The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 136

Fig. 4.4 Spherical
coordinates and rectangular
coordinates

2 V
∂ ˁ

2 V
∂ ʸ

2 V
∂ ʻ

V V

∂ ʸ þ

1
sin 2

2 ∂

ˁ ∂

∂ ˁ þ ∂

ʸ ∂

∂

ˁ

2 þ

2

2 þ

cot

2 ¼

0 : ð 4 : 16 Þ

ʸ

Solution of the above Laplace equation (derivation omitted) is the harmonic
function in the spherical coordinates, which can be represented as (Hofmann and
Moritz 2005; Torge 1989):

1

nþ1 X

n

1

V ð ˁ; ʸ; ʻ Þ¼ ð a nk cos k ʻ þ b nk sin k ʻ Þ P nk cos ð ʸ Þ: ð 4 : 17 Þ

ˁ

n¼0 k¼0

Equation ( 4.17 ) is a series expansion, which indicates that the Earth's gravita-
tional potential at an exterior point can be described by an infinite series. (

)
are the spherical coordinates of the exterior point of the Earth and a nk and b nk are the
coefficients of the Earth's gravity field, which can be determined by the observed
values. Therefore, the gravitational potential problem can be regarded as the
problem of study of the coefficient of the gravitational potential. P nk (cos

ˁ , ʸ , ʻ

) repre-
sents the associated Legendre polynomials (also known as the associated Legendre
functions of the first kind), n is degree, and k is order. The expressions of the
associated Legendre polynomials are given by:

ʸ