Astronomy & Celestial Mechanics
the sky, computed
Concept

Continued fraction — where it appears

An expansion of a number as a whole part plus one over another such expansion, whose truncations are the best rational approximations of their size. Every historical calendar cycle is one of them, and the size of the next term in the expansion is what makes the current approximation good.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

sky · Calendars 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.

sky · Calendars 15 revolutions of a 185 km swath. The band an instrument 185 km wide sweeps over 15 revolutions of a 233/16 repeat orbit at 98.2° and 699.6 km, drawn on an equirectangular map. Each band's edges are placed 92.5 km either side of the track on the sphere, so the bands really are the same width everywhere and only look wider towards the poles because this projection stretches longitude by 1/cos φ. Successive passes are 24.72° of longitude apart, and across the track at the equator that is 2691 km — so neighbouring passes leave strips about 2,506 km wide unimaged between them at the equator, while near ±82° the same bands lie on top of one another many times over. Filling the tropical strips is what the rest of the 16-day cycle is for.

A swath is sized at the equator

An imaging satellite's tracks crowd together towards the poles and spread apart towards the equator, so whether a swath leaves gaps is settled at the equator and nowhere else. The order in which a repeat cycle then closes those gaps is not orbital mechanics at all. It is a theorem about points on a circle.

spaceflight · Ground tracks 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.

sky · Calendars 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.

sky · Calendars

Named alongside it

The objects these essays reach for when they reach for this one.

Tropical yearIntercalationLeap-yearMetonic cycleSynodic monthBest approximationCalendar driftClimatic precessionCommensurabilityComputusConvergentEcclesiastical moon

All concepts