Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 373

q i

a i

3

c i

a i (1 + f i )

2

1

Figure 5.5 The inner core is tilted at angle θ i with respect to the rest of the
Earth. We divide the volume

V 0 outside the inner core into three distinct regions.

Region is bounded on the inside by the inner core and on the outside by a sphere

with radius equal to the equatorial radius a i of the inner core. The polar radius of
the inner core is c i .Region

➀

lies between region and the equipotential with

➁ ➀

polar radius a i and thus equatorial radius a i (1 + f i ), where f i is the flattening of

the inner core. Region is exterior to region .

➂ ➀

Because f i 1/400, our calculations will only be carried out to first order in the
flattening. For the same reason, regions ➀ and ➁ can be treated as surface densities,
using the density just outside the inner core denoted by ρ i .
We begin with the evaluations of the contributions of regions ➀ and ➁ to the
gravitational potential V 0 . For region ➀ , entirely within the sphere of radius a i ,we
can use the expansion (5.117) for

1 1

D . r | =

(5.221)

| r −

For region ➁ , entirely outside the sphere of radius a i , the expansion (5.118) can
be used. Since the mass distribution in region ➀ is symmetrical with respect to the
rotated inner core, no contribution to the restoring torque results, and we focus on
the contribution of region ➁ to the gravitational potential V 0 . This contribution is

G G ρ i

r
r

n

∞

ρ( r )

1
r

V ≈− V . − d P n (cos Θ ) d (5.222)

| r − r |

➁ ➁

n = 0

The equipotential forming the outer boundary of region ➁ , with polar radius a i ,has
mean radius a i = (1 + 2 f i /3). The equipotential surface then is given by