Astronomy & Celestial Mechanics
the sky, computed
Theme

The sky as a clock

Periods that repeat to the second, used to build calendars, find longitude, and detect planets nobody can see.
Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach. Orbits

Equal areas in equal times, which is angular momentum in disguise

Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

 The three anomalies at E = 1.15 rad. The auxiliary circle construction. The eccentric anomaly E is measured at the centre, the true anomaly ν at the focus, and the mean anomaly M is time expressed as an angle. Here E = 1.150, ν = 1.827 and M = 0.602 radians, related by M = E − e sin E. Orbits

The position that has no formula, and is computed anyway

Kepler's equation relates where a body is to when it is there. It cannot be solved in elementary functions, Kepler said so, and nobody has managed it since — which has stopped nothing.

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

The law that links period to size, and weighs everything

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

 Flight time against semi-major axis, for a fixed 135° sweep. Lambert's theorem drawn: the time to fly between two points 1 and 1.524 AU out and 135° apart, against the semi-major axis of the orbit that does it. Nothing else about the orbit enters — not its eccentricity, not where its periapsis is, not how it is oriented — which is the content of the theorem and the reason a two-point transfer is a one-dimensional search rather than a six-dimensional one. Two branches: the lower one is the ellipse whose arc stays short of apoapsis, falling towards the parabolic floor at 103.2 days as a grows without limit; the upper one is the ellipse of the same size whose arc runs through apoapsis, rising without limit. They meet at a = s/2 = 1.2161 AU, 244.2 days, which is the minimum-energy transfer and the slowest ellipse available — every faster one is bigger. Each branch is monotone, checked point by point across the drawn range, so a horizontal line cuts each at most once: for a given pair of points and a given time there is exactly one ellipse, and at 260 days it is a = 1.2189 AU on the upper branch. The freedom a mission designer has is not in this picture: it is the choice of the two points, which is what a porkchop plot sweeps. Orbits

Two places and a clock decide the path

The time to fly between two points depends on the semi-major axis, the chord between them, and the sum of their distances — and on nothing else about the orbit. Not the eccentricity, not where periapsis is, not the orientation. Lambert's theorem is why an interplanetary launch date is the root of one equation.

 Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap. Orbits

No planet has an eccentricity of its own

Strip the short-period terms out of the planetary equations and what is left is a linear system. Its eigenvectors are modes of the whole solar system, and the number a catalogue quotes for a planet's eccentricity turns out to be a reading of a clock rather than a property of the planet.

 5 transfers through the same two points in the same 1400 days. Time of flight against semi-major axis for every transfer through two points 135° apart at 1 and 1.524 AU, with the revolution count running from 0 to 2. Each count contributes two branches, and for N ≥ 1 the pair folds: the time has a minimum at a = 1.2426 AU for one revolution — only 2.2 per cent above the minimum-energy value of 1.2161, which is why the horizontal axis is the excess over that value and logarithmic — so a flight time above it is met twice and below it not at all. Reading the crossings of the 1400-day line off the drawn curves gives 5 of them — 0 revs high, 1 rev low, 1 rev high, 2 revs low, 2 revs high — which is 2N + 1 with N = 2, and the count is a property of the time rather than of the geometry. That is the practical content: a root-finder started from a single guess returns one of these 5 and gives no sign that the other 4 exist, and the cheapest of them is often not the one nearest the guess. Orbits

One time of flight and five ways round

Lambert's theorem says two positions and an interval fix the transfer. Allow the transfer to complete whole revolutions and that stops being true: the flight time folds, one number admits five arcs, and the cheapest of them is usually not the one a solver started nearest to.

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

The table that is a fit

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

 After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging. Orbits

An error that is nearly all in one direction

A fitted orbit's uncertainty is not a ball. Within a few revolutions it has collapsed onto a line along the track, because an error in the size of an orbit is an error in its period and an error in period is a phase that runs away — which is why an impact probability is an integral along a curve rather than over a volume.

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

A surface dated by counting holes in it

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

 An impulse delivered inside 0.5 AU, and a comet 2050 hours early. Above: Marsden's outgassing law, the factor g(r) that scales a comet's non-gravitational acceleration, against distance from the Sun over one orbit of a comet with perihelion at 0.336 AU and aphelion at 4.09. It is close to an inverse square inside the water snow line and then falls off a cliff, because water ice that is not being heated does not sublimate. Half the whole revolution's impulse is delivered inside 0.55 AU — a few weeks out of a 3.3-year orbit — so the force is effectively a kick at perihelion rather than a perturbation spread around the path. Below: what a kick of that kind does to the timekeeping. A transverse component changes the semi-major axis and so the period, by 2.5 hours per revolution here, and a constant change in the period accumulates as the square of the number of revolutions rather than in proportion to it. After 40 returns the comet arrives 2050 hours — more than 85.4 days — before an orbit fitted without the term predicts, and doubling the number of returns multiplies the discrepancy by 3.90. That is why the effect was found in the eighteen-twenties from nothing but arrival times, and a century and a half before anyone photographed a jet. Orbits

A comet that arrives a day early

Encke's comet returned two and a half hours ahead of prediction, every revolution, for decades before anybody could say what was pushing it. The force is a rocket — a few tonnes a second of vapour leaving the sunward side of a rotating nucleus, delivered almost entirely in the few weeks around perihelion, and it accumulates in the arrival time as the square of the number of returns.

 Integrating the measured recession back: the Moon reaches the Earth 1.54 Gyr ago. The Earth–Moon separation and the length of the Earth's day, integrated backwards from the measured present recession rate of 3.83 cm per year. Constant-Q tidal friction makes a^(13/2) linear in time, so the history is a single line in a variable nobody plots, and it is calibrated to the laser-ranging measurement rather than to a modelled k₂/Q — the k₂/Q it implies is 0.0257, or Q = 11.6 for the Earth's k₂ of 0.299, which is a startlingly dissipative Earth. Run back at that rate the separation reaches zero 1.54 Gyr ago and crosses the Roche limit at 2.88 Earth radii only 4 years before it, so the drawing is cut off there rather than extrapolated. The Moon is 4.5 Gyr old, so this is a refutation and not a date: the present rate cannot have been the rate, and a mean Q of 34 — drawn dashed, reaching 4.51 Gyr — is the sort of value the age requires. Tidal rhythmites at 620 Myr put the day at 21.9 h and the Moon at 96.5 per cent of its present distance, and this history reads 20.1 h and 92.4 per cent — too fast and too close, which is the same failure the zero crossing is. Day length follows from total angular momentum, 23.93 h today, 9.84 h at half the present lunar distance and 4.97 h at the Roche limit, and depends on the separation alone: it is the same curve whatever Q is. The rate of lengthening the recession requires is 2.10 ms per century, against a tidal total of about 2.3 including the Sun's tide, which slows the Earth without moving the Moon, and an observed 1.75 from ancient eclipses and occultations — the shortfall being the Earth's moment of inertia falling as the mantle rebounds from the last glaciation. Gravitation

A day five hours long

The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing.

 A thousandth of the field, and all of the precession. Left: the Earth's figure against a sphere of the same equatorial radius, with the flattening drawn 28× its true value. The real difference between the equatorial and polar radii is 21.4 km on 6378 — 1 part in 298 — which at this size would be 0.5 pixels and invisible, so the drawing is a schematic and the number is here instead. Right: the two components of the J₂ perturbation at the surface, each as a fraction of the monopole μ/r², both differentiated from the potential rather than quoted. The radial one strengthens the inward pull by 1.62×10⁻³ over the equator, where the extra mass is, and weakens it by 3.25×10⁻³ over the poles, vanishing at ±35.26° where P₂ does. The transverse one is zero at the equator and at both poles and peaks at ±45°, at 1.62×10⁻³ — and that is the component that does the work. It pulls an inclined orbit back towards the equatorial plane, which is a torque about the line of nodes, and a torque applied to something already turning moves it sideways rather than back. Averaged over an orbit the pair leave a, e and i untouched and turn the whole plane instead, which is why a term a thousandth of the field is the largest single perturbation on almost every satellite ever flown. Gravitation

The Earth's shape, read off a satellite's node

The Earth is a thousandth of a part from being a sphere, and that thousandth turns every satellite's orbital plane. Vanguard 1 measured it in 1959 — and one retrograde inclination turns the plane at exactly the rate the Sun moves, which is a perturbation used as a design constraint rather than corrected for.

 255 km sees degree 154; 35786 km sees degree 6. The same field spectrum, multiplied by the upward continuation factor (R/r)ˡ⁺¹ for orbits at 255 km, 450 km, 800 km, 35786 km. A harmonic of degree ℓ has ℓ bumps around the planet, so it falls off with height like a wave of that wavelength — fast, and faster the finer it is. The horizontal rule is a measurement floor; where each curve crosses it is the highest degree that orbit can feel at all, and the answer is 154 at 255 km, 101 at 450 km, 65 at 800 km, 6 at 35786 km, or 130 km, 198 km, 308 km, 3340 km of horizontal resolution on the ground. Nothing in that arithmetic is an instrument. It is why GOCE flew at 255 kilometres with drag compensation rather than at a comfortable altitude with a better gradiometer, why the Moon's mascons were not seen until something orbited low over them, and why a geostationary satellite's ephemeris needs a field with four terms in it. Gravitation

The field a satellite is allowed to feel

Past the flattening, a planet's gravity is a sum of harmonics whose sizes follow a rule with no physics in it. How much of that sum a spacecraft can measure is decided by its altitude and by nothing else — which is why one mission flew at 255 kilometres and had to push itself along.

 Io's measured heat needs k₂/Q = 0.016, and Enceladus's needs 0.011. Tidal surface heat flux against orbital eccentricity, from Ė = (21/2)(k₂/Q)GM_p²R⁵ne²/a⁶ evaluated at each satellite's own orbit, drawn at a common k₂/Q of 0.015. Every line has slope 2 because the dissipation is quadratic in e and nothing else on this axis varies. The filled marks are each body at its actual eccentricity; the two ringed ones are the bodies with a measured surface heat flux, and they are the only points here that are observations. Solving each of those for k₂/Q gives 0.016 for Io and 0.011 for Enceladus — within a factor of 1.5 of one another, for a warm silicate body and a 500-kilometre ball of ice, which ought to be a coincidence and is instead the sharpest problem in the subject: nothing about Enceladus's ice can plausibly dissipate at 0.011, and the number is what the heat requires all the same. The dashed line is the Earth's measured surface heat flux, 0.087 W/m², which Io exceeds by a factor of 28 — the most volcanically active body in the solar system is the fourth largest moon of the fifth planet, and the reason is entirely in the orbit. Gravitation

A moon heated by not being allowed to relax

Tidal dissipation goes as the square of an eccentricity that tides themselves destroy, so a moon radiating tidal heat is spending something it cannot have saved. Io's would be gone in a hundred and forty thousand years, and the resonance that keeps putting it back is the reason there are volcanoes.

 Nothing visible in any pulsar, and a quadrupole in the angle between them. Above: 4 millisecond pulsars' timing residuals over 15 years, at the few hundred nanoseconds a good one reaches. Each wanders, and none of them shows anything a reader could call a signal; a gravitational-wave background of amplitude 2.4·10⁻¹⁵ at one cycle per year contributes a common part to all of them that is smaller than each pulsar's own red noise. Below: the correlation between pairs, against the angle on the sky between them. 2211 pairs out of 67 pulsars, binned into 15 angles, against three curves with no free parameters between them. A quadrupolar background gives the Hellings–Downs shape — positive for nearby pulsars, negative near 83°, and back up to exactly half its zero-separation value at 180° because a background looks the same in opposite directions. An error in the observatory clock would give a flat line, because it shifts every pulsar identically. An error in the solar-system ephemeris would give a cosine, because it moves the barycentre in one direction. The drawn points prefer the quadrupole over the flat line by Δχ² = 358. That is the detection: not a waveform, not an event, not a moment — a shape in an angle, accumulated over fifteen years, on data taken for another purpose entirely. Gravitation

A detector the size of the galaxy

At a nanohertz no instrument can be built, so the clocks already in the sky are used instead. The signal is in no single pulsar's data — it is in the correlation between pairs as a function of the angle between them, and that curve has no free parameters at all.

 The sky from latitude 52°. The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination 20 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint. The observed sky

The sphere that is not there, and why it is still the right model

The stars are at wildly different distances and the celestial sphere is a fiction. It is also the most useful fiction in observational astronomy, because for pointing at things, distance is exactly the information to throw away.

 The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day. The observed sky

The Sun's path, and the tilt that makes the seasons

Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.

 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.

 The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close. The observed sky

The pole star has a shelf life, and the sky has a slow hand

The Earth's axis traces a circle among the stars once every 25,772 years. Polaris is at the pole now, was not four thousand years ago, and will not be in two thousand more.

 The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes. The observed sky

The Sun is a bad clock, by up to sixteen minutes

Solar noon and twelve o'clock are not the same instant, and the discrepancy runs through a fixed annual cycle. It has two causes, one from the shape of the orbit and one from the tilt of the axis.

 One star, two coordinate systems, at latitude 52°. The equatorial grid and the horizon grid drawn on the same sphere for an observer at latitude 52°. The star marked has declination 20° and hour angle -40° in the first, and altitude 45.5° and azimuth 239.4° in the second. The two frames differ by a single rotation through the co-latitude 38°, which is why the celestial pole stands 52° above the northern horizon. The observed sky

Where a star is depends on who is asking

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

 Three periods that nearly share a multiple. How far the draconic and anomalistic months are from a whole number, after a whole number of synodic months, in hours. At 223 synodic months — 6585.321 days — both are within an hour of closing, which is what makes an eclipse repeat. The draconic residual is 0.87 hours and the anomalistic 5.19 hours. The observed sky

The eclipse that repeats a third of a world away

Three lunar periods nearly share a multiple after 6,585 days. The word "nearly" is what makes eclipses predictable, and the leftover third of a day is what moves each repeat a third of the way round the Earth.

