Astronomy & Celestial Mechanics
the sky, computed
Series

Precession — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close.

    The pole star has a shelf life, and the sky has a slow hand

    The Earth's axis traces a circle among the stars once every 25,772 years. Polaris is at the pole now, was not four thousand years ago, and will not be in two thousand more.

    part 1 · sky
  2. The wobble inside the wobble. Left: the two components of nutation over 40 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one.

    The wobble inside the wobble

    Precession and nutation are the same torque. The difference is that the Moon's orbital plane turns once in 18.6 years, so part of the pull oscillates instead of accumulating — and a catalogue position is not a direction until it says which pole it is measured from.

    part 2 · sky
  3. 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.

    part 3 · sky
  4. The nutation lines and the core's resonance. The resonance factor |σ₀ / (σ − σ₀)|, σ₀ being the free core nutation's frequency, on a logarithmic scale, against frequency in the celestial frame in cycles per sidereal day, negative for retrograde motion, with the free core nutation at a period of 430.2 days retrograde. The vertical lines are the largest forced nutations the Sun and Moon drive: the 18.6-year, annual, semiannual and fortnightly terms, each prograde and retrograde, and precession at zero. Any part of the Earth's response that involves the core is multiplied by this factor. The retrograde annual term, 365 days, is 5.6 on this scale; the prograde annual term 0.46; the retrograde 18.6-year term 1.07. The annual retrograde nutation is the one the core's resonance distorts most, because a year is the forced period nearest the core's own.

    A core that wobbles at its own period

    The Earth's liquid core can turn about an axis slightly tilted from the mantle's, and the flattened boundary between them pushes it back. That gives the planet a second free mode of rotation: in space a slow retrograde nutation of about 430 days, on the ground a wobble just faster than once a day. Its period is a measurement of the shape of a surface under 2,900 kilometres of rock, and it is read twice — in the nodding of the Earth's axis against the quasars, and in the rise and fall of the solid ground under the daily tides.

    part 4 · sky
  5. 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.

    part 5 · sky
  6. 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.

    part 6 · sky
  7. The Moon's reach north and south, over two cycles of its node. The greatest declination the Moon reaches in a month, north or south, from 2010 to 2045: about 23.44° plus the orbit's 5.145° tilt times the cosine of the node's longitude, which regresses once in 18.61 years. At the major standstill (2025.1, and 2006.5, 2043.7) the Moon reaches 28.6°, five degrees beyond the Sun's solstice declination (dashed); at the minor standstill, 18.3°, five degrees short of it. Near each extreme the reach changes slowly for two or three years, which is the standstill.

    A Moon that rises further north than the Sun

    The Sun's rising point swings north and south with the seasons between limits that do not change from one year to the next. The Moon's swings with the month, and its limits themselves open and close over 18.6 years as the plane of its orbit turns. At the widest, in 2024 and 2025, the Moon rose five degrees of declination further north than the midsummer Sun and skimmed the southern horizon of northern Scotland; at the narrowest, nine years later, it stays well inside the Sun's range.

    part 7 · sky

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