Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 317 / 3

Because $H\ll R$ and $M_{\mathrm{env}}\ll M$, the envelope has nearly constant area $4\pi R^2$ and nearly constant surface gravity of a star $g=GM/R^2$. Plane-parallel hydrostatic equilibrium isSince $d\Sigma /dz=-\rho$, one has $dP/d\Sigma =g$. Taking the pressure to vanish with $\Sigma$ at the surface gives

The spherical stellar energy equation is $d{L}_r/dr=4\pi r^2\rho ϵ$. With $r\simeq R$ and $F=L_r/(4\pi R^2)$,Division by $d\Sigma /dr=-\rho$ givesLikewise, radiative diffusion in a star givesand therefore

Write the ideal gas law as $P=\mathcal{R}\rho T/\mu$. Since $P=g\Sigma$,The prescribed opacity and burning laws becomeThe radiative equation is consequentlyWith $y=T^7$ and $x={\Sigma }^2/2$,The energy equation similarly givesDifferentiating the first relation therefore produces the nonlinear ordinary differential equation

At the idealized zero-temperature surface, $x=0$ and $y=0$. The outward flux there is $F_s=L/(4\pi R^2)$, soAt the base, $x_0={\Sigma }_0^2/2$ and $y=T_0^7$. The core supplies no luminosity in this model, so all flux has been generated in the overlying hydrogen envelope and $F(x_0)=0$. Hence

Multiplying $y^{\prime \prime }=-{\omega }^2{y}^2$ by $y^{\prime }$ and integrating giveswhere the base conditions fixed the constant. At the surface, $y=0$ and $y^{\prime }=A{F}_s$, with $F_s=L/(4\pi R^2)$. Thereforeand

Separating variables in the first integral givesSet $y=T_0^7\eta$. ThenSince $x_0={\Sigma }_0^2/2$ and ${\Sigma }_0=M_{\mathrm{env}}/(4\pi R^2)$,Combining this with $F_s\propto T_0^{21/2}$ yields

This inverse relation signals thin-shell instability. Hydrogen burning has extreme temperature sensitivity, $ϵ\propto \rho T^{15}$, while the weight of the thin envelope fixes its pressure and limits thermostatic expansion. As burning consumes envelope mass, the equilibrium luminosity rises sharply, accelerating consumption rather than restoring the original state. A thermal runaway can therefore lead to a shell flash or classical nova rather than steady stable burning.