Hill Sphere Calculator

Estimate the Hill sphere — the region where a planet or moon's own gravity dominates over its host's tidal pull — from its mass, its host's mass, orbital distance, and eccentricity, via r_H ≈ a(1-e)(m/3M)^(1/3), in km, AU, and body radii, with a scaled orbit + Hill-sphere diagram.

Hill sphere calculator

The Hill sphere is roughly how far a body's own gravity can hold onto a satellite despite the pull of whatever it orbits: r_H ≈ a(m/3M)^(1/3), or the more conservative r_H ≈ a(1−e)(m/3M)^(1/3) at periapsis, where tidal stress is strongest. It's an approximation, not a hard boundary — see below for why.

Orbiting body's mass (m)

Host body's mass (M)

Semi-major axis (a)

Eccentricity (e) — optional

Orbiting body's radius — optional

r_H ≈ 1.472 × 10⁶ km = 0.00984 AU

Circular-orbit estimate: 1.497 × 10⁶ km · 231× the orbiting body's own radius

Left: the orbit (host body and orbiting body at periapsis), to its own scale. Right: the orbiting body and its Hill sphere (dashed), to a separate, much more zoomed-in scale — the Hill sphere is almost always far too small to see next to the full orbit.

Log scale — Hill radii across real systems span more than three orders of magnitude, from a moon's own modest sphere to a giant planet's sphere millions of km across.

The Moon orbits Earth, not the Sun, even though the Sun’s gravity at Earth’s distance is more than 300,000 times stronger than the Sun’s pull on a small nearby object would suggest matters — because what actually decides who “keeps” a satellite isn’t the absolute strength of either pull, it’s how the difference in that pull across the orbit compares to the smaller body’s own gravity. The region where a planet or moon wins that comparison is its Hill sphere.

The formula

r_H \approx a\left(\frac{m}{3M}\right)^{1/3}

a is the orbiting body’s own semi-major axis (its distance from what it orbits), m its mass, and M the mass of the larger body it orbits. Earth: m = 1 M⊕, M = 1 M☉, a = 1 AU, gives r_H ≈ 1.5 million km — the rough outer edge of the region where Earth’s gravity, not the Sun’s, dominates. The Moon sits comfortably inside that at 384,400 km.

For an eccentric orbit, use the more conservative form instead:

r_H \approx a(1-e)\left(\frac{m}{3M}\right)^{1/3}

Tidal stress from the host is strongest at periapsis (closest approach), so a satellite that would be fine at the circular-orbit r_H can still get stripped away if the orbiting body’s own path is eccentric enough to bring it well inside that at closest approach.

Why the Hill radius isn’t a hard boundary

It’s tempting to treat r_H as a wall — safe just inside, doomed just outside — but the real picture is softer in both directions:

A satellite can be unstable well inside r_H. The Hill sphere marks where gravity could still dominate; it says nothing about whether a particular orbit is actually stable over many orbital periods. In practice, long-term-stable prograde satellite orbits generally need to stay within roughly a third to a half of r_H, and retrograde orbits (orbiting opposite to the planet’s own motion around its star) tend to remain stable somewhat farther out than prograde ones do, for reasons rooted in the same three-body dynamics the Hill approximation simplifies away.
The formula itself is an approximation. It comes from a simplified (circular, restricted three-body) version of a genuinely more complicated problem; the exact boundary from full orbital dynamics is close to, but not identical to, r_H.

Treat the number this calculator returns as “roughly how much room there is to work with,” not a precise line a satellite either survives or doesn’t.

Why hot Jupiters are cramped

Since r_H scales directly with a, a hot Jupiter parked at 0.05 AU from its star has a Hill sphere only a small fraction the size of an ordinary, Jupiter-distance gas giant’s — despite having the same mass. This is a genuine part of why close-in giant planets are thought to struggle to hold onto large moons: there’s simply much less gravitationally “safe” territory available that close to an overwhelmingly more massive star.

Reading the visuals

The orbit + Hill sphere diagram shows two panels at two completely different, independently auto-scaled sizes: the orbit itself (host body, orbital ellipse, and the orbiting body at periapsis) on the left, and a zoomed-in view of the orbiting body and its Hill sphere on the right — because at the orbit’s own scale, the Hill sphere is almost always far too small to draw at all.
The comparison ladder places the current Hill radius on a log scale next to the Moon’s own (much smaller) Hill sphere, Earth’s, and Jupiter’s — a direct sense of just how many orders of magnitude this quantity spans across real solar-system bodies.