector sends light straight back the way it came — in its own frame. On a satellite crossing the sky the return is aberrated by twice the satellite's sideways speed over the speed of light, fifty microradians for a low orbit, which is further than a small perfect cube spreads its light. So the cubes on geodetic satellites are made imperfect on purpose, their angles offset by about an arcsecond to throw the returning beam out to where the telescope will be.

Starlight

A rotation found where the transform goes dark

A star's rotation and the turbulent flows on its surface both widen its spectral lines, and a profile fitted for one can be refitted for the other with almost no change. Convolution becomes multiplication in Fourier space, though, and the transform of a rotating disc has zeros — frequencies where it is exactly nothing — while the transforms of turbulence and of the spectrograph have none. Where the transform of a line goes dark is a measurement of rotation that turbulence cannot fake.

Starlight

A periodogram shaped like a shadow

A periodogram that fits sinusoids sees a transiting planet only through the fundamental harmonic of its box-shaped dip, which for a short transit is a few per cent of the signal. Folding the data at each trial period and fitting a box recovers all of it — five times the signal-to-noise for a transit two per cent of the orbit long, eighteen times for an Earth crossing the Sun. The box pays for its extra parameters in trials, and the price is about one unit of signal-to-noise.

Stars

A star that distils itself as it freezes

A carbon–oxygen white dwarf does not freeze at one composition. The crystals that form at its centre take more oxygen than the liquid they form from, the carbon-rich liquid left behind is lighter and rises, and the star slowly sorts its own interior, oxygen down and carbon up. The gravitational energy that releases is radiated at a luminosity so faint that it holds the star up for about a billion years — a delay comparable to the latent heat of freezing, and one of the larger uncertainties in every white-dwarf age.

Stars

A scatter that points the way dust does

A Cepheid's period fixes its luminosity only to within the width of the instability strip, and at a given period the hotter stars are the brighter ones. Dust moves stars in the colour–magnitude diagram along a line too, and the two lines point nearly the same way. A magnitude built to cancel the dust — the Wesenheit index — therefore cancels most of the strip's width as well, which is why the tightest period–luminosity relations are the ones that ignore the extinction law's own advice.

Cosmology

A null that removes a model and not its family

The plateau potentials that survive every measurement of inflation predict a tensor-to-scalar ratio a few thousandths high, and the next polarisation experiments are designed to reach it. A null result there would exclude Starobinsky's model. It would not exclude the family it belongs to: the α-attractors stretch one shape by one parameter, their tilt stays where the measurement put it, and their ratio slides down as α does, without a floor.

One likelihood, read with three priors. A Gaussian likelihood in the effective neutrino mass sum, peaking at −0.08 eV with a width of 0.05 eV — of the shape that combinations of microwave-background and baryon-acoustic data have been reported to give. Read with no prior it says the data prefer a negative sum. Read with the physical prior, sum ≥ 0 (shaded right of zero), its 95 per cent upper limit is 0.059 eV, at the normal-ordering floor. Read with the oscillation prior, sum ≥ 0.059 eV, its limit is 0.102 eV — but only 0.3 per cent of the likelihood lies above the floor at all, which is the more informative number: the data and the oscillation experiments disagree at about 2.8 standard deviations.

Cosmology

A bound below the floor is not a mass

Oscillation experiments prove that neutrinos weigh at least 0.059 eV between them. Cosmology's latest upper bound on their total is 0.064 eV, and a bound computed differently would sit below the floor. The usual reading is that cosmology has nearly weighed the neutrino. The better one is that the fit's likelihood peaks at a negative mass: the data see more clustering than any real neutrino allows, and the bound is a symptom of that tension rather than a measurement of anything.

$More pairs on the target list than in the sky. The fraction of a magnitude-limited survey's targets that are unresolved binaries, against the fraction of stars in the volume that are, for companions with mass ratios spread evenly between a tenth and one. For solar-type stars, of which about 45 per cent have a companion, the target list holds 53 per cent binaries if the stars fill space uniformly, 50 if the survey reaches beyond the disc. The excess is modest on average because most companions are faint; it is concentrated in the near-equal pairs, which a magnitude limit over-represents by up to a factor of 2.8.$ Exoplanets

A target list that counts pairs twice

A planet survey chooses its stars by how bright they look, and an unresolved pair of stars looks brighter than either alone. A twin pair is seen from 2√2 times the volume a single star is, so a magnitude-limited target list holds more binaries than the Galaxy does, weighted towards exactly the near-equal pairs that dilute a transit most. The survey then counts its planets per target, and a target is not a star.

A size–luminosity relation with a population below it. The lag of the broad Hβ line behind the continuum, in light-days, against the continuum luminosity at 5100 ångströms, for a synthetic population of active nuclei with accretion rates spread from slow to well above the Eddington rate. The line is the radius–luminosity relation calibrated on slowly accreting nuclei, log R = 1.527 + 0.533 log L₄₄. Nuclei accreting at dimensionless rates below 3 scatter about it by about 0.13 dex; those above (filled) lie systematically below it, by 0.41 dex on average in this model, in which the shortening grows as the accretion rate to the power −0.4. A mass computed from the line for one of them takes a radius that is too large.

