# How to calculate the sphere of influence of a planet?

I'm making an orbit simulator, and to make it simpler, I'm only simulating one celestial object(planet, moon, sun) acting on each object(sattelite).

So that the sattelites and rockets can switch between what planet/moon they are orbiting, I need to figure out the sphere of influence of each planet.

I've found some formulas online, but they all unclude the mass of the smaller object, or the Semi-major axis, which would indicate that a rocket of a mass of, say $$100kg$$, would have a different SOI(Sphere-Of-Influence) of the earth than a rocket of $$1kg$$ orbiting at the same height.

This confused me since I thought the SOI was always the same no matter what was orbiting (or how fast, or how far away).

I would be happy if someone explained why this is, or if there is some other information I'm missing, thx!

• What's wrong with the formula at the top of en.wikipedia.org/wiki/Sphere_of_influence_(astrodynamics) ? Feb 1, 2021 at 11:04
• @PM2Ring like i said in the post: I've found some formulas online, but they all unclude the mass of the smaller object, or the Semi-major axis, which would indicate that a rocket of a mass of, say 100kg, would have a different SOI(Sphere-Of-Influence) of the earth than a rocket of 1kg orbiting at the same height. This confused me since I thought the SOI was always the same no matter what was orbiting (or how fast, or how far away). Feb 1, 2021 at 11:47

I think you misinterpred the formulas you have read. The wikipedia article, states that the SOI is computed for one planet RELATIVE to another. For example, for the SOI of the earth, relative to the sun, we have that : $$r_{SOI} = a (\frac{m}{M})^{2/5}$$
Where $$a$$ is the semi major axis of the earths orbit around the sun. Naturally, it both depends on the mass of the earth and the sun, since we are comparing their influence. It has to depend on two masses, because we are comparing gravitational strength. You cannot define the SOI of a celestial body alone, you have the SOI comparing two celestial bodies, where one orbits the other.
But your intuition is correct, for a satellite of $$100 kg$$ and a satellite of $$1kg$$, you would use the same $$r_{SOI}$$ to see if it should orbit the Earth, or the Sun. If you would then like to know if it should orbit the moon instead, you would look at the SOI of the moon, compared to the earth, and so on.
• The Google Calculator is handy for things like this. Eg, put (149598023 km)*((1 earth mass)/(1 solar mass))^0.4 into the Google search bar & it returns 924 520.73 kilometers. Feb 1, 2021 at 12:11