Neutron Stars and Dense Matter

Above the Chandrasekhar mass, electron capture removes the electrons that hold a white dwarf up, and the core collapses until the density reaches that of an atomic nucleus. There the strong interaction and the degeneracy pressure of a new fermion, the neutron, arrest the collapse. The result is a neutron star: a body of one to two solar masses compressed into a radius of about ten kilometres, with a mean density comparable to nuclear matter. The Fermi-gas calculation that fixed the white dwarf carries over with the neutron mass in place of the electron mass, but general relativity now enters at leading order.

Neutron degeneracy pressure

At nuclear density the reaction $p + e^{-} \to n + ν_{e}$ runs to near completion: the matter is almost pure neutrons, with a small residual proton-electron fraction set by beta equilibrium. Neutrons are spin- $\frac{1}{2}$ fermions, so the degeneracy analysis of the ideal Fermi gas applies with degeneracy $g_{s} = 2$ . The decisive change from the white dwarf is that the neutrons now supply both the pressure and the mass. There is no separate heavy species, so the composition factor $μ_{e}$ disappears and the mass density is

ρ = m_{n} n,

with $n$ the neutron number density. The nonrelativistic degeneracy pressure is the standard result with the neutron mass,

P = \frac{( 3 π ^{2} ) ^{2/3}}{5} \frac{ℏ ^{2}}{m _{n}} n^{5/3} = K_{n} ρ^{5/3}, K_{n} = \frac{( 3 π ^{2} ) ^{2/3}}{5} \frac{ℏ ^{2}}{m _{n}^{8/3}} .

At a fixed number density the neutron pressure is smaller than the electron pressure by the mass ratio $m_{e} / m_{n} \approx 1/1839$ , because degeneracy pressure scales as $1/ m$ . On logarithmic axes the two pressures are parallel lines of slope $\frac{5}{3}$ , the neutron line lying below the electron line.

Degeneracy pressure scales as

1/ m

, so at a fixed number density the electron gas (upper line) is far stiffer than the neutron gas (lower, dashed); both grow as

n^{5/3}

until their particles turn relativistic.

Rescaling the white-dwarf calculation

The Newtonian mass-radius relation transfers directly. Balancing the nonrelativistic neutron pressure against gravity gives

R \sim \frac{ℏ ^{2}}{G m _{n}^{8/3}} M^{- 1/3},

the white-dwarf result with $m_{e} (μ_{e} m_{u})^{5/3}$ replaced by $m_{n}^{8/3}$ . At a fixed mass the radius ratio between the two objects is

\frac{R _{NS}}{R _{WD}} = \frac{m _{e} ( μ _{e} m _{u} ) ^{5/3}}{m _{n}^{8/3}} \approx \frac{m _{e}}{m _{n}} (\frac{2 m _{u}}{m _{n}})^{5/3} \approx 1.7 \times 1 0^{- 3} .

A neutron star is about a thousand times smaller than a white dwarf of equal mass: the same solar mass that fills an Earth-sized white dwarf sits in a radius near $10$ – $12 km$ . The mean density then reaches $ρ \sim 1 0^{17}$ – $1 0^{18} kg m^{- 3}$ , comparable to the nuclear saturation density $ρ_{0} \approx 2.3 \times 1 0^{17} kg m^{- 3}$ . A neutron star is, to order of magnitude, a single nucleus with the mass of a star.¹

A neutron star has roughly the density of an atomic nucleus,

\sim 1 0^{17} kg m^{- 3}

, some fourteen orders of magnitude above ordinary matter; it is a nucleus-density object with the mass of a star and a radius of order ten kilometres.

Where the ideal-gas argument breaks

Two effects make the Newtonian ideal-neutron-gas estimate quantitatively wrong, even as it gets the scale right.

General relativity. The compactness of a neutron star, $GM / R c^{2} \sim 0.2$ , is not small; for a white dwarf it is $\sim 1 0^{- 4}$ and for the Sun $\sim 1 0^{- 6}$ . Newtonian hydrostatic equilibrium is replaced by the Tolman-Oppenheimer-Volkoff equation, in which pressure itself gravitates and the effective gravity is stronger than the Newtonian value. General relativity is destabilizing: it lowers the maximum mass relative to the Newtonian prediction.
Nuclear interactions. At and above $ρ_{0}$ the neutrons are not free. The strong interaction is repulsive at short range and stiffens the equation of state well beyond the ideal $P \propto ρ^{5/3}$ , while its attractive part and the appearance of new particles (hyperons, possibly deconfined quarks) soften it. The pressure-density relation at these densities is not known from first principles.

