Neutron star

A neutron star represents the gravitationally collapsed remnant core of a massive supergiant star. Its formation occurs during a supernova explosion of a massive star, where gravitational collapse compresses the core beyond the density of a white dwarf star to that of atomic nuclei. These stellar objects are recognized as the second smallest and densest known class, exceeded only by black holes. Neutron stars typically possess a radius of approximately 10 kilometers (6 miles) and a mass around 1.4 solar masses (M_☉). The progenitor stars that evolve into neutron stars generally exhibit an initial total mass ranging from 10 to 25 M_☉, with potentially higher masses for those particularly abundant in elements heavier than hydrogen and helium.

Estimates suggest the presence of approximately one billion neutron stars within the Milky Way galaxy, with a minimum of several hundred million, a figure derived from calculations of stars that have undergone supernova explosions. Nevertheless, a significant number of these objects have existed for extended durations and have consequently cooled substantially. Initially, the detection of neutron stars was considered challenging due to their low emission rates. However, subsequent discoveries revealed that rotating neutron stars emit detectable radiation. The majority of observed neutron stars are identified either as pulsars or as components within binary systems.

Within a binary system, a neutron star paired with a main sequence star can draw substantial quantities of gas from its companion, a phenomenon known as accretion. These binary systems undergo continuous evolution, often leading to the companion star's eventual transformation into another compact object, such as a white dwarf or even another neutron star. Alternative outcomes include the complete destruction of the companion through processes like ablation or collision.

The investigation of neutron star systems constitutes a fundamental aspect of gravitational wave astronomy. The coalescence of binary neutron stars generates gravitational waves and is linked to kilonovae and short gamma-ray bursts. Notably, in 2017, the LIGO and Virgo interferometer facilities recorded GW170817, marking the inaugural direct observation of gravitational waves originating from such an event. Previously, indirect evidence for gravitational waves had been deduced through examining the gravitational radiation emitted during the orbital decay of a distinct, unmerged binary neutron system, specifically the Hulse–Taylor pulsar.

Formation

Any main-sequence star possessing an initial mass exceeding 8 M_☉ (equivalent to eight solar masses) is capable of evolving into a neutron star. As the star departs from the main sequence, stellar nucleosynthesis processes generate an iron-rich core. Upon the depletion of all nuclear fuel within this core, its stability becomes solely reliant on degeneracy pressure. Subsequent accumulation of mass from shell burning causes the core to surpass the Chandrasekhar limit. Consequently, electron-degeneracy pressure is overcome, leading to further core collapse and an increase in temperatures to beyond §10_11§×§121314§ K. At these extreme temperatures, photodisintegration—the dissociation of iron nuclei into alpha particles by high-energy gamma rays—takes place. As the core's temperature continues to escalate, electrons and protons merge through electron capture, forming neutrons and releasing a torrent of neutrinos. When densities attain a nuclear density of §1819§×§202122§ kg/m§2324§, the combined effects of strong force repulsion and neutron degeneracy pressure arrest the contraction. The star's collapsing outer envelope is then abruptly halted and propelled outward by the neutrino flux generated during neutron formation, culminating in a supernova and leaving a neutron star remnant. Nevertheless, if the remnant's mass exceeds approximately 3 M_☉, it will instead form a black hole. Recent observations of gravitational waves from the neutron star merger GW170817, which is believed to have produced a black hole shortly thereafter, have refined this mass limit estimate to approximately 2.17 M_☉.

When the core of a massive star undergoes compression during a Type II, Type Ib, or Type Ic supernova and subsequently collapses into a neutron star, it largely preserves its angular momentum. Due to possessing only a minute fraction of its progenitor's radius, which significantly diminishes its moment of inertia, a neutron star initially forms with an exceptionally high rotational velocity that gradually decreases over an extended duration. Observed neutron stars exhibit rotation periods spanning approximately 1.4 milliseconds to 30 seconds. The extreme density of a neutron star also results in exceptionally high surface gravity, typically ranging from 10¹² to 10¹³ m/s§1112§, which is over §1617^18§ times greater than Earth's gravity. A testament to this immense gravitational pull is the fact that neutron stars possess an escape velocity exceeding half the speed of light. The gravitational field of a neutron star accelerates incoming matter to immense speeds, and tidal forces close to its surface are capable of inducing spaghettification.

Properties

Following their formation, neutron stars cease to actively generate heat and consequently cool over time, though they can continue to evolve through processes such as collisions or accretion. Fundamental models for these celestial bodies suggest they are predominantly composed of neutrons, as the immense pressure compels electrons and protons in ordinary matter to merge, forming additional neutrons. These stars are partially sustained against further gravitational collapse by neutron degeneracy pressure, analogous to how electron degeneracy pressure supports white dwarfs. Nevertheless, neutron degeneracy pressure alone is inadequate to support an object exceeding 0.7 M_☉; repulsive nuclear forces progressively contribute to the stability of more massive neutron stars. Should the remnant star's mass surpass the Tolman–Oppenheimer–Volkoff limit, which is approximately 2.2 to 2.9 M_☉, the combined effect of degeneracy pressure and nuclear forces becomes insufficient to prevent collapse, leading to the formation of a black hole. The most massive neutron star identified to date, PSR J0952–0607, has an estimated mass of 2.35±0.17 M_☉.

Newly formed neutron stars can exhibit surface temperatures of ten million Kelvin or higher. However, as neutron stars do not produce new heat through nuclear fusion, they inevitably cool following their genesis. Despite this, surface temperatures typically remain around one million Kelvin after one thousand to one million years, and even older, cooler neutron stars remain readily detectable. For instance, the extensively studied neutron star RX J1856.5−3754 possesses an average surface temperature of approximately 434000 K. In contrast, the Sun's effective surface temperature is merely 5780 K.

The material composing a neutron star is extraordinarily dense: a standard matchbox filled with neutron star matter would possess a mass of approximately 3 billion tonnes, equivalent to the mass of a 0.5-cubic-kilometer segment of Earth's crust (a cube with edges measuring around 800 meters).

During the collapse of a stellar core, its rotational velocity escalates due to the conservation of angular momentum, resulting in newly formed neutron stars typically rotating at several hundred times per second. Certain neutron stars emit focused beams of electromagnetic radiation, rendering them observable as pulsars; the discovery of pulsars by Jocelyn Bell Burnell and Antony Hewish in 1967 provided the initial observational evidence for the existence of neutron stars. The most rapidly rotating neutron star identified is PSR J1748−2446ad, which spins at a rate of 716 times per second, or 42960 revolutions per minute, imparting a linear (tangential) surface speed approaching 0.24‍c (i.e., nearly one-quarter the speed of light).

