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 2*M*.

Thus, note that if *d*<*d*_{eff}, 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 *d*_{eff} in terms of *ρ*_{M} 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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Tags: Friday Physics, Gravitation, physics, Roche Limit, Tidal Force

This entry was posted on January 23, 2009 at 1:27 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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