Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A Sun that stops and runs backwards

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

Assumes Equation of time and Resonance.

14 min read 6 figures The sky as a clockTwo bodies, and no more

The equation of time is what happens when a planet’s rotation, which is uniform, is measured against the Sun’s apparent motion, which is not. On the Earth the unevenness is worth sixteen minutes, and on Mars, with its more eccentric orbit, about fifty. In both cases the Sun always moves the same way across the sky. It merely moves at a rate that varies by a few per cent, because the planet spins hundreds of times for each orbit and the orbital motion is a small correction to the spin.

Mercury spins slowly enough that the correction is not small. It turns once on its axis every 58.65 days and goes round the Sun every 87.97 days, so the rotation is only one and a half times as fast as the orbit. The orbital motion is not a small correction to that. For a few days around each perihelion it is larger than the spin, and the Sun, rather than slowing, reverses.

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.
Fig. 1 The rate at which the Sun moves across Mercury’s sky, in degrees of hour angle per Earth day, against days from perihelion over one 87.97-day orbit, for the planet’s 3:2 spin and eccentricity of 0.2056. The rate is the spin minus the rate at which the Sun’s direction turns as the planet moves along its orbit, and that orbital rate peaks at perihelion at 1.551 times its mean — above the spin’s 1.5. From 4.0 days before perihelion to 4.0 days after, the rate is negative and the Sun moves backwards by 1.11 degrees; at perihelion it is moving 0.21 degrees a day the wrong way. Elsewhere it crosses the sky at up to 3.4 degrees a day.

Two rates, and which one wins

The Sun’s position in a planet’s sky is set by the difference between two angles. One is how far the planet has rotated, which grows at a constant rate. The other is how far round its orbit the planet has moved, because an observer who has gone part of the way round the Sun sees it in a different direction against the stars. Noon comes when the rotation has carried the observer to face the Sun’s current direction, and the time from one noon to the next — the solar day — is set by how fast the first angle outruns the second.

For the Earth that difference is what makes the solar day four minutes longer than a rotation. For Mercury the arithmetic is more dramatic. If the rotation takes 58.65 days and the orbit 87.97, the rotation gains a whole turn on the orbital motion every 1/(1/58.65 − 1/87.97) days, which is 175.9 days: a single Mercury day from noon to noon lasts two of its years, and three of its rotations.

That is the average. The instantaneous rate at which the Sun’s direction turns is not constant, because a planet sweeps equal areas in equal times and so moves fastest at perihelion. The angular rate there exceeds the mean by a factor of (1+e)2/(1−e2)3/2(1+e)^2/(1-e^2)^{3/2}, which for Mercury’s eccentricity of 0.2056 is 1.551. The spin, meanwhile, is exactly 1.5 times the mean orbital rate, because the rotation is locked in a 3:2 resonance with the orbit. For a few days around perihelion the orbital term wins by three per cent, the difference changes sign, and the Sun’s motion across the sky runs the other way.

The same subtraction of two motions, one of them the observer’s own, is what makes the planets appear to loop backwards among the stars. Mars does not reverse its orbit when it retrogrades; the Earth overtakes it. The Sun does not reverse anything when it runs backwards over Mercury; for eight days Mercury’s orbital motion turns the line to the Sun faster than Mercury’s rotation turns the ground.

What eccentricity it takes

A reversal needs the orbital rate at perihelion to exceed the spin. For a 3:2 spin that happens only above a particular eccentricity, and Mercury is not far above it.

A Sun that slows at perihelion and never turns back. The rate at which the Sun moves across the sky of a planet with a 3:2 spin–orbit ratio and eccentricity 0.15, 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 1.368 times its mean. The spin is 1.5 times the mean orbital rate, so the Sun slows near perihelion to 0.54 degrees a day and never reverses. Away from perihelion the Sun crosses the sky at up to 3.1 degrees a day, and a whole solar day, noon to noon, takes 175.9 days — two orbits.
Fig. 2 The same 3:2 spin with the orbit’s eccentricity lowered from 0.2056 to 0.15. The orbital rate at perihelion now peaks at 1.368 times its mean, below the spin’s 1.5, so the Sun slows near perihelion to 0.54 degrees a day and never reverses. The solar day is the same 175.9 days, because that depends only on the two periods; what changed is only how unevenly the day is spent.

