Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 341

Tabl e 5 . 2 Comparison of gravity formulae.

cient of sin 2 cient of sin 2 2φ

g e (m s − 2 )

Data sets Coe φ Coe

10 − 6 10 − 6

Rapp (1974) 9.780340 5,301.1 × − 5.8 ×

10 − 6 10 − 6 G.R.S.1980 9.780326 5,302.5 × − 5.8 ×

10 − 6 10 − 6 G.R.S.1967 9.780310 5,302.4 × − 5.8 ×

10 − 6 10 − 6 1924 / 30 formula 9.780490 5,288.4 × − 5.9 ×

These allow construction of gravity formulae from the three basic data sets given
in Table 5.1. Table 5.2 shows the three resulting formulae in contrast with the
International Gravity Formula adopted in 1924 and 1930.
The level reference surface described in this section is not, of course, the geoid.
In ground-based geodesy, the geoidal surface is referred to mean sea level. Satellite
geodesy uses as geoid the equipotential surface with the same gravity potential
value as the reference surface described here. Displacement of the satellite geoid
from the reference surface is then easily calculated using the satellite determined
gravitational coe cients, permitting a map to be made of geoidal undulations. For

geophysical purposes, though, it is more useful to display the geoidal undulations
with respect to the equilibrium equipotential surface, considered in the next section.

5.3 Equilibrium theory of the internal figure
There are many reasons why the internal figure of the Earth should depart from
fluid equilibrium form. The most obvious is that the solid parts of the Earth are
capable of sustaining non-hydrostatic stress for long periods of time. It is unlikely
that they are capable of sustaining such stresses indefinitely, but dynamical pro-
cesses could easily be acting to continually renew the disequilibrium. Similarly,
flow in the fluid parts of the Earth prevents these more mobile regions from achiev-
ing equilibrium. At a more subtle level, it can be shown theoretically that, even
in the absence of flow, a rotating fluid body can have no static equilibrium unless
it has a particular distribution of heat sources unlikely to be met exactly in the
Earth. Nonetheless, it is important to determine just how far the Earth does depart
from equilibrium form, for any theory of its static or dynamic behaviour must be
in accord with the observed departure.
If the adjustment to fluid equilibrium is complete, the pressure p is related to the
density and the geopotential by

∇ p =− ρ ∇ U . (5.104)

In general, the density is related to the pressure and temperature T by an equation
of state,

ρ = ρ( p , T ). (5.105)