Astronomy & Celestial Mechanics
the sky, computed
Theme

Light is the only messenger

One thin stream of photons carries distance, temperature, composition, motion and age — and everything visible is already old.
Phases are a viewing angle, not a shadow. A satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse. The observed sky

Phases are not shadows, and eclipses are

Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy.

 Refraction against apparent altitude. How far the air lifts an object, in arcminutes, against where the object appears to be. The lift is 34.5′ at the horizon — larger than the Sun's own diameter — and falls below one arcminute above 45°. At the horizon it is also the least reliable number in positional astronomy, because it depends on the temperature profile of the air the sight line crosses. The observed sky

The Sun sets before it sets

At the moment the Sun's lower edge appears to touch the horizon, the whole of it is already below. Refraction lifts it by more than its own diameter, squashes it while it is there, and makes every sunrise and sunset time in every almanac a statement about the air rather than about the sky.

 The same ellipse at 955 times the size, and a quarter of a year out of step. The aberration ellipse (solid) and the parallax ellipse (dashed) for γ Draconis at four ecliptic latitudes, drawn to one scale. Aberration is v/c towards the Earth's own direction of travel, so its ellipse has semi-major axis 20.49551″ for every star in the sky; parallax is 1/d towards the Sun, so its semi-major axis is that star's own 0.02147″ — 955 times smaller, and drawn 955 times smaller here. At the ecliptic pole both are circles; on the ecliptic both collapse to lines; in between both are ellipses of semi-minor axis sin β times the major, and the ratio is identical at every latitude because both effects project the same way. The marks are the same four dates on each: they are a quarter of a year apart between the two, because the aberration displacement follows the Earth's velocity and the parallax displacement follows its position, and velocity leads position by 90° on a circular orbit. That quarter-year is the only thing distinguishing the two phenomena on the sky, and it is why Bradley, looking for the dashed ellipse in 1728, spent months unable to interpret the solid one he had found instead. The observed sky

The other twenty arcseconds

Every star in the sky traces a small ellipse over a year, of the same shape as its parallax ellipse and 90° out of step with it — and the same size for all of them, near or far. Bradley found it in 1728 while hunting for parallax, and it proved the Earth moves a century before anything's distance was known.

 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.

 7 clocks, and a body 233 kilometres across. A stellar occultation reduced. Each horizontal segment is one observer's chord: the star vanished, the star came back, and the interval multiplied by the shadow's 21.4 km/s across the ground is the length drawn. The longest, at an offset of -28 km, is 252.3 kilometres. The dashed ellipse is the limb fitted to the chords by least squares in the half-length squared, and its equivalent-area diameter is 233.9 km against the silhouette's true 232.9 — the residual is the shape the fitted ellipse cannot hold, not an error in any timing. The two open marks are observers inside the predicted path who saw nothing, and they are measurements: they bound the limb inside their own offsets, which is what fixes the extent when the positive chords all fall on one side. At 0.12 s per contact the length precision is 2.6 km, 1.10 per cent of the body — a size measured with a clock rather than with an angle, on an object no telescope resolves. The observed sky

A shape measured by the edge of a shadow

When a small body passes in front of a star the observable is two times. Multiply the interval by the body's speed across the sky and it becomes a chord across a silhouette no telescope can resolve — and enough chords give a profile to a kilometre or two, for an object a few hundred across.

 A shadow edge 10.3 metres wide, and a stellar diameter read off how blurred it is. A star disappearing behind the Moon, drawn as intensity against position across the shadow. The horizontal axis is in Fresnel scales of √(λD/2) = 10.3 metres at 550 nm and 3.844e+5 km, which is the only length the problem has; at a limb speed of 0.62 km s⁻¹ one of them takes 16.6 milliseconds to pass, so the whole event is over in a tenth of a second and needs photometry at a kilohertz. The Moon has no atmosphere and its limb is a knife edge, and a knife edge does not cast a shadow with an edge: the intensity at the geometric boundary is 0.250, a quarter rather than a half, and outside it the light overshoots to 1.37 before ringing down. Every one of those numbers is a property of the wave and of nothing else. What the star contributes is the blurring. Each point of the stellar disc casts its own copy of the pattern, displaced by its own position, so the observed trace is the pattern convolved with the star's projected disc — 22.4 metres wide for the 12 milliarcsecond curve, against a 10.3-metre fringe. The contrast falls from 0.28 to 0.04 across the four curves drawn, and inverting that fall is how several hundred stellar diameters were measured with a single telescope, no interferometer, and no resolution at all. The picture cannot show the limitation that ended the technique's dominance: the Moon goes where it goes, so only stars within a few degrees of the ecliptic are ever occulted, and each is occulted at whatever position angle the geometry happens to offer. 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 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 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.

 Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 10,343 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,400 K, where hydrogen is 78% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all. Starlight

The surface that is a depth

A star has no surface. What looks like one is the level at which the optical depth reaches about two-thirds — and the edge is sharp only because the opacity climbs so steeply that the transition takes a ten-thousandth of the radius.

 The extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V. 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.

 The curve of growth. The equivalent width of an absorption line against the number of absorbers along the sight line, both logarithmic, computed by integrating a Voigt profile with damping parameter 0.005. Three regimes: the width grows in proportion to the abundance while the line is weak, then almost not at all for two decades once the core saturates, then as the square root once the damping wings dominate. A line measured in the middle stretch carries almost no information about the abundance, and most strong lines in a stellar spectrum are there. 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.

 B − V against temperature, computed and measured. B − V against effective temperature. The curve is the colour index of a blackbody, obtained by integrating Planck's law against B and V response functions and subtracting a constant so that the index is exactly zero at 9600 K — the convention that an A0V star has every colour index zero, which is a choice and not a measurement. The 15 points are the main sequence as it is actually measured, and they do not lie on the curve: at the Sun's temperature the blackbody gives 0.446 where the sky gives 0.653, and at M0V 0.995 against 1.40. The model is too blue almost everywhere, and least wrong near 9600 K — which is why the zero point is put where it is. A colour index is a difference of two magnitudes, so it is a difference of two integrals, and changing either filter changes the number. 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.

 Fringe visibility for a 47 mas disc at 575 nm. Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the same observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star. 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.

 Three profiles of equal equivalent width, 28.6 mÅ. Three absorption profiles with the same equivalent width — 28.6 milliångström, 1.72 km/s at 500 nm, matched to better than 0.1% by root-finding over the quadrature — differing in nothing but shape. Left: the cores, on a common velocity axis. Right: the same three normalised to their own half widths, on a logarithmic depth scale. The thermal profile is a Gaussian set by Fe's mass at 6000 K, 1.34 km/s; the collisional one a Lorentzian of γ = 2.28e-3 nm; the rotational one the classical kernel of a disc turning at v sin i = 1.73 km/s, which is exactly zero beyond 1.28 half widths and is the only one of the three with an edge. Matching the areas does not match the widths: the half widths are 1.11 km/s (thermal), 1.35 km/s (rotational), 0.68 km/s (collisional), a factor of 1.98 between the widest and the narrowest. At three half widths the collisional wing is 49 times the thermal one and at five it is 1.27·10⁶ times — six decades, which is why a line's shape stays diagnostic long after its width has stopped being so. 11% of the Lorentzian's own equivalent width lies beyond the right-hand panel's edge and is not drawn anywhere. Starlight

The same width for three different reasons

Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another.

 The Balmer maximum is at 9,870 K, and it is a maximum in temperature. What two absorption lines actually count, each normalised to its own maximum. The first curve is the fraction of all hydrogen sitting in n = 2 — the only hydrogen a Balmer line can absorb — which is a Boltzmann factor climbing with temperature multiplied by the neutral fraction falling with it. The product peaks at 9,870 K, where 35.2% of the hydrogen is still neutral and only 8.73·10⁻⁶ of all of it is in n = 2 at all. Below the peak there is plenty of hydrogen and almost none of it excited; above it there is plenty excited and almost none of it neutral. The second curve is the fraction of calcium that is singly ionised, which is what the Ca II K line counts, and it peaks at 6,335 K — cooler, because calcium gives up its first electron at 6.113 eV. At 6,000 K the Balmer curve is at 0.1% of its own maximum while Ca II is near its peak, and calcium is 4.5·10⁵ times rarer than hydrogen in the same gas. A spectrum in which Ca II K is the strongest line is not a spectrum of a calcium star. Reading it as one is precisely the error that had the Sun made of iron until 1925. Starlight

Why hydrogen's lines are strongest where hydrogen is not

A Balmer line counts the hydrogen atoms sitting in one particular excited state, and that population peaks near 10,000 K — where a third of the hydrogen has already been ionised away. The strength of a line is a thermometer, and reading it as an abundance put hydrogen at one per cent of the Sun until 1925.

 The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in. 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.

 A strong line at three gravities, 6000 K. Left, one strong absorption line of Fe computed at three surface gravities and drawn on the same wavelength scale — no normalisation to each profile's own width, which is exactly what the comparison is about. The thermal core is identical in all three, because the Doppler width is 0.00223 nm at 6000 K whatever the star's size. The wings are not: collisional damping is proportional to the density of perturbers, that density is proportional to the gas pressure, and in a grey atmosphere the pressure at the photosphere is proportional to g — so log g = 4.4 carries a damping parameter 794 times that of log g = 1.5. Right, the width at a tenth of the core depth against gravity, measured off those curves: a slope of 0.499, against the half a Lorentzian wing forces. That is the second dimension of a spectral classification. The first is the temperature, read from which lines are present; this one is the pressure, read from how wide they are, and it is the whole reason a spectrum can be turned into an absolute magnitude and then into a distance. 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.

 The Bouguer line, and the intercept nobody observed. Instrumental magnitude against airmass for one star of magnitude 10 outside the atmosphere, observed at 5 airmasses in 5 bands. Each slope is that band's extinction coefficient, computed from Rayleigh scattering, an aerosol term and ozone rather than assumed: U 0.493, B 0.246, V 0.126, R 0.061, I 0.027 magnitudes per airmass. Every line is fitted through its points and extended to X = 0, and the intercept there is the published magnitude — a measurement made at an airmass no observation is ever taken at, because the smallest airmass available is 1 and that is already a whole atmosphere. Two consequences follow and neither is a detail. The slope has to be re-measured every night, because the aerosol term changes with the weather and is not a property of the site. And the U-band line is 3.9 times steeper than the V-band one, so the extrapolation is 3.9 times longer in exactly the band where photons are scarcest — which is why ultraviolet photometry from the ground was always the least trustworthy part of a magnitude system. 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.

 The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them. Starlight

Resolution without a mirror

Two telescopes a kilometre apart do not make a kilometre-wide telescope. They measure one number — the Fourier component of the sky at the spatial frequency their separation sets — and an image is what you get by collecting enough of those.

 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 peak at 0.55 µm, and therefore a grain size. Interstellar polarisation against wavelength — the Serkowski law, p(λ) = p_max exp[−K ln²(λ_max/λ)] with K = 1.66 λ_max. The heavy curve peaks at 0.550 µm, measured off the drawing rather than read back from the parameter, and falls to half its peak at 1.314 µm on the red side, against the closed form λ_max exp√(ln2/K) = 1.315. That peak wavelength is the measurement. It is set by the size of the grains doing the aligning — bigger grains, longer λ_max — and it is tied to the shape of the extinction curve along the same sight line by R_V ≈ 5.5 λ_max, which gives 3.03 here against the diffuse-medium value of 3.1. The two faint curves are populations peaking at 0.35 µm and 0.75 µm: the same amount of polarisation, distributed differently, and a different dust. Nothing in a photometric measurement of the same star distinguishes them. Starlight

The direction a photon count throws away

A photometer records how many photons arrived. It discards a two-component quantity that survives every attenuation on the way, and that quantity carries a magnetic field direction, a grain size, and the shape of an exploding star nobody can resolve.

 A colour term of -0.059 magnitudes per magnitude, and one star that will not obey it. Synthetic photometry of blackbodies from 3,000 to 42,000 K through two V bands: the standard one, and a natural system whose effective wavelength is 9 nm longer and whose width is 1.12 times as large. The vertical axis is the difference between the two magnitudes for the same star — not a constant, because a wider redder band collects a different fraction of a hot spectrum than of a cool one. Fitting a straight line against B − V gives a colour term of -0.0585 magnitudes per magnitude and leaves a residual of 3.1 millimagnitudes, which is why a linear transformation is the standard reduction and why it works. The mark off the line is a cool star with molecular absorption bands in the red, drawn from the same blackbody with three synthetic bites taken out of it: it sits 27 millimagnitudes from the fit, 9 times the blackbody scatter. A colour term knows one number about a star and a spectrum has a shape, and that gap is the reason all-sky photometry stops at a per cent while differential photometry on one field reaches a millimagnitude. Starlight

The same star through two telescopes

A magnitude is defined by a response curve, and no two telescopes have the same one. The difference between two observatories' measurements of one star is not a constant to be subtracted but a function of the star's colour — and for a star whose spectrum has structure, not even that.

 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.

 20 closure phases, unmoved by an atmosphere that ruins every baseline. Closure phase measured against closure phase true, for all 20 triangles of a 6-antenna array observing a binary 3.2 mas apart with a flux ratio of 0.35. Each antenna has been given an independent atmospheric phase of 65° rms, which corrupts the individual baseline phases by 102° rms — several times the 31.2° the source itself produces, so no single visibility phase in this simulation carries usable information. Every point here nonetheless sits exactly on the diagonal: the per-antenna terms cancel identically round any triangle, and the largest departure over all 20 is 2.5e-14 degrees, which is round-off. The price is in the counting. 6 antennas give 15 baseline phases of which 5 are consumed by the unknowns, so of the 20 triangles only 10 closures are independent — a fraction (N−2)/N = 0.667 of the phase information. For two antennas that fraction is zero and there is no closure at all; the Event Horizon Telescope's image rests on quantities of this kind and on no absolute phase whatever. Starlight

A phase that survives what corrupts it

An atmosphere over each antenna adds an unknown to the phase of every baseline that antenna takes part in. Sum the phases round a triangle and every one of those unknowns cancels identically — which is the reason an image can be made across ten thousand kilometres, and the reason it has no position on the sky.

 A true peak at 0.3103 and a false one at 0.6897, from the same data. The Lomb–Scargle periodogram of 162 simulated observations taken over 419 nights from one site, of a star carrying a 4.2-unit sinusoid at 0.3103 cycles per day under 3 units of Gaussian noise per point. The injected signal is recovered at 0.3103 cycles per day, within the 0.0024 resolution element the baseline allows. The second peak, at 0.6899, is 87 per cent as tall and corresponds to nothing: it is 1 − f, the signal reflected in the one-cycle-per-day spike of the sampling. Neither peak is more real than the other in this picture — deciding between them needs a second site at a different longitude, or a run long enough for the seasonal window to separate them. The dashed line is the power a pure-noise series would exceed once in 1000 trials, computed from 586 independent frequencies rather than from the 4691 grid points searched; using the grid count instead would put the line 2.08 higher and reject a real detection. Starlight

A period found in the gaps

Astronomical time series are sampled when the sky is dark and clear and the target is up, which is a schedule with a spectrum of its own. That spectrum is convolved with the real one, so a single sinusoid produces several peaks — and the tallest is not always the true one.

 A 3-gauss field moves the line by 0.33% of its width and is measured anyway. Above: the Fe I 6173 Å line, Landé factor 2.5, at a Doppler width of 0.041 Å. The solid curve is the unmagnetised profile; the dashed curve is the same line in a longitudinal field of 3 gauss, which splits it by 1.3·10⁻⁴ Å — 0.33 per cent of its own width — and is drawn on top of it because the two are not distinguishable. The double-lobed curve underneath them is the Stokes V profile of that same field, magnified 100 times: the two σ components are circularly polarised with opposite handedness, so what is lost in the sum survives in the difference, and the difference of two profiles a hair apart is the derivative of one of them. Below: what that buys. The V amplitude is linear in the field, because it is a first derivative; every signature of the same field in the intensity is quadratic, because a symmetric splitting can only broaden. At a polarimetric precision of 10⁻⁴ the first reaches 0.1 gauss; at a line-width accuracy of 0.001 the second reaches 38, a factor of 420 worse. A quantity far too small to resolve is measured because it is the only thing in the signal that carries a sign — and the same argument run the other way says what polarimetry cannot do: a field of mixed polarity inside the resolution element cancels in V and does not cancel in I, so the 3000-gauss field of a sunspot, which does resolve, is measured the other way round. Starlight

A field strength read off a line that will not split

A few hundred gauss splits a spectral line by a ten-thousandth of its own Doppler width, which no spectrograph will ever resolve. The two components are circularly polarised with opposite handedness, so the split survives in the difference of two polarisations — where it is linear in the field rather than quadratic.

 Three slopes: 2.5, −0.75 and −1.25, and only the middle one is about the electrons. A radio source's spectrum across five decades of frequency, computed for an electron population with index p = 2.5 in a uniform field. Nothing in this curve is a temperature, because a power law has no scale and therefore nothing a thermometer could read. The straight section between the two bends has slope −0.75, measured here off the drawn curve, and the electron index follows from it and from nothing else: α = (p − 1)/2, so a flux ratio between two frequencies is a measurement of the energy distribution of particles in a place no detector will ever visit. The two bends are the other two measurements. Below 80 MHz the source is opaque to its own radiation and rises as ν^2.5 — a slope fixed at 5/2 by the geometry alone, whatever the electrons are doing — and the frequency at which that happens gives an angular size for a source nothing has resolved, because the turnover is where the brightness temperature meets the electrons' own. Above 12 GHz the spectrum steepens by 0.50: the electrons that radiate at high frequency lose their energy fastest, so the top of the distribution has already emptied, and the frequency of the break is a clock. What no part of this curve gives is the magnetic field. Only the field and the particle density together enter the emission, and every field strength ever quoted for a radio source comes from assuming the two share the energy equally. 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.

 Two means of one opacity, 75 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 6 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 75 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.76 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.13, and then turns up sharply, slope 0.99, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps. Starlight

A mean dominated by the gaps

The opacity in the equation of radiative transport is not an average of the opacity. It is a harmonic average weighted by the temperature derivative of the Planck function, which makes it a measurement of the transparent windows between the lines rather than of the lines — and that single fact decides what adding a million spectral lines to a table does.

 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 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.

 Energy generation against core temperature. The proton–proton chain and the CNO cycle, in solar units, against core temperature on logarithmic axes. The CNO curve is far steeper, so the two cross at 18.8 million kelvin — above that temperature a star runs mostly on CNO, and below it mostly on pp. Stars

The furnace that runs cooler than a compost heap

The Sun's core produces 276 watts per cubic metre. A human body produces more. The Sun is bright because it is enormous, and its fusion is astonishingly slow — which is exactly why it lasts.

 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 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.

 Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it. Stars

A star held up by a rule about counting

A white dwarf makes no energy and does not collapse. What holds it up is not heat or pressure in any ordinary sense — it is a quantum rule forbidding two electrons from occupying the same state.

 Inside a star that is holding itself up (polytrope n = 3). Temperature, density and pressure through a star, as fractions of their central values, against fractional radius. All three come from one numerical integration of the Lane–Emden equation at index 3, which is the hydrostatic balance written for a gas whose pressure is a power of its density. The inner half of the radius holds 90% of the mass, and the outer half is nearly weightless — which is why the load, and so the temperature, is concentrated where the burning is. Stars

A star is held up by its own weight

A star has a central temperature because it has a central pressure, and it has a central pressure because everything above is pressing down. The nuclear reactions do not set that temperature — they obey it.

