Field

The observed sky

Coordinates, seasons, phases and shadows — geometry seen from inside it.

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 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.

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 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 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.

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.

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 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 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.

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

Three definitions of night

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

The 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.

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

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.

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 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 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 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.

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.

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

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.

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

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

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

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

The correction has to be faster than the air

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

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.

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

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.

The edge of a shadow is a wave

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

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.

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

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 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 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.

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

A boundary that hardly moves

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

The flare that arrives from somewhere else

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

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

The atmosphere is a prism as well as a lens

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

A second that depends on where the clock is

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

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

A frame made of things that are not points

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

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

An error budget added in quadrature

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

A solar radius measured past a mountain range

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

Why the Moon holds the tilt still. Spin-axis precession rate against obliquity, with the solar system's own secular frequencies drawn across it. The rising-and-falling curves are α cos ε for three precession constants: the Earth as it is, with the Moon supplying about two thirds of the torque; and a Moonless Earth at two plausible rotation rates. The horizontal lines are the nodal eigenfrequencies of the planetary secular system — s₆, s₃, s₄, s₂, s₁, s₇, s₈ — which are the rates at which the Earth's own orbit plane wobbles. A crossing is a resonance: the axis is precessing about a plane that is itself turning at the same rate, the resonant angle stops circulating, and the obliquity librates instead of holding still. The present Earth precesses at 50.3 arcseconds a year, faster than every secular frequency in the list, and its nearest crossing is at 61.3°, which is 37.8 degrees from the obliquity drawn. Take the Moon away and α falls by a factor of three; the crossings come down with it, 3 of them land below 85°, and where several overlap the obliquity has no stable value at all. The Moon does not hold the axis by pulling on it. It holds it by making the precession too fast to resonate with anything.

A tilt held still by being too fast to resonate

The Moon does not hold the Earth's axis by pulling on it. It triples the rate at which the axis precesses, which lifts that rate clear of every frequency at which the Earth's own orbit plane wobbles — and a precession with nothing to resonate with cannot wander.

A factor of 100 in the wind, 8 degrees in the aurora. The latitude of the last closed field line against the solar wind's dynamic pressure, for a dipole of 0.31 gauss at the equator. The chain is short and every link is exact. Pressure balance puts the magnetopause at a distance going as the sixth root of the field pressure over the ram pressure; a dipole line reaching equatorial distance L returns to the surface at colatitude arcsin(1/√L); so the polar cap's edge is that, and the aurora sits just equatorward of it on the outermost closed lines. Across the 100-fold range of pressure drawn — which covers everything from a quiet wind to a severe storm — the magnetopause moves from 13.1 to 6.1 planetary radii and the oval from 74.0 to 66.1 degrees. The shaded band is where the oval is actually observed. The open magnetic flux is exactly the reciprocal of the standoff distance, so it rises by a factor of 2.15 over the same range — which is the quantity that actually matters for a storm, and it is the only one of the three that changes by much.

An aurora that is a sixth root inside an arcsine

The magnetopause distance fixes which field lines are open, the open ones map to a cap around each pole, and the cap's edge is where the aurora is. A hundredfold change in the solar wind moves that edge by eight degrees — and across planets whose fields differ by four orders of magnitude it moves by ten.

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 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.

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.

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.

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.

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

The shadow that climbs the zenith

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

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

Ozone keeps the twilight zenith blue

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

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

The air's height against the planet sets the twilight

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

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.

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.

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.

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 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 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.

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

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.

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.

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

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 sunniest place on Earth is the summer pole. Daily-mean insolation against latitude, on the June solstice, an equinox, the December solstice, computed from Q = (S₀/π)(ā/r)²[H₀ sin φ sin δ + cos φ cos δ sin H₀] with the half-day angle clipped to a whole day inside the polar circle. On the June solstice the north pole receives 524 W/m² and the equator receives 385 — a third more, at a place where the Sun never climbs above 23.4°. Both halves of the product move: the Sun is low, so each square metre catches little; and it never sets, so the catching goes on for twenty-four hours. The second wins. That is why the daily total and the noon altitude disagree about where summer is strongest, and the daily total is the one an ice sheet answers to. The two hemispheres are not equal either. The Earth is nearest the Sun in early January, so the December solstice delivers 559 W/m² to the south pole against the north pole's 524 in June — 6.7 per cent more. What this cannot show is any temperature. Antarctica is the coldest place on Earth and receives the most sunlight of anywhere on any day; between the insolation and the temperature sit an albedo, an altitude and an ocean.

