Astronomy & Celestial Mechanics
the sky, computed
The observed sky

On Mars the orbit outweighs the tilt

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

Assumes Equation of time and Equation of time.

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

On the Earth, a sundial and a clock disagree by up to sixteen minutes, and the disagreement has two causes that can be taken apart. One is the shape of the orbit: the planet moves fastest near perihelion, so the Sun’s eastward drift against the stars is uneven, and noon comes early for half the year and late for the other half. The other is the tilt: even a Sun moving evenly along the ecliptic projects unevenly onto the equator, which is where the hour angle that a sundial reads is measured, and that term runs through two cycles a year. On the Earth the two are comparable in size — seven and a half minutes against ten — and the curve they add up to has the familiar two unequal humps.

Nothing in that reasoning is about the Earth. Every planet with a tilted axis and an eccentric orbit has the same two terms, and their sizes are set by two numbers each planet has: the eccentricity for the first and the obliquity for the second. Mars has an axis tilted almost exactly like the Earth’s, 25.19 degrees against 23.44, and an orbit more than five times as eccentric. The same two terms, at its numbers, make a different kind of curve.

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.
Fig. 1 The equation of time on Mars over one Martian year of 668.6 sols, against sols since the northern spring equinox, computed from Kepler’s equation for an orbit of eccentricity 0.0934 and a tilt of 25.19°, with perihelion at solar longitude 250.87°. The sum runs from −51.1 local minutes, 611 sols after the equinox, to +39.9, 385 sols after it. The eccentricity term alone swings by ±42.9 minutes and the obliquity term by ±11.4 — the first is 3.74 times the second — and perihelion falls 485 sols after the equinox.

Two terms, and what sets their size

The eccentricity term is the difference between where a planet is and where it would be if it moved uniformly — between its true anomaly and its mean anomaly. That difference has a name older than Kepler, the equation of centre, and for a nearly circular orbit its amplitude is close to twice the eccentricity in radians. Converted to time at four minutes of hour angle per degree, that is about 458 minutes multiplied by the eccentricity: 7.7 minutes for the Earth’s 0.0167, and 42.8 for Mars’s 0.0934. The drawn amplitude, 42.9, is that arithmetic carried through Kepler’s equation exactly.

The same equation of centre is what lets the Moon show more than half its surface. A body that rotates at a uniform rate while moving round an ellipse at a non-uniform one turns its face slightly away from its partner and back again each orbit, and the angle it turns through is the equation of centre. A Martian sundial fifty minutes slow and the extra slivers of the Moon’s limb visible from the Earth are the same quantity, measured once as a time and once as an angle.

The obliquity term does not depend on the orbit at all. It comes from projecting a circle tilted by ε onto the equator, and its amplitude grows roughly as the square of the tangent of half the tilt: 9.9 minutes for the Earth, 11.4 for Mars, a difference of fifteen per cent from a tilt larger by less than two degrees. So the two planets’ obliquity terms are close cousins, and the whole difference between their equations of time comes from the orbit.

The Earth: a sundial 16 minutes ahead and 14 behind. The equation of time on the Earth over one of its years, 365.2 days long, against days since the northern spring equinox, computed from Kepler's equation for an orbit of eccentricity 0.01671 and an axis tilted 23.439°, with perihelion at solar longitude 282.94°. The solid curve is the difference between true and mean solar time: it runs from −14.2 local minutes, 328 days after the equinox, to +16.4, 228 days after it. The dashed curves are its two parts. The eccentricity term, one cycle a year, swings by ±7.7 minutes; the obliquity term, two cycles, by ±9.9; the first is 0.78 times the second. Perihelion falls 289 days after the equinox, marked, and the vertical lines are the equinoxes and solstices.
Fig. 2 The Earth drawn by exactly the same reduction, against days since the March equinox. The equation of time runs from −14.2 minutes, 328 days after the equinox, to +16.4, 228 days after it. The eccentricity term swings by ±7.7 minutes and the obliquity term by ±9.9, so here the orbit’s term is the smaller, 0.78 times the tilt’s. Perihelion falls 289 days after the equinox, in early January.

That second figure is a check as much as a comparison. The reduction used for Mars has the year length, the day length, the eccentricity, the tilt and the longitude of perihelion as arguments, and the Earth’s own numbers put through it must return the Earth’s familiar curve — sixteen minutes at the start of November, fourteen in the other direction in mid-February. They do, and only because they do can the Martian curve be read as the same physics at different numbers rather than as a different calculation.

Set side by side, the two curves differ in kind as well as in size. The Earth’s has four turning points a year of comparable height, because two cycles of one term and one cycle of another, at similar amplitudes, interfere. The Martian curve is dominated by one broad swing a year, with the obliquity term putting a modest ripple on each side of it. At a ratio of 3.7 to one, the second harmonic has stopped competing.

