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Next: Free precession of Earth Up: Darwin-Radau equation Previous: Derivation of Darwin-Radau equation


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

$\displaystyle \rho(a) =\rho_0\,a^{-\alpha},$ (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),

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

where

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

Thus, Equation (D.33) yields

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

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

    $\displaystyle {\cal C}$ $\displaystyle = \frac{2}{3}\left(\frac{3-\alpha}{5-\alpha}\right),$ (D.58)
    $\displaystyle \epsilon_0$ $\displaystyle = \frac{15}{2}\left[\frac{1}{1+25/(5-\alpha)^2}\right]\zeta,$ (D.59)
and   $\displaystyle J_2$ $\displaystyle = \left[\frac{4-25/(5-\alpha)^2}{1+25/(5-\alpha)^2}\right]\zeta,$ (D.60)

respectively.

For the case of the Earth (Yoder 1995),

    $\displaystyle \zeta$ $\displaystyle =1.15\times 10^{-3},$ (D.61)
and   $\displaystyle \epsilon_0$ $\displaystyle = 3.35\times 10^{-3}.$     (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

    $\displaystyle {\cal C}$ $\displaystyle =0.332,$ (D.63)
and   $\displaystyle J_2$ $\displaystyle = 1.08\times 10^{-3},$     (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),

    $\displaystyle \zeta$ $\displaystyle =2.70\times 10^{-2},$ (D.65)
and   $\displaystyle \epsilon_0$ $\displaystyle = 6.49\times 10^{-2}.$     (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

    $\displaystyle {\cal C}$ $\displaystyle =0.28,$ (D.67)
and   $\displaystyle J_2$ $\displaystyle = 1.64\times 10^{-2},$     (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.
next up previous
Next: Free precession of Earth Up: Darwin-Radau equation Previous: Derivation of Darwin-Radau equation
Richard Fitzpatrick 2016-03-31