Simple application of DarwinRadau theory
As a simple application of the preceding analysis, let us assume that the density distribution inside the rotating body is such that

(D.54) 
for
, where
. Of course, for . According to Equations (D.26) and (D.42),

(D.55) 
where

(D.56) 
Thus, Equation (D.33) yields

(D.57) 
Finally, it follows from Equations (D.48), (D.49), and (D.51) that
respectively.
For the case of the Earth (Yoder 1995),
These values are consistent with formula (D.59) provided
, which implies that
. Equations (D.58) and (D.60) then give
respectively. It turns out that these values for and are fairly accurate [the true values are
and
(Yoder 1995)]. This suggests that the response of the Earth to its centrifugal potential is essentially fluidlike, and, also, that the Earth's core is significantly denser than
its crust.
For the case of Jupiter (Yoder 1995),
These values are consistent with formula (D.59) provided
, which implies that
. Equations (D.58) and (D.60) then give
respectively. These values for and are somewhat inaccurate [the true values are
and
(Yoder 1995).] One possible reason for the disagrement is the fact that the values of and
are
sufficiently large for Jupiter that it is not a good approximation to neglect terms that are second order in these quantities in the analysis (Cook 1980).
However, the fact that for Jupiter is significantly greater than for the Earth suggests that the Jupiter's mass distribution is much more
centrally condensed than the Earth's.