The same unevenness sets how long each individual solar day is, because the slope of the equation of time is the difference between one true solar day and the mean. On the Earth the steepest slope is half a minute a day, which is why the days around the December solstice, measured from one noon to the next, run about thirty seconds longer than the clock’s day. On Mars the steepest slope is 0.71 local minutes a sol, so the longest true sols exceed the mean by about forty-three seconds, and they come not near a solstice but a few weeks after perihelion, where the eccentricity term changes fastest. The Earth’s longest solar days are set by the tilt and the orbit together; Mars’s are set by the orbit almost alone.

The day of the extremes follows the perihelion

On the Earth the dates of the equation of time’s extremes are learned as facts — early November, mid-February, mid-May, late July. On Mars they fall where the orbit puts them, and the orbit puts perihelion at a solar longitude of 251 degrees, about 485 sols into the northern year and just before the southern summer solstice.

The largest excursion, a sundial fifty-one local minutes behind the clock, comes 611 sols after the northern spring equinox, as the planet climbs away from perihelion and the eccentricity term is near its extreme. The largest excursion the other way, forty minutes ahead, comes near sol 385, before perihelion, on the approach. Because the eccentricity term dominates, the two extremes sit 226 sols apart, a third of a Martian year, and bracket perihelion, where on the Earth the four extremes are arranged by the interference of two terms and do not bracket anything in particular.

The unevenness of the orbit also makes the Martian seasons unequal in a way the equation of time shows directly. By the long-established count, northern spring lasts 194 sols and northern autumn 142, because equal areas are swept in equal times and an orbit of eccentricity 0.09 takes a good deal longer over its far half than its near half. The equation of time is the same bookkeeping kept in minutes: the Sun falls behind mean time through the slow half of the orbit and catches up through the fast half.

A figure-of-eight that cannot close

The analemma — the position of the Sun at the same clock time each day, plotted as declination against the equation of time — is the curve most people know the equation of time by, and on the Earth it is a lopsided figure-of-eight. On Mars it is not.

The analemma of Mars, a teardrop. The Sun's declination against the equation of time over one year of Mars — where the Sun stands at the same clock time each sol. It is 91 local minutes wide and 50.4° tall, the height being twice the tilt of the axis. The curve does not cross itself, so it is a teardrop rather than a figure-of-eight: the eccentricity term's single yearly cycle is large enough to pull the whole loop to one side before the obliquity term's second cycle can fold it back. The four marked points are the equinoxes and solstices, and their horizontal positions are the equation of time on those dates.
Fig. 3 The Sun’s declination against the equation of time over one Martian year — where the Sun stands at the same local clock time each sol. It is 91 local minutes wide and 50.4° tall, the height being twice the tilt of the axis. The curve does not cross itself: it is a teardrop, because the eccentricity term’s single yearly cycle pulls the whole loop to one side before the obliquity term’s second cycle can fold it back. The four marked points are the equinoxes and solstices.

Reading the teardrop is the same as reading the Earth’s figure-of-eight. Each point is where the Sun stands at one fixed local mean time on one sol: its height is the Sun’s declination, and its sideways displacement is how far the Sun is ahead of or behind the clock on that sol. The height is set by the tilt alone, so Mars’s analemma is only seven per cent taller than the Earth’s; its width is set almost entirely by the eccentricity term, so it is nearly three times as wide. An observer on Mars photographing the Sun at the same clock time every sol would need 669 sols to close the curve, and would end up with a shape whose proportions no photograph taken on the Earth has ever had.

The analemma of the Earth, a figure-of-eight. The Sun's declination against the equation of time over one year of the Earth — where the Sun stands at the same clock time each day. It is 31 local minutes wide and 46.9° tall, the height being twice the tilt of the axis. The curve crosses itself once, so it is a figure-of-eight: the obliquity term's two cycles a year are strong enough to fold the loop over. The four marked points are the equinoxes and solstices, and their horizontal positions are the equation of time on those dates.
Fig. 4 The Earth’s analemma by the same reduction: 31 minutes wide and 46.9° tall. The curve crosses itself once, making the familiar figure-of-eight, because at the Earth’s numbers the obliquity term’s two cycles a year are strong enough to fold the loop over. The lower lobe is the larger because perihelion falls in the northern winter.

The crossing is what a figure-of-eight is, and it needs the horizontal coordinate — the equation of time — to reverse direction twice while the declination runs once from one solstice to the other. The obliquity term supplies those reversals, because it runs through two cycles in a year. The eccentricity term supplies only one cycle and pushes the whole curve sideways, and when it is large enough the reversals it would need to cancel are overwhelmed: the curve swings out and back without ever crossing its own path.