Galaxies

A lag too short for the light

Most black-hole masses beyond the nearby universe come from one spectrum and one relation: the size of the broad-line region scales with the quasar's luminosity, calibrated on nuclei that accrete slowly. Nuclei accreting near or above the Eddington rate have lags shorter than the relation predicts, so their masses are overestimated — and since the accretion rate is computed from the mass, it is underestimated, which hides the very shortening that caused the error.

Velocities that stop pointing along the stream. The angle between the stream's direction on the sky, as seen from the Galactic centre, and the mean motion of its stars, along the stream, for Large Magellanic Cloud masses of 0.0, 0.5, 1.5, 2.5 × 10¹¹ solar masses. In a static halo the misalignment stays at 1° or below — the small offset any stream has from its progenitor's orbit. With the Cloud it grows, peaking at 4°, 13°, 32° for the three masses, on the stretch of stream that passed nearest the Cloud. The misalignment is a measurement: with the halo's own potential known, its size along the stream gives the Cloud's mass.

Galaxies

A stream that points the wrong way

In a halo that holds still, the stars of a tidal stream move along it. The Large Magellanic Cloud, falling into the Milky Way for the first time, pulls on a passing stream directly and drags the Galaxy's own centre towards itself, and the stream is bent sideways faster than its stars' motions can follow. Their velocities stop pointing along the track — and how far they stop is a measurement of the Cloud's mass.

An electroweak field decays toward one line, and helicity decides where it meets it. The comoving strength of a magnetic field against its coherence length, both on logarithmic scales. A field made at the electroweak transition, at 100 GeV, starts on a hundredth of the Hubble length then (1 AU today) with a tenth of the radiation's energy density (1 × 10⁻⁶ G), and decays as its turbulence is processed until the largest eddy is the one an Alfvén wave can cross by recombination — the straight line, 10⁻⁸ G at 1 Mpc. With no net helicity it falls as λ^−5/2 and stops at 0.038 pc and 4 × 10⁻¹⁶ G. Maximally helical, it falls only as λ^−½ and stops at 4.7 kpc and 5 × 10⁻¹¹ G. The shaded floor is what the missing gamma-ray halos exclude: a field coherent over more than a megaparsec must exceed 9 × 10⁻¹⁷ G, and the bound rises as the inverse square root of the coherence length below the leptons' cooling length of 0.6 Mpc. The non-helical field ends 3.0 decades below it; the helical one 4.6 decades above it. Above 10⁻⁹ G at a megaparsec the microwave background excludes the field.

Cosmology

A twist that decides where the field ends

A magnetic field made in the first trillionth of a second starts strong and absurdly small, and its turbulence spends the next three hundred thousand years trading strength for size. How far it gets depends on one conserved quantity — how twisted its field lines are — and for a field made at the electroweak transition, that twist is the difference between failing the void bound by three decades and clearing it by five.

A sightline through the web, and where its field would be. The density of matter along a 300-megaparsec sightline through a model cosmic web, relative to the mean, on a logarithmic scale: a lognormal field whose logarithm has a width of 1.2. Galaxies sit at the 8 density peaks above the mean, and each has blown a magnetised bubble 1.4 Mpc in radius (shaded columns). Together they cover 7% of the sightline. The dots along the bottom are where 60 teraelectronvolt photons, stopped at random along the path, made their pairs: a photon stops where it stops, regardless of what is there, so the pairs fall on the empty stretches as often as on the crowded ones. Four of the 60 are made inside a bubble and are turned by its field. The other 56 are made in unmagnetised space, and their light forms a halo at full strength whatever the bubbles hold.

Cosmology

A field the galaxies cannot spread far enough

The missing gamma-ray halos are read as a field older than the galaxies because the galaxies seem unable to have put one there. The reason is not strength. A galaxy's wind carries a field that, spread over a megaparsec, is still ten million times stronger than the halos require. What it cannot do is reach the empty volume where most of the pairs are made — and a halo is dimmed only by the share of space the field fills.

Exoplanets

A neighbour an eccentric arrival cannot leave

A hot Jupiter that arrived on an orbit of eccentricity near one swept every distance between its closest and farthest point, thousands of times, while tides shrank its orbit — and anything small in that range was scattered, swallowed or thrown into the star. A hot Jupiter carried in by the disc can keep its neighbours. So a hot Jupiter with a close companion came through the disc, and the share of them that have one puts a floor under how many did — a floor whose height is set less by the count than by three chances nobody has measured.