Equation of State

The equation of state for neutron stars remains presently unknown. This uncertainty stems from their extreme density, ranking them as the second densest known objects in the universe, surpassed only by black holes. Such extreme densities preclude the laboratory replication of their constituent matter on Earth, a standard method for testing equations of state for substances like ideal gases. Furthermore, the considerable distance to the nearest neutron star renders direct observational study impractical. Although neutron stars are understood to resemble a degenerate gas, their immense gravitational fields prevent them from being modeled strictly as such, unlike white dwarfs. Consequently, the neutron star equation of state necessitates the application of general relativity, as Newtonian gravity proves inadequate under these extreme conditions. Moreover, phenomena like quantum chromodynamics (QCD), superconductivity, and superfluidity require incorporation into these models.

Within the exceptionally high densities characteristic of neutron stars, ordinary matter undergoes compression to nuclear densities. This matter exhibits a layered composition: from nuclei immersed in an electron sea within the low-density outer crust, transitioning to progressively neutron-rich structures in the inner crust, then to uniform, extremely neutron-rich matter in the outer core, and potentially culminating in exotic states of matter at the highest densities within the inner core.

Elucidating the composition of matter within the diverse layers of neutron stars, alongside the phase transitions occurring at their interfaces, represents a significant unresolved challenge in fundamental physics. A hypothetical neutron star equation of state would encapsulate structural information and elucidate the behavior of matter under the extreme densities prevalent within these stellar remnants. Such constraints would, in turn, inform our understanding of the strong interaction within the Standard Model, yielding profound implications for nuclear and atomic physics. Consequently, neutron stars serve as invaluable natural laboratories for investigating fundamental physical principles.

For instance, the exotic states potentially residing in neutron star cores are classified as quantum chromodynamics (QCD) matter. Under the extreme densities found at the centers of neutron stars, neutrons undergo disruption, leading to the formation of a quark sea. The equation of state for this matter is dictated by the principles of quantum chromodynamics; however, as QCD matter cannot be synthesized in terrestrial laboratories, current understanding remains predominantly theoretical.

Variations in the equation of state result in divergent values for observable astrophysical quantities. Although the equation of state fundamentally links density and pressure, these parameters subsequently enable the calculation of observables such as the speed of sound, mass, radius, and Love numbers. Numerous neutron star equations of state have been proposed, including FPS, UU, APR, L, and SLy, representing an active field of ongoing research.

A further characteristic of the equation of state concerns its classification as either 'soft' or 'stiff'. This distinction pertains to the pressure exerted at a given energy density and frequently correlates with phase transitions. As a material approaches a phase transition, its pressure typically rises until it transitions into a more stable state. A soft equation of state is characterized by a gradual increase in pressure with energy density, whereas a stiff one exhibits a more abrupt pressure increase. For neutron stars, nuclear physicists continue to investigate whether the equation of state is predominantly stiff or soft; moreover, its character can vary within specific models, contingent on internal phase transitions. Such variations are termed 'stiffening' or 'softening' of the equation of state, reflecting its preceding behavior. Given the unknown precise composition of neutron stars, the equation of state allows for the exploration of various potential phases of matter.

Density and pressure

The overall density of neutron stars ranges from 3.7×10¹⁷ to 5.9ק1213§17 kg/m§1516§, which is equivalent to 2.6ק222324§ to 4.1ק303132§ times the Sun's density. This density is comparable to that of an atomic nucleus, approximately §3637§×§383940§ kg/m§4142§. Density within a neutron star increases with depth, ranging from approximately §4647§×§484950§ kg/m§5152§ at the crust to an estimated §5657§×§585960§ or §6465§×§666768§ kg/m§6970§ in deeper regions. Correspondingly, pressure escalates from about 3.2ק7677^78§ Pa (32 QPa) in the inner crust to 1.6ק848586§ Pa at the core.

The extreme density of a neutron star implies that a mere 5 milliliters (one teaspoon) of its constituent material would possess a mass exceeding 5.5×10¹² kg, which is approximately 900 times the mass of the Great Pyramid of Giza. If the Earth's entire mass were compressed to neutron star density, it would occupy a sphere with a diameter of 305 meters, roughly equivalent to the size of the Arecibo Telescope.

In popular science discourse, neutron stars are occasionally characterized as macroscopic atomic nuclei. Both entities are fundamentally composed of nucleons and exhibit comparable densities, differing by approximately an order of magnitude. Nevertheless, significant distinctions exist between neutron stars and atomic nuclei in several key aspects. Atomic nuclei are bound by the strong nuclear force, whereas neutron stars are gravitationally bound. While nuclear density is uniform, neutron stars are theorized to comprise multiple stratified layers, each possessing distinct compositions and densities.

Current Observational Constraints

Given that various equations of state for neutron stars yield distinct observable properties, such as differing mass-radius relationships, numerous astronomical constraints are applied to these theoretical models. These constraints are primarily derived from data collected by the LIGO gravitational wave observatory and the NICER X-ray telescope.

Observations of pulsars within binary systems by NICER, which enable the estimation of pulsar mass and radius, provide valuable constraints on the neutron star equation of state. For instance, a 2021 measurement of the pulsar PSR J0740+6620 successfully constrained the radius of a 1.4 M_☉ neutron star to 12.33+0.76
−0.8 km, with a 95% confidence level. The integration of these mass-radius constraints with chiral effective field theory calculations further refines the limitations on the neutron star equation of state.

Constraints on the equation of state derived from LIGO gravitational wave detections originate with nuclear and atomic physics researchers, who develop theoretical equations of state, including models such as FPS, UU, APR, L, and SLy. These proposed equations are subsequently utilized by astrophysics researchers to conduct simulations of binary neutron star mergers. Such simulations enable the extraction of gravitational waveforms, facilitating the investigation of the correlation between the equation of state and the gravitational waves produced during binary neutron star mergers. By leveraging these relationships, the neutron star equation of state can be constrained upon the observation of gravitational waves originating from binary neutron star mergers. Numerical relativity simulations of binary neutron star mergers have identified correlations between the equation of state and the frequency-dependent peaks within the gravitational wave signal, which must align with LIGO observations. For instance, the LIGO detection of the binary neutron star merger GW170817 established limits on the tidal deformability of neutron star binaries, thereby excluding entire families of equations of state. Forthcoming gravitational wave signals, detectable by next-generation instruments such as Cosmic Explorer, are anticipated to impose even more stringent constraints.

