Blog #25 - Sphere of Influence vs. Hill Sphere

Sphere of Influence vs. Hill Sphere

Beyond the Hill Sphere, which dictates the maximum orbital distance a satellite body can be around its host planet, planets have another astrodynamic factor called the Sphere of Influence (SOI).

The Sphere of Influence has a similar definition to the Hill Sphere but dictates how far an orbiting body from a planet can be for the planetary body to still have the dominant gravitational effect on it, compared to the much larger, further stellar body.

The Sphere of Influence has a derivation based on the 3-body problem between a star, planet, and orbiting body. The 3-body problem dictates how three orbiting bodies will move with one another based on patched conics, or the interplay of eccentricity and orbit shape.

The SOI has a complicated derivation but ends up with a very similar form to the Hill Sphere at \[R_{SOI} = a * (\frac{m}{M})^{\frac{2}{5}}\] Where a is the planet’s semi-major orbital axis, m is the mass of the planet, and M is the mass of the star. This equation is very similar to the Hill Sphere which holds as \[R_{Hill } = a * (\frac{m}{3M})^{\frac{1}{3}} \] The only difference as we see is the exponential factor and a constant in the denominator.

While the Hill Sphere tells how far an orbiting satellite or moon can sit around a central planet, the SOI dictates which body (the planet or the star) should dictate the orbiting body’s motion. Within the SOI of a planet, the patched conics orbital mechanics approach will be based on the mass and distance to the planet, without considering the massive star, as it is not the dictating gravitational player in the system.