There’s an interesting science puff piece that’s been circulating around various media outlets about the length of the day after the earthquake in Chile. At random, here’s the NY Daily News version:

The quake that rocked the South American nation may have also knocked the Earth off its axis.

The 8.8-magnitude earthquake near Chile may have also made our planet’s days shorter, according to NASA scientist Richard Gross.

A minor change in the Earth’s axis isn’t expected to alter much in terms of weather. The planet’s tilt influences the seasons, allowing for winter, spring, summer and fall, and it would take a far greater change in the Earth’s axis to affect them.

The Chile quake may have moved the Earth’s axis by about 3 inches, Gross said.

The quake also shortened the day by 1.26 microseconds, the scientist determined, using a complex model he and others developed.

It’s not a bad article, and in fact I really don’t see much to quibble with except maybe for using “isn’t expected to” instead of “won’t”. It’s sort of like saying continental drift isn’t expected to alter the time of London to New York flights. Various other publications’s versions of the story go into more detail, like Bloomberg’s version which correctly notes that the axis in question isn’t quite the rotation axis but “axis about which the Earth’s mass is balanced”, which is almost exactly but not quite the same thing.

The article goes on to make a comparison with figure skaters. As a spinning skater pulls her arms in, she rotates faster. While the shifting of the crust here isn’t nearly so symmetric, it’s the same concept. It would be interesting to do a rough calculation with these numbers to see how much the earth’s “arms” – ie, its diameter – were pulled in.

To begin with, we write down the angular momentum of the earth. It’s:

Where I is the moment of inertia and omega is the angular frequency of the earth’s rotation – ie, one per day, with a factor of two pi multiplied in for reasons that don’t matter in this context. We can pretend that the earth is a perfect uniform sphere and write down the expression for I in terms of the mass and radius:

While we’re at it, we might as well write down omega in terms of the period of rotation, which is of course T = 1 day:

Slap those down in the expression for L:

All right, from there we can actually get a number for the angular momentum of the earth. Plugging in T = 1 day and m = 5.9742 x 10^{24}kg, I get L = 1.4139 x 10^{34} in the rather unwieldy units of m^2 kg/s.

But angular momentum is conserved, so it’s the same both before and after the quake. If T goes up, therefore r has got to go down. The new T is 1 day – 1.26 microseconds. Solve for r, plug in the new T, and I get that r changes by 46.5 microns. That’s roughly the diameter of a human hair, though of course I’m a pretty bad button puncher so you ought to recheck my numbers to be sure.

The actual shift of the local geography involved was much, much larger. This tiny number represents sort of an average over the whole globe, and that only with approximations of limited validity.

Now here’s hoping there’s no more earthquakes to do calculations on.