If two stars have identical size, how much more luminous is one whose surface temperature is twice as high as the other?
Answer
It will be sixteen times greater.
The relationship defining the total energy radiated by a star is quantified by the Stefan-Boltzmann Law, which establishes that energy radiated per unit area is proportional to the fourth power of the absolute temperature ($T^4$). If all other factors, such as radius and the Stefan-Boltzmann constant, are held constant, doubling the surface temperature means the luminosity dependency becomes $2^4$. This results in a luminosity that is sixteen times greater. This extreme dependence on temperature explains why even small temperature variations can cause massive shifts in a star's intrinsic energy output when size is equal.
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