# An Exact Expression for Normal Gravity

At high geodetic heights (>>20 km) the Taylor series expansion
for normal gravity above the reference ellipsoid becomes inaccurate. In
this case normal gravity is better described in an ellipsoidal coordinate
system (u,
,
). The coordinate u is the semi-major axis of an ellipsoid of revolution
which is confocal with the reference ellipsoid and whose surface passes through
the altitude Z of interest. The coordinate
is the reduced altitude and
is the geocentric longitude. In this coordinate system, the normal gravity
is to sub-microgal levels (1 gal = 1 cm/s^{2}):

but since we are dealing with an ellipsoid of revolution the longitude
component vanishes, and

.

These remaining two components are given by:

where we are using WGS-84 values for a, b, e, GM and
.

The Cartesian rectangular coordinates are given in terms of the geodetic
coordinates (
,
, h) by:

And the radius of curvature of the prime vertical is:

[Back to Home Page]