Astronomy & Celestial Mechanics
the sky, computed
Generator

The libration generator

The sub-Earth point over 400 days
The sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats.

Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats.

3 essays call libration. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

The sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats. The observed sky

The face that is not quite fixed

The Moon keeps one face turned toward the Earth, and the sentence is exactly true only of a fictitious Moon on a circular orbit. The real one rocks by a few degrees each month, in two directions and for two unrelated reasons, and the rocking has shown 59 per cent of its surface to people who never left the ground.

 Mercury's forced libration: 38.5″ against the 16.2″ a solid body would give. The measurement that finds a liquid core from a distance. A body on an eccentric orbit does not feel a steady torque: the pull on its equatorial bulge swings back and forth through the orbit, and the body rocks about its mean rotation by a small angle. How small depends on how much moment of inertia has to be rocked, and on nothing else — every other factor in the problem belongs to the orbit or to the body's own measured gravity field. The horizontal axis is therefore the fraction of the total polar moment that participates, one if the whole body turns rigidly together, and the vertical axis is the resulting amplitude. The curve is a rectangular hyperbola, because the same torque applied to less moment produces proportionally more angle. A Mercury turning in one piece would librate by 16.2 arcseconds. Radar measurements of the actual rocking give 38.5 ± 1.6, which is 2.38 times larger and many standard deviations away, so only 42 per cent of the moment is being rocked at all. The other 58 per cent is not following the mantle on an eighty-eight-day timescale, and the only way for an interior not to follow its own mantle is for the two to be separated by a liquid. That is how a planet nobody has landed on was shown to have a molten core — by watching, from Earth, the tiny irregularity of its turning. The figure treats the librating shell as rigid, which is right for a rocky mantle and wrong for an ice shell floating on an ocean, where the shell's own elasticity enters at the same level as the effect. The observed sky

Four numbers that weigh a planet's core

Mercury rocks about its mean rotation by thirty-eight arcseconds, which is more than twice what a planet turning in one piece could manage. The excess says that most of the planet's moment of inertia is not following the mantle on an eighty-eight-day timescale, and the only thing that does not follow a mantle is a liquid.

 A circular orbit has one lock, and an eccentric one has several. The strength of each spin–orbit resonance against orbital eccentricity, as the Hansen coefficient H(p, e) that multiplies the restoring torque on a permanently non-spherical body. At zero eccentricity every curve but the synchronous one is exactly zero — the figure checks that rather than showing it — so a body on a circular orbit can lock only by turning once per orbit. Away from zero the others switch on: at Mercury's eccentricity of 0.2056 the 3:2 resonance has 73 per cent of the synchronous one's strength and more than twice the 2:1's. A planet spinning down through this family therefore meets the 3:2 before the 1:1 and has a real chance of being caught there, which is what happened — Mercury turns three times for every two orbits, a fact discovered by radar in 1965 after a century of assuming it was locked. The free libration of that locked state follows from the same coefficient and the measured 2.03e-4 for (B − A)/C: 12.1 years, against a measured period near twelve. What the figure cannot show is the capture probability itself, which depends on how the tide dissipates and ranges from a few per cent for a simple constant-lag tide to more than half once friction between a liquid core and the mantle is included. Gravitation

A rotation locked to the orbit, but not one to one

Mercury turns exactly three times for every two circuits of the Sun. That was not what anybody expected, and it is not an accident — on a circular orbit a tidally despun body has exactly one place to lock, and on an eccentric one it has several — with the strength of each set by a coefficient that vanishes when the eccentricity does.

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