 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 sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats. The observed sky

The face that is not quite fixed

The Moon keeps one face turned toward the Earth, and the sentence is exactly true only of a fictitious Moon on a circular orbit. The real one rocks by a few degrees each month, in two directions and for two unrelated reasons, and the rocking has shown 59 per cent of its surface to people who never left the ground.

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

Three definitions of night

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

 Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops. The observed sky

The loop a planet does not make

Mars stops in the sky, backs up for ten weeks, and goes on. Neither orbit reverses anywhere — the loop belongs to the difference of two position vectors seen from one of them, and its width and duration are fixed by the ratio of the two radii and by nothing else.

 Why the solar day is 3m 55.91s longer than the sidereal one. Two positions of the Earth one day apart, with the direction of a fixed star and the direction of the Sun marked at each. After one sidereal day — 23h 56m 4.0905s — the Earth has turned through exactly 360° and the star is back on the meridian; the Sun is not, because the Earth has moved 0.9856° along its orbit, and the further 0.9856° of turning takes 3m 55.91s. That is the whole of the difference: 366.2422 turns against the stars in the 365.2422 solar days of a year, one more turn than the 365.2422 against the Sun. Nothing here is to scale. The orbital arc is drawn at 40°, which exaggerates the real 0.9856° by 41 times, and the Earth's disc is about 4073 times too large for its orbit. The two star sight lines are drawn parallel because they are: a star's distance cannot be put on the same page as an orbit. The observed sky

The day that is four minutes short

A star crosses the meridian 3 minutes 55.91 seconds earlier each night, and the same star slips 3 minutes 56.56 seconds a day against a civil clock. Those are two different numbers, and the gap between them is the one extra turn the Earth makes against the stars in every year.

 The wobble inside the wobble. Left: the two components of nutation over 40 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one. The observed sky

The wobble inside the wobble

Precession and nutation are the same torque. The difference is that the Moon's orbital plane turns once in 18.6 years, so part of the pull oscillates instead of accumulating — and a catalogue position is not a direction until it says which pole it is measured from.

 One path, five numbers. Left: the apparent path of a star over 4 years, with a proper motion of 193 mas a year and a parallax of 50 mas, at ecliptic latitude 42°. It is one curve and there is nothing in the sky it can be compared against — the reference stars have paths of their own. Right: the same path with a straight line taken out of it. What is left is an ellipse of semi-major axis 50.0 mas and semi-minor axis 33.5 mas, closed and repeating once a year. The two are separated by their time signatures and by nothing else: proper motion is secular and parallax is annual, in a phase the Earth's position fixes in advance. That is why the five parameters can be told apart at all, and why an astrometric catalogue quotes five rather than two — a position without them is a position at one instant, which is not a direction to anything. The observed sky

Five numbers from one wiggle

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

 The 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.

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

The solar system measured from inside one orbit

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

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

The whole sky drifting towards one point

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

 A clock that has lost 5.7 hours in 2720 years. ΔT = TT − UT1, the accumulated difference between a uniform time scale and the Earth's own rotation, from 700 BC to 2020, on a logarithmic scale. The points are the published record; the smooth curve is the parabola 31.94·u² fitted to the entries at or before 1000, with u in centuries from 1820. It is a parabola and not a line because the Earth is not merely slow, it is slowing: a day lengthening at a constant rate makes a clock fall behind by the integral of the lag. That coefficient is a length-of-day rate and nothing else — 31.94 = ½ × r × 36525 days per century gives r = 1.75 ms per century, against the 1.78 ms computed from tidal angular momentum with no eclipse anywhere in the derivation. At 700 BC the offset is 5.7 hours, which is 85° of the Earth's rotation, and that is why the ancient measurements are records of the place a total eclipse was seen from rather than of the hour: the hour was never written down accurately enough to matter, and the shadow's track on the ground was. The observed sky

Six kinds of second

The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock.

 Five rules, and how fast each one leaves the seasons behind. The accumulated difference between a calendar and the seasons, for five leap-year rules, over 4000 years. Each is a straight line whose slope is the rule's fraction minus the tropical year's 0.2421897, and nothing else about a calendar matters to this plot — not which months are long, not where the year starts, not what anything is called. The Julian quarter is out by 11.25 minutes a year, which is one day every 128 years and is why ten days had to be removed in 1582. The Gregorian rule takes 3223 years to lose a day and the Persian 4264, in the other direction — the older rule is the better one, at a denominator of 33 against 400. The 128-year rule is level on this scale: 454545 years to a day, which is longer than any calendar has been kept — and it is not a separate invention but the Julian rule with its own error subtracted, since 1/4 − 1/128 = 31/128 exactly and the Julian drift is exactly one day per 128 years. What the figure cannot show is that the tropical year is itself shortening, by about half a second a century, so the slopes drawn here are the ones at the present epoch and every line is very slightly curved. The observed sky

A year that is not a whole number of days

The tropical year is 365.24219 days, so every calendar is a fraction chosen to approximate 0.24219. The continued fraction says which fractions are best, and the one in use is not among them.

 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.

 The Metonic cycle slips a day in 219 years, and the next rule needs 334. Accumulated disagreement between a lunisolar rule and the sky, against elapsed time, for the continued-fraction convergents of the 12.368266 synodic months in a tropical year. Each line is one historical cycle: 2 years to 25 months, 3 years to 37 months, 8 years to 99 months, 11 years to 136 months, 19 years to 235 months, 334 years to 4131 months. Every line has slope exactly 1, because an error made once per cycle accumulates linearly, so the only thing that distinguishes the rules is where they start. The two that were actually used are the octaeteris — eight years, 1.59 days out per cycle, useless within a generation — and Meton's nineteen, which is out by 2.1 hours per cycle and therefore takes 219 years to slip a single day. The reason nineteen is so much better than eleven is a number in the continued fraction rather than anything about the Moon: the partial quotient that follows 235/19 is 17, and a convergent's error is bounded by one over the next quotient times the square of its denominator — so a large quotient there is exactly a good approximation here, and the next improvement costs 334 years of cycle for a rule nobody could keep. What follows from the 219 years is the whole character of a lunisolar calendar: it is a table rather than an observation. The ecclesiastical moon that fixes Easter is computed from a Metonic cycle, not looked at, and the Julian version of that computus — still used to date Easter in the Eastern churches — has slipped four to five days from the sky since it was fixed in the fourth century, exactly as this plot says it must. A calendar's job is agreement, and agreement and accuracy are different requirements that diverge at a rate the arithmetic predicts. The observed sky

A month that has to be tabulated

A lunisolar calendar reconciles two periods that share no common multiple, so every historical cycle is a convergent of one continued fraction. Meton's nineteen years is wrong by two hours, which is a day in two hundred and nineteen years — and that is why the moon that fixes Easter is a table rather than the sky.

 Two dips a side, 36.5 seconds apart, symmetric about a body 256 km across. One observer's light curve across a small body with two narrow rings, at a chord 44 km from the centre. The body itself removes the star for 11.2 seconds; the four brief dips, two either side of it, are ring crossings, at 391 km and 405 km from the centre and 7 km and 3 km wide radially. The evidence that they are rings and not two more objects is the symmetry: each pair sits at equal times before and after mid-event, to within 0.07 s here, and two independent bodies have no reason to do that. Each dip is drawn wider than the ring is thick because the chord crosses obliquely — the path length through the material goes as 1/cos, which is also why a ring is deeper near the ansae. Nothing in this light curve was looked for: Chariklo's rings turned up in 2013 in a run recorded to measure a diameter, and every ring system found since has been found the same way. The observed sky

A star that blinked before it should have

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

 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.

 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.

 How long a star lasts, against its mass. Main-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years. Stars

Why the biggest stars die first, and take the galaxy with them

A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of 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.

 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.

 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.

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

A clock that runs down and says what it is

A pulsar hands over two measured numbers, a period and its derivative. Everything else usually quoted about it — an age, a magnetic field, a luminosity — is derived from those two through a model, and the one measurement that tests the model refutes it.

 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.

 A Hohmann transfer, 2.6 to 1 in radius. Two circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation. Spaceflight

The cheapest way between two orbits, and why it is so slow

Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait.

 Catching a target 40° ahead. A phasing manoeuvre. Dropping into an orbit 6% lower shortens the period to 0.9553 of the target's, so the chaser gains 16.1° each lap and closes 40° in 3 revolutions. Speeding up would have lost ground instead. Spaceflight

Catching up by slowing down, which cost Gemini 4 its fuel

To reach something ahead in the same orbit, a spacecraft must fire backwards. Pointing at the target and thrusting makes the gap grow, and a crew found that out in orbit before anyone had flown the correct manoeuvre.

 Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between. Spaceflight

The same planet, three times

A flyby cannot change the encounter speed, only its direction, so a tour has to be designed in the space of what is conserved. One pass moves a spacecraft along a single curve and no further than the planet can bend it — and reaching a distant target means walking that curve, returning to the same planet again and again.

 The cost of going to Mars, against when to leave and how long to take. Contours of departure energy C₃ over a grid of 64 × 56 solved Lambert problems: each point is a departure date, a flight time, and the unique single-revolution conic that connects the two planets between them. The cheapest transfer on this grid costs 5.1 km²/s², leaving in Mar 2001 with a flight time of 220 days, against 8.7 for the idealised Hohmann transfer between circular orbits of the same radii. A launch window is a region on this plane, not a moment, and its shape is what a launch period is negotiated against. Spaceflight

Two dates decide a mission

Given where a spacecraft leaves from, where it is going, and how long it may take, there is exactly one orbit joining the two. Solving that problem over every pair of departure and arrival dates produces a contour map, and the shape of the contours is what a launch window actually is.

 three revolutions, on a turning Earth. The ground track of a circular orbit at 420 km and 51.64° inclination, over 3 revolutions, on an equirectangular graticule. The latitude is a sine wave bounded by ±51.64° exactly — sin φ = sin i sin u, so the inclination is the highest latitude the orbit ever passes over, and it is reached twice per revolution. Each successive pass is displaced west by (ω⊕ − Ω̇) × 92.90 min = 23.61°, of which 0.32° is the orbital plane's own regression and the rest is the planet turning underneath: the vehicle comes back to nearly the same place in inertial space and the place has moved. The period used is the nodal one, 92.899 min against the Keplerian 92.970: J₂ makes the two differ by 4.31 s, which is 0.018° of walk per revolution and 102° in a year — the difference between a repeat track and a track that used to repeat. The map is equirectangular and therefore wrong about area everywhere; what it is right about is longitude difference, which is the whole of what this figure measures. Spaceflight

The line under a satellite

A ground track is an orbit seen from a frame that is turning, so every pass lands west of the last one. The track closes only when two periods are commensurable — which turns "look at the same place every day" into a condition on the altitude.

 Where the fuel goes, and it is not where a satellite points. Left, the orbit pole of a geostationary satellite, in degrees from the Earth's. The Sun and the Moon between them carry it round a circle of radius 7.4° in 53 years, and a satellite launched into the equatorial plane starts on the rim of that circle rather than at its centre — so its inclination climbs from zero at 0.88° a year, reaches 14.8° after 27 years, and comes back. Right, what holding it costs. A plane change of 0.88° at 3.07 km/s is 47.1 m/s a year; holding the longitude against the equatorial bulge, computed from the same resonant term that makes the longitude a pendulum, is 1.8 m/s a year. North–south is 96% of the budget, and a satellite that gives up on it does not fail — it starts tracing a figure of eight on the sky 1.8° tall in the first year, which a fixed dish cannot follow and a steerable one can. Retiring at the end of the propellant is therefore a choice about which service ends first. Spaceflight

The orbit that has to be paid for every year

A geostationary satellite is not in equilibrium in any direction. The Sun and Moon tilt its plane by 0.85 degrees a year, the Earth's equatorial ellipticity makes two longitudes stable and two unstable, and the end of a satellite's life is the end of its propellant.

 Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between. Spaceflight

A map of the transfers that are free

Drawn as contours of perihelion against aphelion, the invariant that survives an encounter becomes a map. A flyby slides a spacecraft along its own contour and costs nothing; a burn is the only thing that moves it between contours — so tour design is reading a graph.

 An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 23 nanoradians — which is 4.7 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit. Spaceflight

A position measured from a frequency

A spacecraft is unresolvable and unreachable, and everything known about where it is comes from two scalars — a round-trip light time and a Doppler shift. Neither is an angle. The orbit solution returns two angles anyway, because the antenna is bolted to a rotating planet.

 A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky. Spaceflight

An angle measured against a quasar

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

 A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the limb-darkened stellar disc over the region the planet covers. The depth is 1.26%, deeper than (Rp/R⋆)² = 0.01055 because the planet crosses a limb-darkened disc whose centre is brighter than its average. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii. Exoplanets

Four contact points, and what they fix

The depth of a transit gives a radius ratio. The shape gives the impact parameter, and then — through nothing but Kepler's third law — the mean density of the star being crossed.

 A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10. Exoplanets

Planets found by a transit running late

A planet on a fixed orbit transits like a clock. A second planet pulling on it makes the clock run fast and slow by minutes — and fitting that wander weighs a planet that may never cross the star at all.

