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In a thin nondegenerate hydrogen-burning white-dwarf envelope, fix gravity $g$ and use $P=g\Sigma$, $\rho =B\Sigma /T$. With opacity $\kappa ={\kappa }_0\rho T^{-2}$ and specific heating $ϵ={ϵ}_0\rho T^{15}$, set $y=T^7$, $x={\Sigma }^2/2$. Then $y^{\prime }=AF$ and $F^{\prime }=-{ϵ}_{0}B{y}^2$, where $A=21{\kappa }_{0}B/(4ac)$ and ${\omega }^2=A{ϵ}_{0}B$. Radiative-zero surface and zero core-flux base conditions give $(y^{\prime }{)}^2=(2{\omega }^2/3)(y_b^3-y^3)$. Thus $T_b\propto F_s^{2/21}$ and $F_s=12I^3/(A{\omega }^2{\Sigma }_b^6)$, with $I={\int }_0^1(1-t^3{)}^{-1/2}dt$. At fixed column the specific heating varies as $T^{14}$ while radiative cooling varies as $T^7$, giving thin-shell instability in this approximation.

The envelope is light and geometrically thin, so neglect its self-gravity and replace $m(r)$ and $r$ by the core's fixed $M,R$. The stellar hydrostatic equation becomes $dP/dz=-\rho g$, where $g=GM/R^2$. Since $d\Sigma /dz=-\rho$, it gives $dP/d\Sigma =g$. Neglecting the surface pressure yields $\boxed{P=g\Sigma }$.

With $F=L_r/(4\pi R^2)$, the luminosity equation gives $dF/dz=\rho ϵ$, and radiative diffusion in a star gives $dT/dz=-3\kappa \rho F/(4ac{T}^3)$. Dividing each by $d\Sigma /dz$ givesThe signs reflect that column mass increases inward: the temperature rises inward while outward flux decreases toward its source-free inner boundary.

Let $B=\mu g/\mathcal{R}$ for constant mean molecular weight. The gas equation of state and $P=g\Sigma$ give $\rho =B\Sigma /T$. With the supplied opacity and heating laws, put $y=T^7$ and $x={\Sigma }^2/2$. ThenDefine $A=21{\kappa }_{0}B/(4ac)>0$. Since $dx/d\Sigma =\Sigma$, the equations become $dy/dx=AF$ and $dF/dx=-{ϵ}_{0}B{y}^2$. Thus the inverse-square-opacity burning envelope satisfies

At the outer boundary $x=0$, the radiative-zero approximation neglects photospheric temperature relative to the burning layer, so $y=0$ and $y^{\prime }=A{F}_s=AL/(4\pi R^2)$. This is an approximation, not an exactly zero physical photospheric temperature. At the base $x_b={\Sigma }_0^2/2$, $y=y_b=T_0^7$. There is no luminosity entering from the helium core because all the luminosity is generated in the envelope, so $F(x_b)=0$ and $y^{\prime }(x_b)=0$.

Multiply the second-order equation by $y^{\prime }$ and integrate using the base conditions:Therefore the base temperature obeysThe physical branch has $y^{\prime }\ge 0$. A second integration givesThe integral is finite because its endpoint singularity is proportional to $(1-t{)}^{-1/2}$. Hence $y_b\propto x_b^{-2}\propto {\Sigma }_0^{-4}$ and $F_s\propto y_b^{3/2}\propto {\Sigma }_0^{-6}$. More explicitly,This envelope-mass comparison holds the core and the composition/opacity/heating normalizations fixed; the coefficient depends on $g$.

The steady solutions are susceptible to thin-shell instability. At fixed overlying column, base pressure is constrained by $g\Sigma$, so $\rho \propto T^{-1}$. The nuclear heating per unit mass scales as $T^{14}$, whereas the radiative cooling through a fixed-column profile scales as $T^7$: the opacity decreases as $T^{-3}$ and diffusion adds the fourth-power temperature dependence. A positive temperature perturbation therefore increases heating faster than cooling; thin-layer expansion does not supply the strong pressure relief of a whole-star expansion. The inverse relation $F_s\propto {\Sigma }_0^{-6}$ also means consuming the envelope raises the corresponding steady luminosity rather than damping the burning. This idealized model predicts runaway or flashes, despite nondegenerate gas. It is not a claim that all accreting white dwarfs are unstable: sufficiently thick burning layers, altered reaction regimes or other transport/accretion conditions can stabilize burning outside these assumptions.

A $5M_{\odot }$ star of roughly solar stellar composition begins on the zero-age main sequence with a hot convective core, a mostly radiative envelope, and hydrogen burning dominated by the CNO cycle. The strong temperature dependence of the CNO cycle concentrates the stellar energy-generation rate near the centre, helping produce the convective core. Core mixing replenishes hydrogen within that region; it does not homogenize the entire star. The nuclear fuel supply gives a stellar nuclear timescale of order $10^8$ years. For a numerical comparison, Table 2 of Ekström et al.'s solar-metallicity models gives core hydrogen burning lifetimes of about $88$ and $109$ million years for nonrotating and rotating $5M_{\odot }$ models, respectively.

