The Hertzsprung-Russell Diagram

Luminosity vs temperature reveals stellar evolution

Plot the luminosity of 300 nearby stars against their surface temperature and something remarkable emerges: they do not scatter randomly across the graph. About 90% fall along a single diagonal band running from top-left (hot, luminous) to bottom-right (cool, dim). A smaller group of cool but extremely luminous stars clusters in the upper right, and a group of hot but very faint stars sits in the lower left. This is the Hertzsprung-Russell (HR) diagram, which plots luminosity on the vertical axis against surface temperature on the horizontal axis — with temperature increasing to the left.

Stars are not randomly distributed. They cluster into distinct regions that reflect their evolutionary state:

• Main sequence: A diagonal band running from top-left (hot, luminous, massive O/B stars) to bottom-right (cool, dim, low-mass M stars). Stars spend roughly 90% of their lives here, fusing hydrogen in their cores. Mass increases from bottom-right to top-left.
• Red giants and supergiants: Above and to the right of the main sequence. These are evolved stars that have exhausted their core hydrogen and expanded enormously. They are cool but very luminous, which — by the Stefan-Boltzmann law — requires a very large radius.
• White dwarfs: Below and to the left. Hot but very dim, implying a tiny radius. These are the dense remnants of low-to-intermediate-mass stars after they shed their outer layers.

The Stefan-Boltzmann law connects these three quantities directly:

At the same temperature, a red giant's enormous radius $R$ makes it orders of magnitude more luminous than the Sun. A white dwarf's tiny radius makes it dim despite a high $T$.

The HR diagram plots luminosity (y) vs temperature (x) with temperature increasing to the left. Main sequence = diagonal H-fusion band (OBAFGKM: hottest→coolest; Sun = G2V); red giants top-right (cool, huge $R$); white dwarfs bottom-left (hot, tiny $R$). Stefan-Boltzmann: $L = 4\pi R^2\sigma T^4$, so at the same $T$, higher $L$ means larger $R$.

Pause — copy the highlighted structure and law into your book before moving on.

From observation to physical properties

We just saw the HR diagram's structure — main sequence, red giants, and white dwarfs — and the Stefan-Boltzmann law linking luminosity, temperature, and radius. That raises a question: how do we actually use the HR diagram to find a star's mass, lifetime, or distance? This card answers it → via the mass-luminosity relation $L \propto M^{3.5}$, the lifetime relation $t_{MS} \propto M^{-2.5}$, and spectroscopic parallax.

The HR diagram is more than a classification tool — it reveals fundamental physical properties of stars that cannot be directly observed.

• Mass: Along the main sequence, mass increases from bottom-right to top-left. The mass-luminosity relation $L \propto M^{3.5}$ means a $10\,M_\odot$ star is roughly $10^{3.5} \approx 3\,000$ times more luminous than the Sun.
• Radius: From $L = 4\pi R^2\sigma T^4$ we get $R \propto \sqrt{L}/T^2$. Stars of the same temperature at different luminosities must have different radii — this is what separates giants from dwarfs.
• Age (cluster turn-off): In a star cluster all stars formed simultaneously. More massive stars have shorter main-sequence lifetimes ($t_{MS} \propto M^{-2.5}$) and evolve off the main sequence first. The turn-off point — the most massive star still on the main sequence — directly indicates the cluster's age.
• Distance (spectroscopic parallax): Measure a star's spectral type (hence $T$), use the HR diagram to read off its absolute magnitude, then compare to its apparent magnitude. The difference gives the distance modulus and hence distance.

Mass-luminosity: $L \propto M^{3.5}$ (a $10\,M_\odot$ star is ~3000× more luminous). Main sequence lifetime: $t_{MS} \propto M^{-2.5}$, so cluster age $t = 10\,(M/M_\odot)^{-2.5}$ Gyr. Spectroscopic parallax: spectral type → HR diagram absolute magnitude → distance modulus → distance. Radius: $R \propto \sqrt{L}/T^2$ from Stefan-Boltzmann.

Add the highlighted relations to your notes before the check below.

Applying Stefan-Boltzmann to estimate stellar radii

We just saw the key HR diagram relations for mass, lifetime, and distance. That raises a question: how do we chain those formulas together in a single worked problem to calculate, say, a red giant's radius and a cluster's age? This card answers it → step by step: use $R/R_\odot = \sqrt{L/L_\odot} \times (T_\odot/T)^2$, and $t = 10\,(M/M_\odot)^{-2.5}$ Gyr.

Radius ratio: $R_X/R_\odot = \sqrt{L_X/L_\odot} \times (T_\odot/T_X)^2$. Red giants (cool, ~3000–5000 K, but $10^2$–$10^4\,L_\odot$) have radii tens–hundreds of $R_\odot$; white dwarfs (hot, dim, $L \sim 10^{-3}$–$10^{-1}\,L_\odot$) have radii $\sim 0.01\,R_\odot \approx$ Earth's radius. For $5\,M_\odot$ turn-off: $t \approx 180$ Myr.

Pause — write the highlighted formula and worked results into your book before the check below.