In the case of satellites with their orbits degrading in closer to the planet they orbit, there is an interesting consequence of the differential forces.
It would seem logical that the satellite would even eventually have its orbit degrade until it runs into the body it orbits. This eventuality is, however thwarted by forces that come into play at a radius known as the "Roche limit".
The Roche limit is a radius at which an orbiting satellite assumed to be in an ideal liquid form would be torn apart as a result of differential forces and turbulence.
Liquid composition is assumed so that mass of the satellite may be considered to be held together by the satellite's internal gravitational pull. Under normal circumstances it must be understood that a satellite also has some tensile property holding it together along with the gravitational attraction so that a normal satellite is capable of passing safely within the Roche limit before it would run the risk of being torn apart by the differential force and resulting turbulence.
The most common derivation of the expression for the Roche limit assumes that the radius for the Roche limit is found simply where the differential force overcomes a satellite's internal gravitational force holding it together. This derivation serves well to illustrate what the Roche limit is. There are however more elements at work than simply the differential forces.
Edouard Roche, after whom the Roche limit takes its name, took a more sophisticated approach analyzing the satellite's potential energies under the condition that the satellite had some kind of equilibrium shape. He found an expression that gives Roche limits significantly further out than the standard simplistic derivation gives.
In the final analysis, a fairly accurate and simple expression for the Roche limit is given as follows:
RRoche<2.456*RM*(ρM/ρm)1/3 (Eq. 5)
When a satellite passes within the Roche limit and is torn up, the result would be orbiting debris. The rings of Jupiter as well as other ring systems are thought to be partially made up of moons that had passed through their Roche limit and been torn to pieces.
