The limiting distance to which a
satellite can approach without breaking up depends on the rigidity of the satellite. At one extreme, a completely rigid satellite will maintain its shape until tidal forces break it apart. At the other extreme, a highly fluid satellite gradually deforms leading to increased tidal forces, causing the satellite to elongate, further compounding the tidal forces and causing it to break apart more readily. Most real satellites would lie somewhere between these two extremes, with tensile strength rendering the satellite neither perfectly rigid nor perfectly fluid. For example, a
rubble-pile asteroid will behave more like a fluid than a solid rocky one; an icy body will behave quite rigidly at first but become more fluid as tidal heating accumulates and its ices begin to melt. But note that, as defined above, the Roche limit refers to a body held together solely by the gravitational forces which cause otherwise unconnected particles to coalesce, thus forming the body in question. The Roche limit is also usually calculated for the case of a circular orbit, although it is straightforward to modify the calculation to apply to the case (for example) of a body passing the primary on a parabolic or hyperbolic trajectory.
Rigid satellites 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, 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: : d = R_M\left(2 \frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}} where R_M is the
radius of the primary, \rho_M is the
density of the primary, and \rho_m is the density of the satellite. This represents 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. Since the limit depends only on the density, this implies, that the satellite will be torn entirely, if it consists of loose dust or separate rocks bound only by gravity.
Fluid satellites 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: : d \approx 2.44 \ R\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: : d \approx 2.423 \ R\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 c/R is the
oblateness of the primary. 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. == Modern revisions and exceptions to the Roche limit ==