Hill Sphere: A Planet's Gravitational Reach

Planet

Choose a planet

Distance from Sun 1.00 AU

Test Moon

Start distance 1.00 × r_H

Orbital speed 1.00 × circular

Live Readouts

Hill radius r_H

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Moon distance

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Status

Bound

Planet's year

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How it works

A planet doesn't own all the space around it — the Sun's gravity always competes for the same territory. The Hill sphere is the region where the planet's own gravity dominates over the Sun's tidal pull, so a moon can stay bound to it:

r_H = a · (m / 3M)^(1/3)

a = planet's orbital distance from the Sun
m = planet mass, M = Sun's mass

This simulator does not just draw a sphere and check a formula — it numerically integrates the real restricted three-body problem with a 4th-order Runge-Kutta method: the moon feels genuine Newtonian gravity from both the Sun and the planet every step, while the planet follows its own unperturbed Keplerian orbit. Position and velocity are advanced with:

a_moon = -GM_sun · r/|r|³ − GM_planet · (r−R_planet)/|r−R_planet|³

Push the speed slider well above 1.0× and the moon can be flung past the Hill boundary, where the Sun's tidal stretching wins and it drifts away for good — an escape you can watch happen, not just predict. One extra real effect worth trying: flip between prograde and retrograde. Numerical studies (Hamilton & Burns, 1992) show retrograde moons stay bound out to roughly twice the distance of prograde ones, because a retrograde orbit fights the Sun's tidal torque instead of resonating with it — you can reproduce that asymmetry yourself by comparing how far out each direction survives.

Planet — sets the mass ratio m/M that drives r_H.
Distance from Sun — r_H grows linearly with a, so the same planet has a bigger gravitational reach farther out.
Start distance / Orbital speed — the moon's initial condition, in Hill-radius units and multiples of the local circular velocity.
Prograde / Retrograde — direction of the moon's orbit around the planet relative to the planet's own orbit around the Sun.

Planet Test moon Toward the Sun

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