equation

neutron star equation of state

The equation of state describes the relationship between pressure and density in neutron stars.
48 related articles · 4 formulas

Description

The neutron star equation of state is a mathematical relationship that describes how pressure and density are linked inside a neutron star — the ultra-compact remnant of a supernova explosion. These stars pack more mass than the Sun into a sphere the size of a city, with densities exceeding that of an atomic nucleus. The equation of state determines the star's maximum mass, its radius, and how it deforms under tidal forces. It is essentially the 'material law' for matter at extreme densities, where neutrons, protons, and possibly exotic particles like hyperons or quarks coexist. Understanding it is key to explaining what happens when neutron stars collide and merge, and to interpreting the gravitational waves they emit.

History

The concept of the equation of state for neutron stars was introduced in the 1930s, shortly after the neutron's discovery. Modern constraints began in the 1960s with the first pulsar observations, and the field was revolutionized in 2017 by the gravitational-wave detection of a neutron star merger.

How it works

The equation of state is not derived from first principles but is constrained by observations and nuclear physics experiments. On the theoretical side, physicists model the strong nuclear force at high densities using many-body calculations, often incorporating data from terrestrial experiments that probe nuclear matter. On the observational side, precise measurements of neutron star masses and radii — obtained from X-ray observations, pulsar timing, and gravitational-wave signals from mergers — provide direct constraints. For example, the maximum observed neutron star mass sets a lower limit on the stiffness of the equation of state, while the tidal deformation measured in a merger reveals how compressible the star is. The 'nuclear pasta' phases — exotic shapes of nuclei that resemble spaghetti and lasagna — are predicted to exist in the inner crust and affect the star's thermal and mechanical properties.

In practice

The equation of state is used to interpret gravitational-wave signals from neutron star mergers and to model the cooling and structure of these stars, which is essential for understanding the origin of heavy elements like gold and platinum.

Fun fact

The matter inside a neutron star is so dense that a sugar-cube-sized amount would weigh about as much as all of humanity — roughly 400 million tons.

Concept graph →

Formulas

$$w = \frac{P}{\rho}$$
Relates pressure to energy density in cosmology.
What is in the formula
wvariableequation of state parameter — equation of state parameter
Pvariablepressure — pressure pascal
ρvariableenergy density — energy density joule per cubic metre
φ=p/\varepsilon · 2601.02980v1
w=-1 · 2601.06656v1
w=-1 · 2605.20060
w=\frac{p}{ρ} · 2605.13705v1
$$w_{\rm eff} = -1 - \frac{1}{3} \frac{d \ln \rho}{d \ln a}$$
Parameter describing the effective pressure-to-density ratio in cosmology.
What is in the formula
w_effvariableEffective equation of state parameter — equation of state parameter
rhovariableEnergy density of the cosmic fluid — energy density joule per cubic metre
avariableScale factor of the universe — scale factor
d/d ln aoperatorDerivative with respect to the logarithm of the scale factor — logarithmic derivative
w_{\rmeff}=-1-\frac{1}{3}\frac{y_h'}{y_h} · 2607.12745
$$\eta = \frac{n_B}{s}$$
Ratio of baryon number density to entropy density.
What is in the formula
\etavariableBaryon to entropy ratio — baryon to entropy ratio
n_BvariableBaryon number density — baryon number density per cubic metre
svariableEntropy density — entropy density joule per cubic metre per kelvin
\eta=\frac{n_b}{s} · 2605.02008
$$\frac{dP}{dr} = -\frac{(P+\varepsilon)(m+4\pi r^3 P)}{r(r-2m)}$$
Describes hydrostatic equilibrium in general relativistic stars.
What is in the formula
PvariablePressure at radius r — pressure pascal
rvariableRadial coordinate — radius metre
εvariableEnergy density — energy density joule per cubic metre
mvariableMass enclosed within radius r — mass kilogram
d/droperatorDerivative with respect to radial coordinate — ordinary derivative
\frac{dp_i}{dr}=-\frac{(p_i+\varepsilon_i)(m+4πr^3(p_{{nm}}+p_{{dm}}))}{r(r-2m)} · 2607.03840v1

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