E.5.7—Stellar radii

Model a star as a spherical black body

Its luminosity $L$ is the total power radiated from surface area $4\pi R^2$ at absolute surface temperature $T$. This approximation connects observable luminosity and spectrum-derived temperature to radius.

Form the ratio

$\frac{R_1}{R_2}=\sqrt{\frac{L_1}{L_2}\left(\frac{T_2}{T_1}\right)^4}$. A hotter star can have a smaller radius at the same luminosity, while a very luminous cool star must be large.

Worked example — Canopus

For $L=10700L_\odot=4.12\times10^{30}\,\mathrm{W}$, $T=7400\,\mathrm{K}$ and $\sigma=5.67\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$, $R=\sqrt{L/(4\pi\sigma T^4)}=4.4\times10^{10}\,\mathrm{m}$. The large radius explains high luminosity despite a moderate surface temperature.

Check powers

Temperature enters to the fourth power and radius enters squared. Keep the temperature ratio in the inverse order shown before taking the square root.

Common trap

Do not use $R\propto L/T$; the correct scaling is $R\propto\sqrt L/T^2$.