Earthquakes and Earth’s rotation

 Updates: 1. Every article other than the one I linked to says the change in the length of the day is in microseconds, not milliseconds.  Much more plausible.  2.  The Onion’s on the case.

Ashley pointed out this article on the Chile earthquake’s effect on Earth’s rotation.

The massive 8.8 earthquake that struck Chile may have changed the entire Earth’s rotation and shortened the length of days on our planet, a NASA scientist said Monday.

The quake, the seventh strongest earthquake in recorded history, hit Chile Saturday and should have shortened the length of an Earth day by 1.26 milliseconds, according to research scientist Richard Gross at NASA’s Jet Propulsion Laboratory in Pasadena, Calif.

“Perhaps more impressive is how much the quake shifted Earth’s axis,” NASA officials said in a Monday update.

The change in the length of the day is good first-year physics stuff.  Angular momentum is conserved, and is equal to moment of inertia times rotation rate.  The moment of inertia of a body depends on how its mass is distributed.  If you change the distribution you change the moment of inertia, and the rotation rate has to change to compensate.  Think of the standard-issue spinning figure skater pulling in his arms, or diver going into a tuck position, and starting to rotate faster. I’m a bit surprised the change is as large as this, but I guess it’s possible.

Here’s an embarrassing confession. I can’t make sense of this:

The Earth’s figure axis is not the same as its north-south axis, which it spins around once every day at a speed of about 1,000 mph (1,604 kph).

The figure axis is the axis around which the Earth’s mass is balanced. It is offset from the Earth’s north-south axis by about 33 feet (10 meters).

I don’t think I know what “figure axis” means in this context.  The Earth at any instant has an axis about which it’s rotating, and that axis will always pass through the center of mass, which is my best guess at the meaning of the phrase “around which the Earth’s mass is balanced.”  But is that the figure axis or the north-south axis?  What’s the difference between the two?  (North-south axis could in principle be defined by the magnetic field, but that would be different by much more than 10 meter, so it’s not that.)

There’s one other thing I don’t understand:

Over the course of a year, the length of a day normally changes gradually by one millisecond. It increases in the winter, when the Earth rotates more slowly, and decreases in the summer, Gross has said in the past.

Why would Earth’s rotation vary over the course of a year?  I can think of two possibilities:

Possibility 1. Annual changes in wind speed and/or direction.  The total angular momentum of Earth-plus-atmosphere is what’s conserved, so when the wind is blowing west to east, the Earth will rotate slower than when it’s blowing east to west.  Do winds blow more west to east in the (northern-hemisphere) winter?  Paging my brother Andy for the answer to this.

Possibility 2. The article’s made a mistake.  It’s not that the rotation rate changes, but rather that the Earth’s orbital speed around the Sun changes.  If the rotation rate is fixed, then the length of a sidereal day (a day measured relative to the stars)  remains the same.  But a solar day (measured relative to the Sun, of course) is a bit longer than a sidereal day, and the difference depends on the orbital speed.  In the (northern-hemisphere) winter, the orbital speed is faster, which means that the length of a solar day is longer, and vice versa in the summer.  So that effect has the right sign to be what Gross is talking about.  But it’s much too large an effect: I think it’s a few seconds, not milliseconds.

After the jump, I’ll try a back-of-the-envelope calculation to see if Possibility 2 makes sense.

Take an extreme, exaggerated case, where the entire atmosphere is blowing from east to west in the summer and west to east six months later.  How fast would the wind have to be blowing (or rather how much of a shift in speed is required) for this to cause a 1-ms change in the Earth’s rotation?

The Earth’s moment of inertia is something like MR2 (actually, it’s more like .4 times this, but close enough), where M is the mass of the Earth and R is its radius.  The atmosphere’s contribution to this quantity is mR2, where m is the mass of the atmosphere.  The atmosphere weighs about 5 x 1018 kg, which is about 10-6 times the mass of the Earth.  We want to cause a change in the Earth’s rotation rate of 1 ms/day, which is 10-8.  So the atmosphere would need to change its motion by only about 1% of the Earth’s rotation rate.  That is, if the wind speed changed by an amount equivalent to  1 circuit around the earth per 100 days (10 mph), that’d do it.

Of course, the whole atmosphere doesn’t shift directions, so the part that (hypothetically) does would have to shift by correspondingly more.  Considering that probably a smallish fraction of the atmosphere is involved, that sounds like a lot to ask, but I guess it’s not obviously outrageous.

