A day longer than the year

A calendar on the Earth starts from two numbers it is given: the day and the year. The year is not a whole number of days, and every solar calendar is a choice of fraction to approximate the remainder. But the problem only has that form because the two numbers are so far apart — 365 of one in the other — that the day can be treated as a unit and the year as a long count of it. The day is the Earth’s rotation measured against the Sun, and it differs from the rotation measured against the stars by four minutes, which is the year’s share: the Earth has moved a little round its orbit during each turn, so it must turn a little further to bring the Sun back.

That four minutes is small only because the Earth spins fast. On a world that turns slowly, the orbital motion’s share of each day is not a correction but the main term, and the question of what a day is stops having an obvious answer. The planets of the solar system include every case, and each of them is a calendar problem of its own.

A day is a beat #

The solar day is a beat frequency. A world turns at one rate against the stars and goes round its star at another, and the rate at which the star appears to circle its sky is the difference:

$$\frac{1}{P_{\rm sol}} = \frac{1}{P_{\rm rot}} - \frac{1}{P_{\rm orb}},$$

with the rotation period counted negative if the world spins backwards. For the Earth the second term is a three-hundred-and-sixty-fifth of the first and makes the day four minutes longer than the rotation. As the spin slows towards the orbital rate the two terms approach each other and their difference goes to zero, so the solar day grows without limit, which is the divergence in the figure. At exactly one rotation per orbit the world keeps one face to its star and there is no day at all — permanent noon on one side, permanent night on the other.

Past the divergence the solar day is negative, which means the star moves backwards across the sky; a world spinning slower than it orbits sees its sun rise in the west. A world spinning backwards always has a positive day, shorter than its rotation, because the orbit and the spin add rather than subtract — the Sun goes round the sky faster than the world turns. The figure’s two halves are the two ways a slow spin and an orbit can combine, and the solar system has a planet on each.

Mercury: a day of two years, and a Sun that backs up #

Mercury turns on its axis once every 58.65 days and orbits the Sun once every 87.97, three turns for every two orbits. That ratio is not a coincidence but a resonance its own tides settled into: Mercury’s orbit is eccentric, and the tidal torque from the Sun is strongest at perihelion, where the planet is moving fastest, so the spin was pulled towards the orbital rate at perihelion rather than the average rate, and the three-to-two state is where it caught. The beat formula then gives a solar day of exactly 175.94 days — two Mercury years. From any place on Mercury, the Sun rises, crosses the sky in one year, and spends the next year below the horizon.

The eccentricity does something more striking to the Sun’s motion within that day.

Near perihelion Mercury’s orbital angular velocity rises to more than one and a half times its average, and its spin rate is exactly one and a half times the average. For about eight days around each perihelion the orbital motion outruns the spin, and the Sun, which had been crossing the sky in the ordinary direction, stops, moves backwards, and stops again before resuming. Seen from the longitude where perihelion comes at noon, the Sun climbs to overhead, hangs there, backs a little way towards where it rose, and then goes on to set. Seen from the longitude ninety degrees away, perihelion comes at sunrise: the Sun rises, sinks back below the horizon, and rises a second time. Because of the three-to-two lock, the same two longitudes have these experiences at every perihelion, and they are called the hot poles and warm poles of Mercury, because the first are under the Sun when it is closest.

A calendar on Mercury therefore has no single answer to when a day begins. The sunrise at the warm-pole longitudes happens twice, and the noon at the hot-pole longitudes lasts a week. A day defined by the Sun’s motion would have to choose between three crossings of the horizon or treat one of them as not counting.

Venus: a day shorter than the year, and the face it shows the Earth #

Venus turns backwards, once every 243.02 days, and goes round the Sun in 224.70. Its rotation is the slowest of any planet, slower than its year, and on the beat formula the retrograde spin and the orbit add: the solar day is 116.75 days, a little more than half a Venus year. The Sun rises in the west and sets in the east, twice a year.

The Venus day connects to the Earth’s calendar through a near-coincidence that was once taken for a law.