$How a cloud's glow loses its polarisation as it thickens. The polarisation fraction of dust emission against the column density through a cloud, both on logarithmic scales, for four clouds built from cells of 10²¹ hydrogen atoms per square centimetre, each polarising its glow by 20%. A single uniform field with every grain aligned keeps 20% at every column. A field tangled from cell to cell by 40 degrees about a mean direction loses polarisation at first and then levels off at 7.5%, the share its mean direction carries: between 10²² and 10²³ its slope is −0.05. A field with no mean direction at all adds its cells as a pure random walk and falls with a slope of −0.55. Grains that align only in the outer 3 × 10²¹ cm⁻² on each side, where starlight still reaches, contribute nothing from the interior, and the fraction falls with a slope of −1.00.$ Galaxies

A falling polarisation and the angle that says why

The glow of dust in a molecular cloud is less polarised the thicker the cloud, almost everywhere it has been measured. Two explanations fit that fall equally well — a magnetic field tangled along the line of sight, or grains in the dark interior that have stopped lining up with it — and between them they can produce any slope a cloud has shown. What separates them is not how much polarisation is lost but whether the direction of what remains wanders.

Orbits

A clock checked by the craters it is making now

Every young age on Mars is a crater count divided by a rate borrowed from the Moon. Since 2006 the orbiting cameras have been watching that rate happen: hundreds of fresh craters, metres across, that were not there in the previous image. They form several times less often than the borrowed rate predicts at the smallest sizes, and more often than it at a few tens of metres — and the youngest surfaces on Mars are dated from exactly the sizes where the two disagree most.

The observed sky

Two exponents a solar cycle cannot pull apart

Whether a magnetic field protected the early Earth's air depends on what limits the escape of ions from a planet without one — the ultraviolet that makes them or the solar wind that carries them off. Measured at Mars through a solar cycle, the two causes rise and fall together, and the fit that tries to separate them leaves each exponent uncertain by half its value. Remarkably, the young Sun's loss is predicted well anyway. What spoils the prediction is not the confusion but a switch between the two limits that the present-day data cannot see coming.

Starlight

A halo the telescope paints on every galaxy

A telescope puts almost all of a star's light into a dot and a few per cent of it into faint wings that reach arcminutes away. Around a galaxy those wings add up. The scattered light from the bright centre lands in the outskirts, and below about twenty-nine magnitudes per square arcsecond it outshines the galaxy's own — bending the profile up into a halo, or an edge-on disc out into a thick one, at the same depth around every galaxy whatever its size.

Starlight

A hundred kelvin under every radius

The temperatures of cool stars are quoted to tens of kelvin, and the scale they are all measured on is uncertain by about a hundred. That hundred is under almost everything else: a star's radius moves by twice its fractional error, a transiting planet's density by six times, a main-sequence age by ten per cent and an iron abundance by a fifth. And because a scale error moves every star of a sample the same way, no amount of averaging removes it.

Stars

An expansion clock read from the stars' motions

The masses of a young cluster's brown dwarfs are read from their brightness, and a brown dwarf's brightness is a mass multiplied by an age. The age can now be measured without any model of a star: the cluster is flying apart, and the time since it began is its size divided by its speed. But the expansion clock counts only the coasting, not the stars' lives, and it runs fast when the stars began spread out — two biases of opposite sign, and nothing in the motions themselves to say which one wins.

Stars

A correction that grows as the fifth power of the frequency

Every model of a star gets its outermost layers wrong, because one-dimensional models cannot follow the turbulent convection that fills them. The error shows up in the star's oscillations as an offset that is nothing at low frequency and twenty microhertz at high, two hundred times a frequency's measurement error. It is corrected with a power law nobody can derive, and the correction recovers the star only when its shape happens to be right — which is why the ages of stars are fitted not to their frequencies but to ratios of differences the surface cannot reach.

A running the measurement cannot yet tell from zero. The running of the spectral tilt, dnₛ/d ln k, against the tilt itself. The ellipses are the microwave-background measurement, nₛ = 0.9649 ± 0.0042 and a running of −0.0045 ± 0.0067, at one and two standard deviations. The three curves are what slow-roll inflation predicts once the tilt is fixed: a running of −(1 − nₛ)²/β, with β = 2 for the plateaus and every α-attractor (and for φ²), 4/3 for the steepest power law still drawn and 4 for a cubic hilltop. At the measured tilt they give −0.0009 to −0.0003, all within a band thinner than a line at this scale, 11 times smaller than the measurement's error.

Cosmology

A drift in the tilt that is the tilt squared

The primordial spectrum's tilt has been measured to half a per cent. Whether the tilt itself changes with scale has not, and slow-roll inflation says why it will be hard: the running is fixed by the tilt, at about six ten-thousandths, ten times below the present error. The one measurement that reaches far enough along the spectrum to feel it — the μ-distortion of the microwave background's colour — would need a precision no planned instrument has.

Cosmology

A ratio in which the galaxies cancel

Matter falls in one gravitational potential and light is bent by the sum of two, and relativity says the two are equal. A survey that measures both the infall of galaxies and the lensing around them can divide one by the other so that the galaxies' bias and the fluctuations' amplitude both cancel — leaving a ratio that relativity predicts from the matter density alone, and that a gravity which bends light differently from the way it pulls matter cannot fake.

Exoplanets

A peak that should move with the star

Round Sun-like stars, giant planets are commonest at about three astronomical units, close to where water first froze in the disc. If the snow line put them there, the peak should sit closer in round smaller stars — three times closer round a star of a third of the Sun's mass. Every hypothesis agrees about the Sun, which is why the Sun-like stars cannot test them; and one of them makes a Jupiter at the peak wobble its star by the same angle whatever the star's mass.

