Astronomy & Celestial Mechanics
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Obliquity — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two equilibria become four, and three worlds sit near the join. The Cassini equilibria of a spin axis, drawn against the ratio of its own precession rate to the rate at which its orbit plane turns, for an orbit inclination of 1.5 degrees. Each column of dots is the full set of obliquities at which the two precessions keep step at that ratio, found by root-finding rather than by tracing a remembered curve. Below α cos ε/|g| = 1.135 there are two such obliquities and above it there are four, and the figure checks both counts on either side of the join. The three marked bodies are placed by their own measured precession constants: the Earth with the Moon at 2.67, safely on the four-state side; the Earth without it at 0.86; and Mars at 1.06. Two of the three sit within a few tenths of the bifurcation, which is the whole reason their obliquities are not constants: near the join the equilibria are close together, the libration around them is wide, and a body pushed between neighbouring resonances wanders. The Moon's contribution to the Earth's precession constant is what moves the first mark away from that region, and the second mark is the same planet with that contribution removed. This is a two-frequency model of a many-frequency system, and the real chaos comes from the overlap of resonances it does not contain.

    A tilt that is not a constant

    The Earth's axis leans by 23.4 degrees, and that lean is what makes the seasons. It is also a dynamical variable with its own equilibria, its own resonances and its own chaos — and on Mars the same variable has swung between nearly zero and sixty degrees without anything having to happen.

    part 1 · sky
  2. Why the Moon holds the tilt still. Spin-axis precession rate against obliquity, with the solar system's own secular frequencies drawn across it. The rising-and-falling curves are α cos ε for three precession constants: the Earth as it is, with the Moon supplying about two thirds of the torque; and a Moonless Earth at two plausible rotation rates. The horizontal lines are the nodal eigenfrequencies of the planetary secular system — s₆, s₃, s₄, s₂, s₁, s₇, s₈ — which are the rates at which the Earth's own orbit plane wobbles. A crossing is a resonance: the axis is precessing about a plane that is itself turning at the same rate, the resonant angle stops circulating, and the obliquity librates instead of holding still. The present Earth precesses at 50.3 arcseconds a year, faster than every secular frequency in the list, and its nearest crossing is at 61.3°, which is 37.8 degrees from the obliquity drawn. Take the Moon away and α falls by a factor of three; the crossings come down with it, 3 of them land below 85°, and where several overlap the obliquity has no stable value at all. The Moon does not hold the axis by pulling on it. It holds it by making the precession too fast to resonate with anything.

    A tilt held still by being too fast to resonate

    The Moon does not hold the Earth's axis by pulling on it. It triples the rate at which the axis precesses, which lifts that rate clear of every frequency at which the Earth's own orbit plane wobbles — and a precession with nothing to resonate with cannot wander.

    part 2 · sky
  3. Mars's tilt over fifty million years, and a steadier twin. The obliquity of a Mars-like planet against time over 50 million years, integrated from 25 degrees with its orbital plane wobbling in five secular modes. With Mars's precession constant, 8.26 arcseconds a year, the axis wanders between 1 and 70 degrees, lingering near some values and lurching between them. The same planet precessing 6 times faster — about the Earth's rate, which the Moon's pull makes possible — stays between 23.2 and 30.8 degrees, because its precession is too fast to resonate with any of the orbital-plane frequencies. The orbital perturbations are illustrative, of the size Mars's own are; the chaos is computed from them.

    A past that can only be given as odds

    Mars's axis is tilted by twenty-five degrees today, and nothing holds it there. Its slow precession falls among the frequencies at which its orbital plane wobbles, the resonances overlap, and the tilt wanders chaotically between nearly zero and sixty degrees. Two calculations that start a hair apart part company within a few tens of millions of years — so Mars's obliquity a billion years ago is not a number anybody will compute. What can be computed is how much of its history the planet spent at each tilt, and that distribution is a prediction the ice at its poles can test.

    part 3 · sky
  4. The summer sunshine at Mars's north pole over the last million years. The sunshine reaching Mars's north pole at the summer solstice, relative to its average, over the last 1,000 thousand years (blue), from a forcing built from the main periods of Mars's recent orbit: the tilt varying by about 3.5 degrees in 120,000 years and 1 degree in 99,000, the eccentricity by 0.03 in 95,000 years and 0.02 in 2.4 million, and the perihelion going round the seasons once in 51,000 years. The part due to the tilt alone is dashed. The sunshine swings by a factor of 2.06 between its extremes: the tilt sets a slow swell with a period of 120,000 years, and the precession, whose strength the eccentricity sets, cuts a faster ripple of 51,000 years into it, strongest when the orbit is most eccentric. These two periods, and their ratio of 2.35, are what a record of the sunshine should carry.

    Layers that keep the rhythm and lose the dates

    Mars's polar caps are stacks of ice and dust laid down while the planet's tilt and orbit changed the summer sunshine, and the question is whether they can be read as a record of that sunshine. They can — for the rhythm. Even when the ice accumulates faster under weak sunshine than under strong, the cycles of the tilt and the precession survive in the depth of the stack in their correct proportion. What does not survive is the date of any one layer.

    part 4 · sky

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