Gravitational potential

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In physics, gravitational potential is the potential energy per unit mass of an object due to its position in a gravitational field. The gravitational potential due to a point mass M is

$U(r) = \frac{-GM}{r}\!\,$

where:

$G\!\,$ is the universal gravitational constant,
$r\!\,$ is the distance to the center of the object,
$M\!\,$ is the mass of the object.

If the structure of an object is spherically symmetrical, e.g. the Earth, then the potential is the same as that of a point object having the same mass.

As a result, the gravitational potential energy of an object of mass m relative to a point mass M is

$U(r,m) = \frac{-GMm}{r}$

where r is the distance between the centers of the two objects.

In astrodynamics the gravitational potential function has to take into effect effects of non-sperical and non-homogenous nature of typical sources of gravitational potential. In this case a gravitational potential may depend on polar $\phi\!\,$ and azimuth $\lambda\!\,$ direction of vector $r\!\,$ .

The most widely used form of the gravitational potential function depends on $\phi\!\,$ (latitude) and potential coefficients, Jn, called the zonal coefficients:

$U(r,\phi) = \frac{GM}{r} \left [1 - \sum_{n=2}^N J_{n} \left (\frac{R}{r} \right)^2 P_n (\sin \phi) \right ]$