Exoplanets

A transit that says which star it crossed

When a planet transits one star of an unresolved pair, the catalogue usually assumes the brighter star. The light curve itself can say otherwise, in two ways: the transit's shape measures the density of the star crossed, and a red dwarf is three times denser than the Sun; and the centre of the pair's light moves during the transit, away from whichever star was dimmed. Each test works where the other fails, and between them they leave a corner of close, nearly equal pairs that no measurement of the transit reaches.

Galaxies

A flare rate a hole cannot afford for long

Tidal disruption flares are found in galaxies that stopped forming stars within the last billion years far more often than those galaxies' numbers allow — by factors estimated between thirty and two hundred. Each proposed cause can be priced. A dense cluster from the burst would need ten times the hole's mass of new stars inside a parsec; a triaxial nucleus would give twenty times too much; and any excess of a hundred, kept for the length of the phase, would feed a million-solar-mass hole several times its own mass.

Galaxies

A lifetime read from how gas and stars sit apart

No molecular cloud has been watched for long enough to see it age, and every cloud looks the same whether it lives for one free-fall time or for ten. But a galaxy holds thousands of clouds at every stage at once, and how often molecular gas and young stars appear in the same place — measured in apertures of shrinking size, centred first on one and then on the other — turns a single snapshot into a clock, calibrated by how long the young stars shine.

Oceans that a changing G would have boiled. The Earth's surface temperature in a crude climate — equilibrium temperature for a fixed albedo, scaled as the received flux to the quarter power, plus a fixed greenhouse warming of 33 K — against time before the present, if the Sun's luminosity goes as G⁷ and the Earth's orbit as 1/G, so the flux goes as G⁹, on top of the Sun's own brightening. With Dirac's G ∝ 1/t and an age of 13.8 Gyr, the oceans boil 1.9 Gyr ago; with G falling by 10⁻¹¹ of itself a year, the surface is 292 K at 3.8 Gyr. Rocks of that age were laid down in water. Constant G gives the faint young Sun's cooling, 270 K in this climate — a separate problem, which the real atmosphere solved.

Gravitation

A constant bounded by the oceans

In 1937 Dirac proposed that the gravitational constant falls as the universe ages. Eleven years later Teller pointed out what that would have done to the Earth: a star's luminosity depends on G to about the seventh power, the Earth's orbit shrinks as G grows, and the sunlight reaching the Earth goes as the ninth power. Run backwards, Dirac's G boils the oceans less than two billion years ago. The rocks were laid down in water, so G cannot have changed that fast — the first bound on a changing constant came from geology.

$One impact, three ways to fade. The fractional luminosity of the dust from a giant impact at 1 AU against time since the impact, for 0.01 Earth masses of debris whose largest pieces are centimetre spherules condensed from the vapour, kilometre fragments, or hundred-kilometre fragments, each grinding down as a collisional cascade. The condensate is bright at once and gone within 286 years; the fragments keep the dust above the dashed detection threshold of 0.0001 for 8 thousand years and 9.3 million years. One impact carries two clocks: a flash measured in years and a glow measured in millennia.$ Orbits

Two clocks in the dust of one collision

A collision between two young planets fills the inner system with dust that shines in the infrared far above anything a steady belt could sustain. The dust carries two clocks. The spherules that condense from the vapour grind themselves away within a few centuries, so a fresh impact flashes and fades on a timescale an astronomer can watch. The fragments last thousands to millions of years, and how long depends on the size of the largest pieces — not on how much was thrown out. The share of young stars that glow counts impacts times fragment size; only the flashes count impacts.

The Sun's height, against the radius it is measured over. The Sun's height above the midplane as a survey would measure it, fitting the midplane to the stars within a projected radius R, against R, for a corrugation of 15 pc on a 2500-pc wavelength with the Sun 15 pc above the midplane beneath it. Nearby the answer is 15 pc whatever the phase; over several kiloparsecs it is the height above the mean plane: 30 pc with the Sun on a crest, 0 pc with it in a trough. Between them every intermediate value is a correct answer to a different question — a spread of 31 pc, comparable with the five to thirty parsecs the published measurements span.

The observed sky

A midplane that depends on how far it is averaged

The Sun sits a little above the middle of the Galaxy's disc — twenty parsecs, by the most careful recent counts, and anywhere from five to thirty by others. If the disc were flat that scatter would be error. It is not flat: the disc near the Sun ripples vertically by tens of parsecs on kiloparsec wavelengths, and on a rippled disc the Sun's height above 'the' midplane depends on how far around it the midplane is averaged. Every value in the scatter can be a correct answer to a different question.

