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Derivation

The Chandrasekhar Mass Limit

D-335 Home PU-308 Threads matter · chance · force · energy Depends on The Lane-Emden Equation for Polytropes, degenerate-fermi-gas-pressure
Statement

For a cold white dwarf supported by the pressure of a fully degenerate, ultrarelativistic electron gas, the equation of state is the polytrope \(P=K\rho^{4/3}\) with \(K=\dfrac{(3\pi^2)^{1/3}}{4}\dfrac{\hbar c}{(\mu_e m_H)^{4/3}}\). Because \(\gamma=\tfrac43\) corresponds to polytropic index \(n=3\), hydrostatic equilibrium (the Lane–Emden equation) admits a self-gravitating solution only at a single mass, independent of central density and hence of radius: \[ M_{\mathrm{Ch}} = 4\pi\,\big(-\xi_1^2\theta'_1\big)_{n=3}\left(\frac{K}{\pi G}\right)^{3/2} = \frac{\sqrt{3\pi}}{2}\,\omega_3^{0}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2}\approx 1.44\left(\frac{2}{\mu_e}\right)^{2} M_\odot, \] where \(\omega_3^{0}=(-\xi_1^2\theta'_1)_{n=3}=2.018\) is a pure number fixed by the \(n=3\) Lane–Emden solution.

Why it matters

This is the sharpest quantitative statement in stellar astrophysics: two of Nature's fundamental constants (\(\hbar c/G\)) and one composition number (\(\mu_e\)) fix, with no free parameters, the heaviest possible cold star supported by electrons. A white dwarf that accretes or is born above \(\approx 1.4\,M_\odot\) cannot find a static equilibrium; it collapses, igniting the runaway carbon burning of a Type Ia supernova or forming a neutron star. Because the limit is nearly universal, Type Ia explosions are near-standard candles — the very rulers that revealed cosmic acceleration.

Conceptually it is the moment quantum statistics loses to gravity. Non-relativistic degeneracy pressure (\(P\propto\rho^{5/3}\)) always wins at small radius and yields a stable star for any mass; the relativistic softening to \(P\propto\rho^{4/3}\) removes that safety margin, and the delicate cancellation that follows is the physical content of the Chandrasekhar mass.

