The HR Diagram

Section 3 of 6

The Theorist's HR Diagram

Part 3: The Theorist’s HR Diagram — Overlaying Physics

Same Patterns, Physical Axes

The observer’s diagram uses $M_V$ and spectral type. Theorists prefer physical quantities: luminosity $L$ (in $L_\odot$ ) and effective temperature $T_{\text{eff}}$ (in K), related through calibrations — $M_V \to L$ via the magnitude-luminosity relation, spectral type $\to T_{\text{eff}}$ (e.g. G2 → $5{,}800~\text{K}$ ). The conversion is:

$M - M_\odot = -2.5\log_{10}\!\left(\frac{L}{L_\odot}\right)$

(treating the $V$ -band as a proxy for bolometric here). The theorist’s HR diagram plots $\log(L/L_\odot)$ against $\log T_{\text{eff}}$ , temperature decreasing rightward. The remarkable fact: the same patterns appear on both versions — so they are real features of stellar physics, not artifacts of the measurement system.

Lines of Constant Radius

The theorist’s diagram lets you overlay theoretical relationships, the most important from the Stefan-Boltzmann law:

This connects $L$ , $R$ , and $T$ . On the HR diagram $L$ and $T$ are the axes, so fixing $R$ defines a line: $L = 4\pi R^2\sigma T^4 \Rightarrow \log L = \text{const} + 4\log T$ (at fixed $R$ ).

These lines of constant radius are diagonals on the theorist’s HR diagram — each represents all combinations of $L$ and $T$ for a star of a given radius.

Numeric answer

At fixed radius, if a star’s temperature $T$ doubles, by what factor does its luminosity change?

Your answer

HR diagram with log luminosity in solar units on the vertical axis and log effective temperature on the horizontal axis (reversed, hotter on left). Dashed diagonal lines show constant stellar radii from 0.01 to 1000 solar radii. Main sequence stars form a diagonal band, giants cluster in the upper right near the 10-100 solar radii lines, and white dwarfs cluster in the lower left near the 0.01 solar radii line. The Sun is marked at log T = 3.76, log L = 0. — Figure 4The theorist's HR diagram uses physical axes — log(L/Lsun) vs. log T_eff — and overlays lines of constant radius from the Stefan-Boltzmann law. Giants sit on lines of R ~ 10-100 Rsun; white dwarfs on R ~ 0.01 Rsun (Earth-sized). The same patterns appear as in the observer diagram, confirming they are real features of stellar physics.ASTR 201 (generated)

Reading the diagram with radius lines: main-sequence stars span ${\sim}0.1\,R_\odot$ (M dwarfs) to ${\sim}10\,R_\odot$ (O stars); giants sit on $R \sim 10\text{–}100\,R_\odot$ lines (luminous despite being cool); supergiants reach $R \sim 100\text{–}1{,}000\,R_\odot$ ; white dwarfs sit on $R \sim 0.01\,R_\odot$ lines (faint despite being hot).

Worked Example 3Radius from the HR Diagram

Problem

A red giant has $L = 400\,L_\odot$ and $T_{\text{eff}} = 4{,}000~\text{K}$ . What is its radius compared to the Sun ( $T_\odot = 5{,}800~\text{K}$ )?

StepUse the ratio form (no CGS constants)

$\frac{L}{L_\odot} = \left(\frac{R}{R_\odot}\right)^2\left(\frac{T}{T_\odot}\right)^4 \Rightarrow \left(\frac{R}{R_\odot}\right)^2 = \frac{L/L_\odot}{(T/T_\odot)^4} = \frac{400}{(4{,}000/5{,}800)^4}$

StepEvaluate

$\frac{4{,}000}{5{,}800} = 0.690,\quad 0.690^4 = 0.226 \Rightarrow \left(\frac{R}{R_\odot}\right)^2 = \frac{400}{0.226} = 1{,}770 \Rightarrow \frac{R}{R_\odot} = 42$

Dimensional check

All quantities are solar ratios — units cancel; the answer is in solar radii ✓.

Result

The red giant has $R \approx 42\,R_\odot$ — typical for the red giant branch. Placed at the Sun’s center it would reach about halfway to Mercury ✓.

Problem

On the theorist’s HR diagram, which way do lines of constant radius slope?
A white dwarf has $T_{\text{eff}} = 2 \times 10^4~\text{K}$ and $L = 0.01\,L_\odot$ . Estimate its radius in $R_\odot$ .
Where would a star with $R = R_\odot$ and $T = 2 \times 10^4~\text{K}$ appear? Is such a star observed?

Part 2 built a map from measurements alone; Part 3 added radius. But neither tells us what orders stars along the main sequence. To explain that, we need the hidden parameter that sets both luminosity and temperature: mass.