A year that depends on the season counting it

Every calendar is a fraction. The year is not a whole number of days, so a calendar inserts leap days at a rate that approximates the remainder, and the quality of a leap rule is measured by how slowly its average year drifts against the real one. The Gregorian rule — a leap day every four years, except in century years not divisible by four hundred — gives an average year of 365.2425 days, and the tropical year is usually quoted as 365.24219. The difference, 0.00031 days, is about twenty-seven seconds a year, and the Gregorian calendar is said to drift by a day in about three thousand two hundred years.

That statement contains an assumption that does not hold. The tropical year is defined as the time for the Sun to return to the same position relative to the equinoxes and solstices, and it depends on which position is chosen. The hero figure draws it, for every point of the Earth’s orbit, as the difference from the mean value. Measured from the March equinox, the year is twelve seconds longer than the mean; from the June solstice, fifty seconds shorter; from the December solstice, forty-nine seconds longer. The mean tropical year is the average over all season points, and no particular season keeps it.

Why the seasons disagree #

If the Earth’s orbit were a fixed ellipse and the equinoxes did not move, the year measured from any season point would be the same: the time to go once round. Neither is fixed. The equinoxes precess westward along the ecliptic by about fifty arcseconds a year — the slow turn of the axis that moves the pole star — and the perihelion of the orbit advances eastward by about twelve, so relative to the equinoxes the perihelion moves by about sixty-two arcseconds a year — a full circuit of the seasons in about twenty-one thousand years. This is the climatic precession, the cycle that moves the Earth’s closest approach to the Sun from January toward summer and back.

Between two passages of the Sun through the same season point, then, the Earth must cover not a full orbit in true anomaly but a full orbit minus those sixty-two arcseconds. How long it takes to cover the missing sliver depends on how fast it is moving where the sliver is — that is, at the season point being used. Near perihelion, where the Earth moves fastest, the sliver is crossed quickly and the year is short of a full orbit by a small time; near aphelion it is crossed slowly. Turned around: the season points the Earth reaches near perihelion return later than average, and those it reaches near aphelion return earlier.

To first order in the drift, the year measured from a season point at true anomaly $\nu$ differs from the mean by

$$T(\nu) - \bar T = -\frac{\Delta\varpi}{n}\left[\frac{(1-e^2)^{3/2}}{(1+e\cos\nu)^2} - 1\right],$$

with $\Delta\varpi$ the drift per year and $n$ the mean motion. For the Earth’s eccentricity of 0.0167 the bracket swings by about ±3.3 per cent of the orbit’s average rate, and with a drift of sixty-two arcseconds the year swings by about ±51 seconds. The December solstice falls two weeks before perihelion, so the year counted from it is near the maximum; the June solstice falls near aphelion, so its year is near the minimum. The equinoxes fall between, on the rising and falling parts of the curve.

The first-order formula is an approximation, and the year counted from the March equinox in the published ephemerides is about sixteen seconds longer than the mean rather than the twelve the formula gives here; the difference comes from terms the formula omits, such as the perihelion’s own nonuniform motion and the small inequalities the planets impose on the Earth’s orbit. The pattern — a fifty-second swing, longest in northern winter — is the formula’s.

A disagreement older than the explanation #

Astronomers measured the year from the seasons long before anyone could say why the seasons should disagree, and the disagreement left a trail. Hipparchus, in the second century BC, timed equinoxes and solstices over a century and a half of records and found the year shorter than 365¼ days by a three-hundredth of a day, nearly five minutes; Ptolemy adopted his value. Medieval astronomers working in Baghdad, Cairo and Toledo, comparing their own equinox timings with Ptolemy’s, found the year shorter again, by amounts that differed from one observer to the next — and the differences were large enough that some concluded the year itself was changing, oscillating in a cycle of “trepidation” of the equinoxes that persisted in astronomical tables until the seventeenth century.

Part of that apparent variation was error in the old observations, which were difficult to time to better than a day. Part was real: the year measured from equinoxes and the year measured from solstices genuinely differ, and the difference itself changes over centuries as the perihelion moves. A measurement based mostly on equinox timings and one based mostly on solstice timings, made a thousand years apart, compare two different years at two different epochs. The trepidation theories were wrong about the mechanism and right that the year being measured was not a fixed number.

