Editing Neutron star (section)

==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 [[Newton's law of universal gravitation|Newtonian gravity]] is no longer sufficient in those conditions. Effects such as [[Quantum chromodynamics|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.<ref name=":1">{{Cite journal |last1=Hebeler |first1=K. |last2=Lattimer |first2=J. M. |last3=Pethick |first3=C. J. |last4=Schwenk |first4=A. |date=2013-07-19 |title=Equation of State and Neutron Star Properties Constrained by Nuclear Physics and Observation |url=https://iopscience.iop.org/article/10.1088/0004-637X/773/1/11 |journal=The Astrophysical Journal |volume=773 |issue=1 |pages=11 |doi=10.1088/0004-637X/773/1/11 |arxiv=1303.4662 |bibcode=2013ApJ...773...11H |issn=0004-637X}}</ref>

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 interaction|strong force]] of the [[Standard Model|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 number]]s. 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===
[[File:White dwarf vs neutron star.svg|thumb|right|Comparison of a 10 km radius neutron star (top left corner) and a 6000 km radius [[white dwarf]], the latter roughly the size of [[Earth]].]]
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),<ref group="lower-alpha">{{val|3.7|e=17|u=kg/m3}} derives from mass {{val|2.68|e=30|u=kg}} / volume of star of radius 12&nbsp;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&nbsp;km</ref> which is comparable to the approximate density of an atomic nucleus of {{val|3|e=17|u=kg/m3}}.<ref>{{cite web |url=http://heasarc.gsfc.nasa.gov/docs/xte/learning_center/ASM/ns.html |title=Calculating a Neutron Star's Density |access-date=2006-03-11 |archive-date=2006-02-24 |archive-url=https://web.archive.org/web/20060224011955/http://heasarc.gsfc.nasa.gov/docs/xte/learning_center/ASM/ns.html |url-status=live }} NB {{val|3|e=17|u=kg/m3}} is {{val|3|e=14|u=g/cm3}}</ref> The density increases with depth, varying from about {{val|1|e=9|u=kg/m3}} at the crust to an estimated {{val|6|e=17}} or {{val|8|e=17|u=kg/m3}} deeper inside.<ref name="Miller">{{cite journal |title=Introduction to neutron stars |journal=American Institute of Physics Conference Series |volume=1645 |issue=1 |pages=61–78 |bibcode=2015AIPC.1645...61L |last1=Lattimer |first1=James M. |year=2015 |doi=10.1063/1.4909560 |series=AIP Conference Proceedings |doi-access=free }}</ref> Pressure increases accordingly, from about {{val|3.2|u=Pa|e=31}} (32&nbsp;[[quetta-|Q]]Pa) at the inner crust to {{val|1.6|e=34|u=Pa}} in the center.<ref>{{Cite journal |last1=Ozel |first1=Feryal |last2=Freire |first2=Paulo |title=Masses, Radii, and the Equation of State of Neutron Stars |journal=Annu. Rev. Astron. Astrophys. |volume=54 |issue=1 |pages=401–440 |date=2016 |doi=10.1146/annurev-astro-081915-023322 |bibcode=2016ARA&A..54..401O |arxiv = 1603.02698 |s2cid=119226325 }}</ref>

A neutron star is so dense that one teaspoon (5 [[milliliter]]s) 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]].<ref group="lower-alpha">The average density of material in a neutron star of radius 10&nbsp;km is {{val|1.1|e=12|u=kg/cm3}}. Therefore, 5&nbsp;ml of such material is {{val|5.5|e=12|u=kg}}, or 5,500,000,000 [[metric ton]]s. This is about 15 times the total mass of the human world population. Alternatively, 5&nbsp;ml from a neutron star of radius 20&nbsp;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 1''g''.</ref> The entire mass of the Earth at neutron star density would fit into a sphere 305&nbsp;m in diameter, about the size of the [[Arecibo Telescope]].