To assess the viability of a proposed equation of state, nuclear physicists typically compare its predictions for neutron star masses and radii against established observational constraints. Additionally, recent research endeavors focus on constraining the equation of state by analyzing the speed of sound within hydrodynamic models.

Tolman–Oppenheimer–Volkoff equation

The Tolman–Oppenheimer–Volkoff (TOV) equation provides a framework for describing the structure of neutron stars. This equation constitutes a specific solution to Einstein's field equations in general relativity, applicable under the conditions of a spherically symmetric and time-invariant metric. By incorporating a defined equation of state, the solution of the TOV equation enables the determination of observable properties, including stellar mass and radius. Numerous computational programs are employed to numerically solve the TOV equation for various equations of state, thereby establishing the mass-radius relationship and other observable attributes specific to each state.

The subsequent differential equations can be numerically solved to ascertain the observable characteristics of neutron stars:

Mass–Radius Relationship

The derivation of a mass–radius curve is achievable through the application of the Tolman–Oppenheimer–Volkoff (TOV) equations in conjunction with a specific equation of state. Theoretically, any neutron star conforming to a given, accurate equation of state would be situated on its corresponding mass–radius curve. The generation of these curves necessitates solving the TOV equations across a spectrum of central densities. For each distinct central density, the mass and pressure equations require numerical integration until the pressure diminishes to zero, signifying the stellar surface. Each such solution yields a unique pair of mass and radius values associated with that specific central density.

Mass–radius curves, derived from various equations of state, invariably exhibit a maximum value at particular radii. This apex corresponds to the maximum mass. Exceeding this maximum mass renders the star gravitationally unstable, leading to an inevitable collapse into a black hole. Consequently, distinct equations of state generate unique mass–radius curves, each associated with a specific maximum mass. For instance, Oppenheimer and Volkoff established the Tolman–Oppenheimer–Volkoff limit at approximately 0.7 M_☉, predicated on a non-interacting degenerate neutron gas equation of state. Subsequent theoretical advancements, incorporating neutron interactions, elevate the gas pressure and, consequently, raise the mass limit beyond 2.0 M_☉. Furthermore, rapid stellar rotation can extend this limit to approximately 2.9 M_☉. Observational data for the neutron star PSR J0952-0607 indicate a mass of 2.35±0.17 M_☉.

A notable astrophysical phenomenon concerning the maximum mass of neutron stars is termed the "mass gap." This mass gap denotes a specific range, approximately between 2 and 5 solar masses, within which the detection of compact objects has historically been sparse. This interval is predicated on the currently accepted maximum mass for neutron stars (approximately 2 M_☉) and the inferred minimum mass for black holes (around 5 M_☉). Recent gravitational wave observations have, however, revealed the existence of objects within this previously underpopulated mass range. A precise determination of the true maximum mass of neutron stars would significantly aid in classifying compact objects within this mass range as either neutron stars or black holes.

I-Love-Q Relationships

Beyond mass and radius, three additional properties of neutron stars are both dependent on the equation of state and amenable to astronomical observation: the moment of inertia, the quadrupole moment, and the Love number. The moment of inertia quantifies a neutron star's rotational velocity given a constant spin angular momentum. The quadrupole moment characterizes the degree of deviation from a perfect spherical symmetry in a neutron star. The Love number indicates the susceptibility of a neutron star to deformation by tidal forces, a factor particularly relevant in binary star systems.

Although these properties are intrinsically linked to the stellar composition and, consequently, to the equation of state, a fundamental relationship exists among these three quantities that is independent of the specific equation of state. This relationship is derived within the framework of general relativity, assuming slowly and uniformly rotating stars. Despite its independence from the equation of state, which precludes its use in constraining the latter, this relationship offers several other significant applications. Should one of these three parameters be empirically determined for a given neutron star, this relationship enables the inference of the remaining two. Furthermore, this relationship proves instrumental in resolving degeneracies in gravitational wave detections concerning the quadrupole moment and spin, thereby facilitating the determination of the average spin within a specified confidence interval.

Temperature

The internal temperature of a nascent neutron star typically ranges from approximately 10¹¹ to 10¹² kelvin. Nevertheless, a substantial portion of this thermal energy is rapidly dissipated by an immense flux of residual neutrinos, causing the temperature of an isolated neutron star to decrease to approximately §141516§ K within a few years. Subsequent to attaining this reduced temperature, the majority of the residual radiation emitted by the gradually cooling star manifests as X-rays.

A classification system for neutron stars, employing Roman numerals to categorize them by mass and cooling rates, has been proposed by some researchers. This system, distinct from the Yerkes luminosity classes for non-degenerate stars, designates Type I for neutron stars exhibiting low mass and cooling rates, Type II for those with greater mass and cooling rates, and a hypothesized Type III for neutron stars possessing even higher masses, nearing 2 M_☉, coupled with elevated cooling rates, potentially identifying them as candidates for exotic stars.

Magnetic Field

Surface magnetic field strengths of neutron stars typically range from approximately 10⁴ to 10¹¹ tesla (T). These magnitudes significantly surpass those observed in any other celestial object; for context, a sustained 16 T field, achievable in laboratory settings, is powerful enough to induce diamagnetic levitation in a living frog. Discrepancies in magnetic field intensities are posited as the primary determinant for differentiating various neutron star types based on their spectra and for elucidating the periodic emissions characteristic of pulsars.

Magnetars, a distinct subclass of neutron stars, possess the most intense magnetic fields, typically spanning from 10⁸ to 10¹¹ T. It is broadly accepted that these objects are the progenitors of soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs). The magnetic energy density associated with a §141516§ T field is extraordinarily high, substantially surpassing the mass-energy density of conventional matter. Such powerful fields can polarize the vacuum, rendering it birefringent, which allows photons to merge or split and facilitates the creation of virtual particle–antiparticle pairs. Furthermore, these fields alter electron energy levels and compel atoms into elongated cylindrical configurations. In contrast to typical pulsars, the spin-down of a magnetar can be directly driven by its magnetic field, which is sufficiently potent to induce crustal fractures. These crustal fractures precipitate "starquakes," manifested as exceptionally luminous millisecond hard gamma-ray bursts. The resulting fireball becomes confined by the magnetic field, appearing and disappearing from view as the star rotates, a phenomenon observed as periodic soft gamma repeater (SGR) emissions with durations of a few minutes and periods ranging from 5 to 8 seconds.