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Ted Bunn

I am chair of the physics department at the University of Richmond. In addition to teaching a variety of undergraduate physics courses, I work on a variety of research projects in cosmology, the study of the origin, structure, and evolution of the Universe. University of Richmond undergraduates are involved in all aspects of this research. If you want to know more about my research, ask me!

12 thoughts on “Earthquakes and Earth’s rotation”

  1. “The Earth at any instant has an axis about which it's rotating, and that axis will always pass through the center of mass, which is my best guess at the meaning of the phrase "around which the Earth's mass is balanced." But is that the figure axis or the north-south axis? What's the difference between the two?”

    I’m guessing now, but here goes: Imagine something which is NOT rotating around its centre of mass. Obviously, this is possible if there are additional forces involved. What about the gravity of the Moon and Sun? Presumably, these could cause the axis of rotation to be offset from the “mass axis” defined as the axis around which the Earth would spin in the absence of external forces.

    Possibility 3 is correct, and is covered in the course Radio Astronomy 101: Since there is more land in the northern hemisphere, there are more trees, and the leaves increase the moment of inertia, slowing down the Earth in the spring and summer, and conversely speeding it up in the winter. This is the reverse of what is quoted above; maybe THAT’S a mistake (increasing speed is decreasing day length and vice versa). By the way, why is this important for radio astronomy, as opposed to other types of astronomy? Because it has to be taken into account in VLBI, along with continental drift.

  2. Figure axis is the Earth’s center of gravity – the media reports tended to omit “figure.” I think the seasonal difference is due to the mass of water that sits on top of the planet during boreal (N Hemispheric) winter. I can find some numbers for it but the shift in the hydrosphere’s mass over the year is much greater vs the shift in the atmosphere’s mass over the year. My guess.

  3. Still confused about the figure axis. When they say that the figure axis differs from the north-south axis, what do they mean by the latter? Does that not go through the center of mass? Why not?

    I’m kicking myself for not realizing that it’s the water, not the air, that moves around seasonally enough to change the rotation rate. The hydrosphere has 1000 times more mass than the atmosphere (I’m actually surprised it’s not more than that), so you don’t have to move it around nearly as much to get the same effect.

  4. The “figure axis” is just the principal axis of inertia, which does not quite agree with the rotation axis. In fact, the rotation axis precesses around the principal axis of inertia, as predicted by Euler’s eqns for the free rigid body. This is usually called the Chandler precession:

  5. Umm – what’s the margin for error in calculations involving milli or microseconds?

    Have they considered there’s been no shift or change at all but it’s due to their methodology?

  6. The short answer is that measurement accuracy’s fine for this sort of thing. Timekeeping is phenomenally precise these days. Atomic clocks are accurate to something like nanoseconds over the course of a day, I think.

    I presume the hard part of Earth-rotation timing is position measurement, rather than time. If you wanted to measure the length of any one particular day to microsecond precision, you’d need to be able to tell when the Earth had undergone precisely 1.00000000000 rotations. If I’ve done the arithmetic right, that would mean measuring Earth’s orientation to an accuracy of something like 10 micro-arc-seconds. (An arc-second is 1/3600 of a degree.) You can probably do that with the technique of very long baseline interferometry.

    In fact, though, the figures quoted for changes in the length of the day are surely not based on measurements of a single day. Rather, they’re based on somewhat less precise measurements taken over many days and then fit to a model.

    Some astronomical applications (precise timing of pulsars, for instance) require very precise knowledge of the Earth’s position and orientation, so quite a bit of effort over the years has gone into understanding these measurements, and especially sources of error in the measurements.

  7. Will there be a new 0 degree if the north-south rotation has been shifted by approx 10 meters? Since, latitude is the angular distance, in degrees, minutes, and seconds of a point north or south of the Equator. Lines of latitude are often referred to as parallels.

  8. I find it interesting that you have mentioned the 2010 earthquake in Chile. I was in one of the hardest hit areas in Chile. We had no communication with the rest of the country for two days and had no idea that it was as bad as it was. The scientific events of that earthquake have drawn geologists from all over the world to study the effects it has had on the world.

    I wrote shortly after that earthquake because it is so important to be prepared when one hits where you are at. Believe me, they hit when you least expect it!

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