In this post we find an expression for the gravitational energy of an isotropic sphere. An isotropic sphere is a sphere such that the mass density can be written as .

The gravitational energy of two point masses and is given by

The gravitational potential of a point mass is given by

Thus we can write

.

When we use a mass density we can write

where . Let then it is clear that

Therefore

where and Now

So

Thus

Then it is clear that

So

For the total energy we have

In case is constant we obtain

Then

as expected and well known.

Let us define and . Then we can write

So

Thus

But so

and using the definition of we can write

Again the case of a constant then we get

The gravitational potential of an isotropic sphere is given by

where is the radial mass density.

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