Astronomy & Celestial Mechanics
the sky, computed
Series

Angular momentum — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.

    Equal areas in equal times, which is angular momentum in disguise

    Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

    part 1 · orbits
  2. Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 61.1 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.

    Ninety-nine per cent of the mass and none of the spin

    The Sun holds 99.87 per cent of the solar system's mass and 0.6 per cent of its angular momentum. Jupiter holds a thousandth of the mass and three-fifths of the spin. That is not a curiosity of accounting — it is the record of the single operation that had to succeed before a star could form at all.

    part 2 · orbits
  3. The plane a moon's orbit turns about, between equator and ecliptic. The inclination of the Laplace plane — the plane about which a satellite's orbit precesses — to the Earth's equator, against distance from the Earth in Earth radii, for an obliquity of 23.44°. Close in, the Earth's equatorial bulge dominates and the plane is the equator; far out, the Sun dominates and it is the ecliptic, 23.44° away. The dashed curve uses today's bulge, which makes the transition at 8.4 Earth radii. The solid curve lets the bulge grow with the faster spin the Earth had when the Moon was closer — a 5.3-hour day at 5 Earth radii, from the system's angular momentum alone — which moves the transition out to about 14. The Moon today, at 60.3 Earth radii, has a Laplace plane within 0.0022° of the ecliptic, so its 5.15° inclination is entirely free: nothing forces it.

    A tilt the Moon should not have

    A moon's orbit precesses about a plane that sits on its planet's equator close in and on the planet's own orbit far out. A Moon born in the Earth's equator and carried outward by the tides should have followed that plane all the way to the ecliptic, because the angle it keeps to it is an adiabatic invariant. It is inclined by five degrees instead, and slow recession cannot have put them there.

    part 3 · orbits
  4. A tilt that wanders up, one passing body at a time. The inclination of the Moon's orbit to the plane it started in, against the number of lunar-mass bodies that have passed the Earth–Moon pair at 5 km/s within 300 Earth radii, with the Moon at 30 Earth radii, for four different sequences of passages. Each passage tilts the orbit by a fraction of a degree in a random direction, so the inclination performs a random walk and grows as the square root of the number of passages: after 1,000 the four runs stand at 3.8, 1.8, 3.9, 0.2 degrees. The dashed line is the Moon's present inclination, 5.145 degrees. Whether a given history reaches it is a matter of chance; how far a typical one gets is set by the passages' number, mass and speed.

    A tilt built from a thousand near misses

    The Moon's orbit is tilted five degrees to the ecliptic, and slow tidal recession cannot have done it. One candidate is the leftover debris of planet formation: bodies the size of the Moon, passing the young Earth–Moon pair, each giving its orbit a kick of a fifth of a degree in a random direction. A random walk of a few thousand such kicks reaches five degrees — and the bodies that would supply them are the same population whose impacts left the Earth's mantle its gold.

    part 4 · orbits

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