# Hill sphere

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A Hill sphere approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from another heavier body around which it orbits. It was defined by the American astronomer George William Hill. It is also called the Roche sphere because the French astronomer Édouard Roche independently described it.

Considering a central body and a second body in orbit around it (for example the Sun and Jupiter), the Hill sphere is derived from consideration of the following three vector fields:

• gravity due to the central body
• gravity due to the second body
• the centrifugal force in a frame of reference rotating about the central body with the same angular frequency as the second body

The Hill sphere is the largest sphere, centered at the second body, within which the sum of the three fields is directed towards the second body. A small third body can orbit the second within the Hill sphere, with this resultant force as centripetal force.

The Hill sphere extends between the Lagrangian points L1 and L2, which lie along the line of centers of the two bodies. The region of influence of the second body is shortest in that direction, and so it acts as the limiting factor for the size of the Hill sphere. Beyond that distance, a third object in orbit around the second (Jupiter) would spend at least part of its orbit outside the Hill sphere, and would be progressively perturbed by the tidal forces of the central body (the Sun) and would end up orbiting the latter.

The Roche sphere is not to be confused with the Roche lobe and Roche limit which were also described by Roche. The Roche limit which is the distance at which an object held together only by gravity begins to break up due to tidal forces. The Roche lobe describes the limits at which an object which is in orbit around two other objects will be captured by one or the other.

## Formula and examples

If the mass of the smaller body is m, and it orbits a heavier body of mass M with a semi-major axis a and an eccentricity of e, then the radius r of the Hill sphere for the smaller body is [1]

$r \approx a (1-e) \sqrt[3]{\frac{m}{3 M}}$

When eccentricity is negligible (the most favourable case for orbital stability), this becomes

$r \approx a \sqrt[3]{\frac{m}{3 M}}$

For example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 Gm. The Hill sphere for Earth thus extends out to about 1.5 Gm (0.01 AU). The Moon's orbit, at a distance of 0.370 Gm from Earth, is comfortably within the gravitational sphere of influence of Earth and is therefore not at risk of being pulled into an independent orbit around the Sun. In terms of orbital period: the Moon has to be within the sphere where the orbital period is not more than 7 months.

The formula can be re-stated as follows:

$3\frac{r^3}{a^3} \approx \frac{m}{M}$

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 these two spheres.

A quick way of estimating the radius of the Hill sphere comes from replacing mass with density in the above equation:

$\frac{r}{R_{secondary}} \approx \frac{a}{R_{primary}} \sqrt[3]{\frac{\rho_{secondary}}{3 \rho_{primary}}} \approx \frac{a}{R_{primary}}$

where $\rho_{second}$ and $\rho_{primary}$ are the densities of the primary and secondary bodies, and $\frac{r}{R_{secondary}}$ and $\frac{r}{R_{primary}}$ are their radii. The second approximation is justified by the fact that, for most cases in the solar system, $\sqrt[3]{\frac{\rho_{secondary}}{3 \rho_{primary}}}$ happens to be close to one. (The Earth-Moon system is the largest exception, and this approximation is within 20% for most of Saturn's satellites.) This is also convenient, since many planetary astronomers work in and remember distances in units of planetary radii.

An astronaut could not orbit the Space Shuttle (mass = 104 tonnes), if the orbit is 300 km above the Earth, since the Hill sphere is only 120 cm in radius, much smaller than the shuttle itself. In fact, in any low Earth orbit, a spherical body must be 800 times denser than lead in order to fit inside its own Hill sphere, or else it will be incapable of supporting an orbit. A spherical geostationary satellite would need to be more than 5 times denser than lead to support satellites of its own; such a satellite would be 2.5 times denser than iridium, the densest naturally-occurring material on Earth. Only at twice the geostationary distance could a lead sphere possibly support its own satellite; the moon itself must be at least 3 times the geostationary distance, or 2/7 its present distance, to make lunar orbits possible.

The Hill sphere is but an approximation, and other forces (such as radiation pressure) can make an object deviate from within the sphere. The third object must also be of small enough mass that it introduces no additional complications through its own gravity. Orbits at or just within the Hill sphere are not stable in the long term; from numerical methods it appears that stable satellite orbits are inside 1/2 to 1/3 of the Hill radius (with retrograde orbits being more stable than prograde orbits).

Within the solar system, the planet with the largest Hill sphere is Neptune, with 116 Gm, or 0.775 AU; its great distance from the Sun amply compensates for its small mass relative to Jupiter (whose own Hill sphere measures 53 Gm). An asteroid from the main belt will have a Hill sphere that can reach 220 Mm (for 1 Ceres), diminishing rapidly with its mass. In the case of (66391) 1999 KW₄, a Mercury-crosser asteroid which has a moon (S/2001 (66391) 1), the Hill sphere varies between 120 and 22 km in radius depending on whether the asteroid is at its aphelion or perihelion!