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{{short description|Collapsed core of a massive star}}{{other uses|Neutron Star (disambiguation)}}File:Moving heart of the Crab Nebula.jpg|thumb|upright=1.4|Central neutron star at the heart of the Crab NebulaCrab NebulaFile:PIA18848-PSRB1509-58-ChandraXRay-WiseIR-20141023.jpg|thumb|upright=1.4|Radiation from the rapidly spinning pulsar PSR B1509-58 makes nearby gas emit X-rays (gold) and illuminates the rest of the nebula, here seen in infraredinfraredA neutron star is a collapsed core of a massive supergiant star. The stars that later collapse into neutron stars have a total mass of between 10 and 25 solar masses ({{solar mass}}), possibly more if the star was especially rich in elements heavier than hydrogen and helium.JOURNAL, Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., Hartmann, D. H., 2003, How Massive Single Stars End Their Life, Astrophysical Journal, 591, 1, 288–300, astro-ph/0212469, 2003ApJ...591..288H, 10.1086/375341, 59065632, Except for black holes, neutron stars are the smallest and densest known class of stellar objects.BOOK, Compact Stars: Nuclear Physics, Particle Physics and General Relativity, illustrated, Norman K., Glendenning, Springer Science & Business Media, 2012, 978-1-4684-0491-3, 1,weblink 2016-03-21, 2017-01-31,weblink live, Neutron stars have a radius on the order of {{convert|10|km|sigfig=1|sp=us}} and a mass of about {{solar mass|1.4}}.BOOK, Astronomy: The Solar System and Beyond, 6th, Michael, Seeds, Dana, Backman, Cengage Learning, 2009, 978-0-495-56203-0, 339,weblink 2018-02-22, 2021-02-06,weblink live, They result from the supernova explosion of a massive star, combined with gravitational collapse, that compresses the core past white dwarf star density to that of atomic nuclei.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 producing 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 {{Solar mass|0.7|link=y}}JOURNAL, R. C., Tolman, 1939, Static Solutions of Einstein's Field Equations for Spheres of Fluid, Physical Review, 55, 4, 364–373, 10.1103/PhysRev.55.364, 1939PhRv...55..364T,weblink 2019-06-30, 2018-07-22,weblink live, JOURNAL, J. R., Oppenheimer, G. M., Volkoff, 1939, On Massive Neutron Cores, Physical Review, 55, 4, 374–381, 10.1103/PhysRev.55.374, 1939PhRv...55..374O, and repulsive nuclear forces play a larger role in supporting more massive neutron stars.WEB, Neutron Stars,weblink www.astro.princeton.edu, 14 December 2018, 9 September 2021,weblink live, JOURNAL, Douchin, F., Haensel, P., December 2001, A unified equation of state of dense matter and neutron star structure, Astronomy & Astrophysics, 380, 1, 151–167, 10.1051/0004-6361:20011402, 0004-6361, astro-ph/0111092, 2001A&A...380..151D, 17516814, If the remnant star has a mass exceeding the Tolman–Oppenheimer–Volkoff limit, which ranges from {{nowrap|2.2–2.9 {{solar mass}},}} 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 {{val|2.35|0.17|u=solar mass}}.Newly formed neutron stars may have surface temperatures of ten million K or more. However, since neutron stars generate no new heat through fusion, they inexorably cool down after their formation. Consequently, a given neutron star reaches a surface temperature of one million degrees K when it is between one thousand and one million years old."Q&A: Supernova Remnants and Neutron Stars", Chandra.harvard.edu (September 5, 2008) Older and even-cooler neutron stars are still easy to discover. For example, the well-studied neutron star, {{nowrap|RX J1856.5−3754,}} has an average surface temperature of about 434,000 K."Magnetic Hydrogen Atmosphere Models and the Neutron Star RX J1856.5−3754" (PDF), Wynn C. G. Ho et al., Monthly Notices of the Royal Astronomical Society, 375, pp. 821-830 (2007), submitted December 6, 2006, ArXiv:astro-ph/0612145. The authors calculated what they considered to be "a more realistic model, which accounts for magnetic field and temperature variations over the neutron star surface as well as general relativistic