Astronomy & Celestial Mechanics
the sky, computed
Concept

Length of day — where it appears

The Earth's rotation period, which varies by milliseconds on timescales from days to decades. The seasonal part is atmospheric; the decadal part is angular momentum exchanged with the liquid core, which makes a clock comparison into a probe of a flow nobody can see.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

A clock that has lost 5.7 hours in 2720 years. ΔT = TT − UT1, the accumulated difference between a uniform time scale and the Earth's own rotation, from 700 BC to 2020, on a logarithmic scale. The points are the published record; the smooth curve is the parabola 31.94·u² fitted to the entries at or before 1000, with u in centuries from 1820. It is a parabola and not a line because the Earth is not merely slow, it is slowing: a day lengthening at a constant rate makes a clock fall behind by the integral of the lag. That coefficient is a length-of-day rate and nothing else — 31.94 = ½ × r × 36525 days per century gives r = 1.75 ms per century, against the 1.78 ms computed from tidal angular momentum with no eclipse anywhere in the derivation. At 700 BC the offset is 5.7 hours, which is 85° of the Earth's rotation, and that is why the ancient measurements are records of the place a total eclipse was seen from rather than of the hour: the hour was never written down accurately enough to matter, and the shadow's track on the ground was.

Six kinds of second

The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock.

sky · Timescales A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 150 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.

A wobble that should have stopped

The Earth's rotation pole wanders across its own crust in a circle a few metres wide. A rigid Earth would do it in 305 days; it takes 433, and the difference is a measurement of the planet's elasticity. At the observed damping it should have died out within a human lifetime, and it has not.

sky · Precession How far the clock drifts from the Sun once leap seconds stop. The accumulated difference between the Earth's rotation and atomic clock time, in seconds, that builds up after leap seconds are abandoned, against years since then, for a day that now exceeds 86,400 SI seconds by −0.3 ms, 0.5 ms, 1.5 ms, each lengthening by 1.7 milliseconds a century as the tides brake the Earth. The drift is the integral of the excess: 20 s after 100 years, a minute after 158 for −0.3 ms; 49 s after 100 years, a minute after 113 for 0.5 ms; 86 s after 100 years, a minute after 76 for 1.5 ms. The present excess is close to zero — the day has recently been slightly shorter than 86,400 seconds — so the drift starts slowly or even backwards, and the tidal term then takes over and bends every curve upward. The time for the difference to reach a minute is set far more by the next century's core-driven fluctuations, which have swung the excess by two or three milliseconds within a few decades before, than by anything that can be predicted.

A clock allowed to leave the Sun

Twenty-seven leap seconds have kept civil time within a second of the Earth's rotation since 1972. By 2035 they will stop, and the clock will be allowed to drift from the Sun. How far it drifts, and how soon a minute accumulates, depends on a length of day that no one can forecast.

sky · Timescales 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 The rhythms in the rock, set by the Moon's distance. The periods of the climatic precession cycles — the precession constant plus the perihelion frequency of each of the planets that dominate the Earth's eccentricity — and of the obliquity cycle, the precession constant plus the node frequency s₃, against the Earth–Moon distance. Today they are 23.8, 22.4, 19.0 thousand years for precession and 41 for obliquity. At 341,000 km they were 14.7, 14.2, 12.8 and 20.0. The planetary frequencies are held at today's values, which the chaos of the inner solar system makes uncertain beyond a few tens of millions of years; the precession constant is the term that moves.

A rhythm in the rock that measures the Moon

The Earth's axis precesses because the Sun and Moon pull on its equatorial bulge, and two-thirds of the pull is the Moon's. When the Moon was nearer and the day shorter, the axis turned faster, and the climate cycles that precession drives were shorter. Sediments laid down 1.4 billion years ago record those cycles — and with nothing in the rock but layer thicknesses, the ratio of two rhythms gives the Moon's distance at the time.

sky · Precession Two old points, and the history they draw back to the Moon's birth. The Earth–Moon distance against time before the present, with the estimates from tidal rhythmites (620 million years), cyclic sediments (1.4 billion) and banded iron formation (2.46 billion, 321.8 thousand km, ±6.2). Today's tidal efficiency, scaled as the inverse sixth power of the distance, brings the Moon to the Earth 1.54 billion years ago. The efficiency fitted between the two oldest points — 21% of today's — held further back brings it out from the Earth 4.78 billion years ago, close to the Moon's age of about 4.5 billion. The constant law is an assumption; what the points measure is the average between them.

A Moon measured in banded iron

Layers of iron-rich sediment laid down 2.46 billion years ago in what is now Western Australia keep time with the Earth's precession, and the precession's period says how far away the Moon was: about 321,800 kilometres, with a day of seventeen hours. Together with an older-is-closer point from 1.4 billion years ago, it measures how hard the tides worked between them — about a fifth as hard as today. Held further back, that efficiency brings the Moon out from the Earth a few hundred million years too early, which says the young tides worked harder than the middle-aged ones.

sky · Precession

Named alongside it

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

Angular momentumCyclostratigraphyΔTDynamical ellipticityLeap secondLunar recessionMilankovitch cyclesTidal frictionAtomic clockBeat frequencyChandler wobbleClimatic precession

All concepts