After central hydrogen exhaustion, an inert helium core contracts and heats through Kelvin-Helmholtz contraction, while a hydrogen-burning shell supplies the growing luminosity. The envelope expands and cools: the track crosses the Hertzsprung gap toward the red giant region. Its deepening convective envelope causes first dredge-up, bringing material altered by hydrogen burning to the surface. The rapid structural crossing has a thermal timescale, roughly $10^5$–$10^6$ years for a star of this mass, rather than the full stellar nuclear timescale; an estimate follows from the Kelvin-Helmholtz cooling time $G{M}^2/(RL)$ with the evolving radius and luminosity.

The helium core reaches ignition before becoming strongly supported by electron degeneracy pressure. Thus a normal $5M_{\odot }$ star starts core helium burning without the helium flash characteristic of lower-mass stars. The Triple-alpha process produces carbon-12, and carbon-12 alpha capture produces oxygen-16; these reactions build a carbon-oxygen core. There is a convective core and continued hydrogen-burning shell activity. Core helium burning lasts order $10^7$ years: the same model table gives approximately $19$ and $18$ million years. The changing core and envelope structure can produce a blue loop in the Hertzsprung-Russell diagram; its extent depends on stellar composition, mixing and stellar rotation.

After central helium exhaustion, the carbon-oxygen core contracts and becomes increasingly supported by electron degeneracy pressure. A helium-burning shell forms above it, beneath the hydrogen-burning shell. The star returns to a cool, luminous Asymptotic giant branch configuration. On this early branch, helium-burning shell activity is an important energy source; the overlying hydrogen-burning shell can weaken during readjustment and resume later. At this mass, a deepening convective envelope can cause second dredge-up. This shell-supported evolution to the first AGB thermal pulse takes order $10^6$ years, with model-dependent duration. As the helium-burning shell becomes sufficiently thin, the thin-shell instability ends steady shell burning and marks the onset of AGB thermal pulses.

The original sketch shows the direction of stellar evolution, with higher effective temperature to the left. It illustrates a possible blue loop; its coordinates and loop shape are schematic, rather than a computed stellar model. The stages before the first AGB thermal pulse are the requested track; the later remnant evolution is described below.

In the usual final evolution, stellar winds and an AGB superwind remove the envelope before the carbon-oxygen core can ignite substantial carbon burning. Once shell fuel is exhausted, the exposed core contracts and heats, potentially ionizing the expelled gas as a planetary nebula, and finally becomes a carbon-oxygen white dwarf that loses its stored thermal energy by cooling. The relevant competition is between the AGB envelope-loss and core-growth timescales, not just the total initial mass.

Much higher late-stage mass loss strips the envelope sooner, shortens the Asymptotic giant branch phase, and allows less core growth, usually producing a smaller carbon-oxygen white dwarf. Much lower mass loss prolongs shell burning and permits greater core growth. In an extreme hypothetical case, retention of the envelope could let a degenerate carbon-oxygen core approach the Chandrasekhar mass and undergo explosive carbon burning: the proposed type 1.5 supernova channel. This is not an inevitable fate for a $5M_{\odot }$ star. Doherty et al.'s evolutionary calculations emphasize the uncertainty of that channel and find that their adopted mass loss removes carbon-oxygen envelopes before such ignition. If carbon burning instead proceeds without disruption, the possible further evolution depends on the resulting core and continued envelope retention; lowering mass loss alone does not establish an electron-capture supernova outcome.

In a circular binary star, the distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesEquivalently, $J=\mu \sqrt{GMa}$ by Kepler third law.

First consider a rapid conservative perturbation. Both $M=M_1+M_2$ and $J$ are fixed, while $d{M}_1=-d{M}_2$. With $q=M_2/M_1$,soThe Roche lobe formula then gives its mass-radius exponentAfter mass loss, dynamical stability of binary mass transfer requires the donor to shrink relative to its lobe. Since $d\log M_2<0$, this means ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, or

For stable secular conservative binary mass transfer, put $m={\dot{M}}_2/M_2$. The angular-momentum and contact conditions areElimination of $\dot{a}/a$ givesHere magnetic braking of a binary star removes orbital angular momentum, while the donor's expansion maintains Roche-lobe overflow.

In a cataclysmic variable, hydrogen-rich material accumulates on a degenerate white dwarf. Degeneracy prevents initial expansion from regulating its temperature, so nuclear ignition produces the thin-shell instability and a classical nova. Nuclear burning of hydrogen to helium releases about $0.007m{c}^2$, whereas the binding energy at a white-dwarf surface is only of order $G{M}_{1}m/R_1$, typically a few $10^{-4}m{c}^2$. Even modest coupling can therefore eject all the newly accreted envelope without disrupting the white dwarf.

Finally suppose every transferred mass element is expelled by isotropic re-emission from a binary star. Then ${\dot{M}}_1=0$, $\dot{M}={\dot{M}}_2$, and expelled matter carries the white dwarf's specific angular momentum $j_1=(M_2/M{)}^2{a}^2\Omega$. HenceOn the other hand, logarithmic differentiation of $J=M_1{M}_2{M}^{-1/2}{G}^{1/2}{a}^{1/2}$ and of the Roche-lobe radius givesEliminating the separation produces

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.