 Leavitt's law, through the Milky Way calibrators. Mean absolute magnitude against the logarithm of the pulsation period, for classical Cepheids with independently known distances, and RR Lyrae itself for comparison. The line is M = −2.81 log P − 1.43: a tenfold longer period is 2.81 magnitudes brighter, a factor of 13 in luminosity. The vertical axis runs the astronomers' way, with brighter upward. Stars

A star that tells its distance by how slowly it blinks

Some stars pulsate, and the slow ones are the bright ones. That single correlation turns a clock into a ruler, and it is how the size of the universe was first measured.

 The main sequence against its own ceiling. Luminosity against mass, in solar units, with the Eddington limit drawn on the same axes. The limit is exactly linear in mass. The main sequence is not — until it is: above about 55 solar masses the relation flattens to a straight proportionality as well, and the two lines then run parallel at a fixed ratio of 83 per cent. They never cross on this range, which is the more interesting outcome: the most massive stars do not approach their limit gradually, they are built at a fixed fraction of it. The Sun, by contrast, radiates 0.0026 per cent of its own ceiling. Stars

A brightness that would blow the star apart

Light pushes. For a star of a given mass there is a luminosity at which the outward push on its own outer layers equals the inward pull of gravity, and it depends on nothing but the mass and the opacity — not on the star's structure, its composition or its age.

 A furnace with the thermostat taken out. Left, the two pressures in a red giant's helium core at 10⁶ g/cm³. The ideal-gas pressure rises with temperature, as the whole regulation of a star depends on it doing; the degenerate electron pressure is a horizontal line, because the exclusion principle does not know the temperature. At the ignition point near 10⁸ K the degenerate term is 5.1 times the gas term, so a temperature rise there raises the total pressure by almost nothing and the core does not expand. Right, what that costs. Both curves start at the same temperature with the same triple-alpha rate, whose logarithmic slope is 41 — the famous T⁴⁰, differentiated here rather than quoted. The regulated core settles; the degenerate one, with nowhere to put the extra energy, goes vertical. The flash reaches 10¹⁰ solar luminosities for a few seconds and not one photon of it is seen: every erg goes into lifting the degeneracy, and the star's visible response is to become fainter and settle on the horizontal branch. Stars

The resonance that had to exist

Three alpha particles cannot meet at once, and the two-body intermediate falls apart in 8×10⁻¹⁷ seconds. That is a hundred thousand times longer than a crossing time, which is just enough — provided a nuclear level sits at 7.65 MeV. Hoyle argued from the existence of carbon to the existence of the level, and it was found where he said.

 Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has. Stars

The part of a star that boils

Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite.

 The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number. Stars

The star that swells because its centre shrank

A helium core contracts, and the envelope around it expands by a factor of sixty. The central density rises at the same time as the radius, which is the opposite of what a self-gravitating body is expected to do, and a burning shell between the two is the whole reason it happens.

 14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here. Stars

The interior read from a comb of frequencies

A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

 One population, three integrals, two ends. The same mass function weighted three ways, each normalised so that the area under it is one, against mass on a logarithmic axis — so a share of the page is a share of the total. The light is a zero-age population's, every star still on the main sequence, which is the only age at which the top of the range is present at all. The steep curve on the left is the number of stars, which peaks at 0.08 M☉ and falls away because the function is steeper than m⁻¹; the middle one is the mass they carry; the one on the right is the light they emit, computed from this file's own main-sequence relation and peaking at 55.0 M☉ — 682 times further up the axis. The population that is counted and the population that is seen are two different populations. Above 8 M☉ there are 0.63% of the stars, 21% of the mass and 99% of the light. The slopes are α = 1.3 above 0.08 M☉, α = 2.3 above 0.5 M☉, and the low-mass end is where the honesty runs out: it has never been measured in another galaxy, and every mass inferred from a luminosity assumes it. Stars

Most stars are small, and most of the light is not

The mass function is steep and the mass–luminosity relation is steeper, so counting stars and measuring their output are integrals of the same function dominated by opposite ends of it. Every mass inferred from a brightness passes through that mismatch.

 A light curve that is a decay chain. Bolometric luminosity against days since explosion, for three Type Ia models synthesising 0.3 M☉, 0.6 M☉, 0.9 M☉ of ⁵⁶Ni, from Arnett's one-zone diffusion model. Nothing in the shape is fitted: the two timescales are the laboratory half-lives of ⁵⁶Ni and ⁵⁶Co, 8.8 and 111.3 days, and the rise is those decays seen through an envelope that takes 11.3, 15.5, 18.6 days to leak. Each peaks at 14, 18, 21 days, and at maximum the luminosity equals the instantaneous deposited power to within 0.6% — Arnett's rule, and the only reason a peak brightness can be read as a mass of nickel. After maximum the curves settle to nearly straight lines on this logarithmic axis, declining at 0.0360 magnitudes a day against ⁵⁶Co's own 0.0098. The difference is gamma-ray escape: the ejecta become transparent to the very photons that are supposed to be heating them, and the tail is therefore steeper than the isotope. The one thing the tail is not is a property of the star — it is a half-life, plus a column density that is falling as t⁻². 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 furnace with the thermostat taken out. Left, the two pressures in a red giant's helium core at 10⁶ g/cm³. The ideal-gas pressure rises with temperature, as the whole regulation of a star depends on it doing; the degenerate electron pressure is a horizontal line, because the exclusion principle does not know the temperature. At the ignition point near 10⁸ K the degenerate term is 5.1 times the gas term, so a temperature rise there raises the total pressure by almost nothing and the core does not expand. Right, what that costs. Both curves start at the same temperature with the same triple-alpha rate, whose logarithmic slope is 41 — the famous T⁴⁰, differentiated here rather than quoted. The regulated core settles; the degenerate one, with nowhere to put the extra energy, goes vertical. The flash reaches 10¹⁰ solar luminosities for a few seconds and not one photon of it is seen: every erg goes into lifting the degeneracy, and the star's visible response is to become fainter and settle on the horizontal branch. Stars

The explosion that never reaches the surface

When helium ignites in a degenerate core the thermostat every other burning stage runs on is disconnected. The runaway reaches ten billion solar luminosities for a few seconds, and the star's visible response is to get fainter.

 The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round. Stars

The disc that has to throw angular momentum away

Matter cannot simply fall onto a compact object. At the energy it arrives with, it has far too much angular momentum, and the only way in is for some of it to be carried outwards — which is what a disc is for.

 Two stars at the same point of every other diagram, 4.1 times apart in one. Above: the gravity-mode period spacing against the large frequency separation, for 90 shell-burning giants and 55 core-burning ones. These are the same stars in every other measurement. They have the same luminosity, the same temperature, the same colour, the same surface gravity and — inside the band drawn — the same Δν, which means the same mean density; the global scaling relations return the same mass and the same radius for both. What separates them is a quantity that comes from nowhere near the surface: ΔΠ₁ is set by the buoyancy frequency integrated across the core, and a core that has ignited helium is expanded and convective, so its integral is smaller and its period spacing larger. The two sequences do not touch — 61 seconds against 248 — and the gap sorts a catalogue of tens of thousands of giants into stars burning hydrogen in a shell and stars burning helium in a core, by a Fourier transform of a light curve. Below: what that looks like in the spectrum itself, drawn over two radial orders. The number of mixed ℓ = 1 modes between consecutive radial modes is Δν/(ΔΠ₁ν²), so the shell burner has about 12 of them and the core burner about 3: the star with the larger period spacing has the sparser spectrum. The picture cannot show what the same modes are also used for and cannot settle — the splitting of each mixed mode gives the rotation rate of the core separately from the envelope, and the cores come out spinning some ten times faster than the surface and a hundred times slower than any model of angular-momentum transport predicts. Stars

Two stars only a Fourier transform can tell apart

A giant burning hydrogen in a shell and one burning helium in its core sit at the same luminosity, the same temperature and the same mean density. Every scaling relation returns the same mass and radius for both. The gravity-mode period spacing is fifty seconds for one and three hundred for the other.

 The period spans 492× and the pulsation constant 2.1×. The pulsation constant Q = P√(ρ̄/ρ̄_⊙) for 7 radial pulsators, against their periods. A radial pulsation is a standing sound wave across a star and the sound speed in a self-gravitating body is set by its own gravity, so the period should go as (Gρ̄)^−1/2 and Q should be the same for every star pulsating in the same mode. Across a factor of 492 in period — from 1.3 hours to 27 days — Q varies by 2.07, with a mean of 0.0393 days. That is the whole reason a period–luminosity relation can exist: a period is a density, a density is a mass and a radius, and a radius with a temperature is a luminosity. The masses here are the weak link and the caption should say so — for the Cepheids they come from evolutionary models rather than from a dynamical measurement, and the drift of Q with period is where that assumption is showing. Stars

The valve that has to sit at the right depth

A period–luminosity relation only exists because a period is a density in disguise. Which stars have a period at all is decided by whether a partial ionisation zone sits deep enough to have mass behind it and shallow enough that convection has not taken it over.

 Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find. Stars

A clock read against a model of everything in between

A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image.

 4 cycles of wings, and the polarity reverses at every boundary. Sunspot latitude against date, one mark per spot group, over 4 cycles of 11 years. The pattern is the reason the plot is called a butterfly diagram, and both of its features are laws with names. Spörer's law is the downward slope: spots emerge near ±28° at the start of a cycle and near ±7° at the end, so each wing narrows towards the equator and never crosses it. Hale's law is what the two mark shapes say: the leading spot of a pair has one magnetic polarity in the north and the other in the south, and both reverse when a new cycle starts — so a diagram that repeats every 11 years in appearance repeats only every 22 in magnetism. The wings overlap: the first high-latitude spots of a cycle appear about 1.6 years before the last low-latitude spots of the one before, which is why counting spots gives a cycle length slightly different from measuring one between polarity reversals. The dot density in time is the sunspot number itself, drawn from the standard skewed fitting function — the rise to maximum takes about four years and the decline about seven, in every cycle ever recorded. Stars

A magnetic clock read off a butterfly

Plot sunspot latitude against date and the marks form wings that open at thirty degrees and march to the equator. The polarities reverse between wings, so the magnetic period is twenty-two years and the famous eleven is an artefact of counting spots rather than fields.

 7 solar masses of progenitor become 0.56 of white dwarf. Above: the initial–final mass relation. The steep line is what a star would leave if it kept everything; the shallow one is what it actually leaves, M_f = 0.08M_i + 0.489, fitted to white dwarfs in open clusters. Everything between 1 and 8 solar masses ends up between 0.57 and 1.13 — a range of 7 compressed by a factor of 12 — and the fraction thrown away rises from 43% at the bottom to 86% at the top. A star spends most of its mass in the last per cent of its life, at rates no theory predicts from first principles, and this relation is measured by running stellar evolution backwards: a cluster's turn-off gives the age, a white dwarf's temperature gives its cooling time, the difference gives how long its progenitor lived, and a model turns that into the mass it was born with. Below: the white dwarf mass distribution that follows, from a Salpeter initial mass function pushed through the relation above. It peaks at 0.61 solar masses, which is what every survey of white dwarfs measures. Two things make that peak and the flatness of the relation is only one of them: the unsmoothed distribution (the stepped curve) falls monotonically and is largest at its low-mass edge, because the initial mass function is steep. The edge is where it is because no star lighter than about 1.0 solar masses has had time to die yet, and the scatter of the measurement rounds that cliff into a maximum a little above it. A flat relation produces a narrow range; it takes the age of the Galaxy to produce a peak. Stars

The mass a star does not keep

The initial–final mass relation is nearly flat, so everything from one to eight solar masses ends as a white dwarf between 0.57 and 1.13 — and the observed distribution piles up at 0.6 almost regardless of what went in. The relation is measured by running stellar evolution backwards, because nothing predicts the loss.

 The light curve of AI Phoenicis, computed from its elements. Total light against orbital phase, computed by overlapping two discs of radius 1.805 and 2.9303 solar radii at an inclination of 88.5°, each weighted by its own surface brightness. The two eclipses hide the same area of sky and have different depths — 48.0 and 19.1 per cent — because what is lost is the light of whichever star is behind, and the ratio of the depths is therefore the ratio of the two surface brightnesses. Two things are left out and both matter to a real solution: this is the bolometric light rather than the light in a filter, and the discs are uniform, where a real one is limb-darkened and so has a deeper, rounder eclipse than the flat-bottomed one drawn here. 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.

 Four kinds of explosion, told apart by which line is missing. Spectra of 4 supernova types — Ia, II, Ib, Ic — near maximum light, stacked, over the same wavelength range and drawn with the same photospheric expansion speed of 10,000 kilometres per second. Every feature is a P Cygni profile: a trough blueshifted by the material expanding towards the observer and a peak at rest wavelength from the material moving across the line of sight, and the width of both is the expansion speed. Reading the silicon trough of the type Ia off the drawing puts it at 614.0 nanometres, which recovers 10,131 km/s from a rest wavelength of 635.5. The classification is a decision tree on absences. Hydrogen present makes it a type II; no hydrogen but silicon makes it a Ia; no hydrogen and no silicon but helium makes it a Ib; none of the three makes it a Ic. That tree was written down before anybody knew what these objects were, and it separates a detonating white dwarf from a collapsing massive core almost perfectly — because what the letters are actually reading is how much of the star's outer envelope was still there when it exploded. The type II kept its hydrogen, the Ib had lost it and shows the helium beneath, the Ic had lost that too. 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.

 A fold 0.06 Einstein times wide, and two configurations that make the same one. A binary lens of mass ratio 0.003 at a projected separation of 1.5 Einstein radii. Top left: the source plane, with the caustic — the set of source positions at which the magnification is formally infinite — and the track of a background star across it. A single lens has no such curve; it magnifies smoothly and diverges only at one point. A second mass makes the lens mapping fold, and the fold has edges: crossing one, the number of images changes from 3 to 5, because a pair is created out of nothing on the critical curve. The small closed curve near the origin is the central caustic, always there; the larger one at 0.83 Einstein radii is the planetary caustic, and its distance from the origin is s − 1/s, which is where the planet's own image lies. Top right: the central caustic drawn twice, once for s = 1.5 and once for s = 0.667. They are 0.0184 and 0.0169 Einstein radii across and they lie on top of each other. That is not a coincidence of these numbers: to the order that a central-caustic anomaly is measured, a close binary and a wide one with the reciprocal separation produce the same perturbation, so an event with only a central anomaly returns two separations and no way to choose. Below: the light curve along the track. The smooth part is what a single lens of the same total mass would do; the spikes are the two crossings, 0.06 Einstein times apart, so a few hours inside an event lasting a month. The two curves differ in one thing only — the size of the source. A point source diverges at each fold and reaches 27; a source of angular radius 0.006 Einstein radii averages over its own disc and reaches 9 — an eighth of a source radius inside the fold the two are 8 and 4, with the divergence replaced by a rounded shoulder whose width is the source's own diameter. Everywhere else in this collection the finite size of a star is a nuisance that degrades a measurement. Here it is the ruler: the fold is a straight edge of known sharpness sweeping across a disc, so the shape of that shoulder gives the source's angular radius, and dividing it by the crossing time gives the angular Einstein radius — which is the one quantity a light curve otherwise cannot supply. Exoplanets

A light curve with a fold in it

A single lens magnifies smoothly. A second mass makes the lens mapping fold, and a fold has an edge — a curve across which two images appear out of nothing and the magnification formally diverges. Crossing it turns the finite size of the source star from a nuisance into a ruler.

 A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the uniform stellar disc over the region the planet covers. The depth is 1.05%, which is exactly (Rp/R⋆)² = 0.01055. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii. 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.

 A magnification, and a spike inside it. The brightness of a background star as a foreground one passes in front of it. The smooth curve is exact: a point mass magnifies a point source by (u² + 2)/(u√(u² + 4)), which peaks at 6.7 for this track. The spike is the planet, of mass ratio 0.001, lensing one of the two images. Its duration is the Einstein time scaled by √q — about 23 hours against 30 days — so the whole planetary signal is a few hours in an event lasting a month, and it never repeats. The deviation is drawn in the approximation that the planet lenses the image in isolation; a real caustic crossing has structure this smooths over, and its true height is set by the source's size rather than by the geometry. 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.

 Contrast against separation, which is where direct imaging lives. Planet-to-star brightness ratio against apparent separation, both logarithmic, for a system 10 parsecs away. The reflected-light curves are A_g(R_p/a)² and fall as the inverse square of the orbit; the thermal curve is the ratio of two Planck functions at 10 µm and does not, which is why every imaged planet so far is young and hot rather than merely large. The vertical lines are diffraction limits λ/D — nothing inside a telescope's own line is reachable by it at any contrast at all. An Earth at ten parsecs sits at 6.3e-10, which is 4 orders of magnitude below the faintest planet yet imaged. The four imaged planets are plotted at their measured near-infrared contrasts rather than at 10 µm, because the near infrared is the band they were found in — which is itself part of the argument, since a young planet is hot enough to be bright where its star is not. Exoplanets

Nine orders of magnitude, half an arcsecond apart

Photographing a planet is not a resolution problem. It is a contrast problem, and the contrast is set by an inverse square that punishes exactly the planets a telescope can most easily separate.

 A gap where planets should be. The number of planets per star per interval of log radius, for orbital periods under a hundred days, corrected for detection efficiency. There are two peaks — super-Earths near 1.3 R⊕ and sub-Neptunes near 2.4 R⊕ — and a deficit between them at 1.89 R⊕, where the occurrence falls to 33% of the peak. The gap is not a gap in what can be detected: detection efficiency rises smoothly through it, so a smooth underlying distribution could not produce a dip there. Exoplanets

A gap in a histogram that says how planets are built

Between the super-Earths and the sub-Neptunes there is a radius at which planets are markedly rarer. The gap is not a gap in what can be detected, and its slope with orbital period names the process that made it.

 A radius that depends on the colour it is measured in. Transit depth against wavelength for a planet of 1.38 Jupiter radii at 1400 K, whose atmosphere has a scale height of 538 km — computed from H = kT/µg, not assumed. One scale height of extra opacity adds 153 parts per million to a transit of 1.40 per cent, so the whole spectral signal is 862 ppm at its strongest: one part in 16 of the transit that carries it. The features are at real band centres — sodium at 0.589 µm, water at 1.4 µm, carbon dioxide at 4.3 µm — with the rise at the blue end the Rayleigh slope of scattering off the smallest particles. 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.

 One whole orbit. The system's total brightness through one orbit: the transit at phase 0, the slow rise and fall of the planet's illuminated hemisphere between, and the secondary eclipse at phase 0.5 where the planet's own light is removed. The transit is 1.05%; the secondary eclipse is 1800 ppm, about 6 times shallower. Exoplanets

The planet is seen when it disappears

Half an orbit after the transit the planet passes behind its star, and the light that vanishes is the planet's own. Subtracting two brightnesses taken hours apart isolates a body nothing has ever resolved.