In popular scientific writing, neutron stars are sometimes described as macroscopic [[atomic nucleus|atomic nuclei]]. Indeed, both states are composed of [[nucleon]]s, and they share a similar density to within an order of magnitude. However, in other respects, neutron stars and atomic nuclei are quite different. A nucleus is held together by the [[strong interaction]], whereas a neutron star is held together by [[gravity]]. The density of a nucleus is uniform, while neutron stars are [[#Structure|predicted to consist of multiple layers]] with varying compositions and densities.<ref>{{Cite journal |last1=Baym |first1=G |last2=Pethick |first2=C |date=December 1975 |title=Neutron Stars |journal=Annual Review of Nuclear Science |language=en |volume=25 |issue=1 |pages=27–77 |doi=10.1146/annurev.ns.25.120175.000331 |issn=0066-4243 |bibcode=1975ARNPS..25...27B |doi-access=free }}</ref>

=== 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 [[LIGO]],<ref>{{Cite web |title=LIGO Lab {{!}} Caltech {{!}} MIT |url=https://www.ligo.caltech.edu/ |access-date=2024-05-10 |website=LIGO Lab {{!}} Caltech}}</ref> which is a gravitational wave observatory, and [[Neutron Star Interior Composition Explorer|NICER]],<ref>{{Cite web |title=NICER - NASA Science |url=https://science.nasa.gov/mission/nicer/ |access-date=2024-05-10 |website=science.nasa.gov |language=en-US}}</ref> which is an X-ray telescope.

NICER's observations of [[pulsar]]s 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.<ref>{{Cite journal |last1=Raaijmakers |first1=G. |last2=Greif |first2=S. K. |last3=Hebeler |first3=K. |last4=Hinderer |first4=T. |last5=Nissanke |first5=S. |last6=Schwenk |first6=A. |last7=Riley |first7=T. E. |last8=Watts |first8=A. L. |last9=Lattimer |first9=J. M. |last10=Ho |first10=W. C. G. |date=2021-09-01 |title=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 |journal=The Astrophysical Journal Letters |volume=918 |issue=2 |pages=L29 |doi=10.3847/2041-8213/ac089a |doi-access=free |arxiv=2105.06981 |bibcode=2021ApJ...918L..29R |issn=2041-8205}}</ref> These mass-radius constraints, combined with [[Chiral perturbation theory|chiral effective field theory]] calculations, tightens constraints on the neutron star equation of state.<ref name=":1" />

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 [[Neutron star merger|binary neutron star mergers]]. From these simulations, researchers can extract [[gravitational wave]]forms, 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.<ref>{{Cite journal |last1=Takami |first1=Kentaro |last2=Rezzolla |first2=Luciano |last3=Baiotti |first3=Luca |date=2014-08-28 |title=Constraining the Equation of State of Neutron Stars from Binary Mergers |url=https://link.aps.org/doi/10.1103/PhysRevLett.113.091104 |journal=Physical Review Letters |language=en |volume=113 |issue=9 |page=091104 |doi=10.1103/PhysRevLett.113.091104 |pmid=25215972 |arxiv=1403.5672 |bibcode=2014PhRvL.113i1104T |issn=0031-9007}}</ref> 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.<ref>{{Cite journal |last1=Annala |first1=Eemeli |last2=Gorda |first2=Tyler |last3=Kurkela |first3=Aleksi |last4=Vuorinen |first4=Aleksi |date=2018-04-25 |title=Gravitational-Wave Constraints on the Neutron-Star-Matter Equation of State |url=https://link.aps.org/doi/10.1103/PhysRevLett.120.172703 |journal=Physical Review Letters |language=en |volume=120 |issue=17 |page=172703 |doi=10.1103/PhysRevLett.120.172703 |pmid=29756823 |arxiv=1711.02644 |bibcode=2018PhRvL.120q2703A |issn=0031-9007}}</ref> Future gravitational wave signals with next generation detectors like [[Cosmic Explorer (gravitational wave observatory)|Cosmic Explorer]] can impose further constraints.<ref>{{Cite journal |last1=Finstad |first1=Daniel |last2=White |first2=Laurel V. |last3=Brown |first3=Duncan A. |date=2023-09-01 |title=Prospects for a Precise Equation of State Measurement from Advanced LIGO and Cosmic Explorer |journal=The Astrophysical Journal |volume=955 |issue=1 |pages=45 |doi=10.3847/1538-4357/acf12f |doi-access=free |arxiv=2211.01396 |bibcode=2023ApJ...955...45F |issn=0004-637X}}</ref>