The modern resolution came with the recognition that the “year” in a theory of the Sun’s motion is a mean quantity — the period of the Sun’s mean longitude relative to the moving equinox — and that the time between particular passages differs from it by periodic terms. Simon Newcomb’s tables of the Sun, at the end of the nineteenth century, gave the mean tropical year as a slowly varying polynomial in time, and the periodic departures as separate terms. That separation is why the figure usually quoted for the Gregorian calendar’s error compares it with the mean: the mean is the number the tables printed.

A tilt that is not a constant and an orbit whose perihelion wanders are the same story told for the climate: the precession that moves the perihelion through the seasons is the one whose cycle appears in the ice-age record. On the timescale of a calendar it is a correction of seconds; on the timescale of the climate it decides which hemisphere’s summer comes at perihelion.

The Gregorian calendar’s real target #

The Gregorian reform of 1582 had a specific purpose: to keep the March equinox near the twenty-first of March, because the date of Easter is computed from it. Its arithmetic is a finite machine acting on the year number, and the leap rule had to hold the equinox in place. The year it was built to approximate was therefore not the mean tropical year but the year counted from the March equinox.

The hero figure shows how well it does. The Gregorian average year, 365.2425 days, is twenty-seven seconds longer than the mean tropical year, but only about fifteen seconds longer than the year measured from the March equinox in this approximation — and in the more exact ephemeris value about eleven. The calendar drifts against the equinox it was built for by a day in about eight thousand years, not three thousand. The figure usually quoted is a comparison with the wrong year.

The reformers did not know any of this explicitly. The tables they used gave a single tropical year, and the rule was chosen as a convenient approximation to it. That the rule happens to match the equinox year more closely than the mean is partly an accident of the tables’ errors and partly a consequence of having calibrated against the equinox, which is the season point the astronomers of the time measured.

A target that moves #

The year counted from any one season is not a constant either, because the perihelion keeps moving through the seasons.

As the perihelion advances, the equinox’s position relative to it changes, and the equinox year swings through the same ±51 seconds over the twenty-one-thousand-year cycle. Today it is rising: the March equinox is moving toward the part of the orbit where the Earth is fastest, which lengthens the year counted from it, and in about eleven hundred years it will equal the Gregorian average exactly. After that it will exceed it for several thousand years, and the calendar will drift the other way.

A calendar can be fitted to the year counted from one season at one epoch. It cannot be fitted to all seasons at once, because they disagree by a minute and a half, and it cannot be fitted to any one season for long, because that season’s year changes by most of a minute over a few thousand years. A leap rule more precise than about ten seconds a year is a rule fitted to a moment.

Ten thousand years of equinox #

Two slower effects compound the moving target. The mean tropical year is itself getting shorter, by about half a second a century, as the rate of precession changes. And the day is getting longer: the tides raised by the Moon brake the Earth’s rotation, partly offset by the crust’s slow rebound since the last ice age, and the average day has lengthened by about 1.8 milliseconds a century over the last few thousand years. A calendar counts days, so a lengthening day means fewer of them in each year.

Over the next ten thousand years the three accounts diverge. The textbook comparison with a fixed mean year gives a drift of three days. The year the calendar actually holds, the equinox year, keeps the equinox within a day for the next eight thousand years, because it is close to the calendar’s year now and rises toward it — and then pulls away as the tropical year’s own shortening accumulates. The lengthening day is the largest effect of all over that span, because its error grows quadratically: each century’s millisecond and eight-tenths adds to every subsequent year. By the year twelve thousand, the equinox will fall about six days early.

How much of the six days is the day’s fault depends on a rate that has to be measured, not computed, and the figure is sensitive to it.

The tidal part of the lengthening is the better understood: it follows from the Moon’s measured recession, about 3.8 centimetres a year, and the angular momentum it carries away. The part that offsets it is the Earth’s shape relaxing after the last ice sheets melted, which makes the Earth rounder and spins it up, and that relaxation is decaying on a timescale of thousands of years. Over ten thousand years the difference between the two rates is a day in the equinox.