The genesis of these intense magnetic fields remains an unresolved question. One prevailing hypothesis is "flux freezing," which postulates the conservation of the initial magnetic flux during the neutron star's formation. According to this principle, if an object maintains a constant magnetic flux across its surface while its surface area contracts, the magnetic field strength would proportionally intensify. Similarly, a collapsing progenitor star, possessing a significantly larger surface area than the nascent neutron star, would, through magnetic flux conservation, yield a substantially more powerful magnetic field. Nevertheless, this straightforward explanation does not entirely account for the observed magnetic field strengths of neutron stars.

Gravity

The gravitational field at the surface of a neutron star is approximately 2×10¹¹ times more potent than Earth's, registering around 2.0ק121314§ m/s§1516§. This immense gravitational force functions as a gravitational lens, deflecting the radiation emitted by the neutron star to such an extent that portions of its ordinarily obscured far side become observable. Should the neutron star's radius be equal to or less than 3GM/c§2223§, photons could become gravitationally bound in orbit, rendering the entire stellar surface visible from a singular observation point, while simultaneously destabilizing photon orbits at or within one stellar radius.

During the supernova explosion that precedes the formation of a neutron star, a portion of the progenitor star's mass is liberated, consistent with the principle of mass–energy equivalence, E = mc§56§. This released energy originates from the gravitational binding energy inherent to the neutron star.

Consequently, the gravitational field of a characteristic neutron star is immense. For instance, an object falling from a mere height of 1 m onto a neutron star with a 12 km radius would attain an impact velocity of approximately 1400 km/s. Nevertheless, prior to impact, the extreme tidal forces would induce spaghettification, disintegrating any conventional object into a linear stream of constituent material.

The immense gravitational field of a neutron star induces significant time dilation relative to Earth. For instance, eight years elapsed on a neutron star's surface would correspond to ten years on Earth, even without considering the additional time-dilation effects from the star's rapid rotation.

Relativistic equations of state for neutron stars delineate the relationship between radius and mass across various theoretical models. For a given neutron star mass, the most probable radii are bounded by the AP4 model (representing the smallest radius) and the MS2 model (representing the largest radius). The gravitational binding energy, denoted as E_B, for an observed neutron star with mass M and radius R, is expressed by the following equation: E B M c §31_32§ = 0.60 β §5253§ − β / §67_68§ {\displaystyle {\frac {E_{\text{B}}}{Mc^{2}}}={\frac {0.60\,\beta }{1-{\beta }/{2}}}} where β<!-- β --> = G M / R c §108_109§ {\displaystyle \beta =GM/R{c}^{2}}

According to the AP4 model, a neutron star with a mass of 2M_☉ would not exhibit a radius smaller than 10970 m. For such a star, its mass fraction gravitational binding energy, E_B/Mc§16_{17§, would be 0.187, corresponding to an exothermic value of −18.7%. This value deviates significantly from 0.6/2 = 0.3, or −30%.}

Internal Structure

The current comprehension of neutron star structure is primarily derived from established mathematical models. However, specific structural details may be deduced through investigations of neutron-star oscillations. Asteroseismology, a technique employed for conventional stars, offers a method to elucidate the internal composition of neutron stars by scrutinizing the observed spectra of their stellar oscillations.

Contemporary models suggest that the surface matter of a neutron star consists of atomic nuclei compressed into a solid lattice, permeated by a "sea" of electrons occupying the interstitial spaces. Iron nuclei are hypothesized to be present at the surface, owing to their high binding energy per nucleon. Alternatively, heavier elements like iron might gravitate below the surface, leaving lighter nuclei such as helium and hydrogen predominant. Should the surface temperature surpass 10⁶ K (characteristic of a nascent pulsar), the surface is expected to be in a fluid state, contrasting with the solid phase potentially found in cooler neutron stars (temperatures below 10§910§ K).

The outermost strata of a neutron star comprise an atmosphere, ranging from several millimeters to centimeters in altitude, which gradually transitions into an "ocean" of Coulomb liquid, extending from meters to tens of meters in "depth." The dynamics within these layers are governed by the neutron star's rotational period and magnetic field. Below this ocean lies a solid crust, characterized by extreme hardness and exceptional smoothness (with surface irregularities typically measuring millimeters or less), a direct consequence of the intense gravitational field.

Progressing deeper into the neutron star, nuclei are encountered that possess progressively higher neutron counts. While these nuclei would rapidly decay under terrestrial conditions, they are stabilized by the immense pressures within the star. With increasing depth, this process intensifies, leading to an overwhelming neutron drip and a swift escalation in the concentration of free neutrons.

Neutron stars originate from the remnants following the supernova explosion of a supergiant star. These celestial objects primarily consist of neutrons (neutral particles), alongside a minor proportion of protons (positively charged particles), electrons (negatively charged particles), and atomic nuclei. Within the extreme density characteristic of a neutron star, a significant number of neutrons exist as free particles, unbound to atomic nuclei and capable of unrestricted movement throughout the star's dense material, particularly within its innermost regions, specifically the inner crust and core. Throughout the star's evolutionary lifespan, escalating density correlates with an increase in electron energy, thereby facilitating the production of additional neutrons.

Within neutron stars, the phenomenon known as neutron drip signifies a transitional state where atomic nuclei become so saturated with neutrons that they can no longer retain additional ones, resulting in the formation of a pervasive 'sea' of free neutrons. This emergent neutron sea contributes crucial supplementary pressure, which is instrumental in preserving the star's structural stability and counteracting gravitational collapse. The onset of neutron drip occurs within the inner crust of the neutron star, triggered by a density threshold at which nuclei are incapable of accommodating further neutrons.

Initially, at the commencement of neutron drip, the internal pressure contributions from neutrons and electrons, along with the total stellar pressure, are approximately equivalent. With increasing neutron star density, atomic nuclei begin to disintegrate, leading to the ascendancy of neutron pressure as the primary force. Upon reaching a critical density where nuclei come into contact and subsequently coalesce, they form a fluid primarily composed of neutrons, interspersed with a minor presence of electrons and protons. This specific transition point defines the neutron drip, signifying a fundamental shift in the neutron star's dominant internal pressure from degenerate electrons to neutrons.

