Astronomy & Celestial Mechanics
the sky, computed
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Hill sphere — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The Hill radius against the mass ratio, and where four planets keep their outermost moon. The Hill radius R_H = a(μ/3)^⅓ and the exact L₁ distance from the same quintic, both in units of the planet's own semi-major axis, against the mass ratio μ = m/(M+m) from 10⁻⁸ to 0.12. The two are the same line to the width of the stroke across the whole planetary range — the cube-root formula runs 0.33% short of the exact L₁ distance at the Earth's mass ratio and 10.0% short at the right-hand edge — so the departure between them is drawn on its own percentage scale at the right, which is the only place it can be seen. Below the curves are the numerically determined stability limits, 0.5 R_H for a prograde satellite and 0.7 R_H for a retrograde one, and the outermost known satellite of each of four planets, each at a Hill fraction computed from that planet's own mass and orbit: Moon 0.257, Sinope 0.451, Phoebe 0.198, Neso 0.424. Of the four, three are retrograde — Sinope, Phoebe, Neso — and they reach 0.451 of their planet's Hill radius against 0.257 for the one prograde satellite: 1.8 times as far out, in a diagram whose two stability limits stand in the ratio 0.7 to 0.5. No equilibrium argument predicts that, and it is what the drawing makes plain.

    The region a planet may keep a moon in

    A satellite is not held by orbiting but by orbiting inside the radius at which the Sun's tidal field would take it away. That radius is the same distance the innermost Lagrange point sits at, asked a different question — and the boundary the sky actually respects is smaller, by a factor that depends on which way the moon is going round.

    part 1 · gravitation
  2. Three orders of magnitude in astronomical units, and a factor of 9.6 in mutual Hill radii. Every adjacent pair of planets in three systems, plotted by the separation between them measured in their own mutual Hill radius, ((m₁+m₂)/3M⋆)^⅓ · (a₁+a₂)/2. In astronomical units the same 17 separations span a factor of 2554 — from 0.0043 AU between two TRAPPIST-1 planets to 10.9 between Uranus and Neptune — and carry no visible structure at all. In this unit they span 9.6, with a median of 11.8. The solid line at 2√3 = 3.46 is a theorem: two planets on circular coplanar orbits wider apart than that can never have a close encounter, whatever else happens, and every pair here is clear of it. The dashed line at 10 is a fit, to integrations of systems of several planets over billions of years, and it is the one that bites — a system packed tighter than about ten does not survive, which is why the observed distribution has a floor there and not at the theorem. A planetary system's spacing is not measured in kilometres. It is measured in a unit the planets define.

    A feeding zone, and the spacing it forces

    The radius that decides what a planet may keep also decides what it could reach while it was growing — and measuring the gaps between planets in that unit turns a distribution spanning three orders of magnitude in astronomical units into a band a factor of ten wide, with a floor that is partly a theorem and partly a fit.

    part 2 · gravitation

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