Gravity field of the earth - Physical Geodesy - page 53

On the other hand, since n is the unit vector corresponding to g but of
opposite direction, it is given by

g
g

g
g ,

n = − = − (2-60)

so that

g = − g n . (2-61)

This equation, together with (2-58) and (2-59), gives

− W x = g cos Φ cos Λ ,

− W y = g cos Φ sin Λ ,

−W z = g sin Φ .

(2-62)

Solving for Φ and Λ, we finally obtain

−W z

Φ=tan − 1

W 2

,

x + W 2 y

(2-63)

Λ=tan − 1 W y
W x

,

W = W ( x, y, z ) .

These three equations relate the natural coordinates Φ , Λ ,W to the rectan-
gular coordinates x, y, z , provided the function W = W ( x, y, z )isknown.
We see that Φ , Λ ,H are related to x, y, z in a considerably more com-
plicated way than the spherical coordinates r, ϑ, λ ofSect.1.4.Notealso
the conceptual difference between the astronomical longitude Λ and the geo-
centric longitude λ .

2.5 The potential of the earth in terms of spherical

harmonics

Looking at the expression (2-7) for the gravity potential W , we see that the
part most dicult to handle is the gravitational potential V , the centrifugal
potential being a simple analytic function.
The gravitational potential V can be made more manageable for many
purposes if we keep in mind the fact that outside the attracting masses it is
a harmonic function and can therefore be expanded into a series of spherical
harmonics.