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
- <math>U(r) = \frac{-GM}{r}\!\,</math>
where:
- <math>G\!\,</math> is the universal gravitational constant,
- <math>r\!\,</math> is the distance to the center of mass of the object,
- <math>M\!\,</math> 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
- <math>U(r,m) = \frac{-GMm}{r}</math>
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-spherical and non-homogeneous nature of typical sources of gravitational potential. In this case a gravitational potential may depend on polar <math>\phi\!\,</math> and azimuth <math>\lambda\!\,</math> direction of vector <math>r\!\,</math>.
The most widely used form of the gravitational potential function depends on <math>\phi\!\,</math> (latitude) and potential coefficients, Jn, called the zonal coefficients:
- <math> 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 ] </math>
where:
- <math> \phi\,\!</math> ...
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