Acheson - Irene Burgess - Jessica K.
To apply this relationship to the problem of finding the gravitational binding energy of the Earth, we imagine removing one infinitesimally thick outer shell after the other from a spherical distribution of mass that gets smaller and smaller as shells are removed. Each mass shell is moved an infinite distance away through the diminishing force of gravity between the shell and sphere. It will take some energy to move all these shells; that energy is the negative of the binding energy of the system. To find energy needed to remove all these shells, we will find the change in the system's potential energy. To find the change in potential energy, we need to know the mass of the shell, the mass of the remainder of the sphere, and the force through which we're moving the shell.
The mass of any volume is simply density times volume. Each shell has a volume given by its area times its thickness and a mass given by its volume times its density. In short, we have
where is the radius of the sphere.
If we assume that the density of the sphere is the same everywhere, then each shell contributes an infinitesimal portion of potential energy as described by
where the average density is given by
where the angle brackets indicate an average value and
Substituting, we obtain
To find the total change in potential energy as all the shells are removed by summing all the infinitesimal bits of potential energy, we integrate over the radius of all the shells, from the outside to the inside, like this
Thus for any point within the sphere, the change in potential energy using average density is
Therefore the total change in potential energy as the Earth is disassembled using average density becomes
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