Neutron Degeneracy Pressure

From electrons to neutrons

The simplest mental model is the electron degeneracy story with one substitution: replace m_e by m_n in the Fermi pressure. Pauli's exclusion principle applies just as well to neutrons as to electrons because both are spin-1/2 fermions. A cold dense neutron gas fills momentum states up to a neutron Fermi momentum p_F, and the pressure follows the same algebra:

P_NR ≈ (1/5) (3π²)^(2/3) (ℏ²/m_n) n_n^(5/3), P_UR ≈ (ℏ c / 4) (3π²)^(1/3) n_n^(4/3).

Because m_n ≈ 1839 m_e, the non-relativistic neutron pressure at a given number density is far smaller than the electron pressure at the same density. The compensating point is that neutron stars sit at densities about a billion times higher than white dwarfs (~10¹⁷ kg/m³ vs ~10⁹ kg/m³), so neutron number density is enormous and the support is still adequate — but only barely.

When is degeneracy alone enough?

Oppenheimer and Volkoff in 1939 solved the relativistic hydrostatic-equilibrium equation for a non-interacting neutron gas and found a maximum mass of about 0.7 M_sun. Observed neutron stars are typically 1.4 M_sun or more, with the most massive at 2.35 M_sun (PSR J0952−0607). The free-gas estimate is therefore inadequate by a factor of two or three.

The missing physics is the strong nuclear force. At nucleon spacing comparable to the size of a nucleon (about 1 fm), the interaction between nucleons is strongly repulsive — this is the same hard core that prevents normal nuclei from collapsing. In neutron-star matter at densities a few times ρ_0 ≈ 2.7 × 10¹⁷ kg/m³, the strong-force repulsion supplies the bulk of the pressure and the Pauli pressure becomes a floor on top of which everything else is built. Modern equations of state model this via realistic nucleon-nucleon potentials, chiral effective theory, and increasingly direct constraints from neutron-star observations.

The TOV equation and its limit

In Newtonian gravity, hydrostatic equilibrium is dP/dr = −G m(r) ρ(r) / r². In general relativity, the same equation acquires three correction factors that all push toward stronger effective gravity:

dP/dr = − G m(r) ρ(r) / r² × [1 + P/(ρ c²)] × [1 + 4π r³ P / (m c²)] × [1 − 2 G m / (r c²)]⁻¹.

This is the Tolman-Oppenheimer-Volkoff equation. The first factor expresses that pressure itself contributes to the energy density that sources gravity; the second adds the pressure inside the enclosed mass to the gravitational pull; the third is the metric correction that strengthens local gravity in the strong-field regime. Together, they shrink the maximum mass below the Newtonian polytropic estimate and give a definite TOV mass for any chosen equation of state.

Observed neutron-star masses, gravitational-wave constraints from GW170817, and X-ray pulse-profile measurements from NICER converge on M_TOV ≈ 2.0–2.3 M_sun. The best single-number summary in recent literature is M_TOV ≈ 2.16 M_sun.

Anatomy of neutron-star matter

Region	Density (kg/m³)	Composition	Dominant pressure
Outer crust	10⁹ – 4×10¹⁴	Ions + degenerate electrons	Electron degeneracy
Inner crust	4×10¹⁴ – 1.6×10¹⁷	Neutron-rich nuclei + free neutrons	Mixed: electron + free-neutron degeneracy
Outer core	1.6×10¹⁷ – ρ0	Uniform n + p + e + μ matter	Nuclear interactions + degeneracy
Saturation density	2.7×10¹⁷ (ρ0)	~ 90% n, 10% p, leptons	Strong-force repulsion dominant
Inner core	(2 – 8) ρ0	Possibly hyperons, quark matter	Model-dependent
Beyond TOV mass	—	—	No EoS suffices → black hole

Worked example — pressure of a free-neutron gas at nuclear density

Take ρ = ρ_0 = 2.7 × 10¹⁷ kg/m³ and treat neutrons as free and non-relativistic. The number density is n_n = ρ / m_n ≈ 2.7 × 10¹⁷ / 1.675 × 10⁻²⁷ ≈ 1.6 × 10⁴⁴ m⁻³. The Fermi momentum is p_F = ℏ (3π² n_n)^(1/3) ≈ 1.05 × 10⁻³⁴ × (3π² × 1.6 × 10⁴⁴)^(1/3) ≈ 3.5 × 10⁻¹⁹ kg·m/s, giving p_F / m_n c ≈ 0.70 — mildly relativistic. The non-relativistic Fermi pressure is

P_NR = (1/5)(3π²)^(2/3)(ℏ²/m_n) n_n^(5/3) ≈ 5 × 10³³ Pa.

This is gigantic by terrestrial standards (1 atm ≈ 10⁵ Pa) but only about a quarter of what a 1.4 M_sun neutron star needs at its core. The shortfall is exactly what the strong nuclear repulsion makes up. At 2 × ρ_0 the strong force already contributes most of the pressure; the Pauli pressure is the foundation, not the structure.

