Neutron Star Properties

• Neutron star properties are defined by extreme densities, compact sizes, and strong relativistic effects, as described by the TOV equations.
• Their microphysics involve layered compositions with degenerate matter, exotic states, and transitions governed by dense-matter equations of state.
• Observations across radio, X-ray, and gravitational waves tightly constrain mass, radius, and tidal deformability, advancing dense-matter research.

A neutron star is a gravitationally bound remnant of massive stellar evolution, distinguished by extreme densities ($\gtrsim 10^{14}$–$10^{15}\, \mathrm{g/cm}^3$), compact radii ($\sim 10$–$15\ \mathrm{km}$), and the emergence of quantum, nuclear, and relativistic effects at macroscopic scale. Its astrophysical and microphysical properties are interrelated through general relativity, dense-matter equations of state (EoS), and multi-messenger observations, making neutron stars unique laboratories for supranuclear matter and strong-field gravity.

1. Equations of State and Macroscopic Structure

The equilibrium structure of a nonrotating, spherically symmetric neutron star is governed by the general relativistic Tolman–Oppenheimer–Volkoff (TOV) equations:

$\frac{dm}{dr} = 4\pi r^2 \rho(r)$

$$\frac{dP}{dr} = -\frac{G\,\bigl[\rho(r)+P(r)/c^2\bigr]\,\bigl[m(r)+4\pi r^3 P(r)/c^2\bigr]}{r\bigl[r-2G m(r)/c^2\bigr]}$$

where $r$ is the radial coordinate, $m(r)$ is the enclosed gravitational mass, $P(r)$ is the pressure, $\rho(r)$ is the energy density, $G$ is the gravitational constant, and $c$ is the speed of light. A physically meaningful stellar sequence is obtained by supplying a cold, β-equilibrated EoS $P(\rho)$ and integrating outward from a central density $\rho_c$ until $P(R)=0$ defines the stellar radius.

The solution space produces a one-parameter family of mass-radius $(M, R)$ configurations, typically with $M$ increasing with $\rho_c$ up to a critical maximum ($M_\mathrm{max}\sim 1.5$–$2.5\,M_\odot$ for realistic nuclear EoS), beyond which $dM/d\rho_c<0$ and stable equilibrium breaks down (Pizzochero, 2010). General relativistic corrections (e.g., pressure as a source of gravity) reduce $M_\mathrm{max}$ compared to Newtonian and Chandrasekhar–Stoner–Landau limits, enforcing an upper bound set by causality, $M_\mathrm{max}\lesssim 3\,M_\odot$ (Ekşi, 2015, Reisenegger et al., 2015).

Principal macroscopic properties:

Parameter	Typical Value	Notes
Mass, $M$	$1.1$–$2.3\,M_\odot$	Observed up to $2.08\pm0.07\,M_\odot$ (e.g., PSR J0740+6620)
Radius, $R$	$10$–$14$ km	Canonical star: $R_{1.4} \approx 11$–$13$ km
Central density, $\rho_c$	$2$–$8\,\rho_\mathrm{sat}$	$\rho_\mathrm{sat}=2.8\times10^{14}$ g cm$^{-3}$
Surface gravitational redshift, $z$	$0.2$–$0.4$	$z = (1-2GM/Rc^2)^{-1/2} - 1$ (Pizzochero, 2010, Zhao, 2017)

2. Microphysical Composition and Equation of State

At the core of a neutron star, matter is arranged in distinct zones defined by density:

• Outer crust ($\rho<4\times10^{11}$ g cm$^{-3}$): Fully ionized nuclei in a Coulomb lattice, degenerate electrons.
• Inner crust ($4\times10^{11}<\rho<\rho_\mathrm{sat}$ g cm$^{-3}$): Neutron-rich nuclei with a free neutron superfluid and electrons.
• Outer core ($\rho\gtrsim\rho_\mathrm{sat}$): Homogeneous nuclear matter, mainly neutrons with a fraction of protons, electrons, and muons in beta equilibrium.
• Inner core (few$\times\rho_\mathrm{sat}$): Potential presence of hyperons (e.g., $\Lambda$, $\Xi$), meson condensates, deconfined quarks, depending on the EoS (Miyatsu et al., 2012, Zhao, 2017, Zhu et al., 27 Oct 2025).

Degenerate Fermi pressure is central:

$$P_{\rm deg} \simeq \frac{\hbar c}{12\pi^2} (3\pi^2 n)^{4/3}$$

for ultra-relativistic particles at $T \ll T_F$.