Under conditions of very high density, neutron pressure emerges as the principal force sustaining the star, with neutrons existing in a non-relativistic state (moving at a small fraction of the speed of light) and subjected to extreme compression. Nevertheless, at exceptionally high densities, neutrons commence movement at relativistic velocities, approaching the speed of light. Such elevated velocities substantially augment the star's total pressure, thereby modifying its equilibrium state and potentially facilitating the emergence of exotic matter phases.

Within this specific region, a composition of atomic nuclei, free electrons, and free neutrons is observed. Atomic nuclei progressively diminish in size as gravitational pressure overcomes the strong nuclear force, culminating in the core, which is fundamentally defined by its predominant neutron composition. The anticipated stratification of nuclear matter phases within the inner crust is metaphorically termed "nuclear pasta," exhibiting a reduction in voids and an increase in structural size under higher pressure conditions. The precise composition of the superdense matter constituting the core, however, remains a subject of ongoing scientific inquiry. One theoretical model posits the core as superfluid neutron-degenerate matter, predominantly comprising neutrons alongside a minority of protons and electrons. Further exotic matter configurations are also hypothesized, such as degenerate strange matter (incorporating strange quarks in addition to up and down quarks), matter containing high-energy pions and kaons alongside neutrons, or ultra-dense quark-degenerate matter.

Radiation

Pulsars

The detection of neutron stars is primarily accomplished through the observation of their electromagnetic radiation. These celestial bodies typically emit pulsed radio waves and other forms of electromagnetic radiation; those exhibiting such pulsed emissions are designated as pulsars.

The radiation emitted by pulsars is theorized to originate from particle acceleration occurring in proximity to their magnetic poles, which are not necessarily co-aligned with the neutron star's rotational axis. A substantial electrostatic field is believed to accumulate near these magnetic poles, precipitating the emission of electrons. Subsequently, these electrons undergo magnetic acceleration along the field lines, generating curvature radiation that exhibits strong polarization aligned with the plane of curvature. Furthermore, high-energy photons are capable of interacting with lower-energy photons and the ambient magnetic field, facilitating electron-positron pair production; the subsequent annihilation of these pairs then yields additional high-energy photons.

The radiation originating from the magnetic poles of neutron stars is appropriately termed magnetospheric radiation, a nomenclature derived from the neutron star's magnetosphere. This phenomenon should not be conflated with magnetic dipole radiation, which arises from the misalignment between the magnetic and rotational axes, and possesses a radiation frequency identical to the neutron star's rotational frequency.

When the rotational axis of a neutron star diverges from its magnetic axis, observers detect radiation beams exclusively when the magnetic axis aligns with their line of sight during the star's rotation. Consequently, periodic pulses are observed, occurring at a frequency identical to the neutron star's rotational rate.

In May 2022, astronomers documented the discovery of PSR J0901-4046, an ultra-long-period radio-emitting neutron star exhibiting spin characteristics that diverge significantly from those of previously identified neutron stars. The mechanism responsible for its radio emission remains undetermined, posing a challenge to existing models of pulsar evolution.

Non-Pulsating Neutron Stars

Beyond pulsars, non-pulsating neutron stars have also been identified, occasionally exhibiting minor periodic fluctuations in luminosity. This characteristic appears prevalent among X-ray sources designated as Central Compact Objects in supernova remnants (CCOs in SNRs), which are hypothesized to be nascent, radio-quiet, isolated neutron stars.

Spectra

Neutron stars are detectable across various segments of the electromagnetic spectrum, encompassing visible light, near-infrared, ultraviolet, X-rays, and gamma rays, in addition to their radio emissions. Pulsars observed in the X-ray range are classified as X-ray pulsars when powered by accretion, whereas those identified in visible light are termed optical pulsars. While most detected neutron stars, including those observed optically, in X-rays, and in gamma rays, also emit radio waves—exemplified by the Crab Pulsar's pan-spectral electromagnetic output—a distinct category known as radio-quiet neutron stars exists, characterized by an absence of detectable radio emissions.

Rotation

Following their formation, neutron stars exhibit exceptionally rapid rotation, a phenomenon attributed to the conservation of angular momentum. Analogous to an ice skater drawing in their arms to accelerate a spin, the comparatively slow rotation of the progenitor star's core intensifies as it contracts. A newly formed neutron star can achieve rotational rates of multiple revolutions per second.

Spin-Down

Over extended periods, neutron stars gradually decelerate as their rotating magnetic fields effectively dissipate energy linked to their rotation; consequently, older neutron stars may require several seconds to complete a single revolution. This phenomenon is termed spin-down. The rate of rotational deceleration in a neutron star typically remains constant and is exceedingly minute.

The periodic time (P) represents the rotational period, defined as the duration required for one complete rotation of a neutron star. The spin-down rate, which quantifies the deceleration of rotation, is denoted by the symbol ${\displaystyle {\dot {P}}}$ (P-dot), signifying the derivative of P with respect to time. This rate is formally defined as the increase in periodic time per unit time; although inherently dimensionless, it is often expressed with units of s⋅s⁻¹ (seconds per second).

The spin-down rate (P-dot) for neutron stars typically ranges from approximately 10⁻²² to 10⁻⁹ s⋅s⁻¹. Notably, observable neutron stars with shorter periods (i.e., faster rotation) generally exhibit a smaller P-dot. As a neutron star undergoes aging, its rotational velocity diminishes (indicated by an increase in P); ultimately, the rotational rate becomes insufficient to sustain the radio-emission mechanism, rendering radio emissions from the neutron star undetectable.

The parameters P and P-dot enable the estimation of lower bounds for the magnetic fields of neutron stars. Furthermore, P and P-dot are instrumental in calculating the characteristic age of a pulsar; however, when applied to nascent pulsars, this method tends to yield an age estimate that is marginally greater than the actual age.

The parameters P and P-dot, when integrated with a neutron star's moment of inertia, enable the estimation of spin-down luminosity, symbolized as ${\displaystyle {\dot {E}}}$ (E-dot). This value represents the theoretical rate at which rotational energy is lost and subsequently converted into radiation, rather than a direct measurement of luminosity. Neutron stars exhibiting a spin-down luminosity comparable to their observed luminosity are classified as "rotation-powered." The Crab Pulsar's observed luminosity aligns with its spin-down luminosity, thereby substantiating the hypothesis that its radiation is driven by rotational kinetic energy. Conversely, for magnetars, where the actual luminosity can surpass the spin-down luminosity by approximately two orders of magnitude, the energy source is presumed to be magnetic dissipation, not rotational power.