 A 27.5 m/s velocity the star does not have. Left: a rotating stellar disc, approaching on one side and receding on the other, with the chord a planet of 0.1 stellar radii takes across it at impact parameter 0.5 and a sky-projected obliquity of 0°. Right: the apparent radial velocity that results, computed by covering the disc cell by cell — the flux hidden at each phase and its mean line-of-sight velocity — rather than from a fitted formula. The star's centre of mass does not move at any point in this: the anomaly is entirely a statement about which parts of the line profile are missing. Its amplitude is 27.5 m/s at v sin i = 4.5 km/s, and the two numbers are related by the depth of the transit, since blocking a fraction f of light of mean velocity v shifts a flux-weighted centroid by f·v. The curve is antisymmetric about mid-transit to 0.00% of its own amplitude, which is what an aligned transit gives: equal time on the blue half and the red. Limb darkening is included at u = 0.6, and it matters: it weights the hidden light towards the centre of the disc, where the rotation velocity is smallest. Exoplanets

A velocity measured from a shape

A transiting planet hides part of a rotating disc, so the star's line profile loses a slice at one velocity and its fitted centroid moves. The star has not moved at all — and the lopsidedness of that motion is the whole measurement of whether the orbit lies in the star's own equatorial plane.

 Two objects, the same 1.055 per cent, and only one of them a planet. A planet of 0.1027 stellar radii transiting at impact parameter 0.3, and a background eclipsing binary of radius ratio 0.62 whose 38 per cent eclipse is diluted by the target's light to 1.055 per cent — the same depth to nine decimal places, because the dilution was chosen to make it so. A blend contributing 2.7 per cent of the light in the aperture can manufacture any planetary depth at all, so the depth is not evidence about what produced it. Two things in the same photometry are. The ingress occupies 20.4 per cent of the planet's transit and 80.3 per cent of the blend's, a factor of 3.9: the shape of the shoulders is set by the radius ratio of whatever is actually eclipsing, and dilution scales a curve without changing its shape. And the duration with the period gives the mean density of the star being crossed — 1.41 g/cm³ here against 0.15, a factor of 9 — so a blend usually implies a host of a completely different kind from the one the spectrum shows. Neither test needs an observation the survey did not already make, and neither of them proves a planet: they reject specific alternatives, and what is left is a probability. Exoplanets

A planet that is never confirmed, only validated

A background eclipsing binary diluted by the target's light reproduces a planetary transit depth exactly, and no amount of better photometry separates the two. Most known planets are therefore the output of a probability calculation rather than a detection, and the honest statement about them is a statement about a false-positive rate.

 Pluto and Triton sit on the nitrogen line, and the Earth sits between helium and nitrogen. Escape speed against exospheric temperature, with a criterion line for each molecular species at v_esc = 6 v_th — the speed at which the Jeans loss time is comparable to the age of the solar system. Every line has slope one half, because the thermal speed goes as √T; a body above a line keeps that gas and a body below it does not. The horizontal axis is the temperature at the exobase, which for the Earth is near 1000 K rather than the 255 K of its equilibrium — the difference is extreme-ultraviolet heating, and using the wrong temperature puts the Earth above the hydrogen line, keeping an atmosphere it observably lost. Three readings are worth making. The Earth falls between helium and nitrogen and does exactly that: it loses helium as fast as radioactive decay supplies it, and keeps nitrogen for ever. Titan sits just above nitrogen and just above methane, which is why it has a thick nitrogen atmosphere and is slowly losing its methane. And Pluto and Triton sit on the nitrogen line, within 1 per cent — which is why both have atmospheres that are marginally bound and measurably escaping. Where it fails it fails in one direction only. Mercury, the Moon and the Galilean satellites all plot above lines for gases they do not have, because retention is necessary and not sufficient: a body also needs a source, and needs to survive the non-thermal losses this criterion says nothing about. Venus is the sharpest case — it sits above the hydrogen line and has still lost an ocean, because that hydrogen left by charge exchange with the solar wind rather than by moving fast enough. Exoplanets

The gas a planet cannot keep

Escape is a statement about the tail of a distribution rather than about its mean, so the threshold is not a speed but a dimensionless number near thirty — and past that number the loss rate falls by twelve orders of magnitude. Then, for a hot Jupiter, the whole picture fails and the atmosphere leaves as a wind.

 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 mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection. 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.

 Tully–Fisher, from two assumptions and no fitting. 44 model galaxies spanning two and a half decades in luminosity, each built from a constant disc surface brightness of 380 L☉ per square parsec with 0.13 dex of scatter and a stellar mass-to-light ratio of 1.4 with 0.1 dex. Nothing about a luminosity–speed relation is put in: the scale length follows from the surface brightness, the mass from the light, and the speed from v² = GM/R. The construction makes L ∝ v⁴ exactly, so the true slope is −10.0 magnitudes per decade of speed. Least squares of magnitude on log speed returns -8.45, and the reverse regression -9.15: the scatter here is scatter in the speed at fixed luminosity, and error in the abscissa flattens a fitted slope, so neither fit returns the value the family was built with and the distance between them is the size of the effect. Residual scatter about the drawn line is 0.54 magnitudes. Observed slopes run from about −7.5 in blue light to −10 in the near infrared, where the mass-to-light ratio is steadiest — which is the same statement as the second assumption above. 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.

 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.

 Two populations, and the valley between them. 420 model galaxies, 42 per cent of them drawn from a tight red sequence tilted by -0.08 magnitudes of colour per magnitude of brightness, the rest from a broad blue cloud. Only 3.3 per cent land in the band of width 0.20 magnitudes midway between the two sequences, against 17 per cent in the same band laid on the blue cloud and 27 per cent on the red sequence. The bimodality is the strongest statement the galaxy census makes: a galaxy is usually forming stars or has stopped, rarely in between, and since colour tracks the age of the newest stars, the emptiness of the middle is a statement about a timescale — whatever ends star formation does it in a few hundred million years, not a few billion. Galaxies

Two colours, and almost nothing between

Plot a few hundred thousand galaxies by colour and brightness and they do not fill the plane. They pile into a tight red sequence and a broad blue cloud with a near-empty gap between, and an empty gap is a statement about how fast something happens.

 The star-formation law, and the clock inside it. 46 model discs on the Kennicutt–Schmidt plane, built from Σ_SFR = 2.5e-4 Σ_gas^1.4 with 0.28 dex of scatter, with the exponent then fitted back off the drawn points at 1.42. The diagonal dashed lines are constant depletion times: gas divided by the rate consuming it. Every disc on this plane consumes its gas within a few gigayears, and a galaxy that has been forming stars for ten of them therefore cannot have been working from the gas it started with. The slope steeper than one is what makes the depletion time shorter where the gas is richer, which is the opposite of the intuition that a full tank lasts longer. Galaxies

The gas runs out before the galaxy does

Divide a disc's gas by the rate it is turning that gas into stars and the answer is about two billion years — a seventh of the age of the disc. Every spiral still forming stars is therefore being fed from outside, and the star-formation law says the richer the disc, the sooner it starves.

 A coherent one-per-cent distortion, invisible on every galaxy in the picture. 150 background galaxies behind a lens of Einstein radius 14″, each drawn at its own ellipticity: an intrinsic shape with a dispersion of 0.3 per component, plus the reduced shear the lens adds. The strongest shear on any galaxy here is 0.035, one part in 8 of the intrinsic scatter, so no object in this field is measurably distorted and the tangential alignment cannot be seen by eye at all. Averaged over these 150, the mean tangential ellipticity is 0.0088 ± 0.0245 against the 0.0130 the lens model predicts — consistent with the lens and equally consistent with nothing, because 150 galaxies buy a precision of 0.024 and the signal is 0.013. Detecting it at five sigma takes about 13,275 of them, which is not a picture anybody can draw. The cross component — every shape rotated by 45°, which gravitational lensing cannot produce — averages −0.0016 ± 0.0245, consistent with nothing, and that null is what separates a mass from a badly figured optic. The signal is not in any galaxy. It is in the sum, and the whole design of a lensing survey follows from that. 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.

 A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 51 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic. 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.

 Two densities in thermal balance at one pressure, 121 times apart. Thermal equilibrium for interstellar gas, drawn as pressure against density with both axes logarithmic. Every point on the curve is a temperature between 40 and 9000 K at which cooling exactly balances the 2·10⁻²⁶ erg per second per hydrogen nucleus that grain photoelectrons deliver. The curve is not monotonic: it rises to 5007 K cm⁻³, falls to 1597, and rises again, so a horizontal line anywhere between those two crosses it three times. At the 3000 K cm⁻³ of the local medium the three crossings are a warm phase at 0.47 cm⁻³ and 6354 K, a cold phase at 57 cm⁻³ and 52 K, and one in between drawn dashed because it cannot survive — a parcel there that is squeezed cools and keeps contracting, and one that expands heats and keeps expanding. The two survivors differ by a factor of 121 in density and by exactly the same factor in temperature — necessarily the same, because their product is the pressure both are held at — and yet they press on one another equally, which is why they can share the same volume of the disc indefinitely rather than mixing. 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.

 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.

 The same recession law from two different galaxies. Twenty galaxies at fixed comoving positions after the whole picture has been multiplied by 1.34, with each galaxy's displacement drawn from where it was to where it is. Every arrow's tail sits at the old separation and its tip at the new one, so the arrow is exactly 0.34 times its tail's distance from the highlighted galaxy — proportional to separation for one reason and no other: a uniform scaling moves everything in proportion to its distance from whatever point the scaling is measured about. The right-hand panel measures from a different galaxy and gets the identical law with the identical constant. That is the content of a linear velocity–distance relation. It is the signature of an expansion with no centre, and the observation that every galaxy recedes is therefore not evidence that this one is the centre — it is evidence that none of them is. Cosmology

The shift that is not a Doppler shift

Every galaxy beyond the Local Group has its lines shifted to the red, by an amount proportional to its distance. Read as a velocity that looks like a confession that everything is fleeing from here; read as a change of scale it says the opposite, because a uniform expansion produces exactly the same law measured from any galaxy in it.

 A blackbody at 2.7255 kelvin, filling the sky. The Planck function at 2.7255 K in the units the measurement is reported in, with the peak marked where Wien's law in frequency puts it: x = hν/kT = 2.8214, so ν = 160.2 GHz and the intensity there is 384 MJy per steradian. The points are drawn at the twenty-one frequencies across the FIRAS band, displaced from the curve by a Gaussian of 50 parts per million of the peak, which is the root-mean-square deviation the instrument actually reported. At the scale of this plot that displacement is a fifth of a pixel and the points sit on the line — which is the entire finding. Nothing else in astronomy is a blackbody to a part in twenty thousand: a stellar spectrum is a blackbody with absorption lines cut into it and a continuum that is the wrong shape at both ends. A thermal spectrum this exact requires that the radiation was once in equilibrium with matter, which requires that the universe was once opaque, which requires that it was once hot and dense. 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.

 Four abundances, one free parameter. The abundances big-bang nucleosynthesis predicts, against the one number it is free to choose: η₁₀, the ratio of baryons to photons in units of 10⁻¹⁰. Four curves spanning nine decades, from a helium mass fraction of about a quarter down to a lithium abundance of one atom in ten billion, and they are not four independent predictions — they all come out of the same reaction network run at the same density. The horizontal bands are what is measured in the sky, each at its published one sigma. The measurement that matters is deuterium, because its curve is the steep one: inverting the drawn curve at D/H = 2.527e-5 gives η₁₀ = 6.11, and the ends of the observed interval give 6.06 to 6.15. The vertical band is what the microwave background gives, 6.13 ± 0.04, from the height of the second acoustic peak relative to the first. Those two agree to 0.4 per cent, and they have nothing whatever in common: one is a nuclear-reaction network run in the first three minutes and read off a quasar absorption line, the other is a fluid oscillation at four hundred thousand years read off a sky map. Lithium is the exception and it is not a small one — the network predicts 4.70e-10 at the microwave background's density and the halo stars show 1.60e-10, a factor of 2.9 too much, which is unresolved. Cosmology

Four abundances and one free parameter

In the first three minutes the universe ran a nuclear reaction network with exactly one adjustable number in it. That number predicts four abundances spanning nine orders of magnitude, three of them are observed and match, and the fourth is wrong by a factor of three and has been for twenty-five years.

 Every shell contributes the same. Concentric shells of equal thickness around an observer, with stars scattered uniformly in volume. The number of stars in a shell grows as its radius squared and the flux from each falls as the radius squared, so the two cancel exactly and every shell delivers the same total light — the drawn counts are 6, 15, 31, 55, 88, in proportion to r³ − r₀³, and the drawn sizes fall as 1/r. That cancellation is the paradox, and it is why no amount of dust helps: dust absorbs the light and then re-radiates it, and in a universe old enough for the sum to converge it would come to the same temperature as the stars. The sum diverges linearly with radius, so something has to stop it — and the only two candidates are that the shells eventually overlap, or that there are no shells beyond a certain distance because there has not been time for their light to arrive. Cosmology

Why the sky is dark

In an infinite universe of stars every sight line ends on a stellar surface, so the whole sky should be as bright as the Sun's disc. It is not, and the usual answer — that the expansion redshifts the light away — accounts for a factor of six out of a hundred trillion.

 One ionising photon per atom, and where that happens. The number of photons energetic enough to ionise hydrogen, per baryon, against temperature — with temperature falling to the right, so the universe ages to the right. The quantity is the fraction of a Planck distribution above 13.6 eV, integrated numerically, divided by the 6.13·10⁻¹⁰ baryons there are per photon. It crosses one at 5,844 K, which is z = 2143, and it falls by 19 orders of magnitude across the plot because it is the tail of an exponential. That crossing is the answer to why recombination waits until three thousand kelvin. At hydrogen's own ionisation temperature of 157,803 K there are two billion ionising photons per atom and the gas has no chance; the temperature has to fall by a factor of forty before the supply runs out, and it runs out suddenly because an exponential tail does. The number that sets the factor of forty is not an energy at all — it is the photon-to-baryon ratio, which is to say the entropy per baryon, which is to say a number fixed long before any of this and measured today from the second acoustic peak and from deuterium alike. Cosmology

The surface the background actually is

Hydrogen ionises at 157,800 kelvin and the universe became transparent at 3,000. The factor of fifty between them is not an error, and it is not about energy — it is about there being two billion photons for every atom, so the far tail of the distribution can keep the gas ionised long after the typical photon has become useless.

 A neutral fraction of 2.4·10⁻⁶ is already opaque. The Gunn–Peterson optical depth against the neutral fraction of the intergalactic medium, at z = 3, 5, 6.3, for Ω_b = 0.0493 and h = 0.674. Note the range of the vertical axis. A fully neutral medium at z = 6.3 gives τ = 4.1·10⁵, which is not absorption but extinction of everything; the medium reaches τ = 1 — the point at which it stops transmitting most of the light — at neutral fractions of 6.1·10⁻⁶ at z = 3, 3.3·10⁻⁶ at z = 5, 2.4·10⁻⁶ at z = 6.3. That is why the argument runs from the flux that survives rather than from the flux that does not. A spectrum showing any transmission at all between Lyman α and Lyman β is a measurement that the medium is ionised to better than one part in 164,727, and no fit to any absorption line is needed to establish it. Cosmology

A trough that proves the forest survived

A uniform neutral medium at redshift six would absorb Lyman alpha with an optical depth of four hundred thousand. So the existence of any transmitted light in a quasar's spectrum is a measurement — of a neutral fraction below one part in ten thousand — made from a detection rather than from an absorption.

 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.

 A wind that writes its own terminal speed on the blue edge, at 2017 km/s. A P Cygni profile, computed by integrating a spherical wind on a grid rather than drawn. The wind accelerates outward as one minus the inverse radius to the power 0.8, reaching 2000 kilometres a second, and its density falls as the inverse square of the radius over its own speed; the horizontal axis is the Doppler shift in units of that terminal speed, with blue to the left. The column directly in front of the stellar disc is moving toward the observer at every speed from nearly nothing near the surface up to the terminal value far out, so it takes light out of the beam across that whole range and the absorption trough runs from the rest wavelength to the blue edge at -1.01 of the terminal speed. That edge is the measurement: no model of the star, its distance or its mass enters it, only the geometry of a column seen end-on. The emission comes from everywhere else, where the wind moves across the line of sight and can only add photons; it is Infinity times stronger to the red than to the blue, because the part of the shell receding directly away is hidden behind the star and its blueshifted counterpart is not. Two features, one shell, and between them a velocity and a mass-loss rate. 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.

 Bubbles that meet at z = 5.3, and a scattering depth of 0.047. The fraction of the volume of the universe filled by ionised bubbles, integrated from redshift 20 down to 4.5. The equation has two terms and no others: photons escaping from young galaxies open new volume, at a rate taken from the measured cosmic star formation history with an escape fraction of 0.2; recombinations inside the bubbles close it again, on a timescale that is one over the density times the recombination coefficient times a clumping factor of 3. Early on the density is high and recombination wins almost everything; the curve is nearly flat. As the universe expands the recombination time lengthens as the cube of one plus the redshift while the star formation rate is still rising, the balance tips, and the filling factor runs to one in under half a billion years. It reaches unity at redshift 5.31, which is overlap — the moment the bubbles meet and the last neutral walls between them disappear. The same integration gives an electron-scattering optical depth of 0.0465 for the microwave background, against the 0.054 that is measured, and that agreement is the check: the two observations constrain the same history from opposite ends, one fixing when it finished and the other how long it took. Cosmology

It ends when the walls meet

Reionisation is usually described as the universe becoming transparent, which makes it sound like a change of state. It is not. Each young galaxy opens a bubble of ionised gas around itself and recombination closes it again, and what ends the epoch is geometric — the bubbles meet, and the last neutral walls between them disappear.

 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.

 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.

 A transit depth of 1.200 per cent for a planet of area 1.000 per cent. Three transits of the same planet across the same star, differing only in how the star's brightness falls toward its edge. A planet of radius ratio 0.1 covers 1.000 per cent of the stellar disc's area, and if the disc were uniformly bright that would be the depth. It is not uniformly bright: a sight line near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer, so the edge is dimmer than the centre, and a planet crossing near the middle blocks light that is brighter than average. The transit drawn with realistic coefficients is 1.200 per cent deep — 20 per cent deeper than the area — and it is also rounder, because the covered brightness changes through the crossing instead of staying flat. The consequence is stated in the numbers beside the curves. Each is a least-squares fit of the radius ratio to the realistic curve, performed with a different assumed limb-darkening law, and the recovered radius moves by up to 3.6 per cent depending on which law is assumed. Fitting with the law the curve was made from returns the input to five figures, which is the control: the bias is the mis-specification and not the fitter. Since the coefficients come from a model atmosphere rather than from the light curve, every published planetary radius carries a systematic from stellar physics that no amount of photometric precision removes — and it is the dominant one for the best-measured planets. The picture holds the impact parameter fixed; a grazing transit is worse, because it samples only the limb, where the disagreement between laws is largest. 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.