The contrast is the point of drawing the second figure at all. The length of the solar day, the fact of the resonance and the average rate of the Sun across the sky are identical in both; only the eccentricity differs, and it decides whether the Sun merely dawdles or turns round. Everything unusual about Mercury’s sky is a consequence of an orbit that is eccentric enough to tip one rate past the other, and not of the resonance on its own.

How eccentric an orbit has to be for its Sun to run backwards. The smallest orbital eccentricity at which the Sun reverses its motion across the sky near perihelion, against the ratio of a planet's spin rate to its mean orbital rate. The Sun stops when the orbital angular rate at perihelion, (1 + e)² divided by the 3/2 power of (1 − e²), times the mean, equals the spin rate, so the boundary is that expression set equal to the ratio. Above the curve the Sun runs backwards once per orbit; below it the Sun only slows. A synchronously rotating body, at a ratio of one, crosses the boundary at any eccentricity at all: for a moon locked to its planet the body that runs backwards across its sky is the planet, and from the Moon's near side the Earth, which never rises or sets, wanders back and forth by several degrees each month — the Moon's libration seen from the other end. At Mercury's 3:2 the boundary is e = 0.191, and Mercury's orbit, at 0.2056, clears it by a margin of 0.015: a slightly rounder orbit and the Sun would stall at perihelion without reversing.
Fig. 3 The smallest eccentricity at which the Sun reverses near perihelion, against the ratio of spin rate to mean orbital rate. The boundary is where (1+e)2/(1−e2)3/2(1 + e)^2/(1 - e^2)^{3/2} equals the ratio. Above it the Sun runs backwards once per orbit; below it the Sun only slows. At Mercury’s 3:2 the boundary is e = 0.191, and Mercury’s 0.2056 clears it by 0.015. At a ratio of one, any eccentricity at all is enough.

Mercury clears the line by a margin of a little over a hundredth of eccentricity. That margin is not a fixed fact about the planet. Mercury’s eccentricity is perturbed by the other planets and has ranged, over millions of years, well above and below its present value, so the backward Sun is a feature of the present epoch of Mercury’s orbit rather than of Mercury.

The left-hand end of the boundary has a familiar example. A moon locked in synchronous rotation turns once per orbit, a ratio of one, and any eccentricity at all makes its orbital rate exceed its spin near periapsis and fall short near apoapsis. What runs back and forth in its sky is then not the Sun but the planet it orbits. From the Moon’s near side the Earth never rises or sets; it hangs over one spot and wanders by several degrees in a small figure each month. That wander is the Moon’s libration seen from the other end, and it is the same inequality as Mercury’s backward Sun at the extreme where the spin has no margin at all.

One day, two perihelia

Following the Sun through a whole Mercury day shows how the reversal sits inside the ordinary motion.

One Mercury day, 176 Earth days from noon to noon. The Sun's altitude over one solar day of Mercury, 175.9 Earth days, seen from the equator at two longitudes, for a planet on its 3:2 spin–orbit resonance with eccentricity 0.2056 and no axial tilt. Perihelion comes twice in that day, at 0 and 88.0 days, marked. From the longitude where noon falls at perihelion, the Sun climbs to the zenith, hesitates, backs a little way down the sky and climbs again before setting — and 88 days later that same place has midnight at the next perihelion. From the longitude where sunrise falls at perihelion, the Sun rises, sinks back below the horizon and rises again. Between perihelia the Sun crosses the sky steadily; all of the strangeness is packed into a few days either side of the closest approach, when the planet's orbital motion briefly outruns its spin.
Fig. 4 The Sun’s altitude over one solar day of Mercury, 175.9 Earth days, from the equator at two longitudes, for no axial tilt. Perihelion comes twice, at 0 and 88.0 days. From the longitude where noon falls at perihelion, the Sun climbs to the zenith, hesitates and climbs again before setting, and 88 days later the same place has midnight at the next perihelion. From the longitude where sunrise falls at perihelion, the Sun rises, sinks back and rises again. Between perihelia the Sun crosses the sky steadily.

