--- title: White Dwarfs and the Chandrasekhar Limit draft: false module: Stellar Death and Compact Remnants moduleNumber: 8 lessonNumber: 1 order: 801 summary: > A white dwarf is held up by the degeneracy pressure of its electrons, a quantum-mechanical stiffness that survives to zero temperature. Filling the Fermi sea sets a pressure that scales as density to the five-thirds power when the electrons are slow and only four-thirds when they are relativistic. The softer relativistic law produces the inverted mass-radius relation and a maximum mass, the Chandrasekhar limit near 1.4 solar masses, above which no cold equilibrium exists. Cooling and crystallization then turn the white-dwarf population into a clock for the Galactic disk. topics: [Stellar Death and Compact Remnants] sources: - book: Carroll & Ostlie ref: "Ch. 16 — The Degenerate Remnants of Stars; §16.2 White Dwarfs; §16.3 The Physics of Degenerate Matter" - book: Maoz ref: "Ch. 4 — Stellar Death and Remnants" --- 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 $1.4\,M_\odot$ 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, $$ N( **Definition (Degeneracy).** A fermion gas is **degenerate** when nearly all > states below the Fermi energy are occupied and nearly all above it are empty, so > the occupation approaches a step function. This holds when $k T \ll E_F$, where > $E_F$ is the Fermi energy. The pressure is then set by the exclusion principle > rather than by thermal motion, and it persists at $T = 0$. The electron number density follows from the mass density through the **mean molecular weight per electron** $\mu_e$, the number of nucleons per free electron: $n_e = \rho/(\mu_e m_H)$. For fully ionized helium, carbon, or oxygen — the compositions of real white dwarfs — each nucleus contributes half its mass number in electrons, so $\mu_e = 2$. Hydrogen would give $\mu_e = 1$, but white dwarfs have burned their hydrogen away. $$ % caption: At zero temperature the electrons fill a sphere in momentum space out to % the Fermi momentum; compression raises the density, enlarging the sphere and the % Fermi momentum with it. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % first sphere (lower density) \draw[very thick] (2.0,2.0) circle (1.0); \draw[->, black] (0.6,2.0) -- (3.7,2.0) node[right, black!70] {p x}; \draw[->, black] (2.0,0.6) -- (2.0,3.6) node[above, black!70] {p y}; \draw[densely dashed] (2.0,2.0) -- (2.707,2.707); \node[anchor=south east] at (2.6,2.5) {pF}; \node[black, anchor=north] at (2.0,0.75) {lower density}; % second sphere (higher density) \draw[very thick] (6.4,2.0) circle (1.5); \draw[->, black] (4.5,2.0) -- (8.4,2.0) node[right, black!70] {p x}; \draw[->, black] (6.4,0.3) -- (6.4,3.9) node[above, black!70] {p y}; \draw[densely dashed] (6.4,2.0) -- (7.46,3.06); \node[anchor=south east] at (7.3,2.9) {pF larger}; \node[black, anchor=north] at (6.4,0.75) {compressed}; \end{tikzpicture} $$ ## Degeneracy pressure, non-relativistic and relativistic Pressure is the flux of momentum carried by particles across a surface. For an isotropic gas the kinetic pressure is $$ P = \frac{1}{3}\int_0^{p_F} n(p)\, v\, p \,\d p, \qquad n(p)\,\d p = \frac{8\pi}{h^3} p^2 \,\d p, $$ with $v$ the speed of an electron of momentum $p$. Two limits bracket the physics. **Non-relativistic electrons.** When $p_F \ll m_e c$ the speed is $v = p/m_e$, and $$ P_{\mathrm{NR}} = \frac{8\pi}{3 m_e h^3}\int_0^{p_F} p^4 \,\d p = \frac{8\pi}{15 m_e h^3}\, p_F^5. $$ Substituting $p_F \propto n_e^{1/3}$ collapses this to a pure power law of density, $$ P_{\mathrm{NR}} = \frac{(3\pi^2)^{2/3}}{5}\,\frac{\hbar^2}{m_e}\,n_e^{5/3} = K_{\mathrm{NR}}\left(\frac{\rho}{\mu_e}\right)^{5/3}, $$ a polytrope of index $n = 3/2$. **Relativistic electrons.