 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.

 Four defensible boxes, and a factor of 2.9 between the answers. Above: the occurrence surface in starlight received against planet radius, with four published definitions of "an Earth-size planet in the habitable zone" drawn on it as rectangles. The surface is a stated parameterisation — two lognormal populations at 1.3 and 2.4 Earth radii, with the radius valley at 1.9 between them, normalised so that the whole of it comes to 0.5 planets per star between one and four Earth radii inside a hundred days. Integrating it over the four boxes gives 17%, 8%, 6%, 10% — a factor of 2.9 between the conservative zone and a broad definition, before any error bar is attached to any of them. The dashed curve is what a transit survey can actually see: at one year around a Sun-like star the smallest detectable planet is 1.6 Earth radii, so the lower-left corner of every box contains no detections at all and the rate there is an extrapolation of a fitted surface rather than a count of anything. Below: the same integral with only one corner moved. Holding the flux range fixed and sliding the radius bound from 1.5 to 1.75 Earth radii — a quarter of an Earth radius, well inside the uncertainty of a measured planetary radius — changes the answer by 43 per cent, which is larger than every error bar quoted with any of these numbers. The published values of η⊕ span two per cent to sixty; roughly a factor of 2.9 of that is definition, and the rest is how far each author was willing to extrapolate past the dashed line. Exoplanets

The part of a rate that is a definition

Published values for the frequency of Earth-size planets in habitable zones span two per cent to sixty. The spread is not measurement error — it is where the box was drawn, on a surface that is steepest exactly at the corner every author has to choose, and outside the last detection.

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

A galaxy measured from inside it

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

 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.

 A one-parameter model, and 9 times too many metal-poor stars. The metallicity distribution of long-lived stars near the Sun — the bars — against two models of how a galaxy enriches itself. The closed box is as simple as a model of a galaxy can be: gas turns into stars, stars make metals and return them, nothing enters and nothing leaves. It has exactly one parameter, the yield, and it predicts the whole curve. It gets the peak roughly right and the tail catastrophically wrong: 13.7 per cent of its stars fall below [Fe/H] = -1 against an observed 1.6 per cent, a factor of 9. This is the G-dwarf problem, and the reason it is an argument rather than a discrepancy is that the failure cannot be fixed by changing the yield: the yield sets where the peak is, and moving the peak to fix the tail moves it away from the data. What is wrong is a boundary condition. Let gas keep arriving — pristine, at roughly the rate it is being consumed — and the gas is never both abundant and metal-poor for long, so few stars form while it is. That is the second curve, with 3.2 per cent below -1, and it needed no new nucleosynthesis and no new parameter beyond the fact of accretion. The picture cannot show the thing that would settle it directly, which is the infall itself: the gas arriving on the disc now is a few solar masses a year spread over twenty kiloparsecs, and it has never been securely detected. Galaxies

A histogram that says the box was not closed

The simplest model of a galaxy enriching itself has exactly one parameter and predicts the whole metallicity distribution of its surviving stars. The solar neighbourhood's disagrees, in a specific direction — and the failure is not in the nucleosynthesis but in a boundary condition.

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

A distance measured with a stopwatch

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

 Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround. Cosmology

One number that is an age, a size and a density

The slope of the velocity–distance relation has units of inverse time, so it can be read as an age, multiplied by c to give a length, or squared to give a density. All three readings are natural, all three are quoted, and not one of them is the quantity it appears to be.

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

The same ruler measured twice, ten billion years apart

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

 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.

 Beyond redshift 1.87, what a galaxy does today will never be seen. For a galaxy at each redshift, the cosmic time of the last event on it that will ever be visible from here — not the last that has arrived, but the last that ever arrives, integrated to infinite future time. The horizontal line is the present, 13.8 billion years. A galaxy below redshift 1.87 has its curve above that line: its entire future will be seen from here, arriving ever more slowly and ever more redshifted, so it never quite disappears. A galaxy above redshift 1.87 has its curve below the line, and that is the whole content of an event horizon: what such a galaxy is doing today will never be seen, ever, by anybody here. Only a finite slice of its history is coming, and when the last of that light arrives the object stops changing. The redshift at which the curve crosses is 1.87, the comoving distance there is 16.7 billion light years against a particle horizon of 46.1, and the ratio of the volumes says that 95 per cent of the galaxies now observable are already beyond reach. Superluminal recession is not what does this. Everything past about redshift 1.5 has always been receding faster than light and is seen perfectly well, because the Hubble sphere grows to meet the photon; what closes the horizon is that with a cosmological constant the comoving Hubble radius stops growing and begins to shrink, so a photon that has not already been overtaken never will be. Cosmology

The galaxies that are already out of reach

With a cosmological constant the comoving distance a photon can ever cover converges, so there is a redshift beyond which light leaving today never arrives. It is 1.87, and about ninety-five per cent of the galaxies now visible are past it — which superluminal recession has nothing to do with.

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

The number that would say whether it is a constant

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

 A line nothing spun up by accretion can lie above, and the millisecond pulsars beneath it. The period–period-derivative diagram with the spin-up line drawn on it. An accreting neutron star is torqued by the disc until its magnetosphere turns at the same rate as the material arriving there, which fixes an equilibrium period as a function of the magnetic field and the accretion rate. Eliminating the field between that relation and the dipole formula that every point in this diagram is already read through leaves a straight line of slope 1.33, drawn here for accretion at the Eddington rate — the fastest a star can be pushed. The 7 recycled pulsars all sit below it, which is what the figure is for: none of them was spun up faster than the limit allows, and their positions are a record of how much mass each one received rather than of how old it is. The young pulsars are in the opposite corner, above the line and to the right, spinning down from birth. The two populations are not two stages of one life. A star that reaches the bottom left has been fed by a companion for a hundred million years, which is why almost every millisecond pulsar has one and almost no young pulsar does. Stars

A corner of the diagram that has to be earned

A pulsar spinning a thousand times a second cannot have been born that way and stayed that way, because its own radiation would have slowed it in a few million years. It got there by being fed, and the line it cannot lie above is where the accretion torque balances the magnetic one.

 Mestel cooling for white dwarfs of 0.4 to 1 solar masses. Luminosity against age for white dwarfs of 0.4, 0.6, 0.8, 1 solar masses, both axes logarithmic. Each line is the whole of the star's later life: there is no nuclear source, so what is radiated is the thermal energy the ions had when the star was made, leaking out through a thin envelope whose opacity is Kramers'. That one sentence gives L ∝ t^(−7/5), and the lines are straight here because a power law is straight on these axes — the slope is the exponent and can be measured off the picture. Two features are worth more than the numbers. The first is how flat the sequence becomes: a factor of ten in luminosity costs a factor of 5.2 in age, so a white dwarf spends most of its existence faint, and any population of them piles up at the bottom. The second is that the heavier tracks lie above the lighter ones. A heavier white dwarf is smaller, so it has more ions to cool and less surface to lose them through, and at a given age it is the brighter object. The picture does not show what happens at the faint end, where the ions crystallise and the heat capacity stops being constant; that is a separate figure, and it is where this law stops being enough. Stars

A clock with no fuel in it

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

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

The part of a star that never slowed down

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

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

Four numbers that weigh a planet's core

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

 A period, a colour, and an age. Rotation period against colour for stars of 125, 625, 1000, 2500, 4570 million years, under the empirical relation P = t^0.5189 × 0.7725(B−V − 0.4)^0.601. The isochrones do not cross and are separated at every colour by exactly the age ratio raised to 0.5189, which is what allows a single measured period to be inverted for an age once the colour is known. The Sun, at B−V = 0.653 and 4570 million years, is placed by the relation at 26.8 days against the 25.4 days it is observed to have. Three clusters are marked at a common colour to show the spacing directly. The dashed boundary at the left is where the Rossby number — the period divided by the convective turnover time — passes 2 on the oldest isochrone, at B−V = 1.35: past that point the braking weakens and the relation is known to over-predict the age, which is the one place a rotation period stops being a clock. What the figure cannot show is the scatter, which is a few days at fixed colour and age and is the real error bar on any single star. Stars

The clock that starts by forgetting

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

 Two equilibria become four, and three worlds sit near the join. The Cassini equilibria of a spin axis, drawn against the ratio of its own precession rate to the rate at which its orbit plane turns, for an orbit inclination of 1.5 degrees. Each column of dots is the full set of obliquities at which the two precessions keep step at that ratio, found by root-finding rather than by tracing a remembered curve. Below α cos ε/|g| = 1.135 there are two such obliquities and above it there are four, and the figure checks both counts on either side of the join. The three marked bodies are placed by their own measured precession constants: the Earth with the Moon at 2.67, safely on the four-state side; the Earth without it at 0.86; and Mars at 1.06. Two of the three sit within a few tenths of the bifurcation, which is the whole reason their obliquities are not constants: near the join the equilibria are close together, the libration around them is wide, and a body pushed between neighbouring resonances wanders. The Moon's contribution to the Earth's precession constant is what moves the first mark away from that region, and the second mark is the same planet with that contribution removed. This is a two-frequency model of a many-frequency system, and the real chaos comes from the overlap of resonances it does not contain. The observed sky

A tilt that is not a constant

The Earth's axis leans by 23.4 degrees, and that lean is what makes the seasons. It is also a dynamical variable with its own equilibria, its own resonances and its own chaos — and on Mars the same variable has swung between nearly zero and sixty degrees without anything having to happen.

 A period below which every orbit is round. Orbital eccentricity against period for binaries in four clusters of 0.125, 0.625, 6, 4 billion years, with the eccentricities drawn from one seeded distribution and then damped by exp(−age/τ), where τ rises as the 5.333 power of the period. Each cluster shows the same thing: below a boundary period nothing survives eccentric, above it the original distribution is untouched, and there is almost nothing in between because the timescale is so steep. The boundary is a clock. It moves as the three-sixteenths power of the age, which the figure checks against the drawn curves, and the calibration puts it at 6.5 days at 125 million years, 8.8 at 625 million and 12.5 at four billion — against measured cut-offs near 7.2, 8.5 and 12.5 days in the Pleiades, the Hyades and M67. The boundaries are also read back off the plotted points rather than trusted, and required to move outward with age. This is the cleanest measurement of tidal dissipation in ordinary stars that exists, and its cleanliness comes from the ages: a cluster's age is read off its main-sequence turn-off and owes nothing whatever to the tide being measured. Orbits

A cut-off period that is an age

Plot eccentricity against orbital period for the binary stars of one cluster and the picture has a wall in it. Below a certain period every orbit is circular; above it, the original spread survives untouched. The wall moves outward as the cluster ages, and where it stands is a measurement of how stars dissipate a tide.

 A circular orbit has one lock, and an eccentric one has several. The strength of each spin–orbit resonance against orbital eccentricity, as the Hansen coefficient H(p, e) that multiplies the restoring torque on a permanently non-spherical body. At zero eccentricity every curve but the synchronous one is exactly zero — the figure checks that rather than showing it — so a body on a circular orbit can lock only by turning once per orbit. Away from zero the others switch on: at Mercury's eccentricity of 0.2056 the 3:2 resonance has 73 per cent of the synchronous one's strength and more than twice the 2:1's. A planet spinning down through this family therefore meets the 3:2 before the 1:1 and has a real chance of being caught there, which is what happened — Mercury turns three times for every two orbits, a fact discovered by radar in 1965 after a century of assuming it was locked. The free libration of that locked state follows from the same coefficient and the measured 2.03e-4 for (B − A)/C: 12.1 years, against a measured period near twelve. What the figure cannot show is the capture probability itself, which depends on how the tide dissipates and ranges from a few per cent for a simple constant-lag tide to more than half once friction between a liquid core and the mantle is included. Gravitation

A rotation locked to the orbit, but not one to one

Mercury turns exactly three times for every two circuits of the Sun. That was not what anybody expected, and it is not an accident — on a circular orbit a tidally despun body has exactly one place to lock, and on an eccentric one it has several — with the strength of each set by a coefficient that vanishes when the eccentricity does.

 A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 150 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically. The observed sky

A wobble that should have stopped

The Earth's rotation pole wanders across its own crust in a circle a few metres wide. A rigid Earth would do it in 305 days; it takes 433, and the difference is a measurement of the planet's elasticity. At the observed damping it should have died out within a human lifetime, and it has not.

 One step of memory, and it is kept at the poles. Two predictors of a solar cycle's amplitude, each tested against the seven cycles for which both quantities exist, with amplitudes scaled so that cycle 21 is one. Left: the polar field measured at the minimum before the cycle begins, which correlates with what follows at r = 1.00. Right: the amplitude of the previous maximum, which correlates at r = 0.34 — that is, not at all. The cycle is therefore not a pendulum with momentum; it is a process with exactly one state variable, and the variable is the poloidal field that the decay of the previous cycle's spots leaves behind at high latitude. That is the observational core of the flux-transport picture: spots emerge tilted, their trailing polarity drifts poleward, and what accumulates there is the seed the next cycle's shear will wind up. It is also the only prediction in solar physics with a lead time of years, and it was what said, correctly and unpopularly, that cycle 24 would be the weakest in a century. Stars

One step of memory kept at the poles

The amplitude of a solar maximum tells almost nothing about the next one. The polar field measured at the minimum in between tells a great deal — so the cycle is not a pendulum with momentum but a process with exactly one state variable, and the variable is a field of a few gauss at latitudes nobody can see well.

