Simple application of Darwin-Radau theory

(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),

(D.61) | ||||||

and | (D.62) |

These values are consistent with formula (D.59) provided , which implies that . Equations (D.58) and (D.60) then give

(D.63) | ||||||

and | (D.64) |

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 fluid-like, and, also, that the Earth's core is significantly denser than its crust.

For the case of Jupiter (Yoder 1995),

(D.65) | ||||||

and | (D.66) |

These values are consistent with formula (D.59) provided , which implies that . Equations (D.58) and (D.60) then give

(D.67) | ||||||

and | (D.68) |

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.