Venus and the Earth come closest every 583.92 days, the synodic period that sets when Venus can be studied best and when spacecraft can most cheaply be sent. That interval is 5.0015 Venus solar days. At every close approach the Earth sits on Venus’s midnight meridian, so the face Venus turns towards the Earth is nearly the same each time, advancing by about half a degree. When radar first measured Venus’s rotation through its clouds in the early 1960s and found this, it was read as a lock — the Earth’s tides on Venus holding its face, as the Earth holds the Moon’s. The Earth’s tidal torque on Venus is far too weak for that, smaller by orders of magnitude than the Sun’s and smaller still than the torque the massive atmosphere exerts as it is heated and pushed around by the Sun every day. The near-commensurability is a coincidence, and the Venus day drifts slowly against it.

The coincidence is a reminder that runs through this whole subject. Eclipse cycles, the Metonic cycle and the leap rules are all near-commensurabilities between periods that have no physical reason to be related; a few of them are resonances maintained by forces, most are arithmetic, and the difference is not visible in the numbers alone.

Mars: an Earth-like day and an awkward year #

Mars is the case most like the Earth, and that makes it the case where the calendar problem looks most familiar. Its rotation is 24 hours 37.4 minutes; its solar day, the sol, is 24 hours 39.6 minutes, longer by the share of its orbit; and its year is 668.59 sols. A calendar for a crew living by Martian daylight — or for a lander whose power comes from the Sun, as every Mars lander’s schedule already does — has to count sols in a year, and the fraction it has to approximate is 0.5907.

The method is the one the Earth’s fraction gives: expand the remainder as a continued fraction, and the best leap rules for each cycle length are its convergents. For the Earth the fraction 0.2422 sits close to a quarter, and the Julian rule of one leap day in four years is the convergent that made a usable calendar with a four-year cycle. Mars’s 0.5907 has no such luck. Its first useful convergent, three in five, is worse than the Julian rule by a factor of two, and the next good ones need cycles of seventeen or twenty-two years. The calendars proposed for Mars have therefore used nested rules — a leap sol in most odd-numbered years and in years divisible by ten, except those divisible by a hundred, and so on — which are the continued fraction’s later terms written as exceptions, exactly as the Gregorian rule’s century exceptions are for the Earth.

There is one further choice, and it has no counterpart on the Earth: the length of the sol itself does not divide into Earth hours. Mission teams that work on Mars time shift their clocks forty minutes a day and live on a twenty-four-hour-thirty-nine-minute schedule for months at a time, and the calendar’s units have to be either Earth seconds stretched or Martian seconds defined. Whatever a Martian calendar counts, the clock that measures it is already a convention.

Locked moons: a day set by a planet #

The far right of the divergence is the case of a world whose spin has been pulled all the way into step with its orbit — but around a planet, not the Sun. The Moon, Titan, Europa and most large moons in the solar system are tidally locked to their planets, turning once per orbit. Each still has a day relative to the Sun, because its planet carries it round the Sun, and that day is set by the moon’s orbit.

The Moon’s solar day is the synodic month, 29.53 days: the Moon keeps one face to the Earth, not to the Sun, and the Sun goes round its sky once per lunation. A lunar calendar for people on the Moon would therefore count the same month that the Earth’s lunar calendars count, but as a day, and its year would be the Earth’s year — 12.37 lunar days, which is the same awkward fraction that every lunisolar calendar on the Earth has had to tabulate. The problem a lunisolar calendar solves on the Earth is the problem a solar calendar would have to solve on the Moon.

A locked moon has a second clock no unlocked world has. Its planet hangs fixed in the sky of one hemisphere and never rises in the other’s. From Titan’s Saturn-facing side, Saturn does not move across the sky but goes through phases, once per Titan day, and eclipses the Sun for a few hours at the right season; from the far side it is never seen. Half the world’s most prominent astronomical object is a body that never moves.

Two worlds locked to each other #

Pluto and its largest moon, Charon, are the case that completes the set. Charon is locked to Pluto, as moons usually are; Pluto is also locked to Charon, which no planet in the solar system is to its moon. Both turn once every 6.39 days, which is also their orbital period about each other, so Charon hangs motionless over one hemisphere of Pluto and is never seen from the other, and the same is true the other way round. The pair is turning as a single rigid dumbbell.