$How small an outlier fraction the mean redshift can bear. The shift in a tomographic bin's mean redshift, f·Δz, against the outlier fraction f, for outliers displaced by Δz = 0.5, 1.5, 2.5, both logarithmic. The dashed line is the requirement for a bin at z = 0.8, 0.0036. Outliers half a unit of redshift away must be fewer than 0.72% of the bin; outliers two and a half units away, fewer than 0.14% — or known to that precision, since a known fraction can be corrected. The best photometric redshifts reach outlier fractions of a few per cent, ten to a hundred times above these lines.$ Starlight

A mean that a few per cent of strays can drag

A lensing survey sorts its galaxies into slices of redshift by their colours, and what its cosmology needs from each slice is a single number, the mean redshift, to two parts in a thousand. A few per cent of the galaxies in a slice are catastrophic outliers — young galaxies at redshift three mistaken for old ones at one — and each moves the mean by its full error. Three per cent of strays two redshift units away move it twenty times more than the survey can tolerate, and finding them needs spectra of exactly the galaxies whose spectra fail.

The visibility precision a one-per-cent diameter needs. The precision in fringe visibility needed to measure a uniform disc's angular diameter to 1%, on the best baseline up to 330 m, against the diameter, at three wavelengths. Below the resolution limit the required precision falls as the square of the diameter. Even at its best, a visibility measures a diameter only about as precisely as it is itself measured, so a one-per-cent diameter needs visibilities better than a per cent. The shaded band, one to three per cent, is what a single calibrated visibility typically achieves; averaging ten or more measurements brings the effective precision to a few tenths of a per cent (dotted). At 0.3% the limit is about 0.13 mas in the visible, 0.19 mas in the red and 0.38 mas in the near-infrared H band, where most stellar diameters have been measured.

Starlight

A diameter smaller than the fringes can see

The surface-brightness method gives distances good to a per cent for binaries of cool giants, and loses that precision for the hot stars that dominate other galaxies' bright binaries. The repair is an empirical calibration from the angular diameters of nearby hot stars — and those diameters are a few tenths of a milliarcsecond, below what the longest optical baselines comfortably resolve. A diameter measured at the edge of resolution is a small departure from perfect fringe contrast, and it costs visibility precision as the square of how small the star is.

$How far a wrong metallicity term moves the Hubble constant. The fractional change in the Hubble constant against the error in the adopted metallicity term, in magnitudes per dex, for supernova hosts whose Cepheids are 0.1, 0.2, 0.4 dex richer in metals than the geometric calibrators'. A term too weak (too close to zero) by 0.2 mag/dex with hosts 0.2 dex richer moves H₀ by 1.9%. The shaded band is the ±8% disagreement between the local measurement and the microwave background's prediction: reaching it with a 0.2-dex difference needs the term wrong by 0.9 mag/dex.$ Stars

A metallicity term too small to close a gap

Cepheids of the same period are slightly brighter when they are richer in metals, by about a fifth of a magnitude per factor of ten in metal content. The galaxies whose supernovae calibrate the Hubble constant hold Cepheids a little richer than the galaxies whose distances are known geometrically, so the term enters the answer. How much it can move it is simple to compute: a few per cent at most for any plausible error — and the eight per cent separating the local Hubble constant from the early universe's would need the term to be wrong by four times its own size.

$The nebula's line, if the source is beamed. The luminosity of the He II 4686 Å line a surrounding nebula would emit, against the source's beaming fraction b, for sources that appear at 10⁴⁰ and 10⁴¹ erg/s to an observer in the beam, with 10% of their true output in photons able to ionise He⁺. The nebula sees the true luminosity, b times the apparent one, so its line falls in proportion to b: a source apparently at 10⁴¹ that is beamed into a hundredth of the sky lights its nebula like an unbeamed source at 10³⁹. Measured He II luminosities of nebulae round ultraluminous sources are 10³⁵ to 10³⁷ erg/s (shaded). On this conversion a source apparently at 10⁴⁰ whose nebula shows a few 10³⁶ needs no beaming at all; one apparently at 10⁴¹ with the same line is beamed into a tenth of the sky or less.$ Stars

A nebula that cannot be beamed at

An X-ray source a hundred times brighter than a neutron star should be allowed to shine may be lying about its luminosity: if its light leaves through a narrow funnel pointed at the Earth, it looks as bright as a source a hundred times more powerful. A single observer cannot tell. The gas around it can. A nebula surrounding the source is lit from every direction at once and records what the source emits in all of them, and a bubble blown by its wind records the power it has put out over a million years. Each is a calorimeter no beam can fool.

The flow round the Local Group, shell by shell. Radial velocity against distance from the Local Group's centre today, for spherical shells of matter in a standard expanding background with the cosmological constant, round a group whose mass inside the zero-velocity surface is 1×10¹² M☉, 2×10¹² M☉, 4×10¹² M☉. Inside the zero-velocity radius — 0.79 Mpc, 1.00 Mpc, 1.26 Mpc — the shells are falling in; outside it they recede, slowly at first and then at the Hubble rate (dashed, 67.7 km/s per Mpc). At 1.5 Mpc the three groups give 75, 57, 30 km/s against 102 for an undisturbed flow; by 5 Mpc the curves differ by 10 km/s and have nearly rejoined the line. The group's pull is written into the first two or three megaparsecs of the flow, and only there.

