# Regression equation with mean temperature as output variableMay 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)?
posted by clockzero to Science & Nature (4 answers total)

Best answer: If you approximate the earth as a black body, then the Stefan-Boltzmann law says that P∝T4, 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/r2 where r is the orbital radius. So T4=k/r2, 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, 2014 [3 favorites]

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, 2014 [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, 2014 [1 favorite]

Response by poster: Thanks for the great answers, all!
posted by clockzero at 6:49 PM on May 19, 2014

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