Two perihelia fall in each solar day, and because the day is exactly two orbits long, they always fall at the same local times at the same places. The longitudes that face the Sun at one perihelion face away from it at the next, and after a full day they face it again. Mercury therefore has two longitudes, 180 degrees apart, where the Sun is always overhead at perihelion — and two more, halfway between, where it is always on the horizon at perihelion. The first pair are called the hot poles, and the largest impact basin on the planet, Caloris, whose name means heat, sits close to one of them.

Three noons

At a hot pole the reversal happens with the Sun almost overhead, and it cannot be seen in an altitude plot drawn to the horizon. Drawn as the Sun’s angle from the meridian, it is unmistakable.

Three noons in eight days. The Sun's hour angle — its distance from the meridian, in degrees, positive to the west — in the 28 days around perihelion, from the equatorial longitude of Mercury where noon falls at perihelion, for a 3:2 spin and eccentricity 0.2056. The Sun crosses the meridian westwards 7.1 days before it, then crosses the meridian back eastwards at perihelion, then crosses the meridian westwards 7.1 days after it. In the whole window it never strays more than 6.2° from the meridian, standing almost overhead for a fortnight at the largest it ever looks from this planet, so the places on Mercury that face the Sun at perihelion receive more heat than any others — the hottest spots on the planet are fixed in longitude by this arithmetic.
Fig. 5 The Sun’s hour angle — its distance from the meridian, positive to the west — in the 28 days around perihelion, from the longitude where noon falls at perihelion. The Sun crosses the meridian westwards 7.1 days before perihelion, crosses back eastwards at perihelion, and crosses westwards again 7.1 days after. In that fortnight it never strays more than 6.2° from the meridian.

An observer at a hot pole would see the Sun pass noon, drift a few degrees west, stop, come back through noon, drift a few degrees east, stop again, and cross noon a third time before finally heading for the western horizon a month and a half later. For a fortnight the Sun hangs within six degrees of the zenith at its largest apparent size, and that is why these longitudes are hot. A point on Mercury’s equator that faces the Sun at perihelion receives the planet’s most intense sunlight for longer than any other point, and the surface temperatures there reach about 430 degrees Celsius.

The Sun is at its largest there, too. At perihelion Mercury is 0.307 astronomical units from the Sun, at aphelion 0.467, so the Sun’s disc is 1.52 times wider at perihelion than at aphelion — about three and a quarter times the width it has from the Earth. The reversal, the triple noon and the greatest heat all happen in the same week, while the Sun looks biggest.

A sunrise, a sunset and a second sunrise

At the warm poles, ninety degrees away, the same fortnight straddles the horizon.

A sunrise, a sunset and a second sunrise. The Sun's altitude in the 28 days around perihelion, from the equatorial longitude of Mercury where the Sun is on the horizon at perihelion, for a 3:2 spin and eccentricity 0.2056. The altitude never exceeds 6.2° in this window, because the Sun hardly moves: it rises 7.1 days before perihelion, then sets at perihelion, then rises 7.1 days after perihelion. The retreat is the same backward motion that stops the Sun at noon elsewhere on the planet, seen where it happens to straddle the horizon. The Sun is also at its largest here: its disc is 1.52 times wider at perihelion than at aphelion, so the reversal happens while it looks biggest.
Fig. 6 The Sun’s altitude in the 28 days around perihelion, from the longitude where the Sun is on the horizon at perihelion. It rises 7.1 days before perihelion, sets again at perihelion, and rises for good 7.1 days after. Its altitude never exceeds 6.2° in the window. The Sun’s disc is 1.52 times wider at perihelion than at aphelion.

The numbers are identical to the three noons, as they must be: the same backward motion of the same size, seen at a place where it happens to carry the Sun across the horizon rather than across the meridian. What an observer there would see is a Sun that lifts a few degrees above the eastern horizon over a week, sinks back below it, and a week later rises again and keeps going. The same happens in reverse at the opposite warm pole, as a sunset followed by a brief return of the Sun and a second sunset.

None of this involves Mercury’s tilt, which is a few hundredths of a degree and was set to zero in every figure. On the Earth the tilt supplies half of the equation of time. On Mercury it supplies essentially none, and the entire effect is the eccentricity term — the equation of centre — measured against a spin too slow to keep ahead of it.