Assumptions
Fully degenerate electrons at zero temperature (\(k_BT\ll\varepsilon_F\)).If dropped, thermal pressure adds a temperature-dependent term, the star is no longer an exact polytrope, and the clean radius-independent mass is only recovered in the cold limit; hot proto-white-dwarfs are puffier.
Ultrarelativistic electrons, \(p_F\gg m_e c\).If dropped, the exact free-electron equation of state interpolates between \(n=\tfrac32\) (low density, \(P\propto\rho^{5/3}\)) and \(n=3\); the mass limit is then approached only asymptotically as the central density \(\rho_c\to\infty\), and no finite star exactly saturates it.
Electrons supply the pressure, ions supply the mass: \(\rho=\mu_e m_H\,n_e\).If dropped, one would use \(n_e=\rho/m_H\) and overestimate the electron density, changing \(M_{\mathrm{Ch}}\) by the factor \(\mu_e^{2}=4\); the ions are cold and contribute negligible pressure but essentially all the mass.
Newtonian hydrostatic equilibrium; the star is spherical, non-rotating, unmagnetised.If dropped, general-relativistic gravity makes an \(n=3\) polytrope unstable at a finite central density slightly below the Newtonian limit, while rotation and strong magnetic fields can support super-Chandrasekhar configurations.
Ideal, non-interacting electron gas; neglect electrostatic (lattice) corrections and inverse \(\beta\)-decay.If dropped, Coulomb corrections lower the pressure by a few percent (reducing \(M_{\mathrm{Ch}}\) slightly), and at high density electron capture on nuclei removes electrons entirely, softening the equation of state before the ideal limit is reached.
Derivation
1
\[ n_e = 2\int\frac{d^3p}{(2\pi\hbar)^3}\,\Theta(p_F-p)=\frac{p_F^3}{3\pi^2\hbar^3}\ \Longrightarrow\ p_F=\hbar\,(3\pi^2 n_e)^{1/3} \]
Fill every electron momentum state up to the Fermi momentum \(p_F\) (imported: degenerate-fermi-gas-pressure). The factor \(2\) is spin. Symbols only. A
2
\[ P=\frac{1}{3}\,2\int\frac{d^3p}{(2\pi\hbar)^3}\,p\,v\,\Theta(p_F-p),\qquad v=\frac{\partial\varepsilon}{\partial p}\xrightarrow[\text{UR}]{}c \]
The kinetic pressure is \(\tfrac13 n\langle p v\rangle\). In the ultrarelativistic limit \(\varepsilon=pc\), so the group velocity saturates at \(v=c\) — this is the single physical input that softens the equation of state. B
3
\[ P=\frac{c}{3}\,\frac{1}{\pi^2\hbar^3}\int_0^{p_F}\!p^3\,dp=\frac{c\,p_F^4}{12\pi^2\hbar^3} \]
Carry out the isotropic momentum integral with \(v=c\) pulled out. The integrand \(p\cdot p^2\) gives \(p^3\), so the pressure scales as \(p_F^4\) — one power steeper than the number density's \(p_F^3\). B
4
\[ P=\frac{(3\pi^2)^{1/3}}{4}\,\hbar c\,n_e^{4/3} \]
Eliminate \(p_F\) using step 1, \(p_F^4=\hbar^4(3\pi^2 n_e)^{4/3}\), and simplify \((3\pi^2)^{4/3}/(12\pi^2)=(3\pi^2)^{1/3}/4\). The exponent \(\tfrac43\) — not \(\tfrac53\) — is the crux: it is the polytrope that cannot support arbitrary mass. B
5
\[ n_e=\frac{\rho}{\mu_e m_H}\ \Longrightarrow\ P=K\rho^{4/3},\qquad K=\frac{(3\pi^2)^{1/3}}{4}\frac{\hbar c}{(\mu_e m_H)^{4/3}} \]
Trade electron density for mass density: each electron is accompanied by \(\mu_e\) nucleons of mass \(\approx m_H\), which carry the mass while the electrons carry the pressure. The result is a pure polytrope \(P=K\rho^{1+1/n}\) with \(1+\tfrac1n=\tfrac43\), i.e. \(n=3\). A
6
\[ \frac{1}{\xi^2}\frac{d}{d\xi}\!\left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^{\,n},\quad \rho=\rho_c\theta^{\,n},\ \ r=a\xi,\ \ a^2=\frac{(n+1)K\,\rho_c^{\,1/n-1}}{4\pi G} \]
Imported result: hydrostatic equilibrium \(dP/dr=-G m(r)\rho/r^2\) together with \(dm/dr=4\pi r^2\rho\) for a polytrope reduces to the Lane–Emden equation (lane-emden-polytrope). The dimensionless \(\theta\) runs from \(1\) at the centre to its first zero at \(\xi=\xi_1\), which marks the surface. C
7
\[ M=\int_0^{R}4\pi r^2\rho\,dr=4\pi a^3\rho_c\int_0^{\xi_1}\!\xi^2\theta^{\,n}\,d\xi=4\pi a^3\rho_c\,\big(-\xi_1^2\theta'_1\big) \]
Insert \(\rho=\rho_c\theta^n\) and \(r=a\xi\). The Lane–Emden equation lets us replace \(\xi^2\theta^n=-\tfrac{d}{d\xi}(\xi^2\theta')\), so the mass integral collapses to the surface boundary term \(-\xi_1^2\theta'_1\), a pure number for each \(n\). C
8
\[ n=3:\quad a^2=\frac{4K}{4\pi G}\rho_c^{-2/3}=\frac{K}{\pi G}\,\rho_c^{-2/3}\ \Longrightarrow\ a^3\rho_c=\left(\frac{K}{\pi G}\right)^{3/2} \]
Set \(n=3\), so \(n+1=4\) and \(\rho_c^{1/n-1}=\rho_c^{-2/3}\). Then \(a^3=(K/\pi G)^{3/2}\rho_c^{-1}\), and the central density cancels exactly in the product \(a^3\rho_c\). This cancellation — unique to \(n=3\) — is the mathematical heart of the limit. C
9
\[ M_{\mathrm{Ch}}=4\pi\,\big(-\xi_1^2\theta'_1\big)_{n=3}\left(\frac{K}{\pi G}\right)^{3/2} \]
Combine steps 7 and 8. Every reference to \(\rho_c\), \(R\), and \(a\) has vanished: the mass is the same for a compact dense white dwarf and a diffuse one. There is exactly one mass at which an ultrarelativistic degenerate star balances gravity. B
10
\[ M_{\mathrm{Ch}}=4\pi\,\omega_3^{0}\left(\frac{(3\pi^2)^{1/3}}{4\pi}\right)^{3/2}\!\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2}=\frac{\sqrt{3\pi}}{2}\,\omega_3^{0}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2} \]
Substitute \(K\) from step 5. The identity \(\big((3\pi^2)^{1/3}\big)^{3/2}=(3\pi^2)^{1/2}\) reduces the prefactor \(4\pi\big((3\pi^2)^{1/3}/4\pi\big)^{3/2}=\sqrt{3\pi}/2\), giving the compact closed form with \(\omega_3^{0}=2.018\). B
Result
\[ M_{\mathrm{Ch}}=\frac{\sqrt{3\pi}}{2}\,\omega_3^{0}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2}\approx 1.44\left(\frac{2}{\mu_e}\right)^{2} M_\odot,\qquad \omega_3^{0}=2.018 \]

