Not So Oblious

Marika and I have been getting ready to head down to Florida this coming week, and while paying a farewell visit to her grandparents, Papou posed another of his fascinating questions: Given the Earth is an oblate spheroid (flattened at the poles and bulging in the middle), what direction does gravity point relative to the surface?

If we take a cross-section of the Earth through the poles, we get an ellipse, which is given by the equation

where a and b give the half-width and half-height of the ellipse respectively. We want to try varying these two values to see what happens to the gravity. If we assume a constant density, we can keep the total mass constant by fixing the area of the ellipse:

We can also consider the eccentricity of the ellipse, given by

Using these, we can rewrite a and b as

Newton's Law of Gravity says the force between two masses is

where m1 and m2 are the two masses, r is the distance between them, and G is Newton's constant. For our elliptical planet, we can add up little bits of mass throughout the volume to make a complicated integral:

where ρ is the mass density and A is the space covered by the ellipse. Initially, I thought I could avoid evaluating this nasty object by using Gauss's Law, which would suggest that the pull was always toward the center, but the ellipse doesn't offer the necessary symmetry. I decided to prove this to myself by throwing the integral at Python to solve for me:

The red line goes to the origin, and the black arrow points in the direction of gravity, as found from the integral above. You can see that for large eccentricities, the arrow begins to diverge. We can plot the angular difference:

We get up to around 12° difference, which suggests that an assumption of radial gravity won't cut it. Unfortunately, the numerical integration is both slow, and has inaccuracies for certain points. That leads to some uninformative plots:

Here, theta is the angle from the x-axis, and the horizontal gravity is in arbitrary units. The integrator has problems when it gets to the edges of the ellipse, which leads to some of the force arrows going nuts. If we disregard the deviation discussed above, and assume the gravity really does point toward the center, we can get a much smoother graph, which roughly matches the true solution:

As a check on this, it's often useful in Physics to consider the most extreme case (e.g. zero or infinite mass, length, energy). For this system, we could imagine a planet that was flattened into a disk. The mass ends up concentrated in the center, tapering off toward the edges. If you stand in the center, gravity pulls straight down, but moving away, it would begin to pull back toward the center. If you were standing on the edge of the disk, the center would be straight down, so there's no horizontal force. This leads to the 4 zeros on the plot above: the centers of the top and bottom, and the right and left edges. Thanks for another great question, Papou!