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.<ref>{{cite arXiv |last1=Lovato |first1=Alessandro |last2=Dore |first2=Travis |display-authors=1 |title=Long Range Plan: Dense matter theory for heavy-ion collisions and neutron stars |date=2022 |class=nucl-th |eprint=2211.02224}}</ref> There is also recent work on constraining the equation of state with the speed of sound through hydrodynamics.<ref>{{cite journal|last1=Hippert |first1=Mauricio |last2=Noronha |first2=Jorge |last3=Romatschke |first3=Paul |title=Upper Bound on the Speed of Sound in Nuclear Matter from Transport |journal=Physics Letters B |date=2025 |volume=860 |doi=10.1016/j.physletb.2024.139184 |arxiv=2402.14085|bibcode=2025PhLB..86039184H }}</ref>

=== Tolman-Oppenheimer-Volkoff Equation ===
The [[Tolman–Oppenheimer–Volkoff equation|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:<ref>{{cite journal |last1=Silbar |first1=Richard R. |last2=Reddy |first2=Sanjay |title=Neutron stars for undergraduates |journal=American Journal of Physics |date=1 July 2004 |volume=72 |issue=7 |pages=892–905 |doi=10.1119/1.1703544|arxiv=nucl-th/0309041 |bibcode=2004AmJPh..72..892S }}</ref>

<math display="block">\frac{dp}{dr} = - \frac{G\epsilon(r) M(r)}{c^2 r^2} \left(1+\frac{p(r)}{\epsilon(r)}\right) \left(1+\frac{4\pi r^3p(r)}{M(r)c^2}\right) \left(1-\frac{2GM(r)}{c^2r}\right)</math>
<math display="block">\frac{dM}{dr} = \frac{4\pi}{c^2} r^2 \epsilon(r)</math>

where <math>G</math> is the gravitational constant, <math>p(r)</math> is the pressure, <math>\epsilon(r)</math> is the energy density (found from the equation of state), and <math>c</math> 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, one 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|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|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).<ref>{{cite journal |last1=Kumar |first1=N. |last2=Sokolov |first2=V. V. |title=Mass Distribution and "Mass Gap" of Compact Stellar Remnants in Binary Systems |journal=Astrophysical Bulletin |date=June 2022 |volume=77 |issue=2 |pages=197–213 |doi=10.1134/S1990341322020043|arxiv=2204.07632 |bibcode=2022AstBu..77..197K }}</ref> 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 force]]s, 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.<ref>{{cite journal |last1=Yagi |first1=Kent |last2=Yunes |first2=Nicolás |title=I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves, and fundamental physics |journal=Physical Review D |date=19 July 2013 |volume=88 |issue=2 |page=023009 |doi=10.1103/PhysRevD.88.023009|arxiv=1303.1528 |bibcode=2013PhRvD..88b3009Y }}</ref>

===Temperature===

The temperature inside a newly formed neutron star is from around {{val|e=11}} to {{val|e=12|ul=kelvin}}.<ref name="Miller" /> However, the huge number of [[neutrino]]s 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}}.<ref name="Miller" /> 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 [[Stellar classification|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 star]]s.<ref>{{Cite journal |last1=Yakovlev |first1=D. G. |last2=Kaminker |first2=A. D. |last3=Haensel |first3=P. |last4=Gnedin |first4=O. Y. |year=2002 |title=The cooling neutron star in 3C 58 |journal=Astronomy & Astrophysics |volume=389 |pages=L24–L27 |arxiv=astro-ph/0204233 |bibcode=2002A&A...389L..24Y |doi=10.1051/0004-6361:20020699 |s2cid=6247160}}</ref>

===Magnetic field===
The magnetic field strength on the surface of neutron stars ranges from {{circa|{{val|e=4}}}} to {{val|e=11}}&nbsp;[[Tesla (unit)|tesla]] (T).<ref name="reisenegger">{{cite arXiv |first=A. |last=Reisenegger |year=2003 |title=Origin and Evolution of Neutron Star Magnetic Fields |eprint=astro-ph/0307133 }}</ref> These are orders of magnitude higher than in any other object: for comparison, a continuous 16&nbsp;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.<ref name="reisenegger"/>

