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
where and Now
Then it is clear that
For the total energy we have
In case is constant we obtain
as expected and well known.
Let us define and . Then we can write
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.