Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 374

a i 1

2

3 f i 1

2
3 f i P 2 (cosθ)

R = + − +···

a i 1

2

3 f i 1

P 2 (cosθ) +··· = + − . (5.223)

Θ

Replacing P n (cos ) by the addition formula (B.9), we have

G

ρ( r )

− d V

| r − r |

➁

G ρ i ∞

n

r n 1) m P n (cosθ) ≈− ( − ×

m =− n

n = 0

2π

e im (φ − φ ) π
0

R

1

( r ) n − 1 dr P − n (cosθ )sinθ d θ d φ .

(5.224)

0 a i

The inner integral becomes

a i 1 +

3 f i 1 − P 2 (cosθ )

R

2

( r ) 2 − n
2

1

( r ) n − 1 dr =

−

n a i

a i

a 2 − i 1

3 f i 1

P 2 (cosθ ) 2 − n

2

a 2 − n
i

+ − −

=

2 − n

a 2 − i 1 n ) f i 1 P 2 (cosθ ) −

2

a 2 − n
i

+

3 (2

− −

=

2 − n

2

f i 1 P 2 (cosθ ) .

3 a 2 − n

= − (5.225)

i

Using the orthogonality relation (B.7), we find the contribution of region ➁ to the
gravitational potential V 0 to be

G

ρ( r ) 8 3 G ρ i a i f i + 8 15 G ρ i f i r 2 P 2 (cosθ),

V ≈− − d (5.226)

r |

| r −

➁

correct to first order in the flattening.
For region ➂ , entirely outside the sphere of radius a i , once again the expansion
(5.118) for 1/ D can be used. The contribution of this region to the gravitational
potential V 0 is then

G

d

2π π

G ∞

ρ( r )

0 ρ( r 0 ) P n (cos

Θ )

V =− r n V . − d d (5.227)

| r − r | R n + 1

0

➂

n = 0

2
3 f i )

a i (1 +