Archive for November, 2010

The Equatorial Bulge

5 November 2010

A quarter-section of a cross-section of Earth. Not to scale.

In the rotating frame of Earth, the centrifugal pseudoforce generates its own pseudopotential:

\displaystyle U_c = -\int_0^r m\frac{v^2}{r} dr = -\displaystyle\int_0^rmr\omega^2dr = -\frac{1}{2}mr^2\omega^2,

where r is the distance to Earth’s axis, not the distance to the center. The gravitational potential energy of a particle at a distance r from the axis and height h above the plane of the equator is given by

\displaystyle U_g \approx -\frac{GMm}{\sqrt{r^2+h^2}}.

The exact formula for an oblate spheroid like Earth is very complicated and differs, in the case of Earth, only in decimal places beyond what we are concerned with.

A ball placed on the surface of a spinning planet in hydrostatic equilibrium does not roll; thus, the potential energy–including the centrifugal pseudopotential–in the rotating frame must be constant. In particular, the potential energy at the pole must equal that at the equator:

\displaystyle U_p = U_e \hspace{0.5cm} \Longrightarrow \hspace{0.5cm} -\frac{1}{2}m0^2\omega^2-\frac{GMm}{r_p} = -\frac{1}{2}mr_e^2\omega^2-\frac{GMm}{r_e}

We note that we can rewrite the mass of Earth in terms of its density and volume. The volume of an oblate spheroid is \frac{4}{3}\pi r_e^2r_p, so

\displaystyle \frac{G\left(\rho\frac{4}{3}\pi r_e^2r_p\right)}{r_p} = \frac{1}{2}r_e^2\omega^2 + \frac{G\left(\rho\frac{4}{3}\pi r_e^2r_p\right)}{r_e}.

Note that we have taken the opportunity to clear the signs and divide out the mass of the ball. To simplify life somewhat, we define a quantity \epsilon such that \epsilon = G\rho\frac{4}{3}\pi:

\displaystyle \epsilon r_e = r_e\frac{\omega^2}{2} + \epsilon r_p.

Rearranging this, we find that

\displaystyle \frac{r_p}{r_e} = 1-\frac{\omega^2}{2\epsilon}.

Earth’s density is about 5515 \frac{kg}{m^3} and rotates once per day, giving it an angular frequency of \frac{2\pi}{86400} H\!z. Thus,

\frac{r_p}{r_e} \approx 0.998285.

The equatorial radius is 6378.1 km; the calculated polar radius is therefore 6367.2 km. This gives us an equatorial bulge of re-rp = 10.9 km. The polar radius is actually 6358.9 km, making the real equatorial bulge 19.2 km.