Reading. The maximum mass is built entirely from constants: the quantum-relativistic combination \((\hbar c/G)^{3/2}\), which is a mass (the Chandrasekhar mass is a few times this "gravitational mass scale"), divided by the square of the mass per electron. It does not depend on radius or central density — the defining feature of an \(n=3\) polytrope — so as a white dwarf's mass rises toward \(M_{\mathrm{Ch}}\) its radius shrinks toward zero. Composition enters only through \(\mu_e\): carbon–oxygen and helium dwarfs (\(\mu_e=2\)) give \(1.44\,M_\odot\); heavier compositions give less. The scale is set by gravity being weak enough that a macroscopic number \(\sim(\hbar c/Gm_H^2)^{3/2}\approx 2\times10^{57}\) of nucleons is needed before it wins.

Units check. \(\hbar c\) has units \(\mathrm{J\,m}=\mathrm{kg\,m^3\,s^{-2}}\); dividing by \(G=\mathrm{m^3\,kg^{-1}\,s^{-2}}\) gives \(\hbar c/G=\mathrm{kg^2}\). Then \((\hbar c/G)^{3/2}=\mathrm{kg^3}\), and dividing by \((\mu_e m_H)^2=\mathrm{kg^2}\) leaves \(\mathrm{kg}\), a mass. The factor \(\sqrt{3\pi}\,\omega_3^{0}/2\) and \(\mu_e\) are dimensionless. Numerically \((\hbar c/G)^{3/2}=1.03\times10^{-23}\,\mathrm{kg^3}\), \((\mu_e m_H)^2=1.12\times10^{-53}\,\mathrm{kg^2}\), product with \(3.098\) gives \(2.85\times10^{30}\,\mathrm{kg}=1.44\,M_\odot\).

