Post-Main-Sequence Evolution of Low-Mass Stars

A star of one to two solar masses leaves the main sequence when hydrogen is gone from its center. What follows is a sequence of core contractions and envelope expansions driven by the exhaustion and reignition of successive fuels, punctuated by a violent flash when a degenerate core ignites. The star traverses the giant branches of the Hertzsprung–Russell diagram, sheds most of its mass, and ends as a slowly cooling remnant supported by electron degeneracy.¹

The subgiant branch and shell hydrogen burning

Core hydrogen exhaustion leaves a helium core with no energy source. Nuclear reactions continue in a shell around the core where hydrogen remains and the temperature is still high enough for the pp chain or the CNO cycle. The inert helium core is nearly isothermal, since without a source there is no temperature gradient to sustain a flux, and it is supported by ideal-gas pressure from the still-hot gas.

An isothermal core can support only a limited overlying weight. Above the Schönberg–Chandrasekhar limit, roughly

no isothermal-core solution exists in equilibrium. As the hydrogen shell dumps helium ash onto the core, the core mass crosses this limit, and the core begins to contract and heat. By the virial theorem the contraction releases gravitational energy, half radiated and half heating the gas, so the shell burns hotter and faster. The luminosity stays nearly constant while the envelope expands and cools: the star moves almost horizontally to the right across the subgiant branch.

Ascent of the red-giant branch

As the envelope expands and cools, its opacity climbs and it turns fully convective. A convective envelope cannot occupy an arbitrary position in the H-R diagram; the Hayashi line marks the coolest a star in hydrostatic equilibrium can be, and a fully convective star sits against it. Further contraction of the core cannot cool the surface past this line, so the star instead grows more luminous at nearly fixed surface temperature. It ascends the red-giant branch almost vertically.

The core and envelope respond oppositely, a coupling known as the mirror principle: when a burning shell separates them, contraction of the core forces expansion of the envelope and vice versa. The hydrogen shell narrows and intensifies as the core contracts, so the luminosity is set essentially by the core mass alone,

a steep core-mass–luminosity relation. The star reaches radii of tens to a hundred solar radii while its degenerate core holds only $\sim 0.45M_{\odot }$ in a body the size of the Earth.

The helium flash

The contracting helium core of a low-mass star becomes electron-degenerate before its center reaches the $\sim 10^8\text{K}$ needed for the triple-alpha reaction. Degeneracy pressure does not depend on temperature, so when triple-alpha burning finally ignites, the energy released raises the temperature without expanding and cooling the core. A higher temperature drives the extremely temperature-sensitive triple-alpha rate ($ϵ\propto T^{40}$) higher still, and the burning runs away in a thermonuclear helium flash. The peak luminosity of the burning briefly reaches $\sim 10^{10}{L}_{\odot }$, comparable to a galaxy, but almost none escapes: the energy goes into lifting the degeneracy. Once the core expands and becomes a non-degenerate ideal gas, the pressure again responds to temperature, burning becomes stable, and the runaway ends.

Because the flash occurs at a nearly fixed core mass of $0.45M_{\odot }$, the tip of the red-giant branch has a nearly fixed luminosity, $M_{\mathrm{bol}}\approx -3.6$. This makes the tip of the red-giant branch a standard candle, one of the rungs of the cosmic distance ladder. The physics of the triple-alpha reaction and the Hoyle resonance is treated in helium burning and the triple-alpha process.

The horizontal branch

After the flash the star settles into quiescent core helium burning with a hydrogen shell still active above. Its luminosity drops to $\sim 50L_{\odot }$, far below the red-giant tip. The surface temperature at this stage depends on the envelope mass remaining after the giant-branch mass loss, and stars with different envelope masses spread out horizontally at nearly constant luminosity: the horizontal branch. Stars whose temperature places them in the instability strip pulsate as RR Lyrae variables, connecting this phase to stellar pulsation and the instability strip. Core helium burning lasts about $10^8\text{yr}$, roughly a hundredth of the main-sequence lifetime, because the energy yield per gram of the triple-alpha reaction is a tenth that of hydrogen burning and the luminosity is higher.

The asymptotic giant branch

Helium burning leaves a carbon–oxygen core, again inert and now degenerate. Helium burning migrates to a shell, and the star climbs a second giant branch, the asymptotic giant branch, so named because its track approaches the red-giant branch from the blue side. The interior now has two active shells: an inner helium-burning shell and an outer hydrogen-burning shell, separated by a helium-rich layer, with the whole structure wrapped in a vast convective envelope.

Thermal pulses and dredge-up

A thin burning shell is thermally unstable. If the shell heats and expands slightly, the reduced pressure lets it produce energy faster rather than slower, because the geometry of a thin shell decouples its temperature from the envelope weight above. The helium shell therefore does not burn steadily; it ignites in brief thermal pulses, flashing to high luminosity for a few hundred years every $10^4$ to $10^5\text{yr}$, while the hydrogen shell supplies the star between pulses.

Each pulse drives a brief convective episode that can reach into the intershell region and mix freshly synthesized material outward, the third dredge-up. This carries carbon and slow-neutron-capture (s-process) elements to the surface; when the dredged-up carbon abundance exceeds oxygen, the star becomes a carbon star. The s-process nucleosynthesis in the intershell is developed in advanced burning and neutron-capture nucleosynthesis.

Mass loss and the planetary nebula

Throughout the giant branches the star loses mass in a slow, dense wind, and on the AGB the loss accelerates to $10^{-5}{M}_{\odot }{\text{yr}}^{-1}$, driven by radiation pressure on dust grains that condense in the cool, extended atmosphere. The wind strips the hydrogen envelope down to a thin remnant layer. As the envelope thins, the hot core is progressively exposed and the surface temperature climbs at nearly constant luminosity, moving the star to the left across the top of the diagram.

When the exposed core reaches $\sim 30,000\text{K}$, its ultraviolet radiation ionizes the surrounding ejected shell, which glows as a planetary nebula. The nebula expands and disperses over $\sim 10^4\text{yr}$, leaving the bare core: a carbon–oxygen white dwarf of $\sim 0.6M_{\odot }$, supported by electron degeneracy pressure, with no remaining fuel. It cools and fades down the white-dwarf sequence. The relation between initial mass and final white-dwarf mass, together with the degenerate equation of state, is developed in white dwarfs and the Chandrasekhar limit.

The whole sequence, from a two-solar-mass main-sequence star to a half-solar-mass white dwarf, returns more than half the original mass to the interstellar medium, enriched in carbon, nitrogen, and s-process elements. Stars above about $8M_{\odot }$ never form a degenerate carbon–oxygen core; their carbon ignites, and they continue to the heavier burning stages and core collapse treated in the evolution of massive stars.