Astronomy & Celestial Mechanics
the sky, computed
Concept

Synodic month — where it appears

The interval from one new moon to the next, 29.53 days, which is longer than the Moon's orbital period because the Sun has moved on. It is the period a lunar calendar has to track, since a phase is a relative configuration rather than a position.

Named by 7 essays across 3 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 A right angle short by 0.147°, and a ratio of 389 hanging on it. The ratio of the Sun's distance to the Moon's implied by the angle between them at the moment the Moon is exactly half lit, on a logarithmic scale, for angles from 80° to 89.95°. At that moment the angle at the Moon between the directions to the Sun and the Earth is a right angle, so the ratio is the secant of the observed angle — a construction with no distance in it, the same right triangle that gives an inferior planet's orbit from its greatest elongation. Aristarchus measured 87°, which gives 19.1. The mean distances give 389, which corresponds to 89.853°. The curve's steepness is the whole story: across a tenth of a degree centred on each marked angle the ratio changes by 3.4 per cent at 87°, 10.5 per cent at 89° and 103 per cent at the true angle. The method was exact and it asked for a right angle measured to a hundredth of a degree, at an instant the eye can judge only to within hours, on a terminator that is never quite straight.

A right angle short by a seventh of a degree

When the Moon is exactly half lit, the angle at the Moon between the Sun and the Earth is a right angle, so the angle seen from the Earth gives the Sun's distance in units of the Moon's. Aristarchus measured 87° and concluded the Sun was nineteen times further away. The construction was exact; the angle he needed was 89.85°, and at that angle a tenth of a degree is the whole answer.

sky · Apparent motion 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.

sky · Calendars The tide's spectrum, with every frequency set by the sky. The equilibrium tide-raising potential broken into its largest constituents, each a sinusoid whose frequency is a sum of whole-number multiples of six astronomical rates — the Earth's turning relative to the Moon, the Moon's and the Sun's motions round the sky, and the slow turning of the lunar perigee and node — and whose height is its equilibrium amplitude relative to M₂. They fall in three bands: near two cycles a day (M₂ at 1.9323, S₂ at exactly 2, N₂, K₂), near one (K₁, O₁, P₁, Q₁), and near zero (fortnightly, monthly, semi-annual). Every frequency is known to many decimal places from astronomy; what a harbour does to each, its own amplitude and phase, is not, and is the only thing a tide table has to measure.

A tide table is astronomy with the harbour fitted

Nobody can compute the tide at a harbour from the Moon and the Sun, because the ocean's response depends on coastlines and depths nobody models well enough. Tides are predicted to centimetres years ahead anyway, because the frequencies are pure astronomy and only the response to each one has to be measured. The method needs a record long enough to tell neighbouring frequencies apart, and a correction for a cycle of 18.6 years.

gravitation · Tides 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.

sky · Calendars The Moon pushed towards the Sun, once a month. The change in the Earth–Moon distance, in millimetres, if the Earth's gravitational self-energy fell towards the Sun differently from the rest of its mass by a fraction η = 0.001 — the term that would be present at the synodic period, the cycle of lunar phases, in an integration of the Moon's orbit in the Sun's full field. The amplitude is 11.4 mm, which is 11.4 metres times η: the Moon's orbit would be stretched towards the Sun at new moon and squeezed at full, or the reverse, depending on the sign. The shaded band is ±2 mm, roughly the precision of a modern lunar laser ranging normal point. A violation a thousandth of the size of the effect it tests would be twice that precision.

A Moon that would lean towards the Sun

Part of the Earth's mass is its own gravitational binding energy — a few parts in ten billion. If that part fell towards the Sun even slightly differently from the rest, the Earth and the Moon would fall at different rates and the Moon's orbit would be pushed sunward once a month by about eleven metres for every unit of the difference. Fifty years of laser ranging to the Moon have looked for that push at the millimetre level, against a term the Sun's ordinary gravity puts at the same frequency a hundred kilometres high.

orbits · Ephemerides

Named alongside it

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

Tropical yearLunisolar calendarMetonic cycleComputusContinued fractionEpactIntercalationAnomalistic monthAstronomical unit (AU)Best approximationCalendar driftConvergent

All concepts