Limiting cases
  • Non-relativistic degeneracy (\(p_F\ll m_e c\), low-mass dwarfs): \(P=\dfrac{(3\pi^2)^{2/3}}{5}\dfrac{\hbar^2}{m_e}n_e^{5/3}\propto\rho^{5/3}\), an \(n=\tfrac32\) polytrope. Now \(M\propto\rho_c^{1/2}\) does not cancel; instead \(R\propto M^{-1/3}\) and every mass has a stable equilibrium — no limit exists.
  • Composition dependence: \(M_{\mathrm{Ch}}\propto\mu_e^{-2}\). Iron cores (\(\mu_e=56/26=2.15\)) give \(1.24\,M_\odot\); pure hydrogen (\(\mu_e=1\)) would give \(5.8\,M_\odot\), but hydrogen ignites long before, so real dwarfs are C/O with \(\mu_e=2\).
  • \(\rho_c\to\infty\): the fully relativistic star has \(R\to0\) and \(M\to M_{\mathrm{Ch}}\) from below; the limit is the asymptote of the mass–central-density curve, saturated only in the idealised massless-electron limit.
  • Restore electron rest mass: the exact equation of state stiffens at low \(p_F\), so the true maximum mass sits just under \(M_{\mathrm{Ch}}\) at a large but finite central density (\(\rho_c\sim10^{10}\,\mathrm{g\,cm^{-3}}\)).
  • General-relativistic correction: including the \(\mathrm{O}(GM/Rc^2)\) term destabilises the star at a finite \(\rho_c\), lowering the maximum by \(\sim1\%\) and turning the marginal \(n=3\) neutrality into genuine instability.
Breaks when
  • Inverse \(\beta\)-decay / neutronisation at high density. When \(\varepsilon_F\) exceeds the electron-capture threshold of the nuclei (\(\sim1.4\times10^{9}\,\mathrm{g\,cm^{-3}}\) for \(^{56}\mathrm{Fe}\)), electrons are absorbed, \(n_e\) and the pressure drop, and the star collapses before the ideal-gas \(M_{\mathrm{Ch}}\) is reached. The equation of state is no longer the pure \(n=3\) polytrope.
  • General relativity. An exactly \(n=3\) polytrope is marginally (neutrally) stable in Newtonian gravity; the first post-Newtonian correction is destabilising, so the real maximum is a genuine turning point at finite \(\rho_c\), slightly below the Newtonian value. Near a neutron-star-forming collapse the Newtonian treatment fails entirely.
  • Rotation and magnetic fields. Rapid differential rotation or ultra-strong fields add support absent from the static spherical model, permitting "super-Chandrasekhar" white dwarfs up to \(\sim2\,M_\odot\) — invoked to explain over-luminous Type Ia events.
  • Finite temperature (young/hot dwarfs). When \(k_BT\) is not negligible against \(\varepsilon_F\), thermal pressure adds support and the star is not a cold polytrope; the strict radius-independent mass only emerges after the dwarf cools.
Failure modes
  • Using the non-relativistic pressure \(P\propto n_e^{5/3}\) (the \(n=\tfrac32\) polytrope) and still concluding there is a mass limit. There is none: only the ultrarelativistic \(\tfrac43\) exponent produces the \(\rho_c\)-independent mass. The limit is a relativistic effect.
  • Writing \(n_e=\rho/m_H\), forgetting the mean molecular weight per electron \(\mu_e\). This inflates \(n_e\) by \(\mu_e\), the pressure by \(\mu_e^{4/3}\), and gives a mass wrong by the factor \(\mu_e^{2}=4\).
  • Confusing the polytropic index \(n=3\) with the electron number density \(n_e\); they are unrelated symbols that both appear in the derivation.
  • Claiming the mass depends on radius or central density. The entire content of the result is that at \(n=3\) it does not — the \(a^3\rho_c\) product is \(\rho_c\)-free.
  • Putting the electron mass \(m_e\) into \(\rho\). The mass is carried by nucleons; \(m_e/m_H\sim1/1836\) is negligible for the mass but the electrons still provide all the pressure.
  • Dropping the factor \(\tfrac13\) in \(P=\tfrac13 n\langle pv\rangle\) or forgetting \(v\to c\) in the ultrarelativistic limit, corrupting the coefficient of \(K\) and hence the numerical \(1.44\,M_\odot\).
  • Taking \(-\xi_1^2\theta'_1\) from the wrong polytrope (e.g. the \(n=\tfrac32\) value) instead of the \(n=3\) value \(2.018\).
Discussion

The physical origin of the limit is a scaling coincidence between two energies. Gravitational binding scales as \(E_{\mathrm{grav}}\sim -GM^2/R\), while the internal energy of an ultrarelativistic degenerate gas scales as \(E_{\mathrm{int}}\sim \hbar c\,N_e^{4/3}/R\) — both as \(1/R\). Their sum is therefore \(E\sim(\,\text{const})/R\) with a coefficient that is either positive (pressure wins, the star expands until non-relativistic stiffening halts it) or negative (gravity wins, unbounded collapse). The sign flips at a single critical mass, and radius drops out because both energies share the same \(1/R\) dependence. Non-relativistically the internal energy scales as \(1/R^2\), always beating gravity's \(1/R\) at small \(R\), which is why every mass then has a stable minimum. The Chandrasekhar limit is the mass at which this competition becomes scale-invariant.

The appearance of \((\hbar c/G)^{3/2}/m_H^2\) is profound: it is (up to \(\mu_e\) and \(\mathrm{O}(1)\) numbers) the Planck mass cubed over the nucleon mass squared, \(M_{\mathrm{Pl}}^3/m_H^2\). Equivalently \(M_{\mathrm{Ch}}\sim m_H\,\alpha_G^{-3/2}\), where \(\alpha_G=Gm_H^2/\hbar c\approx6\times10^{-39}\) is the gravitational fine-structure constant for nucleons. The staggering number \(\alpha_G^{-3/2}\approx2\times10^{57}\) is simply how many nucleons must be assembled before their collective gravity overwhelms quantum degeneracy — a stellar mass expressed purely in fundamental constants. That a star's fate is written in \(\hbar\), \(c\), \(G\) and \(m_H\) is one of physics' great unifications of the quantum, the relativistic, and the gravitational.

