Gravitational Self Energy

The gravitational potential energy of a composite system of n particles, given is called the self-energy of the system:



The above energy represents the (negative of) work done in assembling the system from individual particles that were initially infinitely apart. Hence, self-energy of a body is the potential energy stored within the body as the body is assembled from free individual particles.

Uniform sphere: Let us evaluate gravitational self-energy of a uniform solid sphere of mass M and radius R. Since we are now considering a continuous mass distribution, the summations go over to integrals. The calculation goes as following.

Suppose at any moment, we have already assembled a sphere at its surface is given by



If we now bring an additional mass element dm to the sphere, the work done in bringing dm from ∞ to r is




Hence, the total potential energy of the system as it is built up from m = 0 to m = M is given by



Now, note that m and r are related to each other through the density ρ, which is constant (sphere is uniform). We have,



which is the self energy of a uniform sphere. This is same as the positive energy required to dissociate the sphere into individual particles and take them infinitely apart.

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