** As the density climbs, $p_F$ approaches and then exceeds $m_e c$, and the fastest electrons move at nearly $c$. In the ultra-relativistic limit $v \to c$, $$ P_{\mathrm{UR}} = \frac{8\pi c}{3 h^3}\int_0^{p_F} p^3 \,\d p = \frac{2\pi c}{3 h^3}\, p_F^4 = \frac{(3\pi^2)^{1/3}}{4}\,\hbar c\, n_e^{4/3} = K_{\mathrm{UR}}\left(\frac{\rho}{\mu_e}\right)^{4/3}, $$ a polytrope of index $n = 3$. The exponent softens from $5/3$ to $4/3$: at fixed compression the relativistic gas resists less. This softening is the whole story of the Chandrasekhar limit. > **Worked example.** Compare the two pressures at the density where they cross, > $p_F = m_e c$. The crossover electron density is > $n_e = \tfrac{8\pi}{3}(m_e c/h)^3 \approx 6 \times 10^{35}\ \mathrm{m^{-3}}$, > corresponding to $\rho = \mu_e m_H n_e \approx 2 \times 10^{9}\ \mathrm{kg\,m^{-3}}$ > for $\mu_e = 2$. Below this the $5/3$ law governs and the star behaves like an > $n = 3/2$ polytrope; above it the $4/3$ law takes over and the star approaches the > $n = 3$ marginal configuration. Central densities of massive white dwarfs sit right > in this transition, which is why neither limit alone describes them and the full > Fermi-Dirac pressure must be integrated. $$ % caption: The degenerate pressure follows the five-thirds law at low density and % bends to the softer four-thirds law once the electrons turn relativistic near a % density of order ten to the nine kilograms per cubic metre. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {log density}; \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log pressure}; % NR slope 5/3 (steeper) then continues dashed \draw[very thick] (0.4,0.4) -- (4.4,3.2); \draw[densely dashed] (4.4,3.2) -- (6.0,4.3); \node[anchor=north west] at (1.0,1.4) {slope 5/3}; % UR slope 4/3 (shallower) branch after crossover \draw[black, very thick] (4.4,3.2) -- (8.2,4.3); \draw[black, densely dashed] (2.6,1.95) -- (4.4,3.2); \node[black, anchor=north west] at (6.0,3.7) {slope 4/3}; % crossover marker \fill[acc] (4.4,3.2) circle (2pt); \draw[black, densely dotted] (4.4,0) -- (4.4,3.2); \node[black, anchor=north] at (4.4,-0.05) {pF = me c}; \end{tikzpicture} $$ ## The inverted mass-radius relation The equilibrium radius follows from balancing degeneracy pressure against gravity without solving the full structure. Dimensional (homology) estimates suffice for the scalings. Hydrostatic equilibrium sets the central pressure at order $$ P_c \sim \frac{G M^2}{R^4}, $$ while the non-relativistic degeneracy pressure at the mean density $\rho \sim M/R^3$ is $$ P_{\mathrm{NR}} \sim \left(\frac{M}{R^3}\right)^{5/3} = \frac{M^{5/3}}{R^5}. $$ Equating the two and solving for $R$, $$ \frac{G M^2}{R^4} \sim \frac{M^{5/3}}{R^5} \quad\Longrightarrow\quad R \propto M^{-1/3}. $$ More massive white dwarfs are **smaller**. Gravity from the extra mass compresses the star until the stiffer, denser electron gas can support it. This inverted relation is the observational fingerprint of degeneracy, opposite to the $R \propto M$ behavior of ordinary stars. A carbon-oxygen white dwarf of $0.6\,M_\odot$ has a radius near $0.013\,R_\odot$, comparable to the Earth, at a mean density of order $10^{9}\ \mathrm{kg\,m^{-3}}$.[^co-mr] $$ % caption: The degenerate mass-radius relation: radius falls as mass to the minus % one-third and the curve turns down to zero radius as the mass approaches the % Chandrasekhar value, where relativistic softening removes all support. