Equilibrium or rupture, under the effect of the self-gravitational field, which must ensure cohesion, and of the gradient terms of the gravitational field, which tears the satellite apart (in the frame of reference of the center of mass of the satellite).
Roche's argument is as follows : although the satellite is spherical, let us suppose that it is made up of two spheres of radius r and mass m. Let us think about two snow balls of radius r, held together by the gravitation force that each sphere exerts on the other. That force, , is given by Newton's relation :
Now, let us consider that each satellite is at a distance D of a planet of mass M and radius R. The attraction force F, between the planet and the nearest snow ball, will be greater than the force F' between the planet and the more distant ball. The force F is given by the following relation :
And force F' is given by :
The two balls will feel the resulting force , which tends to separate them. This force is equivalent to the difference between the forces F and F'. Therefore we have :
And, because D>>r :
The two masses will be separated if the force is greater than the force , that is if :
Let us now replace the mass M by , where is the density of the planet, and the mass m by , where is the density of the satellite.
Therefore, the two masses are separated if the distance D is less than
This is a quite satisfactory approximation since is equal to 2.51, whereas the exact value is 2,456.