Cosmology

A flow too quiet for the lumps it runs between

The galaxies within ten megaparsecs of the Milky Way recede almost exactly as the Hubble law says, scattering about it by a few tens of kilometres a second. The matter they move through is lumpy at the factor-of-several level, and elsewhere galaxies move relative to their neighbours at three hundred. The quiet is a measurement, it limits what the group's own pull can be read from, and it says something about the place rather than about the law.

Cosmology

A clock that flickers without a tick

If the universe expands, every process seen at redshift z runs (1+z) times slow. A supernova carries a calibrated clock and shows it. A quasar's light wanders with no period at all, and its characteristic timescale grows with the mass of its black hole and shrinks with the colour it is seen in, both of which change with redshift in any real survey. Reading the dilation from quasars means holding those fixed first, and on a short enough baseline the measurement hides the effect it is looking for.

$Rise, steady state and decline. The mass of magnesium (solid) and iron (dashed) in a white dwarf's mixing layer, each as a fraction of its steady-state value for magnesium, during an accretion episode that delivers both at equal rates for 5 magnesium sinking times and then stops; time in units of the magnesium sinking time. Iron sinks 1.6 times faster. While they rise, both grow together and their ratio is the accreted ratio; once accretion has run for a few sinking times each element saturates at its rate times its own sinking time, so the layer holds 0.63 times the accreted iron-to-magnesium ratio; after accretion stops iron vanishes first, and 7 sinking times later the layer holds 0.009 of the accreted ratio. The same body seen at three moments reads as three compositions.$ Exoplanets

A planet read in the ashes of its star

A white dwarf's gravity drags every heavy element out of its visible surface, so a white dwarf showing magnesium, iron and silicon is swallowing rock now. Its spectrum is the only place where the bulk composition of a rocky body round another star can be read element by element. But each element sinks at its own rate, so the ratio seen depends on when in the meal the spectrum was taken — and only the hottest hydrogen white dwarfs say.

The bands of sky from which each planet would be seen to transit. Ecliptic latitude against ecliptic longitude, with the vertical scale stretched 11 times: the band from which each of Mercury, Venus, Earth, Mars, Jupiter would be seen to cross the Sun's disc. The Earth's band lies along the ecliptic, ±0.266° wide. Mercury's is the widest, ±0.69°, and tilted by its 7° inclination; the giants' are narrow, ±0.051° for Jupiter. Each crosses the Earth's at the two nodes of its orbit, and only near those crossings can an observer see two planets transit. From nowhere on the sky can more than 3 be seen.

Exoplanets

The stars that could catch the Earth in transit

The geometry that lets one Earth twin in two hundred be seen crossing its star works in reverse: the Earth crosses the Sun for observers in a band of sky half a degree wide, under half a per cent of the sky. About two thousand stars within a hundred parsecs sit in it. The band is fixed to the Earth's orbit and the stars are not, so the list changes on the timescale of recorded history — and from no direction can more than three of the eight planets be seen transiting at once.

Where the quasar counts turn over, the signal changes sign. The local slope s of quasar number counts against limiting magnitude (solid), for counts that steepen to 0.90 at the bright end and flatten to 0.29 at the faint end with a turnover near magnitude 19, and the factor 2.5s − 1 that multiplies twice the convergence in the change of counts (dashed). The factor crosses zero at magnitude 20.1: brighter quasars are found in excess round foreground galaxies, fainter ones in deficit, and quasars near the crossing show no correlation at all. The predicted reversal, observed, is what identifies the correlation as lensing rather than as dust or as quasars and galaxies sitting in the same structures.

Galaxies

A lens counted instead of measured

A foreground mass makes the sources behind it brighter and spreads them over a larger patch of sky. Whether the patch then holds more of them or fewer depends only on how steeply their numbers rise towards faint fluxes, and at one slope the two effects cancel exactly. So a lens can be weighed without measuring a single shape — by counting — and the signal changes sign from bright quasars to faint ones in a way that nothing but lensing can imitate.

Galaxies

A halo weighed by counting its oldest clusters

The globular clusters round a galaxy add up to about four parts in a hundred thousand of the mass of its dark-matter halo — in a dwarf, in the Milky Way, in a giant elliptical and in a cluster of galaxies, across five decades of halo mass. The galaxy's own stars follow no such rule. So a count of a few dozen old star clusters weighs a halo that its stars cannot, and the strangest thing about the rule may be that it is less strange than it looks.

How long a wide pair survives a halo of heavy objects. The time for a binary of one solar mass to be dissolved by passing compact objects, against its separation, both logarithmic, if the whole local dark-matter halo — 0.01 M☉ per cubic parsec, moving past at 300 km/s — is made of objects of 10, 100, 1000 M☉. A pair 0.1 pc apart (about 20,000 AU) lasts 477, 48 and 4.8 billion years respectively; at 1 pc, 47.7, 4.8 and 0.48. The heavier the objects at fixed total density, the fewer and harder the kicks, and the sooner the widest pairs go. The dashed line is 10 billion years, the age of the halo's oldest binaries.

