White dwarf

A white dwarf is a compact stellar remnant supported mainly by electron degeneracy pressure. Its ions carry most of the mass, while its electrons provide most of the supporting pressure. The Pauli exclusion principle makes compression fill quantum states with larger momenta; support therefore persists even when thermal motion becomes small. Cooling and hydrostatic equilibrium are separate questions: a white dwarf can radiate stored energy while retaining degeneracy support.

The surprising trend is that a more massive cold white dwarf is smaller. Its higher mass density raises the Fermi momentum, eventually requiring special relativity. The degenerate-electron pressure crossover explains why this response approaches a limiting stiffness and why the Chandrasekhar limit exists. Subrahmanyan Chandrasekhar's 1983 lecture develops this connection between quantum statistics, stellar structure and stellar evolution.^[ref]

With carbon-ion thermal energy $U=M{k}_B{T}_c/(8m_u)$ and luminosity $L=A(M/M_{\odot })T_c^{7/2}$, the time between two core temperatures is $k_B{M}_{\odot }(T_f^{-5/2}-T_i^{-5/2})/(20m_{u}A)$. Late cooling is slow in this approximation. Extrapolating the formula does not prove a categorical observational absence of ultracool remnants, since atmosphere, crystallization and heat-capacity physics alter the simple model.

A white dwarf companion in the nearby Procyon binary system. A measured bolometric luminosity $L$ and effective temperature $T$ give its stellar radius through $R=\sqrt{L/(4\pi \sigma T^4)}$.^[ref]

An age inferred from a cooling model for a white dwarf or a white-dwarf population. The progenitor lifetime must be added to obtain time since stellar formation. Crystallization, composition, envelope properties, population history and observational completeness affect the inference.

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. It bounds the retained remnant's mass, rather than the birth mass of a star that can lose an envelope. The ideal limit assumes fixed mean molecular weight per electron, Newtonian gravity and a noninteracting degenerate electron gas; electron capture, general relativity, stellar rotation and ignition can change the attainable endpoint.

The Chandrasekhar limit belongs to work of the early 1930s. In his 1983 retrospective, Subrahmanyan Chandrasekhar reprints the 1935 mass–radius sequence and then discusses stellar evolution and relativistic stability.^[ref] The 1983 Nobel Prize in Physics recognized his broader theoretical work on stellar structure and stellar evolution; the other half recognized William Alfred Fowler's theoretical and experimental work on nuclear reactions underlying chemical element formation.^[ref]

Consider an ideal degenerate electron gas at zero temperature. Counting two spin states per phase space cell gives number density $n_e=p_F^3/(3{\pi }^2{\hslash }^3)$. The isotropic momentum flux gives pressureHere $m_e$ is electron mass, $c$ is the speed of light, $\hslash =h/(2\pi )$ with Planck constant $h$, and special relativity gives $v(p)=pc/\sqrt{p^2+m_e^2{c}^2}$. The factor $1/3$ averages the directional momentum flux, and the spin factor is already included. This normalization makes $P_{\ast }$ a pressure, not an energy per electron.

For $x\ll 1$, expand the denominator: $I(x)\sim x^5/5$ and $P\simeq n_e{p}_F^2/(5m_e)$. For $x\gg 1$, the integrand approaches $t^3$: $I(x)\sim x^4/4$ and $P\simeq n_e{p}_{F}c/4$. At fixed composition, mass density is $\rho ={\mu }_e{m}_u{n}_e$, with ${\mu }_e$ the mean molecular weight per electron and $m_u$ the unified atomic mass unit. Since $n_e\propto x^3$, the logarithmic stiffness isPressure keeps rising; its fractional gain per fractional compression becomes smaller. Relativistic electrons approach a speed ceiling while their momenta keep increasing. Thus the high-density equation of state loses the extra compression leverage of a nonrelativistic white dwarf. The Chandrasekhar mass from central polytropic constants supplies the corresponding global mass calculation.

For a calculated local example with ${\mu }_e=2$, $x=1$ gives $\rho \simeq 1.95\times 10^9\mathrm{kg}{\mathrm{m}}^{-3}$, $I(1)=0.1537384$ and $P\simeq 7.38\times 10^{21}\mathrm{Pa}$. Its stiffness is $1.53314$, intermediate between the two limits. These are ideal equation of state values at one point, not a measured central mass density or a whole white dwarf model. Finite temperature and interactions are excluded.

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 $2q/[(1+q)(4-3q)]$.