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

The exponent no pulsar has

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

 TCB has gained 23.5 seconds on TT since 1977. How far four of the solar system's time scales have drifted apart, against years since they were set equal. TT is the time a clock on the Earth's geoid keeps, and it is flat here by definition. TCG is the time a clock at rest just outside the Earth's gravitational well would keep, and it runs faster by seven parts in ten thousand million — 0.0220 seconds a year. TCB is the time a clock at rest outside the Sun's well would keep, and it runs faster by fifteen parts in a thousand million, or 0.4893 seconds a year. TDB is TCB with that rate divided out so that it stays within milliseconds of TT, which is what makes it usable as an ephemeris argument and what makes it a coordinate rather than a proper time. The differences are constants and they are not corrections: an ephemeris tabulated against one of these and evaluated against another is wrong by the whole of this plot. The observed sky

A second that depends on where the clock is

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

 A peak worth 10.8 in a narrow search is worth nothing in a wide one. The probability that noise alone produces a peak at least as tall as a given power, for searches over four different numbers of independent frequencies. A single frequency examined in isolation gives a one-per-cent chance at a power of 4.6; searching fifty thousand frequencies for the same one-per-cent chance requires 15.4. The threshold rises as the logarithm of the width of the search, which is why the penalty is survivable — but it is a penalty, it is often not applied, and the number of independent frequencies in an unevenly sampled time series is not the number of frequencies on the grid. Overestimating that count is conservative and underestimating it is not, which is the one asymmetry worth remembering. Starlight

The tallest peak in nothing at all

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

 Three causes, three shapes, one residual. An eclipse-timing residual: the observed minus the computed time of each eclipse, against cycle number, in days. Three effects are superposed and each has its own functional form. A slow change in the orbital period — from mass transfer or from magnetic braking — integrates to a parabola. A third body in a wide orbit moves the whole binary towards and away from the observer, so its light-travel time adds a sinusoid at the third body's period. And an eccentric orbit whose apsides are precessing moves the two eclipses in opposite directions, which is a sinusoid that changes sign between primary and secondary minima. The last is why both eclipses have to be timed: a third body moves them together and apsidal motion moves them apart, and a series of primary minima alone cannot tell the two apart at all. Stars

Three causes with three shapes in one curve

The times of eclipse in a binary star are a clock, and the clock runs late and early. Three completely different things make it do so — a third body, a changing period, and a slowly turning orbit — and they are separable only because each imposes a different shape on the residual.

 Two longitudes a satellite falls towards, and 0.5 m/s a year to stay elsewhere. The along-track potential a geostationary satellite feels, against longitude. The Earth's equator is slightly elliptical — about seventy metres between its long and short axes — and that one harmonic of the gravity field gives the geostationary ring two minima and two maxima. A satellite left unattended drifts towards the nearer minimum at 75° or 255° east, overshoots, and librates about it with a period of a couple of years. The peak acceleration accumulates 0.5 metres a second of velocity change a year, and an operational east–west budget is a small multiple of that once the correction cycle is accounted for — a fixed cost of every commercial slot in the ring, paid forever. The two minima are the graveyard of the geostationary population: uncontrolled satellites accumulate there, which is why they are the most crowded longitudes in the belt and why an uncontrolled object is most likely to be found near one. Spaceflight

A satellite that drifts to one of two longitudes

The Earth's equator is elliptical by about seventy metres. That one harmonic of the gravity field gives geostationary orbit a potential with two minima, so an unattended satellite slides towards the nearer one and stays — and every operational slot in the ring is paid for with fuel, every year, forever.

 One velocity, two distances — 4.7 and 9.5 kiloparsecs. The radial velocity of gas along a line of sight at galactic longitude 30 degrees, against distance from the Sun, for a flat rotation curve. The velocity rises to a maximum at the tangent point — where the line of sight is tangent to a circle of radius R₀ sin l — and falls again beyond it, so every velocity below the maximum corresponds to two distances. A cloud observed at 84 kilometres a second is at either 4.7 or 9.5 kiloparsecs, and nothing about its velocity says which. The two possibilities differ by a factor in distance and by its square in luminosity and mass, so the ambiguity is not a refinement — it decides whether a star-forming region is an ordinary one nearby or a monster on the far side of the Galaxy. Galaxies

One velocity and two distances

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

 Three perturbing masses drawing one curve. Transit-timing residuals for three systems whose perturbing planets are 12, 8, 5 Earth masses — a factor of 2.4 apart — each given the inner-planet eccentricity that the near-resonant theory says will compensate: 0.0000, 0.0166, 0.0363. The three curves have amplitudes of 1.7, 1.7, 1.7 minutes, within 0 per cent of each other, and they are drawn by three separate integrations that were told nothing about the theory used to pick the eccentricities. The eccentricity enters the near-resonant term divided by Δ, the fractional distance from exact resonance — here 0.0267 — so a hundredth of an eccentricity does the work of a factor of two in mass. This is why a transit-timing mass is not a mass until something else fixes the eccentricity, and why the masses that came out of the first years of such fits were systematically lower than the radial-velocity masses of the same planets. Exoplanets

A mass that is only a mass once the eccentricity is known

The near-resonant part of a transit-timing signal carries the perturber's mass and the pair's free eccentricity in the same bracket, divided by the distance from resonance. A hundredth of an eccentricity therefore does the work of a factor of two in mass, and three quite different systems draw one curve.

 June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is. Orbits

An average that precession cannot move

Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent.

 Windows every 780 days, costing between 5 and 13. The cheapest departure energy available in each of 8 consecutive launch opportunities, each found by solving a grid of Lambert problems around the window and taking the minimum. Opportunities recur every 780 days — the synodic period of Earth and Mars, which is exact and which is why the interval between missions is always about twenty-six months. Their cost is not periodic on that interval. The cheapest here is 5.0 km²/s² and the dearest 12.9, a factor of 2.59, and the pattern repeats on a period of about fifteen years rather than on the synodic one. The cause is Mars's eccentricity of 0.093: a transfer that arrives near Mars's perihelion has less distance to cover and meets a faster-moving planet, and whether an opportunity does that depends on where Mars is in its own orbit — which drifts relative to the synodic cycle by a fixed amount each time and comes back into phase after seven windows. In launch mass the factor is larger than it looks: departure energy enters the rocket equation through an exponential, so a C₃ of 13 rather than 5 costs roughly 1.08 times the propellant at departure. Spaceflight

The window that comes back and the cost that does not

Launch opportunities to Mars recur every 780 days exactly, because that is the synodic period and a synodic period is arithmetic. What they cost does not repeat on that interval at all — the cheapest window is a factor of two and a half below the dearest, and the pattern comes back every fifteen years rather than every two.

 A 1.1 solar-mass white dwarf held up for 2.8 extra billion years. The time a white dwarf takes to reach the crystallisation luminosity, and the two delays that follow, against mass. The lower band is bare Mestel cooling — thermal energy of the ions leaking out through an envelope whose opacity is Kramers'. On top of it sits the latent heat of crystallisation, 0.85 kT per ion released when the liquid interior freezes into a lattice; and on top of that the gravitational energy of ²²Ne settling through what is left, taken here as 0.6 kT per ion. Neither is fuel: both are energy the star already had, released late and radiated at the low luminosity it has by then, which is why so little of it buys so much time. The delay rises from 1.88 billion years at 0.5 solar masses to 2.81 at 1.1 — 58 per cent of the cooling already done. A white dwarf age computed from the bare law is too young, and it is too young by more the heavier the star is, which is exactly the direction that matters, because the massive white dwarfs are the ones used to date the oldest populations. Stars

A clock that stops while its interior freezes

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling — until the interior crystallises. The latent heat of that phase change, and the settling of a heavy isotope through what is left, hold a massive white dwarf up for nearly three extra billion years.

 Which inclinations a launch site can reach, and which it cannot. Orbital inclination against launch azimuth for three sites, from cos i = sin A cos φ. Due east is the only azimuth that gives the minimum, and that minimum is the latitude itself: Kourou 5.2°, Kennedy 28.5°, Baikonur 45.6°. Everything below the shaded line is unreachable from the highest-latitude site by any azimuth at all, and getting there costs a plane change afterwards — 5.94 km/s from a 400 km orbit to reach the equator from 45.6°. Baikonur in fact flies no lower than 51.6° rather than its 45.6°, and the extra 6.0° is overflight constraint rather than mechanics: the azimuth that would give 45.6° sends the spent stages over places they may not fall on. Spaceflight

A plane change paid at the worst speed there is

A launch reaches an inclination fixed by its latitude and its azimuth, and no azimuth reaches an inclination below the latitude. Getting there afterwards means turning at orbital speed, which is the most expensive place available — so a site's latitude is a floor no trajectory removes.

 The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes. The observed sky

Two fictitious suns in sequence

The equation of time is usually described as one quantity with two causes. It is better read as two separate reductions, each with its own imaginary body — one that moves uniformly along the ecliptic and one that moves uniformly along the equator — and the second is not the first.

 The extreme sunset and sunrise, neither of them on the December solstice. Sunset and sunrise in local mean solar time through the ninety days around the December solstice, at latitude 52°, computed from Kepler's equation and the tilt. Each is the solar noon — which is 12:00 minus the equation of time — plus or minus half the day arc, and the two ingredients have different stationary points. The day length is stationary at the solstice, exactly, because the declination is. The equation of time is not: it is changing at about 0.50 minutes a day there, moving solar noon steadily. So the sum is still changing after the day length has stopped, and the earliest sunset comes on 14 Dec — 9 days before the solstice on 23 Dec — while the latest sunrise comes on 31 Dec, 9 days after it. The shortest day is the solstice, and neither of its ends is extreme there. Nothing in this is an approximation or a correction: it is a sum of two functions with different stationary points, and the stationary point of a sum is not the stationary point of either. The observed sky

The earliest sunset is not the shortest day

A sunset time is solar noon plus half the day length, and the two have different stationary points. The day length stops shortening at the solstice; the equation of time does not stop moving, so the sum keeps falling — and the earliest sunset comes days before the shortest day.

 15 revolutions of a 185 km swath. The band an instrument 185 km wide sweeps over 15 revolutions of a 233/16 repeat orbit at 98.2° and 699.6 km, drawn on an equirectangular map. Each band's edges are placed 92.5 km either side of the track on the sphere, so the bands really are the same width everywhere and only look wider towards the poles because this projection stretches longitude by 1/cos φ. Successive passes are 24.72° of longitude apart, and across the track at the equator that is 2691 km — so neighbouring passes leave strips about 2,506 km wide unimaged between them at the equator, while near ±82° the same bands lie on top of one another many times over. Filling the tropical strips is what the rest of the 16-day cycle is for. Spaceflight

A swath is sized at the equator

An imaging satellite's tracks crowd together towards the poles and spread apart towards the equator, so whether a swath leaves gaps is settled at the equator and nowhere else. The order in which a repeat cycle then closes those gaps is not orbital mechanics at all. It is a theorem about points on a circle.

 How far across the ground a satellite can be seen from, at 0, 10, 30° masks. The footprint of a satellite — the largest angle at the Earth's centre between the point beneath it and a station that can still see it above an elevation mask — against altitude, on a logarithmic axis. λ = arccos(R cos ε / (R + h)) − ε. At low altitude it grows as the square root of the height, λ ≈ √(2h/R), drawn dashed for the horizon mask, so doubling a low orbit's altitude widens its footprint by only about forty per cent; far out it saturates, and no altitude sees past 90° − ε. At 420 km (a space station) the 0° footprint is 20.2°, a circle 2,254 km in radius holding 3.1 per cent of the Earth's surface. At 20,180 km (a navigation satellite) the 0° footprint is 76.1°, a circle 8,472 km in radius holding 38.0 per cent of the Earth's surface. At 35,786 km (geostationary) the 0° footprint is 81.3°, a circle 9,050 km in radius holding 42.4 per cent of the Earth's surface. Raising the mask costs most at low altitude: at 420 km a 10° mask takes 38 per cent off the footprint's radius and 62 per cent off its area, because most of what a low satellite could see is near the horizon, while at 35,786 km the same mask takes 20 per cent of the area. Spaceflight

The circle a station can see

A ground station can talk to a satellite only while the satellite is above its horizon, and the region it can do that from is a circle drawn round the point beneath the spacecraft. The circle grows as the square root of the altitude and then stops growing, the time spent inside it diverges at one altitude, and the stations that get the most contact are not where anyone would first put them.

 The figure of eight a geosynchronous orbit draws at 5°, 15°, 30° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 5, 15, 30°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15°, ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.109°, ±0.982°, ±3.927°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 8 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west. Spaceflight

A stationary satellite that draws a figure of eight

A satellite with a period of exactly one sidereal day returns over the same ground every day, but only an orbit in the equator with no eccentricity returns over a single point. A few degrees of tilt draw a figure of eight, a little eccentricity a swing in longitude, and the two together draw the figure the Sun draws in the sky over a year.

 A sidereal clock that runs up to 1.15 seconds ahead or behind, with the Moon's node. The equation of the equinoxes from 1990 to 2030: apparent sidereal time, counted from the true equinox, minus mean sidereal time, counted from an equinox that only precesses. It is the nutation in longitude times the cosine of the obliquity, and it is drawn here from the four largest nutation terms, which carry it to about a hundredth of a second. The dominant term follows the Moon's node round its 18.61-year cycle with an amplitude of ±1.052 s; riding on it are a half-yearly term of ±0.081 s from the Sun and a fortnightly one of ±0.014 s from the Moon. Over this span the sum runs from −1.147 to 1.146 s. None of it is the Earth's rotation: it is the zero point of the clock moving, because the zero point is the intersection of the equator with the ecliptic and the equator nods. The observed sky

A clock whose zero is moving

Sidereal time is counted from the equinox, and the equinox does not stay put. Its steady drift makes the sidereal day eight milliseconds short, its acceleration puts a quadratic term into the formula for sidereal time, and the Moon makes it nod by a second every nineteen years. The Earth rotation angle removes all three by counting from a point defined not to move along the equator.