 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 damping tail measures a thickness: ℓ_D = 1400 for a shell 80 deep in redshift. The suppression of small-scale structure in the microwave background, against multipole, on logarithmic axes. Every other feature of the power spectrum measures the last-scattering surface as a surface — its distance, the sound horizon written on it, the ruler it provides. This one measures how thick it is. Recombination takes time: the ionised fraction falls over a range of redshift rather than all at once, and while it is falling the photons are still scattering, so each one random-walks. A photon taking N steps of a mean free path λ diffuses √N λ, which is much further than a single step and much less than the whole interval, and any temperature fluctuation smaller than that distance is mixed away before it can be frozen in. What is left is a Gaussian cut-off, drawn here for three shell thicknesses. A thicker shell means more steps and a longer walk, so it damps at a smaller multipole: Δz = 40 gives ℓ_D = 1980, Δz = 80 gives ℓ_D = 1400, Δz = 160 gives ℓ_D = 990. The observed cut-off is near ℓ = 1400, corresponding to a diffusion length of about 0.03 proper megaparsecs at the time — a scale the reader should compare with the sound horizon, some hundred and fifty comoving megaparsecs, which is what the peaks measure. The tail is therefore a genuine probe of the inside of the transition rather than of its position, and because the damping depends on the free-electron density it is also one of the cleanest constraints on anything that changes it: extra relativistic species, a varying fine-structure constant, or an early energy injection all move ℓ_D while leaving the peak positions nearly alone. The normalisation here is set once by the observed value, so what the figure claims is the scaling with thickness and the shape of the cut-off, not the absolute number. Cosmology

A blur that measures a depth

Every other feature of the microwave background measures the last-scattering surface as a surface — its distance, the ruler written on it, the geometry between. The damping of its small-scale structure measures how thick it is, because a photon random-walking through a finite transition smears away anything smaller than its walk.

 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.

 A spot is dark because it is squeezed. Pressure against depth below the quiet photosphere's optical surface, drawn as a ratio to the pressure there. The rising curve is the surrounding gas, which grows exponentially with a 140-kilometre scale height because that is what hydrostatic equilibrium in an ideal gas produces. The flat pair of bands is the spot's own budget: a magnetic pressure of 3.58·10⁵ dyn/cm² from a 3000-gauss field, which is 25 per cent of the total, plus the gas pressure left over. Horizontal balance requires the two columns to reach the same total at the same geometric level, and the level at which they do is 348 kilometres below the quiet surface — so the spot's own optical surface sits in a hollow. Measured Wilson depressions, obtained from the foreshortening of a spot near the limb, are four to six hundred kilometres. Nothing about the darkness was assumed: a field strength read off a Zeeman splitting fixes the magnetic share, the share fixes the depression, and the depression fixes the temperature the deeper layer must have to carry the reduced flux. 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 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 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.

 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.

 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 star drawn out into 3.0 arcseconds of spectrum. Atmospheric refraction relative to its value at 550 nanometres, against wavelength, at four zenith angles. The air's refractive index rises towards the blue, so the blue image of a star sits above the red one and the object is smeared into a short vertical spectrum. At 60 degrees from the zenith the separation across an optical band is 2.96 arcseconds — several times the size of the image at a good site, and comparable to the width of a spectrograph slit. Every curve here is the same curve multiplied by the tangent of the zenith angle, which is why one corrector with an adjustable strength works at every airmass. The practical consequences are three: a slit aligned other than vertically loses blue light or red light depending on where it was centred, a photometric aperture contains a different fraction of the light in each band, and an astrometric position depends on the colour of the star it is measured from. The observed sky

The atmosphere is a prism as well as a lens

Refraction lifts a star towards the zenith, and everybody corrects for that. It lifts blue light further than red, and the difference is a short vertical spectrum a few arcseconds long — larger than the image, larger than a spectrograph's slit, and quietly present in every ground-based measurement not taken straight overhead.

 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.

 4000 lines averaged into one profile, and 2.3 m/s out of it. A cross-correlation function: the average absorption profile obtained by shifting a mask of 4000 line positions across a spectrum and summing what falls under it. The faint curves behind are individual lines, each with its own depth, its own width and its own small offset; the heavy curve is what averaging them produces. The velocity is the position of the peak, and its precision is the width divided by the contrast, the signal-to-noise and the square root of the number of lines — 2.3 metres a second here. Nothing about this construction is a measurement of any one line. It is a measurement of where a weighted average of thousands of them sits, and the weights are a choice: a mask built for one spectral type applied to another weights the disagreement between the lines differently, and moves the peak. 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.

 A dome flat that is 3.5 per cent wrong leaves 0.042 magnitudes across the field. Two things called the flat field. The left panel is the true illumination of the focal plane, falling by 12 per cent from centre to corner because of vignetting and the filter's own radial transmission. The middle panel is what a dome flat measures, which is the illumination produced by a screen at a finite distance lit by lamps — a different angular distribution, and therefore a different fall-off by a few per cent. The right-hand plot is what survives dividing one by the other: a smooth radial gradient of 0.042 magnitudes from centre to edge. Pixel-to-pixel response scatter, which is what most people mean by a flat field, is 1.8 per cent per pixel and averages down to 0.255 per cent inside a photometric aperture. The term everybody removes is the one that does not matter, and the term that matters is smooth, is different for every flat-fielding method, and looks exactly like a real gradient in the sky. 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.

 A continuum drawn 4.3 per cent below the real one. A short stretch of spectrum with one strong line in it and 150 weak ones scattered across the same interval. The upper dashed line is the true continuum — the flux the star would emit with no lines at all — and it is not observable. The lower one is what a fit through the highest points of the spectrum returns, which is 4.3 per cent lower, because the weak lines have depressed the gaps between the strong ones. Measuring the strong line's equivalent width against the apparent continuum instead of the real one makes it 11.2 per cent too small. The error has a sign, it is worse in spectra with more lines, and it therefore correlates with metallicity — which is exactly the quantity being measured. 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.

 A sample that gets brighter with distance because the faint ones drop out. The mean absolute magnitude of a magnitude-limited sample, relative to the population it is drawn from, against distance. The population has a spread of 0.5 magnitudes about a mean of -4, and the survey stops at apparent magnitude 20. Nearby, everything is detected and the sample is unbiased. Beyond about 316228 parsecs the faint end of the distribution starts falling below the limit and the survivors are brighter than average; further out the bias deepens without limit, because eventually only the extreme tail is detectable. The horizontal line is the classical Malmquist value of 1.382 times the square of the spread, which is what the bias averages to over a magnitude-limited sample as a whole — it is a property of the sample rather than of any one object, and using it as a correction for an individual star is a common and specific mistake. 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.

 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.

 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 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.

 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.

 A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous. 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.

 A bump that crosses the line from -41 to 25 km/s. The residual of a rotationally broadened line profile during a transit, drawn at five epochs and offset vertically. The planet covers a strip of the stellar disc whose radial velocity is the projected rotation at that position, so it removes light from one velocity and leaves a bump in the residual there. As the planet crosses, the bump travels across the profile — and where it starts and ends is set by the geometry of the chord. An orbit aligned with the star's equator gives a track symmetric about the line centre; this one, tilted by 30 degrees, runs from -41 to 25 kilometres a second and is not. The measurement is of a path rather than of a centroid, which is why it works on rapidly rotating stars where the velocity anomaly is swamped by the line's own width. Exoplanets

A shadow crossing a rotating line

A transiting planet hides a strip of a rotating star, and that strip has a definite velocity. So the planet removes light from one place in the line profile and leaves a bump there — a bump that travels across the line as the transit proceeds, tracing the path the planet took across the disc.

 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.

 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.

 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.

 A single-epoch mass, good to a factor of 2.6. The error budget of a black-hole mass obtained from one spectrum, as probability densities in the logarithm of the ratio of the estimate to the truth. Four independent widths go in. The radius comes from the luminosity through a relation with 0.19 dex of scatter about it. The line width has to be measured on a profile that is blended with narrow lines and with iron, and it enters the mass squared, so 0.12 dex. The luminosity was not measured at the epoch the relation was calibrated for and the object has varied since, which is 0.053 dex. And the term this figure exists for is orientation: the virial factor is one number fitted to a sample, every individual object has its own according to how its broad-line region is inclined, and the spread of that is 0.35 dex — the largest of the four by some margin. In quadrature the total is 0.42 dex — a factor of 2.6 — and the drawn total's half-width is checked against that combination rather than assumed. The bracket at the foot is a different kind of error and is not in the quadrature: the absolute calibration of the virial factor, uncertain by about a factor of 3, which moves every quasar mass ever published by the same 0.24 dex and widens nothing. 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.

 A few per cent of diameter, hidden in the second lobe. Visibility against baseline in the natural variable πθB/λ, for stars with linear limb-darkening coefficients of 0, 0.3, 0.6, 0.9, each rescaled to the uniform disc that best fits its own first lobe. In the first lobe the four curves are within 0.72 per cent of one another; past the first null they differ by up to 4.2 per cent. That is the whole difficulty of measuring a stellar diameter. A limb-darkened star has a faint edge, so a uniform-disc fit returns a diameter 9.7 per cent too small at u = 0.9 and 2.5 per cent too small at u = 0.3 — and the information needed to tell which is in a region where the visibility is under five per cent and the calibration errors of a real interferometer are comparable to the signal. The correction from what is measured to what is wanted is taken from a model atmosphere, because the observation that would supply it is the hardest one there is. The fit here is done by least squares on the drawn curves rather than read from a conversion table, and the zero-coefficient case returns the uniform disc to 0.00 per cent, which is the fit checking itself. 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.

 An exponent between 2.3 and 4, not a number. The local logarithmic slope of the mass–luminosity relation against mass — the exponent that "L ∝ M^3.5" stands in for — read off the same fit to eclipsing binaries that the relation itself is drawn from. It is a step function with two corners, and the steps are 2.3 below 0.43 M☉, 4 0.43 to 2 M☉, 3.5 2 to 55 M☉. The horizontal lines are what homology predicts: a star in radiative equilibrium has L ∝ μ⁴M³/κ̄, which gives an exponent of 3 when the opacity is electron scattering and a constant, 5.5 when it is Kramers and depends on density and temperature, and 1 where radiation pressure holds the star up and the luminosity is pinned to the Eddington value. The measured steps sit between those predictions rather than on them, which is what a real star does — no star is homologous, the low-mass ones are convective throughout and the heavy ones have convective cores, and the fitted exponents are what is left after all of that. The useful statement is not that the exponent is 3.5 but that it is between 2 and 5 everywhere and 4 in the middle, and that the reason it moves is a change of which opacity is carrying the energy out. Stars

An exponent that is a slope, not a law

The phrase “L goes as M to the three and a half” stands in for a curve with three straight pieces and two corners. The exponent is 2.3 below half a solar mass, 4 in the middle and falls towards 1 at the top — and each change of exponent is a change in which opacity is carrying the energy out.

 The deuterium burning rate against the expansion, at three baryon densities. The rate at which a deuteron is destroyed, divided by the expansion rate, against temperature. Temperature falls to the right, so the picture reads left to right as time, and the horizontal line at one is where the burning stops mattering: above it a deuteron is destroyed many times over before the universe doubles in size, below it the reaction has effectively ceased. The rate drawn is D(p,γ)³He at the NACRE parameterisation, multiplied by the free-proton density — three quarters of the baryons by number once ⁴He has taken the rest — and the expansion rate is the same 1.66√g* T²/mPl the freeze-out calculation raced the weak interactions against. The three curves differ in one number and one only: the baryons per photon, 3, 6, 12 in units of 10⁻¹⁰. They are therefore vertical translations of each other, exactly in proportion to η, because the reaction is two-body and the expansion is not. That is the entire mechanism by which an abundance measures a density. A denser universe crosses the line later — 21.4 keV at η₁₀ = 3, 16.0 keV at η₁₀ = 6, 12.3 keV at η₁₀ = 12 — and every extra second below the crossing is deuterium that does not survive. What the figure does not do is predict the abundance itself: the residue depends on the whole reaction network and on the ⁷Be and ³He channels that feed back into it, and the curve the abundance is read off is a fit to that network rather than to this. 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.

 Two signals that peak at different separations. The astrometric centroid shift and the photometric magnification against the source–lens separation in Einstein radii, on a common horizontal axis and their own vertical ones, for θ_E = 1 milliarcseconds. The shift is u/(u²+2) times θ_E: it vanishes at u = 0, because the two images are then symmetric about the lens and their centroid is the lens itself; it vanishes as u grows, because the minor image fades and the major one approaches the source; and it is largest in between, at u = √2 exactly, where it is θ_E/2√2 = 0.3536 mas. The magnification at that separation is only 1.15, which is a nineteen per cent brightening — a signal a survey would barely flag. The two observables are therefore complementary rather than redundant. The photometric event is short, bright and centred on the closest approach; the astrometric one is broad, largest on the wings, and lasts several times longer, falling only as 1/u once the source is well away. That slow decline is why astrometric microlensing needs years of monitoring and why it was a prediction for sixty years before it was a measurement. 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.

 The zenith sky after sunset, as single scattering predicts it. The brightness of the zenith sky at 550 nm, in magnitudes per square arcsecond with brighter upward, against the Sun's depression below the horizon, computed by single scattering in a spherical atmosphere: US Standard Atmosphere densities, Rayleigh scattering, a 300 Dobson-unit ozone layer, and every photon scattered exactly once. The Earth's shadow climbs the zenith as R(sec d − 1) — 8.7 km at 3°, 35.1 at 6°, 142 at 12° — through air that thins by a factor of e every eight kilometres or so, so the brightness does not fall steadily: it falls 1.4 magnitudes a degree between 3° and 6°, and 2.7 a degree between 7° and 10°. The model gives 14.3 at the end of civil twilight and 27.0 at the end of nautical. The dashed line is the natural night sky, 21.9 magnitudes per square arcsecond. Single scattering reaches it at 9.25° of depression — 8.8 degrees before the 18° at which astronomical twilight is observed to end. The difference is not an error in the arithmetic; it is the light this model leaves out, scattered more than once. The observed sky

The shadow that climbs the zenith

After sunset the sky overhead is lit only above the Earth's own shadow, and the shadow climbs as the square of the Sun's depression. Scattering computed once from that geometry predicts a sky that dims faster and faster and is as dark as night by nine degrees. The real sky takes eighteen, and the difference is light that has been scattered more than once.

 How much a galaxy's redshift changes in 10 years, and which way. The change in a galaxy's apparent recession velocity over 10 years of the observer's time, c ż/(1 + z), against the galaxy's redshift, for three universes with the same present expansion rate of 67.36 km/s/Mpc. The drift is (1 + z) H₀ − H(z): positive if the expansion rate at the galaxy's epoch was less than (1 + z) times today's, which is to say if the expansion has been accelerating since. In the empty universe the expansion rate is exactly (1 + z) H₀ at every epoch and nothing drifts at all. In the matter-only universe the expansion has only ever slowed, and every redshift falls: −8.6 cm/s over 10 years at z = 1 and −25.5 at z = 4. In ΛCDM the drift is positive nearby, largest at z = 0.63 where it reaches 2.51 cm/s, changes sign at z = 1.91, and is −5.5 cm/s at z = 4. The whole signal is a few centimetres per second in a decade, against the thirty kilometres per second of the Earth's own orbital motion that has to be removed from every spectrum first. Cosmology

A redshift that changes while it is watched

A galaxy's redshift is a ratio of two sizes of the universe, and the second one is still growing while the light is being collected. So every redshift drifts, by a few centimetres per second in a decade, and the direction of the drift says whether the expansion has been speeding up since the light left — the one test of that question that needs no distance and no model of any source.

 How long a planet's hydrogen lasts, against how much of it there is. The time a young Sun-like star's saturated X-ray and ultraviolet output would take to remove a planet's whole hydrogen envelope, at 100 times the Earth's insolation, against the envelope's share of the planet's mass, for cores of 3, 5, 8 Earth masses. It is computed with energy-limited escape and an interior fit for the envelope's thickness, and it is not monotonic. A heavy envelope takes long to remove because there is a lot of it. A very light one takes long because the planet is small and intercepts little light. In between the time peaks, and it peaks where the envelope has swollen the planet most for its mass: 3 Earth masses at an envelope of 1.9 per cent, 129 Myr, where the envelope is 1.31 times as thick as the core's radius; 5 Earth masses at an envelope of 2.8 per cent, 364 Myr, where the envelope is 1.30 times as thick as the core's radius; 8 Earth masses at an envelope of 4.0 per cent, 946 Myr, where the envelope is 1.30 times as thick as the core's radius. The peak is what makes a valley. A planet above it losing gas moves up the curve, its remaining envelope lasting longer and longer, and settles; a planet below it moves down, lasting less and less, and loses everything. The horizontal lines are the saturated phase, 100 Myr, and the 300 Myr of saturated-equivalent exposure the whole history delivers: a core whose peak lies below the second line cannot keep any envelope at all. Exoplanets

The envelope that doubles a planet lasts longest

The gap in the radii of small planets is not merely a place where planets are rare — it is nearly empty, and the reason is a peak. The time a young star needs to strip a planet's hydrogen is longest for the envelope that swells the planet to a little over twice its core's size. Anything thinner runs away to nothing, and anything thicker settles back towards the peak.

 The colour of the zenith at twilight, with and without ozone. The colour of the zenith sky relative to sunlight — the 450 nm brightness against the 650 nm brightness, in magnitudes, bluer upward — against the Sun's depression, computed by single scattering with a 300 Dobson-unit ozone layer and again with none. Scattering alone favours blue by λ⁻⁴, which would make the sky 1.60 magnitudes bluer than sunlight if nothing were removed on the way; that is the dotted line. But after sunset every ray has travelled a long grazing path, and Rayleigh scattering removes blue from that path faster than red, so the two effects fight. Without ozone they very nearly cancel: the zenith is 0.01 magnitudes from sunlight's own colour at sunset and 0.06 at 6° — a pale, colourless sky — and only turns bluer, −0.13 at 10°, once the lit layer has climbed above most of the air the grazing ray used to cross. With ozone the zenith is −0.34 at sunset, −0.67 at 6° and −0.76 at 10°: bluer than without by 0.72 magnitudes at 6°, because the grazing ray also crosses the ozone layer near its tangent point, and ozone's Chappuis band absorbs orange and red rather than blue. The ozone cross-sections are approximate, and the conclusion does not depend on them to better than a factor of two. The observed sky

Ozone keeps the twilight zenith blue

After sunset the sky overhead turns a deep blue, and scattering alone cannot explain it. The light that reaches the zenith has first grazed hundreds of kilometres of air, which strips blue out of the sunlight as fast as scattering puts it back, and the two very nearly cancel. What tips the balance is a gas that makes up a few parts in ten million of the atmosphere and absorbs the orange and red end of the spectrum.

 How far the Sun has to sink before the sky darkens, on four worlds. The dimming of the zenith sky relative to sunset, in magnitudes, against the Sun's depression, for single scattering in an isothermal atmosphere with the stated scale height and vertical optical depth, scattering isotropically: the Earth (R 6371 km, H 8.5 km, τ 0.097); Mars (dust) (R 3389.5 km, H 11.1 km, τ 0.5); Titan (haze) (R 2574.7 km, H 50 km, τ 4); Pluto (haze) (R 1188 km, H 50 km, τ 0.02). These are representative values for the layer that does the scattering — air on the Earth, dust on Mars, haze on Titan and Pluto — not full models of those atmospheres. The angle over which a sky darkens is set by the height of the scatterers against the size of the planet, √(2H/R), because that is the depression at which the shadow over the zenith has risen one scale height. The Earth dims by ten magnitudes at 8.6° of depression, which at the equator of a world whose solar day is 24 hours takes the Sun 35 minutes. Mars (dust) dims by ten magnitudes at 13.4° of depression, which at the equator of a world whose solar day is 24.66 hours takes the Sun 55 minutes. Titan (haze) dims by ten magnitudes at 29.8° of depression, which at the equator of a world whose solar day is 382.7 hours takes the Sun 31.7 hours. Pluto (haze) dims by ten magnitudes at 43.9° of depression, which at the equator of a world whose solar day is 153.3 hours takes the Sun 18.7 hours. The observed sky

The air's height against the planet sets the twilight

On the Earth the sky overhead fades over about nine degrees of the Sun's descent. That angle is not a property of air or of sunlight. It is the square root of twice the height of the scattering layer divided by the radius of the planet, and on a small world with a tall haze it grows to tens of degrees — so that twilight on Titan lasts more than a day.

