limiting zero-velocity surface around the second object is basically its Hill "sphere". Formula and examples If the mass of the smaller body (e.g. the Earth) is , and it orbits a heavier body (e.g. the Sun) of mass with a semi-major axis and an eccentricity of , then the radius of the Hill sphere of the smaller body, calculated at the pericenter, is approximately When eccentricity is negligible (the most favourable case for orbital stability), this becomes In the Earth-Sun example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU). The Moon's orbit, at a distance of 0.384 million km from Earth, is comfortably within the gravitational sphere of influence of Earth and it is therefore not at risk of being pulled into an independent orbit around the Sun. All stable satellites of the Earth (those within the Earth's Hill sphere) must have an orbital period shorter than seven months. The previous (eccentricity-ignoring) formula can be re-stated as follows: This expresses the relation in terms of the volume of the Hill sphere compared with the volume of the second body's orbit around the first; specifically, the ratio of the masses is three times the ratio of the volume of these two spheres. Derivation The expression for the Hill radius can be found by equating gravitational and centrifugal forces acting on a test particle (of mass much smaller than ) orbiting the secondary body. Assume that the distance between masses and is , and that the test particle is orbiting at a distance from the secondary. When the test particle is on the line connecting the primary and the secondary body, the force balance requires that where is the gravitational constant and is the (Keplerian) angular velocity of the secondary about the primary (assuming that ). The above equation can also be written as which, through a binomial expansion to leading order in , can be written as Hence, the relation stated above If the orbit of the secondary about the primary is elliptical, the Hill radius is maximum at the apocenter, where is largest, and minimum at the pericenter of the orbit. Therefore, for purposes of stability of test particles (for example, of small satellites), the Hill radius at the pericenter distance needs