The equation of state describes the relationship between pressure and density in neutron stars.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.
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.
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.
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.
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.
| w | variable | equation of state parameter — equation of state parameter |
| P | variable | pressure — pressure pascal |
| ρ | variable | energy density — energy density joule per cubic metre |
| w_eff | variable | Effective equation of state parameter — equation of state parameter |
| rho | variable | Energy density of the cosmic fluid — energy density joule per cubic metre |
| a | variable | Scale factor of the universe — scale factor |
| d/d ln a | operator | Derivative with respect to the logarithm of the scale factor — logarithmic derivative |
| \eta | variable | Baryon to entropy ratio — baryon to entropy ratio |
| n_B | variable | Baryon number density — baryon number density per cubic metre |
| s | variable | Entropy density — entropy density joule per cubic metre per kelvin |
| P | variable | Pressure at radius r — pressure pascal |
| r | variable | Radial coordinate — radius metre |
| ε | variable | Energy density — energy density joule per cubic metre |
| m | variable | Mass enclosed within radius r — mass kilogram |
| d/dr | operator | Derivative with respect to radial coordinate — ordinary derivative |















