PSR B1509-58 makes nearby gas emit
X-rays (gold) and illuminates the rest of the
nebula, here seen in
infrared (blue and red). Once formed, neutron stars no longer actively generate heat and cool over time, but they may still evolve further through
collisions or
accretion. Most of the basic models for these objects imply that they are composed almost entirely of
neutrons, as the extreme pressure causes the
electrons and
protons present in normal matter to combine into additional neutrons. These stars are partially supported against further collapse by
neutron degeneracy pressure, just as
white dwarfs are supported against collapse by
electron degeneracy pressure. However, this is not by itself sufficient to hold up an object beyond and repulsive nuclear forces increasingly contribute to supporting more massive neutron stars. If the remnant star has a
mass exceeding the
Tolman–Oppenheimer–Volkoff limit, approximately , the combination of degeneracy pressure and nuclear forces is insufficient to support the neutron star, causing it to collapse and form a
black hole. The most massive neutron star detected so far,
PSR J0952–0607, is estimated to be . and older, even cooler neutron stars are still easy to discover. For example, the well-studied neutron star, , has an average surface temperature of about . By comparison, the effective surface temperature of the Sun is only . Neutron star material is remarkably
dense: a normal-sized
matchbox containing neutron-star material would have a weight of approximately 3
billion tonnes, the same weight as a 0.5-cubic-kilometer chunk of the Earth (a cube with edges of about 800 meters3 = (0.8 km)3 = 0.512 km3-->) from Earth's surface. As a star's core collapses, its rotation rate increases due to
conservation of angular momentum, so newly formed neutron stars typically rotate at up to several hundred times per second. Some neutron stars emit beams of electromagnetic radiation that make them detectable as pulsars, and the discovery of pulsars by
Jocelyn Bell Burnell and
Antony Hewish in 1967 was the first observational suggestion that neutron stars exist. The fastest-spinning neutron star known is
PSR J1748−2446ad, rotating at a rate of 716 times per second or
revolutions per minute, giving a linear (tangential) speed at the surface on the order of 0.24
c (i.e., nearly a quarter the
speed of light).
Equation of state The
equation of state of neutron stars is not currently known. This is because neutron stars are the second most dense known object in the universe, only less dense than black holes. The extreme density means there is no way to replicate the material on Earth in laboratories, which is how equations of state for other things like ideal gases are tested. The closest neutron star is many parsecs away, meaning there is no feasible way to study it directly. While it is known neutron stars should be similar to a
degenerate gas, it cannot be modeled strictly like one (as white dwarfs are) because of the extreme gravity.
General relativity must be considered for the neutron star equation of state because
Newtonian gravity is no longer sufficient in those conditions. Effects such as
quantum chromodynamics (QCD),
superconductivity, and
superfluidity must also be considered. At the extraordinarily high densities of neutron stars, ordinary matter is squeezed to nuclear densities. Specifically, the matter ranges from nuclei embedded in a sea of electrons at low densities in the outer crust, to increasingly neutron-rich structures in the inner crust, to the extremely neutron-rich uniform matter in the outer core, and possibly exotic states of matter at high densities in the inner core. Understanding the nature of the matter present in the various layers of neutron stars, and the phase transitions that occur at the boundaries of the layers is a major unsolved problem in fundamental physics. A presumptive neutron star equation of state would encode information about the structure of a neutron star and would explain how matter behaves at the extreme densities found inside neutron stars. Constraints on the neutron star equation of state would then provide constraints on how the
strong interaction of the
Standard Model works, which would have profound implications for nuclear and atomic physics. This would make neutron stars natural laboratories for probing fundamental physics. For example, the exotic states that may be found at the cores of neutron stars are types of
QCD matter. At the extreme densities at the centers of neutron stars, neutrons become disrupted giving rise to a sea of quarks. This matter's equation of state is governed by the laws of
quantum chromodynamics and since QCD matter cannot be produced in any laboratory on Earth, most of the current knowledge about it is only theoretical. Different equations of state lead to different values of observable quantities. While the equation of state relates directly only to density and pressure, these in turn lead to calculating observables like the speed of sound, mass, radius, and
Love numbers. There are many proposed neutron star equations of state, such as FPS, UU, APR, L, and SLy, and it is an active area of research. Another aspect of the equation of state is whether it is a soft or stiff equation of state. This relates to how much pressure there is at a certain energy density, and often corresponds to phase transitions. When the material is about to go through a phase transition, the pressure will tend to increase until it shifts into a more comfortable state of matter. A soft equation of state would have a gently rising pressure versus energy density while a stiff one would have a sharper rise in pressure. In neutron stars, nuclear physicists are still testing whether the equation of state should be stiff or soft, and sometimes it changes within individual equations of state depending on the phase transitions within the model. This is referred to as the equation of state stiffening or softening, depending on the previous behavior. Since it is unknown what neutron stars are made of, there is room for different phases of matter to be explored within the equation of state.
