Neutron stars are the core remnants of moderately massive stars that have undergone violent stellar death via tremendous supernova explosions after leaving the Main Sequence. The cores of such large stars are too massive to exist as white dwarfs. If the remnant has more mass than the critical 1.44 times the Sun's mass (called Chandrasekhar limit) than it can not exist as a white dwarf but collapses further to a smaller and even denser neutron star.  A teaspoonful of neutron star matter may weigh as much as 10 billion tonnes! There's really nothing even to compare it to try and visualize it. The core collapses to such immense densities because the gravitational field is so enormously strong that normal matter is crushed and destroyed and even extremely dense white dwarf matter is crushed and destroyed. Very massive stars leave heavier remnants that can not even exist as neutron stars, but collapse into mysterious entities known as black holes. During its Main Sequence lifetime the pressure of the radiation emitted by nuclear reactions stops the core from collapsing, but when the star runs out of fuel, the core collapses until it reaches a state of matter that can resist further collapse.

In the case of a neutron star, this matter is very strange and not entirely understood, but appears to be mostly composed of neutrons. Atoms are composed of a nucleus of one or more protons and neutrons and one or more electrons in shells around the nucleus. Free neutrons, when removed from the nucleus, are very unstable and decay into electrons, protons and anti-electron neutrinos:

n → p + e^– + ν_e

In some radioactive nuclei, there are too many neutrons which makes the nucleus unstable. In this case, neutrons inside the nucleus may also decay into a proton, electron and anti-electron neutrino in a process called beta-decay, which releases the highly energetic electron from the nucleus as ionising radiation. The number of free neutrons halves every ten minutes as the neutrons decay. (This is the principle of the neutron bomb - irradiate an area with energetic neutrons, killing all the inhabitants, and within a day it is safe to move in and occupy the area). However, in a neutron star the immense pressure forces this equation to the left, as electrons (e) and protons (p) are squeezed into neutrons (n), so that neutrons become stable and a neutron star is made up mostly of neutrons (there are about 200 neutrons for every electron). This material can be conveniently called neutronium.

How does a neutron star get turned from ordinary matter into neutronium?

The neutron star is enormously compressed by its immense gravitational field, and is only about 10 to 20 kilometres in diameter (about the size of a city) but is still very hot and luminous. To see what is happening we shall examine the best physical model we have of its structure. Its atmosphere is only about one metre thick and beneath this is a solid crust which is one to two kilometers thick and composed of atomic nuclei and electrons, and so is relatively normal matter which is nevertheless extremely dense and heavy and very hot. The surface gravity is about 2 hundred billion to 3 thousand billion times that on the surface of the Earth. Thus, an average man standing on the surface of a neutron star would weigh as much as 200 billion tonnes or more! Clearly, his body would be squashed to nothing!

The interior, beneath the crust, is thought to be liquid and is composed of neutron rich nuclei as the increasing pressure starts to convert electrons and protons into neutrons. As we move deeper into the neutron star the pressure rises considerably. The enormous pressure starts to pull neutrons from the neutron rich atomic nuclei (a process called neutron drip) so we have a liquid of nuclei, neutrons and electrons which is very hot and dense. Deeper in, the nuclei completely dissolve into a sea of neutrons, with some electrons and protons (there are about 200 neutrons to each proton, so the bulk of the neutron star is essentially a superhot and super-dense neutron fluid. Superhot by our standards that is, but compared to the high density the neutrons are expected to behave as if they are cold, and so may resemble a solid in many respects. It is possible that the neutrons form degenerate matter, which is a special state of matter not normally encountered, but comprises particles that are squeezed into their least energetic states. Only the fact that two neutrons cannot have the exact same set of quantum numbers (they belong to a type of particle called fermions) prevents them from merging together, in fact this purely quantum mechanical phenomenon maybe all that stops the neutron star from collapsing further - in other words the neutrons don't want to be any closer together and so they exert a pressure which holds the star up against gravity which is trying to pull it in.

