White Dwarfs and the Chandrasekhar Limit

A white dwarf is the exposed core left when a low-mass star exhausts its nuclear fuel. With no fusion to supply pressure, gravity compresses the core until the electrons are squeezed into a degenerate gas, and their zero-temperature pressure halts the collapse. The star is then a cold ball of electron-degenerate matter, roughly the mass of the Sun packed into the volume of the Earth. The balance between degeneracy pressure and gravity fixes its size, and the relativistic softening of that pressure sets an upper limit on its mass.

Degeneracy pressure against gravity

The matter is fully ionized: bare nuclei sit in a sea of free electrons. The electrons are degenerate because the density is enormous, while the nuclei stay classical because their large mass gives them a Fermi temperature far below the stellar temperature. The pressure is therefore the electron degeneracy pressure of the previous lessons, evaluated at the local electron density $n_e$.

Mass is carried by the nuclei. With ${\mu }_e$ nucleons per electron (so ${\mu }_e\approx 2$ for helium, carbon, or oxygen, where each electron is balanced by two nucleons), the mass density is

where $m_u$ is the atomic mass unit. The nonrelativistic degeneracy pressure from the zero-temperature Fermi gas is

The star is in hydrostatic equilibrium: at every radius the outward pressure gradient balances the inward pull of the mass interior to that radius,

The nonrelativistic mass-radius relation

An order-of-magnitude balance gives the mass-radius relation without solving the differential equation. The gravitational pressure scale is $P_{\mathrm{grav}}\sim G{M}^2/R^4$, obtained by integrating $\mathbf{d}P/\mathbf{d}r$ across the star with $\rho \sim M/R^3$ and $m\sim M$. The degeneracy pressure scale is $P\sim K_{\mathrm{nr}}{\rho }^{5/3}\sim K_{\mathrm{nr}}(M/R^3{)}^{5/3}$. Setting the two equal,

More massive white dwarfs are smaller: adding mass strengthens gravity, which compresses the star until the stiffer degeneracy pressure at higher density restores balance. The product $R{M}^{1/3}$ is a constant fixed only by fundamental constants and the composition ${\mu }_e$. Numerically, a solar-mass white dwarf has a radius near $7\times 10^3\mathrm{km}$, comparable to the Earth, at a mean density of order $10^9\mathrm{kg}{\mathrm{m}}^{-3}$.¹

Relativistic softening of the pressure

The mass-radius relation predicts unbounded density as $M$ grows, but the electrons cannot supply the required pressure indefinitely. Compression raises the Fermi momentum $p_F=\hslash (3{\pi }^2{n}_e{)}^{1/3}$, and once $p_F\gtrsim m_{e}c$ the electrons at the Fermi surface become relativistic. The dispersion crosses over from $ϵ=p^2/2m_e$ to $ϵ\approx pc$, and the pressure changes its density scaling.

The onset density follows from $p_F=m_{e}c$, or $n_e=(m_{e}c/\hslash {)}^3/3{\pi }^2\approx 6\times 10^{35}{\mathrm{m}}^{-3}$, corresponding to a mass density $\rho ={\mu }_e{m}_u{n}_e\approx 2\times 10^9\mathrm{kg}{\mathrm{m}}^{-3}$. This is reached in the core of a white dwarf whose mass approaches a solar mass, so the relativistic correction is not academic: it governs the most massive white dwarfs, precisely those near the stability limit.

In the ultrarelativistic limit the Fermi energy is ${ϵ}_F=\hslash c{k}_F$, the ground-state energy is $U=\frac{3}{4}N{ϵ}_F$, and the pressure of an ultrarelativistic gas obeys $PV=\frac{1}{3}U$. This gives

The exponent has softened from $\frac{5}{3}$ to $\frac{4}{3}$. On logarithmic axes the pressure-density relation is a line whose slope decreases from $\frac{5}{3}$ to $\frac{4}{3}$ as the gas turns relativistic.

The Chandrasekhar mass

The softening removes the stabilizing feedback. Repeat the equilibrium balance with the ultrarelativistic pressure:

The radius $R$ cancels from both sides. Instead of fixing $R$ for each $M$, the balance fixes a single mass at which degeneracy pressure and gravity scale together at every radius. Solving for that mass,

The full polytrope calculation supplies the numerical coefficient,

with ${\omega }_3\approx 2.018$ the constant of the $n=3$ Lane-Emden solution.² The Chandrasekhar mass is built entirely from $\hslash$, $c$, $G$, and the nucleon mass: the combination $(\hslash c/G{)}^{3/2}/m_u^2\approx 1.9M_{\odot }$ is a fundamental stellar mass scale, and the composition factor ${\mu }_e^{-2}$ and the $O(1)$ coefficient bring it to $1.4M_{\odot }$. A white dwarf below this mass sits stably on the mass-radius curve; as $M\to M_{\mathrm{Ch}}$ the equilibrium radius shrinks toward zero and no cold configuration exists above it.

Beyond the limit

A white dwarf pushed over $M_{\mathrm{Ch}}$, by accretion from a companion or by the merger of two dwarfs, cannot regain equilibrium. Two outcomes follow. If the carbon and oxygen ignite under the rising density, runaway fusion unbinds the star as a type Ia supernova. If instead the density climbs first, the electrons are forced onto the nuclei by inverse beta decay (electron capture), $p+e^{-}\to n+{\nu }_e$, which removes the very electrons that supply the pressure. The loss of pressure accelerates the collapse until the matter reaches nuclear density and neutron degeneracy takes over. That neutron-supported remnant is the subject of the next lesson.

Summary

• White-dwarf matter is a cold electron-degenerate gas; the electrons supply the pressure and the nuclei (${\mu }_e$ nucleons per electron) supply the mass, $\rho ={\mu }_e{m}_u{n}_e$.
• Balancing nonrelativistic degeneracy pressure $P\propto {\rho }^{5/3}$ against gravity gives $R\propto M^{-1/3}$: heavier white dwarfs are smaller and denser, a solar mass in an Earth-sized volume.
• At high density the electrons turn relativistic and the pressure softens to $P\propto {\rho }^{4/3}$, removing the radius from the equilibrium balance and fixing a maximum mass.
• The Chandrasekhar mass $M_{\mathrm{Ch}}\approx 1.44(2/{\mu }_e{)}^2{M}_{\odot }$ is set by $(\hslash c/G{)}^{3/2}/({\mu }_e{m}_u{)}^2$; beyond it the star collapses toward a neutron star or a supernova.