A P–P-dot diagram can be constructed by plotting the P and P-dot values for neutron stars. This diagram provides extensive data regarding the characteristics and population dynamics of pulsars, and its significance for neutron star astrophysics is often compared to that of the Hertzsprung–Russell diagram.

Spin-up

Neutron stars can experience an acceleration in their rotational velocity, a phenomenon termed spin-up. This process can occur when a neutron star accretes orbital material from a companion star, leading to an augmented rotation rate and a transformation of the neutron star's morphology into an oblate spheroid. For millisecond pulsars, this accretion can result in an increase in the neutron star's rotational speed exceeding one hundred revolutions per second.

The neutron star PSR J1748-2446ad currently holds the record for the fastest rotation, achieving 716 revolutions per second. In 2007, a publication documented the observation of an X-ray burst oscillation, an indirect indicator of spin, at 1122 Hz originating from the neutron star XTE J1739-285, implying a rotational rate of 1122 rotations per second. Nevertheless, this particular signal has been detected only once to date and is considered provisional, pending corroboration from subsequent bursts from the same celestial object.

Glitches and starquakes

Neutron stars occasionally experience a "glitch," characterized by an abrupt, minor increase in rotational speed. These glitches are hypothesized to result from starquakes, which occur as the neutron star's rotation decelerates, causing its shape to become more spherical. Given the rigidity of the neutron star's crust, this shape adjustment manifests as discrete events where the crust fractures, generating a starquake analogous to terrestrial earthquakes. Subsequent to a starquake, the star's equatorial radius diminishes, and, owing to the conservation of angular momentum, its rotational velocity consequently increases.

Starquakes in magnetars, which produce associated glitches, constitute the primary hypothesis explaining the gamma-ray emissions from objects known as soft gamma repeaters.

More recent research, however, indicates that a starquake might not liberate adequate energy to account for a neutron star glitch. An alternative proposition suggests that glitches could arise from the transition of vortices within the neutron star's theoretical superfluid core from a metastable energy state to a lower one, thereby releasing energy that manifests as an accelerated rotation rate.

Anti-glitches

The occurrence of an "anti-glitch," defined as an abrupt, minor deceleration in a neutron star's rotation, has also been documented. One such event was observed in the magnetar 1E 2259+586, where it resulted in a twenty-fold increase in X-ray luminosity and a notable alteration in the spin-down rate. Existing neutron star models do not account for this observed phenomenon. Should the origin be internal, this observation implies differential rotation between the solid outer crust and the superfluid interior components of the magnetar.

Population and distances

Currently, approximately 3,200 neutron stars have been identified within the Milky Way and the Magellanic Clouds, with most being detected as radio pulsars. While neutron stars predominantly reside within the Milky Way's disk, their distribution perpendicular to the disk is extensive, attributed to the high translational velocities (up to 400 km/s) imparted to nascent neutron stars during supernova explosions.

Among the nearest identified neutron stars are RX J1856.5−3754, situated approximately 400 light-years from Earth, and PSR J0108−1431, located around 424 light-years away. RX J1856.5-3754 belongs to a proximate cluster of neutron stars known as The Magnificent Seven. Additionally, a proximate neutron star, observed traversing the Ursa Minor constellation, was designated Calvera by its Canadian and American discoverers, referencing the antagonist from the 1960 film The Magnificent Seven. This high-velocity celestial body was identified through the ROSAT Bright Source Catalog.

Neutron stars are primarily observable using contemporary technology during their initial evolutionary phases (typically spanning less than one million years) and are significantly outnumbered by more ancient neutron stars, which would solely be discernible via their blackbody radiation and gravitational interactions with other stellar bodies.

Binary Neutron Star Systems

Approximately five percent of all identified neutron stars are constituents of binary systems. The genesis and subsequent evolution of binary and double neutron star configurations represent intricate astrophysical processes. Observations have confirmed neutron stars coexisting in binary arrangements with typical main-sequence stars, red giants, white dwarfs, and other neutron stars. Contemporary theories of binary stellar evolution postulate the existence of neutron stars within binary systems alongside black hole companions. The coalescence events involving either two neutron stars or a neutron star and a black hole within binary systems have been detected via the emission of gravitational waves.

X-ray Binaries

Neutron star binary systems frequently exhibit X-ray emissions, originating from superheated gas as it descends onto the neutron star's surface. This gas originates from the companion star, whose outer atmospheric layers can be gravitationally stripped by the neutron star if their proximity is adequate. The accretion of this gaseous material by the neutron star can lead to a mass increase; should a sufficient quantity of mass be accreted, the neutron star possesses the potential to collapse into a black hole.

Neutron Star Binary Mergers and Nucleosynthesis

In a close binary system, the orbital separation between two neutron stars is observed to diminish due to the emission of gravitational waves. Eventually, these neutron stars will converge and coalesce. The merger of binary neutron stars constitutes a prominent theoretical framework explaining the genesis of short gamma-ray bursts. Substantial empirical support for this model emerged from the detection of a kilonova linked to the short-duration gamma-ray burst GRB 130603B, with definitive confirmation provided by the observation of gravitational wave GW170817 and short GRB 170817A by LIGO, Virgo, and 70 observatories spanning the electromagnetic spectrum during the event's observation. The luminosity generated during a kilonova is posited to result from the radioactive decay of matter expelled during the binary neutron star merger. This merger transiently establishes conditions of such intense neutron flux that the r-process is facilitated; this mechanism, distinct from supernova nucleosynthesis, is potentially accountable for synthesizing approximately half of the isotopes found in chemical elements heavier than iron.

