Geopotential

Remarks

Example: for a non-rotating ball of mass $M$ with spherically symmetric density, $\omega = 0$ and $W(\mathbf{x}) = GM/\norm{\mathbf{x}}$ outside the ball; then $\nabla W$ has magnitude $GM/\norm{\mathbf{x}}^2$, as in Newton's law of universal gravitation. Non-example: the potential $V$ alone is not the geopotential of the rotating Earth, since it omits the centrifugal force. Geodesy uses the sign convention $W > 0$ and $\mathbf{g} = \nabla W$; physics often uses the opposite sign. Outside the Earth $V$ is harmonic, but $\Delta W = 2\omega^2$, because of the centrifugal part. The geopotential number of a point $P$ is $W_0 - W(P)$, where $W_0$ is the value on the geoid.