In the neutron star core the pressures far exceed the maximum pressures that can be produced in a laboratory, so we have to speculate. The core might consist of neutrons, or the neutrons may  collapse into particles called pions, muons and hyperons (and possibly kaons). Hyperons contain strange quarks (normal protons and neutrons contain quarks but no strange quarks) and are referred to as a type of strange matter. At even greater pressures this strange matter may collapse into a quark liquid.

The neutrons inside a neutron star are expected to be ultrarelativistic, which means that they are moving close to the speed of light! (Light moves at about 2.998 hundred million metres per second in a vacuum!). The liquid is also predicted to be a superfluid. Superfluids are very strange things, they move without friction and can flow uphill and also behave as if they are a single particle (if you lift a portion out in a ladle, then it will flow up over the sides of the ladle to join up with the rest, as if it likes to be a single entity!).

There is speculation that especially heavy neutron stars may contain predominantly strange matter (a so-called strange star) and neutron stars more than 2 to 3 times the Sun's mass (the Tolman-Oppenheimer-Volkoff limit) may be composed almost entirely of free quarks, a so-called quark star.

When a neutron star is formed in a massive supernova explosion, the neutron star is typically slung across space at immense speeds and often rotates at tremendous rates. A rotating neutron star is a pulsar and is considered in more detail in the pulsar section.

Degenerate Matter

Neutron stars are, like white dwarfs, degenerate stars. Degenerate means that a number of particles have the same energy value. In particular, degenerate matter consists of particles called fermions, such as electron, protons and neutrons. Fermions generally have spin 1/2. Spin refers to the rotational angular momentum of a particle, in classical terms this is due to a particle rotating on its axis, in quantum mechanics (QM) this is not the case, though it is helpful to think of it as rotation or spin. Angular momentum, like other properties of confined quantum systems, is quantised, meaning that only a few very discrete values are possible, as in, for example, the energy levels of an atom (see atomic spectra). In a spin 1/2 particle, the angular momentum due to rotation can take one of only two possible values: +1/2 or -1/2 (which you can imagine to be like spinning clockwise and spinning counterclockwise) also referred to as spin-up and spin-down. According to the Pauli Exclusion Principle, fermions cannot coexist in the same region of space with the same values of their quantum numbers (the values of their quantised energy and momenta), such as energy or spin. This can also be viewed in terms of Heisenberg's Uncertainty Principle, in which both the spatial position and momentum of a particle cannot be simultaneously known to an arbitrary degree of accuracy. Particles are systems of quantum-mechanical waves and they don't like to be made to sit still with perfectly defined exact values of momentum. The more you attempt to confine the momentum of a particle to a certain specific value, the more uncertain its spatial position becomes, and likewise the more you confine its spatial position, the more its momentum becomes uncertain.

In neutron stars, the neutrons are highly compressed by an immense gravitational field, such that they almost  exclusively occupy the lowest available energy levels or ground states. In normal matter, thermal energy excites ] particles to lie above their ground states (they move about with thermal kinetic energy) and gaps are left in the ] then many available energy levels as particles move up-and-down between the many available states. In  degenerate matter, thermal energy can barely excite the particles at all and they sit neatly huddled together as tightly as possible. This restricts their energy and momenta, such that many particles occupy each available  energy and momentum level. For this reason, degenerate matter is called cold matter, though it is still very hot  as it stores immense potential heat energy. However, confining their momentum in this way, causes the spatial  positions of each particle to become more uncertain, and being fermions no two neighbouring particles in the  same energy level can occupy the same region of space, therefore, QM prevents the particles being pressed  any more closely together. The particles resist compression by applying pressure. In normal matter, pressure,  say in a gas, is caused by the thermal motion of the particles jostling them about so that they collide with the  walls of their container and impart energy and momentum to the walls - this is thermal pressure. In degenerate  matter, the pressure is purely quantum-mechanical and is called the Fermi pressure. Matter in this state is  extremely dense!

In white dwarfs, the Fermi pressure is provided by degenerate electrons, whilst in neutron stars it is provided by degenerate neutrons.

