Astronomy & Celestial Mechanics
the sky, computed
Concept

Core mantle boundary — where it appears

The interface between a body's liquid core and its solid mantle. Its existence is inferred from how much of the body's moment of inertia fails to follow the mantle on an orbital timescale, and friction across it couples the two imperfectly.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

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.

sky · Libration A free libration does not last. How long a free libration survives before tidal dissipation removes it, against the tidal quality factor, for four locked bodies — with the age of the solar system marked. A lightly damped oscillator loses its amplitude in about Q of its own periods, and a free libration's period is years, so the decay time is thousands to hundreds of thousands of years for any plausible Q. Against 4.5 billion years that is instantaneous. A free libration that is observed today was excited within the last geological instant, and one that is not observed bounds how much has happened. The Moon's is detected, at a few arcseconds, and requires something recent — an impact, a core–mantle interaction, or a resonance passage. Mercury's has been searched for and not found, and the upper limit is the constraint.

A rocking nobody started

A locked body is a pendulum, and a pendulum has a period of its own. That free libration is damped by tides in thousands of years, so any body still oscillating at its own frequency was disturbed in the last geological instant — and the Moon is, while Mercury is not.

sky · Libration 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.

sky · Precession

Named alongside it

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

Spin orbit resonanceCassini stateDiurnal tidesDynamical ellipticityEarth orientationExcitationForced eccentricityForced librationFree core nutationFree librationGravity scienceLibration

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