Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 317 (Structure and Evolution of Stars)

A $6$--$8M_{\odot }$ main sequence star burns hydrogen mainly through the CNO cycle. Its strong temperature sensitivity produces a convective core, which leaves a helium-rich core with a relatively sharp composition discontinuity as it retreats. After central hydrogen exhaustion, hydrogen burning continues in a shell while the helium core contracts and the envelope expands toward the red-giant branch.

When the core becomes hot enough, the Triple-alpha process starts core helium burning. Alpha capture on the newly made carbon also produces oxygen, so central helium exhaustion leaves a carbon-oxygen core. The star then enters the Asymptotic giant branch with an inert core, a helium-burning shell, a hydrogen-burning shell, and a deep convective envelope. At the high-mass end, off-centre carbon burning converts the carbon-oxygen core into an oxygen-neon-magnesium core.

The Schönberg-Chandrasekhar limit is the largest mass fraction that an approximately isothermal inert core can have while remaining matched in hydrostatic and thermal equilibrium to a hydrogen-rich envelope. For a simple isothermal core and polytropic envelope,Because a helium core has larger mean molecular weight than its hydrogen-rich envelope, the limiting fraction is typically near ten per cent. Once shell burning grows the core beyond this limit, no neighboring equilibrium with an isothermal core exists: the core contracts and heats by Kelvin-Helmholtz contraction, the hydrogen-burning shell brightens, and the envelope expands rapidly toward a red giant.

During first dredge-up, the convective envelope deepens on the red-giant branch and brings CNO cycle-processed material to the surface, increasing helium and nitrogen while reducing carbon and changing isotopic ratios. After central helium exhaustion, second dredge-up occurs in this intermediate-mass range: the envelope penetrates into layers processed by hydrogen and sometimes helium burning, lowers the hydrogen-exhausted core mass, and further enriches the surface in helium and nitrogen.

During the thermally pulsing Asymptotic giant branch, a helium-shell flash can be followed by third dredge-up. The envelope then reaches the intershell and may expose newly synthesized carbon and slow-neutron-capture products. In the more massive objects, hot-bottom burning at the base of the convective envelope can convert some dredged-up carbon into nitrogen.

Hydrogen-shell burning deposits helium onto a geometrically thin helium layer. Once helium ignites, the strong temperature dependence of the Triple-alpha process and the shell's initially weak expansion response produce the Härm–Schwarzschild instability. The resulting AGB thermal pulse drives a short-lived intershell convection zone, expands the layers above it, and temporarily extinguishes the hydrogen-burning shell before the cycle restarts.

Toward the upper end of the mass range, neutrino cooling keeps the centre cooler than an off-centre shell, so carbon burning can ignite off-centre under partial electron degeneracy pressure. Repeated flashes and an inward-moving carbon flame consume most carbon and leave a degenerate oxygen-neon-magnesium core, surrounded by helium- and hydrogen-burning shells.

The exact initial-mass boundaries depend on metallicity, convective overshooting, rotation, and mass loss, but stars near the upper intermediate-mass range, roughly $6.5$--$9M_{\odot }$, can become Super-AGB stars. They ignite carbon but do not immediately ignite neon hydrostatically throughout the core.

Their final fate is set by competition between shell-driven core growth and envelope loss. If a stellar wind removes the envelope first, the remnant is an oxygen-neon, or oxygen-neon-magnesium, white dwarf. If the degenerate core grows toward the Chandrasekhar mass, electron captures on magnesium and neon reduce the electron pressure and can trigger an electron-capture supernova, leaving a neutron star. Slightly higher-mass stars can ignite further fuels and proceed to ordinary iron-core collapse.

For a homologous stellar sequence, mass conservation, hydrostatic equilibrium, and the ideal gas equation giveIntegrating the given stellar energy-generation rate over a fixed dimensionless profile givesTherefore

The radiative diffusion in a star equation gives the homology scalingUsing the central scalings from part (i) and the electron-scattering opacity $\kappa \propto 1+X$ yieldsThe radius cancels because an electron-scattering opacity is independent of density and temperature.

