# A Determination of the Earth's Gravity Field in Spheroidal Coordinates

5-1961

Thesis

## Degree Name:

Master of Science (MS)

## Department:

Mathematics and Statistics

Joe Elich

Joe Elich

## Abstract

The earth's gravity field G * at a point P in the region surrounding the earth's surface is defined as the force acting on a unit mass concentrated at P. This is a force resulting from two components: (1) G1 due to the gravitational attraction of the earth's mass, and (2) G2 due to the earth's rotation.

As a result of Newton's law of gravitation, G1 can be written in integral form as follows:

G1 = k ( (V ( rdm/r3

where r = PQ, r = |r|, Q is a point which ranges over the region V bounded by the earth's surface, k is the gravitational constant, dm = p dv, and p is the density at Q.

The density function, p, is not known, and so equation (1) is not suitable for the determination of G1. However, it can be shown that the earth's gravity field is conservative, and therefore, there exists a scalar function U such that G is determined by the gradient of U; that is

G = -∇U

The function U is called the total potential and can be written as the sum of two functions, U1 and U2, where U1 is the potential corresponding to G1, and U2 is that associated with G2.

An important property of the function U1 is that it satisfies Laplace's differential equation, that is

2U1 = 0

Various mathematical expressions have been determined which represent the gravity field of the earth with varying degrees of accuracy. The main problem in obtaining these results is that of solving Laplace's equation with appropriate boundary conditions to determine U1. The boundary conditions involve the shape of the surface of the earth. By assuming the earth's surface to be ellipsoidal and using spherical coordinates, a reasonably accurate solution in infinite series form can be found. Based on the first few terms of this series and on certain accepted measurements related to the size and shape of the earth's surface, the International Gravity Formula is obtained:

|G| = g = go(1+0.0052884 sin 2ϴ-

0.0000059 sin 22ϴ)

where ϴ is the geographical latitude and go represents the measured acceleration of gravity at the equator.

## Checksum

33111b96407a3c33c8c148f5777dd9dd

## Share

COinS

#### DOI

https://doi.org/10.26076/8528-a173