[part 3 of 3]

In part 2, we showed how if the distant point mass *M* orbits our oblate spheroid (or our ellipsoid orbits it) in a circular orbit, with an orbital frequency *Ω* much faster than the torque produced, one may time-average the torque to get a result where the rotation axis of our ellipsoid, the *z’* axis of the body frame, maintains the same angle with, and preccesses about, the axis perpendicular to the plane of the orbit, the *z* axis of the inertial frame. In particular, we found that

, where is the angular frequency of the precession, *ω _{z}‘* is the angular velocity of the ellipsoid’s rotation,

*θ*

_{0}is the angle between

*z’*and

*z*, and

*d*is the distance between our ellipsoid and the mass

*M*.

So far we’ve kept all this in general, abstract terms. Now we will move this to a concrete example: the shape of the planet Earth may be approximated as an oblate spheroid of equatorial radius

*a*=6378.1 km and polar radius

*c*=6356.8. More specifically, if we use either pair of values given here for the moment of inertia of the earth, we get a value for of about 0.00323.

The other major parameters for the Earth are

*θ*

_{0}and

*ω*. The tilt of the earth’s axis,

_{z}‘*θ*

_{0}, is approximately 23.44°=0.4091 radians, so that cos

*θ*

_{0}=0.9175. The angular frequency of the earth’s rotation about its axis,

*ω*is found to be 7.2921×10

_{z}‘^{-5}s

^{-1}, using as rotational period one sidereal day.

Now, we consider first the effect of the sun. The mass of the sun is 1.99892×10

^{30}kg, so

*GM*for the sun is 1.3272×10

^{20}m

^{3}s

^{-2}. The average distance between the earth and the sun is

*d*=1.496×10

^{11}m. Plugging these into our precession formula,

gives a precession frequency of about 2.417×10

^{-12}s

^{-1}, which corresponds to a period of 2.600×10

^{12}s, or about 82,000 years.

Now, for the moon. The mass of the moon is is 7.3477×10

^{22}kg, so

*GM*for the moon is 4.903×10

^{12}m

^{3}s

^{-2}. The average earth-moon distance is 3.844×10

^{8}m. Ignoring the tilt of the moon’s orbit relative to the ecliptic, we can use our precession formula as before, giving a precession frequency of about 5.262×10

^{-12}s

^{-1}, which corresponds to a period of 1.194×10

^{12}s, or about 38,000 years.

As noted before, we have made many simplifying assumptions and approximations; the actual rate of the precession of Earth’s axis, historically called the “precession of the equinoxes,” has a period of about 25,700 years. Further, there are added complications, such as nutation, due to factors we have ignored (such as eccentricity of the orbits of the moon and Earth, the tilt of the moon’s orbit relative to the ecliptic, the time-varying nature of the torque, and so on). See here for a deeper treatment.