effects," which yielded an average surface temperature of {{val|4.34|e=5|+0.02|-0.06|u=K}} at a confidence level of 2𝜎 (95%); see §4, Fig. 6 in their paper for details. For comparison, the Sun has an effective surface temperature of 5,780 K."The Sun is less active than other solar-like stars" (PDF), Timo Reinhold et al., ArXiv:astro-ph.SR (May 4, 2020) ArXiv:2005.01401Neutron 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 meters) from Earth's surface.WEB,weblink heasarc.gsfc.nasa.gov, Tour the ASM Sky, 2016-05-23, 2021-10-01,weblink live, WEB,weblink Density of the Earth, 2009-03-10, 2016-05-23, 2013-11-12,weblink" title="web.archive.org/web/20131112145139weblink">weblink live, As a star's core collapses, its rotation rate increases due to conservation of angular momentum, and newly formed neutron stars 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 secondJOURNAL, Hessels, Jason, 4, Ransom, Scott M., Stairs, Ingrid H., Freire, Paulo C. C., Victoria Kaspi, Kaspi, Victoria M., Camilo, Fernando, A Radio Pulsar Spinning at 716 Hz, Science (journal), Science, 311, 5769, 1901–1904, 2006, 10.1126/science.1123430, 16410486, 2006Sci...311.1901H, astro-ph/0601337, 10.1.1.257.5174, 14945340, NEWS, Naeye, Robert, 2006-01-13, Spinning Pulsar Smashes Record, Sky & Telescope,weblink 2008-01-18,weblink" title="web.archive.org/web/20071229113749weblink">weblink 2007-12-29, dead, or 43,000 revolutions per minute, giving a linear (tangential) speed at the surface on the order of 0.24c (i.e., nearly a quarter the speed of light).There are thought to be around one billion neutron stars in the Milky Way,WEB,weblink NASA.gov, 2020-08-05, 2018-09-08,weblink live, and at a minimum several hundred million, a figure obtained by estimating the number of stars that have undergone supernova explosions.BOOK, Camenzind, Max, Compact Objects in Astrophysics: White Dwarfs, Neutron Stars and Black Holes, 24 February 2007, Springer Science & Business Media, 978-3-540-49912-1, 269,weblink 2007coaw.book.....C, 6 September 2017, 29 April 2021,weblink live, However, many of them have existed for a long period of time and have cooled down considerably. These stars radiate very little electromagnetic radiation; most neutron stars that have been detected occur only in certain situations in which they do radiate, such as if they are a pulsar or a part of a binary system. Slow-rotating and non-accreting neutron stars are difficult to detect, due to the absence of electromagnetic radiation; however, since the Hubble Space Telescope's detection of RX J1856.5−3754 in the 1990s, a few nearby neutron stars that appear to emit only thermal radiation have been detected.Neutron stars in binary systems can undergo accretion, in which case they emit large amounts of X-rays. During this process, matter is deposited on the surface of the stars, forming "hotspots" that can be sporadically identified as X-ray pulsar systems. Additionally, such accretions are able to "recycle" old pulsars, causing them to gain mass and rotate extremely quickly, forming millisecond pulsars. Furthermore, binary systems such as these continue to evolve, with many companions eventually becoming compact objects such as white dwarfs or neutron stars themselves, though other possibilities include a complete destruction of the companion through ablation or collision. The merger of binary neutron stars may be the source of short-duration gamma-ray bursts and are likely strong sources of gravitational waves. In 2017, a direct detection (GW170817) of the gravitational waves from such an event was observed,JOURNAL, 2017, Multi-messenger Observations of a Binary Neutron Star Merger, The Astrophysical Journal Letters, 848, 2, L12, 1710.05833, 2017ApJ...848L..12A, 10.3847/2041-8213/aa91c9, Abbott, B. P., Abbott, R., Abbott, T. D., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Richard, Howard, Adhikari, R. X., Huang-Wei, 217162243, free, along with indirect observation of gravitational waves from the Hulse-Taylor pulsar.

