Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 329

obeys Laplace's equation. Hence, for a reference level surface that is a surface of
revolution, symmetrical under reflection in the equatorial plane, it is represented
by a series with terms P n (cosθ)/ r n + 1 , where the Legendre polynomials are of even
degree n . With the expression of R in powers of m (5.8), and again using the bino-
mial expansion, in dimensionless variables we have

1
R +

α m
R 3 P 2 (cosθ)

1 1

3 mR 2 3 mR 2 P 2 (cosθ)

U 0 =− − +

− mR 1 − m 2 R 2 − R 1 =− 1 +···

+ α m { 1 − 3 mR 1 +···} P 2 (cosθ)

1
3 m

1
3 m

− { 1 + 2 mR 1 +···}+ { 1 + 2 mR 1 +···} P 2 (cosθ), (5.19)

with αacoe cient to be determined. With substitution for R 1 from (5.18), correct

to first order in m we have that

1

m P 2 (cosθ)

2
3

f

U 0 =− + m · + α mP 2 (cosθ)

(5.20)

1
3 m

1
3 mP 2 (cosθ).

− +

Recognising that U 0 is a constant, this equation, on equating coe cients of

Legendre polynomials to terms correct to first order in m , yields

1
3 m ,

U 0 =− 1 − (5.21)

with α found to be

1

1
3

2 f
m

α =−

−

. (5.22)

The reference surface (5.19) is an equipotential of the gravity potential outside
the Earth, given by

1

r + α m

1

3 mr 2 P 2 (cosθ)

1 , U =−

r 3 P 2 (cosθ)

+ − (5.23)

correct to first order in m . Normally, ground-based gravimeters measure normal
gravity, or the gravity intensity,

∂ U
∂ r

2

r 2 ∂ U

2

1

g =|∇ U |= + , (5.24)

∂θ

with

∂ U
∂ r =

1
r 2 −

3 α m

2
3 mr ( P 2

r 4 P 2

+ − 1), (5.25)