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {mass}; \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {radius}; % R ~ M^{-1/3}, steepening to zero at M_Ch \draw[acc, very thick] (0.5,4.0) .. controls (1.8,2.5) and (3.4,1.7) .. (5.4,1.1) .. controls (6.6,0.8) and (7.2,0.35) .. (7.5,0.0); \node[acc, anchor=south west] at (1.4,1.7) {heavier dwarfs are smaller}; \draw[black, densely dashed] (7.5,0) -- (7.5,4.2); \node[black, anchor=south, rotate=90] at (7.72,2.1) {Chandrasekhar mass}; \node[black, anchor=west] at (0.7,0.55) {Earth-sized at 0.6 solar masses}; \end{tikzpicture} $$ ## The Chandrasekhar mass The relativistic case removes the radius entirely. When the electrons are ultra-relativistic, $P_{\mathrm{UR}} \sim (M/R^3)^{4/3} = M^{4/3}/R^4$, which carries the **same** $R^{-4}$ dependence as the gravitational pressure $G M^2/R^4$. Setting them equal, $$ \frac{G M^2}{R^4} \sim \frac{M^{4/3}}{R^4}, $$ the radius cancels and leaves a single mass, independent of $R$. Only one mass admits relativistic-degenerate equilibrium; below it the star settles at finite radius on the $5/3$ branch, and above it no cold configuration balances gravity at any radius. The exact coefficient comes from the $n = 3$ Lane-Emden solution, worked out in the [polytrope lesson](/astrophysics-cosmology/stellar-structure/the-equation-of-state-and-polytropes), and evaluates to $$ M_{\mathrm{Ch}} = \frac{5.83}{\mu_e^2}\,M_\odot \approx 1.44\,M_\odot \qquad (\mu_e = 2). $$ > **Theorem (Chandrasekhar limiting mass).** A body supported by > electron-degeneracy pressure has a maximum mass > $M_{\mathrm{Ch}} \approx 0.20\,(\hbar c/G)^{3/2} m_H^{-2} \mu_e^{-2}$. Above this > mass no static equilibrium exists for cold degenerate matter. > **Proof.** Support requires the internal energy to defeat gravity. For $N_e$ > electrons in radius $R$ the number density is $n_e \sim N_e/R^3$ and the Fermi > momentum $p_F \sim \hbar n_e^{1/3} \sim \hbar N_e^{1/3}/R$. In the relativistic > regime each electron carries energy $\sim c\,p_F$, so the total kinetic energy is > $E_K \sim N_e\, c\, p_F \sim \hbar c\, N_e^{4/3}/R$. The gravitational energy of a > body of mass $M \sim N_e \mu_e m_H$ is > $E_G \sim -G M^2/R \sim -G(\mu_e m_H)^2 N_e^2/R$. Both energies scale as $1/R$, so > the total $E = E_K + E_G$ has a fixed sign set by $N_e$. For small $N_e$ the > positive kinetic term dominates and $E > 0$: shrinking $R$ costs energy, so a > minimum exists at finite radius. For large $N_e$ the negative gravitational term > wins and $E \to -\infty$ as $R \to 0$: the body collapses. The threshold is > $N_e \sim (\hbar c/G)^{3/2}(\mu_e m_H)^{-2}$, giving the quoted $M_{\mathrm{Ch}}$. > The vanishing of the $R$-dependence at the threshold reflects the relativistic > softening from $5/3$ to $4/3$. $\qquad\blacksquare$ The limit is built from $\hbar$, $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](/astrophysics-cosmology/stellar-death-and-compact-remnants/thermonuclear-supernovae-type-ia), and a collapsing iron core that exceeds it cannot stop at the white-dwarf stage, proceeding instead to a [neutron star](/astrophysics-cosmology/stellar-death-and-compact-remnants/neutron-stars-and-pulsars) 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 as[^co-cool] $$ L \propto M\, T_c^{7/2}. $$ The heat reservoir is the thermal energy of the **ions** — the electrons, being degenerate, contribute almost nothing that varies with temperature. With $U_{\mathrm{ion}} \propto M T_c$, energy conservation $L = -\,\d U/\d t$ integrates to the **Mestel cooling law**, a cooling time that lengthens steeply as the star fades, $$ t_{\mathrm{cool}} \propto \left(\frac{M}{L}\right)^{5/7}. $$ Reaching $L \sim 10^{-4}\,L_\odot$ takes of order $10^{10}$ years. Because faint white dwarfs cool slowly, they pile up: the **white-dwarf luminosity function**, the number per unit luminosity interval, rises toward low luminosity and then drops sharply at the luminosity the oldest white dwarfs have had time to reach. > **Definition (White-dwarf luminosity function as a clock).