Hertzsprung-Russell Diagram
The Hertzsprung-Russell (HR) diagram plots absolute luminosity $L$ (or absolute magnitude $M_V$) on the vertical axis against effective temperature $T_\text{eff}$ (increasing rightward to leftward) or spectral type. The key features:
- Main sequence: a diagonal strip where hydrogen-burning stars reside ($90\%$ of catalogued stars). Luminosity scales steeply with mass: $L \propto M^{3.5\text{–}4}$.
- Red giant branch (RGB): stars evolving off the main sequence with degenerate helium cores.
- Horizontal branch (HB): helium-burning stars at roughly constant luminosity.
- Asymptotic giant branch (AGB): double shell-burning stars ascending again.
- White dwarf sequence: cooling degenerate remnants in the lower left.
The Stefan-Boltzmann law connects luminosity, radius, and temperature:
\[L = 4\pi R^2 \sigma T_\text{eff}^4\]Lines of constant radius are diagonal in the HR diagram. The Sun: $L_\odot = 3.828 \times 10^{33}\ \text{erg/s}$, $T_{\text{eff},\odot} = 5778\ \text{K}$, $R_\odot = 6.96 \times 10^{10}\ \text{cm}$.
| Star type | Spectral class | $T_\text{eff}$ (K) | $L/L_\odot$ | $M/M_\odot$ |
|---|---|---|---|---|
| O supergiant | O | $>30{,}000$ | $10^5$–$10^6$ | $20$–$120$ |
| A main sequence | A | $7{,}500$–$10{,}000$ | $5$–$80$ | $1.5$–$3$ |
| G dwarf (Sun) | G2V | $5{,}500$–$6{,}000$ | $0.6$–$1.5$ | $0.8$–$1.1$ |
| M dwarf | M | $2{,}500$–$3{,}800$ | $0.001$–$0.08$ | $0.08$–$0.5$ |
Pre-Main Sequence Evolution
Stars form by gravitational collapse of molecular cloud cores. The Jeans mass for a cloud of temperature $T$ and density $\rho$ is
\[M_J = \left(\frac{5 k_B T}{G \mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho}\right)^{1/2} \approx 1 M_\odot \left(\frac{T}{10\ \text{K}}\right)^{3/2} \left(\frac{n_H}{10^3\ \text{cm}^{-3}}\right)^{-1/2}\]A collapsing protostar follows the Hayashi track—a nearly vertical descent in the HR diagram—while fully convective. The luminosity is set by the Kelvin-Helmholtz contraction:
\[L = -\frac{dE_\text{grav}}{dt} = \frac{GM^2}{2R^2}\dot{R}\]Once radiative transport develops in the core, the star moves onto the Henyey track (nearly horizontal at roughly constant luminosity) before settling on the zero-age main sequence (ZAMS) when central nuclear burning ignites. The pre-main sequence timescale:
\[t_\text{PMS} \approx \frac{G M^2}{R_\text{ZAMS} L_\text{ZAMS}} \approx 30\ \text{Myr}\ (M/M_\odot)^{-2.5}\]Hydrogen Burning and Main Sequence Lifetime
The pp chain dominates for $M < 1.5 M_\odot$ (lower core temperatures $T_c < 1.7 \times 10^7\ \text{K}$):
\({}^1\text{H} + {}^1\text{H} \to {}^2\text{H} + e^+ + \nu_e \quad \text{(slow, governs rate)}\) \({}^2\text{H} + {}^1\text{H} \to {}^3\text{He} + \gamma\) \({}^3\text{He} + {}^3\text{He} \to {}^4\text{He} + 2{}^1\text{H}\)
Net: $4{}^1\text{H} \to {}^4\text{He} + 2e^+ + 2\nu_e + \text{energy}$ ($Q_\text{eff} \approx 25\ \text{MeV}$)
The CNO cycle dominates for $M > 1.5 M_\odot$ (energy generation $\varepsilon_\text{CNO} \propto T^{18}$), driving convective cores. The main sequence lifetime is
\[t_\text{MS} = \frac{\eta Q f_H M}{L} \approx 10^{10}\ \text{yr} \times \left(\frac{M}{M_\odot}\right)^{-2.5}\]where $f_H \approx 0.1$ (core hydrogen fraction burned), $\eta = 0.007$. This creates a fundamental tension between stellar lifetimes and galactic ages for stars more massive than $\sim 10 M_\odot$.
The main sequence turnoff in a stellar cluster—the point where stars have just exhausted core hydrogen—gives the cluster age directly from the HR diagram.
