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,Beta ,Lambda ). 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 Beta is the reduced altitude and Lambda is the geocentric longitude. In this coordinate system, the normal gravity is to sub-microgal levels (1 gal = 1 cm/s2):
GAMMA 3
but since we are dealing with an ellipsoid of revolution the longitude component vanishes, and
GAMMA 2 .
These remaining two components are given by:

1

1

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

1

1

1

1

1

1

1

1

1

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

1

1

1

And the radius of curvature of the prime vertical is:
1






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