Hill sphere

The Hill sphere (or Roche sphere, not to be confused with the Roche limit) applies to objects such as planets that (1) are in orbit around a more massive object such as a star, and (2) are massive enough themselves that smaller objects (moons or satellites) can be in orbit around it. For a planet, the Hill sphere is the imaginary sphere within which a satellite or moon can be in orbit around the planet, and outside of which the Sun or star's gravity prevents the smaller body from orbiting the planet. In other words, the radius of the Hill sphere (Hill radius) is the maximum distance a satellite can be from a planet and still orbit the planet.

As an example, since the Moon orbits Earth it must lie within Earth's Hill sphere. However, if the Moon were far enough away from Earth -- outside Earth's Hill sphere -- it would then orbit the Sun rather than Earth. The gravitational force from Earth must dominate that of the Sun in order for a satellite to orbit it, which only happens if the satellite is close enough to Earth.

Formulas

For a planet orbiting a star in an elliptical orbit, the Hill radius (radius of the Hill sphere) r_Hill is given approximately by

$r_{Hill}=a(1-e)\left({\frac {m}{3M}}\right)^{1/3}$

where a and e are the semimajor axis and eccentricity, respectively, of the planet's elliptical orbit, and m and M are the masses of the planet and star, respectively. In the above formula, the quantity a(1-e) is the distance of closest approach of the planet to the star, e.g., the perihelion distance for planets orbiting the Sun or the perigee distance for satellites orbiting Earth.

For circular orbits, e is zero and a is the radius r_orbit of the orbit. In this case, the Hill radius is approximately

$r_{Hill}=r_{orbit}\left({\frac {m}{3M}}\right)^{1/3}$.

The formulas for r_Hill make intuitive sense for the following reasons. A larger r_Hill implies a greater tendency for satellites to orbit the planet rather than the star. It can be expected that this would happen for either a larger planet orbital radius (the planet is farther away from the star), a larger planet mass, or a smaller star mass. These all result in an increased influence of the planet's gravity relative to the star's. The formula is entirely consistent with these intuitive claims. Moreover, a highly elliptical orbit (greater e) brings the planet closer to the star for the same semimajor axis a, and so one would expect a smaller Hill radius for larger e, again consistent with the formula.

Comparison to distance of equal gravitational forces

The Hill radius is not the same as the distance at which the gravitational forces exerted on the satellite by the star and by the planet are equal, but is in fact generally larger than this distance by an additional factor of $\left(M/9m\right)^{1/6}$. This works out to roughly a factor of 6 in the case of Earth and the Sun.

Hill sphere of objects that orbit Earth

The Moon

The Moon's Hill radius is about 360,000 km, calculated by applying the above formulas to its orbit about Earth. Applying the same formulas to its orbit about the Sun gives a much larger Hill radius of 340 million km, so the Moon's actual Hill radius is limited by the effects of Earth rather than the Sun.

Artificial satellites in low-Earth orbit

Satellites in low-Earth orbit are too close to Earth to have smaller objects orbiting it. As a typical example, take a satellite of mass 1000 kg orbiting Earth at an altitude of 400 km, corresponding to an orbital radius of about 6800 km. The Hill radius is then calculated to be only 26 cm, much smaller than the typical size of such a satellite. In order for an object to be smaller than its Hill radius at this altitude, it would have to have a density greater than around 14 g/cc, which, while possible in principle, does not generally happen.

Hill sphere of the Sun

The formulas listed earlier are not applicable to calculating a Hill radius for the Sun, even though the Sun must have a Hill sphere owing to the presence of other stars in the galaxy that will significantly perturb the orbit of any object that is far enough away.

It is reasonable to consider two phenomena that could determine the Sun's Hill radius. One is the effect of the galaxy as a whole, as the Sun orbits the galactic center every xxx years. However, the distribution of the matter in the galaxy is such that the net gravitational force on the Sun is not inversely proportional to the distance from the galaxy's center. An inverse-square dependence is assumed in deriving the above formulas for the Hill radius.

Rather than considering the effect of the galaxy as a whole on objects orbiting the Sun, one can consider the effect of individual stars. In particular, that of the nearby Alpha Centauri star system. However, as the Sun and Alpha Centauri are not in bound orbits about each other, the earlier Hill radius formulas do not apply.