White Dwarfs and Neutron Stars

White dwarfs and neutron stars are supported by quantum matter, but they probe very different regimes of it. Electron degeneracy explains the white-dwarf sequence and its limiting mass; neutron stars require relativistic gravity, nuclear interactions, layered phases, superfluid dynamics, and long-term thermal and magnetic evolution. This chapter builds those connections without turning a clean limiting model into an overconfident description of a real star. Each observable is traced back to the assumptions that make it diagnostic.

Enter this chapter

Read the six pages in the following sequence:

1. Degenerate Fermion Support derives the zero-temperature Fermi-gas pressure and its nonrelativistic and ultrarelativistic limits.
2. The Chandrasekhar Mass and White-Dwarf Sequences turns the ultrarelativistic electron gas into a global $n=3$ mass scale and then restores the microphysics that determines real endpoints.
3. Neutron-Star Composition maps the crust and core using charge neutrality and weak equilibrium while separating robust constraints from candidate high-density phases.
4. Maximum Neutron-Star Mass defines the cold nonrotating limit, distinguishes rotating and transient support, and shows how mass likelihoods constrain EOS families.
5. Crust, Superfluidity, and Glitches connects lattice elasticity, quantized vortices, entrainment, and coupling to sudden angular-momentum transfer.
6. Cooling and Magnetic Evolution closes the chapter with the redshifted energy budget, magnetothermal feedback, and a calibrated X-ray inference example.

The first two pages form one derivation. Page 3 changes from an almost ideal gas to interacting matter, and page 4 inserts that matter into relativistic structure. Pages 5 and 6 ask how an equilibrium star evolves and how its hidden interior can be inferred. Readers interested only in cold hydrostatic sequences should still read the observational cautions on page 4 before using a measured mass as a sharp cutoff.

One quantum principle, two stellar regimes

At zero temperature, Pauli exclusion fills momentum states up to a Fermi momentum $p_F$. For a free spin-$1/2$ species,

$$n=\frac{p_F^3}{3\pi^2}.$$

When $p_F\ll m$, the pressure scales as $n^{5/3}/m$; when $p_F\gg m$, it scales as $n^{4/3}$. In a white dwarf, nuclei carry most of the mass and electrons provide most of the pressure, so charge neutrality introduces the composition factor $\mu_e$. The ultrarelativistic limit produces an $n=3$ polytrope whose mass no longer depends on central density. Chandrasekhar’s original limiting calculation displays the mass scale in Chandrasekhar 1931, Eqs. (2)–(6), pp. 81–82.

That limit is an organizing scale, not a promise that every white dwarf reaches it. Electron capture, Coulomb corrections, general relativity, composition, temperature, crystallization, and rotation alter a physical sequence. The radius shrinks as electrons become relativistic, so a large mass and a small radius are correlated consequences of the same change in pressure law.

In a neutron star, gravity compresses matter beyond neutron drip and nuclear saturation. The pressure is no longer the kinetic pressure of one free species: strong interactions and composition derivatives of the energy density are essential. Degeneracy remains part of the physics, but it cannot by itself predict a neutron-star radius, maximum mass, or core inventory.

From an EOS to observables

A candidate cold EOS supplies $p(\varepsilon)$ for the stellar-structure equations. Charge neutrality and beta equilibrium constrain the fractions, while nuclear masses and interactions select crustal phases. Chamel and Haensel develop the relation among crust composition, elasticity, superfluidity, and transport in Chamel and Haensel 2008, §§3–4 and 8. Uncertainty grows with density: nuclear calculations are best controlled near saturation, whereas the center of a massive star may sample several times that density.

The resulting stable sequence has a largest nonrotating mass $M_{\rm TOV}$, defined across all admissible stable branches. A measured mass supplies a probabilistic lower constraint on the maximum mass at that star’s spin. It constrains $M_{\rm TOV}$ after rotational support has been modeled or shown to be negligible. It does not identify one pressure or composition because many EOS curves can support the same mass. Uniform rotation, differential rotation, and thermal pressure define different ceilings and must not be substituted for the old, cold pulsar problem.

Timing and thermal data access different dynamics. A glitch constrains angular-momentum storage and coupling but not a unique vortex trigger. An X-ray spectrum constrains surface emission after atmosphere, redshift, absorption, and detector response; mapping it to core pairing requires a cooling model. The coupled energy and induction equations, together with their numerical assumptions, are reviewed in Pons, Dehman, and Viganò 2026, §§2.2–2.3, 3.1–3.2, and 4.2–4.4.

Checks that travel across the chapter

Several checks recur. Thermodynamic derivatives used for pressure, chemical potentials, and sound speed must come from the same energy model. Every stellar profile must be regular at the center and matched consistently at interfaces and the surface. A mass extremum must be interpreted with the stability hypotheses stated. Angular momentum must close across a glitch, and redshifted energy must close across a cooling calculation.

There are also limiting tests. A degenerate-gas calculation should reproduce both $p_F/m\ll1$ and $p_F/m\gg1$ asymptotics. Removing interactions from a composition model should recover the corresponding free chemical potentials. Suppressing pinning removes one way to sustain a rotational lag, although weak coupling may also allow relative rotation; suppressing magnetic feedback should recover ordinary cooling. These checks can reject a model implementation without pretending to validate the uncertain high-density microphysics.

Connections to other volumes

Relativistic Stellar Structure provides the TOV, baryon-number, sequence, and radial-stability machinery used here. Rotating and Magnetized Stars develops the two-dimensional equilibria needed beyond spherical mass limits. Pulsars and Magnetospheres follows the observable spin and magnetic torque, while later chapters treat stellar modes, collapse, and binaries. Nuclear priors, mass–radius likelihoods, and multimessenger EOS inference are assembled in the final chapter rather than silently combined on these microphysics pages.

The chapter deliberately stops short of treating a fit as a detection of a phase. Existence of a thermodynamic phase, stability of a star containing it, formation of that branch, and observational preference are four separate claims. Keeping them separate is the main defense against reading a compact star more precisely than the data and theory allow.

Review the chapter

An idealized cold star has electron number density $n_e$, baryon density $n_b=2n_e$, and ultrarelativistic electrons. A second model doubles $n_b$ without changing the electron fraction. By what factor do $p_F$ and the ideal electron pressure change, and does this determine the maximum neutron-star mass?

Solution

At fixed electron fraction, doubling $n_b$ doubles $n_e$. Since $p_F=(3\pi^2n_e)^{1/3}$, the Fermi momentum increases by $2^{1/3}$. In the ultrarelativistic limit $p\propto n_e^{4/3}$, so the electron pressure increases by $2^{4/3}$. This scaling helps explain electron support in white dwarfs but does not determine the neutron-star maximum mass: strong-interaction pressure, beta-equilibrated composition, causality, and the full relativistic stellar sequence are required.

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.” The Astrophysical Journal 74 (1931): 81–82. DOI.
• Pons, José A., Clara Dehman, and Daniele Viganò. “Magnetic, Thermal and Rotational Evolution of Isolated Neutron Stars.” Living Reviews in Computational Astrophysics 12 (2026): 5. DOI.