The neutron stars known as [[magnetar]]s have the strongest magnetic fields, in the range of {{val|e=8}} to {{val|e=11|u=T}},<ref name="mcgill">{{cite web |title=McGill SGR/AXP Online Catalog |url=http://www.physics.mcgill.ca/~pulsar/magnetar/main.html |access-date=2 Jan 2014 |archive-date=23 July 2020 |archive-url=https://web.archive.org/web/20200723080137/http://www.physics.mcgill.ca/~pulsar/magnetar/main.html |url-status=live }}</ref> and have become the widely accepted hypothesis for neutron star types [[soft gamma repeater]]s (SGRs)<ref name="sa">{{cite journal |first1=Chryssa |last1=Kouveliotou |first2=Robert C. |last2=Duncan |first3=Christopher |last3=Thompson |date=February 2003 |title=Magnetars |journal=Scientific American |volume=288 |issue=2 |pages=34–41 |doi=10.1038/scientificamerican0203-34 |pmid=12561456 |bibcode=2003SciAm.288b..34K }}</ref> and [[anomalous X-ray pulsar]]s (AXPs).<ref>{{cite journal |first1=V.M. |last1=Kaspi |first2=F.P. |last2=Gavriil |year=2004 |title=(Anomalous) X-ray pulsars |journal=Nuclear Physics B |series=Proceedings Supplements |volume=132 |pages=456–465 |doi=10.1016/j.nuclphysbps.2004.04.080 |arxiv=astro-ph/0402176 |bibcode=2004NuPhS.132..456K|s2cid=15906305 }}</ref> The magnetic [[energy density]] of a {{val|e=8|u=T}} field is extreme, greatly exceeding the [[theoretical total mass-energy|mass-energy]] density of ordinary matter.{{efn|Magnetic [[energy density]] for a [[magnetic field|field B]] is {{nowrap| U {{=}} {{frac|[[Vacuum permeability|μ<sub>0</sub>]] B<sup>2</sup>|2}} .}}<ref>{{cite web |url=http://scienceworld.wolfram.com/physics/MagneticFieldEnergyDensity.html |title=Eric Weisstein's World of Physics |website=scienceworld.wolfram.com |archive-url=https://web.archive.org/web/20190423232524/http://scienceworld.wolfram.com/physics/MagneticFieldEnergyDensity.html |archive-date=2019-04-23}}</ref> Substituting {{nowrap| B {{=}} {{val|e=8|u=T}} ,}} get {{nowrap|U {{=}} {{val|4|e=21|u=J|up=m3}} .}} Dividing by c<sup>2</sup> 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 [[Vacuum polarization|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 [[Starquake (astrophysics)#Starquake|starquake]]s, 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&nbsp;seconds and which lasts for a few minutes.<ref>{{cite web |url=http://solomon.as.utexas.edu/magnetar.html |title='Magnetars', soft gamma repeaters & very strong magnetic fields |first=Robert C. |last=Duncan |date=March 2003 |access-date=2018-04-17 |archive-date=2020-01-19 |archive-url=https://web.archive.org/web/20200119142438/http://solomon.as.utexas.edu/magnetar.html |url-status=live }}</ref>

The origins of the strong magnetic field are as yet unclear.<ref name="reisenegger"/> One hypothesis is that of "flux freezing", or conservation of the original [[magnetic flux]] during the formation of the neutron star.<ref name="reisenegger"/> 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.<ref name="reisenegger"/>

===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).<ref name="Zahn" /> In [[Geometrized unit system|natural units]], this star's mass is 1 and its radius is 4, or twice its [[Schwarzschild radius]].<ref name="Zahn" />]]