The marginal stability of the \(n=3\) polytrope repays closer scrutiny. For a polytrope the total energy behaves as \(E\propto (3\gamma-4)\); at \(\gamma=\tfrac43\) this vanishes and the star is neutrally stable — infinitesimally perturbing \(R\) costs no energy to leading order. The Newtonian \(M_{\mathrm{Ch}}\) is thus not a true maximum but a degenerate ridge; it is the higher-order corrections (finite electron mass stiffening it below, general relativity softening it) that resolve the ridge into a genuine turning point in the mass–radius relation, \(dM/d\rho_c=0\), which is where dynamical instability actually sets in. This is why realistic maximum-mass white dwarfs sit at \(\approx1.4\,M_\odot\) with central densities \(\sim10^{9}\text{–}10^{10}\,\mathrm{g\,cm^{-3}}\) rather than at literally infinite density.

Common misconceptions. The limit is often described as "the mass at which electrons move at the speed of light" — but electrons are already relativistic well below it; the limit is where the relativistic equation of state can no longer scale to support more mass. It is also not a statement that degeneracy pressure "runs out" or has a maximum value — pressure keeps rising with density; rather, it rises too slowly (as \(\rho^{4/3}\)) to keep pace with the gravitational demand \(\propto\rho^{4/3}\) at fixed mass, so no larger mass finds equilibrium. Finally, \(M_{\mathrm{Ch}}\) is a property of cold matter, independent of the star's history; a Type Ia progenitor reaches it by accretion, but the number itself is thermodynamic, not evolutionary.

Worked examples
1
\[ \text{Evaluate } M_{\mathrm{Ch}} \text{ for a carbon–oxygen white dwarf},\quad \mu_e=2 \]
Use \(M_{\mathrm{Ch}}=\tfrac{\sqrt{3\pi}}{2}\omega_3^{0}(\hbar c/G)^{3/2}/(\mu_e m_H)^2\) with \(\omega_3^{0}=2.018\), \(\hbar=1.0546\times10^{-34}\,\mathrm{J\,s}\), \(c=2.998\times10^{8}\,\mathrm{m\,s^{-1}}\), \(G=6.674\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}\), \(m_H=1.673\times10^{-27}\,\mathrm{kg}\). Symbols first, then numbers. A
\[ \frac{\sqrt{3\pi}}{2}\,\omega_3^{0}=\frac{\sqrt{9.4248}}{2}\,(2.018)=(1.535)(2.018)=3.098 \]
\[ \frac{\hbar c}{G}=\frac{(1.0546\times10^{-34})(2.998\times10^{8})}{6.674\times10^{-11}}=4.737\times10^{-16}\,\mathrm{kg^2} \]
\[ \left(\frac{\hbar c}{G}\right)^{3/2}=(4.737\times10^{-16})^{3/2}=1.031\times10^{-23}\,\mathrm{kg^3} \]
\[ (\mu_e m_H)^2=(2\times1.673\times10^{-27})^2=1.119\times10^{-53}\,\mathrm{kg^2} \]
\[ M_{\mathrm{Ch}}=3.098\times\frac{1.031\times10^{-23}}{1.119\times10^{-53}}=2.854\times10^{30}\,\mathrm{kg} \]
\[ M_{\mathrm{Ch}}=2.85\times10^{30}\,\mathrm{kg}=1.44\,M_\odot \]

Reading. With \(M_\odot=1.989\times10^{30}\,\mathrm{kg}\), the result is \(1.44\,M_\odot\), the canonical Chandrasekhar mass. Every input was a fundamental constant plus the composition number \(\mu_e=2\); no stellar-structure detail entered. Observed massive white dwarfs cluster just below this value, and Type Ia progenitors are believed to detonate on approaching it.

