Hill Sphere & Sphere of Influence Calculator

Understanding the Hill sphere and sphere of influence calculator — concept and principle

A small body orbiting a much larger one has a region around it where its own gravity dominates. This calculator gives two standard measures of that region: the Hill radius, which marks where satellites can stay bound, and the Laplace sphere of influence.

Inside the blue sphere Earth can hold moons; the green sphere is where spacecraft switch to an Earth-centred orbit.

How it is calculated

Hill radius: r_H = a × (m / 3M)^(1/3)
Sphere of influence: r_SOI = a × (m / M)^(2/5)

Here a is the semi-major axis of the small body's orbit (km), m is its mass and M is the central mass (both in kg; only the ratio matters). The radius comes out in the same unit as a. The formulas assume m is much smaller than M and the orbit is close to circular.

The principle

The Hill radius comes from the restricted three-body problem. In a frame rotating with the planet, a test particle feels the planet's pull, the star's pull and the centrifugal effect. Along the star–planet line these balance at the Lagrange points L1 and L2, whose distance from the planet is approximately a(m/3M)^(1/3). Inside that distance the planet's gravity wins over the star's tidal pull, so it can hold moons; in practice long-term stable orbits lie well inside it, often quoted as about half the Hill radius or less.

The Laplace sphere of influence answers a different question: where is it more accurate to treat the motion as an orbit around the planet, disturbed by the star, rather than the reverse? Patched-conic trajectory planning switches from a heliocentric to a planet-centred orbit at this boundary.

Worked example

Inputs: a = 149,597,870.7 km (1 au), m = 5.972 × 10²⁴ kg (Earth), M = 1.98847 × 10³⁰ kg (Sun).
m/(3M) = 1.0011 × 10⁻⁶; its cube root is 0.010004.
Hill radius = 149,597,870.7 × 0.010004 = 1,496,529.374 km, about 1% of an au.
m/M = 3.0033 × 10⁻⁶; raised to 0.4 it is 0.0061808.
Sphere of influence = 924,625.176 km.

Each point is the calculator re-run with only “Orbit semi-major axis (km)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Orbit semi-major axis (km)	Hill radius (km)
3.74e+07	374,132.667
7.48e+07	748,264.3336
1.12e+08	1,122,392.999
1.5e+08	1,496,530.668
1.87e+08	1,870,658.333
2.24e+08	2,244,796.002
2.99e+08	2,993,061.336

How to read the result

For Earth, r_H ≈ 1.5 million km, which is also roughly the distance to the Sun–Earth L1 and L2 points used by space telescopes. The Moon at about 384,400 km sits at roughly a quarter of the Hill radius, comfortably bound. The sphere of influence is always smaller than the Hill radius for planets because (m/M)^(2/5) shrinks faster than (m/3M)^(1/3).

Common mistakes

Entering a in metres or au while reading the result as km; the output unit always matches a.
Treating every orbit inside r_H as stable for billions of years.

Keep in mind: Both radii are first-order approximations for near-circular orbits with m ≪ M. For eccentric orbits use the periapsis distance a(1 − e) for a conservative estimate, and use numerical integration for real stability studies.

Frequently asked questions

Is the Hill sphere the same as the Roche limit?

No. The Roche limit is the inner distance where tides from a body tear apart a satellite; the Hill radius is the outer distance where the body can still hold a satellite.

Why do spacecraft use the sphere of influence instead of the Hill sphere?

The patched-conic method needs a single switching boundary that minimises the error of ignoring one body, and the Laplace criterion is defined for exactly that.

Sources: C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press · R. R. Bate, D. D. Mueller and J. E. White, Fundamentals of Astrodynamics, Dover

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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