{"id":208,"date":"2010-03-04T11:42:31","date_gmt":"2010-03-04T16:42:31","guid":{"rendered":"http:\/\/blog.richmond.edu\/physicsbunn\/2010\/03\/04\/earthquakes-and-earths-rotation\/"},"modified":"2010-03-04T18:41:54","modified_gmt":"2010-03-04T23:41:54","slug":"earthquakes-and-earths-rotation","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/physicsbunn\/2010\/03\/04\/earthquakes-and-earths-rotation\/","title":{"rendered":"Earthquakes and Earth’s rotation"},"content":{"rendered":"

\u00a0Updates<\/strong>: 1. Every article other than the one I linked to says the change in the length of the day is in microseconds, not milliseconds.\u00a0 Much more plausible.\u00a0 2.\u00a0 The Onion’s on the case<\/a>.<\/p>\n

Ashley pointed out this article<\/a> on the Chile earthquake’s effect on Earth’s rotation.<\/p>\n

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.<\/p>\n

The quake, the seventh strongest earthquake<\/a> 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.<\/p>\n

“Perhaps more impressive is how much the quake shifted Earth’s axis,” NASA officials said in a Monday update.<\/p><\/blockquote>\n

The change in the length of the day is good first-year physics stuff.\u00a0 Angular momentum is conserved, and is equal to moment of inertia times rotation rate.\u00a0 The moment of inertia of a body depends on how its mass is distributed.\u00a0 If you change the distribution you change the moment of inertia, and the rotation rate has to change to compensate.\u00a0 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.<\/p>\n

Here’s an embarrassing confession. I can’t make sense of this:<\/p>\n

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).<\/p>\n

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).<\/p><\/blockquote>\n

I don’t think I know what “figure axis” means in this context.\u00a0 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.”\u00a0 But is that the figure axis or the north-south axis?\u00a0 What’s the difference between the two?\u00a0 (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.)<\/p>\n

There’s one other thing I don’t understand:<\/p>\n

Over the course of a year, the length of a day<\/a> 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.<\/p><\/blockquote>\n

Why would Earth’s rotation vary over the course of a year?\u00a0 I can think of two possibilities:<\/p>\n

Possibility 1<\/strong>. Annual changes in wind speed and\/or direction.\u00a0 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.\u00a0 Do winds blow more west to east in the (northern-hemisphere) winter?\u00a0 Paging my brother Andy<\/a> for the answer to this.<\/p>\n

Possibility 2<\/strong>. The article’s made a mistake.\u00a0 It’s not that the rotation rate changes, but rather that the Earth’s orbital speed around the Sun changes.\u00a0 If the rotation rate is fixed, then the length of a sidereal day<\/em> (a day measured relative to the stars)\u00a0 remains the same.\u00a0 But a solar day<\/em> (measured relative to the Sun, of course) is a bit longer than a sidereal day, and the difference depends on the orbital speed.\u00a0 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.\u00a0 So that effect has the right sign to be what Gross is talking about.\u00a0 But it’s much too large an effect: I think it’s a few seconds, not milliseconds.<\/p>\n

After the jump, I’ll try a back-of-the-envelope calculation to see if Possibility 2 makes sense.<\/p>\n

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.\u00a0 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?<\/p>\n

The Earth’s moment of inertia is something like MR2<\/sup> (actually, it’s more like .4 times this, but close enough), where M is the mass of the Earth and R is its radius.\u00a0 The atmosphere’s contribution to this quantity is mR2<\/sup>, where m is the mass of the atmosphere.\u00a0 The atmosphere weighs about 5 x 1018<\/sup> kg, which is about 10-6<\/sup> times the mass of the Earth.\u00a0 We want to cause a change in the Earth’s rotation rate of 1 ms\/day, which is 10-8<\/sup>.\u00a0 So the atmosphere would need to change its motion by only about 1% of the Earth’s rotation rate.\u00a0 That is, if the wind speed changed by an amount equivalent to\u00a0 1 circuit around the earth per 100 days (10 mph), that’d do it.<\/p>\n

Of course, the whole atmosphere doesn’t shift directions, so the part that (hypothetically) does would have to shift by correspondingly more.\u00a0 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u00a0Updates: 1. Every article other than the one I linked to says the change in the length of the day is in microseconds, not milliseconds.\u00a0 Much more plausible.\u00a0 2.\u00a0 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 … Continue reading Earthquakes and Earth’s rotation<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-208","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/posts\/208","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/comments?post=208"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/posts\/208\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/media?parent=208"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/categories?post=208"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/physicsbunn\/wp-json\/wp\/v2\/tags?post=208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}