1
\[ \text{At what density do the electrons become ultrarelativistic?}\quad \text{Set } p_F=m_e c \]
The polytrope assumption requires \(p_F\gtrsim m_e c\). Find the electron density and mass density at the crossover from \(p_F=\hbar(3\pi^2 n_e)^{1/3}=m_e c\), with \(\mu_e=2\), \(m_e=9.109\times10^{-31}\,\mathrm{kg}\). B
\[ n_e=\frac{1}{3\pi^2}\left(\frac{m_e c}{\hbar}\right)^3 \]
\[ \frac{m_e c}{\hbar}=\frac{(9.109\times10^{-31})(2.998\times10^{8})}{1.0546\times10^{-34}}=2.590\times10^{12}\,\mathrm{m^{-1}} \]
\[ n_e=\frac{(2.590\times10^{12})^3}{3\pi^2}=\frac{1.737\times10^{37}}{29.61}=5.87\times10^{35}\,\mathrm{m^{-3}} \]
\[ \rho=\mu_e m_H\,n_e=(2)(1.673\times10^{-27})(5.87\times10^{35})=1.96\times10^{9}\,\mathrm{kg\,m^{-3}} \]
\[ \rho_{\text{rel}}\approx2.0\times10^{9}\,\mathrm{kg\,m^{-3}}=2.0\times10^{6}\,\mathrm{g\,cm^{-3}} \]

Reading. Above a few times \(10^{6}\,\mathrm{g\,cm^{-3}}\) the electrons are relativistic and the \(n=3\) polytrope applies. Low-mass white dwarfs (central density \(\sim10^{5}\,\mathrm{g\,cm^{-3}}\)) are non-relativistic and lie on the stable \(n=\tfrac32\) branch; only the massive dwarfs approaching \(M_{\mathrm{Ch}}\), with \(\rho_c\gtrsim10^{6}\text{–}10^{9}\,\mathrm{g\,cm^{-3}}\), are governed by the relativistic softening. This is why the mass limit is a property of the densest, most massive white dwarfs.