What a clock on Mercury would keep

A mean-time clock on Mercury would keep a day of 175.9 Earth days, and its fictitious Sun would move across the sky at a steady 2.05 degrees of hour angle per Earth day. The real Sun departs from that fictitious one by the equation of centre — the angle between where the planet is on its orbit and where it would be at a uniform rate — and for Mercury’s orbit that angle reaches 23.7 degrees, about eighteen days before and after each perihelion. Converted at the mean rate of the Sun across the sky, it puts a sundial up to 11.6 Earth days ahead of or behind the clock. The Earth’s sixteen minutes become more than a week and a half.

Measured as a share of the day it is correcting, the progression is steep. The Earth’s largest equation of time is 1.1 per cent of a solar day, Mars’s 3.5 per cent, and Mercury’s 6.6 per cent. The two fictitious suns that separate the Earth’s equation of time collapse into one on Mercury, because with no tilt the second reduction changes nothing: the whole of the effect is the first, the uniform Sun set against the true one, and a mean anomaly is exactly the average that fictitious Sun follows.

A sundial on Mercury would show the reversal as a shadow walking backwards. At a hot pole the gnomon’s shadow would shorten towards noon, pass it, creep a little way into the afternoon, stop, return through noon into the morning and then set off into the afternoon for good — a week-long hesitation traced by the tip of a shadow a few degrees long. Divided into twelve unequal hours of daylight, Mercury’s day of eighty-eight Earth days of sunshine would give hours averaging more than seven Earth days, and at a hot pole the sixth of them would end three times.

A rotation that was measured wrongly for eighty years

For most of the twentieth century Mercury was believed to rotate once per orbit, keeping one face to the Sun. Giovanni Schiaparelli had concluded so from drawings of faint surface markings in the 1880s, and it seemed physically reasonable: tides raised by the Sun should lock a planet that close. If it had been true, Mercury would have had a permanently hot side and a permanently frozen one, and its Sun would have rocked back and forth over the day side by the planet’s libration rather than ever setting.

The error was one of sampling. Mercury is best observed from the Earth at intervals that are close to a whole number of its rotations and orbits combined, so the same markings were seen repeatedly in the same places, as they would have been for a locked planet. In 1965 radar pulses bounced off the planet from the Arecibo telescope measured the Doppler spread of the echo and found a rotation period of about 59 days, not 88. Giuseppe Colombo pointed out almost at once that 59 days was two thirds of the orbital period, and that a planet on so eccentric an orbit could be held by the Sun’s tides in a resonance that is not one to one, because the torque at perihelion is what locks it and a 3:2 spin faces the Sun the same way at every perihelion.

That locking mechanism and the backward Sun are the same inequality seen from the two sides. The resonance is stable because, at perihelion, the planet’s rotation and the Sun’s direction are nearly matched, so the tide can grip; the Sun reverses because, at perihelion, the orbital rate slightly exceeds the rotation. The planet’s bulge is aligned with the Sun at the moment when the Sun hardly moves across the sky, which is what gives the tide time to hold it there. The slight rocking this leaves in Mercury’s rotation is small enough to need radar to see, and large enough that its size weighed the planet’s liquid core.

What was actually measured

The rotation period is known to a fraction of a second from radar and from the MESSENGER spacecraft’s four years in orbit, and it matches three halves of the orbital period to within the precision of either. The eccentricity and the orbital period are among the best-determined numbers in the solar system. Everything drawn here follows from those three numbers with no free parameter, which is why the timing of the triple noon and the double sunrise can be stated to a tenth of a day.

The surface temperatures that follow from the geometry have been measured too. The hot longitudes near 0° and 180° reach the highest daytime temperatures on the planet, and the pattern of heating — two hot and two warm longitudes rather than a uniform band — is visible in the thermal emission mapped from the Earth at radio wavelengths, where the ground’s warmth is seen through its upper layers. Nobody has stood at a warm pole and watched the Sun set and rise again, but the prediction that it does is as secure as Kepler’s second law.

What the figures leave out

The tilt is zero. Mercury’s axis is tilted by about two minutes of arc, far too little to change anything drawn here and exactly enough to matter somewhere else. With so little tilt the Sun never rises more than a few arcminutes above the horizon at Mercury’s poles, so the floors of polar craters never see it at all, and in the closest planet to the Sun they stay colder than about minus 170 degrees Celsius indefinitely. Radar echoes from those crater floors, first recorded from the Earth in 1991, were unusually bright, and the spacecraft that orbited Mercury from 2011 confirmed that the deposits are water ice. The same absence of tilt that leaves the equation of time on Mercury to its orbit alone keeps ice a few kilometres from ground that reaches four hundred degrees.

