Degenerate Fermion Support

A cold compact star can resist compression even when its thermal pressure is negligible. Pauli exclusion fills available momentum states up to a Fermi momentum even at zero temperature. The resulting momentum flux remains nonzero when thermal excitations disappear. The same phase- space argument produces two different polytropic limits: nonrelativistic fermions give $p\propto n^{5/3}$, whereas ultrarelativistic fermions soften to $p\propto n^{4/3}$. That change of exponent is the essential reason an ideal electron-supported star develops a limiting mass.

Required background. The Einstein Equation supplies the gravitational equilibrium in which the pressure is used.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies energy and pressure conventions; Relativistic Matter and Compact Objects supplies thermodynamics and EOS context; Computational Relativity supplies numerical sequence checks.

Filling the Fermi sphere

Assume a uniform, zero-temperature gas of spin-$1/2$ fermions with two spin states and no interactions. In units $\hbar=c=1$, all momentum states inside a sphere of radius $p_F$ are occupied. The number density is

$$n=2\int_{\lvert\boldsymbol p\rvert\le p_F} \frac{d^3p}{(2\pi)^3} =\frac{p_F^3}{3\pi^2}.$$

The isotropic pressure is one third of the momentum flux,

$$p=\frac{1}{3\pi^2} \int_0^{p_F}\frac{p^4}{\sqrt{p^2+m_f^2}}\,dp,$$

and the total energy density is

$$\epsilon=\frac{1}{\pi^2} \int_0^{p_F}p^2\sqrt{p^2+m_f^2}\,dp.$$

These expressions include rest energy. Subtracting $m_fn$ defines internal energy but must not be substituted for $\epsilon$ in the Einstein equation. The electron-gas specialization and composition conventions are summarized in Chamel and Haensel 2008, §2.1, Eqs. (1)–(9), and §3.1, Eq. (22).

Nonrelativistic and relativistic limits

When $p_F\ll m_f$, expand the single-particle energy and obtain

$$p_{\rm NR} =\frac{(3\pi^2)^{2/3}}{5m_f}n^{5/3}.$$

When $p_F\gg m_f$, the velocity approaches unity and

$$p_{\rm UR} =\frac{(3\pi^2)^{1/3}}{4}n^{4/3}, \qquad \epsilon\simeq3p.$$

An independent check uses the zero-temperature chemical potential $\mu=\sqrt{p_F^2+m_f^2}$. The first law and Gibbs–Duhem relation give $d\epsilon=\mu\,dn$ and $dp=n\,d\mu$, so

$$c_s^2=\frac{dp}{d\epsilon} =\frac{n}{\mu}\frac{d\mu}{dn} =\frac{p_F^2}{3(p_F^2+m_f^2)}.$$

It approaches $p_F^2/(3m_f^2)$ in the nonrelativistic limit and $1/3$ in the ultrarelativistic limit. Thus the ideal gas remains acoustically causal through the crossover, and its sound speed is not the ratio $p/\epsilon$ except in the ultrarelativistic limit.

The dimensions provide an immediate check: number density has dimension energy cubed, so each pressure scales as energy to the fourth power. The ultrarelativistic expression loses explicit dependence on $m_f$ because the rest mass no longer controls particle velocity.

For white-dwarf matter, charge neutrality gives $n_e=Y_e n_B=\rho/(\mu_e m_u)$ to the accuracy that ionic rest mass dominates the mass density. Here $Y_e$ is electrons per baryon and $\mu_e=1/Y_e$. Consequently the pressure constant carries composition: $p_{\rm NR}\propto\mu_e^{-5/3}\rho^{5/3}$ and $p_{\rm UR}\propto\mu_e^{-4/3}\rho^{4/3}$.

Temperature, interactions, and validity

Degeneracy is controlled by $T/T_F$, where the Fermi temperature is the Fermi energy measured from rest mass. A Sommerfeld expansion gives leading thermal corrections of order $(T/T_F)^2$ for a strongly degenerate gas. “Cold” means $T\ll T_F$, not absolute zero. The ions can still provide entropy, heat capacity, and crystallization physics while electrons dominate pressure.

The ideal gas omits Coulomb interactions, lattice energy, exchange, finite nuclear size, inverse beta decay, and general-relativistic gravity. In white dwarfs these are corrections or termination mechanisms whose importance grows with density. In neutron stars, interacting baryons rather than free electrons carry most pressure, so reusing the electron formula would be a model error. The neutron-star crust nevertheless contains highly degenerate electrons and offers a controlled domain for the same phase-space calculation.

The derivation also clarifies what “degeneracy support” does not mean. The Pauli principle fixes occupation, but pressure is the derivative of the many- particle energy with respect to volume. A fermion species contributes little support if its number fraction is tiny, and a large Fermi energy does not make interactions negligible. For neutrons near and above nuclear density, strong forces and pairing reorganize both energy and response even though the matter remains highly degenerate.

Support and the approach to a limiting mass

For a Newtonian polytrope $p=K\rho^{1+1/n_p}$, the nonrelativistic exponent $5/3$ corresponds to $n_p=3/2$ and produces a radius that decreases as mass increases. The ultrarelativistic exponent $4/3$ corresponds to $n_p=3$; its mass scale is independent of central density. Chandrasekhar identifies this limiting behavior in Chandrasekhar 1931, Eqs. (2)–(5), pp. 81–82. The numerical coefficient requires the stellar density profile, not merely dimensional analysis.

This ideal limit is not automatically the maximum mass of every real white dwarf. Composition changes, capture thresholds, Coulomb corrections, rotation, magnetic fields, and relativistic gravity alter the endpoint. The robust conclusion here is the loss of increasing pressure stiffness as electrons become relativistic.

Check the polytropic exponents

Use $p\propto p_F^5/m_e$ in the nonrelativistic limit and $p\propto p_F^4$ in the ultrarelativistic limit to recover the density exponents.

Solution

The filled-state count gives $n_e\propto p_F^3$, hence $p_F\propto n_e^{1/3}$. Therefore $p_{\rm NR}\propto p_F^5/m_e\propto n_e^{5/3}$, while $p_{\rm UR}\propto p_F^4\propto n_e^{4/3}$. At fixed composition $n_e\propto\rho$, so the same exponents apply to mass density. The change is caused by the velocity factor: $v\simeq p/m_e$ when nonrelativistic but $v\simeq1$ when ultrarelativistic.

References

• Chamel, Nicolas, and Pawel Haensel. “Physics of Neutron Star Crusts.” Living Reviews in Relativity 11 (2008): 10. DOI.
• Chandrasekhar, Subrahmanyan. “The Maximum Mass of Ideal White Dwarfs.” Astrophysical Journal 74 (1931): 81–82. DOI.