Density and pressure , the latter roughly the size of
Earth. Neutron stars have overall densities of to ( to times the density of the Sun), which is comparable to the approximate density of an atomic nucleus of . The density increases with depth, varying from about at the crust to an estimated or deeper inside. Pressure increases accordingly, from about (32
QPa) at the inner crust to in the center. A neutron star is so dense that one teaspoon (4.929
milliliters) of its material would have a mass over , about 900 times the mass of the
Great Pyramid of Giza. The entire mass of the Earth at neutron star density would fit into a sphere 305 m in diameter, about the size of the
Arecibo Telescope. In popular scientific writing, neutron stars are sometimes described as macroscopic
atomic nuclei. Indeed, both states are composed of
nucleons, and they share a similar density to within an order of magnitude. However, in other respects, neutron stars and atomic nuclei are quite different. A nucleus is held together by the
strong interaction, whereas a neutron star is held together by
gravity. The density of a nucleus is uniform, while neutron stars are
predicted to consist of multiple layers with varying compositions and densities.
Current constraints Because equations of state for neutron stars lead to different observables, such as different mass-radius relations, there are many astronomical constraints on equations of state. These come mostly from the
LIGO gravitational wave observatory and the
NICER X-ray telescope. NICER's observations of
pulsars in binary systems, from which the pulsar mass and radius can be estimated, can constrain the neutron star equation of state. A 2021 measurement of the pulsar
PSR J0740+6620 was able to constrain the radius of a neutron star to with 95% confidence. These mass-radius constraints, combined with
chiral effective field theory calculations, tighten constraints on the neutron star equation of state. For example, the LIGO detection of the binary neutron star merger
GW170817 provided limits on the tidal deformability of neutron star binaries, ruling out whole families equations of state. Future gravitational wave signals with next generation detectors like
Cosmic Explorer can impose further constraints. When nuclear physicists are trying to understand the likelihood of their equation of state, it is good to compare with these constraints to see if it predicts neutron stars of these masses and radii. There is also recent work on constraining the equation of state with the speed of sound through hydrodynamics.
Tolman–Oppenheimer–Volkoff equation The
Tolman–Oppenheimer–Volkoff (TOV) equation can be used to describe a neutron star. The equation is a solution to Einstein's equations from general relativity for a spherically symmetric, time invariant metric. With a given equation of state, solving the equation leads to observables such as the mass and radius. There are many codes that numerically solve the TOV equation for a given equation of state to find the mass-radius relation and other observables for that equation of state. The following differential equations can be solved numerically to find the neutron star observables: \frac{dp}{dr} = - \frac{G\epsilon(r) M(r)}{c^2 r^2} \left(1+\frac{p(r)}{\epsilon(r)}\right) \left(1+\frac{4\pi r^3p(r)}{M(r)c^2}\right) \left(1-\frac{2GM(r)}{c^2r}\right) \frac{dM}{dr} = \frac{4\pi}{c^2} r^2 \epsilon(r) where
G is the gravitational constant,
p(
r) is the pressure,
ϵ(
r) is the energy density (found from the equation of state), and
c is the speed of light.
Mass–radius relation Using the
TOV equations and an
equation of state, a mass–radius curve can be found. In theory, for a correct equation of state, every neutron star that could possibly exist would lie along that curve. To create these curves, the TOV equations must be solved for different central
densities. For each central density, the
mass and
pressure equations must be numerically solved until the pressure goes to zero, which represents the outside of the star. Each solution gives a corresponding mass and
radius for that central density. Mass-radius curves for different equations of state all reach a maximum value at specific radii. This maximum point is known as the maximum mass. Beyond that mass, the star will no longer be stable, i.e. no longer be able to hold itself up against
gravity, and would collapse into a
black hole. Since each equation of state leads to a different mass-radius curve, each equation also leads to a unique maximum mass value. For example,
Oppenheimer and
Volkoff came up with the
Tolman–Oppenheimer–Volkoff limit of ~ using a non-interacting degenerate neutron gas equation of state. More advanced theories include interactions between the neutrons which increases the pressure of the gas and thus increases the mass limit above . If, in addition the star is rotating rapidly this limit can be as high as . Observations of the neutron star
PSR J0952-0607 suggest it has a mass of . One phenomenon in this area of astrophysics relating to the maximum mass of neutron stars is known as the "mass gap". The mass gap refers to a range of masses from roughly 2 to 5 solar masses where very few compact objects were observed. This range is based on the current assumed maximum mass of neutron stars (~) and the minimum black hole mass (~). Recently, some objects have been discovered that fall in that mass gap from gravitational wave detections. If the true maximum mass of neutron stars was known, it would help characterize compact objects in that mass range as either neutron stars or black holes.