 A 5/5/1 constellation at one instant: where no satellite is above 0°. The points beneath the 5 satellites of a 5/5/1 Walker delta pattern at 43.5°, at 11,605 km, at one instant, with each satellite's 0° footprint — a circle 69.2° in radius at the Earth's centre — drawn round it, on an equirectangular map that swells the circles towards the poles. The shaded cells are ground with no satellite above the mask: none at this instant. Everywhere else is seen by between 1 and 3 at once. The largest empty circle at this instant is 67.6° in radius, computed exactly from the satellites' directions, inside the footprint, which is why there are none. Spaceflight

Five satellites and not four

No single orbit keeps a satellite above every point on the Earth. How many are needed is a question about the largest empty circle among their directions at the worst instant, and it has sharp answers. Four can never do it, five can from 11,605 kilometres up, and past that the arrangement of the orbits matters as much as their number.

 When the Orion Nebula can be observed from latitude 52°, through a year. Every night of 2027, from local noon to the following noon, for a station at latitude 52°. The shaded cells are the times at which the Orion Nebula (right ascension 5.59 h, declination −5.4°) stands at least 20° above the horizon while the Sun is more than 18° below it. The two outer curves are the start and end of astronomical darkness, which never comes on 64 nights around midsummer; the diagonal line is the object's transit, which arrives four minutes earlier each night and wraps through the whole day once a year. It crosses local midnight on 15 Dec, when the object stands opposite the Sun. The shaded season is the stretch of the year either side of that date in which the transit falls inside the dark hours. From this latitude the object transits at 32.6° altitude; it is observable on 181 nights for at least an hour, for 6.3 hours on the best of them (3 Jan), and for 860 hours in the year. The observed sky

A right ascension is a date

An object can be observed only when two clocks agree — the sidereal clock that brings it high in the sky and the solar clock that makes the sky dark. The two drift apart by one turn a year, so every right ascension has a season, every latitude gives that season a different length, and an object that never sets is best observed at the opposite time of year from the one its right ascension names.

 Starobinsky: e-folds before the end, against how reheating went. N, the number of e-folds between the pivot scale leaving the Hubble radius and the end of inflation, for the Starobinsky potential, against the temperature at which reheating finished, for 3 equations of state during it. All the lines meet on the right at instant reheating, 2.6 × 10¹⁵ GeV, where N = 55.6. The left edge is 5 MeV, below which nucleosynthesis would not have happened. w = 0, oscillating field: 42.0 at 5 MeV; w = ⅓, like radiation: 55.6 at 5 MeV; w = 1, kination: 69.0 at 5 MeV. The reason is how far the universe stretches while the energy density falls: an oscillating field dilutes like matter, as a⁻³, so for the same fall in density it expands further than radiation would, more of the growth of today's scales happens after inflation, and fewer e-folds of inflation are needed to put them where they are. A stiff epoch with w = 1 dilutes as a⁻⁶, stretches less, and needs more. A radiation-like epoch changes nothing. None of this epoch has been observed; the lines are the arithmetic of energy and entropy, and the spread between them is how much an unobserved history moves a quantity the spectral index depends on. Cosmology

The epoch nobody saw moves the tilt

Between the end of inflation and the hot universe that made the light elements lies an interval nothing has observed, in which the energy of the inflaton became radiation. How long that took changes how many e-folds before the end the observed scales left — by as many as fourteen — and that moves every model's predicted spectral index by more than the measurement's uncertainty. A potential is never tested by the tilt alone; a potential and a reheating history are tested together.

 An ephemeris fitted to 120 days, 43 minutes wrong within a year against a band of ±5.1. Transit times of a 6 Earth-mass planet on a 10-day orbit, perturbed by a 14 Earth-mass planet at 15.24 days, integrated for 1460 days and compared with a straight-line ephemeris fitted only to the transits in the first 120 days — the shaded window. Inside the window the line fits to 2.1 minutes. Outside it the pair's 317-day super-period carries the transits away from the line, and within a year of the window closing the prediction is 42.8 minutes early of the observed transit, 347 days after the last fitted one. The narrow band is the formal three-sigma uncertainty of the same line for a timing precision of 0.5 minutes per transit, which at that date is ±5.1 minutes: the error is 8.4 times the band. A statistical uncertainty assumes the residuals are noise, and these are a signal, so the band describes a planet that does not exist. Exoplanets

A forecast that fails on a schedule

A transiting planet perturbed near a resonance keeps a clock that wanders, and a straight-line ephemeris fitted to part of the wander predicts the next transit with a confidence the wander does not deserve. The error is not noise and does not average down; it grows on the pair's super-period, it is many times the formal uncertainty within a year, and how soon it appears depends on which stretch of the wander happened to be observed. When a model that includes the known perturber still fails, the failure has a period, and the period is a planet.

 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.

 Mars: a sundial 40 minutes ahead and 51 behind. The equation of time on Mars over one of its years, 668.6 sols long, against sols since the northern spring equinox, computed from Kepler's equation for an orbit of eccentricity 0.0934 and an axis tilted 25.19°, with perihelion at solar longitude 250.87°. The solid curve is the difference between true and mean solar time: it runs from −51.1 local minutes, 611 sols after the equinox, to +39.9, 385 sols after it — −52.5 to +41.0 in Earth minutes, since a local minute is a 1,440th of a sol. The dashed curves are its two parts. The eccentricity term, one cycle a year, swings by ±42.9 minutes; the obliquity term, two cycles, by ±11.4; the first is 3.74 times the second. Perihelion falls 485 sols after the equinox, marked, and the vertical lines are the equinoxes and solstices. The observed sky

On Mars the orbit outweighs the tilt

The equation of time is the sum of two terms, one from the shape of the orbit and one from the tilt of the axis, and on the Earth they are nearly the same size. On Mars the orbit's term is almost four times the tilt's, a sundial runs from fifty-one minutes behind the clock to forty ahead, and the figure-of-eight the Sun traces in the Earth's sky becomes a teardrop. Nothing about the two terms is different; only their ratio is.

 A Sun that runs backwards for 8.1 days. The rate at which the Sun moves across the sky of a planet with a 3:2 spin–orbit ratio and eccentricity 0.2056, in degrees of hour angle per Earth day, against days from perihelion, over one 87.97-day orbit. The rate is the spin rate minus the rate at which the Sun's direction turns because the planet moves along its orbit, and by Kepler's second law that orbital rate peaks at perihelion, at 1.551 times its mean. The spin is 1.5 times the mean orbital rate, so for 8.1 days around perihelion, from −4.0 to +4.0 days, the orbital rate wins, the rate is negative, and the Sun moves backwards across the sky by 1.11 degrees before resuming. At perihelion it is moving at 0.21 degrees a day in the wrong direction. Away from perihelion the Sun crosses the sky at up to 3.4 degrees a day, and a whole solar day, noon to noon, takes 175.9 days — two orbits. The observed sky

A Sun that stops and runs backwards

On Mercury the equation of time is not a correction but a reversal. The planet turns three times for every two orbits, and near perihelion its orbital motion briefly outruns its spin, so the Sun halts, backs up by a degree over eight days and resumes. From one longitude that is three noons in a week; from another, a sunrise, a sunset and a second sunrise — and a slightly rounder orbit would have stopped it happening at all.

 Solar noon by the clock, in four places on ordinary time. The clock time at which the Sun crosses the meridian, through a year, for London (0.13°W, clocks on UTC+0), Madrid (3.70°W, clocks on UTC+1), Vigo (8.72°W, clocks on UTC+1), Kashgar (75.99°E, clocks on UTC+8), with summer time where it is kept, from the end of March to the end of October. In London noon falls between 11:44 (4 November) and 13:07 (27 July); in Madrid noon falls between 12:58 (4 November) and 14:21 (27 July); in Vigo noon falls between 13:18 (4 November) and 14:41 (27 July); in Kashgar noon falls between 14:40 (4 November) and 15:10 (12 February). The smooth wave on each curve is the equation of time, the same ±16 minutes everywhere. The steps are summer time, a whole hour. And the vertical offset of each curve is where the place sits inside its time zone, which for a city in the far west of a wide zone is larger than both: noon near three in the afternoon, by the clock, is an ordinary consequence of one time zone spanning a large country. The observed sky

The smallest term between a clock and the Sun

A clock and a sundial disagree for three reasons, and the one astronomy supplies — the equation of time, sixteen minutes at most — is usually the smallest of them. Where a place sits inside its time zone can put noon three hours after twelve, and summer time adds a whole hour on top. The equation of time is visible in ordinary life only where the other two happen to vanish.

 An hour that was a twelfth of the daylight. The length of a daylight hour when the time from sunrise to sunset is divided into twelve, through a year, at Alexandria (31.2°N), Rome (41.9°N), London (51.5°N), Stockholm (59.3°N). At Alexandria the hour runs from 51 minutes at the winter solstice to 71 at the summer solstice; at Rome the hour runs from 46 minutes at the winter solstice to 76 at the summer solstice; at London the hour runs from 39 minutes at the winter solstice to 83 at the summer solstice; at Stockholm the hour runs from 30 minutes at the winter solstice to 93 at the summer solstice. The dashed line is sixty minutes, the length it has at both equinoxes everywhere. This is the hour of the ancient Mediterranean world and of medieval Europe until mechanical clocks: an hour defined by the Sun, which a sundial with suitably drawn lines reads exactly, and in which the equation of time does not exist, because nothing is being compared with a uniform clock. Sunrise, sunset and noon are each defined by the Sun, and the clock that would disagree with them had not been built. The observed sky

An hour that stretched with the season

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

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

The brightest instant of an occultation is its middle

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

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

A population counted by shadows that never repeat

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

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

Each event pays for the prediction of the next

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

 Assimilation buys a factor of 3.6 at 2 hours and 1.03 at 14 days. The along-track position error of a low-orbit object against how far ahead the prediction reaches, on logarithmic axes. The upper curve uses a climatological density model, whose error stays at 15 per cent however long it is run — the limitation is the functional form and the proxies driving it rather than a shortage of data. The others assimilate the observed drag on objects already in orbit, which replaces that with an observation error of 3 per cent and then lets the thermosphere forget, with memories of 0.5, 1.5, 4 days. Every curve rises as the square of the time, because an error in a drag acceleration integrates twice into a position. The advantage is a factor of 3.6 at 2 hours and 1.03 at 14 days, so assimilation changes what a conjunction screening can do and changes nothing about a re-entry date — and the dashed line is the kilometre at which a close approach becomes a manoeuvre decision. Spaceflight

A weather forecast made out of orbits

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

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

The threshold that is not a threshold

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

 Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 23.4393°. The upper curve is noon on the summer solstice and the lower is noon on the winter one; they are the same function of latitude displaced by 23.4393° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 23.44° — the Sun is overhead at noon there on exactly one day, and somewhere inside it on every other day of the year. Where the lower curve reaches 0° is the polar circle, at 66.56° — the Sun fails to clear the horizon on the winter solstice, and on more days the further poleward one goes. They are the same inequality: |φ| ≤ ε for the first and |φ| ≥ 90° − ε for the second, and an obliquity of zero would collapse the tropics to the equator and push the polar circles to the poles, leaving one zone. The areas are the part that is not intuitive. The fraction of a sphere between two latitudes is the difference of their sines, so the tropics — a band a quarter of the way to the pole — hold 39.8 per cent of the Earth's surface, the temperate zones 52.0 per cent, and the polar caps only 8.3. The zone where the Sun can be overhead is 4.8 times the area of the zone where it can fail to rise, and both boundaries are the same 23.44°. The observed sky

Five zones, and one angle

The tropics are where the Sun can stand overhead; the polar circles are where it can fail to rise. Both boundaries are 23.44° measured from opposite ends, and the zone the Sun can reach is nearly five times the area of the zone it can miss.

 The hottest month is not the sunniest. The annual cycle of insolation at latitude 45°, and the temperature three surfaces of different heat capacity answer it with — each curve scaled to its own peak, because what is being compared is timing and not degrees. A surface losing 2 W m⁻² per kelvin above equilibrium obeys C dT/dt = Q − λT, and for a sinusoidal forcing that is one line of algebra: the response lags by arctan(ωC/λ) and is reduced by (1 + (ωC/λ)²)^(−½). a continental interior, 2.4 m of water, lags by 45.6 days and swings by 71 per cent of what a massless surface would; a shallow shelf sea, 10 m of water, lags by 77.6 days and swings by 23 per cent of what a massless surface would; a deep ocean mixed layer, 50 m of water, lags by 88.5 days and swings by 4.8 per cent of what a massless surface would. The lag can never reach a quarter of a cycle — three months — because an arctangent cannot reach 90°, which is why no inhabited place has its warmest month in December in the north, and why the sea comes closest. And the two numbers are one number: the same ωC/λ that carries the phase towards the quarter cycle divides the amplitude, so a long lag is bought with a small swing and cannot be had any other way. The solstice falls on day 170 and the peak insolation with it; the warmest day at this latitude follows between 46 and 89 days later depending on what is underneath. What no curve here can show is the atmosphere's own transport, which carries heat sideways between the surfaces drawn and makes each one's effective capacity partly its neighbour's. The observed sky

The hottest month is not the sunniest

A surface with a heat capacity answers a sinusoid late and small, and the two are the same number. The lag can never reach a quarter of a cycle — three months for a year, six hours for a day — because an arctangent cannot reach ninety degrees.

