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I have a star with given temperature in Kelvin and radius in solar radii. I tried to calculate the luminosity of the star using Stefan Boltzmann's law, and got an absurd number (over 1 million). What am I doing wrong, and are there any units that I should use instead of Kelvin and solar units?

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The Stefan-Boltzmann constant $\sigma$ is not a dimensionless quantity, it comes with units. So whatever units you use, you must ensure that the value you use for the Stefan-Boltzmann constant is consistent with them.

So using the value expressed in terms of SI units:

$$\sigma = 5.670\,374\,419\ldots \times 10^{-8}\,\rm W\,m^{-2}\,K^{-4}$$

you would either have to work with radius, luminosity and temperature in metres, watts and kelvins, or convert $\sigma$ to the units you are actually using.

For example, if you want to work in terms of solar radii and luminosities you have to account for the conversion factors $L_\odot = 3.828 \times 10^{26}\,\rm W$ and $R_\odot = 6.957 \times 10^8\,\rm m$, giving:

$$\sigma = 7.169\ldots \times 10^{-17}\ L_\odot\, R_\odot^{-2}\, \rm K^{-4}$$

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