Astronomy & Celestial Mechanics
the sky, computed
Concept

The mutual Hill radius — where it appears

The Hill radius formed from the combined mass of two neighbouring planets and their mean separation. Measuring the gaps between planets in this unit turns a structureless list into a distribution with a floor near ten.

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

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.

gravitation · Hill sphere A limit with a notch cut in the middle of it. The stability limit for a satellite, in units of its planet's Hill radius, against the inclination of its orbit to the planet's. The two flat stretches are the coplanar answers — 0.5 prograde and 0.7 retrograde, the difference between them being the Coriolis asymmetry that a rotating frame imposes and no potential can hold. Between them the limit is not an interpolation. Inside the shaded band, from 39.23° to 140.77°, the Sun's averaged perturbation exchanges the orbit's inclination for its eccentricity, and a satellite whose semi-major axis is comfortably inside the lobe has an apocentre that is not — so the boundary is set by a(1 + eₘₐₓ) rather than by a, and it falls to exactly half the coplanar value at ninety degrees. The band edges are cos²i = 3/5 exactly, with no fitted quantity anywhere in them, and the notch is the reason a quoted stability limit in Hill radii is a statement about a coplanar orbit and not about a satellite.

A stability limit with a notch cut in it

The two numbers usually quoted for how far out a moon may orbit are coplanar numbers. A tilted orbit is not somewhere between them: inside a band of inclination fifty-six degrees wide the Sun converts the tilt into an eccentricity, the apocentre rather than the semi-major axis has to stay inside the lobe, and the limit falls to half. The band is empty in the sky.

gravitation · Hill sphere

Named alongside it

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

Apsidal precessionCritical inclinationDynamical spacingFeeding zoneHill sphereHill stabilityIrregular satellitesIsolation massJacobi constantKozai lidov mechanismLaplace planeOligarchic growth

All concepts