 A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism. Galaxies

The iron clock has no single delay

The α-element knee is drawn as though Type Ia supernovae switched on a billion years after the stars that made them. Measured rates say otherwise — the delays are spread evenly over every decade from forty million years to a Hubble time, as a power law with no timescale in it — and a clock with no timescale bends where a clock with one would break.

 Jupiter and Saturn meet every 19.86 years, tracing a three-cornered figure that turns 8.5° each round. The heliocentric longitudes at which Jupiter and Saturn are in conjunction — the same longitude seen from the Sun — for 21 successive conjunctions from 1800 to 2200, computed from Keplerian elements and dotted in three colours for the first, middle and last thirds of the span. The mean interval is 19.857 years, the synodic period the two mean motions give. Each conjunction falls 242.8° further round the orbit of Saturn than the one before, so 3 of them come back within 8.5° of where they started: the conjunctions sit near the corners of a 3-sided figure, and the figure itself rotates by 8.5° every 59.6 years. At that rate it returns to its starting orientation — a figure with 3 identical corners only needs to turn by a third of a turn — after about 838 years. The drawn corners are not exactly repeated because the orbits are ellipses: the planets move faster near perihelion, and the conjunction longitudes cluster where both are slow. That near-return is not a coincidence of dates. It is the statement that 3 synodic periods are close to a whole number of each planet's years, which is a near-commensurability of the two mean motions — and near-commensurabilities are where planets perturb one another most. The elements are a fit valid between 1800 and 2050; outside those years they are carried as fixed ellipses turning at their mean rates, which is right for the pattern and not for any individual date. The observed sky

A triangle of meetings that turns in eight centuries

Jupiter and Saturn meet every twenty years, and each meeting falls about two-thirds of the way round the sky from the last, so the meetings trace a triangle. The triangle turns a third of a turn in 838 years because five of Jupiter's years almost equal two of Saturn's — and that same near-fit is the largest perturbation in the solar system, the one that made Saturn appear to be slowing down.

 A limit with a notch cut in the middle of it. The stability limit for a satellite, in units of its planet's Hill radius, against the inclination of its orbit to the planet's. The two flat stretches are the coplanar answers — 0.5 prograde and 0.7 retrograde, the difference between them being the Coriolis asymmetry that a rotating frame imposes and no potential can hold. Between them the limit is not an interpolation. Inside the shaded band, from 39.23° to 140.77°, the Sun's averaged perturbation exchanges the orbit's inclination for its eccentricity, and a satellite whose semi-major axis is comfortably inside the lobe has an apocentre that is not — so the boundary is set by a(1 + eₘₐₓ) rather than by a, and it falls to exactly half the coplanar value at ninety degrees. The band edges are cos²i = 3/5 exactly, with no fitted quantity anywhere in them, and the notch is the reason a quoted stability limit in Hill radii is a statement about a coplanar orbit and not about a satellite. Gravitation

A stability limit with a notch cut in it

The two numbers usually quoted for how far out a moon may orbit are coplanar numbers. A tilted orbit is not somewhere between them: inside a band of inclination fifty-six degrees wide the Sun converts the tilt into an eccentricity, the apocentre rather than the semi-major axis has to stay inside the lobe, and the limit falls to half. The band is empty in the sky.

 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.

 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.

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

A rocking nobody started

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

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

A constant measured through a Sun that is getting lighter

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

 A hotter summer that is also a shorter one. Daily-mean sunlight at 65°N through one year beginning at the March equinox, for an orbit of eccentricity 0.0167 and obliquity 23.44°, drawn twice: with perihelion at the June solstice, when the northern summer is closest to the Sun, and with perihelion at the December solstice, half a precession cycle later. The first summer peaks 6.9% higher, and it is also shorter, because the Earth moves fastest near perihelion: days above 275 W m⁻² number 139 against 144. The energy delivered on those days — the shaded area above the line — is 5.055 against 4.989 GJ m⁻², a difference of 1.3%. The second law makes the intensity and the duration trade against each other almost exactly, so a summer measured by how much energy it delivers barely feels precession at all. Orbits

A hotter summer that is also a shorter one

When the Earth passes closest to the Sun in northern summer, the northern summer is brighter — and, by Kepler's second law, shorter by exactly the right amount to deliver almost the same energy. Measured by the energy it delivers above a melting threshold rather than by its brightest day, a summer barely feels the precession cycle at all and feels the tilt of the axis several times more strongly.

 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.

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

A time scale that is finished after it is used

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

 The rate at which the solar wind opens the Earth's field. The rate at which reconnection at the dayside magnetopause turns closed field lines into open ones, as a voltage in kilovolts — webers of flux per second — against the clock angle of the interplanetary field in the plane perpendicular to the Sun–Earth line, 0° pointing north and 180° south, for a wind at 450 km/s carrying transverse fields of 5, 10, 20 nT. The law is empirical: the rate rises as the speed to the four-thirds, in proportion to the field, and as the sine of half the clock angle to the power 4.5. Due south, a 5 nT field opens flux at 57 kV; turned to within 20° of north, at 22 volts. The steepness in angle is the whole of the solar wind's leverage on the magnetosphere: the wind's pressure decides how large the cavity is, and one component of its field decides whether anything gets in. The observed sky

A substorm timed by webers and volts

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

 When a target can be observed from a telescope that cannot look near the Sun. The days of the year (across) on which a target at ecliptic longitude 0° and a given ecliptic latitude (up) can be observed by a telescope allowed to point only between 85° and 135° from the Sun, the constraint of a large infrared telescope at the Sun–Earth L2 point. The Sun moves along the ecliptic once a year, and the allowed band sweeps round with it. A target on the ecliptic is inside the band twice a year, for about 51 days each time, a little before and a little after its opposition — never at it, since opposition is 180° from the Sun. The two windows widen with latitude and merge into one above 45°, where the elongation at opposition no longer exceeds 135°; within 5° of either ecliptic pole, where the elongation at conjunction no longer falls below 85°, the target is never outside the band at all, which is the continuous viewing zone. The pattern depends on ecliptic, not equatorial, coordinates: the season of a space telescope is set by the Sun's path, and the Earth's axis has no part in it. The observed sky

A season without a night

A telescope in space has no horizon and no night, and it can still only look at most of the sky for part of the year. Its season is a band of solar elongation swept round the ecliptic by the Sun, so an object on the ecliptic can be seen for two stretches of about seven weeks and never at opposition, while the few degrees round the ecliptic poles can be watched all year. From low orbit the Earth takes the night's place, and the zones it never hides are carried round the sky every eight weeks by the planet's equatorial bulge.

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

A crossing that may belong to the fitting form

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

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

The first three minutes use two lifetimes

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

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

The part of a delay no orbit can imitate

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

 The eccentricity the planets force on an asteroid, and where it diverges. The forced eccentricity of a body in the inner Solar System against its semi-major axis, from the linear secular theory with all eight planets — its two giant-planet frequencies set to the measured 4.257 and 28.245 arcseconds a year — on a logarithmic axis: the largest value it reaches over the planetary cycles (solid) and its present value (dashed). Across most of the asteroid belt it is a few hundredths — 0.048 at 2.8 AU — the part of every asteroid's eccentricity that belongs to Jupiter and Saturn rather than to the asteroid. Near 2.04 AU it diverges: there the asteroid's own apsidal precession rate equals the frequency of the planetary mode dominated by Saturn, the denominator A − g₆ vanishes, and the theory's forced eccentricity goes to infinity. That is the ν₆ secular resonance. Real asteroids near it do not reach infinite eccentricity; they reach eccentricities large enough to cross the orbits of Mars and the Earth, and are removed. The location is for a body of low eccentricity and inclination; the full theory curves the resonance to larger distances at higher inclination, which is why the belt's inner edge is not a single number. Orbits

A denominator that emptied the inner belt

The eccentricity Jupiter and Saturn force on an asteroid is a fraction, and its denominator is the difference between the asteroid's own precession rate and one of the planets'. At two astronomical units that difference is zero. Where the zero sits depends on where Saturn is — and the belt's quiet orbits say Saturn did not take long to get there.

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

An age read from angles that meet

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

 How often Easter falls on each date, over the Gregorian computus's whole cycle. The fraction of years in which Gregorian Easter falls on each of its 35 possible dates, 22 March to 25 April, computed year by year over 5.7 million years — the length after which the computus repeats exactly, because the golden number, the weekday and the two century corrections only then return to the same combination. The distribution rises from 0.48 per cent on 22 March through a plateau near 3.3 per cent a date to a spike of 3.87 per cent on 19 April, then falls to 0.74 per cent on 25 April. The ramps at each end come from adding a Sunday to a full moon: the earliest dates need both the earliest full moon and the right weekday. The spike on 19 April has no astronomical meaning at all; it is made by a rule in the table that forbids the paschal full moon to fall on 19 April and moves it to the 18th, so the 18th is a full-moon date half as often again as its neighbours. The observed sky

A full moon computed from the year number

The date of Easter is defined by the Moon and computed without looking at it — ten lines of integer arithmetic on the year number, a finite machine that repeats itself exactly every 5.7 million years. Its output has a spike on 19 April that no sky produces, a table that has kept pace with the real Moon for four centuries, and a Julian twin that is now four days behind it.

 How long a star lives on the main sequence, down to the smallest stars. Main-sequence lifetime against mass, on logarithmic axes, as the hydrogen a star burns divided by its luminosity, scaled so that the Sun lives ten billion years. The dashed line assumes every star burns a tenth of its hydrogen, as the Sun does; the solid line lets stars below about a third of a solar mass, which are convective throughout, burn nearly all of it. The horizontal line is the age of the universe, 13.8 billion years: every star lighter than 0.90 solar masses that has ever formed is still on the main sequence. A star of a tenth of a solar mass lives about 8 trillion years — 597 times the present age of the universe — and its life so far is a few thousandths of one per cent of it. Stars

A lifetime that is still a forecast

No star lighter than nine-tenths of the Sun has ever died, because none has had time. The smallest are convective to their centres, burn almost all their hydrogen rather than a tenth of it, and will shine for trillions of years. Most of the stars in the Galaxy therefore have a life history no observation has tested, and only one in thirteen stars ever born has finished.

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

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

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

 Mars's tilt over fifty million years, and a steadier twin. The obliquity of a Mars-like planet against time over 50 million years, integrated from 25 degrees with its orbital plane wobbling in five secular modes. With Mars's precession constant, 8.26 arcseconds a year, the axis wanders between 1 and 70 degrees, lingering near some values and lurching between them. The same planet precessing 6 times faster — about the Earth's rate, which the Moon's pull makes possible — stays between 23.2 and 30.8 degrees, because its precession is too fast to resonate with any of the orbital-plane frequencies. The orbital perturbations are illustrative, of the size Mars's own are; the chaos is computed from them. The observed sky

A past that can only be given as odds

Mars's axis is tilted by twenty-five degrees today, and nothing holds it there. Its slow precession falls among the frequencies at which its orbital plane wobbles, the resonances overlap, and the tilt wanders chaotically between nearly zero and sixty degrees. Two calculations that start a hair apart part company within a few tens of millions of years — so Mars's obliquity a billion years ago is not a number anybody will compute. What can be computed is how much of its history the planet spent at each tilt, and that distribution is a prediction the ice at its poles can test.

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

A core that wobbles at its own period

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

 How far the clock drifts from the Sun once leap seconds stop. The accumulated difference between the Earth's rotation and atomic clock time, in seconds, that builds up after leap seconds are abandoned, against years since then, for a day that now exceeds 86,400 SI seconds by −0.3 ms, 0.5 ms, 1.5 ms, each lengthening by 1.7 milliseconds a century as the tides brake the Earth. The drift is the integral of the excess: 20 s after 100 years, a minute after 158 for −0.3 ms; 49 s after 100 years, a minute after 113 for 0.5 ms; 86 s after 100 years, a minute after 76 for 1.5 ms. The present excess is close to zero — the day has recently been slightly shorter than 86,400 seconds — so the drift starts slowly or even backwards, and the tidal term then takes over and bends every curve upward. The time for the difference to reach a minute is set far more by the next century's core-driven fluctuations, which have swung the excess by two or three milliseconds within a few decades before, than by anything that can be predicted. The observed sky

A clock allowed to leave the Sun

Twenty-seven leap seconds have kept civil time within a second of the Earth's rotation since 1972. By 2035 they will stop, and the clock will be allowed to drift from the Sun. How far it drifts, and how soon a minute accumulates, depends on a length of day that no one can forecast.