Gravitation

Pairs of stars too loosely bound to survive heavy dark matter

Two stars a tenth of a parsec apart are held together by a force so weak that a passing mass of a few hundred suns, at a distance of a parsec, can loosen them. If the dark matter round the Milky Way were made of heavy compact objects, such pairs would have been unbound long ago. Some have survived ten billion years in the halo. The widest of them sets a ceiling on how heavy the dark matter's pieces can be — and the ceiling has moved twice, both times because of one pair.

A moonlet's period measures its primary's density. The orbital period of a small satellite against the radius of its orbit in units of the primary's radius, both logarithmic, for primaries of bulk density 1, 2, 3.5 g/cm³. The size of the asteroid cancels: a moonlet at five radii from a body of 2 g/cm³ takes 26 hours whether the body is a kilometre or a hundred across. At one radius the curves give the spin period at which a strengthless rubble pile flies apart — 3.3, 2.3 and 1.8 hours. Didymos–Dimorphos, at 3.1 radii and 11.92 hours, gives 2.4 g/cm³; 1999 KW4, at 3.9 radii and 17.42 hours, gives 2.1 g/cm³.

Orbits

A moonlet that weighs a pile of rubble

Put Kepler's third law round an asteroid and its size cancels: a moonlet's period depends only on how many of the asteroid's radii away it orbits and on the asteroid's density. So the first asteroid moon ever seen, a mile-wide speck beside Ida, turned a picture into a density — and the density came out too low for solid rock. A fifth to a half of the asteroids weighed this way is empty space.

Starlight

A wavelength coincidence that weighs a neutron star

An ionised helium atom and a twice-ionised oxygen atom happen to have lines a sixtieth of an ångström apart in the far ultraviolet, close enough that one absorbs the other's light. The oxygen then glows in a cascade of visible lines, and so, by a second coincidence, does nitrogen. In a bright X-ray binary that glow comes from the face of the companion star turned towards the X-rays — a star otherwise lost in the light of its accretion disc — and its Doppler shifts trace the companion's orbit, weighing the neutron star it circles.

Starlight

A polarisation that a round star cancels

In 1946 Chandrasekhar worked out that light leaving a hot star's atmosphere, where scattering by free electrons dominates, is polarised parallel to the star's edge — not at all at the centre of the disc, 11.7 per cent at the very limb. Over a round, unresolved star the pattern adds up to exactly nothing. It took seventy years to see it in the light of a whole star, because something has to break the circle first: a companion crossing the disc, a planet, or a star spun so fast that its outline is no longer round.

A clump that does not grow until the radiation is outweighed. The amplitude of a dark-matter overdensity inside the horizon against the scale factor in units of its value at matter–radiation equality, both logarithmic, from y = 10⁻⁴ to today near y = 3400. The solution of the growth equation, δ = 1 + 3y/2, stays flat until equality and then rises in proportion to the scale factor; matter domination alone would have grown it in proportion to the scale factor from the start. Over the whole range the true growth is 5.1e+3 against 3.4e+7 — radiation withheld a factor of 6666, which is the interval from y = 10⁻⁴ to about y = 0.67.

Cosmology

A clump that waits for the radiation to thin

Dark matter feels no pressure from light, so nothing should have stopped it clumping in the radiation era. Almost nothing did stop it, and yet a clump inside the horizon at that time barely grows. The radiation does not push on it. It only makes the universe expand too fast for the clump's own gravity to keep up — and the length of that wait is the bend in the spectrum of every structure in the sky.

A shift in the time and a change in the length, a quarter-cycle apart. The transit timing shift and duration change of a planet of 0.5 Jupiter masses in a 287-day orbit, carrying a moon of 10 Earth masses at 20 planetary radii, against the moon's orbital phase at the moment of transit, for two cycles. The timing swings by ±42 minutes and the duration by ±10.1 minutes. The moon's period is 12.9 days. Timing is largest when the moon is farthest along the track, and duration when the moon is moving fastest along it, so they are in quadrature; a perturbing planet would produce nothing of that kind in the duration.

Exoplanets

A moon read from how late and how long

A moon pulls its planet round their common centre, and at each transit the planet is somewhere slightly different along that small orbit, moving slightly differently. So the transit arrives early or late, and it lasts longer or shorter. The two effects depend on the moon in different ways and they run a quarter of a cycle apart. Neither alone says anything about the moon. Their ratio gives its period, which a series of transits sampled once an orbit cannot otherwise reach.

Exoplanets

Water that is on the star and not the planet

A transit depth is the planet's shadow divided by the brightness of the whole star, and the shadow falls on only one strip of it. If the strip is cleaner than the rest of the disc, the depth comes out too large, by an amount that changes with colour. The star's own spots hold water, so the error has water's fingerprint. On the smallest red dwarfs, whose rocky planets are the best targets there are, that false signal is larger than the real atmosphere being looked for.

The middle of a cluster cools faster than the universe ages. The time the gas of a cool-core cluster, of central electron density 0.05 per cubic centimetre, would take to radiate its heat at constant pressure, against radius, both logarithmic. Inside 76 kiloparsecs it is less than 7.7 billion years, the time since a redshift of about one when the cluster's core is thought to have settled, and at 10 kiloparsecs it is 0.8 billion years. Beyond a few hundred kiloparsecs it is longer than the age of the universe, and the gas is simply hot.