Problems
  1. An iron white dwarf has \(\mu_e=A/Z=56/26=2.15\). Compute its Chandrasekhar mass and compare with the carbon–oxygen value.
    Solution Since \(M_{\mathrm{Ch}}\propto\mu_e^{-2}\), \(M_{\mathrm{Ch}}(\mathrm{Fe})=1.44\,(2/2.15)^2\,M_\odot=1.44\times0.865\,M_\odot=1.24\,M_\odot\). An iron core is a fifteen-percent lighter limit than a C/O dwarf because each electron drags along more nucleon mass (\(\mu_e\) larger), so gravity reaches the critical balance at a smaller total mass. This lower iron limit is relevant to the collapsing cores of massive stars.
  2. Show that a non-relativistic degenerate electron gas gives an \(n=\tfrac32\) polytrope, and hence a mass–radius relation \(R\propto M^{-1/3}\) with no mass limit.
    Solution Non-relativistically \(\varepsilon=p^2/2m_e\), \(v=p/m_e\), so \(P=\tfrac13\cdot\tfrac{2}{(2\pi\hbar)^3}\!\int_0^{p_F}\!\tfrac{p}{m_e}p\,4\pi p^2dp=\tfrac{(3\pi^2)^{2/3}}{5}\tfrac{\hbar^2}{m_e}n_e^{5/3}\propto\rho^{5/3}\). Thus \(1+\tfrac1n=\tfrac53\Rightarrow n=\tfrac32\). For a polytrope \(M\propto\rho_c^{(3-n)/2n}a^3\rho_c\); with \(a\propto\rho_c^{(1-n)/2n}\) one finds \(M\propto\rho_c^{(3-n)/(2n)}\) and \(R\propto\rho_c^{(1-n)/(2n)}\). Eliminating \(\rho_c\) for \(n=\tfrac32\) gives \(M\propto R^{-3}\), i.e. \(R\propto M^{-1/3}\): heavier dwarfs are smaller, and every mass has a stable equilibrium. The mass never cancels, so there is no limit — the limit is a purely relativistic phenomenon.
  3. Estimate the electron Fermi momentum (in units of \(m_e c\)) and \(p_F c\) in MeV at \(\rho=10^{6}\,\mathrm{g\,cm^{-3}}\), \(\mu_e=2\), and comment on whether the ultrarelativistic approximation is good there.
    Solution \(\rho=10^{9}\,\mathrm{kg\,m^{-3}}\Rightarrow n_e=\rho/(\mu_e m_H)=10^{9}/(2\times1.673\times10^{-27})=2.99\times10^{35}\,\mathrm{m^{-3}}\). Then \(p_F=\hbar(3\pi^2 n_e)^{1/3}=(1.0546\times10^{-34})(3\pi^2\times2.99\times10^{35})^{1/3}=(1.0546\times10^{-34})(2.07\times10^{12})=2.18\times10^{-22}\,\mathrm{kg\,m\,s^{-1}}\). So \(p_F c=(2.18\times10^{-22})(2.998\times10^{8})=6.54\times10^{-14}\,\mathrm{J}=0.41\,\mathrm{MeV}\), and \(p_F/m_e c=0.41/0.511=0.80\). The electrons are mildly relativistic but not yet ultrarelativistic; the pure \(n=3\) polytrope is a good approximation only at higher densities (\(\gtrsim10^{7}\,\mathrm{g\,cm^{-3}}\)), consistent with Worked Example 2. This is why the exact maximum mass sits slightly below the idealised \(1.44\,M_\odot\).
  4. Derive the composition scaling \(M_{\mathrm{Ch}}\propto\mu_e^{-2}\) directly from the polytrope constant, and compute the (hypothetical) hydrogen limit \(\mu_e=1\).
    Solution From \(K=\tfrac{(3\pi^2)^{1/3}}{4}\hbar c/(\mu_e m_H)^{4/3}\), we have \(K\propto\mu_e^{-4/3}\). Since \(M_{\mathrm{Ch}}\propto K^{3/2}\) (step 9), \(M_{\mathrm{Ch}}\propto(\mu_e^{-4/3})^{3/2}=\mu_e^{-2}\). Hence \(M_{\mathrm{Ch}}(\mu_e)=1.44\,(2/\mu_e)^2\,M_\odot\). For \(\mu_e=1\): \(M_{\mathrm{Ch}}=1.44\times4=5.76\,M_\odot\). This is only hypothetical: a hydrogen white dwarf would ignite thermonuclear hydrogen burning long before reaching such a mass, so real degenerate dwarfs are helium, carbon, or oxygen with \(\mu_e=2\).
  5. Advanced (energy method). Model the star as a uniform sphere with gravitational energy \(E_{\mathrm{grav}}=-\tfrac35 GM^2/R\) and ultrarelativistic degeneracy energy \(E_{\mathrm{int}}=\tfrac34 N_e\,p_F c\) with \(p_F=\hbar(3\pi^2 n_e)^{1/3}\), \(n_e=N_e/(\tfrac43\pi R^3)\), \(M=\mu_e m_H N_e\). Show both energies scale as \(1/R\), find the critical mass where they balance, and compare the coefficient to the exact \(3.098\).
    Solution Mean UR energy per electron is \(\tfrac34 p_F c\), so \(E_{\mathrm{int}}=\tfrac34 N_e\hbar c(3\pi^2 n_e)^{1/3}\). With \(n_e=N_e/(\tfrac43\pi R^3)\), \((3\pi^2 n_e)^{1/3}=(9\pi/4)^{1/3}N_e^{1/3}/R\), giving \(E_{\mathrm{int}}=\tfrac34\hbar c(9\pi/4)^{1/3}N_e^{4/3}/R\propto1/R\), the same \(1/R\) as \(E_{\mathrm{grav}}=-\tfrac35 GM^2/R\). The total \(E=[\tfrac34\hbar c(9\pi/4)^{1/3}N_e^{4/3}-\tfrac35 GM^2]/R\). If the bracket is positive, \(E>0\) falls as \(R\) grows — the star expands (and non-relativistic stiffening eventually halts it, giving a stable dwarf); if negative, \(E\to-\infty\) as \(R\to0\) — collapse. The critical mass sets the bracket to zero: \(\tfrac34\hbar c(9\pi/4)^{1/3}N_e^{4/3}=\tfrac35 G(\mu_e m_H N_e)^2\), so \(N_e^{2/3}=\tfrac{5}{4}(9\pi/4)^{1/3}\hbar c/[G(\mu_e m_H)^2]\), whence \(M=\mu_e m_H N_e=\left[\tfrac54(9\pi/4)^{1/3}\right]^{3/2}(\hbar c/G)^{3/2}/(\mu_e m_H)^2\). The coefficient is \(\left[\tfrac54(1.919)\right]^{3/2}=(2.399)^{3/2}=3.72\), versus the exact \(3.098\) — the same fundamental-constant combination and only \(20\%\) high, the error coming entirely from the crude uniform-density assumption. Numerically \(M\approx1.72\,M_\odot\), confirming the order of the Chandrasekhar mass from energy balance alone.