 Where photoevaporation puts the valley, round stars of different mass. The radius of the largest core stripped bare, against orbital period, round stars of 0.5 M☉, 0.75 M☉, 1 M☉, 1.25 M☉, in the same energy-limited model, with each star's luminosity taken as its mass to the fourth power and its saturated phase lengthened for smaller stars as the mass to the −1.5, 100 Myr for the Sun. At ten days the valley is at 1.22 Earth radii round the 0.5 M☉ star, 1.38 Earth radii round the 0.75 M☉ star, 1.50 Earth radii round the 1 M☉ star, 1.60 Earth radii round the 1.25 M☉ star, and every one of them tilts as the −0.176 power of the orbital period, because the tilt comes from the exponents of the escape law and the interior fit, which do not depend on the star. A lower-mass star is far fainter, so at a given period its planets receive much less light, and even its longer active phase does not make up the difference: this model puts the valley at smaller radii round smaller stars while keeping the same sign of tilt. The stellar scalings are rough, and they are the least certain part of the calculation; what is robust is that photoevaporation ties the valley to the XUV energy a planet received, which falls with the star's mass at fixed period, and a mechanism tied to something else would tie it differently. Exoplanets

A smaller star puts the valley lower

Round stars of half the Sun's mass, the gap between bare rocky cores and sub-Neptunes should sit at a smaller radius than round the Sun, because at the same orbital period their planets receive a tenth of the light. Their stars also stay young and active far longer, which pushes the other way. Photoevaporation weighs those two against each other in a definite proportion, and the answer is a valley that scales as the star's mass to about the power three tenths.

 The dipole a 370 km/s motion has to put into counts of distant sources. The amplitude of the dipole in the number of sources per unit solid angle expected from the Sun's motion at 369.82 km/s relative to the microwave background, β = 1.234e-3, split into its two parts: aberration, 2β, which crowds sources towards the direction of motion, and the Doppler boost, x(1 + α)β, which brightens sources there and lifts fainter ones above the flux limit. For radio sources, with counts steepening as S^(−1) and spectra falling as ν^(−0.75), the expected dipole is 0.0046; for mid-infrared quasars, with counts steepening as S^(−1.7) and spectra falling as ν^(−1.26), the expected dipole is 0.0072, and the measured one is 0.0155 — 2.16 times larger, which would need a speed of 797 km/s. A dipole of a few parts in a thousand needs a catalogue of more than a million sources to see at all. The disagreement is not with the direction, which lies close to the microwave background's, but with the size, and it is not yet explained: either the samples carry a systematic nobody has found, or the matter and the radiation do not share one rest frame on these scales — in which case the assumption that the universe looks the same from everywhere is wrong in a way it has never been caught being wrong before. 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.

 Io's eclipses, early at opposition and late at conjunction. The delay in the timing of Io's eclipses by Jupiter's shadow caused by the changing distance between the Earth and Jupiter, against days from an opposition, for circular orbits at 1 and 5.2026 AU. At opposition the two planets are closest and the eclipses arrive 8.3 minutes early against the average; 199 days later, near conjunction, they are furthest apart and arrive 8.3 minutes late. The whole swing, 16.6 minutes, is the time light takes to cross the diameter of the Earth's orbit, and the pattern repeats every 399 days, the synodic period. This is transit timing done on a moon in 1676, with a clock that ran on Io's 42.5-hour orbit and a residual that had nothing to do with Io. Exoplanets

A transit late by the width of an orbit

A transiting planet's clock can run fast and slow for a reason that has nothing to do with gravity acting on the planet. If its star is itself in orbit about a distant companion, each transit's light has further or less far to travel, and the timing wanders by the light-travel time across the star's orbit. It is the measurement that first showed light has a speed, made again on a different kind of clock.

 Five speeds from the same galaxies, and the relation each one gives. The exponent and the scatter of the baryonic Tully–Fisher relation fitted to the same 90 model galaxies with five different speeds, fitted as speed on mass. outer speed: exponent 3.98, scatter 0.080 dex in mass; peak speed: exponent 3.86, scatter 0.110 dex in mass; at 2.2 scale lengths: exponent 3.29, scatter 0.167 dex in mass; W50 ÷ 2: exponent 3.73, scatter 0.119 dex in mass; W20 ÷ 2: exponent 4.05, scatter 0.118 dex in mass. The galaxies were built with the relation in their outer halo speed, and the other four speeds each lose some of it: the ones read from the inner curve inherit how concentrated each galaxy's stars are, and the line widths add the turbulence of the gas, which matters most in the slowest galaxies. 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.

 Where the axes of rapid rotators point, in a sample chosen by brightness, at a darkening exponent of 0.19. The distribution of rotation-axis inclinations — 0° pole-on, 90° equator-on — for gravity-darkened stars whose axes point at random in space, surveyed down to a limit in apparent brightness, with the surface temperature following the local gravity to the power 0.19. The dashed curve is random orientation, which puts 13.4 per cent of stars within 30° of pole-on. At 0.9 of the critical rotation rate a star looks 1.093 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 14.4 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.09 times as common as random orientation would make them. At 0.98 of the critical rotation rate a star looks 1.188 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 15.3 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.19 times as common as random orientation would make them. Averaged over random orientations the apparent luminosity of each star equals its true luminosity to better than half a per cent, as it must; the tilt towards pole-on comes entirely from choosing stars by how bright they look. 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.

 How fast the circulation a rotating star drives would stir it, against its lifetime. The classical Eddington–Sweet circulation time — the star's Kelvin–Helmholtz time divided by the ratio of centrifugal to gravitational acceleration at its surface — as a multiple of its main-sequence lifetime, against rotation rate as a fraction of critical, for stars of 2, 5, 15 solar masses, on logarithmic axes. Below the line at one, the circulation would turn the star over within its life. For 2 solar masses that happens above 0.101 of critical; for 5 solar masses that happens above 0.123 of critical; for 15 solar masses that happens above 0.115 of critical. The dot is the Sun, turning at 0.0084 of its critical rate, whose circulation would take 150 times its main-sequence life. By this estimate almost every star turning at more than a tenth of its critical rate should be stirred from core to surface; the measured surface compositions of such stars show that it is not, because the composition gradient left by core burning resists the circulation, and how strongly it resists is what the nitrogen at their surfaces measures. Starlight

The circulation that should have stirred every fast rotator

A rotating star in radiative equilibrium cannot be balanced in pressure and in heat at the same time, and the mismatch drives a slow circulation from pole to equator. The classical estimate of its speed says that any star turning at more than a tenth of its break-up rate should be stirred from core to surface within its life, bringing the nitrogen of hydrogen burning up with it. Some fast rotators show that nitrogen and some do not, and some slow rotators show it when they should not — including more than the angle of their axes can explain.

 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.

 Where a photosphere's free electrons come from. The share of the free electrons donated by metals rather than by hydrogen, against temperature, at a gas pressure of 12000 N/m² and at solar abundance, 1 dex below solar, 2 dex below solar. The electron pressure is not assumed here — it is solved for, as the value at which the ionisation of the gas supplies exactly the electrons the gas contains. Below about 5897 K at solar abundance every free electron in the gas comes from an element present at one part in ten thousand, because hydrogen's 13.6 eV keeps it neutral while magnesium's 7.6 does not. The handover is fast: solar crosses a half at 5897 K, 1 dex down crosses a half at 5088 K, 2 dex down crosses a half at 4467 K. At 5,772 K and this pressure the solved electron pressure is 1.78 N/m². 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.

 Turbulence stops being supersonic at 0.04 parsecs. The velocity dispersion of molecular gas against the size of the region it is measured over, from σ = 1 (R/pc)^0.5 km/s, with the isothermal sound speed of 10 K gas drawn flat beneath it. The two cross at 0.035 parsecs, which is solved for here rather than quoted: below that scale the motions are subsonic and the gas is supported by its own pressure, above it they are supersonic and nothing thermal is relevant. At ten parsecs the Mach number is 17 and at a hundredth of a parsec it is 0.53. The crossing is the scale at which turbulent support runs out, and it is within a factor of two of the size of the dense cores that actually form stars — which is either the most important coincidence in the subject or the reason cores are the size they are. Galaxies

The support and the seed are the same motions

A molecular cloud's lines are ten times wider than its temperature allows, and the width grows with the size of the region measured. The motions that widen them hold the cloud up as a whole and make the dense lumps inside it — so the same turbulence that delays star formation is what decides where it happens.

 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 carbon-to-oxygen ratio that steps at every front. The carbon-to-oxygen ratio of the gas and of the solids against distance from a 1 solar-mass star, each in units of the star's own ratio. Crossing a condensation front moves one element or both out of the gas and into the solids, so the two curves step in opposite directions at water at 2.7 AU, carbon dioxide at 16.0 AU, carbon monoxide at 196.0 AU. Water takes oxygen and no carbon, so beyond it the gas is carbon-rich — 3.20 times the stellar ratio — and the solids are oxygen-rich. Carbon monoxide takes both, in a ratio of one to one, so beyond that front the gas ratio rises again. A giant planet's atmosphere is made mostly of gas it accreted, so measuring its ratio and inverting this staircase gives a formation radius — and the inversion is not unique, because more than one interval returns the same value once the solids a planet also swallowed are allowed for. 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.

 The darkness, itemised. The extragalactic background light as a spectrum: νIᵥ against wavelength, in nanowatts per square metre per steradian, with wavelength logarithmic. The total under the two humps is not read off a plot — it is the integral (c/4π)∫ε(z)(1+z)⁻¹|dt/dz|dz of the Madau–Dickinson star formation history, taking a continuously star-forming population to radiate 10¹⁰ solar luminosities per solar mass per year, and it comes to 37.5 nW m⁻² sr⁻¹. Half of it was emitted beyond z = 0.91, which is the figure's real content: the dark sky is dominated by light released when the universe was 45 per cent of its present age. The split between the two humps is an assumption rather than a result — 50 per cent of the starlight is taken to be absorbed by dust and re-radiated near 140 µm, and the shapes are lognormals of about the observed widths — but the AREA under each is the computed quantity. Measured, the two come to about 24 and 26: the calculation falls 25 per cent short, and whether the missing light is real or a residual foreground is unsettled. Cosmology

The darkness has a number in it

The night sky is not black. It carries about sixty nanowatts per square metre per steradian, and that number is the sum of every photon every star has ever emitted, redshifted and added up over thirteen billion years.

 The sky's darkness, measured as an opacity. Optical depth to electron–positron pair production against gamma-ray energy, for sources at redshifts 0.03, 0.1, 0.3, 1, both axes logarithmic. A gamma ray of energy E is absorbed most readily by background photons near twice the square of the electron rest energy divided by E: at 1 TeV that is 2.37 µm and at 100 GeV it is 0.24 µm, so the energy axis is a wavelength axis for the background light, running backwards — and the background it is evaluated against is the same two-component spectrum the star formation history produced, not a flat number. Above the marked τ = 1 the universe is opaque. The depth is computed in the delta-function approximation, the cross-section replaced by 0.2 of the Thomson value over a bandwidth of order the energy, with the background's comoving density evolving as (1+z)^1.2. That is a factor-of-two calculation and the shape is what it gets right: the horizon closes from z = 0.59 at 100 GeV to z = 0.16 at 1 TeV. The measurement runs the other way. A blazar's spectrum is observed, the absorbed part is the difference between it and the spectrum the source is believed to have emitted, and that difference gives the background — in the near infrared, where no direct measurement can subtract the zodiacal light well enough to compete. 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.

 Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one. Cosmology

Two skies where the paradox comes out right

Olbers argued that every sight line should end on a source and the sky should blaze. In neutrinos and in gravitational waves it does — the walls are at one second and at no time at all — and both of those skies have now been detected.

 The count theory predicts, and the inference it costs. The galaxy stellar mass function: galaxies per cubic megaparsec per dex of stellar mass, both axes logarithmic. Two Schechter components share a characteristic mass of 10^10.66 M☉ — one of slope -0.35 carrying the quenched galaxies at the knee, one of slope -1.47 carrying the star-forming ones below it — and the dashed line is the single component a luminosity function is usually fitted with. Integrated over the range drawn it gives 0.0487 galaxies per cubic megaparsec holding 2.22·10⁸ solar masses of stars, of which 51 per cent sits above the knee. This function is not measured. What is measured is a luminosity function; turning one into the other needs a mass-to-light ratio for every galaxy in the sample, and that ratio is not a constant — it runs by a factor of about five from the bluest galaxies to the reddest, so the conversion moves the red end of the distribution further than the blue end and changes the SHAPE rather than the units. A stellar mass function is a luminosity function plus a stellar population model, and the second half is where its disagreements live. 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.

 A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: IKAROS, 2010 at 1607 g/m² gives β = 9.5e-4; LightSail 2, 2019 at 156 g/m² gives β = 9.8e-3; a 5 µm film with no structure at 7 g/m² gives β = 0.219, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat. Spaceflight

A drive with no rocket equation

Radiation pressure and solar gravity both fall as the inverse square, so their ratio is a constant of the vehicle. A sun-facing sail therefore only rescales the central mass — it has to be tilted to do anything, and the best tilt throws away sixty-two per cent of the thrust.

 The classical law gives every star the same colour. Planck's law and the Rayleigh–Jeans law at 3,000 K, 5,772 K, 10,000 K, both normalised to the 5,772 K Planck peak, on logarithmic axes. The classical law comes from counting standing waves in a cavity — 8πλ⁻⁴ of them per unit volume per unit wavelength — and giving each the kT that equipartition allows. It agrees with Planck's where the modes are crowded and each holds much less than kT, and it runs away where they are not: at 80 nm it exceeds the real spectrum by a factor of 1.1·10¹² while agreeing to within 55.9 per cent at 3000 nm, and the integral under it does not converge at all. That is the ultraviolet catastrophe, and it is the half everybody knows. The quieter half is that in 2ckT/λ⁴ the temperature is an overall factor, so the ratio of the law at two wavelengths is independent of it: the B − V index of a classical star comes out identical at 3,000 K, 5,772 K, 10,000 K — the same -0.968 magnitudes, to the last digit the quadrature carries — while Planck's law spreads the same three stars over 1.47 magnitudes. A classical universe has stars of every brightness and one colour. Colour is a thermometer only because the exponential in the denominator does not cancel, and the quantum of energy that put it there was fitted to this shape before anybody knew what it meant. 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.

 Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the V light reddens the index by 0.093 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5331 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.005 magnitudes against 0.405 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5744 K, which is 28 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap. Starlight

Three shifts larger than the error bar, and two that cancel

An unresolved companion, a reddening and a metallicity each move a colour index by more than any modern photometer's precision. Two of them move it in opposite directions, so the three together can return the right temperature by cancellation rather than by accuracy.

 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.

 At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral. Cosmology

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

 A planet in the middle of a 0.08 M☉ star's zone spends 94 Myr too hot to keep an ocean. How long a planet spends receiving more flux than the runaway-greenhouse limit while its star contracts onto the main sequence, against stellar mass, for planets at the inner edge, in the middle, at the outer edge of the zone the star will have once it settles. The luminosity is the contraction law of a fully convective star, falling as t^(−2/3) from an age of 1 Myr until arrival; the limit is the runaway flux for the star's temperature. A planet at the inner edge is too hot for 313 Myr round a 0.08 M☉ star and 17.8 Myr round a 0.6 M☉ one; a planet in the middle is too hot for 94 Myr round a 0.08 M☉ star and 5.5 Myr round a 0.6 M☉ one; a planet at the outer edge is too hot for 39 Myr round a 0.08 M☉ star and 2.0 Myr round a 0.6 M☉ one. A runaway greenhouse is not a hot climate; it is a state in which the ocean is entirely in the atmosphere as steam, where ultraviolet light splits it and hydrogen escapes. A planet in the eventual zone of the commonest stars in the galaxy begins its life in that state for tens to hundreds of millions of years — a span comparable with the whole assembly of the Earth. The durations are measured from 1 Myr; a rocky planet may take tens of millions of years to finish forming, and one that formed later misses the start of its exposure, while the smallest stars' arrival times are somewhat short in this model, which lengthens the end of it. Exoplanets

Steam before the zone existed

The smallest stars take hundreds of millions of years to contract onto the main sequence, shining at many times the luminosity they will settle at. A planet in the habitable zone such a star will eventually have spends that time with its ocean in the air as steam, while starlight splits the water and the hydrogen leaves — so the zone of the commonest star in the galaxy is a place that had to survive being too hot first.

 A slope of 0.57 at the low-mass end, against the 0.59 and 0.24 two winds give there. Gas-phase oxygen abundance against stellar mass for star-forming galaxies. The solid measured curve is the relation found from electron-temperature abundances in stacked spectra, which rises as a power of mass below a turnover near 10^8.9 solar masses and saturates above it at 12 + log(O/H) = 8.798. The two model curves are a galaxy in equilibrium with its gas supply, whose metallicity is the yield divided by one plus the mass it ejects per unit mass of stars and one plus the dilution by the gas it must keep accreting. Only the ejection changes with mass. A wind driven by the energy of supernovae must lift gas out of a potential whose depth goes as v², so its loading goes as v⁻² and, with v ∝ M^(1/3), as M^(−2/3); a wind driven by momentum goes as v⁻¹ and M^(−1/3). Those set the limiting slopes far below the turnover, 0.67 and 0.33; at log M = 7.5, where dilution still matters, the drawn curves have slopes of 0.59 and 0.24. The measured slope there is 0.57. The dashed measured curve uses a strong-line calibration of the same galaxies' spectra; it sits higher, flattens sooner and has a slope of only 0.40 at log M = 9 — so which wind the data prefer is decided as much by the choice of abundance calibration as by the galaxies. Above the turnover every curve saturates, because a galaxy that ejects almost nothing keeps what it makes and its abundance is the yield, diluted. Galaxies

The metals a galaxy keeps measure what it threw away

Small star-forming galaxies are metal-poor and large ones are not, in a relation tight enough to be a law. Read as an equilibrium between inflow, star formation and wind, its slope says how the wind is driven, its turnover says where galaxies stop losing what they make — and its redshift evolution says galaxies were poorer because they were still being filled.

