Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 352

d

r 0 ρ( r 0 ) f +

7 κ dr 0

¯ρ( r 0 ) m ( r 0 ) 10
9 9 r 0 f

9 4 2

7 ff +

+ f +

3 ¯ρ( r 0 ) f

r 0 f 2

2 1 2

21

35 47

4 2 r 0 f + 4 r 0 ff +

f 2 3 r 0 κ + − + + + 6κ

5
6 ¯ρ( r 0 ) m ( r 0 )

=− (5.160)

and

d

r 0 ρ( r 0 ) r 0 κ −

1

2κ dr 0

r 0

(5.161)

¯ρ( r 0 ) r 0 f 2 16κ

1
12 r 0

2 r 0 ff − 4 r 0 κ − + + = 0.

Di ff erentiation of the mean density expression (5.156) yields

d ¯ρ
dr 0 =−

3 ρ
r 0

(1 − ρ/¯ρ). (5.162)

Di erentiation of expression (5.160) with respect to r 0 , and substitution from rela-

tion (5.162), gives

ff

− ρ( r 0 ) f +

7 κ

¯ρ( r 0 ) m ( r 0 ) 12
9 9 r 0 f

9 4 2

7 ff +

f +

+

3 ¯ρ( r 0 ) 1

2 r 0 f 2

r 0 f f

2 3 f +

r 0 f +

2 2

21

− + (5.163)

6κ

4 r 0 f 2

ff

35 47

3 κ + r 0 κ +

r 0 ff +

2 ff +

+ + +

− ρ/¯ρ) f

r 0 f 2

2
r 0 ¯ρ(1

1 2

21

35 47

4 2 r 0 f + 4 r 0 ff +

f 2 3 r 0 κ + + + + + 6κ

= 0.

2 r 0 / GM i

Note that the product ¯ρ( r 0 ) m ( r 0 ) is independent of r 0 ,since m ( r 0 ) =Ω

4π r 0 ¯ρ( r 0 )/3, hence ¯ρ( r 0 ) m ( r 0 )

2

and M i = = 3 Ω /4π G .

erentiating (5.161) with respect to r 0 , and again substituting from
(5.162), produces

Similarly, di ff