Degenerate matter can consist of non-relativistic, relativistic or ultrarelativistic particles. Non-relativistic particles are like those encountered in ordinary matter - they move at speeds well below the speed of light. Relativistic particles have enough energy to move at speeds a substantial fraction of the speed of light, and ultrarelativistic ] (or extreme relativistic) particles have enough energy to move at speeds very close to the speed of light. The ] degenerate neutrons in a neutron star are ultrarelativistic. This affects the approximate equations that are
used to determine the energy per particle and the Fermi pressure. The matter in any star can be modeled by a ] state equation (equation of state) which describes the relationship between the various properties of the  system like pressure, volume, temperature, mass, energy and chemical composition. (Specifically a state  variable, like temperature, is one that depends only on the current state of the system and not on its history).  Finding a reliable state equation that accurately describes neutron stars is an ongoing area of research.

In a vacuum, the least energetic and most stable atomic nucleus is iron-56 (iron atoms with 26 protons and 30  neutrons, or a total of 56 nucleons in their nucleus). This is the stable end-result of normal nuclear fusion  reactions in the cores of massive giant stars. The immense density inside a neutron star, however, shifts the  point of stability to more neutron-rich nuclei and as density increases, the process of neutronisation occurs,  in which neutrons add on to the atomic nuclei as they form by reverse beta-decay. The presence of electrons,  tightly squeezed into a small volume too, helps reverse the beta-decay by blocking the normal forward process  of beta-decay by preventing the emission of electrons (there are very few spare energy levels for extra  electrons to move into, so extra electrons tend not to be produced). Neutronisation occurs (at zero degrees K)  at a density between about 2E+07 (20 million) g/cubic cm to 4E+11 (400 billion) g/cubic cm. (Recall that the  normal density of water is 1 g/cubic cm). This is expected to result in the formation of large clusters of nucleons  (neutrons and protons) embedded in a neutron fluid.

As density increases still further, deeper inside the neutron star, the energy of a neutron inside a nucleon  cluster exceeds that of a neutron in the surrounding fluid, resulting in a phase change resulting in the formation  of nucleon clusters embedded in a non-interacting neutron fluid (the neutrons in the fluid do not interact with the clusters but prefer to stay outside). As density approaches that of an atomic nucleus, the clusters grow and overlap, resulting in another phase change, which results in a fluid of neutrons, protons and electrons, but mostly neutrons (the number of protons roughly equals the number of electrons which roughly equals one tenth the number of neutrons, so 80% of the fermions are neutrons). The presence of the electrons stabilises the neutrons, by inhibiting beta-decay.

When the density reaches about half that of nuclear matter, the average kinetic energy of an electron exceeds the rest-mass energy of muon (mu leptons) and many electrons transform into muons:

e^– → μ^– + ν_e + ν_μ

electron → muon + electron neutrino + anti-muon neutrino

At twice the density of nuclear matter, electron-capture occurs - neutrons capture electrons to form hyperons (particles composed of three quarks (baryons), like neutrons and protons, but with a strange quark incorporated) such as the negatively-charged sigma particle (dds, made of two down and one strange quark) and the neutral lambda particle (uds, one up, one down and one strange quark), the neutral sigma particle (also uds) and also negatively charged delta particles (ddd). Hyperon formation:

