Rotational Flattening

Let us transform to a non-inertial frame of reference which co-rotates with the spheroid about the -axis, and in which the spheroid consequently appears to be stationary. From Chapter 7,
the problem is now analogous to that of a non-rotating spheroid, except that
the surface acceleration is written
,
where
is the gravitational acceleration, and
the *centrifugal* acceleration. The latter acceleration
is of magnitude
, and is everywhere directed
away from the axis of rotation (see Figure 40 and Chapter 7).
The acceleration thus has components

(912) |

in spherical polar coordinates. It follows that , where

can be thought of as a sort of centrifugal potential. Hence, the total surface acceleration is

(914) |

As before, the criterion for an equilibrium state is that the surface lie at
a constant total potential, so as to eliminate tangential surface forces which
cannot be balanced by internal pressure. Hence, assuming that the
surface satisfies Equation (901), the equilibrium configuration is specified by

(915) |

(916) |

(917) |

We conclude, from the above expression, that the equilibrium configuration
of a (relatively slowly) rotating self-gravitating mass distribution is an *oblate spheroid*: *i.e.*, a sphere
which is slightly flattened along its axis of rotation. The degree of flattening is proportional
to the square of the rotation rate. Now, from (901), the mean radius
of the spheroid is
, the radius at the poles (*i.e.*, along the axis of rotation) is
, and the radius at the
equator (*i.e.*, perpendicular to the axis of rotation) is
--see Figure 40. Hence, the degree of rotational flattening
can be written

(918) |

Now, for the Earth,
,
, and
.
Thus, we predict that

(919) |

(920) |

For Jupiter,
,
, and
. Hence,
we predict that

(921) |