 Venus is brightest 51 days from inferior conjunction, 38 per cent lit. The brightness of Venus through one synodic period of 584 days, in magnitudes below its brightest, against days from superior conjunction — inferior conjunction at the two ends of the axis. The planet is treated as a matte, Lambert-scattering sphere on a circular orbit of 0.723 AU, so its flux is its phase function divided by the square of its distance from the Earth. The two factors fight: near inferior conjunction the planet is closest but shows only a thin crescent, and near superior conjunction it is fully lit but 1.72 AU away. For this orbit the contest has an interior winner. The brightest moment is 51 days either side of inferior conjunction, at an elongation of 44.6° from the Sun, with 38 per cent of the disc lit and the planet 0.539 AU away; greatest elongation, at 46.3°, comes 71 days from inferior conjunction, after the brightness peak on the way out from inferior conjunction. At superior conjunction the planet is 0.82 magnitudes fainter than its best. The marked point is the observed greatest brilliancy, about 36 days from inferior conjunction at an elongation near 39° — closer to conjunction and to a thinner crescent than any matte sphere on this orbit can be brightest at. A surface that sends more light forward, towards large phase angles, would move the peak exactly that way. Very near either conjunction the planet is lost in the Sun's glare, and the curve there describes light nobody sees. The observed sky

Brightest as a crescent, and not as a disc

Venus is fully lit when it is furthest away and nearest when it is barely lit, and it is brightest in between, as a crescent weeks from inferior conjunction. A matte planet only has such a peak if its orbit is wider than about 0.46 of the Earth's — Mercury's is not, and Mercury is brightest full — and Venus's real peak sits closer to conjunction than any matte sphere allows, which is its clouds throwing light forward.

 A bright star wants a wide aperture and a faint one wants 0.68 of the seeing. Signal-to-noise of simple aperture photometry against the aperture radius, in units of the seeing's full width at half maximum (1″), each divided by what optimal pixel weighting achieves for the same star, for stars of V = 12, 17, 20, 23 observed for 60 s through a 1 m telescope under a sky of 21 mag/arcsec². A small aperture loses starlight; a large one admits sky, and the balance depends on which dominates. For a bright star its own photons are most of the noise, so a wider aperture keeps gaining light almost for free and the best radius is large — 1.63 FWHM at V = 12, reaching 100.0 per cent of the optimum. For a star fainter than its sky the best radius shrinks to 0.680 FWHM and the best aperture reaches only 90.5 per cent of what weighting each pixel by its share of starlight divided by its variance achieves. That residual is exact in the background-limited limit: the best aperture captures 71.5 per cent of the light and 0.902 of the optimal signal-to-noise, so optimal weighting is worth 11 per cent in signal-to-noise, or 23 per cent in exposure time, and no more. The image is taken to be Gaussian; a real point-spread function has broader wings, which makes a fixed aperture a little worse and the optimal weights harder to know. 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.

 Ten comparison stars as bright as the target cost 5 per cent in precision; ten 2 magnitudes fainter cost 23. The precision of a V = 12 target measured relative to an ensemble of comparison stars on the same 60-second frames, against the number of comparison stars, on a logarithmic precision axis, for comparisons 1 mag brighter, as bright as the target, 1 mag fainter, 2 mag fainter. A change in the atmosphere's transparency of 2.0 per cent — which would put the target's raw brightness out by 20.0 mmag — multiplies every star by the same factor and vanishes from the ratio. What is left is the target's own noise, 0.89 mmag, plus the ensemble's, which falls as the inverse square root of the number of stars in it. With comparisons as bright as the target the result is σ√(1 + 1/N): one comparison costs 41 per cent, ten cost 5. Fainter comparisons are noisier and need many more to reach the same point; brighter ones help, but the target's own noise is a floor the ensemble can only approach. Scintillation is treated as independent from star to star, which is right for stars more than a few arcseconds apart on a large telescope and makes it part of the noise that does not cancel. The figure also cannot show the defining weakness: every comparison star is assumed constant, and a variable among them injects its variability into every measurement made against the ensemble. Starlight

The comparison stars are part of the measurement

Measuring a star against others on the same frame cancels everything the atmosphere and the instrument do to all of them at once — a two per cent change in transparency vanishes completely. What does not vanish is the comparison stars' own noise, which the target inherits, and the variability of any comparison that is not constant, which the target reports as its own.

 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.

 The position closure discards, and what it costs to buy back. The astrometric error left after phase referencing, against how far away the calibrator is, for three switching cadences, on a 8000-kilometre baseline at 1.3 mm. Referencing gives up the immunity closure quantities have: instead of forming a combination the atmosphere cannot enter, it measures the atmosphere on a nearby source and subtracts it, which recovers the absolute position and leaves behind whatever differs between the two lines of sight. That difference is governed by the Kolmogorov structure function, which rises as the five-thirds power of a separation, so the phase residual and therefore the position error rise as the five-sixths power of the separation, which the drawn spatial term reproduces exactly. Below about 1.7 degrees at the fastest cadence the curves flatten, because there the atmosphere's change between one visit to the calibrator and the next is the larger of the two differences and the separation has stopped mattering. The consequence is a premium on finding a close calibrator: at one degree the error is 2 microarcseconds and at eight it is 5, and the sky is not dense in compact bright sources. The three curves are the other half of the trade — a faster cycle freezes the atmosphere better and spends more of the observation looking at the calibrator, so the optimum is where the two losses meet rather than as fast as the hardware allows. 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.

 The sign of one gradient decides which way the disc is shaded. The emergent intensity across a stellar disc, against the fractional radius, for four gradients of the source function with depth. Every curve is the Eddington–Barbier relation evaluated at the appropriate μ and checked against a numerical integration of the transfer equation. The contrast between centre and limb is the gradient, and its sign is the gradient's sign. A source function rising inward — the ordinary case, since temperature rises inward and the source function follows it — darkens the limb: a sight line at the edge leaves from higher up, where the gas is cooler. A flat source function produces a uniformly bright disc whatever the geometry. And a source function that falls inward brightens the limb, because now the shallower ray is reading a hotter layer. That last case is not hypothetical: above a star's photosphere the temperature stops falling and begins to rise, so at wavelengths that see those layers — millimetre continuum, the cores of strong lines, the ultraviolet — the Sun's limb is brighter than its centre. Limb darkening is therefore not a property of stars but of a gradient, and the same atmosphere shows both signs at different wavelengths. Starlight

A limb brighter than the middle

Limb darkening is not a property of stars. It is a property of a gradient, and the same relation that produces it produces the opposite whenever the source function falls inward — which it does above every stellar photosphere. The Sun's limb is darker in the visible and brighter at a millimetre, and both are the same equation.

 Two ways to stop, and the same curve shifted. The fraction of galaxies that have stopped forming stars, against their stellar mass, at four local densities. Two things are happening at once. Mass quenching turns the curve upward above about 10^10.6 solar masses, and it does so at the same mass whatever the environment — the curves rise in step. Environment quenching lifts the whole curve, and it does so by the same fraction of what is left at every mass, which is why the four curves look like one curve with its floor raised rather than like four different shapes. At 10⁹·⁵ solar masses the red fraction runs from 13 per cent in the field to 97 in a cluster core, and almost all of that difference is environmental; at 10¹¹·⁵ almost everything is red everywhere and the environment has nothing left to do. The two mechanisms are separable, which is the census's central quantitative result and the thing any account of either one has to explain. Galaxies

Two ways to stop that do not interact

A galaxy's chance of having stopped forming stars depends on its mass and on where it is, and the census says the two dependences multiply. The efficiency with which environment quenches a galaxy is the same number at every mass, across two and a half decades of it — which is surprising, because both mechanisms work on the same gas.

 Three colours watching one galaxy stop. One star-formation history — steady, then shut off with an e-folding time of 0.4 billion years — followed in three colour indices, each plotted as a fraction of the gap its own sequences actually span. The three curves are the same galaxy. Each band's luminosity is the history convolved with that band's own decline after a burst: NUV at 0.1 Gyr, u at 0.35 Gyr, g at 0.9 Gyr, r at 3 Gyr, because the ultraviolet comes from stars that live a hundred million years and the red from stars that live ten billion. In the first three hundred million years the ultraviolet index reddens at 1.56 magnitudes per billion years and the optical one at 0.10 — a factor of 15 — so a galaxy that has just stopped is already conspicuous in one and indistinguishable from a star-forming one in the other. That is what an ultraviolet band buys: not a better measurement of the same thing, but sensitivity to a phase the optical colours arrive too late to see. Galaxies

A crossing time that is the filter's

The emptiness of the gap between the two galaxy populations was read as a timescale, and the reading used one colour. Each passband remembers star formation for a different length of time, so the same galaxies give a crossing time of one billion years in the ultraviolet and three in the optical — and the disagreement between two filters is the measurement the single filter could not make.

 Fine along one axis, coarse across the other two. The smallest scales a forest survey can measure, against how many quasars a square degree it has sightlines to, at redshift 2.33. Along a sightline the sampling is set by the spectrograph: a 69 km/s pixel is 0.29 comoving megaparsecs at this redshift, so the radial wavenumber reaches 10.8 Mpc⁻¹ and it does not depend on the survey's size at all. Across the sky it is set by how densely the quasars sit: at 30 a square degree the mean separation is 18 megaparsecs and the transverse wavenumber reaches 0.17. The ratio is 63, and improving it means finding more quasars, which are a finite population on the sky. That anisotropy is the inverse of a galaxy survey's, where the transverse direction is sampled by a wide sky and the radial one is limited by a redshift shell and contaminated by peculiar velocities — so the forest measures the expansion rate at its redshift better than it measures the distance to it, which is the reverse of every galaxy survey. 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.

 Two references, failing in opposite places. The faintest companion each technique can detect, against separation in resolution elements, in units of the speckle noise a perfectly matched reference would leave. Angular differential imaging builds its reference from the target's own exposures, so the reference matches the instrument's state exactly and the residual speckle is small — and the companion is in every frame of it, so close in the companion subtracts most of itself and the achievable contrast collapses. Reference-star differential imaging builds the reference from archival exposures of other stars. The companion is in none of them, so the throughput is one at every separation and the curve is flat; what is lost is the match, because two stars observed at different times through different air with a differently flexed telescope have different speckles. The two cross, and where they cross depends on how large the library is: 1.1 resolution elements at 20 frames, 1.4 resolution elements at 200 frames, 1.8 resolution elements at 2000 frames. A library of thousands of archival exposures is therefore worth building, and the thing it buys is the inner working angle rather than the depth. 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.

 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.

 One eccentric planet, drawn as two circular ones. Top: the velocity of a star with one companion on an orbit of eccentricity 0.25 (argument of periastron 0.9 radians), in units of its semi-amplitude K, over two orbits — and on top of it the sum of two circular orbits, one at the same period and one at half of it, fitted by taking the first two terms of the exact curve's Fourier series. The second orbit's amplitude comes out at 0.246 of the first, which is the eccentricity to first order. Bottom: the difference between the two, on a scale 12 times magnified; its largest excursion is 0.0889 K and its rms 0.0473 K, second order in the eccentricity. For a planet with K = 30 m/s that is 1.42 m/s rms — below the noise of most instruments — so an eccentric single planet and a pair of planets in a 2:1 period ratio are the same measurement unless the data are good enough to see a term of order e². 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.

 A sky wrong by a thousandth, and the disc that ends or does not. The surface-brightness profile of an exponential disc with a central surface brightness of 21.65 magnitudes per square arcsecond, against radius in units of its scale length, drawn straight — a pure exponential is a straight line in magnitudes — and then as it would be measured after subtracting a sky of 21.5 magnitudes per square arcsecond wrong by +0.1%, −0.1%, +0.3%, −0.3%. A sky left in bends the profile up into a shallower outer part, the shape called an anti-truncation; a sky taken out too hard bends it down into a break and then drops it out of existence where the residual goes negative. The +0.1% error departs by 0.2 magnitudes at 5.2 scale lengths, 27.3 magnitudes per square arcsecond; The −0.1% error departs by 0.2 magnitudes at 5.0 scale lengths, 27.1 magnitudes per square arcsecond; The +0.3% error departs by 0.2 magnitudes at 4.1 scale lengths, 26.1 magnitudes per square arcsecond; The −0.3% error departs by 0.2 magnitudes at 3.9 scale lengths, 25.9 magnitudes per square arcsecond. Real discs are observed to break both ways, at 24 to 27 magnitudes; every one of those breaks lies in the range where a sky error of a few tenths of a per cent would make one. 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.

 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.

 Rays that follow the curve of the Earth. Rays of light leaving an observer five metres above a cold surface at elevations between −0.2° and +0.45°, traced through an atmosphere whose lowest 150 metres warm upwards by 0.15 K per metre, drawn as height above the ground against distance along it — so the Earth's curvature has been removed and a ray that follows the curve runs flat. Rays that start downwards or only slightly upwards are bent back by the inversion before they can leave it, glance off the surface, and are guided along the layer in shallow hops: 5 of the 8 traced remain inside it for the whole 300 km. Only the steeper rays escape upwards. Traced backwards, the trapped rays are lines of sight: an observer inside the duct sees objects hundreds of kilometres away that lie far below the geometric horizon, lifted into view. The observed sky

An atmosphere that bends light round the Earth

A horizontal ray of light curves towards denser air, and in an ordinary atmosphere it curves at a fifth of the rate the ground falls away. Put eleven degrees of warmer air above every hundred metres of cold air over sea ice and the ray curves as fast as the Earth does. It follows the curve, and a Sun five degrees below the horizon rises two weeks early.

 Where every star appears to an observer moving fast. The apparent angle of a star from the direction of motion, against its angle in the rest frame of the stars, for observers moving at 0.3c, 0.6c, 0.9c. The diagonal is a stationary observer. Every star except the two on the line of motion is displaced forward, and the displacement grows with speed until, at 0.9c, a star at right angles to the motion appears 26° from the direction of travel and one directly behind still appears behind. A star at a true angle of 90° is seen at the angle whose cosine is β — 72.5°, 53.1°, 25.8°. At the Earth's orbital speed, ten thousandths of c, the curve is indistinguishable from the diagonal, and the whole of annual aberration is its first-order departure from it. 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.

 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.

 A survival probability with a step in the middle of the spectrum. The probability that an electron neutrino made in the core of the Sun is still an electron neutrino at the Earth, against its energy, from the resonant conversion it undergoes in the Sun's own electrons: mixing angle sin²θ₁₂ = 0.307, splitting Δm² = 7.4·10⁻⁵ eV², and an electron density at the point of production of 90 moles per cubic centimetre. At low energy the matter potential is small beside the vacuum term and the probability is the vacuum average, 0.54 at 0.3 MeV (the flat limit is 0.55). At high energy the matter term dominates, the neutrino is born almost entirely in the heavier mass state, and the probability falls to 0.32 at 10 MeV, close to sin²θ₁₂. The two terms are equal at 2.1 MeV, which falls in the gap between the ⁷Be line and the bulk of the ⁸B spectrum. The points are the measured survival probabilities for each source separately — pp, ⁷Be, pep and ⁸B from Borexino, and ⁸B from SNO — and every one of them lies within about two standard deviations of the curve: low-energy neutrinos survive about half the time, high-energy ones about a third. 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.

 A push that depends on how fast the wind is already going. The force multiplier — the radiative push on a line-driven wind as a multiple of the push on its free electrons — against t, the electron-scattering optical depth across one thermal Doppler width, which is small where the wind accelerates steeply. The solid curve is an ensemble of spectral lines whose strengths follow a power law, the number of lines stronger than a given strength falling as that strength to the power 1 − α, with α = 0.6. If every line were optically thin the push would be the flat line at 2000, independent of the flow; if every line were thick it would fall as 1/t, set only by the velocity gradient. The ensemble lies between: over most of the range it falls as t to the power −0.59, because as t rises lines cross from thin to thick one after another. That single exponent is the whole of the theory — it fixes the terminal speed, the mass-loss rate and how both scale with the star — and it is a property of the atomic physics of the ions in the wind, not of the star. 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.

 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.

 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.

 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.

 How sharp a single short exposure is, for three sizes of telescope. The distribution of the Strehl ratio — the peak brightness of a star's image as a fraction of a perfect telescope's — for 600 independent short exposures through simulated Kolmogorov turbulence, with image motion removed, for telescopes 4, 7, 10 times the Fried parameter r₀ across. The screens' mean phase variance matches the analytic value for tilt-removed turbulence to within 2 per cent. At D/r₀ = 4 the exposures spread over the whole range and 31 per cent are better than 0.37 — a phase error under one radian, a nearly diffraction-limited image. At 7 the distribution has slid to low Strehl ratios and none of the 600 are that good; at 10, none. The best frames at the larger ratios are still several times sharper than the average, which is what practical selection keeps. The atmosphere is not uniformly bad: every so often, for a few milliseconds, the turbulence over a small enough aperture happens to be nearly flat, and those instants are a diffraction-limited telescope for free. The observed sky

A sharp image kept by throwing the rest away

The atmosphere does not blur every instant equally. Now and then, for a few milliseconds, the air over a small telescope is almost flat, and a short exposure taken then is as sharp as the mirror allows. How often that happens falls as the exponential of the square of the telescope's size — which is why the method works on a two-metre mirror in the red and nowhere else.

 A Cepheid's velocity, radius and angular size through two pulsation cycles. Three curves for a model Cepheid with δ Cephei's period of 5.37 days and a mean radius of 44 solar radii, at 272 parsecs, through two cycles. Top: the radial velocity of its spectral lines, in km/s, oscillating about the star's own velocity of -16.8 km/s — negative is towards the observer, so the surface is expanding when the curve is lowest. Middle: the change in radius, from integrating that velocity over time and multiplying by the projection factor p = 1.3: a swing of 4.9 solar radii, 11 per cent of the star. Bottom: the angular diameter the same star presents, in milliarcseconds, swinging by 168 microarcseconds about 1.505. The middle and bottom curves have the same shape because they are the same motion — one in kilometres, one in angle — and their ratio is the distance. 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.

 A colour that predicts an angle. The surface-brightness parameter Fᵥ — a combination of the V magnitude and the angular diameter that depends only on how bright each unit of the star's surface is — against the colour V − K, the ratio of the star's light at 0.55 and 2.2 microns. The line is the relation log θ = 0.5175 + 0.2753 (V − K) − 0.2 V, fitted to interferometric diameters of dwarfs and subgiants: Fᵥ = 3.962 − 0.1376 (V − K). The dots are 40 calibrating stars drawn about it with a 1 per cent scatter in angular diameter, the precision of a good interferometric measurement; refitting them returns a slope of −0.1374 and a scatter of 0.0023 in Fᵥ, which is 1.1 per cent in angle. The dashed curve is what blackbodies of 3,500 to 9,000 K would give, pinned to the Sun: it departs from the stars' line by up to 0.035 over the calibrated range, because real atmospheres have molecular bands and limb darkening a blackbody lacks — which is why the relation is calibrated on measured stars rather than derived. Given a star's V and V − K, the line returns its angular diameter without resolving it. 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.

 The matter power spectrum with galaxies' feedback taken out. The ratio of the matter power spectrum with baryonic feedback to the spectrum without it, against wavenumber in h per megaparsec on a logarithmic scale, for three strengths of feedback that span the range the large hydrodynamic simulations disagree over. Gas blown out of haloes by supernovae and active nuclei removes mass from scales below a few megaparsecs, so the ratio falls below one from about 0.5 h per megaparsec and is deepest near 4 h per megaparsec, at 9, 17 and 26 per cent; at still smaller scales gas that has cooled into galaxy centres adds mass back. The marks show the wavenumber that dominates a lensing measurement at multipoles 100, 1000, 5000, taken where the lensing kernel for sources at redshift 1 peaks: 0.09, 0.87, 4.35 h per megaparsec. The shape is a parametric stand-in placed where simulations put the effect, not a simulation. 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.

 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.