None of this is a defect in the Gregorian rule in the sense the reformers worried about. Their concern was the Julian calendar’s drift of a day every hundred and twenty-eight years, which had moved the equinox ten days in thirteen centuries. The Gregorian rule reduced that by a factor of sixty, and the remaining errors are set by the physics of the Earth and its orbit, not by the arithmetic of the rule. A more precise rule would buy nothing without a prediction of how the day will lengthen, and the day’s length is now measured rather than predicted, its irregular variations from the core and the oceans larger than the trend over decades.

Mars, where the spread is fourteen minutes #

The effect is proportional to the orbit’s eccentricity and to the rate of climatic precession. Mars has an eccentricity of 0.093, more than five times the Earth’s, and its orbit outweighs its tilt in shaping its seasons.

The Martian year swings by fourteen minutes from one season point to another. A Martian calendar faces the leap-rule problem the Earth’s calendar solved: its year is 668.5907 sols, and the best leap rules come from the continued fraction of the remainder. The rule of thirteen leap sols in twenty-two years is wrong by a third of a minute a year — a sol every forty-eight centuries.

But the year it is approximating is uncertain by fourteen minutes depending on which season it is anchored to. The leap rule is forty times more precise than the definition of the year it holds. A Martian calendar designed to keep the northern spring equinox fixed would need a different rule from one designed to keep the start of the dust-storm season, which falls near perihelion, fixed — and the difference between the two rules would be larger than the error of either.

The Martian spread also moves, and faster in proportion than the Earth’s, because Mars’s eccentricity is large and its climatic precession cycle, about fifty-one thousand Earth years, carries the perihelion through every season.

Over a few thousand Martian years the spring-equinox year changes by several minutes, so a leap rule fitted to it now would be out by a sol in a few centuries even if its arithmetic were perfect. The Martian seasons themselves are unequal by weeks — northern spring is fifty sols longer than northern autumn — for the same reason, and the inequality moves round the year with the perihelion. A Martian calendar is a calendar for a planet whose seasons do not stay where a calendar puts them.

The practical answer for a calendar on Mars is the one the Gregorian reform arrived at by accident: choose the season that matters, measure the year from it, and accept that every other season drifts. For Mars the season that matters is plausibly the one tied to perihelion, when dust storms arise and solar power is most at risk, and the year measured from there is about six minutes longer than the mean. A rule fitted to the mean would let that season slip a sol every two centuries.

What the figures leave out #

The drift is first-order. The formula treats the perihelion’s motion relative to the equinox as uniform and small within a year, which it is, but omits the planetary perturbations that make the Earth’s actual orbit depart from a precessing ellipse. Its values differ from ephemeris values by a few seconds for the Earth.

The secular terms are extrapolations. The shortening of the mean tropical year by about half a second a century is a polynomial fit valid for a few thousand years; the lengthening of the day by 1.8 milliseconds a century is the average of the last two and a half millennia, inferred from ancient eclipse records, and the rate over the next ten thousand years is not known to better than a few tenths of a millisecond a century.

The Martian precession rate is approximate. Mars’s spin axis precesses in about 170,000 years and its perihelion advances, giving a climatic precession of about 51,000 Earth years; the rate is uncertain by several per cent, which scales the Martian spread proportionally.

Calendars count days at a place, not at the centre. The equinox’s calendar date also depends on the time zone in which it is reckoned, and on whether the day boundary is set by mean or apparent time — the Sun being a bad clock — which moves individual years’ equinox dates by up to a day and is unrelated to the long-term drift.

Still open: what a calendar should anchor to #

Every calendar so far has been anchored, explicitly or not, to a season chosen by a religious or agricultural purpose: the spring equinox for Easter, the winter solstice for the solar new year in some traditions, the rising of a star for the Nile flood. The choice was never examined as a choice of which year to hold, because the differences between the seasons’ years were smaller than every other error in the calendar.

They are not smaller now. The Gregorian rule’s arithmetic error is about eleven seconds a year against its own target, and that target moves by fifty seconds over the precession cycle; the day’s lengthening dominates on the scale of millennia. Whether the calendar of a civilisation planning on those scales should be anchored to a season, to the mean tropical year, to the atomic second, or should simply be revised every few thousand years as the Gregorian was revised from the Julian, is a question that astronomy can set out exactly and cannot answer. The fraction is easy. Deciding what it is a fraction of is not.