Surface Flux & Colors of Stars

Part 4: Dimensionless Form — Scaling with the Sun

Working in absolute CGS units means juggling $\sigma = 5.67 \times 10^{-5}$ and $R_\odot = 6.96 \times 10^{10}\,\mathrm{cm}$ — easy places for arithmetic errors. Express everything relative to the Sun instead: dividing the Stefan-Boltzmann law for any star by the same law for the Sun cancels the constants.

$\frac{L}{L_\odot} = \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4$

The $4\pi$ and $\sigma$ appear in both numerator and denominator, so they cancel — leaving only ratios. Solving for radius in solar units:

$\frac{R}{R_\odot} = \left(\frac{L/L_\odot}{(T/T_\odot)^4}\right)^{1/2}$

No physical constants appear — just ratios. Solar values: $T_\odot \approx 5800\,\mathrm{K}$, $R_\odot = 6.96 \times 10^{10}\,\mathrm{cm} \approx 696{,}000\,\mathrm{km}$.