 Hayashi lines: a luminosity that moves and a temperature that does not. The effective temperature a fully convective star must have, against its luminosity, for 0.5, 1, 2 solar masses, drawn as a Hertzsprung–Russell diagram with temperature falling to the right. The temperature follows from matching the star's convective adiabat to a photosphere whose opacity, from the negative hydrogen ion, rises as the ninth power of temperature. Over a factor of thirty thousand in luminosity each line moves by only 363 K, 398 K, 436 K respectively: the lines are nearly vertical, at 3650 K, 4000 K, 4383 K for one solar luminosity. A contracting young star, or a swelling giant, moves up or down its line almost without changing colour. To the right of the lines no star in hydrostatic equilibrium can sit. Starlight

A wall built by a steep opacity

A star that is convective all the way to its surface cannot choose its surface temperature. The interior fixes a relation between pressure and temperature, the photosphere fixes another through its opacity, and the negative hydrogen ion's opacity rises so steeply with temperature that the two meet at nearly the same temperature whatever the star's size. Young stars contracting and old stars swelling slide up and down a line in the diagram almost without changing colour — and nothing in equilibrium can sit to its right.

 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.

 Brightness temperature across the infrared, for three profiles. The brightness temperature of the planet's emission against wavelength from 1 to 12 microns, for the three temperature profiles, with water and carbon monoxide bands in the opacity. In a band the opacity is high, so the light comes from higher, lower-pressure layers; in the windows between bands it comes from deeper. Over a profile falling outward the bands are cooler than the windows: at 6.3 microns 1457 K against 1570 K at 3.6. Over the inverted profile they are hotter: 1851 against 1538. Over the isothermal profile every wavelength has the same brightness temperature and the spectrum is a blackbody, whatever molecules are present. 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.

 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.

 The histogram of a confused map, for three faint-end slopes. The distribution of pixel values in a simulated map in which every beam holds 0.2 sources brighter than the break in the counts, for counts identical above the break (falling as S^−3.8) and falling as S^−b below it with b = 1.3, 1.5, 1.7. Pixel values are in units of the break flux per beam, measured from the map's own mean as a real map's are and no instrumental noise has been added. No single source below the break can be picked out of any of these maps, and yet the histograms differ: b = 1.3 gives a peak 0.15 below the mean, a width of 0.566 and a skewness of 2.5; b = 1.5 gives a peak 0.15 below the mean, a width of 0.629 and a skewness of 1.6; b = 1.7 gives a peak 0.13 below the mean, a width of 0.617 and a skewness of 1.5. Shallower faint counts leave a map dominated by the rarer bright sources, with a sharp peak and a long positive tail; steeper ones fill each beam with many faint sources and the histogram moves towards a Gaussian. The shape of the histogram is therefore a measurement of the counts below the level at which anything can be counted. 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.

 Where the velocity information in a spectrum lies. Above: a 600 km/s stretch of two synthetic spectra with the same continuum of 10000 photons per pixel at a resolving power of 100,000 — a cool, slowly rotating star, with 12 lines per nanometre, and a hot star rotating at 25 km/s, with 3 lines per nanometre. Below: the velocity information each pixel carries, (dA/dv)²/A, which is zero in the continuum and at line centres and largest on the steep flanks of deep, narrow lines. Summed over 100 nanometres of spectrum, the first star measures its velocity to 1.61 m/s and the second to 39.17 m/s — 24 times worse from the same number of photons. The photons that measure a velocity are the ones on the sides of lines; the continuum, however bright, contributes nothing. 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.

 A lucky frame stays lucky further from the star when the high air was quiet. The mean Strehl ratio of short exposures of a target at a separation from an on-axis reference star, against the shift of the high turbulent layer between the two lines of sight in units of the aperture, s = θh/D — 0.50 of an aperture is 32 arcseconds for a 2.5-metre telescope and a layer at 8 km. The atmosphere has 50 per cent of its turbulence at the ground and the rest high, with D/r₀ = 7, and 1600 simulated frames. Frames are selected by the reference star's sharpness. When the two layers' strengths vary from frame to frame (log-normal, width 0.6), the best 2 per cent start at a mean Strehl of 0.523 and keep 0.339 at the largest separation; when the strengths are steady, the best frames start at 0.321 and fall to 0.105. In the first case the Strehl never falls to six-tenths of its on-axis value within the range drawn; in the second it does so at s = 0.208, 13 arcseconds. Selecting frames by one star selects, preferentially, the moments when the high layer was weak — and a weak high layer is weak for every star in the field. The observed sky

Luck that reaches as far as the quiet air

Lucky imaging keeps the rare short exposures in which a reference star happens to be sharp, and a sharp star says the air was flat along one line of sight. A target a few arcseconds away looks through different air. It turns out to be sharper in those frames than the geometry of a single turbulent layer allows, because choosing a lucky instant chooses, preferentially, an instant when the high turbulence was weak — and weakness, unlike the pattern of the turbulence, is shared across the field.

 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.

 The zenith sky after sunset, with light scattered twice. The brightness of the zenith sky at 550 nm in magnitudes per square arcsecond, brighter upward, against the Sun's depression: light scattered once (dashed), light scattered twice (dash-dot), and their sum (solid), in a spherical atmosphere with US Standard densities, Rayleigh scattering and 300 Dobson units of ozone. The dashed horizontal line is the natural night sky, 21.9 magnitudes per square arcsecond. Once-scattered light reaches it at 9.2°; the sum, dominated beyond about seven degrees by light scattered twice, reaches it at 13.2°, and it falls to a hundredth of the night sky — five magnitudes fainter, about the smallest excess a careful measurement of the sky can detect — at 17.5°, where once-scattered light alone would have reached that level at 11.9°. The second order falls more slowly than the first because it can reach the zenith by two grazing hops — sunlight to the lit air low towards the Sun, and that air to the shadowed air overhead — each of which needs only part of the depression a single hop would. Orders beyond the second are left out, so the solid curve is a lower bound on the sky. The observed sky

The night is late because the light bounces twice

Light scattered once can reach the zenith only while the Earth's shadow over the observer is still low in the air, and by that reckoning the sky should reach the darkness of night about nine degrees after sunset. It does not until eighteen. The difference is light scattered twice — first by the sunlit air hundreds of kilometres towards the sunset, then again by the shadowed air overhead — and adding it to the calculation moves the end of twilight almost exactly to where observers have always put it.

 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.

 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.

 Two models through the same points, one with a planet. 60 simulated velocities over 200 days of a star rotating every 25 days, with spot activity drawn from a quasi-periodic process of 3 m/s amplitude and a 40-day decay, white noise of 1 m/s, and a planet of 2.5 m/s semi-amplitude on a 27.0-day orbit — 1.08 times the rotation period. Two models are fitted, each with its activity amplitude and decay time chosen by maximum likelihood: activity plus the planet (solid), whose planet comes out at 3.3 m/s, and activity alone (dashed), which has no planet in it. Both pass through the points; the likelihoods differ by 3.6 in the logarithm, a ratio of 34.9, which is not decisive. Near the rotation period a flexible model of the star can carry the planet's signal by itself. 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.

 The red velocity against the blue one. Simulated velocities of two stars, each measured simultaneously in a blue channel at 0.45 micrometres (across) and a red one at 0.8 (up), with 0.6 m/s of noise in each. One star (filled) has a 4 m/s planet on a 9.3-day orbit: its points lie along the diagonal, slope 0.97, because a Doppler shift is the same velocity in every colour. The other (open) has no planet and a spot 900 K cooler than its 5100 K photosphere, rotating every 12 days and evolving: its points lie along a shallower line, slope 0.64, which is the ratio of the spot's contrasts in the two channels. Neither star's blue velocities alone say which it is. The flux-deficit spot model is the simplest case; a spot whose signal is mostly suppressed blueshift lies closer to the diagonal. 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.

 The black holes that can be weighed are the heavy ones. 4000 synthetic galaxies spread evenly through a sphere of 120 megaparsecs, with black-hole masses scattered by 0.45 dex about an M–σ relation of slope 4.24 (faint points), and the 511 whose spheres of influence exceed 0.1 arcseconds (filled), which are the ones a dynamical measurement can resolve. At a given dispersion a heavier black hole has a larger sphere and is resolvable further away, in a larger volume, so the resolved sample is dominated by holes above the relation. Fitted to the resolved galaxies alone, the relation comes out with normalisation 8.74 and slope 2.95 (dashed), against 8.11 and 4.20 for all of them (solid) — 0.63 dex too high at 200 km/s, and more at low dispersions. 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.

 The Sun bends light from every direction in the sky. The gravitational deflection of starlight by the Sun, as seen from the Earth, against the angle between the star and the Sun, as general relativity predicts it (solid) and as a Newtonian particle of light would be bent (dashed), exactly half. At the Sun's limb the deflection is 1.75 arcseconds, the value measured at the eclipse of 1919. It falls only as the cotangent of half the angle, so it is 4.07 milliarcseconds at right angles to the Sun and 9.8 at 45°, the closest a space astrometry mission looks; it reaches zero only directly opposite the Sun. Every star in a modern catalogue has been displaced by an amount far larger than the precision of its position, and the shading marks the angles such a mission actually observes. 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.

 Where on the Earth a month begins on which evening. The line west of which the new crescent can be seen with the naked eye at local sunset, for a new Moon at 0 h universal time near the spring equinox, on the ecliptic. On the first evening (blue) the crescent is seen only west of the curve, because sunset comes later in universal time the further west an observer is and the Moon is that much older; on the second evening it is seen everywhere except beyond the green line, deep in the hemisphere where the evening ecliptic lies shallow. The first-evening line bulges furthest east at about twenty degrees north, where the spring ecliptic stands steepest, and sweeps far west towards the south, where the same season is autumn. An observed calendar that begins each month at first sight begins it on the first evening west of the line and on the second evening east of it — on two different dates within one religious community spread across the map. The observed sky

A month that begins when someone sees it

A purely lunar calendar that starts each month at the first sighting of the crescent has abandoned the Sun to keep the Moon, and has handed the start of the month to the weather, the season and the observer's latitude. The line on the Earth west of which the crescent can first be seen sweeps across the globe, and two cities a few time zones apart begin the same month on different evenings in a measurable share of months.

 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 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.

 The end of the red giant branch, found as an edge. The luminosity function of a synthetic galaxy halo — the number of red giants against apparent magnitude in the I band, smoothed — and its derivative, which is what the edge detector computes, searched over the magnitudes brighter than the branch's crowded lower part, as every application of the method does. 1582 stars are drawn: red giants rising in number towards fainter magnitudes as ten to the 0.3 times the magnitude, ending at the tip at magnitude 24, with a sixth as many asymptotic giants thinning out above it, all measured with errors of 0.08 magnitudes. The derivative peaks sharply where the counts jump, here at 24.09 — 0.09 from the true tip. The red giant branch ends where helium ignites in a degenerate core of almost fixed mass, so the tip's absolute magnitude is close to −4.05 for old, metal-poor stars, and the edge's apparent magnitude is a distance. 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.

 Thin winds cannot cool their own shocks. The ratio of the cooling time of gas behind a shock to the time the wind takes to carry it away, against the wind's mass-loss rate, for shocks of 300, 500, 800 km/s at three stellar radii from a star of 10 solar radii. The ratio goes as the fourth power of the shock speed and inversely as the density, and so as one over the mass-loss rate. Above the dashed line the shocked gas leaves the star before it can radiate its heat and stays at millions of kelvin. The crossing falls at 2·10⁻⁹, 10⁻⁸, 9·10⁻⁸ solar masses a year. Winds thinner than that stay hot behind their shocks: the measured weak winds, 10⁻¹⁰ to 10⁻⁸, are hot behind any shock faster than about 500 km/s and the thinnest of them behind any shock at all, while the dense winds of the most luminous stars, above 10⁻⁷, cool their shocks except the fastest. The weak-wind problem begins about where the shocks stop cooling. 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.

 A cluster's mass written as a dipole on the microwave background. The change in microwave-background temperature caused by lensing behind a cluster of 3·10¹⁴ solar masses at redshift 0.5, where the unlensed background has a uniform gradient of 13 microkelvin per arcminute running left to right. Contours at ±0.5, ±1, ±1.5, ±2, ±2.5 μK: blue where the sky is warmer than it would have been, orange where it is cooler. The cluster bends light towards itself, so each point on the sky shows the temperature of a point a little closer to the cluster's centre: on the side towards which the gradient rises that point is colder, and the sky there is cooler than it would have been; on the downhill side it is warmer. The pattern is a dipole aligned with the gradient, peaking at 2.9 μK, and it carries the cluster's mass profile in its radial shape. Its orientation is set entirely by the background, not by the cluster, which is what lets it be told apart from the cluster's own emission. 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.

 A star brighter at its poles, and three paths a planet might take across it. A star rotating at 0.6 of its critical angular velocity, seen with its spin axis tipped 30° towards the observer from the plane of the sky, drawn with that axis vertical. Its outline is the Roche surface, 1.05 times as wide as it is tall in projection. The contours are the surface brightness, from von Zeipel's law with exponent 0.2 and a linear limb darkening: gravity at the equator is weaker, so the equator is cooler and its intensity there is 81 per cent of the pole's before limb darkening. Three transit chords at the same impact parameter, 0.3 polar radii, cross at projected spin–orbit angles of 0°, 45°, 90°. Only the first is symmetric about the star's axis; the others run from bright regions to dark ones or the reverse, and the light curve records which. 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.

 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.

 The width of an emission core against a star's luminosity. Absolute visual magnitude against the logarithm of the width of the calcium K emission core in km/s, for the Wilson–Bappu relation Mᵥ = 27.59 − 14.94 log W₀ (line), with 120 illustrative stars scattered about it by 0.5 magnitudes, the relation's typical scatter, and the Sun marked at about 33 km/s and Mᵥ = 4.83. Over a range of fifteen magnitudes in luminosity — from dwarfs fainter than the Sun to supergiants a million times brighter — the logarithm of one width predicts the magnitude to half a magnitude, whatever the star's temperature. A factor of two in width is four and a half magnitudes in brightness, sixty times in luminosity. 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.

 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.

 The companion's mass against the visible star's, for f = 6.08 M☉: never below 6.08, and 7.3 M☉ for a 0.7 M☉ star seen edge-on. The mass of the unseen companion against the mass of the visible star, both in solar masses, for a mass function of 6.08 M☉, at inclinations of 90°, 70°, 55°. Every curve starts at or above 6.08 when the visible star weighs nothing and rises as it gets heavier, because a heavier visible star moves less for the same companion. For a visible star of 0.7 solar masses — V404 Cygni's donor is a cool subgiant of about that mass — the companion is 7.3 M☉ at 90°, 8.6 M☉ at 70°, 12.4 M☉ at 55°. The shaded band is the range of neutron-star masses, up to about 2.3 M☉; no curve comes near it. A mass function above that limit is a proof that the companion is too heavy to be a neutron star, without knowing the inclination or the visible star's mass at all. 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.

 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.

 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.

 Three sails against the two routes that have flown. Inclination reached against years since launch. The Venus staircase and the Jupiter step are the arithmetic of the launch-window figures — repeated Venus flybys at a v∞ of 15 km/s reaching 23°, and a single Jupiter flyby after 1.2 years reaching 90° on a heavy launch. The sails, of lightness number 0.020, 0.040, 0.080, spiral in to 0.48 AU — flat lines, 4.6, 2.3, 1.1 years — and then climb at a constant rate until they are polar. They pass the Venus route after 9.0, 4.5, 2.2 years; only the lightest catches Jupiter inside the decade, and none ever stops climbing. Spaceflight

A climb out of the ecliptic paid for in orbits

A solar sail can tilt its orbit without propellant by leaning against sunlight and changing which way it leans twice an orbit. Each orbit then adds the same few degrees whatever its size, so the climb runs faster the closer to the Sun the sail goes. Nothing in the orbit fixes how close that can be. The sail film's temperature does.

 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.

 How much too small a planet looks when a second star shares its light. The factor by which a planet's true radius exceeds the radius inferred from its transit, against the mass of an unresolved companion star in units of the Sun's, for a planet transiting a Sun-like primary and for one transiting the companion, with the radius computed in both cases from the diluted depth and the primary's radius. Logarithmic vertical axis. A planet on the primary is diluted by the companion's light alone, and the factor is √(1 + F₂/F₁), never more than √2 = 1.41, reached for an equal twin; for a companion of half a solar mass, whose 3943 K surface gives 4.5 per cent of the primary's light in a Kepler-like band, it is 1.022. A planet on the companion is diluted by the primary's much larger light and measured against the wrong star's radius, and the factor is (R₂/R₁)√(1 + F₁/F₂): 2.6 for the half-solar-mass companion and 4.3 for one of 0.3 solar masses. The two curves meet at √2 for twins. The stars are a main-sequence sketch — power-law radii, a smoothed luminosity law and blackbody spectra — good to tens of per cent in flux ratio. 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.

 Where one dispersion and a resolved profile leave the mass and the orbits. The plane of the mass-sheet parameter λ and the orbital anisotropy β. Light shading: the region within Δχ² = 2.3 of a mock lens with λ = 1 and β = 0.1, given one aperture dispersion measured to five per cent — a long diagonal valley, because a larger λ and more tangential orbits trade against each other. Dark shading: the same, given the dispersion in six rings between a quarter and two Einstein radii, each to three per cent. Marginalised over β with a flat prior, and with λ allowed between 0.8 and 1.15, the aperture measurement gives λ = 0.990 ± 0.091 and the resolved profile λ = 1.000 ± 0.040 — the Hubble constant from this lens to 9 and 4 per cent. 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.

 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.

 Where the photons stop mattering and the star starts. Noise in a 13-hour window — the length of an Earth-like planet's transit across a Sun-like star — against the star's brightness, for a space telescope that reaches 17 parts per million in 6.5 hours on a twelfth-magnitude star. Photon noise falls by a factor of ten for every five magnitudes of brightness; the star's granulation, averaged over the window, stays at 5.1 parts per million whatever the brightness. The two are equal at magnitude 10.1; brighter than that, the star is the floor. The dashed line is an Earth's transit depth across the Sun, 84 parts per million: a single transit on a twelfth-magnitude star is detected at 6.4 standard deviations, and on the brightest stars at no better than 17, set by the star alone — and supergranulation, spots and faculae add more on these timescales. 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.

 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.

 A compact galaxy with a faint envelope, seen at redshift two. The surface-brightness profile of a model galaxy at redshift 2, in magnitudes per square arcsecond against radius in kiloparsecs on a logarithmic axis (solid): a compact de Vaucouleurs core of 1 kpc half-light radius and an exponential envelope of 10 kpc holding 30 per cent of the light, dimmed by 4.77 magnitudes by the expansion. The dotted curve is the core alone. Horizontal lines mark survey limits of 27 and 30 magnitudes per square arcsecond. A single Sérsic profile fitted to what is brighter than each limit (dashed) gives a half-light radius of 1.3 kpc and index 4.6 above 27, and a half-light radius of 1.5 kpc and index 5.1 above 30. The galaxy's true half-light radius, core and envelope together, is 2.1 kpc. The envelope is above the deeper limit out to 16 kpc, and the fit still does not follow it: a single profile weighted towards the bright centre cannot bend to two components, and more depth only nudges the compromise it settles on. 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.