n + e^– → Σ^– + ν_e 

neutrino + electron → sigma + electron-neutrino

Thus, the core of a neutron star is expected to consist of neutrons (n), protons (p), hyperons, an electron fluid and a muon fluid. Furthermore, under these pressures, neutrons and protons can pair together, forming nn and pp pairs. About 10% of the nucleons are predicted to do this. The pp pairs are positively charged and superconducting, even at these immense temperatures. A superconductor is a material that conducts electricity with essentially zero resistivity. Some metals become superconducting at atmospheric pressure when cooled to very low temperatures and this superconductivity is due to the formation of electron pairs, ee, called Cooper pairs that can move freely within the metal without being scattered from the ion lattice. (The electrons have like electric charges and so will normally repel one-another, but they form very loosely bound pairs in these conditions by exchanging a phonon, a quasiparticle (pseudoparticle) which is a quantum of vibrational energy). In neutron stars, the pp pairs similarly superconduct. The nn pairs are superfluid, meaning that they can flow around inside the neutron star in very strange ways, without friction, behaving like a single particle in many ways. These superfluid neutrons are excellent thermal conductors, distributing heat around the neutron star. This, coupled with the neutron degeneracy, makes the interior of a neutron star approximately isothermal, that is at the same temperature. The temperature drops in the region of the neutron star crust. Neutron stars slowly cool by losing thermal radiation into space.

Above: a model of a neutron star, showing the composition of matter and the variation in density with fraction of the radius and mass. The crust is a solid, permeated by a fluid of ultrarelativistic particles, whilst the core is fluid and composed mostly of electrons, nucleons (neutrons, n, and protons, p) and more massive baryons and hyperons. The inner core might contain superfluid neutron and superconducting proton fluids, and may be solid in less massive neutron stars. In the most massive neutron stars, the inner core may comprise a fluid of quarks, as the nucleons and other baryons begin to break-down, with their constituent quarks separating, at least partially. The latter hypothetical stars are also called strange stars or quark stars. According to the theory of color confinement (a theory not proven but for which no violations have been observed) the quarks that make-up the nucleons and other hadrons (a hadron is a particle composed of quarks and/or anti-quarks) can not exist in isolation since they possess a color charge.

Color charge is like electric charge, except that whereas electric charge is associated with the electromagnetic force, color charge is associated with the much shorter range strong force which holds hadrons together. Whereas there are two types of electric charge, designated plus (+) and minus (-), there are three color charges, designated red (r), green (g) and blue (b). Color charge has nothing to do with visual color, but since there are three strong force charges and three primary colors, 'color charge' is a suitable term. Color confinement states that the overall color charge must be neutral or colorless. In the case of a nucleon, which contains three quarks, one quark must be red, one green and one blue, since mixing red, green and blue light gives white light - hence the analogy to color is useful here. However, under immense pressures, the nucleons (or their wavefunctions) overlap and they emerge into a continuous system in which the quarks can move around as if free in a gas. The strong force, which acts between the colour charges of quarks, is conveyed by particles called gluons, which quarks constantly exchange, and so the resultant state of nuclear matter is called a quark-gluon plasma. This kind of matter could exist in the inner core of heavy neutron or quark stars.

Atmosphere

Above the surface of the crust is an atmosphere of plasma, which becomes less dense further from the surface, falling from the density of the crust to the density of interstellar matter. With the immense gravitational field of the neutron star, however, the atmosphere is very compressed and only a meter or two in height!

Magnetosphere

Outside the crust is a region dominated by the neutron star's gravitational field, called the magnetosphere, which extends to several times the star's radius into space. Neutron stars typically have very strong magnetic fields, with B > 10⁹ gauss (G) (10⁵ tesla, T) and with typical values in pulsars of 10⁹ T (10¹³ G)! [1 T = 10⁴ G = 10 000 G]. B is the magnetic flux density and is a measure of the intensity of the field (though is not strictly the same thing as magnetic field 'intensity' or 'strength'). In comparison the Earth's magnetic field is 0.3 G to 0.6 G and very strong superconducting magnets are around 13.5 T and pulsed magnets at 72 T. Some neutron stars have unusually strong magnetic fields at ~10¹¹ T (10¹⁵ G) and are then called magnetars. Pulsars are stars that emit periodic pulses of radio energy at a frequency of around once per second or higher. Pulsars (or at least most of them) are thought to be rapidly rotating neutron stars with strong magnetic fields.

