Geopotential is the potential of the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

's gravity field. For convenience it is often defined as minus the potential energy

Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

 per unit mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

, so that the gravity vector is obtained as the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

 of this potential, without the minus.

For geophysical applications, gravity is distinguished from gravitation. Gravity is defined as the resultant of gravitation

Gravitation

Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

 and the centrifugal force caused by the Earth's rotation. The global mean sea surface

Sea level

Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

 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

Geoid

The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...

.

For the purpose of satellite orbital mechanics, the geopotential is typically described by a series expansion into spherical harmonics

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

 (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)
Φ=∫(g)dz from 0 to z.
Φ=∫[Gm/(a+z)²]dz from 0 to z
where G is the gravitational constant, 6.67E-11 Nm²/kg²
m is the mass of the earth, 5.975E24 kg
a is the average radius of the earth, 6.37E6 m
z is the height in meters

Integrate to get
Φ=Gm[(1/a)-(1/(a+z))]
units [m²/s²] or [J/kg]

Solving for geopotential (Φ)

Φ=∫(g)dz from 0 to z.

Φ=∫[Gm/(a+z)²]dz from 0 to z

where G is the gravitational constant, 6.67E-11 Nm²/kg²

m is the mass of the earth, 5.975E24 kg

a is the average radius of the earth, 6.37E6 m

z is the height in meters

Integrate to get Φ=Gm[(1/a)-(1/(a+z))]

units [m²/s²] or [J/kg]

See also

• For further details see Geoid
  Geoid
  The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...
  
   and Physical geodesy
  Physical geodesy
  Physical geodesy is the study of the physical properties of the gravity field of the Earth, the geopotential, with a view to their application in geodesy.-Measurement procedure:...
  
  
• geopotential height
  Geopotential height
  Geopotential height is a vertical coordinate referenced to Earth's mean sea level — an adjustment to geometric height using the variation of gravity with latitude and elevation. Thus it can be considered a "gravity-adjusted height"...