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When modeling the heat spread from a hot object through a vacuum (or a gas), what is the dominant physical process for heat energy transportation? Thermal diffusion, thermal radiation, a mix of the two, or something else? What properties of the object or the gas may influence this?

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Radiation always occurs, and it is proportional to $T^4$, but it works both ways between itself and the environment. If it is not much hotter, in absolute temperature, than its environment, its heat transfer rate is roughly linear in the temperature difference. The rate of radiative transfer also depends on the character of the radiating surface. For example, if it is shiny, it will reduce the rate of heat transfer.

Then if there's gas around, heat can also be transferred to the gas, by conduction. Again it is roughly proportional to temperature difference. If gravity is also present, then the gas may undergo convection, which enhances the conduction by carrying hot gas away, to be replaced by cooler gas.

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