The magnetic field of neutron stars is so immense that it can pull electrons and protons out from the crust, except near to the poles, these charged particles follow the magnetic field lines back down to the star's surface, these field lines forming a toroidal belt around the star's equator. At the poles, charged particles would have to move faster than light to make it back down and so instead the field lines detach and the particles stream away, electrons closest to the pole, emitted in cones, surrounded by protons slightly further from the pole, at least according to one model anyway. As the charged particles spiral around these magnetic field lines at near-light speeds, they emit radio frequency radiation, so-called synchrotron emission.

Furthermore, the magnetic field is so intense that the electron orbitals in the atoms of the crust are distorted, being narrower in the direction Normal (perpendicular) to the magnetic field lines as they emerge from the crust, such that rather than being spherical, the atoms should be prolate spheroids (like rugby balls standing upright):

One consequence of this arrangement is that the electrons are closer to the nucleus and so are more tightly bound and it takes considerable energy to ionize the atoms and remove the electrons (for a magnetic field of B = 10^12 G, it requires a temperature of about 10^9 K to ionize the atoms). The atoms are also tightly packed, resulting in a tight crystal lattice that is very strong. The crust also has anisotropic (different in different directions) electrically conductivity - conducting well in one direction, whilst acting as a good insulator in another direction.

Equations of State

See, for example, the review by Heiselberg and Pandharipande, 2000 (Recent progress in neutron star theory, Annual Reviews of Nuclear and Particle Sci, 50: 481-524) for a good account of recent historical developments in models of neutron star matter.

In order to better model and predict parameters such as the minimum and maximum mass of a neutron star, physicists are working on developing various models of the equation of state for neutron star matter. An equation of state relates observable parameters, such as the temperature, density and pressure of matter inside a neutron star: it literally describes the state of the matter. There are many unknowns when attempting to model the behavior of matter at such extreme densities, though lab-based experiments on very dense matter have shed some light on certain aspects, nevertheless coming up with the equations of state requires ingenuity.

Approaches to finding a reliable equation of state are varied. Ultimately, the most appropriate tool to use is probably quantum field theory (QFT). In QFT the primary components of matter are not particles, per se, but fields. Quantum fields are mathematical constructs that have been used with much success to make predictions in atomic, nuclear and particle physics. Particles and antiparticles exist as vibrational modes of the fields. The effects of special relativity are incorporated. For example, such an equation of state for a neutron star would have to factor in fields for the neutrons, pions and quarks and other particles involved. Usually one or more approximations are used to simplify the equations, for instance the pion field is often omitted when using the mean field approximation. In the mean field approximation, each nucleon (such as a neutron or proton) is modeled as interacting with the average force field (strong force and electrostatic force) generated by the other nucleons. This approximation has been used to model nuclei, however, it is not very accurate and not very satisfactory.

QFT utilizes the methods of the calculus of variations and its functional approach to minimize action when deriving the equations of motion (i.e. equations of state describing particle behavior) from the Lagrangian via the Euler-Lagrange equations. The Lagrangian contains the physics input into the model, such as the interacting particle fields and is essentially a 'best guess'. Predictions can then, however, be compared with observational data on neutron star diameter and mass.

Another approach involves the use of variational methods to approximate the quantum mechanical Hamiltonian. The quantum Hamiltonian is a mathematical operator that allows us to manipulate the mathematical representations of particle systems to extract useful information such as the energy of the system. This may involve adding relativistic corrections to account for large particle momentum.

Some of the results of these models include the prediction that about 10% of the mass of a neutron star is converted into binding energy that holds the particles together. that is, 1.5 solar masses will yield a neutron star of final mass 1.35 solar masses (approximately).

These models also shed light on how the matter may change as density and pressure increase, both in heavier neutron stars and in going from the crust into the core of a neutron star. For example, kaon energy decreases with increasing density, due to the attraction between the nucleons and kaons. The kaon (K meson) is one of a group of four particles containing a strange or anti-strange quark coupled with an up or down quark (or the anti-particle equivalent). Normal (i.e. familiar) matter contains up and down quarks, but the extreme pressures increasingly favor the production of heavy quarks such as the strange quark. When the density reaches a certain threshold, the kaon energy falls below the electron chemical potential and kaon condensation is predicted to occur in which a bose condensate of negative kaons, K^–, may form inside the neutron star core. (Note: chemical potential can be thought of as the potential energy of a chemical species (strictly energy per amount of substance, such as energy per mole) and may give an indication of the amount of energy that may be released or absorbed when a system changes or reacts - systems tend to move from higher to lower chemical potential). Charge is conserved, so kaons compete with electrons for negative charge and so will only condense when their energy falls below that of the electrons in the material.