Planets

Exoplanets are capable of orbiting neutron stars. Such planets may be primordial, circumbinary, gravitationally captured, or formed through a subsequent epoch of planetary genesis. Furthermore, pulsars possess the capacity to ablate a star's atmosphere, resulting in a planetary-mass remnant, which can be categorized as either a chthonian planet or a stellar object, contingent upon astrophysical interpretation. In the context of pulsars, these 'pulsar planets' are detectable using the pulsar timing method, a technique offering exceptional precision and enabling the identification of significantly smaller planets compared to alternative methodologies. Two such systems have received definitive confirmation. The inaugural exoplanets ever identified were Draugr, Poltergeist, and Phobetor, orbiting the pulsar Lich, with their discovery occurring between 1992 and 1994. Among these, Draugr holds the distinction of being the smallest exoplanet yet detected, possessing a mass approximately twice that of Earth's Moon. Another notable system is PSR B1620−26, featuring a circumbinary planet that orbits a binary composed of a neutron star and a white dwarf. Additionally, multiple unconfirmed candidates exist. Pulsar planets receive minimal visible light but are subjected to substantial ionizing radiation and high-energy stellar winds, rendering them considerably inimical environments for life as currently comprehended.

History of Discoveries

In December 1933, during a meeting of the American Physical Society (with proceedings published in January 1934), Walter Baade and Fritz Zwicky posited the existence of neutron stars, a concept introduced less than two years after James Chadwick's discovery of the neutron. To explain the genesis of supernovae, they hypothesized that supernova explosions transform ordinary stars into highly dense objects composed of tightly packed neutrons, which they termed neutron stars. Baade and Zwicky accurately theorized that the gravitational binding energy released from these neutron stars fuels the supernova, stating, "In the supernova process, mass in bulk is annihilated." For decades, neutron stars were considered too dim for detection, leading to minimal research until November 1967. At this time, Franco Pacini proposed that if neutron stars possessed rapid rotation and strong magnetic fields, they would emit electromagnetic waves. Coincidentally, and unbeknownst to Pacini, radio astronomer Antony Hewish and his graduate student Jocelyn Bell at Cambridge were on the verge of detecting radio pulses from celestial bodies now identified as highly magnetized, rapidly rotating neutron stars, commonly known as pulsars.

In 1965, Antony Hewish and Samuel Okoye identified an anomalous source exhibiting high radio brightness temperature within the Crab Nebula. This entity was subsequently recognized as the Crab Pulsar, originating from the significant supernova event of 1054.

Iosif Shklovsky, in 1967, analyzed X-ray and optical observations of Scorpius X-1, accurately deducing that the emitted radiation originated from a neutron star undergoing an accretion phase.

In 1967, Jocelyn Bell Burnell and Antony Hewish detected consistent radio pulses emanating from PSR B1919+21. This pulsar was subsequently identified as an isolated, rotating neutron star, with its energy derived from the neutron star's rotational kinetic energy. As of 2010, the majority of identified neutron stars, approximately 2000, have been discovered through their emission of regular radio pulses as pulsars.

In 1968, Richard V. E. Lovelace and his team, utilizing the Arecibo Observatory, measured the period of the Crab Pulsar to be approximately ${\displaystyle P\!\approx 33}$ milliseconds. This finding led scientists to conclude that pulsars were rotating neutron stars, a departure from the prior widespread belief that they were pulsating white dwarfs.

In 1971, Riccardo Giacconi, Herbert Gursky, Ed Kellogg, R. Levinson, E. Schreier, and H. Tananbaum identified 4.8-second pulsations originating from an X-ray source, Cen X-3, located in the constellation Centaurus. They attributed this phenomenon to a rotating hot neutron star, with the energy source being gravitational, generated by gas accreting onto the neutron star's surface from either a companion star or the interstellar medium.

In 1974, Antony Hewish received the Nobel Prize in Physics for his pivotal contribution to the discovery of pulsars, a recognition that notably excluded Jocelyn Bell, who was a co-discoverer.

Joseph Taylor and Russell Hulse discovered the first binary pulsar, PSR B1913+16, in 1974. This system comprises two neutron stars, one of which is observed as a pulsar, orbiting a common center of mass. Albert Einstein's general theory of relativity posits that massive objects in close binary orbits should emit gravitational waves, leading to a gradual decay of their orbital period. This predicted orbital decay was precisely observed, confirming the tenets of general relativity, and subsequently earned Taylor and Hulse the Nobel Prize in Physics in 1993 for their groundbreaking discovery.

In 1982, Don Backer and his collaborators identified the first millisecond pulsar, PSR B1937+21. This object exhibits a rotational frequency of 642 times per second, a characteristic that provided crucial constraints on the mass and radius parameters of neutron stars. Although numerous other millisecond pulsars were subsequently discovered, PSR B1937+21 held the record as the fastest-spinning known pulsar for 24 years, until the detection of PSR J1748-2446ad, which rotates approximately 716 times per second.

Marta Burgay and her team, in 2003, discovered PSR J0737−3039, the inaugural double neutron star system where both constituent objects are observable as pulsars. The identification of this system facilitates a total of five distinct tests of general relativity, some of which offer unparalleled precision.

In 2010, Paul Demorest and his team determined the mass of the millisecond pulsar PSR J1614−2230 to be 1.97±0.04 M_☉ through the application of Shapiro delay. This measurement significantly exceeded all prior neutron star mass determinations (e.g., 1.67 M_☉ for PSR J1903+0327), thereby imposing stringent constraints on the internal composition of neutron stars.

In 2013, John Antoniadis and his collaborators ascertained the mass of PSR J0348+0432 as 2.01±0.04 M_☉, employing white dwarf spectroscopy. This finding independently corroborated the presence of such massive stars via an alternative methodology. Moreover, it enabled the inaugural test of general relativity utilizing a neutron star of this substantial mass.

In August 2017, the LIGO and Virgo collaborations achieved the initial detection of gravitational waves originating from colliding neutron stars (GW170817), which subsequently facilitated additional insights into neutron star characteristics.

In October 2018, astronomical researchers posited a direct correlation between GRB 150101B, a gamma-ray burst observed in 2015, and the landmark GW170817 event, associating both with the merger of two neutron stars. The pronounced similarities across gamma-ray, optical, and X-ray emissions, alongside the characteristics of their respective host galaxies, were deemed "striking". This suggests that both distinct events likely resulted from neutron star mergers and represent kilonovae, which, according to the researchers, might be more prevalent in the cosmos than previously theorized.

In July 2019, astronomers announced the proposal of a novel methodology for determining the Hubble constant and addressing discrepancies observed in prior methods. This approach is predicated on the mergers of binary neutron stars, specifically following the detection of the GW170817 neutron star merger. Their calculated Hubble constant value is 70.3+5.3
−5.0 (km/s)/Mpc.