The sunniest place is the summer pole

On the June solstice the north pole receives 524 watts per square metre averaged over the day, and the equator 385. The Sun there never climbs above 23.4° and never sets, and the second wins.

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

A right angle short by a seventh of a degree

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

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.

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

Brightest as a crescent, and not as a disc

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

A 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.

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.

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

A square field imaged as a squashed one

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

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

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.

The sky crowded forward by the observer's speed

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

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

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.

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.

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.

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

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.

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

A sharp image kept by throwing the rest away

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

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.

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.

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.

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.

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.

Luck that reaches as far as the quiet air

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

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

A sunset's shape is a sounding of the air

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

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

The night is late because the light bounces twice

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

Every star in the catalogue is bent by the Sun

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

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

The Sun is three hundred kilometres larger seen edge-on

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

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 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.

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

A storm weighed by a compass needle

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

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

A planet that borrows the wind's field

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

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.

A frame fixed before its centre was found

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

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

An eclipse held in place to a millimetre

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

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

Two clouds that shine alike until the crescent

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

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.

A twinkle that knows the high air and not the low

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

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.

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.

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

A state the Moon had to jump to

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

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

A shield that depends on what limits the leak

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

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 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.

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.

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.

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

A polarisation capped by the light the planet catches

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

A reflector made worse so the light lands where it is wanted. The intensity returned by a 3.8 cm fused-silica corner cube against angle from the exact retro-direction, on a logarithmic scale relative to a perfect cube's peak, at 532 nm. A perfect cube returns an Airy pattern whose central lobe is 17 microradians in radius. Offsetting each of its three dihedral angles by 1.25 arcseconds splits the return into six beams, each from a sixth of the face, deflected by (4√6/3)·n·ε = 29 µrad. In the band of offsets a LAGEOS station sees, 32–38 µrad (shaded), the spoiled cube returns 10 times as much light as the perfect one, along the direction of each lobe. Each lobe is drawn here as the Airy pattern of a circular aperture of a sixth of the area, which is an approximation to the true sector shape.

A reflector built wrong so the light comes home

A corner-cube reflector sends light straight back the way it came — in its own frame. On a satellite crossing the sky the return is aberrated by twice the satellite's sideways speed over the speed of light, fifty microradians for a low orbit, which is further than a small perfect cube spreads its light. So the cubes on geodetic satellites are made imperfect on purpose, their angles offset by about an arcsecond to throw the returning beam out to where the telescope will be.

The 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.

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.

A glow counted in square metres of sunlit metal

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

Two exponents a solar cycle cannot pull apart

Whether a magnetic field protected the early Earth's air depends on what limits the escape of ions from a planet without one — the ultraviolet that makes them or the solar wind that carries them off. Measured at Mars through a solar cycle, the two causes rise and fall together, and the fit that tries to separate them leaves each exponent uncertain by half its value. Remarkably, the young Sun's loss is predicted well anyway. What spoils the prediction is not the confusion but a switch between the two limits that the present-day data cannot see coming.

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.

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.

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.

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.

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.

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.

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.

$Rays from a ring of sunsets, bent into the shadow. Sunlight grazing the Earth at heights of 1, 5, 15 km, bent toward the axis by the atmosphere by twice the refraction at that height, reaching the Moon's distance, with the shadow's geometry in the plane through the Sun: the distance axis is compressed by about sixty to one against the vertical. The low rays cross the axis well before the Moon and reach the far side of the shadow; the higher rays only bend enough to enter its edge. The geometric umbra at the Moon has a radius of 4584 km, the two dashed lines, so without air the Moon would disappear inside it. Light arrives from the whole ring round the Earth, which is every sunrise and sunset at that moment.$

A Moon lit by every sunset at once

With no air, the Moon would vanish inside the Earth's shadow. Instead it turns a dull copper, because the atmosphere bends sunlight round the planet, and what reaches the Moon has gone through a ring containing every sunrise and sunset on Earth at that moment. Each height in the air lights one place in the shadow, so the colour across the disc is a map of the atmosphere by altitude. It has sometimes been black, after the largest volcanoes, and that makes a century of eclipse records a measurement of the stratosphere older than any instrument sent up to sample it.

The ladders in this field

19 anchors · one idea each

Celestial sphere Seasons Phases and eclipses Precession Equation of time Refraction Libration Twilight Apparent motion Sidereal time Parallax Aberration Timescales Calendars Seeing Occultations Obliquity Magnetosphere Stellar winds

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