Roche limit

The rigid-body Roche limit is a simplified calculation for a spherical satellite. Irregular shapes such as those of tidal deformation on the body or the primary it orbits are neglected. It is assumed to be in hydrostatic equilibrium. These assumptions, although unrealistic, greatly simplify calculations.

The Roche limit for a rigid spherical satellite is the distance, ${\displaystyle d}$, from the primary at which the gravitational force on a test mass at the surface of the object is exactly equal to the tidal force pulling the mass away from the object:[3][4]

${\displaystyle d=R_{M}\left(2{\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

where ${\displaystyle R_{M}}$ is the radius of the primary, ${\displaystyle \rho _{M}}$ is the density of the primary, and ${\displaystyle \rho _{m}}$ is the density of the satellite. This can be equivalently written as

${\displaystyle d=R_{m}\left(2{\frac {M_{M}}{M_{m}}}\right)^{\frac {1}{3}}}$

where ${\displaystyle R_{m}}$ is the radius of the secondary, ${\displaystyle M_{M}}$ is the mass of the primary, and ${\displaystyle M_{m}}$ is the mass of the secondary.

This does not depend on the size of the objects, but on the ratio of densities. This is the orbital distance inside of which loose material (e.g. regolith) on the surface of the satellite closest to the primary would be pulled away, and likewise material on the side opposite the primary will also go away from, rather than toward, the satellite.

A more accurate approach for calculating the Roche limit takes the deformation of the satellite into account. An extreme example would be a tidally locked liquid satellite orbiting a planet, where any force acting upon the satellite would deform it into a prolate spheroid.

The calculation is complex and its result cannot be represented in an exact algebraic formula. Roche himself derived the following approximate solution for the Roche limit:

${\displaystyle d\approx 2.44R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{1/3}}$

However, a better approximation that takes into account the primary's oblateness and the satellite's mass is:

${\displaystyle d\approx 2.423R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{1/3}\left({\frac {(1+{\frac {m}{3M}})+{\frac {c}{3R}}(1+{\frac {m}{M}})}{1-c/R}}\right)^{1/3}}$

where ${\displaystyle c/R}$ is the oblateness of the primary. The numerical factor is calculated with the aid of a computer.

The fluid solution is appropriate for bodies that are only loosely held together, such as a comet. For instance, comet Shoemaker–Levy 9's decaying orbit around Jupiter passed within its Roche limit in July 1992, causing it to fragment into a number of smaller pieces. On its next approach in 1994 the fragments crashed into the planet. Shoemaker–Levy 9 was first observed in 1993, but its orbit indicated that it had been captured by Jupiter a few decades prior.[5]