Combining Equations (D.2) and (D.3) with the previous three equations, we deduce that, to first order in
, the total potential (i.e., the sum of the gravitational
and centrifugal potentials) can be written
 |
(D.22) |
where
Here, we have assumed that
. However, according to Equation (D.7), if the rotating body is in hydrostatic equilibrium (in the
co-rotating frame) then
is a function of
only. In other words,
, which implies that
![$\displaystyle a^{\,2}\,\epsilon(a)\int_0^a \rho(a')\,a'^{\,2}\,da' -\frac{1}{5}...
...^\infty \rho(a')\,d[\epsilon(a')]
=\frac{{\mit\Omega}^{\,2}\,a^{\,5}}{8\pi\,G}.$](img4357.png) |
(D.24) |
Differentiation with respect to
yields
 |
(D.25) |
where
 |
(D.26) |
is the mean density inside the spheroidal surface
. Note that
 |
(D.27) |
Finally, differentiation of Equation (D.25) with respect to
gives
 |
(D.28) |
This differential equation was first obtained by Clairaut in 1743 (Cook 1980).
Suppose that the outer boundary of the rotating body corresponds to
, where
is the body's mean radius. [In other words,
for
.] It follows that the
total mass of the body is
 |
(D.29) |
The dimensionless parameter
, introduced in Section 6.5, is the typical ratio of the centrifugal acceleration to the gravitational
acceleration at
, and takes the form
 |
(D.30) |
Thus, it follows from Equation (D.25) that
 |
(D.31) |
Now, at an extremum of
, we have
. At such a point, Equation (D.28) yields
 |
(D.32) |
However, if
is a monotonically decreasing function of
, as we would generally expect to be the case, then Equation (D.27)
reveals that
. Hence, the previous equation implies that, at the extremum,
has the same sign as
. In other words, the extremum is a minimum of
. This implies that it is impossible to have a maximum of
. Now, a Taylor expansion of Equation (D.27) about
, assuming that
, where
, reveals that
is an increasing function at small
. We, thus, deduce that
is
a monotonically increasing function. This implies that
has the same sign as
. Hence, Equation (D.31) reveals that
is everywhere positive. In other words, if
is a monotonically decreasing function then
is necessarily a positive, monotonically
increasing function. Thus, we deduce that all density contours in the body are oblate spheroids.