Astronomy & Celestial Mechanics
the sky, computed
Concept

Spin axis precession — where it appears

The slow conical motion of a body's rotation axis, driven by the torque of other bodies on its equatorial bulge. Its rate depends on the body's flattening and spin; when it matches an orbital precession frequency, the axis tilt can be captured or driven.

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

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.

sky · Obliquity The Moon's Cassini states as it recedes. The angle between the Moon's spin axis and its orbit normal in each Cassini state, signed so that the present state is positive, against the Earth–Moon distance in Earth radii, for the Moon's present shape (β = (C − A)/B fixed) and orbital inclination. Today the Moon sits in Cassini state 2 at 6.68°. Closer in, two more states exist, a low-obliquity state 1 and an unstable state 4; they meet and vanish at 33.8 Earth radii, where the spin-axis precession, which weakens as the cube of the distance, falls to the critical fraction of the orbit's regression. A Moon that formed close to the Earth sat in state 1 at small obliquity, growing to 21° as it receded; at 33.8 Earth radii that state disappeared from under it, and it had to move to state 2 — which at that distance is at 46°. Since then, the obliquity has relaxed along state 2 to its present value.

A state the Moon had to jump to

The Moon's spin axis sits a degree and a half from the ecliptic pole, locked in a geometry Cassini described in 1693. The lock is an equilibrium between the Earth's torque on the Moon's bulge and the slow regression of its orbit, and both depend on the Moon's distance. When the Moon was closer than about 34 Earth radii there were two more equilibria — and the one it almost certainly occupied vanished as it receded, leaving it to jump to the present one at an obliquity near 45°.

sky · Libration

Named alongside it

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

Cassini stateLibrationObliquityBifurcationChaosDynamical ellipticityInvariable planeLunar recessionMilankovitch cyclesNodal precessionNode regressionOrbital inclination

All concepts