Formation

(File:Neutronstarsimple.png|thumb|Simplified representation of the formation of neutron stars)Any main-sequence star with an initial mass of greater than {{Solar mass|8|link=y}} (eight times the mass of the Sun) has the potential to become a neutron star. As the star evolves away from the main sequence, stellar nucleosynthesis produces an iron-rich core. When all nuclear fuel in the core has been exhausted, the core must be supported by degeneracy pressure alone. Further deposits of mass from shell burning cause the core to exceed the Chandrasekhar limit. Electron-degeneracy pressure is overcome, and the core collapses further, causing temperatures to rise to over {{val|5|e=9|u=K}} (5 billion K). At these temperatures, photodisintegration (the breakdown of iron nuclei into alpha particles due to high-energy gamma rays) occurs. As the temperature of the core continues to rise, electrons and protons combine to form neutrons via electron capture, releasing a flood of neutrinos. When densities reach a nuclear density of {{val|4|e=17|u=kg/m3}}, a combination of strong force repulsion and neutron degeneracy pressure halts the contraction.JOURNAL, I., Bombaci, 1996, The Maximum Mass of a Neutron Star, Astronomy and Astrophysics, 305, 871–877, 1996A&A...305..871B, The contracting outer envelope of the star is halted and rapidly flung outwards by a flux of neutrinos produced in the creation of the neutrons, resulting in a supernova and leaving behind a neutron star. However, if the remnant has a mass greater than about {{Solar mass|3}}, it instead becomes a black hole.BOOK, The Birth of Stars and Planets, illustrated, John, Bally, Bo, Reipurth, Cambridge University Press, 2006, 978-0-521-80105-8, 207,weblink 2016-06-30, 2017-01-31,weblink live, As the core of a massive star is compressed during a Type II supernova or a Type Ib or Type Ic supernova, and collapses into a neutron star, it retains most of its angular momentum. Because it has only a tiny fraction of its parent's radius (sharply reducing its moment of inertia), a neutron star is formed with very high rotation speed and then, over a very long period, it slows. Neutron stars are known that have rotation periods from about 1.4 ms to 30 s. The neutron star's density also gives it very high surface gravity, with typical values ranging from {{val|e=12}} to {{val|e=13|u=m/s2}} (more than {{val|e=11}} times that of Earth).BOOK, Neutron Stars, PaweÅ‚, Haensel, Alexander Y., Potekhin, Dmitry G., Yakovlev, 978-0-387-33543-8, Springer, 2007, One measure of such immense gravity is the fact that neutron stars have an escape velocity of over half the speed of light.WEB, The Remarkable Properties of Neutron Stars - Fresh Chandra News, ChandraBlog, 2013-03-28,weblink 2022-05-16, The neutron star's gravity accelerates infalling matter to tremendous speed, and tidal forces near the surface can cause spaghettification.

Properties

{{More citations needed section|date=May 2024}}

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 matter#:~:text=Degenerate gases are gases composed,white dwarfs are two examples.|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. Affects 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.JOURNAL, Hebeler, K., Lattimer, J. M., Pethick, C. J., Schwenk, A., 2013-07-19, Equation of State and Neutron Star Properties Constrained by Nuclear Physics and Observation,weblink The Astrophysical Journal, 773, 1, 11, 10.1088/0004-637X/773/1/11, 1303.4662, 2013ApJ...773...11H, 0004-637X, 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. The neutron star equation of state encodes information about the structure of a neutron star and thus tells us 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 force of the standard model works, which would have profound implications for nuclear and atomic physics. This makes 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 is only directly relating the density and pressure, it also leads to calculating observables like the speed of sound, mass, radius, and Love numbers. Because the equation of state is unknown, there are many proposed ones, such as FPS, UU, APR, L, and SLy, and it is an active area of research. Different factors can be considered when creating the equation of state such as phase transitions.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