Galaxies

A cooling flow that never arrives

The gas at the centre of a rich cluster radiates its heat in under a billion years, so it should be sinking inward and turning into stars at hundreds of solar masses a year. For two decades that was the accepted picture and it was treated as a measurement. Then X-ray spectra looked for the gas on its way down and found that it stops at a third of the temperature it began with, while the galaxy in the middle forms stars a hundred times too slowly. Something pays the radiative bill, and bubbles of the right size are visible in the X-ray images.

A valley of limits, and how much of it each experiment holds. The strength α a Yukawa term could have, against its range, with both axes logarithmic and everything above the curve excluded at about 95% confidence. The curve is a rounded envelope of published limits: laboratory torsion balances below a metre, tower and mine gravimetry to a few kilometres, the orbits of geodetic satellites near 10⁷ metres, lunar laser ranging near 4×10⁸ metres at a few parts in 10¹¹, and planetary ranging beyond. It does not rule out a term at ranges under about ten micrometres or above about 10¹³ metres; the limits are weakest where no pair of masses of the right size can be arranged.

Gravitation

A force that can hide at any one distance

The inverse-square law is tested by comparing the strength of gravity at two distances, and every instrument that does so is sensitive over about one decade of range. A Yukawa force with a range outside that decade is invisible to it: much shorter and it never reaches, much longer and it only rescales the constant. So testing the law is not one experiment made more precise but a chain of unrelated ones, from torsion pendulums to the orbits of planets, and the chain has weak links exactly where no pair of masses of the right size can be arranged.

$Rays from a ring of sunsets, bent into the shadow. Sunlight grazing the Earth at heights of 1, 5, 15 km, bent toward the axis by the atmosphere by twice the refraction at that height, reaching the Moon's distance, with the shadow's geometry in the plane through the Sun: the distance axis is compressed by about sixty to one against the vertical. The low rays cross the axis well before the Moon and reach the far side of the shadow; the higher rays only bend enough to enter its edge. The geometric umbra at the Moon has a radius of 4584 km, the two dashed lines, so without air the Moon would disappear inside it. Light arrives from the whole ring round the Earth, which is every sunrise and sunset at that moment.$ The observed sky

A Moon lit by every sunset at once

With no air, the Moon would vanish inside the Earth's shadow. Instead it turns a dull copper, because the atmosphere bends sunlight round the planet, and what reaches the Moon has gone through a ring containing every sunrise and sunset on Earth at that moment. Each height in the air lights one place in the shadow, so the colour across the disc is a map of the atmosphere by altitude. It has sometimes been black, after the largest volcanoes, and that makes a century of eclipse records a measurement of the stratosphere older than any instrument sent up to sample it.

$Almost everything is ionised, and then almost nothing is. The ionised fraction of hydrogen against distance from an O7 star in gas of 10 per cubic centimetre, from integrating photoionisation balance outward, in units of the counted radius of 14.62 parsecs. The fraction is near one until about the edge and then falls from 90% to 10% within 0.192 parsecs, 1.3% of the radius. That width is about 18 mean free paths of an ionising photon in neutral gas (0.011 pc). Nothing about the star sets it. A region with no gradual boundary is a surprising result of a smooth input, and it comes from the fact that a photon that gets as far as the neutral gas is absorbed at once.$ Starlight

An ionised sphere that is a count of photons

A hot star sits in a cloud of hydrogen and ionises it, out to a radius, and there the ionisation stops within a tiny fraction of the distance. That edge was predicted in 1939 from a single count: every recombination uses up one photon, so the star can hold ionised only as many atoms as its light can pay for. By the equation that describes a stellar atmosphere at the same temperature, the same gas would hold ten orders of magnitude fewer neutral atoms than it does. In a nebula the temperature is not what decides the ionisation. A budget is.

Whether an excited ion radiates depends only on what reaches it first. The rate per second at which an oxygen ion in the state that gives the 5007 Å line loses its energy, by radiating and by collision, against electron density, both logarithmic. Radiating takes about 51 seconds whatever the surroundings. Collisions speed up with density and cross at 6.8e+5 per cubic centimetre. The air at the Earth's surface, 2.7×10¹⁹ molecules per cubic centimetre, has a collision rate about 4×10¹³ times the radiative one, so the ion loses its energy without radiating. A nebula at 10⁴ per cubic centimetre is sixty times below the crossing. The line has a tiny transition probability; the atom was never forbidden to make it, and it requires only the emptiness in which there is time.

Starlight

An element that was only empty space

In 1864 William Huggins turned a spectroscope on a planetary nebula and saw a bright green line that matched nothing on Earth. It was given a name, nebulium, and it was treated as a new element for sixty years, until the periodic table had no room left for it. In 1927 Ira Bowen showed it was ordinary oxygen, emitting by a route that collisions close long before any laboratory vacuum could reach it. The line was never forbidden. It simply needs about a minute of being left alone, and nothing on Earth gets that.

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