Computing the shape of planet Earth

It is often noted that the Earth is slightly oblate rather than spherical, but the calculation of the Earth's eccentricity can be challenging. Here, we calculate it by minimizing the sum of the Earth's gravitational potential energy and its centrifugal potential energy. The Earth's gravitational potential energy can be derived with the help of the Green's function of the Laplace operator in oblate spheroidal coordinates. Under the assumption of a homogeneous planet, we obtain an analytic relationship for the Earth's eccentricity that was first derived by Maclaurin in 1742 and is about 13 percent larger than the observed value. Better agreement with observation is obtained by assuming that the Earth's core is about twice the density of the mantle, which reduces the Earth's moment of inertia. This exercise can provide practice in analyzing gravitational systems in spheroidal coordinates; the problem may also be relevant for other gravitating and rotating bodies, such as stars and galaxies.