Neutron stars have overall densities of {{val|3.7|e=17}} to {{val|5.9|e=17|u=kg/m3}} ({{val|2.6|e=14}} to {{val|4.1|e=14}} times the density of the Sun),{{val|3.7|e=17|u=kg/m3}} derives from mass {{val|2.68|e=30|u=kg}} / volume of star of radius 12 km; {{val|5.9|e=17|u=kg/m3}} derives from mass {{val|4.2|e=30|u=kg}} per volume of star radius 11.9 km which is comparable to the approximate density of an atomic nucleus of {{val|3|e=17|u=kg/m3}}.WEB,weblink Calculating a Neutron Star's Density, 2006-03-11, 2006-02-24,weblink" title="web.archive.org/web/20060224011955weblink">weblink live, NB {{val|3|e=17|u=kg/m3}} is {{val|3|e=14|u=g/cm3}} The neutron star's density varies from about {{val|1|e=9|u=kg/m3}} in the crust—increasing with depth—to about {{val|6|e=17}} or {{val|8|e=17|u=kg/m3}} (denser than an atomic nucleus) deeper inside.JOURNAL, Introduction to neutron stars, American Institute of Physics Conference Series, 1645, 1, 61–78, 2015AIPC.1645...61L, Lattimer, James M., 2015, 10.1063/1.4909560, AIP Conference Proceedings, free, A neutron star is so dense that one teaspoon (5 milliliters) of its material would have a mass over {{val|5.5|e=12|u=kg}}, about 900 times the mass of the Great Pyramid of Giza.The average density of material in a neutron star of radius 10 km is {{val|1.1|e=12|u=kg/cm3}}. Therefore, 5 ml of such material is {{val|5.5|e=12|u=kg}}, or 5,500,000,000 metric tons. This is about 15 times the total mass of the human world population. Alternatively, 5 ml from a neutron star of radius 20 km radius (average density {{val|8.35|e=10|u=kg/cm3}}) has a mass of about 400 million metric tons, or about the mass of all humans. The gravitational field is ca. {{val|2|e=11}}g or ca. {{val|2|e=12}} N/kg. Moon weight is calculated at 1g. The entire mass of the Earth at neutron star density would fit into a sphere of 305 m in diameter (the size of the Arecibo Telescope). The pressure increases from {{val|3.2|e=31}} to {{val|1.6|e=34|u=Pa}} from the inner crust to the center.JOURNAL, Ozel, Feryal, Freire, Paulo, Masses, Radii, and the Equation of State of Neutron Stars, Annu. Rev. Astron. Astrophys., 54, 1, 401–440, 2016, 10.1146/annurev-astro-081915-023322, 2016ARA&A..54..401O, 1603.02698, 119226325, A neutron star has some of the properties of an atomic nucleus, including density (within an order of magnitude) and being composed of nucleons. In popular scientific writing, neutron stars are therefore sometimes described as "giant nuclei". 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.JOURNAL, Baym, G, Pethick, C, December 1975, Neutron Stars, Annual Review of Nuclear Science, en, 25, 1, 27–77, 10.1146/annurev.ns.25.120175.000331, 0066-4243, 1975ARNPS..25...27B, free,

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 LIGOWEB, LIGO Lab {{!, Caltech {{!}} MIT |url=https://www.ligo.caltech.edu/ |access-date=2024-05-10 |website=LIGO Lab {{!}} Caltech}}, which is a gravitational wave observatory, and NICER,WEB, NICER - NASA Science,weblink 2024-05-10, science.nasa.gov, en-US, which is an 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 1.4 solar mass neutron star to {{val|12.33|0.76|0.8}} km with 95% confidence.JOURNAL, Raaijmakers, G., Greif, S. K., Hebeler, K., Hinderer, T., Nissanke, S., Schwenk, A., Riley, T. E., Watts, A. L., Lattimer, J. M., Ho, W. C. G., 2021-09-01, Constraints on the Dense Matter Equation of State and Neutron Star Properties from NICER's Mass–Radius Estimate of PSR J0740+6620 and Multimessenger Observations, The Astrophysical Journal Letters, 918, 2, L29, 10.3847/2041-8213/ac089a, free, 2105.06981, 2021ApJ...918L..29R, 2041-8205, These mass-radius constraints, combined with chiral effective field theory calculations, tightens constraints on the neutron star equation of state.Equation of state constraints from LIGO gravitational wave detections start with nuclear and atomic physics researchers, who work to propose theoretical equations of state (such as FPS, UU, APR, L, SLy, and others). The proposed equations of state can then be passed onto astrophysics researchers who run simulations of binary neutron star mergers. From these simulations, researchers can extract gravitational waveforms, thus studying the relationship between the equation of state and gravitational waves emitted by binary neutron star mergers. Using these relations, one can constrain the neutron star equation of state when gravitational waves from binary neutron star mergers are observed. Past numerical relativity simulations of binary neutron star mergers have found relationships between the equation of state and frequency dependent peaks of the gravitational wave signal that can be applied to LIGO detections.JOURNAL, Takami, Kentaro, Rezzolla, Luciano, Baiotti, Luca, 2014-08-28, Constraining the Equation of State of Neutron Stars from Binary Mergers,weblink Physical Review Letters, en, 113, 9, 091104, 10.1103/PhysRevLett.113.091104, 1403.5672, 2014PhRvL.113i1104T, 0031-9007, For example, the LIGO detection of the binary neutron star merger GW170817 provided limits on the tidal deformability of the two neutron stars which dramatically reduced the family of allowed equations of state.JOURNAL, Annala, Eemeli, Gorda, Tyler, Kurkela, Aleksi, Vuorinen, Aleksi, 2018-04-25, Gravitational-Wave Constraints on the Neutron-Star-Matter Equation of State,weblink Physical Review Letters, en, 120, 17, 172703, 10.1103/PhysRevLett.120.172703, 29756823, 1711.02644, 2018PhRvL.120q2703A, 0031-9007, Future gravitational wave signals with next generation detectors like Cosmic Explorer can impose further constraints.JOURNAL, Finstad, Daniel, White, Laurel V., Brown, Duncan A., 2023-09-01, Prospects for a Precise Equation of State Measurement from Advanced LIGO and Cosmic Explorer, The Astrophysical Journal, 955, 1, 45, 10.3847/1538-4357/acf12f, free, 2211.01396, 2023ApJ...955...45F, 0004-637X, 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.ARXIV, Lovato, Alessandro, Dore, Travis, 1, Long Range Plan: Dense matter theory for heavy-ion collisions and neutron stars, 2022, 2211.02224, There is also recent work on constraining the equation of state with the speed of sound through hydrodynamics.ARXIV, Hippert, Mauricio, Noronha, Jorge, Romatschke, Paul, Upper Bound on the Speed of Sound in Nuclear Matter from Transport, 2024, 2402.14085,