A fully convective monatomic ideal gas is an adiabatic stellar polytrope with index $n=3/2$. Its polytropic mass-radius relation givesAt the base of a thin grey atmosphere, the stellar surface boundary condition and electron-scattering opacity giveThe ideal-gas equation evaluated on the convective adiabat givesCombining these relations,The Stefan–Boltzmann law then gives

At fixed initial zero-metal composition, equating parts (i) and (ii) gives $R\propto M^{3/4}$ and $L\propto M^3$. Therefore $T_{\mathrm{eff}}\propto (L/R^2{)}^{1/4}\propto M^{3/8}$, and the fully radiative zero-age line on the Hertzsprung-Russell diagram has slope

For the convective sequence, equating parts (i) and (iii) at fixed composition givesIts slope is consequently

For a fully ionized zero-metal hydrogen-helium mixture with $Y=1-X$, the mean molecular weight obeysFor a radiative star of fixed mass,At the pure-hydrogen point $X=1$,andHenceAs hydrogen burns, $X$ decreases, so both $L$ and $T_{\mathrm{eff}}$ increase. The track moves upward and toward higher temperature from the zero-age main sequence, with initial slope $d\log L/d\log T_{\mathrm{eff}}=256/53\simeq 4.83$.

Write the mean interior density asSincethe assumed outward decrease of $\bar{\rho }$ implies $\rho \le \bar{\rho }$. In mass coordinates, hydrostatic equilibrium isFor every interior mass $m\le m_r$, monotonicity givesIntegrating from the centre and using ${\int }_0^{m_r}{m}^{-1/3}dm=3m_r^{2/3}/2$ proves

Let $\mathcal{R}$ be the gas constant per mole. At the centre,Eliminating $T_c$ gives the Eddington quartic relation in its central form,

At the surface, set $P=0$ and $m_r=M$ in the upper pressure bound. Cubing it givesThe function $(1-\beta )/{\beta }^4$ decreases strictly as $\beta$ increases on $0<\beta <1$. Define ${\beta }^{\ast }$ by equality in the resulting bound:Then ${\beta }_c\ge {\beta }^{\ast }$, orand rearrangement gives exactly

Radiative diffusion in a star can be writtenDividing by hydrostatic equilibrium giveswhich is constant by assumption. Since both $P_{\mathrm{rad}}$ and $P$ vanish at the surface, integration gives $P_{\mathrm{rad}}=(1-\beta )P$ with constant $\beta$.

Eliminating $T$ between the gas and radiation equations of state now givesThus the star is an $n=3$ stellar polytrope. PutThe structure equations become the Lane-Emden equationThe surface is the first zero ${\xi }_1\simeq 6.89685$. Writingthe Lane-Emden mass formula givesThereforewhereThe additional radiative-equilibrium relation is $1-\beta =\nu L/(4\pi cGM)$.

Letbe the local isothermal sound speed squared. Steady spherical mass conservation givesThe momentum equation, with radiative acceleration represented by the radiation-pressure gradient, isRadiative diffusion in a star givesUsing $P_{\mathrm{rad}}/P_g=(1-\beta )/\beta$ andthis becomesSubstitution of $d{P}_g/dr=a_i^{2}d\rho /dr+\rho a_i^{2}d\log T/dr$ and the continuity equation yields the wind equation

Its topology is that of the Parker wind equation. The coefficient of $dv/dr$ vanishes at the sonic line $v=a_i$. A smooth transonic solution must pass through a critical point where the right-hand side also vanishes; generic subsonic solutions are breezes or turn back, while generic supersonic branches cannot be joined smoothly to a quasi-static stellar atmosphere.

At the critical point,The assumed inequality $a_i^2\ll G{m}_r/r$ makes the second factor positive and close to one. Since $0<\beta <1$,Thus acceleration through a regular sonic point requires radiation to cancel most, but not all, of the effective gravity and in particularIf the local luminosity reached or exceeded $L_{\mathrm{crit}}$ in this diffusion model, the numerator would have the wrong sign for the subsonic branch to cross the sonic line smoothly under the cold-wind assumption.