A 2020 investigation conducted by University of Southampton PhD student Fabian Gittins indicated that surface irregularities, termed "mountains," on neutron stars might measure only fractions of a millimeter in height (approximately 0.000003% of the neutron star's diameter). This finding, hundreds of times smaller than earlier predictions, carries significant implications for the observed non-detection of gravitational waves emanating from rotating neutron stars.

Utilizing the James Webb Space Telescope (JWST), astronomers identified a neutron star situated within the remnants of the Supernova 1987A stellar explosion.

Subtypes

Several categories of celestial objects either comprise or incorporate a neutron star:

• Isolated neutron star (INS): A neutron star not gravitationally bound within a binary system.
  • Rotation-powered pulsar (RPP or "radio pulsar"): Neutron stars characterized by the emission of directed radiation pulses at consistent intervals, attributed to their potent magnetic fields.
    • Rotating radio transient (RRATs): These are hypothesized to be pulsars exhibiting more sporadic emissions or greater pulse-to-pulse variability compared to the majority of identified pulsars.
  • Magnetar: A neutron star possessing an exceptionally powerful magnetic field (exceeding that of a typical neutron star by a factor of 1000) and extended rotation periods, typically ranging from 5 to 12 seconds.
    • Soft gamma repeater (SGR).
    • Anomalous X-ray pulsar (AXP).
  • Radio-quiet neutron stars.
    • X-ray dim isolated neutron stars.
    • Central compact objects in supernova remnants (CCOs in SNRs): These are young, radio-quiet, non-pulsating X-ray sources, widely considered to be Isolated Neutron Stars enveloped by supernova remnants.
• X-ray pulsars or "accretion-powered pulsars": A category of X-ray binaries.
  • Low-mass X-ray binary pulsars: A subclass of low-mass X-ray binaries (LMXB), comprising a pulsar paired with a main sequence star, white dwarf, or red giant.
    • Millisecond pulsar (MSP) ("recycled pulsar").
      • "Spider Pulsar": A pulsar characterized by a semi-degenerate companion star.
        • "Black Widow" pulsar: A specific type of "Spider Pulsar" where the companion star possesses an exceptionally low mass (less than 0.1 M_☉).
        • "Redback" pulsar: A type of pulsar where the companion star is more massive.
      • Sub-millisecond pulsar.
    • X-ray burster: A neutron star featuring a low-mass binary companion, from which matter is accreted, leading to intermittent energy bursts from the neutron star's surface.
  • Intermediate-mass X-ray binary pulsars: A class of intermediate-mass X-ray binaries (IMXB), consisting of a pulsar paired with an intermediate-mass star.
  • High-mass X-ray binary pulsars: A class of high-mass X-ray binaries (HMXB), characterized by a pulsar in conjunction with a massive star.
  • A binary pulsar is characterized by a pulsar orbiting a companion, which is frequently a white dwarf or another neutron star.
  • An X-ray tertiary is a theorized celestial object.

Several theorized compact stars exhibit properties analogous to neutron stars but are not classified as such.

• A protoneutron star (PNS) represents a theorized intermediate-stage celestial body that undergoes cooling and contraction, ultimately evolving into either a neutron star or a black hole.
• An exotic star is a hypothetical compact stellar object composed of matter exceeding the density of neutron matter, potentially incorporating quarks, hyperons, or other subatomic particles distinct from neutrons or protons. A Thorne–Żytkow object is currently hypothesized as the result of a neutron star merging with a red giant star. A quark star is presently a hypothetical classification of neutron star, theorized to consist of quark matter or strange matter. As of 2018, three candidate objects have been identified.

Illustrative Examples of Neutron Stars

• The Black Widow Pulsar is a highly massive millisecond pulsar.
• PSR J0952−0607 represents the most massive neutron star identified, possessing a mass of 2.35+0.17
  −0.17 M_☉, and is categorized as a Black Widow Pulsar.
• LGM-1, subsequently designated PSR B1919+21, was the inaugural radio-pulsar to be recognized, discovered by Jocelyn Bell Burnell in 1967.
• PSR B1257+12, also referred to as Lich, holds the distinction of being the first neutron star (a millisecond pulsar) identified with orbiting planets.
• PSR B1509−58 is the celestial source depicted in the "Hand of God" image, captured by the Chandra X-ray Observatory.
• RX J1856.5−3754 is recognized as the nearest neutron star.
• The Magnificent Seven refers to a cluster of proximate, X-ray dim, isolated neutron stars.
• PSR J0348+0432 is the most massive neutron star for which a precisely constrained mass has been determined, measured at 2.01±0.04 M_☉.
• SWIFT J1756.9−2508 is a millisecond pulsar accompanied by a stellar-type companion possessing a mass within the planetary range (i.e., below that of a brown dwarf).
• SWIFT J1818.0−1607 is currently recognized as the youngest magnetar.

Gallery

Notes

Notes

References

Sources

Moustakidis, Charalampos, ed. (2024). The Nuclear Physics of Neutron Stars. MDPI. ISBN 978-3-7258-1600-2.

• Moustakidis, Charalampos, ed. (2024). The Nuclear Physics of Neutron Stars. MDPI. ISBN 978-3-7258-1600-2.Hessels, Jason W. T; Ransom, Scott M; Stairs, Ingrid H; Freire, Paulo C. C; Kaspi, Victoria M; Camilo, Fernando (2003). "Neutron Stars for Undergraduates". American Journal of Physics. 72 (2004): 892–905. arXiv:nucl-th/0309041. Bibcode:2004AmJPh..72..892S. doi:10.1119/1.1703544. S2CID 27807404.Silbar, Richard R; Reddy, Sanjay (2005). "Erratum: "Neutron stars for undergraduates" [Am. J. Phys. 72 (7), 892–905 (2004)]". American Journal of Physics. 73 (3): 286. arXiv:nucl-th/0309041. Bibcode:2005AmJPh..73..286S. doi:10.1119/1.1852544.
• "Mysterious X-ray Sources Potentially Identified as Isolated Neutron Stars." David Shiga. New Scientist. June 23, 2006.
• "Massive Neutron Star Excludes Exotic Matter." New Scientist. A recent analysis indicates that exotic states of matter, including free quarks or Bose-Einstein condensates (BECs), do not form within neutron stars.
• "Neutron Star Recorded at Astonishing Velocity." New Scientist. A neutron star's velocity has been measured exceeding 1500 kilometers per second.