The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 137

P 0 cos ð ʸ Þ¼ 1

P 1 cos ð ʸ Þ¼ cos ʸ

P 11 cos ð ʸ Þ¼ sin ʸ

3
4 cos 2

1
4

P 2 cos ð ʸ Þ¼ ʸ þ

P 21 cos ð ʸ Þ¼ 3 cos ʸ sin ʸ

3
2 cos 2

3
2

P 22 cos ð ʸ Þ¼ ʸ þ

⋮

The P n0 (cos ʸ ) in the above equation is simplified as P n (cos ʸ ). P n (cos ʸ )isan

nth-degree polynomial of cos ʸ , called the Legendre polynomial. The two poly-

nomials P 0 (cos allow the higher degree polynomials

to be generated using the recursion formula below:

ʸ ) ¼ 1 and P 1 (cos ʸ ) ¼ cos ʸ

P nþ1 , k cos ʸ ¼ 2n 1 cos P nk cos ʸ n k P n1 , k cos ʸ

ð n k þ 1 Þ þ ʸ þ

Þ¼ 1 ʸ

1
2 2n

1 P n1, n1 cos ʸ : cos 2 P nn cos ð ʸ

Coefficients a nk and b nk are related to the mass distribution and shape of the
Earth, derived to obtain (Lu 1996):

G ð

8
<

n
1 P n cos

a n0 ¼

Earth ˁ

ð ʸ 1 Þ dm,

G ð

ð n k Þ !

n
1 P nk cos

a nk ¼ 2

Earth ˁ

ð ʸ 1 Þ cos k ʻ 1 dm,

ð 4 : 18 Þ

ð n þ k Þ !

:

G ð

ð n k Þ !

n

b nk ¼ 2

Earth ˁ

1 P nk cos

ð ʸ 1 Þ sin k ʻ 1 dm,

ð n þ k Þ !

where ( ʻ 1 ) denote the coordinates of dm (Fig. 4.5 ). From the above equation,

the meaning of the series expansion coefficients in the spherical harmonics of the
potential of the Earth's gravitational field can obviously be further analyzed. It is
commonly the first several terms of an infinite series that play a predominant role.
The meaning of several coefficients in low-degree terms will be discussed below.
The zero-degree term has only one coefficient, a 00 . Since

ˁ 1 , ʸ 1 ,

0

ˁ

1 ¼

1, P 0 (cos ʸ )

¼ 1, it follows from ( 4.18 ) that:

a 00 ¼ GM,

where M denotes the total mass of the Earth, which corresponds to the potential of
the gravitational field generated by a homogeneous sphere of the Earth with its
center at the coordinate origin.