 Lucky frames chosen by the twinkle instead of the sharpness. The mean Strehl ratio of a target against its separation from the reference star, drawn as the high layer's shift in units of the aperture (32 arcseconds at the right for a 2.5-metre telescope and a layer at 8 km), for the best 2 per cent of 1600 frames chosen three ways: sharpest on the reference (orange), least scintillation of the reference's light (green), and best on both ranks together (blue); half the turbulence at the ground, both layers varying in strength frame to frame. Chosen by sharpness, the frames give 0.52 on axis and 0.34 at the edge. Chosen by scintillation, 0.22 and 0.22: the same everywhere, because the high layer — the only one whose blur differs between directions — has been selected directly, and lower everywhere, because the ground layer, which blurs every direction alike, has not been selected at all. Both ranks together give 0.44 and 0.34, a little more at the edge than sharpness alone at some cost at the centre. All frames average 0.10. The observed sky

A twinkle that knows the high air and not the low

Lucky imaging keeps the frames in which a reference star is sharp, and that choice happens to favour moments when the high turbulence was quiet, which is why it reaches further across the field than it should. A star's twinkling reports the high layer directly. Choosing frames by the twinkle alone gives a field that is equally sharp everywhere and sharp nowhere, because the twinkle cannot see the air at the ground. Choosing by both buys a little more field for a little less centre, and only where the high air comes and goes.

 A spot's mark on a line, and how much of it a shift explains. Above: the disc-averaged profile of one absorption line from a star rotating at 5 km/s, observed at a resolution of 100,000, without (dashed) and with (solid) a dark spot 8° in radius at latitude 20°, 70 per cent darker than the surrounding surface, on the receding side of the disc. Below, magnified 185 times: the change the spot makes (blue), the part of it a pure Doppler shift of the whole line would make (orange, a shift of −36.3 m/s), and what is left over (green). The spot removes light from the velocity at which its own surface is moving, putting a bump into the line where it was darkest. Most of the change, 92 per cent of its square, looks like a shift; 8 per cent does not, and that remainder is the only part of the spot's signal a spectrum could tell apart from a planet. 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.

 Isochrones that crowd on the main sequence and fan out above it. Isochrones for stars of solar metallicity at 1, 2, 4, 6, 8, 10, 12 billion years, luminosity against temperature with temperature falling to the right. On the main sequence they lie almost on top of one another — a star brightens and barely changes colour as it ages — and only where they turn off and cross towards the giants do they separate. Two boxes show the measurement errors of a star with a good parallax and spectrum (twice 70 K and 3 per cent): one around a Sun-like star of 4.6 billion years, whose age they allow anywhere in 4.53 (3.38–5.63) billion years; one around a subgiant of the same mass at 10.5, allowed only 10.5 (10.3–10.7). 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.

 One planet's helium line, around four kinds of star. The excess absorption at the centre of the 1083 nm helium line during transit, against the mass-loss rate, for one sub-Neptune — 26 Earth masses, 4.4 Earth radii, 0.0525 AU — losing an isothermal wind at 8,000 K, around four representative stellar spectra. Both scales logarithmic. The same outflow absorbs very differently: at 1 × 10¹⁰ g s⁻¹ the depth is 0.06%, 0.14%, 0.35%, 0.40% around an F star, a Sun-like star, an active K dwarf, an active M dwarf. Read the other way, a measured depth of 1% means 1.1 × 10¹¹ g s⁻¹ around an F star, 6.3 × 10¹⁰ g s⁻¹ around a Sun-like star, 3.0 × 10¹⁰ g s⁻¹ around an active K dwarf, 2.9 × 10¹⁰ g s⁻¹ around an active M dwarf — a factor of 4 between the extremes, from the star alone. 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.

 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.

 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.

 The same worlds, placed by the light that strips. The bodies of the shoreline plane, each moved from its present instellation to the cumulative X-ray and extreme-ultraviolet energy it has received over its star's life, both relative to the Earth's (arrows). The Sun's planets do not move, by construction. A planet of a small red dwarf moves up by a factor of 7 to 16: its star stayed at its saturated activity level — a thousandth of its light in X-rays and extreme ultraviolet — for 2.3 billion years for TRAPPIST-1 against 0.1 for the Sun, and its planets have absorbed that for longer. TRAPPIST-1 e, which receives 0.65 of the Earth's sunlight, has received 10 times the Earth's lifetime XUV and moves from well inside the line to above it. Every planet measured as bare, TRAPPIST-1 b, TRAPPIST-1 c, LHS 3844 b, now lies above the line. The saturation times are a scaling with stellar mass rather than a fit to any one star. 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 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.

 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.

 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.

 A feature read back from the limb. A bump in the source function at optical depth 0.1, a quarter of a decade wide (black dashed), and what an inversion of the limb profile recovers on average when the profile is measured at 24 angles with errors of 1, 0.1, 0.01, 0.001 per cent, the inversion assuming only that the departure from a grey atmosphere is smooth. At one per cent the recovered bump has 56 per cent of its height and is spread over about twice its width; at a thousandth of a per cent, 82 per cent, and still wider than the truth. A thousandfold gain in precision buys a modest gain in resolution — each factor of ten a step, not a factor. 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.

 A field that makes electrons transparent. The electron-scattering cross-section for photons polarised across a strong magnetic field, as a share of the free-electron value, against photon energy, for fields of 10¹², 10¹³, 10¹⁴ gauss, on logarithmic axes. Below the electron cyclotron energy — 12, 116, 1160 keV for these fields — an electron bound to the field can barely respond to a wave whose electric field is perpendicular to it, and the cross-section falls as the square of the ratio of energies. At the 10 keV of these sources' X-rays, a 10¹⁴ gauss field reduces it 13,456-fold. The Eddington limit is written with the free-electron cross-section; in such a field it is the wrong number. 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.

 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.

 Where a ten-micron zodi comes from. How much each logarithmic interval of radius contributes to the ten-micron glow of a Sun-like star's exozodiacal dust, for blackbody grains in the Solar System's radial profile, seen face-on from 10 pc: with no instrument (dashed) and through a 14.4 m nulling interferometer whose fringes suppress the light near the star (solid, with the averaged transmission dotted). Without the null the glow is concentrated far inside the habitable zone at 1 AU (marked), because the grains' emission at ten microns rises steeply with their temperature: half of it comes from inside 0.34 AU. The null removes the innermost part and pushes the sensitivity outward, but half the nulled signal still comes from inside 0.69 AU, peaking at 0.74. A zodi level measured this way is a measurement of the dust between a few tenths of an AU and the habitable zone, extrapolated to the habitable zone with the Sun's profile. 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.

 An envelope worn down by a young star's ultraviolet. The mass of a primordial hydrogen envelope, as a fraction of the planet's mass, against age, for an Earth-mass planet in the middle of the eventual habitable zone of a 0.1 M☉ star, starting with 0.1%, 0.3%, 1%, 3% of its mass in hydrogen. The star contracts toward the main sequence and stays saturated in X-rays for 1.5 Gyr; the envelope inflates the planet, so it intercepts more of that light and loses mass faster than a bare rock would. The shaded span is the 75 Myr during which the planet receives more light than the runaway-greenhouse limit, when any exposed ocean would be steam. 0.1% is gone at 463 Myr; 0.3% is gone at 1.5 Gyr; 1% outlasts 5 Gyr; 3% outlasts 5 Gyr. An envelope that outlasts the shaded span has shielded the ocean through the whole of the runaway; one that also outlasts the planet's present age has left it a small gas-rich planet rather than a rocky one. Exoplanets

An envelope spent so an ocean could stay

A planet in the eventual habitable zone of a small red dwarf spends its first tens of millions of years too close to a star that is still contracting and still bright, with its ocean boiled into steam and the steam being split and lost. A wisp of hydrogen left over from the nebula — a few hundredths of a per cent of the planet's mass — would change that, because the star's ultraviolet would spend itself removing the hydrogen first. The envelope that shields the ocean and the envelope that never leaves bracket a range a factor of twenty wide, and the amounts a rocky planet is expected to capture fall inside it.

 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.

 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.

 The gradient radiation would need, as a young star shrinks. The radiative temperature gradient at the centre of a contracting, fully convective pre-main-sequence star — the gradient radiation alone would need to carry the luminosity — against the star's radius, which shrinks to the right, for 0.3, 0.5, 1, 2 solar masses, logarithmic. The centre stays convective while the gradient exceeds the adiabatic 0.4 (dashed). As the star shrinks its centre gets hotter and denser, Kramers opacity, ∝ ρT⁻³·⁵, falls, and the gradient falls as the radius to the power 2.5; where it crosses 0.4 a radiative core appears. 0.3 M☉: not before the main sequence; 0.5 M☉: not before the main sequence; 1 M☉: at 1.94 R☉; 2 M☉: at 7.62 R☉. Dots: each mass's main-sequence radius, where contraction stops. A Sun-like star leaves its convective line at 1.9 solar radii; the lightest never do. Starlight

Where a shrinking star stops boiling at its centre

A young star descends the Hertzsprung–Russell diagram at almost fixed temperature, convective from centre to surface, until something at its centre changes and it turns sharply left toward the main sequence. The something is the opacity. As the star shrinks its centre grows hotter and denser, the Kramers opacity there falls, and radiation becomes able to carry the luminosity without the steep gradient that drives convection. A Sun-like star crosses that line at about twice its present radius, two million years old; a star of a third of the Sun's mass never crosses it at all.

 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.

 The same two lines, seen as frequencies. The amplitude of the Fourier transform of the two line profiles of the previous comparison, on a logarithmic scale, against frequency in cycles per kilometre per second. At low frequency — the overall width — they agree, as the profiles did. At higher frequency they separate completely: the first star's transform falls to zero at 0.110 cycles per km/s and the second's at 0.132, because the rotation kernel's transform vanishes at 0.660/v sin i and no Gaussian-like broadening has a zero anywhere. The horizontal dashed line marks a noise level of 0.0005 in the transform, which a high signal-to-noise spectrum averaged over several lines reaches; the dip and the side lobe beyond it rise above it for both stars, so the rotation of each is measured by where its transform dips, independent of the macroturbulence. In a noisier spectrum the second star's weaker side lobe would be lost first. 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.

 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.

 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.

 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.

 Precession cycles grow as the inverse of the lowest frequency heard. The number of times the orbital plane precesses between a detector's lowest frequency and the last stable orbit, against that lowest frequency, both on logarithmic scales, for pairs of 36 + 29 solar masses, 10 + 8 solar masses, 8 + 1.4 solar masses with spins tilted into the orbital plane. The count grows as the inverse of the lowest frequency — the slopes are −1.07, −1.02, −1.01. Above 20 Hz, the limit of the ground-based detectors that have heard every merger so far, the heaviest pair precesses 0.9 times, the middle 4.4 and the lightest 6.2. Above 5 Hz the counts are 4.9, 18.7, 25.5; above 1 Hz, 26, 95, 129. Gravitation

Precessions bought one hertz at a time

A merging pair of black holes whose spins are tilted makes its orbital plane wobble, and the wobble is the only part of the signal that separates one hole's spin from the other's. How many wobbles a detector records is a simple formula: two plus three halves of the mass ratio, divided by the total mass and by the lowest frequency the detector can hear. The detectors that have heard every merger so far stop at twenty hertz, where a typical heavy pair wobbles less than once.

 A glow that switches off when the shadow climbs past the shell. The brightness that 100,000 objects in orbit add to the zenith sky, as a share of a natural moonless sky of 22 magnitudes per square arcsecond, against how far the Sun is below the horizon, for shells at 550 and 1200 km. Each object reflects sunlight with an effective area of 0.5 square metres; together they spread a glow whose brightness does not depend on the shell's distance, because more objects are in each patch of sky the further away the shell is. At the end of astronomical twilight the two shells add 3.0% and 2.5%. The lower shell's zenith goes dark when the Earth's shadow passes it, at 23 degrees of depression; the higher one at 33. The observed sky

A glow counted in square metres of sunlit metal

Satellites too faint to see one at a time still reflect sunlight, and spread across the sky they add a glow that no mask removes. Its brightness does not depend on how far away the satellites are, only on how much sunlit area is in orbit and whether the Earth's shadow has reached it. A hundred thousand satellites as bright as today's add a few per cent to the darkest sky — and the estimate that the sky is already a tenth brighter rests on an area of debris nobody has measured.

 A disc that bends up into a halo only the telescope made. The surface brightness of a face-on exponential disc against radius in units of its scale length, in magnitudes per square arcsecond with the centre at 20, as it is (black, a straight line) and as a telescope sees it if 2% or 5% of each point's light is spread into faint wings falling as the inverse square of distance out to ten arcminutes. Scattered light from the bright centre lands in the outskirts, and once the disc has faded below it the profile stops falling and bends up into a shallow component that looks like a stellar halo: the observed profile rises three-quarters of a magnitude above the disc beyond 10.3 and 9.3 scale lengths, near 29.2 magnitudes per square arcsecond. At ten scale lengths the disc itself is at 30.9 and the image at 30.2 and 29.6. 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.

 Ridges that bend where the model's surface is wrong. An échelle diagram of a Sun-like star's oscillation frequencies — each mode's frequency against its remainder after dividing by the large separation of 135.1 µHz — for radial, dipole and quadrupole modes (red, green, blue), as a one-dimensional model predicts them (open) and as the star produces them (filled), if the model's outer layers are wrong in the way the Sun's models are. Low in the diagram the two coincide. With increasing frequency the star's modes fall behind the model's, by an amount that grows steeply and is the same for every degree, so all three ridges bend to the left together: by 19.3 µHz at 4104 µHz. The bend is the surface effect, and it is larger than the uncertainty of any measured frequency by a factor of a hundred. 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 ratio relativity predicts from one density. The ratio EG — written E with a subscript G in the literature — of the lensing around a set of galaxies to their clustering and infall, against redshift. In relativity it is Ωₘ₀/f(z) and depends on nothing else: with Ωₘ₀ = 0.3153, 0.598 today, 0.455 at z = 0.32 and 0.360 at z = 1, falling because f rises towards one as matter comes to dominate. The dashed curves keep light's bending as in relativity and change how hard matter pulls, by −10% and +10% today, growing with the dark energy's share: a weaker pull slows the infall and raises EG. The point is the first measurement, 0.39 ± 0.06 at z = 0.32 from galaxy–galaxy lensing in the Sloan survey. 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.

 Where each test can name the host. The plane of a pair's separation and brightness difference, divided by which measurement can say which star a 1000-ppm transit crossed. Above Δm ≈ 4 a sub-Neptune on the companion is diluted below detection and there is nothing to assign. Imaging resolves the pair to the right of an illustrative contrast curve, Δm = 5 log(s/0.04″). The centroid's shift, read against the pair's separation and flux ratio, names the host to the right of s = σ(1 + f)/δ for a precision σ of 30 μas, about 0.05″ for near twins. The density from the transit's shape names it between Δm ≈ 1.1, below which the two stars are too alike, and the detection limit — at any separation, but only for strong transits on nearly circular orbits. The corner of close, nearly equal pairs, under about 0.04″ and 1.1 magnitudes, is reached by none. 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.

 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.

 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.

 Three reasons the flicker slows with redshift, only one of them cosmology. The observed flicker timescale of the faintest quasars a flux-limited survey can see, relative to its value at redshift 0.2, logarithmic, separated into its causes: time dilation alone (a factor 5.0 by redshift 4); the bluer rest-frame wavelength a fixed band samples at higher redshift, which shortens the timescale (×0.78); and the heavier black holes the flux limit selects (×15.3). Together, ×49.9. Read as a power of (1+z) the total has exponent 2.74 where dilation alone has 1.00: a test that does not hold mass and wavelength fixed measures the selection, not the expansion. 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.

 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.

 Lines from the lit face follow the star that cannot be seen. Radial velocity against orbital phase, over two orbits, for a binary like Sco X-1 — period 0.787 days, a 1.4 M☉ neutron star, a donor of 0.42 M☉, inclination 40°. The donor itself is drowned by the light of the accretion disc and is not seen. The narrow components of the Bowen blend, emitted by the donor's face lit by X-rays, move in phase with the donor but with 109 km/s rather than the 139 km/s of its centre, because the lit face lies on the side nearer the neutron star. The neutron star moves in antiphase at 42 km/s, a motion the broad lines of the disc track only roughly. Measuring the narrow lines' amplitude puts a lower limit on the donor's velocity. 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.

 A star polarised only at its edge. The degree of linear polarisation of light from an atmosphere where electron scattering is the only opacity, against projected distance from the disc's centre in stellar radii (solid), and the relative brightness of the disc (dashed, scaled so the centre is 12). The polarisation is zero at the centre, 0.2 per cent halfway out, 1.9 per cent at nine-tenths of the radius and 11.7 per cent at the limb, always parallel to the limb. The brightness falls to a third towards the edge, so the most polarised light is also the faintest. 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.

 Elements sorted by how easily they lose an electron. The abundance of each element in the solar wind relative to the photosphere, normalised to oxygen, against its first ionisation potential, logarithmic: in the slow wind (filled) and the fast wind from coronal holes (open). Representative values. Elements easy to ionise — K, Na, Ca, Mg, Fe, Si, all below about 9 eV — are about three times over-abundant in the slow wind and about one and a half in the fast; those harder to ionise sit near one. The step comes at about 10 eV, which is the energy of a photon from the Lyman-α line of hydrogen: the sorting happens where hydrogen's own light sets the ionisation. Stars

A wind that carries the ledger of its loops

The slow solar wind is richer in iron, magnesium and silicon than the Sun's surface is — about three times richer, relative to oxygen — and the elements that are enriched are exactly the ones that lose an electron easily. That sorting happens low in the chromosphere, inside closed magnetic loops, and it takes days. The fast wind from coronal holes barely shows it. So the composition of the wind arriving at the Earth is a record of where it was held, and for how long, before the field let it go.

 A lifetime that stops getting shorter. Main-sequence lifetime against stellar mass, both logarithmic, from a simple model in which luminosity rises as the 3.8th power of mass until radiation pressure caps it at 40% of the Eddington limit (solid), and without the cap (dashed). The Sun lives 10 billion years, a 10 M☉ star 36 million, a 40 M☉ star 3.2 million, and above about 60 M☉ the lifetime stops falling: 2.7 million at 100 M☉, 3.0 million at 300. Without the cap the steep law would give the heaviest stars lifetimes of tens of thousands of years; with it, every very massive star lives a few million years whatever its mass. Stars

A lifetime that stops getting shorter

A star twice the Sun's mass lives a fifth as long, ten times the mass a few hundredths. Extrapolate the steep law that says so to the most massive stars known, a hundred and fifty or two hundred suns, and they should burn out in a few tens of thousands of years. They do not: every star above about sixty solar masses lives between two and a half and three million years, whatever its mass. Radiation pressure caps how bright a star can be per unit of its mass, and once luminosity and fuel both grow in proportion to mass, the mass cancels out of the lifetime.

 Water that is on the star and not on the planet. Excess transit depth, in parts per million, against wavelength, for a planet of depth 7000 ppm around a star of 2600 K. The upper curve is a bare rock with no atmosphere on a star whose disc is 10% covered by 2300 K spots that hold more water than the rest of the photosphere: it shows a broad slope and a band near 1.4 μm, 36 ppm from its shoulder. The lower curve is the signal of an Earth-like air column with water on an Earth-sized planet, about 93 ppm at the band. The band in the spot spectrum is an imposed illustration rather than a model atmosphere, but its sign is measured: it adds where a planet's own water would add. 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.

 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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