Physical Geodesy - The Geopotential

The Geopotential

The Earth's gravity field can be described by a potential as follows:

$\mathbf{g} = \nabla W = \mathrm{grad}\ W = \frac{\partial W}{\partial X}\mathbf{i} +\frac{\partial W}{\partial Y}\mathbf{j}+\frac{\partial W}{\partial Z}\mathbf{k}$

which expresses the gravitational acceleration vector as the gradient of, the potential of gravity. The vector triad is the orthonormal set of base vectors in space, pointing along the coordinate axes.

Note that both gravity and its potential contain a contribution from the centrifugal pseudo-force due to the Earth's rotation. We can write

$W = V + \Phi\,$

where is the potential of the gravitational field, that of the gravity field, and that of the centrifugal force field.

The centrifugal force is given by

$\mathbf{g}_c = \omega^2 \mathbf{p},$

where

$\mathbf{p} = X\mathbf{i}+Y\mathbf{j}+0\cdot\mathbf{k}$

is the vector pointing to the point considered straight from the Earth's rotational axis. It can be shown that this pseudo-force field, in a reference frame co-rotating with the Earth, has a potential associated with it that looks like this:

$\Phi = \frac{1}{2} \omega^2 (X^2+Y^2).$

This can be verified by taking the gradient operator of this expression.

Here, and are geocentric coordinates.

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