White Dwarfs and the Chandrasekhar Limit

A white dwarf is the exposed core left when a low- or intermediate-mass star sheds its envelope. Nuclear burning has ceased; nothing generates heat to replace what radiates away. An ordinary gas would contract and cool without limit, but the electrons in a white dwarf are packed densely enough that the Pauli exclusion principle forbids further compression, producing a pressure that does not vanish as the temperature falls to zero. This lesson derives that electron degeneracy pressure from the filling of momentum space, follows it into the relativistic regime, and extracts the two structural consequences: the inverted mass-radius relation $R\propto M^{-1/3}$ and the Chandrasekhar mass, a maximum near $1.4M_{\odot }$ set entirely by fundamental constants. The final sections treat how a white dwarf cools, crystallizes, and thereby records the age of its parent population.

Degeneracy and the Fermi sea

Electrons are fermions: no two occupy the same quantum state. In a volume $V$ the number of translational states with momentum magnitude below $p$ is the phase-space volume divided by $h^3$, times two for spin,

At zero temperature the electrons fill every state up to a sharp cutoff, the Fermi momentum $p_F$, and none above it. Setting $N(<p_F)$ equal to the electron count $n_{e}V$ inverts to

The electron number density follows from the mass density through the mean molecular weight per electron ${\mu }_e$, the number of nucleons per free electron: $n_e=\rho /({\mu }_e{m}_H)$. For fully ionized helium, carbon, or oxygen — the compositions of real white dwarfs — each nucleus contributes half its mass number in electrons, so ${\mu }_e=2$. Hydrogen would give ${\mu }_e=1$, but white dwarfs have burned their hydrogen away.

Degeneracy pressure, non-relativistic and relativistic

Pressure is the flux of momentum carried by particles across a surface. For an isotropic gas the kinetic pressure is

with $v$ the speed of an electron of momentum $p$. Two limits bracket the physics.

Non-relativistic electrons. When $p_F\ll m_{e}c$ the speed is $v=p/m_e$, and

Substituting $p_F\propto n_e^{1/3}$ collapses this to a pure power law of density,

a polytrope of index $n=3/2$.

Relativistic electrons. As the density climbs, $p_F$ approaches and then exceeds $m_{e}c$, and the fastest electrons move at nearly $c$. In the ultra-relativistic limit $v\to c$,

a polytrope of index $n=3$. The exponent softens from $5/3$ to $4/3$: at fixed compression the relativistic gas resists less. This softening is the whole story of the Chandrasekhar limit.

The inverted mass-radius relation

The equilibrium radius follows from balancing degeneracy pressure against gravity without solving the full structure. Dimensional (homology) estimates suffice for the scalings. Hydrostatic equilibrium sets the central pressure at order

while the non-relativistic degeneracy pressure at the mean density $\rho \sim M/R^3$ is

Equating the two and solving for $R$,

More massive white dwarfs are smaller. Gravity from the extra mass compresses the star until the stiffer, denser electron gas can support it. This inverted relation is the observational fingerprint of degeneracy, opposite to the $R\propto M$ behavior of ordinary stars. A carbon-oxygen white dwarf of $0.6M_{\odot }$ has a radius near $0.013R_{\odot }$, comparable to the Earth, at a mean density of order $10^9\mathrm{kg}{\mathrm{m}}^{-3}$.¹

The Chandrasekhar mass

The relativistic case removes the radius entirely. When the electrons are ultra-relativistic, $P_{\mathrm{UR}}\sim (M/R^3{)}^{4/3}=M^{4/3}/R^4$, which carries the same $R^{-4}$ dependence as the gravitational pressure $G{M}^2/R^4$. Setting them equal,

the radius cancels and leaves a single mass, independent of $R$. Only one mass admits relativistic-degenerate equilibrium; below it the star settles at finite radius on the $5/3$ branch, and above it no cold configuration balances gravity at any radius. The exact coefficient comes from the $n=3$ Lane-Emden solution, worked out in the polytrope lesson, and evaluates to

The limit is built from $\hslash$, $c$, $G$, and the nucleon mass alone. It sets the mass scale of every compact remnant: a Chandrasekhar-mass carbon-oxygen core that reaches ignition detonates as a Type Ia supernova, and a collapsing iron core that exceeds it cannot stop at the white-dwarf stage, proceeding instead to a neutron star or a black hole.

Cooling and the white-dwarf luminosity function

A white dwarf has no nuclear source; it radiates its stored thermal energy and fades. The degenerate electrons conduct heat efficiently and hold the interior nearly isothermal at temperature $T_c$, but a thin non-degenerate surface layer of low conductivity throttles the escaping flux. Matching the radiative envelope to the degenerate core gives a luminosity that scales with the core temperature as²

The heat reservoir is the thermal energy of the ions — the electrons, being degenerate, contribute almost nothing that varies with temperature. With $U_{\mathrm{ion}}\propto M{T}_c$, energy conservation $L=-\mathbf{d}U/\mathbf{d}t$ integrates to the Mestel cooling law, a cooling time that lengthens steeply as the star fades,

Reaching $L\sim 10^{-4}{L}_{\odot }$ takes of order $10^{10}$ years. Because faint white dwarfs cool slowly, they pile up: the white-dwarf luminosity function, the number per unit luminosity interval, rises toward low luminosity and then drops sharply at the luminosity the oldest white dwarfs have had time to reach.

Crystallization

As the interior cools, the ions stop behaving like a gas. Their mutual Coulomb repulsion overwhelms thermal agitation once the Coulomb coupling parameter

the ratio of the nearest-neighbor Coulomb energy to the thermal energy, exceeds a critical value $\Gamma \approx 175$. At that point the ions lock into a body-centered-cubic lattice: the white dwarf crystallizes from the center outward. Crystallization releases latent heat of order $kT$ per ion, delaying the cooling and producing a bump in the luminosity function. Below the crystallization temperature the specific heat follows the Debye law $c_V\propto T^3$, so the very oldest white dwarfs enter a phase of rapid final cooling once Debye freeze-out begins. These effects shift the cooling ages at the few-percent level and are folded into precise white-dwarf dating.³

Summary

Electron degeneracy pressure arises from filling the Fermi sea: with all momentum states occupied to $p_F\propto n_e^{1/3}$, the gas exerts $P_{\mathrm{NR}}=K_{\mathrm{NR}}(\rho /{\mu }_e{)}^{5/3}$ when the electrons are slow and $P_{\mathrm{UR}}=K_{\mathrm{UR}}(\rho /{\mu }_e{)}^{4/3}$ when they are relativistic. The $5/3$ law yields the inverted mass-radius relation $R\propto M^{-1/3}$; the softer $4/3$ law removes the radius from the pressure-gravity balance and fixes a maximum mass, the Chandrasekhar limit $M_{\mathrm{Ch}}\approx 1.44M_{\odot }$, set by $\hslash$, $c$, $G$, and $m_H$. A white dwarf then cools by radiating its ion thermal energy along the Mestel law, crystallizing near $\Gamma \approx 175$, and the faint cutoff of its luminosity function dates the Galactic disk. Cores driven to or past $M_{\mathrm{Ch}}$ leave the white-dwarf branch entirely — by thermonuclear detonation or by core collapse.