White dwarf

A white dwarf is a compact stellar remnant supported mainly by electron degeneracy pressure.

A carbon-oxygen white dwarf is a white dwarf whose core is predominantly carbon and oxygen, retained after core helium burning when the star loses its envelope without sustained carbon burning.

A helium white dwarf is a white dwarf whose core is primarily helium. Stripping a low-mass evolved star before helium ignition can produce one on a much shorter timescale than isolated stellar evolution.

A cold nonrelativistic white dwarf supported by electron degeneracy pressure has a polytropic equation of state $P=K_{\mathrm{NR}}{\rho }^{5/3}$, so its polytropic index is $3/2$. The Lane-Emden mass formula and polytropic mass-radius relation give $R\propto K_{\mathrm{NR}}{M}^{-1/3}$ at fixed composition. As density rises, relativistic corrections soften the exponent toward $4/3$ and lead toward the Chandrasekhar mass regime.

The Chandrasekhar mass is the limiting mass of a cold nonrotating white dwarf supported by relativistic electron degeneracy pressure, about $1.4M_{\odot }$ for ordinary composition.

An ultrarelativistic degenerate electron gas has $P=K{\rho }^{4/3}$, so it forms an index-three stellar polytrope. Insert $P_c=c_{3}G{M}^2/R^4$ and ${\rho }_c=3a_{3}M/(4\pi R^3)$ into that equation of state: the radius cancels, leaving the displayed mass. For fully ionized helium, $n_e=\rho /(2m_p)$ and $K=(3/(2\pi ){)}^{1/3}hc/(16m_p^{4/3})$. This is the ideal limiting Chandrasekhar mass, rather than a claim that every ordinary white dwarf has that mass.

Mestel's cooling law matches a degenerate isothermal white-dwarf core to a nondegenerate radiative envelope. With Kramers opacity it gives $L/M\propto T_c^{7/2}$ in terms of core temperature.

In a negligible-mass white dwarf envelope, take enclosed mass and luminosity constant, neglect radiation pressure, and use an ideal gas with the Kramers' opacity law. Dividing hydrostatic pressure support equation by radiative diffusion in a star gives $PdP\propto (M/L)T^{15/2}dT$. Neglecting the surface integration constant produces the displayed pressure and mass density laws. It is a deep-envelope approximation, not a finite-temperature photospheric boundary condition.

Integrating the hydrostatic pressure support equation with the radiative-zero white-dwarf envelope law yields $T=4GM(1/r-1/R)/(17\mathcal{R})$. Matching $T=T_c$ at $r_b$ gives $s=R-r_b$ as displayed. A strongly degenerate bulk core has thermal energy small compared with its Fermi energy, making $q\ll 1$ and hence $s\ll R$. The inequality $T_c\gg T_{\mathrm{eff}}$ alone does not establish geometrical thinness.

For fixed mass, degenerate radius and composition, the matching of a white-dwarf envelope to a degenerate core gives $L\propto T_c^{7/2}$, while the geometrical thickness of a white-dwarf radiative envelope gives $s\propto T_c$ to leading thin-layer order. Cooling therefore thins this idealized envelope. At finite thickness the ratio additionally contains $(1+q_2)/(1+q_1)$.

Continuity of temperature, pressure and mass density matches the ideal gas envelope pressure $\mathcal{R}{\rho }_b{T}_c$ to the nonrelativistic electron degeneracy pressure $K_{\mathrm{NR}}{\rho }_b^{5/3}$. Substituting this base mass density into the radiative-zero white-dwarf envelope law gives the luminosity-temperature scaling behind Mestel's cooling law. The matched point is a transition approximation; the cold degeneracy law does not represent finite-temperature corrections there exactly.

A hydrogen-burning white-dwarf envelope is a thin nondegenerate surface layer in which accreted or residual hydrogen can undergo nuclear fusion above a degenerate core.

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.

Thin-shell instability is a thermal runaway of nuclear burning in a geometrically thin layer whose pressure is constrained by overlying weight, weakening expansion-driven cooling.

A classical nova is a thermonuclear runaway in hydrogen-rich material accreted onto a white dwarf, producing a sudden luminous outburst and ejecting part of the envelope.

An idealized classical nova that slowly ejects mass $\delta m$ isotropically from the accretor, compared with the orbital period, but quickly compared with the mass-transfer timescale, expands a circular binary star's Roche lobe. The ejecta carry accretor specific angular momentum $j_1/J=M_2/(M_{1}M)=q/M$. Taking $\delta J/J=-q\delta m/M$, $\delta M_1=-\delta m$ and $\delta M_2=0$ in $J=M_1{M}_2\sqrt{Ga/M}$ proves $\delta a/a=\delta m/M$. The lobe law $R_L\propto a(M_2/M{)}^{1/3}$ then gives $\delta R_L/R_L=4\delta m/(3M)$. A donor whose radius is unchanged moves inside its lobe.

For constant external loss $\Gamma =-\dot{J}>0$ and a donor sequence $R_2\propto M_2^{\zeta }$, the fractional gap closes at rate $2\Gamma /J$, giving $t_d=2J\delta m/(3M\Gamma )$. The binary mass-transfer contact equation gives accumulation time $t_s=J\delta m(\zeta +5/3-2q)/(2M_2\Gamma )$. ThereforeThis first-order result requires $\delta m/M\ll 1$, a positive contact denominator and a fixed donor radius during the eruption. At $\zeta =1$ it reduces to $2q/[(1+q)(4-3q)]$.