Built on Facts

Spinning up the earth

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:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

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:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

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:

i-d18063683dcc0d42b9be45451a84d1e3-3.png

Slap those down in the expression for L:

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

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 1024kg, I get L = 1.4139 x 1034 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.

Comments

  1. #1 Michael Varney
    March 2, 2010

    Your hoping isn’t expected to alter much in terms of future earthquake occurrence. =)

  2. #2 Pierce R. Butler
    March 2, 2010

    The quake also shortened the day by 1.26 microseconds…

    The new T is 1 day + 1.26 microseconds.

    Varney’s day is shorter, Springer’s is longer (unless I bobbled the math, for which probability = ~1.0).

    Either way, aren’t astronomical measurements precise enough that these numbers could be checked instrumentally?

  3. #3 Pierce R. Butler
    March 2, 2010

    Damnitall twice, I bobbled the names – it’s Gross who’s going to have to move faster to get the same done in a way.

    Varney probably has a more fatalistic attitude, and won’t even change his expectations…

  4. #4 Matt Springer
    March 2, 2010

    You’re right, the plus sign should be a minus. I’ll fix it. The actual number stays the same, the minus sign of course indicates shrinkage.

    As far as detecting it with astronomical methods, it probably would be possible if there weren’t so many other much larger effects swamping this one. For instance, interaction with the atmosphere creates an effect more than a hundred times larger.

  5. #5 thomas grover
    March 2, 2010

    I love your calculation but what is important is the impulse the rate of change that occurred (momentum changed in a minute or two).
    My problem is that The earth’s axis is changing by 500 ft. because of the melting of the caps.(Science News).also the melting also causes a rebound of the land below(nasa).To make matters worse the spinning up of the polar vortex causes an equal and opposite reaction ,slowing down the planet (Scientific American ,Titan).
    I am Not forcasting doom but living on the Cascadia Subduction Zone I would appreciate it if we didn’t kick the beast.Did you ever push a top….it starts to wobble. Keep up the calculations.

  6. #6 meichenl
    March 3, 2010

    @matt: If you’re a bad button-pusher, don’t use buttons. Don’t use Earth’s mass, or coefficients in formulas, or units.

    1 microsecond is 10^-6 seconds.

    One hour is 3600 seconds; 24 of those is 100,000 = 10^5s. Therefore the length of the day changed by 1 part in 10^11.

    Moment of inertia goes with the square of the radius, so the radius changes fractionally by half as much as the day length, or 5*10^-12. [if this step is not obvious, for small x (1+x)^2 = 1+2x]

    The radius of Earth is 6,000km = 6*10^6m, so multiplying by the fractional change we have the change in the radius is 3*10^-5 m.

    @Pierce: If the probability you got a binary question wrong is near one, just choose the opposite answer and it’ll be near zero. Maximum entropy is at 50%, dude.

  7. #7 Greg Laden
    March 3, 2010

    What I want to know is this: How much did having all those figure skaters spinning in Vancouver alter the rate at which the Canadian Shield is moving, which would in turn affect all sorts of things like Maple Syrup production and so on.

  8. #8 Tim Gaede
    March 3, 2010

    Dams and melting ice caps can slow the Earth’s rotation.

  9. #9 Uncle Al
    March 3, 2010

    Going ’round the Ring of Fire… Alaska, Japan, Malaysia, Chile, and Haiti popped. That leaves two century-overdue ~8-magnitude temblors: California’s Coachella Valley and British Columbia’s Cascadia subduction zone.

    To ratio quake energies, subtract their respective magnitudes and raise 32 to that difference. Thus a 5 and an 8.8 differ by 32^3.8 or a factor of 500,000 in energy. Chile’s large cluster of 5 aftershocks sums to piddles.

    Reduce the sidereal day by about eight hours and the moon will be pushed outward in its orbit by spin-orbit coupling (tides). NASA will have an excuse for failing to Return to the Moon! All will be well with the world.

  10. #10 rob
    March 3, 2010

    a related interesting calculation is how will the length of the day change if we chopped down all the trees on earth.

  11. #11 Michael Varney
    March 3, 2010

    @ Pierce R. Butler

    Way to go, you screw up your references, then cover it up by saying another stupid thing about a person you don’t know. If you practice science at all like you post responses to blogs, you should quit and take up politics.

  12. #13 CCPhysicist
    March 3, 2010

    @7:
    Figure skaters have nothing on NASCAR. However, the effect of running CCW in NASCAR is balanced out somewhat by running CW in Formula 1.

  13. #14 thomas grover
    March 6, 2010

    Any time you guys are done making jokes ..
    What Gross has started is a good start on figuring out how this spaceship works.
    Things fall down not up ,so quakes are a minus T….however
    thermally driven events might not be, Volcanos for example..
    This is such a small magnitude event ( figure skater)that I do not believe they can measure it….
    However They have measured the effect (100x) of the polar vortex on the spin and where as this has no obvious effect on earthquakes…the rise of the land has.
    The math projection of the movement of ice to water on the spin rate assumes an even melting…
    It is not evenly distributed (melting)resulting in a shift of the axis 500ft ….not a figure skater …
    The good news is its not quick so no large impulse.

    In case u guys havent read, lots of people have died
    in earthquakes and the waves they formed..
    Ever seen an S wave coming at you …it is unique
    This IS a physics problem.

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