The figure-of-eight is therefore not a property of analemmas. It is a property of a ratio. Below some eccentricity the obliquity term folds the curve; above it the eccentricity term pulls the fold open. The value where that happens can be found by drawing the curve at every eccentricity and asking whether it crosses itself.

The eccentricity at which a figure-of-eight becomes a teardrop, at a 25.19° tilt. The amplitudes of the equation of time's two terms against the orbit's eccentricity, for an axial tilt of 25.19° and perihelion at solar longitude 250.87°, with the shape of the analemma computed at each value. The obliquity term depends only on the tilt and stays at 11.4 minutes; the eccentricity term grows in proportion to the eccentricity, about 4.60 minutes for every hundredth. The two are equal at e ≈ 0.025. The analemma stops crossing itself — a figure-of-eight becomes a teardrop — at e ≈ 0.054, shaded beyond. The Earth's 0.0167 is far on the figure-of-eight side, and Mars's 0.0934 is on the teardrop side, which is why the two planets' analemmas look like different kinds of curve although they come from the same two terms. The crossing point depends on where perihelion falls against the solstices, so the shaded boundary belongs to this orientation of the orbit.
Fig. 5 The amplitudes of the two terms against eccentricity, for Mars’s tilt of 25.19° and its perihelion at solar longitude 250.87°, with the analemma’s shape computed at each value. The obliquity term stays at 11.4 minutes; the eccentricity term grows by about 4.6 minutes for every hundredth of eccentricity and equals the obliquity term at e ≈ 0.025. The analemma stops crossing itself at e ≈ 0.054, shaded beyond. The Earth, at 0.0167, is well on the figure-of-eight side, and Mars, at 0.0934, well on the teardrop side.

The threshold is not where the two amplitudes are equal. At e = 0.025 the eccentricity term has matched the obliquity term’s amplitude and the curve still crosses itself, because a crossing depends on the shapes and phases of the two terms as well as their sizes; the fold survives until the eccentricity term is nearly five times the Earth’s. Nor is the threshold a property of the eccentricity alone. Where perihelion falls against the solstices decides whether the eccentricity term’s swing reinforces the obliquity term’s reversals or undoes them, and the boundary drawn belongs to Mars’s own orientation. A planet with the same eccentricity and perihelion placed at an equinox would have a differently shaped analemma, and its fold would close at a different value.

The eccentricity at which a figure-of-eight becomes a teardrop, at a 23.439° tilt. The amplitudes of the equation of time's two terms against the orbit's eccentricity, for an axial tilt of 23.439° and perihelion at solar longitude 282.94°, with the shape of the analemma computed at each value. The obliquity term depends only on the tilt and stays at 9.9 minutes; the eccentricity term grows in proportion to the eccentricity, about 4.60 minutes for every hundredth. The two are equal at e ≈ 0.022. The analemma stops crossing itself — a figure-of-eight becomes a teardrop — at e ≈ 0.044, shaded beyond. The Earth's 0.0167 is far on the figure-of-eight side, and Mars's 0.0934 is on the teardrop side, which is why the two planets' analemmas look like different kinds of curve although they come from the same two terms. The crossing point depends on where perihelion falls against the solstices, so the shaded boundary belongs to this orientation of the orbit.
Fig. 6 The same construction at the Earth’s tilt of 23.44° and its perihelion at solar longitude 282.94°. The obliquity term is 9.9 minutes; the eccentricity term again grows by about 4.6 minutes a hundredth and equals it at e ≈ 0.022; the analemma stops crossing itself at e ≈ 0.044. The Earth’s figure-of-eight would open into a teardrop with an orbit about 2.6 times as eccentric as the one it has.

The Earth’s threshold comes out lower than Mars’s, at 0.044 against 0.054, for two reasons that both show in the numbers. The Earth’s smaller tilt gives it a weaker obliquity term, 9.9 minutes against 11.4, so less eccentricity is needed to overpower it. And the Earth’s perihelion falls only thirteen degrees of solar longitude past the December solstice, so the eccentricity term’s swing lines up with the obliquity term’s reversals in a way that opens the fold sooner. The Earth is safely on the figure-of-eight side, but not by the margin its small eccentricity suggests: it sits at a little under two fifths of the value that would change the shape of its analemma.

Running a spacecraft on a planet’s own day

The equation of time on Mars is not a curiosity of the calendar. Every mission that has operated on its surface has had to decide what time it is.