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:JOURNAL, Silbar, Richard R., Reddy, Sanjay, Neutron stars for undergraduates, American Journal of Physics, 1 July 2004, 72, 7, 892–905, 10.1119/1.1703544, frac{dp}{dr} = - frac{Gepsilon(r)M(r)}{c^2r^2}[1+frac{p(r)}{epsilon(r)}][1+frac{4pi r^3p(r)}{M(r)c^2}][1-frac{2GM(r)}{c^2r}]frac{dM}{dr} = frac{4pi r^2epsilon(r)}{c^2}where G is the gravitational constant, p(r) is the pressure, epsilon(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. The idea is that for the correct equation of state, every neutron star that could possibly exist would lie along that curve. This is one of the ways equations of state can be constrained by astronomical observations. To create these curves, one must solve the TOV equations for different central densities. For each central density, you numerically solve the mass and pressure equations until the pressure goes to zero, which is the outside of the star. Each solution gives a corresponding mass and radius for that central density.Mass-radius curves determine what the maximum mass is for a given equation of state. Through most of the mass-radius curve, each radius corresponds to a unique mass value. At a certain point, the curve will reach a maximum and start going back down, leading to repeated mass values for different radii. This maximum point is what 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 the force of gravity, and would collapse into a black hole. Since each equation of state leads to a different mass-radius curve, they also lead to a unique maximum mass value. The maximum mass value is unknown as long as the equation of state remains unknown.This is very important when it comes to constraining the equation of state. Oppenheimer and Volkoff came up with the Tolman-Oppenheimer-Volkoff limit using a degenerate gas equation of state with the TOV equations that was ~0.7 Solar masses Since the neutron stars that have been observed are more massive than that, that maximum mass was discarded. The most recent massive neutron star that was observed was PSR J0952-0607 which was {{val|2.35|0.17}} solar masses. Any equation of state with a mass less than that would not predict that star and thus is much less likely to be correct. An interesting phenomenon in this area of astrophysics relating to the maximum mass of neutron stars is what is called the "mass gap". The mass gap refers to a range of masses from roughly 2-5 solar masses where very few compact objects were observed. This range is based on the current assumed maximum mass of neutron stars (~2 solar masses) and the minimum black hole mass (~5 solar masses).JOURNAL, Kumar, N., Sokolov, V. V., Mass Distribution and “Mass Gap” of Compact Stellar Remnants in Binary Systems, Astrophysical Bulletin, June 2022, 77, 2, 197–213, 10.1134/S1990341322020043, 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.JOURNAL, Yagi, Kent, Yunes, Nicolás, I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves, and fundamental physics, Physical Review D, 19 July 2013, 88, 2, 10.1103/PhysRevD.88.023009,