Beta equilibrium among neutrons, protons, and electrons ($\mu_n = \mu_p + \mu_e$) and charge neutrality ($n_p = n_e$) shift the composition to increasingly neutron-rich matter as density increases (Pizzochero, 2010). Above the neutronization threshold ($E_{F,e}>1.29$ MeV, $\rho_{\rm th}\sim10^7$ g/cm$^3$) electron capture becomes energetically favorable.

At supranuclear densities, the EoS is determined by nucleonic interactions (as modeled by RBHF, chiral EFT, RMF, etc.), three-body forces, and possible exotic components (hyperons, quarks) (Wang et al., 2020, Logoteta et al., 2018, Sen et al., 2018, Miyatsu et al., 2012). Direct incorporation of hyperons or phase transitions softens the EoS and typically reduces the maximum supported mass—a manifestation of the "hyperon puzzle."

3. Superfluidity, Magnetism, and Transport Properties

Several superfluid and superconducting phenomena are realized in neutron stars:

• Superfluid neutrons in the inner crust and core (dominant $^1S_0$ and $^3P_2$ pairing), characterized by energy gaps $\Delta\sim 0.1$–$1$ MeV. Proton superconductivity ($^1S_0$) emerges at lower densities in the core.
• Observable consequences include suppression of the specific heat $C_V$ and of neutrino emissivity $\epsilon_\nu$ below the critical temperature, directly influencing the cooling trajectory and the shape of surface temperature curves.
• Rotational glitches arise from the differential rotation of superfluid neutron vortices pinned to the crustal lattice, facilitating sudden angular momentum transfer during vortex unpinning (Pizzochero, 2010).

The external dipolar magnetic field spans $B\sim 10^{8}$–$10^{15}$ G, with stronger fields ($>10^{13}$ G) in magnetars. The magnetic field structure modifies the charged-particle energy spectra (Landau quantization) and introduces pressure anisotropy (Malik et al., 2018, Menezes, 2021), affecting mass-radius curves by a few percent at $B\sim10^{18}$ G.

Transport and emission properties:

• Thermal evolution is governed by neutrino emission (modified and direct Urca processes) and photon cooling, with early neutrino-dominated stage ($\tau < 10^3$ yr) rapidly reducing core temperatures to $T\sim10^8$ K.
• High-energy emission arises from surface thermal emission (blackbody, $kT\sim0.1$–$0.3$ keV), non-thermal magnetospheric processes (radio, X-ray, gamma-ray), and field decay in magnetars.

4. Observational Constraints and Global Properties

Neutron star macroscopic properties are tightly constrained by multi-messenger data:

• Masses via radio timing and Shapiro delay in binary pulsars (e.g., PSR J0348+0432, $2.01\pm0.04\,M_\odot$).
• Radii from X-ray pulse profiling (NICER) and thermal emission fits ($R_{1.4}\approx11$–$13$ km).
• Tidal deformabilities ($\Lambda$) from gravitational-wave measurement of binary mergers (GW170817): at $M=1.4\,M_\odot$, current bounds $\Lambda_{1.4}=196^{+92}_{-63}$ (90% C.I.) (Kumar et al., 2019, Golomb et al., 18 Oct 2024), restricting the EoS stiffness at $2n_\mathrm{sat}$ (Collazos, 2023, Wang et al., 2020, Zhang et al., 2019).
• Moment of inertia ($I$), gravitational redshift ($z$), and compactness ($C=GM/Rc^2$), which are inferred via pulsar timing and modeling of high-mass neutron stars.
• Maximum mass ($M_\mathrm{TOV}$) inferred jointly from heavy pulsar observations and GW mergers: $M_\mathrm{TOV}=2.28^{+0.41}_{-0.21}M_\odot$ (Golomb et al., 18 Oct 2024), consistent with the $2M_\odot$ lower bound.

Canonical neutron star properties (as constrained by current theory and observation):

Property	Range/Value	Reference Cases
$M_\mathrm{max}$	$2.1$–$2.4\,M_\odot$	PSR J0740+6620, GW170817
$R_{1.4}$	$11$–$13$ km	NICER, GW170817
$\Lambda_{1.4}$	$196^{+92}_{-63}$	GW170817

Precise correlations between nuclear matter parameters and neutron-star observables have been quantified:

$$R_{1.4}\simeq 0.018\,M_0\,[\mathrm{MeV}] + 0.002\,L\,[\mathrm{MeV}] - 3.8,\qquad \Lambda_{1.4}\approx 2.5 - 0.0006M_0 - 0.03 K_{\rm sym}$$

with $M_0 = 9\rho_0^3\partial^3 E_0/\partial\rho^3$, $L = 3\rho_0\partial S/\partial\rho$, $K_{\rm sym} = 9\rho_0^2\partial^2 S/\partial\rho^2$. $R_{1.4}$ and $\Lambda_{1.4}$ thus tightly constrain the EoS parameter space (Providência, 2019).

