Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 331

P

g

g

dR / d qdq

dq

q

R dq

f

Q

O

Figure 5.1 An observer at P on the reference level surface with radius R (θ) meas-
ures geographic latitude φ with respect to the zenith line QP normal to the level
surface. Geocentric latitudeφ is measured with respect to the radius line from the
geocentre at O .

allowing g to be expressed in terms of g e , its value at the equator, to first order in
m as

g = g e 1

5
2 m

f sin 2 φ +··· + − , (5.33)

φ being the geocentric latitude.
Geographic latitude φ is often more accessible observationally than φ .Their
relationship is illustrated in Figure 5.1.
This shows directly that

φ = φ + γ, (5.34)

where γ is determined to be

arctan 1
R

dR
d θ

γ = (5.35)

by considering the e ect of a small increment, δθ, in the geocentric co-latitude

shown by broken lines in Figure 5.1. To first order in m ,wehave

ff

m dR 1

f
m cosθsinθ +···

γ =

d θ +···=

2 m ·

m cos π 2 − φ sin π 2 − φ

f f

m sinφ cosφ +···

= 2 m · +···= 2 m ·

f
m sinφcosφ +···

= 2 m · (5.36)