Temperature

The temperature inside a newly formed neutron star is from around {{val|e=11}} to {{val|e=12|ul=kelvin}}. However, the huge number of neutrinos it emits carries away so much energy that the temperature of an isolated neutron star falls within a few years to around {{val|e=6|u=kelvin}}. At this lower temperature, most of the light generated by a neutron star is in X-rays.Some researchers have proposed a neutron star classification system using Roman numerals (not to be confused with the Yerkes luminosity classes for non-degenerate stars) to sort neutron stars by their mass and cooling rates: type I for neutron stars with low mass and cooling rates, type II for neutron stars with higher mass and cooling rates, and a proposed type III for neutron stars with even higher mass, approaching {{solar mass|2}}, and with higher cooling rates and possibly candidates for exotic stars.JOURNAL, Yakovlev, D. G., Kaminker, A. D., Haensel, P., Gnedin, O. Y., 2002, The cooling neutron star in 3C 58, Astronomy & Astrophysics, 389, L24–L27, astro-ph/0204233, 2002A&A...389L..24Y, 10.1051/0004-6361:20020699, 6247160,

Magnetic field

The magnetic field strength on the surface of neutron stars ranges from {{circa|{{val|e=4}}}} to {{val|e=11}} tesla (T).ARXIV, A., Reisenegger, 2003, Origin and Evolution of Neutron Star Magnetic Fields, astro-ph/0307133, 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.The neutron stars known as magnetars have the strongest magnetic fields, in the range of {{val|e=8}} to {{val|e=11|u=T}},WEB, McGill SGR/AXP Online Catalog,weblink 2 Jan 2014, 23 July 2020,weblink" title="web.archive.org/web/20200723080137weblink">weblink live, and have become the widely accepted hypothesis for neutron star types soft gamma repeaters (SGRs)JOURNAL, Chryssa, Kouveliotou, Robert C., Duncan, Christopher, Thompson, February 2003, Magnetars, Scientific American, 288, 2, 34–41, 10.1038/scientificamerican0203-34, 12561456, 2003SciAm.288b..34K, and anomalous X-ray pulsars (AXPs).JOURNAL, V.M., Kaspi, F.P., Gavriil, 2004, (Anomalous) X-ray pulsars, Nuclear Physics B, Proceedings Supplements, 132, 456–465, 10.1016/j.nuclphysbps.2004.04.080, astro-ph/0402176, 2004NuPhS.132..456K, 15906305, The magnetic energy density of a {{val|e=8|u=T}} field is extreme, greatly exceeding the mass-energy density of ordinary matter.{{efn|Magnetic energy density for a field B is {{nowrap| U {{=}} {{frac|μ0 B2|2}} .}}WEB,weblink Eric Weisstein's World of Physics, scienceworld.wolfram.com,weblink" title="web.archive.org/web/20190423232524weblink">weblink 2019-04-23, Substituting {{nowrap| B {{=}} {{val|e=8|u=T}} ,}} get {{nowrap|U {{=}} {{val|4|e=21|u=J|up=m3}} .}} Dividing by c2 one obtains the equivalent mass density of {{val|44500|u=kg|up=m3}}, which exceeds the standard temperature and pressure density of all known materials. Compare with {{val|22590|u=kg|up=m3}} for osmium, the densest stable element.}} 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.WEB,weblink 'Magnetars', soft gamma repeaters & very strong magnetic fields, Robert C., Duncan, March 2003, 2018-04-17, 2020-01-19,weblink" title="web.archive.org/web/20200119142438weblink">weblink live, The origins of the strong magnetic field are as yet unclear. One hypothesis is that of "flux freezing", or conservation of the original magnetic flux during the formation of the neutron star. If an object has a certain magnetic flux over its surface area, and that area shrinks to a smaller area, but the magnetic flux is conserved, then the magnetic field would correspondingly increase. Likewise, a collapsing star begins with a much larger surface area than the resulting neutron star, and conservation of magnetic flux would result in a far stronger magnetic field. However, this simple explanation does not fully explain magnetic field strengths of neutron stars.

