Geopotential

Geopotential is the potential of the Earth's gravity field.

For convenience it is often defined as the negative of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation.

For geophysical applications, gravity is distinguished from gravitation.

[2] Global mean sea surface is close to one equigeopotential called the geoid.

At latitude 50 deg the off-set between the gravitational force (red line in the figure) and the local vertical (green line in the figure) is in fact 0.098 deg.

For a mass point (atmosphere) in motion the centrifugal force no more matches the gravitational and the vector sum is not exactly orthogonal to the Earth surface.

This is the cause of the coriolis effect for atmospheric motion.

The geoid is a gently undulating surface due to the irregular mass distribution inside the Earth; it may be approximated however by an ellipsoid of revolution called the reference ellipsoid.

The currently most widely used reference ellipsoid, that of the Geodetic Reference System 1980 (GRS80), approximates the geoid to within a little over ±100 m. One can construct a simple model geopotential

that has as one of its equipotential surfaces this reference ellipsoid, with the same model potential

of the geoid; this model is called a normal potential.

Many observable quantities of the gravity field, such as gravity anomalies and deflections of the vertical (plumb-line), can be expressed in this disturbing potential.

is the gravitational potential energy per unit mass.

is the orthonormal set of base vectors in space, pointing along the

Both gravity and its potential contain a contribution from the centrifugal pseudo-force due to the Earth's rotation.

It can be shown that this pseudo-force field, in a reference frame co-rotating with the Earth, has a potential associated with it in terms of Earth's rotation rate, ω: This can be verified by taking the gradient (

The centrifugal potential can also be expressed in terms of spherical latitude φ and geocentric radius r: The Earth is approximately an ellipsoid.

So, it is accurate to approximate the geopotential by a field that has the Earth reference ellipsoid as one of its equipotential surfaces.

[4] It can also be expressed as a series expansion in terms of spherical coordinates; truncating the series results in:[4] where a is semi-major axis and J2 is the second dynamic form factor.

Its geometric parameters are: semi-major axis a = 6378137.0 m, and flattening f = 1/298.257222101.

If we also require that the enclosed mass M is equal to the known mass of the Earth (including atmosphere), as involved in the standard gravitational parameter, GM = 3986005 × 108 m3·s−2, we obtain for the potential at the reference ellipsoid: Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity (

For practical purposes it makes more sense to choose the zero point of normal gravity to be that of the reference ellipsoid, and refer the potentials of other points to this.

has been constructed matching the known GRS80 reference ellipsoid with an equipotential surface (we call such a field a normal potential) we can subtract it from the true (measured) potential

The result is defined as T, the disturbing potential: The disturbing potential T is numerically a great deal smaller than U or W, and captures the detailed, complex variations of the true gravity field of the actually existing Earth from point-to-point, as distinguished from the overall global trend captured by the smooth mathematical ellipsoid of the normal potential.

In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(s); i.e., density depends only on the radial distance These integrals can be evaluated analytically.

This is the shell theorem saying that in this case: with corresponding potential where M = ∫Vρ(s)dxdydz is the total mass of the sphere.

For the purpose of satellite orbital mechanics, the geopotential is typically described by a series expansion into spherical harmonics (spectral representation).

Solving for geopotential in the simple case of a nonrotating sphere, in units of [m2/s2] or [J/kg]:[5]