[Part 2 of ?]

In the previous part, we introduced a spinning oblate spheroid, and showed that a distant point mass *M* at displacement **d** will exert a torque on the spheroid that can be approximated as

. Now, suppose that **d** is in the *xy*-plane of the inertial frame. Further, suppose that our point mass is orbiting our spheroid in a circular orbit of angular frequency *Ω* (or that our spheroid is orbiting our point mass in a circular orbit of angular frequency *Ω*; the model will turn out the same). Then in the inertial frame. Suppose then at a time *t* the *x* and *x’* axes coincide. Then, if we let the angle between *z* and *z’* be *θ*_{0}, we see that **d** has components in the body coordinates of .

Now, if the density of our spheroid is sufficiently symmetric about its axis, then we will have *I _{x’}*=

*I*, and the moment of inertia tensor in the body coordinates will be . Using our results from part one, we find the torque in this situation;

_{y’},

.

Now, we presently have and , so we can rewrite the above in a way independent or our choice of

*x’*and

*y’*axes:

,

giving torque:

Supposing that this torque is small enough that any precession produced is of frequency much slower than

*Ω*, we can then average the torque over time; recalling the time average of trigonometric functions and their products, we get average torque

Recalling that our object has angular momentum along the

*z’*axis, we see then that our average torque is perpendicular to our angular momentum, and to our

*z*axis (as it is along the cross product ). Thus, as in here, we have precession of our spheroid’s rotation about the

*z*axis (so

*θ*

_{0}is constant), and from our previous work on torque-driven precession, we see that the precession has angular frequency

.

(The negative sign indicates that the direction here is opposite the sense of the rotation

*ω*.)

_{z}‘Now, supposing our spheroid has total mass

*M*, then Kepler’s third law for our spheroid-point mass orbit tells us that the period

_{E}*T*of the orbit is . Since , we find , which lets us rewrite the precession frequency in terms of the orbital frequency and the ratio of the masses :

.