Hill radius

An astronomical body's Hill radius is a calculated radius of the surrounding spherical region (Hill sphere aka Roche sphere) within which smaller bodies would tend to orbit the body despite the gravity of some third body that the two are orbiting together. Beyond the Hill radius, the small body may be drawn out of its orbit. For example, the Moon is within Earth's Hill radius and its orbit is stable over billions of years. If it weren't within this radius, perturbations of its orbit caused by the Sun could eventually free it from orbiting Earth and it would independently orbit the Sun. A body's Hill sphere necessarily lies between the L1 and L2 Lagrangian points of the body and its host (Earth and Sun in this example). The formula for the Hill radius for a body orbiting another is:

r ≈ a(1-e)(m/3M)1/3

• r - Hill radius of the orbiting body (e.g., of Earth, in the above example).
• a - semi-major axis of the orbit around the body it is orbiting (Earth around the Sun).
• e - eccentricity of this orbit. (Note that if the orbit is circular, "1-e" becomes "1", and the formula is sometimes seen presuming this.)
• m - mass of the orbiting body (Earth).
• M - mass of the central body (Sun).

(In relation to the Earth example above, this formula describes the Hill radius from Earth's center, formed by the Earth and Sun, affecting the orbits of objects orbiting Earth such as the Moon as well as artificial satellites such as the Hubble Space Telescope.)

The above formula is an approximation of the limit (as m/M → 0) of a calculation of the distances from the smaller body to L1 and L2. It is a good approximation regarding the smaller body's gravitational influence, best if M >> m. A better approximation (useful if m/M is not ≈ zero) substitutes M+m for M in the above equation, but an exact solution is not solvable for r: correction terms or numerical methods are necessary to do better.

The Hill radius is sometimes used as an approximation of how close to a smaller object might be to a body to be accreted, but the Bondi radius is another such calculation based on other factors.

The term tidal radius is often used when discussing gravitationally-bound groups of objects such as an entire stellar cluster or galaxy: the dynamics of a body within the group's tidal radius is determined by that group rather than some other entity.

The term tidal disruption radius (sometimes abbreviated as tidal radius) is used regarding tidal disruption events (TDEs), i.e., stars destroyed by supermassive black holes (SMBHs). Formula:

RTD = Rstar(MSMBH/Mstar)1/3