The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 135

Fig. 4.3 Gravitational
potential of the body with
mass M

G ð
M

:

2
2

dm

r þ ω

x 2 y 2

W ¼ þ ð 4 : 14 Þ

4.1.2 Earth Gravity Field Model

It can be proved that the gravitational potential of a body at an exterior point, given
by ( 4.10 ), that is:

G ð dm
r

V ¼ ,

satisfies the differential equation:

2 V
∂

2 V 2 V

∂

∂

x 2 þ ∂ y 2 þ ∂ z 2 ¼

0 : ð 4 : 15 Þ

∂

Equation ( 4.15 ) is the well-known Laplace equation. The function that satisfies
the Laplace equation is termed the harmonic function (Torge and M¨ ller 2012) or
the spherical harmonics.
In the spherical coordinate system as illustrated in Fig. 4.4 , the relation between
the rectangular coordinates (x, y, z) and the spherical coordinates (

ˁ , ʸ , ʻ ) of point

P is given by:

8
<

x ¼ ˁ sin ʸ cos ʻ ,

y ¼ ˁ sin ʸ sin ʻ ,

:

z ¼ ˁ cos ʸ ,

where is the spherical colatitude. The Laplace equation ( 4.15 ) can be represented

by the spherical variables (derivation omitted), as:

ʸ