Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 377

R dq

g

R

dR / d qdq

dq

Figure 5.6 A small increment δθ in the angle θ results in an increase in the equi-
potential surface radius, R , equal to ( dR / d θ)δθ. A constant radius vector scribes a
circular arc of length R δθ. The ratio of these is equal to the small angle γ between
r and the outward normal to the equipotential surface ν .

with R the equipotential surface given by (5.229). Thus,

1
R

dR
d θ =

γ = 2 f i cosθsinθ. (5.241)

φ ) for spherical polar co-
ordinates in the inner core frame and the system of Cartesian co-ordinate unit vec-
tors ( ı , j ,

θ ,

We adopt both the system of unit vectors ( r ,

k ) in the same frame. Because γ is only a first-order quantity,

2 φ r f i cosθsinθ.

r ×

ν ≈− (5.242)

Since

φ =−

ı sinφ + j cosφ, (5.243)

we have

r × ν = 2 ı r f i cosθ sinθ sinφ − 2 j r f i cosθ sinθ cosφ

ı r f i i 2 iP − 2 (cosθ ) e − i φ

3 P 2 (cosθ ) e i φ

= +

j r f i 2 P − 2 (cosθ ) e − i φ

1

3 P 2 (cosθ ) e i φ

− − + , (5.244)

using the associated Legendre functions,

1

P 2 (cosθ ) 3cosθ sinθ and P − 1

2 cosθ sinθ .

=− = (5.245)

2

Similarly, the volume integral requires

φ r d ρ 0
dr 0

∇ × ( r ρ( r )) =− r ×∇ ρ( r ) = sinγ. (5.246)