The pure ideal-neutron-gas calculation with the TOV equation, carried out by Oppenheimer and Volkoff, gives a maximum mass near $0.7 M_{⊙}$ — below observed neutron-star masses. The discrepancy is the signature that interactions dominate: the measured masses require a stiffer equation of state than the free gas provides.²

The size of the general-relativistic correction is measured by the compactness. At $GM / R c^{2} \sim 0.2$ the surface gravitational redshift is of order $30%$ and the escape velocity is a third of the speed of light. In the TOV equation this enters three ways: the source of gravity is $ρ c^{2} + P$ rather than $ρ c^{2}$ , so pressure itself gravitates; the enclosed mass includes the gravitational binding energy; and the metric factor $(1 - 2 G m / r c^{2})^{- 1}$ steepens the pressure gradient. Each effect strengthens the effective gravity, so a Newtonian star that would be stable becomes unstable in general relativity. Unlike the Chandrasekhar limit, which follows from the relativistic softening of the electron pressure, the neutron-star maximum mass is set jointly by that softening and by relativistic gravity.

The maximum mass

The TOV equation with a given equation of state yields a mass-radius curve with a maximum: a turning point beyond which no stable star exists, the relativistic analog of the Chandrasekhar limit. Its location depends on the uncertain dense-matter physics, so it is quoted as a band rather than a single number. Two observational handles pin it down:

Pulsars with masses near $2 M_{⊙}$ measured from binary timing require the equation of state to be stiff enough to support them, ruling out the softest models.
The tidal deformability inferred from the neutron-star merger GW170817 bounds the stiffness from above, disfavoring the hardest models.

Together these place the maximum neutron-star mass, the TOV limit, at roughly $2.2$ – $2.3 M_{⊙}$ . A remnant heavier than this cannot be supported by any known pressure and collapses to a black hole.

The neutron-star mass-radius relation is nearly vertical over a spread of masses and turns over at the TOV maximum mass; the exact curve is a band because the dense-matter equation of state is uncertain.

The compact-object sequence

The degenerate remnants form a sequence ordered by mass, each supported by a different pressure and each with its own upper bound.

Remnant	Support	Mass scale	Radius
White dwarf	electron degeneracy	up to $M_{Ch} \approx 1.4 M_{⊙}$	$\sim 1 0^{4} km$
Neutron star	neutron degeneracy $+$ nuclear forces	$\sim 1.1$ – $2.2 M_{⊙}$	$\sim 10 km$
Black hole	none	above the TOV limit	horizon $\sim GM / c^{2}$

Each degeneracy pressure caps the mass it can support: electrons at the Chandrasekhar mass, neutrons at the TOV limit. A collapsing core that exceeds the neutron-star bound has no remaining source of pressure and forms a black hole. The progression follows the same physics throughout: a Fermi gas whose zero-temperature pressure holds off gravity until relativistic softening, and finally relativistic gravity itself, exhausts its ability to do so.

The compact remnants lie in order of increasing mass along one axis, each supported by a distinct degeneracy pressure and each ending at the mass where that pressure fails.

Summary

Above the Chandrasekhar mass, electron capture converts the core to neutrons, whose degeneracy pressure ( $ρ = m_{n} n$ , $P \propto n^{5/3}$ ) supports a neutron star; the pressure is smaller than the electron gas by $m_{e} / m_{n}$ at fixed density.
Rescaling the white-dwarf balance with the neutron mass gives a radius near $10 km$ at a solar mass, a density comparable to nuclear matter $ρ_{0} \approx 2.3 \times 1 0^{17} kg m^{- 3}$ .
The ideal-gas estimate is only qualitative: the compactness $GM / R c^{2} \sim 0.2$ forces the general-relativistic TOV equation, and nuclear interactions set the equation of state, so the free-gas Oppenheimer-Volkoff mass ( $\sim 0.7 M_{⊙}$ ) falls short of observed masses.
Observations bracket the TOV maximum mass near $2.2$ – $2.3 M_{⊙}$ ; the remnants form the sequence white dwarf $\to$ neutron star $\to$ black hole, ordered by the mass each degeneracy pressure can hold.

The rescaling and the nuclear-density comparison follow Pathria & Beale, §8.5, and Carroll & Ostlie, Ch. 16. The nuclear saturation density is $ρ_{0} \approx 2.3 \times 1 0^{17} kg m^{- 3}$ ( $n_{0} \approx 0.16 f m^{- 3}$ ). ↩
The Oppenheimer-Volkoff limit for a free neutron gas and the role of the equation of state are developed in Shapiro & Teukolsky, Ch. 9; Pathria & Beale, §8.5, gives the ideal-gas version. ↩