That is the end state the Earth and the Moon are heading towards, very slowly. The tide the Moon raises on the Earth is paying the Earth’s spin into the Moon’s orbit, and if nothing else intervened the Earth’s day would lengthen until it equalled the month, by then about forty-seven of today’s days, and the two would face each other as Pluto and Charon do. The Sun will have changed long before that, so the Earth never gets there; Pluto and Charon got there because Charon is an eighth of Pluto’s mass, so its tides on Pluto were strong and the two despun each other in a time short compared with the age of the solar system.

On Pluto the beat formula gives a solar day almost exactly equal to the rotation, 6.39 days, with the Sun rising in the west because the spin is retrograde relative to Pluto’s orbit round the Sun — and a year of 248 Earth years, 14,000 Pluto days. The calendar problem there is mostly a matter of very large numbers, with Charon fixed in the sky as a clock that does not tick.

The Earth’s own ratio has changed #

The day and the year on the Earth are far apart now, but not by a fixed number. The tides have been lengthening the day for as long as there has been a Moon, by about two milliseconds a century at present, and the year has barely changed. So the number of days in a year has been falling.

It has been measured, in an unexpected record. Corals lay down a fine growth band every day, the bands vary in thickness through the year with the seasons, and fossil corals preserve both. Counting daily bands between annual ones in corals from the Devonian period, about 380 million years ago, John Wells found in 1963 about four hundred days to the year — a day of about twenty-two hours. Tidal sediments that record daily, monthly and annual cycles in their laminations give similar answers from other epochs, back to roughly 620 million years ago, when the year held about four hundred days as well and the month about thirteen of them.

So the Earth’s calendar fraction is itself a drifting quantity on geological timescales. A Gregorian calendar designed in the Devonian would have needed a different leap rule, and one designed a few hundred million years from now will need another. The difference from the slow worlds above is only of degree: on Mercury and Venus the tides finished the work long ago, and on the Earth they are still doing it.

Exoplanets locked to their stars #

Beyond the solar system the right-hand pole of the figure is common. Planets in close orbits around small stars are expected to be tidally locked, one face permanently lit and one permanently dark, and the planets of the TRAPPIST-1 system, in orbits of one and a half to nineteen days, are thought to be locked or nearly so. For such a planet the solar day is infinite and there is no calendar of days at all: the only clock is the orbit, and the only seasonal change is the planet’s slight libration, which rocks the terminator back and forth once an orbit. Whether a thick atmosphere can carry heat from the day side to the night side and make such a world habitable is one of the main questions asked of them — and it is a question about a world with no day to speak of.

Some are not locked but caught in a resonance like Mercury’s, and a planet on an eccentric orbit can settle into a three-to-two or higher state. Then the solar day is a small multiple of the year, the hot-pole geometry of Mercury repeats, and the star stops and reverses in the sky at every periastron.

What the figures assume #

The solar-day formula assumes circular orbits; Mercury’s figure is the one place eccentricity is used, and it changes the Sun’s motion within the day without changing the day’s length. The rotation periods are the present ones; Venus’s is known to vary by several minutes from one measurement campaign to the next, because its atmosphere exchanges angular momentum with the solid planet, and the face it shows at close approach drifts correspondingly. The Martian leap rules are drawn for a constant year length; the Martian tropical year varies slowly as its orbit and axis precess, and a calendar meant for thousands of years would need to follow that, as the Earth’s would. And the locked moons’ solar days use their present orbits, which tides are slowly changing.

Still open: what a calendar should count where the day and year are comparable #

On the Earth the calendar’s question is how to fit a year into days, and it has been settled to a part in a million. On a world where the day is comparable to the year, it is not clear that a calendar of days should exist: Mercury’s day is two years long and its year is half a day, Venus has fewer than two days a year, and a locked world has none. Whatever human activity is eventually carried out on such worlds will be governed by the clock of the Earth it came from or by the light-and-dark cycle of its own surface, and the two cannot both be counted in whole numbers. The arithmetic for the second is a continued fraction like any other; the decision of which to keep is the one this figure cannot make.