 The stability of a pulsar clock against atomic clocks, over averaging times from a day to thirty years. Fractional frequency instability, on a logarithmic axis, against averaging time from a day to thirty years. A single millisecond pulsar timed to 300 nanoseconds every 14 days improves as τ^(−3/2) while its white timing noise averages down, and flattens where its slow spin noise, drawn at 10⁻¹⁵ at ten years, takes over. An ensemble of 20 such pulsars is 4.5 times better if their noises are independent. The atomic curves are representative: a hydrogen maser, excellent at hours and drifting at years, and a steered atomic time scale limited near 3·10⁻¹⁶ at long times. In this representative comparison the pulsar ensemble becomes the steadier of the two beyond about 3 years of averaging. At a day a pulsar is a billion times worse than an atomic clock; over decades it is competitive, because nothing inside a neutron star changes on the timescales on which laboratory clocks are rebuilt. The observed sky

A clock made of neutron stars

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

 The same tilt, and ice ages of two different lengths. Top: the obliquity anomaly in degrees from the five largest terms of the standard series, over 2 million years of model time; it is dominated by the 41,000-year cycle and has no power at all near 100,000 years. Bottom: ice volume in a threshold model driven by it — the ice grows steadily, and it collapses whenever volume plus a multiple of the obliquity anomaly crosses a threshold, marked by dots. For the first 1 million years the threshold is low and the ice collapses at nearly every obliquity maximum, on average every 43 thousand years. At the vertical line the threshold is raised, and nothing else changes: the ice now survives one or two obliquity maxima and collapses at the second or third, every 108 thousand years on average. The forcing is identical on both sides; the ice sheet counts it differently. Orbits

An ice sheet that skips a beat of the tilt

The last eight hundred thousand years of ice ages come roughly every hundred thousand years, and the Earth's orbit has no forcing of any strength at that period. A model in which an ice sheet collapses only when it is both large and warmed by a high-obliquity summer reproduces the rhythm with nothing but the forty-one-thousand-year tilt — the ice sheet grows through one or two maxima and falls on the next — and a slow rise in how large the ice must grow turns the older forty-one-thousand-year world into the newer one without any change in the orbit.

 The tide's spectrum, with every frequency set by the sky. The equilibrium tide-raising potential broken into its largest constituents, each a sinusoid whose frequency is a sum of whole-number multiples of six astronomical rates — the Earth's turning relative to the Moon, the Moon's and the Sun's motions round the sky, and the slow turning of the lunar perigee and node — and whose height is its equilibrium amplitude relative to M₂. They fall in three bands: near two cycles a day (M₂ at 1.9323, S₂ at exactly 2, N₂, K₂), near one (K₁, O₁, P₁, Q₁), and near zero (fortnightly, monthly, semi-annual). Every frequency is known to many decimal places from astronomy; what a harbour does to each, its own amplitude and phase, is not, and is the only thing a tide table has to measure. Gravitation

A tide table is astronomy with the harbour fitted

Nobody can compute the tide at a harbour from the Moon and the Sun, because the ocean's response depends on coastlines and depths nobody models well enough. Tides are predicted to centimetres years ahead anyway, because the frequencies are pure astronomy and only the response to each one has to be measured. The method needs a record long enough to tell neighbouring frequencies apart, and a correction for a cycle of 18.6 years.

 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.

 A solar day that runs off to infinity as the spin approaches the orbit. The length of the solar day — sunrise to sunrise — in units of the world's own year, against its sidereal rotation period in the same units, with retrograde spin to the left. A world that spins much faster than it orbits has a solar day barely longer than its rotation, which is the Earth's case: its rotation is a few thousandths of its year, and it sits at the foot of the curve just right of zero, too close to the axis to mark. As the spin slows towards the orbital rate the solar day diverges, because the world turns to follow the Sun: at exactly one rotation per orbit the Sun stands still in its sky and there is no day at all. Mercury, turning three times in two of its years, has a solar day of 2.000 years — 176 Earth days. Venus, turning slowly backwards, has one of 116.8 days, 0.52 of its years, with the Sun rising in the west. The definitions of a day and of a year that the Earth keeps far apart come within a factor of two of each other, or exchange places. The observed sky

A day longer than the year

On the Earth a day and a year are far apart, so a calendar has only to count one in the other. On a world whose spin is not much faster than its orbit, the solar day runs off towards infinity: Mercury's day lasts two of its years, Venus's sun rises in the west twice a year, and a locked moon has a day set by its planet. Every calendar problem met on the Earth returns, and some of them return inverted.

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

A frame fixed before its centre was found

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

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

Every thawing field falls on one line

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

 What DI Herculis's orbit should do, and what it does. The rate at which the periastron of the eclipsing binary DI Herculis advances, in degrees per century. Top: the prediction for stars whose spins are aligned with the orbit — 2.34 from general relativity, computed from the masses (5.17 and 4.52 solar masses), the 10.550164-day period and the eccentricity of 0.489; and 1.93 from the stars' tidal and rotational distortion, as published stellar models give it, of which the rotational flattening is 1.50 and the tides 0.43. Total 4.27. Bottom: the observed rate, between about 0.6 and 1.4 degrees per century in the published timing analyses — a quarter to a seventh of the prediction. For a quarter of a century after the discrepancy was established it was cited as possible evidence against general relativity, which contributes more than half the predicted total by itself. Stars

A precession slowed by stars lying on their sides

The eclipsing binary DI Herculis turns the long axis of its orbit at a quarter to a seventh of the rate relativity and the stars' own distortion predict, and for a quarter of a century that gap was offered as evidence against general relativity. The answer was in the stars: both spin nearly on their sides, and a flattened star tipped past 54.7 degrees drags the orbit's periastron backwards instead of forwards. Explained, the slow precession becomes a measurement of something no spectrum shows — how far the spins lean towards the Earth.

 One pulsar's stability, limited by the background every pulsar shares. The cubic-fit stability of a single millisecond pulsar, on logarithmic axes, against averaging time from 0.3 to 40 years. White timing noise of 100 ns every 14 days averages down as τ^(−3/2) (falling dashed line). The gravitational-wave background at the amplitude the timing arrays reported in 2023, A = 2.4·10⁻¹⁵, raises the curve as τ^(2/3) (rising dashed line). Their sum (solid) is lowest at 2.8 years, at 2.8·10⁻¹⁵; averaging longer than that makes this pulsar a worse clock, not a better one, because the background grows faster in the residuals than the white noise shrinks. At ten years the background alone is 5.5·10⁻¹⁵. The observed sky

A clock whose best interval is three years

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

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

A young surface dated by a handful of holes

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

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

Layers that keep the rhythm and lose the dates

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

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

A Sun read through a shield that also changed

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

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

The shadow that also switches off the satellites

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

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

A schedule written by where the Sun is

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

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

Four satellites in view, and which four

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

 Errors the identity sees and errors it cannot. Propagations of an ellipse of eccentricity 0.6 through 0.37 of a period, spoiled four ways at sizes from 10⁻¹³ to a hundredth, plotted by their true relative position error across and by how far their coefficients miss f ġ − ḟ g = 1 up (both logarithmic; the bottom row is a miss below the arithmetic's resolution). A solve stopped early and coefficients jostled at random break the identity in proportion to the damage. A wrong time step and a wrong gravitational parameter do not break it at all, at any size — errors up to 0.009 of the position sit on the floor — because each produces an exact Kepler map, for a different question. One equation on four numbers leaves three directions unchecked, and two of them are the two inputs every propagation depends on. Orbits

A check that cannot see the clock

The four Lagrange coefficients that carry an orbit's state forward satisfy one identity exactly, and a propagation that breaks it is wrong by about as much as it breaks it. But one equation on four numbers leaves three directions unchecked, and two of them are the inputs every propagation depends on: the time and the strength of gravity. A state carried forward for the wrong time, or under the wrong gravitational parameter, satisfies the identity perfectly — and a solver can be written so that its own errors take exactly that form.

 Four tropical years, one for each season. How long the Earth takes to return to the same season, measured from each point of its orbit, minus the mean tropical year of 365.24219 days, in seconds, against the season point (the Sun's longitude). The perihelion drifts relative to the equinox, so between two returns of the same season the planet must cover slightly less than a full orbit in true anomaly, and that shortfall takes longest where the planet moves slowest. Measured from the March equinox the year is longer than the mean by 12 seconds; from the June solstice, 50 shorter; from the December solstice, 49 longer. The full range is ±51 seconds. The Gregorian calendar's average year, 365.2425 days (dashed), is 27 seconds longer than the mean and 15 seconds from the year counted from the March equinox, the one it was built to hold. The observed sky

A year that depends on the season counting it

The tropical year is quoted to eight decimal places, and it is not one number. Counted from the June solstice it is fifty seconds shorter than counted from the December solstice, because the Earth's perihelion moves through the seasons and the planet covers the last of each year's shortfall at whatever speed it happens to be moving there. The Gregorian calendar was fitted to the year counted from the March equinox, which is why its average year is the length it is — and that year is itself drifting. On Mars the spread is fourteen minutes, and a calendar's leap rule can be more precise than the year it is meant to hold.

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

Four clocks and the stars none of them reads

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

 The rhythms in the rock, set by the Moon's distance. The periods of the climatic precession cycles — the precession constant plus the perihelion frequency of each of the planets that dominate the Earth's eccentricity — and of the obliquity cycle, the precession constant plus the node frequency s₃, against the Earth–Moon distance. Today they are 23.8, 22.4, 19.0 thousand years for precession and 41 for obliquity. At 341,000 km they were 14.7, 14.2, 12.8 and 20.0. The planetary frequencies are held at today's values, which the chaos of the inner solar system makes uncertain beyond a few tens of millions of years; the precession constant is the term that moves. The observed sky

A rhythm in the rock that measures the Moon

The Earth's axis precesses because the Sun and Moon pull on its equatorial bulge, and two-thirds of the pull is the Moon's. When the Moon was nearer and the day shorter, the axis turned faster, and the climate cycles that precession drives were shorter. Sediments laid down 1.4 billion years ago record those cycles — and with nothing in the rock but layer thicknesses, the ratio of two rhythms gives the Moon's distance at the time.

 The same data searched with a box and with a sine. The signal-to-noise ratio of the best fit at each trial period, for the light curve of the previous figure, searched two ways: with a box of three trial durations at every phase, and with a sinusoid. The box search peaks at 3.85 days with a signal-to-noise ratio of 7.3, against a true period of 3.85. The sinusoid's best peak anywhere is 3.3, at 1.59 days, and at the true period it reaches only 1.5 — no higher than the peaks the noise makes elsewhere in the search. The planet is a clear detection to one method and invisible to the other, in the same numbers. Starlight

A periodogram shaped like a shadow

A periodogram that fits sinusoids sees a transiting planet only through the fundamental harmonic of its box-shaped dip, which for a short transit is a few per cent of the signal. Folding the data at each trial period and fitting a box recovers all of it — five times the signal-to-noise for a transit two per cent of the orbit long, eighteen times for an Earth crossing the Sun. The box pays for its extra parameters in trials, and the price is about one unit of signal-to-noise.

 Three energies a freezing white dwarf radiates late. The age at which a white dwarf has finished crystallising, against its mass, built up from bare Mestel cooling and three energies released at the crystallisation luminosity: the latent heat of freezing (0.85 kT per ion), the gravitational energy of neon-22 settling (0.6), and the gravitational energy of carbon and oxygen separating (0.8). At 0.6 solar masses the three together add 3.20 billion years to 3.56 of cooling; at 1.1 they add 4.36 to 4.85. The separation term is, on these numbers, the second largest of the three. None of them is fuel; all are energy the star already had, released at a luminosity so low that a little of it takes a long time to radiate. Stars

A star that distils itself as it freezes

A carbon–oxygen white dwarf does not freeze at one composition. The crystals that form at its centre take more oxygen than the liquid they form from, the carbon-rich liquid left behind is lighter and rises, and the star slowly sorts its own interior, oxygen down and carbon up. The gravitational energy that releases is radiated at a luminosity so faint that it holds the star up for about a billion years — a delay comparable to the latent heat of freezing, and one of the larger uncertainties in every white-dwarf age.

 One population, two period–luminosity relations. The period–luminosity relation of 400 synthetic Cepheids spread across the instability strip, each behind a random amount of dust up to E(B − V) = 0.6: absolute magnitude in V (upper scatter) and in the Wesenheit index W = V − 2.44(V − I) (lower). In V the scatter about the fitted line is 0.57 magnitudes — 0.20 from the strip's width alone, the rest from the dust. In W it is 0.07: the dust is gone by construction, and most of the strip's width has gone with it, because a star that is hotter at a given period is bluer by about as much as dust would make it redder. That leftover 0.07 magnitudes is 3 per cent in distance per star. Stars

A scatter that points the way dust does

A Cepheid's period fixes its luminosity only to within the width of the instability strip, and at a given period the hotter stars are the brighter ones. Dust moves stars in the colour–magnitude diagram along a line too, and the two lines point nearly the same way. A magnitude built to cancel the dust — the Wesenheit index — therefore cancels most of the strip's width as well, which is why the tightest period–luminosity relations are the ones that ignore the extinction law's own advice.

 One parabola, two causes. Transit times of a planet on a 1.0914-day orbit over 12 years, minus the best straight-line ephemeris, with timing errors of 30 seconds — the shape a period shrinking by 29 milliseconds a year produces, a parabola 116 seconds deep. The curve is drawn once because it is the same curve for both explanations: an orbit decaying at that rate, or a star accelerating towards the observer at 2.9e-6 m/s², which shortens the light-travel time to each successive transit by the same steadily growing amount. Nothing in the transit times can tell them apart. Exoplanets

A clock that slows, or a star that falls

Transit times that curve away from a straight line can mean a planet's orbit is shrinking — the tides draining it into its star — or that the star itself is accelerating along the line of sight under a distant companion's pull. Both draw the same parabola, and the transit times cannot choose. The star's velocities can, but only for the shortest orbits: the acceleration that would fake a given decay grows as one over the period, and beyond a day or two an unseen Saturn can mimic more decay than the tides produce.

