Gravity field of the earth - Physical Geodesy - page 104

4

3

2

1

0

30° 60° 90° 120° 150° 180°

F() Ã

-1

S( ) Ã

-2

Fig. 2.17. Stokes' functions S ( ψ )and F ( ψ )

use ellipsoidal coordinates ϕ, λ . As a spherical approximation, ϑ is the com-
plement of ellipsoidal latitude:

ϑ =90 ◦ − ϕ. (2-315)

Hence, we have

dσ = 2 π
λ =0

π/ 2

cos ϕdϕdλ, (2-316)

ϕ = −π/ 2

σ

so that Stokes' formula now becomes

2 π π/ 2

R
4 πγ 0

∆ g ( ϕ ,λ ) S ( ψ )cos ϕ dϕ dλ , N ( ϕ, λ )= (2-317)

λ =0 ϕ = −π/ 2

where ϕ, λ are the ellipsoidal coordinates of the computation point and ϕ ,λ
are the coordinates of the variable surface element dσ . The spherical distance
ψ is expressed as a function of these coordinates by

cos ψ =sin ϕ sin ϕ +cos ϕ cos ϕ cos( λ − λ ) . (2-318)

Stokes' function in terms of spherical harmonics
In Sect. 2.13, Eq. (2-273), we have found

∞

∆ g ( ϑ, λ )= 1
R

( n − 1) T n ( ϑ, λ ) . (2-319)

n =0