Why the maximum mass is finite

Two related arguments. First, polytropic: in the ultra-relativistic limit, P ∝ ρ^(4/3), which is the marginal polytrope γ = 4/3 that does not yield an R-dependent equilibrium. Mass at this index has a fixed Chandrasekhar-like value. For free neutrons it is ~0.7 M_sun; for realistic equations of state with stiff strong-force support it climbs to ~2.16 M_sun, but a finite ceiling always exists.

Second, GR-specific: pressure gravitates. Stiffer equations of state mean more pressure per density, but each unit of pressure also adds to gravity. There is a self-limit: above some density, increasing P fails to increase the supportable M because the pressure starts pulling the star inward as much as it pushes outward. Any EoS therefore has a maximum mass; the question is just what value it predicts.

Constraints from observation

• PSR J0740+6620. Shapiro-delay mass 2.08 ± 0.07 M_sun. Any EoS must support at least this mass — already ruling out the softest neutron-star equations of state.
• PSR J0952−0607. A 707 Hz "black widow" pulsar with measured mass 2.35 ± 0.17 M_sun, the highest-confidence neutron-star mass to date.
• GW170817. Binary neutron-star merger. Inspiral tidal deformabilities Λ̃ ≲ 720 disfavour stiff EoS; combined with mass measurements, the post-merger collapse timing implies M_TOV ≲ 2.2 M_sun.
• NICER X-ray timing. Pulse-profile modelling has measured radii of R = 12.4 ± 0.5 km for J0740+6620 and R ≈ 12.7 km for J0030+0451, anchoring the radius axis of the M–R diagram.
• GW190814. A 2.6 M_sun secondary in a merger with a 23 M_sun black hole. Is it the lightest BH known or the heaviest NS? If a NS, the TOV mass approaches 2.6 M_sun, in tension with most equations of state.

Exotic phases at the highest densities

At the densest interior, several speculative phases compete:

• Hyperons. Above ~2 ρ_0 it becomes energetically favourable for some neutrons to convert into hyperons (Λ, Σ, Ξ baryons containing strange quarks). Hyperons soften the equation of state, lowering the maximum mass — the "hyperon puzzle" is the tension between this softening and the existence of 2 M_sun pulsars.
• Deconfined quark matter. At a few times ρ_0, quarks may roam freely instead of being confined to nucleons. A hybrid star with a quark core and a hadronic outer core is consistent with current data; pure quark stars are not ruled out.
• Kaon condensation. A negatively charged meson condensate could form at high density, again softening the EoS.
• Colour superconductivity. Quark pairing at ultra-high density may give a coloured superconducting state with distinctive thermal and transport behaviour.

Whether any of these phases occur is the leading open question in neutron-star physics. Future radius measurements with NICER and gravitational-wave detections with LIGO-Virgo-KAGRA at design sensitivity will continue to tighten the constraints.

Brief history

Walter Baade and Fritz Zwicky proposed the existence of neutron stars in 1934, suggesting that supernovae might leave behind such objects. Oppenheimer and Volkoff in 1939 worked out the relativistic equilibrium of a non-interacting neutron gas, deriving the equation that bears their name and the ~0.7 M_sun limit. For three decades neutron stars remained theoretical curiosities — until 1967, when Jocelyn Bell Burnell detected the radio pulses from what became PSR B1919+21, the first pulsar. By the 1970s the basic picture of nuclear-force-stiffened neutron-star matter was in place; subsequent mass and radius measurements have steadily refined the equation of state. Gravitational-wave astronomy, beginning with GW170817 in 2017, added independent constraints that ground the nuclear physics in compact-object spacetime.

Common pitfalls

• Treating a neutron star as a pure free Fermi gas. That gives M_max ≈ 0.7 M_sun, below the observed minimum. Real support requires the strong nuclear repulsion.
• Calling the TOV limit "the Chandrasekhar limit for neutron stars." The label is loose. Chandrasekhar's argument relies on softening of the ultra-relativistic Fermi pressure alone; TOV requires general relativity to make pressure self-gravitate.
• Believing the TOV mass is fixed at exactly 2.16 M_sun. The number depends on the equation of state — current observational constraints give M_TOV in the range ~2.0–2.3 M_sun with central estimates near 2.16.
• Assuming any object above 2 M_sun must be a black hole. The mass gap is no longer empty; objects from 2 to 3 M_sun discovered in gravitational-wave mergers may be heavy neutron stars, light black holes, or something more exotic.
• Identifying neutron-star matter with pure neutrons. A few percent of protons (with charge-balancing electrons and muons) are always present, set by β equilibrium and free-energy minimisation.
• Ignoring rotation. Rapidly spinning neutron stars can support extra mass (~10–20%) via centrifugal support — Keplerian mass limits sit above the static TOV value. A short-lived hypermassive remnant from a merger can briefly exceed M_TOV before collapsing.