 Craters forming on Mars now, against the rate the clock assumes. The rate at which craters larger than a given diameter form on Mars, per square kilometre per year, against that diameter, both on logarithmic scales. The black curve is the production function scaled from the Moon, the rate every crater-count age on Mars assumes. The red curve and its points are the craters found forming between successive orbital images — 248 above 3.9 metres over the survey's coverage, 1.65 × 10⁻⁶ per square kilometre per year, with a cumulative slope of −2.2, shallower than the scaled function's −3. At 4 metres the scaled function predicts 6.4 times what is seen; the two cross near 42 metres; above that the survey has found too few craters to say. Bars are the counts' Poisson errors. Orbits

A clock checked by the craters it is making now

Every young age on Mars is a crater count divided by a rate borrowed from the Moon. Since 2006 the orbiting cameras have been watching that rate happen: hundreds of fresh craters, metres across, that were not there in the previous image. They form several times less often than the borrowed rate predicts at the smallest sizes, and more often than it at a few tens of metres — and the youngest surfaces on Mars are dated from exactly the sizes where the two disagree most.

 Where on the orbit the turning is done, and where the shadow falls. How effectively out-of-plane thrust turns a circular orbit's plane at each point round the orbit, measured from the ascending node: it goes as the absolute value of the cosine of that angle, greatest at the two nodes and zero halfway between them. The shaded bands are the Earth's shadow on an orbit 6,378 km up with the Sun in the orbit's plane, 60 degrees of the circle, placed once over a node and once between the nodes. Over the node the shadow switches the engine off where it would have done the most turning, and the orbit loses 25% of its turning; between the nodes it switches the engine off where it was doing little, and the loss is 7%. Spaceflight

A turn that costs more when the shadow sits on the node

An electric spacecraft turning its orbit's plane does most of the turning near the two nodes, where a push out of the plane tilts it most. Its engine runs on sunlight and stops in the Earth's shadow, and the shadow covers a sixth of the orbit wherever it happens to fall. When it falls on a node it removes a quarter of each orbit's turning, and a law that concentrates its thrust at the nodes loses half. The node drifts one way and the Sun the other, so the shadow slides across the nodes over weeks — and a transfer's length depends on the day it starts.

 An association that has been flying apart for ten million years. The velocity of each of 220 stars along one direction against its position along the same direction, for a model association whose stars were born 10 million years ago inside a region 1 pc across (one standard deviation), with random velocities of 1 km/s, and that began coasting apart at once. The fast stars have gone furthest, so position and velocity are correlated, and the slope of the line through them is the inverse of the time they have been moving: the fit gives 10.1 million years. A star at 10 pc moving at 1 km/s has been travelling for ten million years — which is what an expansion age is. Measuring it needs the stars' distances and their motions across the sky, both of which space astrometry now supplies for associations within a few hundred parsecs. Stars

An expansion clock read from the stars' motions

The masses of a young cluster's brown dwarfs are read from their brightness, and a brown dwarf's brightness is a mass multiplied by an age. The age can now be measured without any model of a star: the cluster is flying apart, and the time since it began is its size divided by its speed. But the expansion clock counts only the coasting, not the stars' lives, and it runs fast when the stars began spread out — two biases of opposite sign, and nothing in the motions themselves to say which one wins.

 Two ways of pointing an aperture, and the fork between them. The gas-to-young-star flux ratio inside apertures of size l, relative to the whole galaxy's, for apertures centred on peaks of molecular gas (upper branch) and on peaks of Hα from young stars (lower branch), if clouds live 20 Myr, the young stars shine in Hα for 4.3 Myr, the two overlap for 2 Myr and independent regions are 150 pc apart. On large scales every aperture averages many regions at all stages and both branches return to one. On small scales an aperture holds one region at one stage, and the branches split: ×2.1 on the gas peaks and ×0.10 on the stars at 10 pc. With the young stars' phase known from stellar evolution, the upper branch gives the overlap, the lower branch the clouds' lifetime, and the scale where the fork closes the regions' separation. Galaxies

A lifetime read from how gas and stars sit apart

No molecular cloud has been watched for long enough to see it age, and every cloud looks the same whether it lives for one free-fall time or for ten. But a galaxy holds thousands of clouds at every stage at once, and how often molecular gas and young stars appear in the same place — measured in apertures of shrinking size, centred first on one and then on the other — turns a single snapshot into a clock, calibrated by how long the young stars shine.

 The Sun's height, against the radius it is measured over. The Sun's height above the midplane as a survey would measure it, fitting the midplane to the stars within a projected radius R, against R, for a corrugation of 15 pc on a 2500-pc wavelength with the Sun 15 pc above the midplane beneath it. Nearby the answer is 15 pc whatever the phase; over several kiloparsecs it is the height above the mean plane: 30 pc with the Sun on a crest, 0 pc with it in a trough. Between them every intermediate value is a correct answer to a different question — a spread of 31 pc, comparable with the five to thirty parsecs the published measurements span. The observed sky

A midplane that depends on how far it is averaged

The Sun sits a little above the middle of the Galaxy's disc — twenty parsecs, by the most careful recent counts, and anywhere from five to thirty by others. If the disc were flat that scatter would be error. It is not flat: the disc near the Sun ripples vertically by tens of parsecs on kiloparsec wavelengths, and on a rippled disc the Sun's height above 'the' midplane depends on how far around it the midplane is averaged. Every value in the scatter can be a correct answer to a different question.

 Two old points, and the history they draw back to the Moon's birth. The Earth–Moon distance against time before the present, with the estimates from tidal rhythmites (620 million years), cyclic sediments (1.4 billion) and banded iron formation (2.46 billion, 321.8 thousand km, ±6.2). Today's tidal efficiency, scaled as the inverse sixth power of the distance, brings the Moon to the Earth 1.54 billion years ago. The efficiency fitted between the two oldest points — 21% of today's — held further back brings it out from the Earth 4.78 billion years ago, close to the Moon's age of about 4.5 billion. The constant law is an assumption; what the points measure is the average between them. The observed sky

A Moon measured in banded iron

Layers of iron-rich sediment laid down 2.46 billion years ago in what is now Western Australia keep time with the Earth's precession, and the precession's period says how far away the Moon was: about 321,800 kilometres, with a day of seventeen hours. Together with an older-is-closer point from 1.4 billion years ago, it measures how hard the tides worked between them — about a fifth as hard as today. Held further back, that efficiency brings the Moon out from the Earth a few hundred million years too early, which says the young tides worked harder than the middle-aged ones.

 How fast a meteor arrives is its orbit's inclination. The speed at which a meteoroid on a near-parabolic orbit with perihelion at 1 AU meets the Earth, against the orbit's inclination to the ecliptic: relative to the Earth far away (dashed) and at the top of the atmosphere after the Earth's gravity has added its pull (solid). A prograde orbit in the ecliptic overtakes the Earth at 12 km/s and enters at 17; a retrograde one meets it head-on at 72 and enters at 73. The points are four showers computed from their parents' elements at the node nearest 1 AU: Draconids 24 km/s at 31.9°, Geminids 38 km/s at 22.3°, Perseids 61 km/s at 113.5°, Leonids 72 km/s at 162.5°. A meteor's speed is a reading of the angle between its orbit and the Earth's. Orbits

A shower that is a reading of an orbit

Every August the Earth runs through the dust of comet Swift–Tuttle, and the meteors arrive at sixty kilometres a second from a point in Perseus. Each of those facts is an orbital element in disguise. The date is the longitude of the stream's node, the speed is its inclination, the radiant is the direction it comes from, and the shower exists at all only because one element happens to put a node within a hair of the Earth's orbit.

 A star that walks through a calendar with no leap day. The day of the 365-day Egyptian civil year on which Sirius first rose in the dawn, seen from latitude 30°, against year, from 3000 BC to AD 501, counting from a coincidence of the rising with the civil New Year in AD 139. The civil year was a quarter of a day short of the star's, so the rising arrived one civil day later every four years and went once round the whole calendar in about 1458 years — the Sothic cycle. The star's own year was 365.2504 days, within a few minutes of the Julian year of 365¼. The observed sky

A year kept by a star seen for one morning

For some seventy days each year Sirius is lost in the Sun's glare, and then one morning it rises just before dawn, for a few minutes, low in the east. The Egyptians kept a calendar of exactly 365 days, and against it that first morning slipped a day every four years, going round the whole calendar in about fourteen and a half centuries. The star's own year turned out to be the Julian year to within minutes — not because it had to be, but because two effects of precession cancel for Sirius seen from Egypt.

 The Moon's reach north and south, over two cycles of its node. The greatest declination the Moon reaches in a month, north or south, from 2010 to 2045: about 23.44° plus the orbit's 5.145° tilt times the cosine of the node's longitude, which regresses once in 18.61 years. At the major standstill (2025.1, and 2006.5, 2043.7) the Moon reaches 28.6°, five degrees beyond the Sun's solstice declination (dashed); at the minor standstill, 18.3°, five degrees short of it. Near each extreme the reach changes slowly for two or three years, which is the standstill. The observed sky

A Moon that rises further north than the Sun

The Sun's rising point swings north and south with the seasons between limits that do not change from one year to the next. The Moon's swings with the month, and its limits themselves open and close over 18.6 years as the plane of its orbit turns. At the widest, in 2024 and 2025, the Moon rose five degrees of declination further north than the midsummer Sun and skimmed the southern horizon of northern Scotland; at the narrowest, nine years later, it stays well inside the Sun's range.

 Jupiter and Saturn trading a degree of longitude over nine centuries. Mean longitudes of Jupiter and Saturn in arcminutes, with their straight trends taken out and a 40-year smoothing applied, from a planar integration of the Sun, Jupiter and Saturn from their elements at the year 2000. Jupiter swings ±20′ and Saturn ±50′, in opposite senses, with a period of about 870 years. Comparing Ptolemy's positions with those of his own day, Kepler could not fit one mean motion to both, and by 1676 Halley reported Jupiter gaining and Saturn losing. Before 1785 that looked as if the Solar System were unstable; the curve shows that it is a very long period. Orbits

A term too small to keep that a denominator kept

Every planetary theory is an expansion, and the order of its terms is meant to say which can be dropped. In the 1770s the largest unexplained motion in the Solar System, Jupiter speeding up and Saturn slowing down by a degree over centuries, came from a term that the ordering called negligible. Five of Saturn's orbits nearly match two of Jupiter's, and that one near-match divides a tiny third-order coefficient twice by a tiny frequency. An integration of the three bodies shows the swap directly, and shows what Laplace saw: it reverses every nine centuries.

 The Moon is late by anything from a quarter of an hour to well over an hour. Minutes between one moonrise and the next at latitude 52° N, against the Moon's longitude along its orbit, with the node at 0°. The daily delay is 10 minutes at longitude 354° and 84 at 222°. The mean of 50 minutes everyone quotes is true only as an average. A full Moon stands opposite the Sun, so it passes the shallow point near the September equinox, and the crescent passes it in spring. The observed sky

A full Moon that stops rising later

The Moon rises about fifty minutes later each day. That number is an average that hardly any night obeys. At fifty-two degrees north, in late September, the full Moon rises only ten or twenty minutes later for several nights running, so that a bright Moon stands in the evening sky for most of a week, just when harvest once needed it. Nothing about the Moon changes. It is the angle at which its path meets the horizon, and that angle goes through its lowest value at one point of the ecliptic, which the autumn's full Moon happens to occupy.

 Seven orbits in fifteen years, each turned from the last. The ellipse of an Aldrin cycler, semi-major axis 1.60 AU and eccentricity 0.390, in the Sun-centred frame, for 7 successive synodic periods, the line of apsides advanced by 48.7° each time. The inner circle is the Earth's orbit and the next is Mars's. Over about fifteen years the pattern has gone once round. The same spacecraft would need to be carried from each ellipse to the next by a flyby of the Earth, or a small burn. The drawing is the circular, coplanar case; the real one has to be recalculated for the actual orbit of Mars. Spaceflight

A timetable that orbits the Sun

Every trip to Mars leaves when the planets line up and accelerates everything it carries. Buzz Aldrin proposed in 1985 that there need be only one departure: a large ship on an orbit that returns to the Earth each synodic period, with crews riding small taxis out to catch it. The orbit almost exists without any propellant. It has a semi-major axis fixed by a single condition, an aphelion that passes Mars with nothing asked for, and one unavoidable bill: it must turn by about fifty degrees every twenty-six months, and the Earth can pay for all but seven of the eighty the turn requires.

 A binary that goes silent for part of its shrinking. The orbital period of a white dwarf of 0.8 M☉ and a donor from 0.5 M☉, in a one-zone evolution, against time. Mass transfer drives the period down fast while magnetic braking acts. At 3.17 hours the donor reaches 0.2 M☉ and loses its radiative core and with it the brake. Its radius, swollen by the fast transfer, relaxes and it falls inside its Roche lobe; transfer stops (dashed, no accretion, nothing to see). Gravitational waves alone close the orbit, until contact returns at 2.30 hours, 0.60 billion years later. The observed gap runs from 2.15 to 3.18 hours. The donor's thermal time at 0.2 M☉, set here to 3×10⁸ years, is chosen to place the upper edge, so that edge is fitted and the lower one is not. The period reaches a minimum of 68 minutes against an observed 80, a shortfall every model of this simplicity shares. Stars

A binary that goes silent for an hour of period

Binaries of a white dwarf feeding off a small star can be found at almost any orbital period down to about eighty minutes, except between about two and three hours, where they are strikingly rare. They have not stopped existing there. They have stopped shining. The moment the donor loses its radiative core, the wind's magnetic brake appears to switch off. The star, swollen by the transfer it was being driven through, shrinks back inside its own Roche lobe, and the feeding stops. Only gravitational waves remain to close the orbit, and it takes them about half a billion years.

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