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.

Why π is irrational (part 5 of 5)

5 August 2010

In this post we investigate the size of the integral. In particular, we look at how we may tweak parameters to arbitrarily lower its magnitude.
Read the rest of this entry »

Why π is irrational (part 4 of 5)

20 July 2010

In this post I show that the integral is an integer.
Read the rest of this entry »

Massive prime numbers

19 July 2010

I participate in PrimeGrid, a distributed computing project to find large prime numbers of various types. It seems that my computer just found such a massive prime–a Proth prime, to be precise. Proth primes have the form

where k and n are integers. Mine was

which, when written out in full base-10 notation, is 157897 digits and can be found beyond the “read the rest of this entry” tag (warning: this particular entry will load slowly in its entirety).
Read the rest of this entry »

I’m back!

17 July 2010

After three weeks of chemistry, humidity, and general nerdiness at CTY Lancaster, I must say it’s great to be home!

Temporary absence

27 June 2010

I will be “out” for the next three weeks – until 16 July – at a nerd camp in Pennsylvania. We won’t be able to access computers there, so until then, don’t expect anything new to come up.

Clustrmaps

21 June 2010

Now I can see where you all are!
Locations of visitors to this page

To keep this current, I’ll periodically move this post up to the top.

Why π is irrational (part 3 of 5)

15 June 2010

In this post I use the derivatives of f(x) to create a new function, F(x), that will help us evaluate the integral from part 1.
Read the rest of this entry »

Earthquake!

15 June 2010

We just had a 5.7 earthquake in my area; its epicenter was on the same general trend of the Easter Sunday earthquake earlier this year. Although this one was 30 times weaker than the Easter quake and not much closer, I felt it much more strongly. Strange.
I felt this earthquake in what could be called “textbook fashion.” Any geology textbook will tell you that there are two main shocks in an earthquake: a sharp initial one followed by a more gradual shaking. The separation between the two shocks grows with distance from the epicenter due to different speeds with which the waves travel through the ground. In all the earthquakes I had felt so far, I had not been able to distinguish between the two shocks. This time, I felt the difference very clearly.
Anyway, detailed info can, of course, be found at the USGS website if you wish to pursue this further. The Wikipedia article for this quake appears to be here.

Why π is irrational (Part 2 of 5)

8 June 2010

In this post I introduce the function f(x) and look at some of its properties.
Read the rest of this entry »