[Part 1 of ?]

Let us consider a rotating oblate spheroid with equatorial radius *a* and polar semi-axis *c*. We choose an inertial frame (*x*,*y*,*z*) and body coordinate system (*x’*,*y’*,*z’*), both with origin at the ellipsoid’s center of mass. Suppose we have a rotation about a principal axis with angular momentum , where *I _{z’}* is the moment of inertia along the

*z’*principal axis and is the

*z’*unit vector).

Now, let us put a point mass

*M*at a displacement

**d**from the center of our ellipsoid. Then a mass element of our ellipsoid at position

**r**( is the density at that point in the object, not necessarily uniform). experiences a gravitational force due to the mass

*M*of

Integrating the torque element due to this force over the volume

*V*of the ellipsoid gives a total torque of

.

Now, suppose our point mass is far from our body, compared to its size, so that , where and . Now using

, we see that approximating to first order in *r*/*d*,

, so we can in this situation approximate our torque by:

.

Now, to find the first integral on the right, we note that

.

For a an object in volume*V* of total mass *m* and density , the center of mass has position (vector)

(see here); as our origin is the center of mass, we thus see that the integral , and the first integral in our approximation is the zero vector, and

.

Consider the moment of inertia tensior **I** of our object (see here):

, where **E**_{3} is the 3×3 identity matrix and is the outer product of **r** with itself. Applying the tensor to **d**, we use and to get

.

Next, consider the cross product of **d** and this vector:

,

which is our remaining integral in the torque approximation:

.