## Physics Friday 56

Consider a rigid spherical body of density ρ. Now, imagine a test mass m located inside this body at a distance r from the center of the sphere. It will experience a gravitational force , with the negative sign indicating a force opposite in direction to the displacement r from the sphere center (an inward force). We see the force varies linearly with r. Thus, the gradient of the gravity is
, a constant. Here, the negative sign indicates that the gradient of the sphere’s gravitational self-attraction compresses the object.

Now, let us consider a mass M with its center of gravity a distance d from the center of our sphere. Let us consider a displacement r from the center of the sphere on the line between the centers of the sphere and M, with positive r being toward the mass M. Then the gravitational force on our test mass due to our external mass is
,
where the positive sign indicates the force is in the direction of positive r.
This, in turn, has a gradient


Where  is the radius of a sphere of density ρ with a mass 2M.
Thus, note that if d<deff, we see that the tidal forces overcome the self-attraction down to the center of the sphere, and thus a stable rigid body held together by gravitational self-attraction alone cannot exist. If our body of mass M is a sphere of radius R, it has density . Thus, writing deff in terms of ρM and R,
.