** The abrupt cutoff at > the faint end of the white-dwarf luminosity function marks the coolest — and hence > oldest — white dwarfs in a population. Its luminosity, converted to a cooling age > through the Mestel law, dates the onset of white-dwarf formation. For the solar > neighborhood the cutoff near $L \approx 3\times10^{-5}\,L_\odot$ implies a disk age > of roughly $10$ billion years. $$ % caption: The white-dwarf luminosity function rises toward faint magnitudes as slow % cooling piles stars up, then cuts off sharply at the luminosity the oldest white % dwarfs have reached, dating the disk. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {faint direction}; \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {number density}; % rising then sharp cutoff \draw[acc, very thick] (0.6,0.5) .. controls (2.4,1.1) and (4.4,2.4) .. (6.0,3.6) -- (6.4,3.9) -- (6.55,1.0) .. controls (6.7,0.5) and (6.9,0.35) .. (7.2,0.3); \draw[black, densely dotted] (6.4,0) -- (6.4,3.9); \node[black, anchor=north west] at (6.05,0.9) {faint end dates the oldest}; \node[acc, anchor=south east] at (5.2,2.6) {slow cooling piles up stars}; \end{tikzpicture} $$ ## Crystallization As the interior cools, the ions stop behaving like a gas. Their mutual Coulomb repulsion overwhelms thermal agitation once the **Coulomb coupling parameter** $$ \Gamma = \frac{(Z e)^2}{4\pi\varepsilon_0\, a\, k T}, \qquad a = \left(\frac{3}{4\pi n_{\mathrm{ion}}}\right)^{1/3}, $$ the ratio of the nearest-neighbor Coulomb energy to the thermal energy, exceeds a critical value $\Gamma \approx 175$. At that point the ions lock into a body-centered-cubic lattice: the white dwarf **crystallizes** from the center outward. Crystallization releases latent heat of order $k T$ per ion, delaying the cooling and producing a bump in the luminosity function. Below the crystallization temperature the specific heat follows the Debye law $c_V \propto T^3$, so the very oldest white dwarfs enter a phase of rapid final cooling once Debye freeze-out begins. These effects shift the cooling ages at the few-percent level and are folded into precise white-dwarf dating.[^maoz-wd] $$ % caption: The cooling track of a white dwarf on the temperature-luminosity plane: % it fades and cools along a nearly fixed radius, slowed by a latent-heat plateau at % crystallization before a final Debye drop. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (8.6,0.4) -- (0.2,0.4) node[left, black!70] {cooler}; \draw[->, black] (0.6,0) -- (0.6,4.4) node[above, black!70] {log luminosity}; % descending cooling track from hot bright to cool faint, right to left \draw[acc, very thick] (8.0,4.0) .. controls (6.4,3.2) and (5.4,2.7) .. (4.6,2.3) -- (3.4,2.05) % crystallization plateau (latent heat) .. controls (2.6,1.7) and (2.0,0.9) .. (1.4,0.55); \node[acc, anchor=south] at (5.8,3.2) {cooling at set radius}; \draw[black, densely dotted] (4.6,0.4) -- (4.6,2.3); \draw[black, densely dotted] (3.4,0.4) -- (3.4,2.05); \node[black, anchor=north] at (4.0,1.9) {crystallization plateau}; \node[black, anchor=north east] at (1.9,0.7) {Debye drop}; \end{tikzpicture} $$ ## 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 $M_{\mathrm{Ch}} \approx 1.44\,M_\odot$, set by $\hbar$, $c$, $G$, 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](/astrophysics-cosmology/stellar-death-and-compact-remnants/core-collapse-supernovae). [^co-mr]: Carroll & Ostlie, §16.2 — White Dwarfs: the mass-radius relation and observed radii and densities. [^co-cool]: Carroll & Ostlie, §16.2 — the cooling of white dwarfs, the surface-to-core temperature relation, and the Mestel cooling law. [^maoz-wd]: Maoz, Ch. 4 — degenerate remnants: crystallization, the Coulomb coupling parameter, and the luminosity function as an age indicator.