Similarly, there are critical pressures at which pions may condense. Neutral pions, π⁰, may condense, which is expected to alter the state of neutron star matter enough to effect its rate of cooling. Nucleons are also predicted to position themselves in harmony with the neutral pion field (residing in the field's nodal planes).

Hyperonic Matter

Hyperons are particles containing one or more strange (s) quarks or anti-strange quarks 9that is they have non-zero strangeness). Negatively charged sigma, Σ^– (quark composition dds, 2 down and one strange quark) and delta Δ^– (ddd) particles will condense if their chemical potential becomes less than that of the neutrons and electrons (they compete with neutrons for d quarks, neutron, n = ddu; and with the electron for charge). Interactions between hyperons and neutrons are not well understood. The hyperons will populate their own energy levels, rather than joining the top of the stack of fully occupied neutron energy levels and so may condense in the core, altering the neutron energies. They are also expected to increase the fraction of neutron star matter in the form of protons.

Kaons, pions, sigma and delta particles are hadrons, as are neutrons and protons, but containing unusual quark constituents. Thus, both the more conventional nuclear or nucleon matter (NM) of neutronium and the more exotic mixtures of hyperonic matter can be called hadronic matter (HM). However, at immense pressures, even this material is thought to become unstable and convert into quark matter (QM).

Quark Matter

When neutrons become so tightly compressed that the cores of the neutrons (each containing 3 quarks) overlap significantly, then the quarks are expected to rearrange into a crystalline lattice as quark matter. At certain densities a mixed phase material is expected to form: containing regions of quark matter (QM) and hadronic matter (HM) whilst at higher densities, the neutron star core may convert into almost pure quark matter (perhaps along with electrons and muons) and be more appropriately called a quark star. These quarks will interact via gluon exchange (whereas in NM the dominant interaction is pion exchange).

Neutron Star Evolution

Neutron stars are born immensely hot (core temperatures around 10¹² K) but despite being in the vacuum of space they cool very rapidly due to neutrino radiation, cooling within minutes to about 10¹⁰ K and 10⁶ K or less in about 100 thousand years. Combined with their small size, this makes neutron stars very hard to detect if they are in isolated systems and radiating mainly thermal radiation.

Similarly, conservation of angular momentum means that neutron stars are generally born with a high rate of rotation (as they formed from the collapse of larger more slowly rotating stellar cores; rather like a spinning ice skater pulling in her arms). In millisecond pulsars, the rate of rotation may reach about 1000 times a second. They also contain powerful compressed magnetic fields. However, misalignment between the magnetic field axis and the spin axis causes energy to beamed away from the poles of the spinning neutron star, in the form of radio waves, and the star spins down (rotates more slowly as it loses energy) due to this magnetic torque.

As it spins down, energy is lost from the outer crust and core of the star, but its inner crust is predicted to contain superfluid neutrons. Superfluids form when a fluid is either extremely cold or under immense pressure (and effectively cold in degenerate matter). Superfluids, as the name suggests, have negligible friction and flow with the utmost ease. This means that the superfluid keeps spinning at the high rate as the rest of the star slows, storing rotational energy as swirling vortices.As the star slows, instabilities are thought to occasionally develop, causing the star to briefly spin faster as angular momentum is transferred to the outer crust and then subsequently lost by the magnetic breaking effect as the star slows down again. these brief periods of faster spin are called glitches.

Pulsars - rapidly rotating, strongly magnetic neutron stars.

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Article updated: 29 March 2021