Simple application of Darwin-Radau theory

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

$$\rho(a) =\rho_0\,a^{-\alpha},$$ (D.54)

for $0\leq a\leq R$, where $0\leq \alpha < 3$. Of course, $\rho(a)=0$ for $a>R$. According to Equations (D.26) and (D.42),

$$\eta(a)=\eta_0,$$ (D.55)

where

$$\eta_0 =\frac{\alpha\,(10-\alpha)}{(5-\alpha)^2}.$$ (D.56)

Thus, Equation (D.33) yields

$$\epsilon(a)= \epsilon_0\left(\frac{a}{R}\right)^{\eta_0}.$$ (D.57)

Finally, it follows from Equations (D.48), (D.49), and (D.51) that

$${\cal C} = \frac{2}{3}\left(\frac{3-\alpha}{5-\alpha}\right),$$ (D.58)

$$\epsilon_0 = \frac{15}{2}\left[\frac{1}{1+25/(5-\alpha)^2}\right]\zeta,$$ (D.59)

$$J_2 = \left[\frac{4-25/(5-\alpha)^2}{1+25/(5-\alpha)^2}\right]\zeta,$$ (D.60)

respectively.

For the case of the Earth (Yoder 1995),

$$\zeta =1.15\times 10^{-3},$$ (D.61)

$$\epsilon_0 = 3.35\times 10^{-3}.$$ (D.62)

These values are consistent with formula (D.59) provided $\alpha=1.02$, which implies that $\eta_0=0.58$. Equations (D.58) and (D.60) then give

$${\cal C} =0.332,$$ (D.63)

$$J_2 = 1.08\times 10^{-3},$$ (D.64)

respectively. It turns out that these values for ${\cal C}$ and $J_2$ are fairly accurate [the true values are ${\cal C} = 0.331$ and $J_2=1.08\times 10^{-3}$ (Yoder 1995)]. This suggests that the response of the Earth to its centrifugal potential is essentially fluid-like, and, also, that the Earth's core is significantly denser than its crust.

For the case of Jupiter (Yoder 1995),

$$\zeta =2.70\times 10^{-2},$$ (D.65)

$$\epsilon_0 = 6.49\times 10^{-2}.$$ (D.66)

These values are consistent with formula (D.59) provided $\alpha=1.57$, which implies that $\eta_0=1.12$. Equations (D.58) and (D.60) then give

$${\cal C} =0.28,$$ (D.67)

$$J_2 = 1.64\times 10^{-2},$$ (D.68)

respectively. These values for ${\cal C}$ and $J_2$ are somewhat inaccurate [the true values are ${\cal C} = 0.25$ and $J_2=1.47\times 10^{-2}$ (Yoder 1995).] One possible reason for the disagrement is the fact that the values of $\zeta$ and $\epsilon_0$ 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 $\alpha$ for Jupiter is significantly greater than $\alpha$ for the Earth suggests that the Jupiter's mass distribution is much more centrally condensed than the Earth's.