The gravitational field at a neutron star's surface is about {{val|2|e=11}} times [[Standard gravity|stronger than on Earth]], at around {{val|2.0|e=12|u=m/s2}}.<ref>{{cite book |title=An Introduction to the Sun and Stars |edition=illustrated |first1=Simon F. |last1=Green |first2=Mark H. |last2=Jones |first3=S. Jocelyn |last3=Burnell |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-54622-5 |page=322 |url=https://books.google.com/books?id=lb5owLGIQGsC&pg=PA322 |access-date=2016-06-09 |archive-date=2017-01-31 |archive-url=https://web.archive.org/web/20170131005503/https://books.google.com/books?id=lb5owLGIQGsC&pg=PA322 |url-status=live }}</ref> 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.<ref name="Zahn">{{cite web |first=Corvin |last=Zahn |title=Tempolimit Lichtgeschwindigkeit |date=1990-10-09 |url=http://www.tempolimit-lichtgeschwindigkeit.de/galerie/galerie.html |language=de |quote=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) |access-date=2009-10-09 |archive-date=2021-01-26 |archive-url=https://web.archive.org/web/20210126171353/https://www.tempolimit-lichtgeschwindigkeit.de/galerie/galerie.html |url-status=live }}</ref>
If the radius of the neutron star is 3''GM''/''c''<sup>2</sup> or less, then the photons may be [[photon sphere|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'' = ''mc''<sup>2</sup>}}). 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 1,400 kilometers per second.<ref>{{cite web |title=Peligroso lugar para jugar tenis |url=http://www.datosfreak.org/datos/slug/Aceleracion-de-superficie-estrella-de-neutrones |website=Datos Freak |access-date=3 June 2016 |language=es |archive-date=11 June 2016 |archive-url=https://web.archive.org/web/20160611022635/http://www.datosfreak.org/datos/slug/Aceleracion-de-superficie-estrella-de-neutrones |url-status=live }}</ref> 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.<ref>{{cite book|author=Marcia Bartusiak | title=Black Hole: How an Idea Abandoned by Newtonians, Hated by Einstein, and Gambled on by Hawking Became Loved| url=https://archive.org/details/blackholehowidea0000bart |url-access=registration |year=2015 | publisher=Yale University Press | isbn=978-0-300-21363-8 |page=[https://archive.org/details/blackholehowidea0000bart/page/130 130]}}</ref>

Neutron star relativistic equations of state describe the relation of radius vs. mass for various models.<ref>[http://www.ns-grb.com/PPT/Lattimer.pdf Neutron Star Masses and Radii] {{Webarchive|url=https://web.archive.org/web/20111217102314/http://www.ns-grb.com/PPT/Lattimer.pdf |date=2011-12-17 }}, p. 9/20, bottom</ref> The most likely radii for a given neutron star mass are bracketed by models AP4 (smallest radius) and MS2 (largest radius). ''E''<sub>B</sub> is the ratio of gravitational binding energy mass equivalent to the observed neutron star gravitational mass of ''M'' kilograms with radius ''R'' meters,<ref>{{Cite journal |arxiv = astro-ph/0002232|last1 = Hessels|first1 = Jason W. T|title = Neutron Star Structure and the Equation of State | journal = The Astrophysical Journal | volume = 550 | issue = 426|pages = 426–442|last2 = Ransom|first2 = Scott M|last3 = Stairs|first3 = Ingrid H|last4 = Freire | first4 = Paulo C. C | last5 = Kaspi|first5 = Victoria M|last6 = Camilo|first6 = Fernando|year = 2001|doi = 10.1086/319702|bibcode = 2001ApJ...550..426L|s2cid = 14782250}}</ref>
<math display="block">E_\text{B} = \frac{0.60\,\beta}{1 - \frac{\beta}{2}}</math><math display="block">\beta \ = G\,M/R\,{c}^{2}</math>
Given current values
*<math>G = 6.67408\times10^{-11}\, \text{m}^3\text{kg}^{-1}\text{s}^{-2}</math><ref name="CODATA 2014">CODATA 2014</ref>
*<math>c = 2.99792458 \times10^{8}\, \text{m}/\text{s}</math><ref name="CODATA 2014" />
*<math>M_\odot = 1.98855\times10^{30}\, \text{kg}</math>
and star masses "M" commonly reported as multiples of one solar mass,
<math display="block">M_x = \frac{M}{M_\odot}</math>
then the relativistic fractional binding energy of a neutron star is
<math display="block">E_\text{B} = \frac{886.0 \,M_x}{R_{\left[\text{in meters}\right]} - 738.3\,M_x}</math>

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%.