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What effect does Earth's size have on the temperature of the north and south poles? If the Earth were larger, or smaller, would this affect those temperatures?

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The primary factor determining the temperature of a planet is the radiation it receives from its sun. If this were the only factor, it would be possible to estimate its temperature using the black body radiation law, as described e.g. on Wikipedia. Based on this calculation, the size of the planet has no effect on its temperature.

However, there are other factors contributing to the temperature of the planet, such as its atmosphere (greenhouse effect), internal heating due to radioactive decay, and the surface reflectivity, affected by such things as plant or snow cover, etc. All these can and have changed in the past.

The black-body approximation gives a temperature of about -18 C, thus the other factors have a smaller but still significant contribution.

So the question is how are the other factors affected by the size of the planet?

  • Effect of the atmosphere: I cannot comment.

  • Internal heating: the heat generated by radioactive decay is proportional to the amount of radioactive material, i.e. the volume of the Earth (proportional to its radius $R$ cubed, $\sim R^3$), assuming a fixed density and composition. The heat loss is proportional to the surface area ($\sim R^2$). So the larger the planet, the higher the temperature increase coming from internal heating.

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