Astronomy & Celestial Mechanics
the sky, computed
Series

Calendars — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · sky
  2. 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.

    part 2 · sky
  3. 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.

    part 3 · sky
  4. Where on the Earth a month begins on which evening. The line west of which the new crescent can be seen with the naked eye at local sunset, for a new Moon at 0 h universal time near the spring equinox, on the ecliptic. On the first evening (blue) the crescent is seen only west of the curve, because sunset comes later in universal time the further west an observer is and the Moon is that much older; on the second evening it is seen everywhere except beyond the green line, deep in the hemisphere where the evening ecliptic lies shallow. The first-evening line bulges furthest east at about twenty degrees north, where the spring ecliptic stands steepest, and sweeps far west towards the south, where the same season is autumn. An observed calendar that begins each month at first sight begins it on the first evening west of the line and on the second evening east of it — on two different dates within one religious community spread across the map.

    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.

    part 4 · sky
  5. A solar day that runs off to infinity as the spin approaches the orbit. The length of the solar day — sunrise to sunrise — in units of the world's own year, against its sidereal rotation period in the same units, with retrograde spin to the left. A world that spins much faster than it orbits has a solar day barely longer than its rotation, which is the Earth's case: its rotation is a few thousandths of its year, and it sits at the foot of the curve just right of zero, too close to the axis to mark. As the spin slows towards the orbital rate the solar day diverges, because the world turns to follow the Sun: at exactly one rotation per orbit the Sun stands still in its sky and there is no day at all. Mercury, turning three times in two of its years, has a solar day of 2.000 years — 176 Earth days. Venus, turning slowly backwards, has one of 116.8 days, 0.52 of its years, with the Sun rising in the west. The definitions of a day and of a year that the Earth keeps far apart come within a factor of two of each other, or exchange places.

    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.

    part 5 · sky
  6. Four tropical years, one for each season. How long the Earth takes to return to the same season, measured from each point of its orbit, minus the mean tropical year of 365.24219 days, in seconds, against the season point (the Sun's longitude). The perihelion drifts relative to the equinox, so between two returns of the same season the planet must cover slightly less than a full orbit in true anomaly, and that shortfall takes longest where the planet moves slowest. Measured from the March equinox the year is longer than the mean by 12 seconds; from the June solstice, 50 shorter; from the December solstice, 49 longer. The full range is ±51 seconds. The Gregorian calendar's average year, 365.2425 days (dashed), is 27 seconds longer than the mean and 15 seconds from the year counted from the March equinox, the one it was built to hold.

    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.

    part 6 · sky
  7. A star that walks through a calendar with no leap day. The day of the 365-day Egyptian civil year on which Sirius first rose in the dawn, seen from latitude 30°, against year, from 3000 BC to AD 501, counting from a coincidence of the rising with the civil New Year in AD 139. The civil year was a quarter of a day short of the star's, so the rising arrived one civil day later every four years and went once round the whole calendar in about 1458 years — the Sothic cycle. The star's own year was 365.2504 days, within a few minutes of the Julian year of 365¼.

    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.

    part 7 · sky

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