Model planetary dominance where orbits transition to satellites. Choose units, include eccentricity, and validate inputs. Save outputs for missions, homework, and quick comparisons today.
Approximate reference cases for quick testing.
| Case | a | e | m | M | SOI (Laplace), km | Hill (circular), km |
|---|---|---|---|---|---|---|
| Earth about Sun | 1 AU | 0.0167 | 1 Earth | 1 Solar | ~ 9.29e5 | ~ 1.50e6 |
| Moon about Earth | 384,400 km | 0.0549 | 0.0123 Earth | 1 Earth | ~ 6.61e4 | ~ 6.16e4 |
| Jupiter about Sun | 5.204 AU | 0.0489 | 1 Jupiter | 1 Solar | ~ 4.82e7 | ~ 5.31e7 |
This tool reports three related radii. The highlighted one depends on your model selection.
Here, a is the semi-major axis, e is eccentricity, m is the secondary mass, and M is the primary mass.
Gravity from the primary and secondary overlaps everywhere, so trajectory work uses a practical boundary to decide which two-body model is dominant. A sphere of influence supports clean reference-frame handoffs, quick sizing of approach arcs, and consistent assumptions in reports and coursework.
The Laplace sphere of influence is commonly used in patched-conic navigation because it scales as r = a (m/M)^{2/5}. It is not a strict stability limit, but it often provides a reasonable distance to switch from heliocentric propagation to planetocentric hyperbolic arrival and departure geometry.
The Hill radius comes from the circular restricted three-body problem and scales as r = a (m/(3M))^{1/3}. It relates to regions where long-lived bound motion around the secondary is more plausible. It is frequently used to screen whether moons, rings, or close satellites can remain bound over many orbits.
The semi-major axis a should describe the secondary’s orbit about the primary. For a planet, use the planet–star semi-major axis. For a moon, use the moon–planet semi-major axis. Mixing distances from different frames is a common mistake that can shift radii by large factors.
Both models depend mainly on the mass ratio m/M, but with different exponents. Because these are power laws, increasing secondary mass by a factor of 10 raises Laplace SOI by about 10^{0.4} and Hill by about 10^{1/3}. The calculator reports the ratio to make sensitivity checks straightforward.
When the secondary orbit is eccentric, the tightest environment occurs at periapsis. A conservative stability estimate scales the Hill radius by (1 − e). This can be helpful for quick safety margins when choosing satellite altitudes, estimating capture robustness, or comparing candidates for temporary moonlets.
As a reference, Earth’s Laplace SOI is roughly 9.3×10^5 km, while its Hill radius is about 1.5×10^6 km. Jupiter’s values are tens of millions of kilometers, enabling large satellite systems. Small bodies can have Hill regions comparable to their orbital separations, limiting stable companions.
Use Laplace SOI for patched-conic switching and flyby planning, and use Hill (especially periapsis Hill) when you need conservative stability intuition. After screening with these radii, validate final trajectories with n-body numerical integration and any needed non-gravitational forces. Export CSV or PDF to document inputs and assumptions.
No. It is an approximation used to simplify calculations. Real gravitational dominance changes gradually and depends on trajectory geometry and perturbations from other bodies.
Use the Hill radius when you care about long-term orbital stability around the secondary, such as satellite retention, ring stability, or approximate stable regions in three-body dynamics.
It comes from a classical scaling argument balancing perturbations in patched-conic contexts. The exponent is not derived from strict stability, but it works well for engineering handoffs.
It multiplies the circular Hill radius by (1 − e), shrinking the stable region to reflect the closest approach to the primary. It is a conservative estimate.
Yes. Treat the moon as the secondary and the planet as the primary, and set a to the moon’s orbital semi-major axis around the planet.
They are based on different approximations and physical goals. Laplace is designed for reference-frame switching in transfers, while Hill is connected to three-body stability.
No. Stability also depends on orbital inclination, resonances, perturbations, and non-gravitational forces. Use these radii for screening, then validate with detailed dynamics.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.