5. Universal Relations and Population Systematics

Dimensionless relations—universal across a wide variety of microphysical EoS—link the moment of inertia ($\bar I=I/M^3$), quadrupole moment ($\bar Q$), and tidal deformability ($\bar\lambda$):

$$\ln y = a + b \ln x + c (\ln x)^2 + d(\ln x)^3 + e(\ln x)^4$$

for $(y,x)$ being $(\bar I, \bar\lambda)$, $(\bar I, \bar Q)$, or $(\bar Q, \bar\lambda)$, with fit coefficients universal to within $1\%$ (Kunz, 2022, Kumar et al., 2019).

These I-Love-Q and binary-Love relations enable the inference of properties (e.g., $R$, $I$, $C$) from measured $\Lambda_{1.4}$, and facilitate joint GW/EM constraints on the EoS without detailed microphysical model dependence (Kumar et al., 2019).

Population synthesis and hierarchical Bayesian analyses now distinguish between Galactic-EM (radio/X-ray) and GW-inferred NS populations, finding consistent maximum masses ($M_{\rm max}$) but statistically distinct mass distributions. The radius inference ($R_{1.4}=12.2^{+0.8}_{-0.9}$ km, 90% HPD) is still primarily determined by microphysics, while details of the mass distribution reflect formation and astrophysical selection effects (Golomb et al., 18 Oct 2024).

6. Bulk, Surface, and Symmetry Properties

Bulk and surface nuclear properties of neutron star matter, such as incompressibility ($K^\mathrm{star}$), symmetry energy ($S^\mathrm{star}$), slope parameter ($L_\mathrm{sym}^\mathrm{star}$), and curvature ($K_\mathrm{sym}^\mathrm{star}$), vary systematically with EoS stiffness and stellar mass (Kumar et al., 2021). Surface-averaged values are calculated via local density approximation and the coherent density fluctuation model:

$K^{\rm star} = \int_0^{\infty} |F(x)|^2\,K^{\rm NSM}[n_0(x)]\,dx \ \ \textrm{(and analogously for %%%%112%%%%, %%%%113%%%%, %%%%114%%%%)}$

For soft EoS (G3), $R_{1.4}\sim 12$ km, $K^{\rm star}\sim25$ MeV, $S^{\rm star}\sim 60$ MeV for $M=1.4\,M_\odot$, rising with stiffer EoS (NL3) and higher mass. Observed trends:

Model	$K^{\rm star}$ (MeV)	$S^{\rm star}$ (MeV)	$L_{\rm sym}^{\rm star}$ (MeV)	$K_{\rm sym}^{\rm star}$ (MeV)
NL3, $M=1.4$	$30$	$80$	$200$	$+20$
G3, $M=1.4$	$25$	$60$	$120$	$-80$
IU-FSU, $M=1.4$	$28$	$55$	$130$	$-30$

High $L_{\rm sym}^{\rm star}$ correlates with larger radii and higher central proton fractions, influencing cooling and Urca threshold conditions (Providência, 2019, Kumar et al., 2021).

7. Open Problems and Future Prospects

The maximum mass and radii of neutron stars continue to serve as critical discriminants among candidate EoSs, especially regarding the transition to exotic phases (hyperonic matter, quark deconfinement). Recent advances employ Gaussian process and spectral approaches to reconstruct the EoS with minimal model bias (Golomb et al., 18 Oct 2024). Model-agnostic multi-messenger inference is expected to constrain $R_{1.4}$ to within $\sim\pm0.5$ km and $M_\mathrm{TOV}$ within $0.1\,M_\odot$ as GW detector catalogs grow.

Key uncertainties persist in the microphysics at $2$–$5\,\rho_\mathrm{sat}$—notably, the high-density symmetry energy, hyperonic and quark matter interactions, and the character of the hadron–quark transition (Zhu et al., 27 Oct 2025). Global properties (mass, radius, tidal response) remain largely insensitive to low-energy nuclear data once EoS models fit symmetry energy and its slope at saturation, highlighting the need for direct constraints at supranuclear densities (Carlson et al., 2022, Golomb et al., 18 Oct 2024).

Astrophysical observations—precision pulse profile modeling, moment of inertia measurements in double pulsars, and next-generation GW events—along with theoretical advances in dense-matter modeling will continue to refine our understanding of the most extreme nuclear-physical environments accessible to experiment.