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# Regression equation with mean temperature as output variable

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# Regression equation with mean temperature as output variable

May 18, 2014 10:29 PM Subscribe

If you had a multivariate equation whose output variable is mean surface temperature on earth, what would be the rough beta value for distance from the sun (in millions of miles or whatever)?

I came here to say something similar to teraflop's comment -- the relationship is not going to be linear. You can still approximate it with a regression equation, but you'll need to add a squared term to model it with a quadratic equation.

posted by mikeand1 at 9:56 AM on May 19 [1 favorite]

posted by mikeand1 at 9:56 AM on May 19 [1 favorite]

The earth's distance varies from 91 million to 94.4 million miles. That's a narrow enough range (+/- 1.5%) that the linear approximation should be pretty good.

posted by SemiSalt at 3:18 PM on May 19 [1 favorite]

posted by SemiSalt at 3:18 PM on May 19 [1 favorite]

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^{4}, where T is temperature (in Kelvin) and P is the outgoing thermal radiation. On average, the earth is (roughly) in thermal equilibrium, so the incoming and outgoing radiation are equal: that is, P falls off proportionally to 1/r^{2}where r is the orbital radius. So T^{4}=k/r^{2}, for some constant k.That's not a linear equation, but we can pretend it is and take the derivative to find a coefficient for a linear approximation. If I'm doing the math right, then differentiating and solving for k gives dT/dr = -T/2r. And by plugging in approximate numbers (T=287K, r=93 million miles) we get a result of about -1.5K for each additional million miles.

(Hopefully that makes sense; if there are any holes in my analysis, corrections are welcome.)

posted by teraflop at 11:02 PM on May 18 [3 favorites]