After the Main Sequence

Section 8 of 8

Reference and Synthesis

Reference Tables

Key Equations and What They Mean Here

Equation	Role in this reading	Core idea
$2K_{\text{th}} + U_{\text{grav}} = 0$	Core response after H exhaustion	Energy loss leads to contraction and heating
$L = 4\pi R^2 \sigma T_{\text{eff}}^4$	Giant radius inference	Luminous + cool implies large radius
$3\,{}^4\text{He} \rightarrow {}^{12}\text{C} + \gamma$	Net triple-alpha reaction	Carbon is built from helium
$\varepsilon_{3\alpha} \propto T^{40}$	Helium flash stability	Small temperature changes strongly amplify burning
$P_{\text{deg}} \propto \rho^{5/3}$	White dwarf support	Degeneracy pressure is density-controlled, not temperature-controlled
$R_{\text{WD}} \propto M^{-1/3}$	White dwarf structure	More massive white dwarfs are smaller

Symbol Legend

Symbol	Meaning	Units
$K_{\text{th}}$	Thermal kinetic energy	`erg`
$U_{\text{grav}}$	Gravitational potential energy	`erg`
$E$	Total energy	`erg`
$\varepsilon$	Energy generation rate per unit mass	`erg g^{-1} s^{-1}`
$T_{\text{eff}}$	Effective surface temperature	`K`
$\rho$	Density	`g cm^{-3}`
$P_{\text{deg}}$	Degeneracy pressure	`dyn cm^{-2}`
RGB	Red giant branch	—
HB	Horizontal branch	—
AGB	Asymptotic giant branch	—

Glossary

Asymptotic giant branch: The late luminous-giant phase of a low-mass star with an inert carbon-oxygen core and two burning shells (helium inside, hydrogen outside). The thin helium shell burns unstably in pulses, driving dredge-up and heavy mass loss that ultimately strips the envelope.
Helium flash: The runaway onset of core helium burning in a low-mass star whose helium core is electron-degenerate. Because degeneracy pressure barely responds to temperature, the steep $\varepsilon_{3\alpha} \propto T^{40}$ burning is not throttled by expansion, so it spikes violently — but the energy is absorbed deep in the interior (lifting degeneracy), not released as a surface explosion.
Horizontal branch: The HR-diagram locus of low-mass stars that are quietly burning helium in their cores (plus hydrogen in a shell) after the helium flash. It is less luminous than the RGB tip but hotter; in metal-rich populations the same core-helium-burning stars instead form a compact “red clump.”
Planetary nebula: A glowing shell of gas ejected by a low-mass star at the end of the AGB, photoionized by the exposed hot post-AGB core at its center. It is a brief ( $\sim 10^4~\text{yr}$ ) transient — the gas thins and fades — not the long-lived remnant itself, and has nothing to do with planets.
Red giant branch: The evolutionary phase (and HR-diagram locus) of a post-main-sequence low-mass star burning hydrogen in a shell around an inert, contracting helium core. The star is cool ( $\sim 3{,}500\text{–}4{,}500~\text{K}$ ) but very luminous, so by the Stefan-Boltzmann law it must be enormous — tens to a hundred times the solar radius.
Triple-alpha process: The reaction sequence that fuses three helium-4 nuclei into carbon-12: $^4\text{He} + {}^4\text{He} \rightleftharpoons {}^8\text{Be}$ , then $^8\text{Be} + {}^4\text{He} \rightarrow {}^{12}\text{C} + \gamma$ . It needs $T \sim 10^8~\text{K}$ and high density, and is fast enough only because of the Hoyle resonance in carbon-12.
White dwarf: The exposed, degenerate carbon-oxygen (or helium) core left after a low-mass star sheds its envelope. It is roughly Earth-sized, holds $\sim 0.5\text{–}1.4\,M_\odot$ , and is supported against gravity by temperature-independent electron degeneracy pressure rather than fusion — so it simply cools over billions of years.