Red Giant Branch and Helium Flash
When central hydrogen is exhausted, the core contracts (Kelvin-Helmholtz) while a hydrogen-burning shell ignites. The envelope expands enormously (radius increases by factor $\sim 100$) and cools—the star becomes a red giant. The core mass grows as the shell burns outward.
For $M < 2 M_\odot$, the helium core becomes electron-degenerate: pressure is independent of temperature. When $M_\text{core} \approx 0.45 M_\odot$ and $T_c \approx 10^8\ \text{K}$, helium ignites in an unstable, runaway flash:
\({}^4\text{He} + {}^4\text{He} \leftrightarrow {}^8\text{Be} \quad \text{(equilibrium)}\) \({}^8\text{Be} + {}^4\text{He} \to {}^{12}\text{C}^* \to {}^{12}\text{C} + 2\gamma \quad \text{(triple-alpha)}\)
In a degenerate gas, increased energy generation raises $T$ but not $P$, so the core cannot expand and cool—the burning is thermally unstable. This helium flash (lasting seconds to minutes) releases $\sim 10^{11} L_\odot$ momentarily but is entirely absorbed by the envelope. After the flash, the core becomes non-degenerate and settles on the horizontal branch (HB) burning helium quietly at $\sim 50 L_\odot$.
Asymptotic Giant Branch, Planetary Nebulae, White Dwarfs
After core helium exhaustion, a carbon-oxygen (CO) core forms surrounded by a double-shell structure: helium burning just above the CO core, and hydrogen burning further out. The star ascends the asymptotic giant branch (AGB):
- Thermal pulses: the helium shell burns unsteadily, flashing quasi-periodically every $10^3$–$10^5\ \text{yr}$
- Third dredge-up: convection mixes carbon to the surface during pulses (carbon stars)
- Stellar winds: mass loss rates $\dot{M} \sim 10^{-8}$–$10^{-4} M_\odot/\text{yr}$ strip the envelope
For $M < 8 M_\odot$, the envelope is ejected as a planetary nebula (ionized shell glowing in forbidden emission lines), leaving a white dwarf (CO core, $0.5$–$1.4 M_\odot$, $R \sim R_\oplus$). The Chandrasekhar mass limit for a degenerate electron-pressure supported object:
\[M_\text{Ch} = \frac{5.87}{\mu_e^2} M_\odot \approx 1.44 M_\odot\]where $\mu_e$ is the mean molecular weight per electron. White dwarfs cool passively over billions of years; the oldest in the Milky Way have $T_\text{eff} \approx 3500\ \text{K}$.
Massive Stars: Supernovae and Compact Remnants
Stars with $M > 8 M_\odot$ ignite carbon, neon, oxygen, and silicon burning successively in their cores, building an onion-shell structure:
| Burning stage | Fuel | $T$ ($10^9$ K) | Duration |
|---|---|---|---|
| H | H | 0.04 | $10^7$ yr |
| He | He | 0.2 | $10^6$ yr |
| C | C | 0.8 | $10^3$ yr |
| Ne | Ne | 1.5 | 1 yr |
| O | O | 2 | 6 months |
| Si | Si | 3.5 | 1 day |
Silicon burning produces an iron peak (Fe, Ni, Co) that cannot release energy by fusion. When the iron core reaches $M_\text{Ch} \approx 1.4 M_\odot$, it collapses in $\sim 0.25\ \text{s}$, reaching nuclear densities ($\rho \sim 3 \times 10^{14}\ \text{g cm}^{-3}$). Neutron degeneracy pressure halts the collapse (core bounce), launching a shock wave that—with neutrino energy deposition—expels the envelope in a Type II core-collapse supernova (energy $\sim 10^{53}\ \text{erg}$, with $\sim 99\%$ as neutrinos).
Type Ia supernovae (no hydrogen lines in spectrum) arise from thermonuclear detonation of a CO white dwarf approaching $M_\text{Ch}$ via accretion or merger—releasing $\sim 10^{51}\ \text{erg}$ as kinetic energy, synthesizing $\sim 0.6 M_\odot$ of ${}^{56}\text{Ni}$.
The remnant left behind depends on mass:
- $M_\text{initial} < 8 M_\odot$: white dwarf
- $8 < M_\text{initial} < 20\text{–}25 M_\odot$: neutron star ($R \approx 10\ \text{km}$, $M \approx 1.4\text{–}2.0 M_\odot$)
- $M_\text{initial} > 20\text{–}25 M_\odot$: black hole
Neutron star cooling is analogous to white dwarf cooling but faster, involving neutrino emission from the dense core (modified Urca process) during the first $\sim 10^5\ \text{yr}$, followed by photon cooling.