# Geopotential

Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as minus the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the minus.

For geophysical applications, gravity is distinguished from gravitation. Gravity is defined as the resultant of gravitation and the centrifugal force caused by the Earth's rotation. The global mean sea surface is close to one of the equipotential surfaces of the geopotential of gravity. This equipotential surface, or surface of constant geopotential, is called the geoid.

For the purpose of satellite orbital mechanics, the geopotential is typically described by a series expansion into spherical harmonics (spectral representation). In this context the geopotential is taken as the potential of the gravitational field of the Earth, that is, leaving out the centrifugal potential.

Solving for geopotential (Φ):

$\Phi(h) = \int_0^h g\,dz\$[1]
$\Phi = \int_0^z \left[ \frac{Gm}{(a+z)^2} \right] dz$

Integrate to get

$\Phi = Gm \left[\frac{1}{a} - \frac{1}{a+z} \right]$

where:

G=6.673x10-11 Nm2/kg2 is the gravitational constant,
m=5.975x1024 kg is the mass of the earth,
a=6.378x106 m is the average radius of the earth,
z is the height in meters
Φ is the geopotential at height z, which is in units of [m2/s2] or [J/kg].