A Martian solar day, a sol, lasts 24 hours 39 minutes and 35 seconds, and the teams that operated the first long-lived rovers in 2004 lived on it, shifting their own working day about forty minutes later every Earth day for months. What they kept was local mean solar time — the time of a fictitious Sun moving uniformly — because a clock cannot keep anything else. What the rovers’ solar panels, cameras and thermal systems responded to was the real Sun, and between the two lay the fifty-minute swing in the opening figure. A command sequence planned for noon by the mission clock executed with the Sun up to fifty minutes from the meridian, depending on the season, and the difference had to be carried in every plan that depended on illumination or temperature.

The algorithms that convert between the two are the same reduction drawn here, carried to higher accuracy: Kepler’s equation solved for Mars’s orbit, with small periodic corrections for the pull of the other planets, and a prime meridian fixed by a small crater named Airy-0. Martian coordinated time is the mean solar time on that meridian, a clock defined by a place exactly as Greenwich time once was, and every Martian local time is referred to it through longitude and then through the equation of time.

The difference from the Earth is one of scale rather than principle. The Earth’s sixteen minutes were small enough that civil life ignored them once clocks arrived; Mars’s fifty would not be. A settlement on Mars keeping mean time would see its noon Sun wander by almost an hour across the year, and one keeping apparent time would need a clock whose rate changed by up to seven tenths of a minute a sol, against at most half a minute a day on the Earth.

A shape that changes over time

The numbers used for Mars are its present ones, and they are not permanent. Mars’s orbit is perturbed by the other planets far more strongly in its eccentricity than the Earth’s is: over the last few million years it has ranged from nearly circular to about 0.12. At its roundest the eccentricity term shrinks below the obliquity term, and the Martian analemma folds back into a figure-of-eight; at its most eccentric the teardrop is wider still.

The orientation changes too. The longitude of perihelion measured from the equinox drifts round the orbit in about fifty thousand years, carrying the extremes of the equation of time through every season and moving the threshold on the share figure with it. And the tilt itself is not fixed: the axis of Mars has swung between nearly upright and sixty degrees, with no large moon to steady it. At sixty degrees the obliquity term would be several times its present size and would dominate even Mars’s eccentric orbit. The equation of time on Mars has been, at different epochs, a small figure-of-eight, a teardrop and a large figure-of-eight, from the same two terms responding to an orbit and an axis that do not stay put.

The Earth’s own curve drifts as well, more slowly. Its perihelion has moved from near the December solstice, where it was about eight centuries ago, to two weeks after it, and that shift of alignment is what makes the earliest sunset fall before the shortest day by the number of days it does now rather than by some other number.

What was actually measured

The Martian equation of time rests on three numbers measured very well and one defined by agreement. The eccentricity and the longitude of perihelion come from Mars’s orbit, fixed to many decimal places by centuries of positions and decades of spacecraft ranging. The obliquity comes from the orientation of the rotation axis, measured by tracking landers whose positions on the surface rotate with the planet; those same measurements detect the axis’s slow precession directly. The length of the sol follows from the rotation period and the orbital motion together, just as the Earth’s solar day exceeds its rotation by four minutes.

What is defined rather than measured is the zero of Martian solar longitude, set at the northern spring equinox, and the prime meridian, set through a crater. Neither changes any number in the figures; both decide the labels on their axes.

What the curves leave out

The orbit is a fixed ellipse. The other planets, Jupiter above all, perturb Mars’s orbit periodically, and the precise conversion between Martian mean and true solar time carries correction terms of a few hundredths of a degree that the figures omit.

The Sun is a point. A sundial reads the centre of a disc that is about two thirds the width it has from the Earth, and whose apparent size changes by nearly twenty per cent between perihelion and aphelion. Neither changes the equation of time; both change what a real sundial’s shadow looks like.

The atmosphere is ignored. Mars’s thin air refracts the Sun near the horizon by much less than the Earth’s does, which matters for sunrise times and not for noon, and the dust in it changes how sharp a shadow is.

And the axis is fixed within the year. The precession of Mars’s axis moves its equinoxes slowly along the orbit; within one Martian year that is negligible, over a few thousand it is not.

Still open: what a calendar on Mars should count

The equation of time is one of several ways Mars’s timekeeping differs from the Earth’s, and it is the easiest to write down. The harder question is what a calendar there should be built from. A calendar is a fraction chosen to approximate a year, and Mars’s year of 668.59 sols needs its own fraction; its seasons are so unequal that months of equal length in sols would drift against them far more than the Earth’s months drift against its seasons; and the planet’s two moons are no use for a month at all, one crossing the sky twice a sol and the other taking more than two sols to cross it once. Proposals exist in abundance and none is used by anyone, because nobody yet lives there. Which of the Martian periods a working calendar would choose to follow, and which it would let drift, is a question that only a settlement will answer.

Where this goes next

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

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AnalemmaEccentricityEquation of centreEquation of timeKepler's second lawMean solar timeObliquityPerihelionSolar daySolar longitude