I-Love-Q relations There are three more properties of neutron stars that are dependent on the equation of state but can also be astronomically observed: the
moment of inertia, the
quadrupole moment, and the
Love number. The moment of inertia of a neutron star describes how fast the star can rotate at a fixed spin momentum. The quadrupole moment of a neutron star specifies how much that star is deformed out of its spherical shape. The Love number of the neutron star represents how easy or difficult it is to deform the star due to
tidal forces, typically important in binary systems. While these properties depend on the material of the star and therefore on the equation of state, there is a relation between these three quantities that is independent of the equation of state. This relation assumes slowly and uniformly rotating stars and uses general relativity to derive the relation. While this relation would not be able to add constraints to the equation of state, since it is independent of the equation of state, it does have other applications. If one of these three quantities can be measured for a particular neutron star, this relation can be used to find the other two. In addition, this relation can be used to break the degeneracies in detections by gravitational wave detectors of the quadrupole moment and spin, allowing the average spin to be determined within a certain confidence level.
Temperature The temperature inside a newly formed neutron star is around to .
Magnetic field The magnetic field strength on the surface of neutron stars ranges from about to
tesla (T). These are orders of magnitude higher than in any other object: for comparison, a continuous 16 T field has been achieved in the laboratory and is sufficient to levitate a living frog due to
diamagnetic levitation. Variations in magnetic field strengths are most likely the main factor that allows different types of neutron stars to be distinguished by their spectra, and explains the periodicity of pulsars. and it has become widely accepted that these magnetars are the source of
soft gamma repeaters (SGRs) and
anomalous X-ray pulsars (AXPs). The magnetic
energy density of a field is extreme, greatly exceeding the
mass-energy density of ordinary matter. Fields of this strength are able to
polarize the vacuum to the point that the vacuum becomes
birefringent: photons can merge or split in two, and virtual particle–antiparticle pairs are produced. The field changes electron energy levels and atoms are forced into thin cylinders. Unlike in an ordinary pulsar, magnetar spin-down can be directly powered by its magnetic field, and the magnetic field is strong enough to stress the crust to the point of fracture. Fractures of the crust cause
starquakes, observed as extremely luminous millisecond hard gamma ray bursts. The fireball is trapped by the magnetic field, and comes in and out of view when the star rotates, which is observed as a periodic soft gamma repeater (SGR) emission with a period of 5–8 seconds and which lasts for a few minutes. The origins of the strong magnetic field are as yet unclear. Such a strong gravitational field acts as a
gravitational lens and bends the radiation emitted by the neutron star such that parts of the normally invisible rear surface become visible. If the radius of the neutron star is 3
GM/
c2 or less, then the photons may be
trapped in an orbit, thus making the whole surface of that neutron star visible from a single vantage point, along with destabilizing photon orbits at or below the 1 radius distance of the star. A fraction of the mass of a star that collapses to form a neutron star is released in the supernova explosion from which it forms (from the law of mass–energy equivalence, ). The energy comes from the
gravitational binding energy of a neutron star. Hence, the gravitational field of a typical neutron star is huge. If an object were to fall from a height of on a neutron star in radius, it would reach the ground at around . However, even before impact, the
tidal force would cause
spaghettification, breaking any sort of an ordinary object into a stream of material. Because of the enormous gravity,
time dilation between a neutron star and Earth is significant. For example, eight years could pass on the surface of a neutron star, yet ten years would have passed on Earth, not including the time-dilation effect of the star's very rapid rotation. Neutron star relativistic equations of state describe the relation of radius vs. mass for various models. The most likely radii for a given neutron star mass are bracketed by models AP4 (smallest radius) and MS2 (largest radius).
EB is the gravitational binding energy of the observed neutron star of mass of
M with radius
R, \frac{E_\text{B}}{Mc^2} = \frac{0.60\,\beta}{1 - {\beta}/{2}} where \beta = GM/R{c}^{2} A neutron star of mass 2 would not be more compact than radius (AP4 model). Its mass fraction gravitational binding energy
EB/
Mc2 would then be 0.187, −18.7% (exothermic). This is not near 0.6/2 = 0.3, −30%. == Structure ==