The observer is on the equator. At higher latitudes the Sun’s path is lower and tilted, but its hour angle behaves identically, so the triple noon happens at every latitude along a hot longitude and the double sunrise at every latitude along a warm one.

The horizon is flat. Local topography on an airless, cratered planet decides the actual moment of sunrise to within hours, and at a warm pole a hill could hide the brief first rise altogether.

And the orbit is fixed. Mercury’s perihelion advances by about 574 arcseconds a century, most of it from the other planets and the famous 43 from relativity, and the eccentricity changes slowly. Neither matters within one solar day; over millions of years the second decides whether the Sun reverses at all.

Still open: how long the Sun has been running backwards

Mercury’s eccentricity is not constant. Driven mainly by Venus, Jupiter and the Earth, it wanders over millions of years, and integrations of the solar system put its excursions well below the 0.191 at which the reversal stops. Whenever it was below that line, Mercury’s Sun only slowed at perihelion, the heating of the hot poles was spread over more days at lower intensity, and the pattern of surface temperature — which decides where volatile ices can survive in shadow and how the regolith weathers — was different. The orbit’s history can be integrated forwards and backwards for a few tens of millions of years with confidence and statistically beyond that, because Mercury’s orbit is the most chaotic of the planets’; for how much of its existence Mercury has had a backward Sun, and whether that has left any trace on its surface, the answer is a distribution rather than a date.

Where this goes next

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 circular orbit has one lock, and an eccentric one has several. The strength of each spin–orbit resonance against orbital eccentricity, as the Hansen coefficient H(p, e) that multiplies the restoring torque on a permanently non-spherical body. At zero eccentricity every curve but the synchronous one is exactly zero — the figure checks that rather than showing it — so a body on a circular orbit can lock only by turning once per orbit. Away from zero the others switch on: at Mercury's eccentricity of 0.2056 the 3:2 resonance has 73 per cent of the synchronous one's strength and more than twice the 2:1's. A planet spinning down through this family therefore meets the 3:2 before the 1:1 and has a real chance of being caught there, which is what happened — Mercury turns three times for every two orbits, a fact discovered by radar in 1965 after a century of assuming it was locked. The free libration of that locked state follows from the same coefficient and the measured 2.03e-4 for (B − A)/C: 12.1 years, against a measured period near twelve. What the figure cannot show is the capture probability itself, which depends on how the tide dissipates and ranges from a few per cent for a simple constant-lag tide to more than half once friction between a liquid core and the mantle is included. A rotation locked to the orbit, but not one to one Mercury turns exactly three times for every two circuits of the Sun. That was not what anybody expected, and it is not an accident — on a circular orbit a tidally despun body has exactly one place to lock, and on an eccentric one it has several — with the strength of each set by a coefficient that vanishes when the eccentricity does. Mars through 8.3 months of sky. The geocentric ecliptic longitude and latitude of Mars over 252 days — 8.3 months — centred on opposition, computed as the direction of P − E with both orbits taken as circles: the Earth's of radius 1 AU, the planet's of 1.5237 AU inclined 1.850°. The motion reverses for 72.7 days and backs up 15.94° of longitude, and that interval is centred on opposition to within 0.0000° of longitude. The track closes on itself: over 149 days the planet visits the same point of the sky twice, and the loop it encloses is 15.9° long and 2.94° tall. Longitude and latitude are at different scales: a degree of latitude is drawn 4.6 times a degree of longitude, because the sweep of 51° in longitude and 4.5° in latitude will not share a scale on one page. Circular orbits mean one loop for every apparition, and that is the real cost of this figure: Mars is 0.525 AU away at the opposition drawn here, its true distance at opposition varies with its eccentricity, and the real loops differ in size from one apparition to the next because of it. The marks are 20 days apart, and they crowd where the motion stops. The 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.

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EccentricityEquation of centreEquation of timeHour angleKepler's second lawLibrationPerihelionRetrograde motionSolar daySpin orbit resonance