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

Here, the positive sign indicates that this tidal gradient puts the object under tension.

Note here that the tidal stretching increases as one approaches the mass M, while the self-attraction compression is constant. In fact, the sum of the two gradients becomes positive, and the tidal force dominates, when

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,
.

A real body would also have a material tensile strength to help hold it together, and also would deform from spherical into tidal bulges under the tidal force; however, there still remains a similar limit, inside which a self-gravitating celestial body (our sphere) orbitting a larger body (the mass M) will disintegrate under the tidal forces. This is called the Roche Limit.

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