How do I calculate the moment of inertia of an idealized Earth's equatorial bulge?
I am working on calculating an idealized Sun-Earth-Moon three-body system. As part of this I want to calculate Earth's axial precession, which requires knowing the torque that is acting on it. Wikipedia gives this equation for the torque caused by a celestial body's gravity acting on the Earth: $$\vec{T} = \frac{3GM}{r^3} (C − A) \sin(δ) \cos(δ) \begin{pmatrix} \sin(α)\ −\cos(α)\ 0\ \end{pmatrix}$$
Where
* $GM$ is the standard gravitational parameter, the product of the gravitational constant $G$ and the mass $M$ of the perturbing body;
* $r$ is the distance between the center of the Earth and the center of the perturbing body;
* $C$ is the moment of inertia around Earth's axis of rotation;
* $A$ is the moment of inertia around any equatorial diameter of Earth;
* $(C − A)$ is the moment of inertia of Earth's equatorial bulge (C > A);
* $δ$ is the declination of the perturbing body (positive for north of the equator, negative for south of the equator); and
* $α$ is the right ascension of the perturbing body (east from vernal equinox)
I have all the numbers needed except for the moments of inertia around Earth's axis of rotation and around "any equatorial diameter of Earth". I don't know how to calculate these, or even what the difference between them is, and Googling hasn't been much help. 483
For the value of $C$ I found one source that says Earth's moment of inertia is $8.04 × 10^{37} \text{ kg m}^2$; and another that says the moment of inertia of an oblate spheroid around its shorter axis is $\frac{2}{5} M r^2$, where $r$ is the major radius, which with my numbers gives me $9.699 × 10^{37} \text{ kg m}^2$. These two numbers are at least the same order of magnitude, which makes me think I'm on the right track, but there's still a large error factor I'd like to eliminate which I'm assuming is related to Earth not being a point mass or of uniform density. That also still leaves me without any idea of what the value of $A$ represents and how it differs from $C$.
In my idealized system, the Earth is an oblate spheroid whose shape is given by the equation $(\frac{x}{6372})^2 + (\frac{y}{6372})^2 + (\frac{z}{6349.875})^2 = 1$, making the equator 22.125 km farther from the center of the Earth than the poles. For density I'm using the Preliminary Reference Earth Model (PREM, en.wikipedia.org/wiki/File:RadialDensityPREM.jpg) and assuming both that the mass is evenly distributed and that the distances in the PREM scale evenly with the radius at any given latitude. 15