* The gravitational potential energy is zero at r = infinity. This is equivalent to choosing the base level of the potential energy to be located very far away from the body.

* One could choose the surface of the body as a base point, but then the potential energy at this reference point would not be zero.

* The gravitational PE is negative for all values of r less than infinity. This is can be visualized by looking at the large mass M as creating a gravitational energy well around it that goes down as you get closer to it.

Derivation of Gravitational Potential:
The gravitational potential energy is defined as the negative of the work it takes to move an object from some reference point to a height h above the surface of the Earth. Near the surface of the earth the force of gravity is nearly constant.

If we do not neglect the force of gravity's dependence upon distance r, we get the following integral to evaluate,

If we let r = R_E + y then dr = dy and

Thus we can see that the change in potential energy is the difference between two negative numbers which turns out to be a positive number.

Let us start with the definition of gravitational potential energy and show that it gives the same equation as above. Starting with

the change in potential is

Reference Level
Had we defined the base level for the potential energy to be the surface of the body, the potential energy could be defined as

Observe that the change in potential energy DPE would still be the same, except our definition of potential energy would also depend upon the size of the body as well as its mass. Moreover it will not be zero at the base level - the surface of the body.

Altitude Correction Factor
It is sometimes useful to use the body's surface as a base point.

Here g is the normal acceleration of gravity at the surface of the body. The term (1 - h/R_E)^-1 represents an altitude correction factor. If h is small compared to the radius of the Earth then this formula goes over to familiar mgh expression for potential energy.