Sphere of Influence Calculator

Example data table

These examples are typical patched-conics use cases. Your inputs may differ by reference model.

Formula used

• Laplace sphere of influence: r = a × (m/M)^(2/5)
• Hill radius (approx.): r = a × (1 − e) × (m/(3M))^(1/3)

Here, a is the semi-major axis, m is the orbiting body mass, M is the central body mass, and e is orbital eccentricity. Both are idealized and assume two-body inputs with a small perturbation.

How to use this calculator

1. Pick a method: Laplace for patched-conics, Hill for stability estimates.
2. Enter the central body mass and select its unit.
3. Enter the orbiting body mass and select its unit.
4. Enter the semi-major axis and choose the distance unit.
5. Set eccentricity. For near-circular orbits, use 0.
6. Optionally enter the orbiting body radius to get “body radii” output.
7. Press Calculate. Results appear above the form.
8. Use the CSV/PDF buttons to download the latest result.

Practical note: For mission design, use consistent reference values (same epoch and model). This tool is best for quick comparisons, sanity checks, and educational use.

What the sphere of influence represents

The sphere of influence (SOI) is a working boundary used to decide when a spacecraft is mainly controlled by one body rather than another. It is not a physical wall. It is a modeling switch that keeps patched‑conics calculations manageable and repeatable.

Inputs that drive the radius

This calculator uses the central mass M, the orbiting mass m, and the orbital scale a. A larger m expands the region, while a larger M compresses it. The semi‑major axis acts like a lever: changing a changes the SOI proportionally.

Laplace SOI for patched-conics transfers

The Laplace SOI, r = a × (m/M)^(2/5), is widely used for interplanetary transfers. It provides a practical handoff distance between a planet-centered arc and a Sun-centered arc. It is most appropriate when the secondary is far smaller than the primary and the orbit is not strongly perturbed.

Hill radius for orbital stability

The Hill radius, r = a × (1 − e) × (m/(3M))^(1/3), is tied to three‑body dynamics. It estimates where satellites can remain bound for long periods. Because it includes (1 − e), it shrinks for eccentric orbits where perturbations peak near periapsis, which is often the limiting case.

Typical scales in the Solar System

Earth’s Laplace SOI is roughly on the scale of a million kilometers, while the Moon’s Hill region around Earth is only tens of thousands of kilometers. Giant planets have much larger SOIs due to their mass, so their moon systems occupy a broader stable zone. For small bodies like asteroids, the SOI can be tiny, making flyby geometry more sensitive.

Why eccentricity matters

Eccentricity changes the minimum separation during the orbit. At periapsis, third‑body effects are strongest, so the stable region contracts. For quick circular estimates, e = 0 is fine, but measured values improve comparisons for real moons and planets.

Practical uses for trajectory design

SOI estimates help decide when to switch reference frames, select encounter boundaries for targeting, and choose numerical step sizes. They also help interpret burns: deep inside the SOI, body-centered approximations behave better; far outside, they can mislead. When planning a capture, compare periapsis distance to the SOI radius.

Common pitfalls and best practices

Keep units consistent and use masses from the same reference set. Remember that SOI is approximate; high‑fidelity navigation uses full n‑body propagation and measured ephemerides. Treat the output as a planning range, then refine with perturbations like oblateness, radiation pressure, and atmospheric drag when relevant.

FAQs

1) Which method should I choose?

Use Laplace for patched‑conics handoffs in interplanetary work. Use Hill when you care about long‑term satellite stability or when three‑body effects near periapsis are important.

2) Is the sphere of influence a real boundary?

No. It is a convenient modeling boundary. Gravity acts everywhere, but SOI helps decide which body-centered approximation is most useful for a given distance range.

3) Why does the Hill formula include eccentricity?

Eccentricity changes the closest approach distance. Near periapsis, perturbations from the central body grow, so the stable region around the orbiting body becomes smaller.

4) What does “SOI in body radii” mean?

If you enter the orbiting body’s radius, the tool divides the SOI radius by that value. It shows how many body radii fit from the center to the SOI boundary.

5) Can I use this for asteroids or exoplanets?

Yes. Enter the masses and semi‑major axis in any supported units. The formulas are general, but the result is still an approximation and should be validated for unusual systems.

6) Why do I sometimes get very large or tiny numbers?

Mass ratios can be extreme. The calculator formats very large or small results using scientific notation to keep values readable without losing scale information.

7) Does SOI guarantee capture or escape?

No. Capture depends on energy, velocity, and encounter geometry. SOI only suggests a distance range where one body’s gravity is the dominant influence for simplified modeling.