Gravity

{{See also|Tolman–Oppenheimer–Volkoff equation|White dwarf#Mass–radius relationship}}File:Neutronstar 2Rs.svg|thumb|Gravitational light deflection at a neutron star. Due to relativistic light deflection over half the surface is visible (each grid patch represents 30 by 30 degrees). In natural units, this star's mass is 1 and its radius is 4, or twice its Schwarzschild radiusSchwarzschild radiusThe gravitational field at a neutron star's surface is about {{val|2|e=11}} times stronger than on Earth, at around {{val|2.0|e=12|u=m/s2}}.BOOK, An Introduction to the Sun and Stars, illustrated, Simon F., Green, Mark H., Jones, S. Jocelyn, Burnell, Cambridge University Press, 2004, 978-0-521-54622-5, 322,weblink 2016-06-09, 2017-01-31,weblink live, 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.WEB, Corvin, Zahn, Tempolimit Lichtgeschwindigkeit, 1990-10-09,weblink de, Durch die gravitative Lichtablenkung ist mehr als die Hälfte der Oberfläche sichtbar. Masse des Neutronensterns: 1, Radius des Neutronensterns: 4, ... dimensionslosen Einheiten (c, G = 1), 2009-10-09, 2021-01-26,weblink live, If the radius of the neutron star is 3GM/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, {{nowrap|1=E = mc2}}). The energy comes from the gravitational binding energy of a neutron star.Hence, the gravitational force of a typical neutron star is huge. If an object were to fall from a height of one meter on a neutron star 12 kilometers in radius, it would reach the ground at around 1400 kilometers per second.WEB, Peligroso lugar para jugar tenis,weblink Datos Freak, 3 June 2016, es, 11 June 2016,weblink" title="web.archive.org/web/20160611022635weblink">weblink live, 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.BOOK, Marcia Bartusiak, Black Hole: How an Idea Abandoned by Newtonians, Hated by Einstein, and Gambled on by Hawking Became Loved,weblink registration, 2015, Yale University Press, 978-0-300-21363-8, 130, Neutron star relativistic equations of state describe the relation of radius vs. mass for various models.Neutron Star Masses and Radii {{Webarchive|url=https://web.archive.org/web/20111217102314weblink |date=2011-12-17 }}, p. 9/20, bottom The most likely radii for a given neutron star mass are bracketed by models AP4 (smallest radius) and MS2 (largest radius). EB is the ratio of gravitational binding energy mass equivalent to the observed neutron star gravitational mass of M kilograms with radius R meters,JOURNAL, astro-ph/0002232, Hessels, Jason W. T, Neutron Star Structure and the Equation of State, The Astrophysical Journal, 550, 426, 426–442, Ransom, Scott M, Stairs, Ingrid H, Freire, Paulo C. C, Kaspi, Victoria M, Camilo, Fernando, 2001, 10.1086/319702, 2001ApJ...550..426L, 14782250, E_text{B} = frac{0.60,beta}{1 - frac{beta}{2}}beta = G,M/R,{c}^{2}Given current values

• G = 6.67408times10^{-11}, text{m}^3text{kg}^{-1}text{s}^{-2}CODATA 2014
• c = 2.99792458 times10^{8}, text{m}/text{s}
• M_odot = 1.98855times10^{30}, text{kg}

and star masses "M" commonly reported as multiples of one solar mass,M_x = frac{M}{M_odot}then the relativistic fractional binding energy of a neutron star isE_text{B} = frac{886.0 ,M_x}{R_{left[text{in meters}right]} - 738.3,M_x}A {{Solar mass|2}} neutron star would not be more compact than 10,970 meters radius (AP4 model). Its mass fraction gravitational binding energy would then be 0.187, −18.7% (exothermic). This is not near 0.6/2 = 0.3, −30%.

Structure

Image:Neutron star cross section.svg|thumb|Cross-section of neutron star. Densities are in terms of ρ0 the saturation nuclear matter density, where nucleons begin to touch.]]Current understanding of the structure of neutron stars is defined by existing mathematical models, but it might be possible to infer some details through studies of neutron-star oscillations. Asteroseismology, a study applied to ordinary stars, can reveal the inner structure of neutron stars by analyzing observed spectra of stellar oscillations.Current models indicate that matter at the surface of a neutron star is composed of ordinary atomic nuclei crushed into a solid lattice with a sea of electrons flowing through the gaps between them. It is possible that the nuclei at the surface are iron, due to iron's high binding energy per nucleon.JOURNAL, 10.1070/pu1999v042n11ABEH000665, 1173–1174, Radio pulsars, 1999, Beskin, Vasilii S., Physics-Uspekhi, 42, 11, 1999PhyU...42.1071B, 250831196, It is also possible that heavy elements, such as iron, simply sink beneath the surface, leaving only light nuclei like helium and hydrogen. If the surface temperature exceeds {{val|e=6|u=kelvins}} (as in the case of a young pulsar), the surface should be fluid instead of the solid phase that might exist in cooler neutron stars (temperature