How the H-R diagram connects to blackbody radiation

The H-R diagram maps thermal state ($T_{\rm eff}$) against total luminosity ($L$). This demo uses log-log axes because stars span many orders of magnitude in both quantities.

Surface flux vs total luminosity

Use these as linked, not interchangeable, statements:

$$F=\sigma T^4$$

$$L=4\pi R^2\sigma T^4$$

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

• At fixed $T_{\rm eff}$, larger $R$ means larger $L$ (vertical offset in H-R space).
• At fixed $R$, hotter $T_{\rm eff}$ means much larger $L$.

Why constant-$R$ lines help

Turning on constant-$R$ guides shows families of stars with the same radius. This makes the geometric factor in $L=4\pi R^2\sigma T^4$ visible directly on the plot.