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I learned it during my environment class. My professor introduced the concept--- ratio of surface area and volume. The numerator has connections with the radiative loss, while the denominator is associated with the heat content. The ratio is inversely proportional to the radius of the planet. The conclusion is if I get smaller ratio, the cooling is more slowly. But I cannot find the relationship between the radius with the cooling speed because I think the longer radius, the larger surface area and bigger volume, which leads to greater radiative loss and more heat content. They are varying in the same direction...

So I hope someone can give me an explanation to correct my idea.

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    $\begingroup$ Think how much surface area each kilogram of planet has to dispose of its heat. $\endgroup$ – DJohnM Oct 6 '13 at 20:41
  • $\begingroup$ Why are there no mice in Antarctica? Why are the only warm-blooded things that live there large, round, and blubbery? It's a similar concept. You have mass $m\propto R^3$ and surface area $S\propto R^2$. The rate of cooling is proportional to $S/m\propto 1/R$. Therefore large objects (like walruses and large planets) cool slower than small objects (like mice and small planets). $\endgroup$ – DumpsterDoofus Oct 9 '13 at 1:12
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Note that the surface area of a sphere is proportional to $R^2$ if $R$ is the radius of said sphere, while the volume of that same sphere is proportional to $R^3$.

This means that the volume will increase more than the surface area when $R$ is increased. In terms of heat this means the heat content increases more than the radiative losses as $R$ grows.

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    $\begingroup$ And of course, planets are not in special in this. Given two things of the same type and material but different sizes, the larger one will cool more slowly. $\endgroup$ – dmckee --- ex-moderator kitten Oct 6 '13 at 16:45

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