Sphere of influence calculator

The sphere of influence (SOI) is the classical patched-conic boundary around a secondary body—planet, moon, or other mass—inside which that body’s gravity is treated as dominant for trajectory design, and outside which the primary’s gravity takes over. Mission planners, orbital mechanics students, and contest authors need a transparent closed-form radius from orbital distance and the two-fifths power of the secondary-to-primary mass ratio, not a black-box n-body dump. This calculator accepts orbital distance a, secondary mass m, and primary mass M, returns r_SOI equals a times (m over M) to the power two-fifths, and exposes the mass ratio and dimensionless r_SOI over a for audit.

What the patched-conic sphere of influence means

In the patched-conic approximation used throughout interplanetary trajectory design, a spacecraft’s path is stitched from successive two-body conics: heliocentric Keplerian arcs between planets, then planet-centred hyperbolas inside each planet’s sphere of influence. The sphere of influence radius marks where the gravitational perturbation of the secondary relative to the primary is of order unity under a simple Laplace-style scaling. The textbook closed form is r_SOI equals a times the secondary-to-primary mass ratio raised to the two-fifths power: r_SOI = a · (m / M)^(2/5).

Here a is the orbital distance of the secondary about the primary—usually the semi-major axis of the planet about the Sun or of a moon about its planet—m is the secondary mass, and M is the primary mass. The exponent two-fifths is fixed by balancing the primary’s tidal differential against the secondary’s inverse-square pull in the classical derivation; it is not an arbitrary fitting constant.

When m is much smaller than M the mass ratio is much less than one, so (m/M)^(2/5) is still small but larger than m/M itself, and r_SOI sits well inside the secondary’s orbit. Equal masses would push the formal SOI out toward a, which is rarely physical for the patched-conic use case but remains mathematically defined. This page implements only that closed-form SOI: it does not integrate the circular restricted three-body problem, does not compute Hill spheres with different exponents, and does not replace full ephemeris-based gravity models.

Those are separate search intents and separate landings. Use this tool when you need the standard patched-conic SOI number that textbooks, contest keys, and preliminary mission trades still cite by name.

What to enter and how to read the result

Orbital_distance is the semi-major axis or centre-to-centre orbital distance of the secondary about the primary in kilometres, metres, or any single consistent length unit; it must be strictly positive so the length scale of the sphere is well defined. Secondary_mass is the mass of the planet, moon, or other secondary in kilograms or any consistent mass unit and must be strictly positive.

Classic checks include Earth–Moon with a about three hundred eighty-four thousand four hundred kilometres and m/M about one point two three percent, recovering an SOI of order sixty-six thousand kilometres; and a toy equal-mass case with a equals one recovering r_SOI equals one. Wire the free browser form for homework and quick sanity checks, then call the identical contract from scripts when you need bulk answer keys or trajectory scaffolding.

Keep units consistent: convert both masses to the same unit and keep length in one unit before calling so the numeric product matches the interpretation your report claims.

Where sphere-of-influence calculators show up in real work

Introductory orbital-mechanics and spacecraft-design courses assign patched-conic drills that ask for the SOI of Earth about the Sun, of the Moon about Earth, or of Jupiter about the Sun so students can decide where to switch frames in a multi-leg tour. Preliminary mission design teams need a fast SOI radius when sketching transfer legs, flybys, and capture corridors before full n-body optimization. Contest authors and LMS plugins need a deterministic oracle that rejects non-positive primary mass instead of silently returning Infinity or NaN from a bare division.

Science communicators convert mass tables into SOI radii in kilometres so audiences can see that Earth’s SOI around the Sun is still a tiny fraction of one astronomical unit. Game and simulation authors seed gravity-well toy models with closed-form SOI spheres when they want a transparent ground truth without floating-point surprises from ad-hoc constants.

If you later need Roche tidal breakup distances, Newtonian force between two masses, orbital period from semi-major axis, or Hill-sphere variants with different exponents, treat those as separate search intents and landings rather than overloading this sphere of influence calculator page.