In a co-rotating reference frame, the shape of a self-gravitating, rotating, liquid planet is determined by a competition between fluid pressure, gravity, and the fictitious centrifugal force. The latter force opposes gravity in the plane perpendicular to the axis of rotation. Of course, in the absence of rotation, the planet would be spherical. Thus, we would expect rotation to cause the planet to expand in the plane perpendicular to the rotation axis, and to contract along the rotation axis (in order to conserve volume).

For the sake of simplicity, we shall restrict our investigation to a rotating planet of uniform density whose outer
boundary is ellipsoidal. An *ellipsoid* is the three-dimensional generalization of an ellipse. Let us adopt the right-handed Cartesian
coordinate system
,
,
.
An ellipse whose principal axes are aligned along the
- and
-axes satisfies

(2.94) |

where and are the corresponding principal radii. Moreover, as is easily demonstrated,

where is the area, an element of , and the integrals are taken over the whole interior of the ellipse. Likewise, an ellipsoid whose principal axes are aligned along the -, -, and -axes satisfies

(2.98) |

where , , and are the corresponding principal radii. Moreover, as is easily demonstrated,

where is the volume, an element of , and the integrals are taken over the whole interior of the ellipsoid.

Suppose that the planet is rotating uniformly about the -axis at the fixed angular velocity . The planet's moment of inertia about this axis is [cf., Equation (2.100)]

(2.102) |

where is its mass. Thus, the planet's angular momentum is

(2.103) |

and its rotational kinetic energy becomes

(2.104) |

According to Equations (2.83) and (2.84), the fluid pressure distribution within the planet takes the form

(2.105) |

where is the gravitational potential (i.e., the gravitational potential energy of a unit test mass) due to the planet, the uniform planetary mass density, and a constant. However, it is demonstrated in Appendix D that the gravitational potential inside a homogeneous self-gravitating ellipsoidal body can be written (Chandrasekhar 1969; Lamb 1993)

(2.106) |

where is the universal gravitational constant (Yoder 1995), and

(2.107) | ||

(2.108) | ||

(2.109) |

Thus, we obtain

(2.110) |

where is the central fluid pressure. The pressure at the planet's outer boundary must be zero, otherwise there would be a force imbalance across the boundary. In other words, we require

(2.111) |

whenever

(2.112) |

The previous two equations can only be simultaneously satisfied if

(2.113) |

Rearranging the previous expression, we obtain

subject to the constraint

where use has been made of Equation (2.99).

Finally, according to Appendix D, the net gravitational potential energy of the planet is

Hence, the body's total mechanical energy becomes

(2.117) |