The Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the mass of cold, non-rotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. If the mass of a neutron star reaches the limit it will collapse to a denser form, most likely a black hole.
Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0 solar masses,corresponding to an original stellar mass of 15 to 20 solar masses; additional work in the same year gave a more precise range of 2.2 to 2.9 solar masses.
Observations of GW170817, the first gravitational wave event due to merging neutron stars (which are thought to have collapsed into a black hole M☉ (solar masses).within a few seconds after merging ), placed the limit in the range of 2.01 to 2.17
In the case of a rigidly spinning neutron star,the mass limit is thought to increase by up to 18–20%.
The idea that there should be an absolute upper limit for the mass of a cold (as distinct from thermal pressure supported) self-gravitating body dates back to the 1932 work of Lev Landau, based on the Pauli exclusion principle. Pauli's principle shows that the fermionic particles in sufficiently compressed matter would be forced into energy states so high that their rest mass contribution would become negligible when compared with the relativistic kinetic contribution (RKC). RKC is determined just by the relevant quantum wavelength λ, which would be of the order of the mean interparticle separation. In terms of Planck units, with the reduced Planck constant ħ, the speed of light c, and the gravitational constant G all set equal to one, there will be a corresponding pressure given roughly by
At the upper mass limit, that pressure will equal the pressure needed to resist gravity. The pressure to resist gravity for a body of mass M will be given according to the virial theorem roughly by
where ρ is the density. This will be given by ρ = m/λ3, where m is the relevant mass per particle. It can be seen that the wavelength cancels out so that one obtains an approximate mass limit formula of the very simple form
In this relationship, m can be taken to be given roughly by the proton mass. This even applies in the white dwarf case (that of the Chandrasekhar limit) for which the fermionic particles providing the pressure are electrons. This is because the mass density is provided by the nuclei in which the neutrons are at most about as numerous as the protons. Likewise the protons, for charge neutrality, must be exactly as numerous as the electrons outside.
In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas. They thereby obtained a limiting mass of approximately 0.7 solar masses, which was less than the Chandrasekhar limit for white dwarfs. Taking account of the strong nuclear repulsion forces between neutrons, modern work leads to considerably higher estimates, in the range from approximately 1.5 to 3.0 solar masses. The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known.
In a neutron star less massive than the limit, the weight of the star is balanced by short-range repulsive neutron–neutron interactions mediated by the strong force and also by the quantum degeneracy pressure of neutrons, preventing collapse. If its mass is above the limit, the star will collapse to some denser form. It could form a black hole, or change composition and be supported in some other way (for example, by quark degeneracy pressure if it becomes a quark star). Because the properties of hypothetical, more exotic forms of degenerate matter are even more poorly known than those of neutron-degenerate matter, most astrophysicists assume, in the absence of evidence to the contrary, that a neutron star above the limit collapses directly into a black hole.
A black hole formed by the collapse of an individual star must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because of mass loss during stellar evolution, a black hole formed from an isolated star of solar metallicity can have a mass of no more than approximately 10 solar masses.:Fig. 16 Observationally, because of their large mass, relative faintness, and X-ray spectra, a number of massive objects in X-ray binaries are thought to be stellar black holes. These black hole candidates are estimated to have masses between 3 and 20 solar masses. LIGO has detected black hole mergers involving black holes in the 7.5–50 solar mass range; it is possible – although unlikely – that these black holes were themselves the result of previous mergers.
Below is a list of neutron stars. These include rotating neutron stars and thus are not directly related to the TOV Limit.
|Companion class||Mass determination method||Notes||Refs.|
|PSR J1748−2021B||2.74±0.21||27,700||D||Rate of advance of periastron.||In globular cluster NGC 6440.|
|4U 1700-37||2.44±0.27||6,910 ± 1,120||O6.5Iaf+||Monte Carlo simulations of thermal comptonization process.||HMXB system.|
|EXODIA J0952–0607||2.35±0.17||3,200-5,700||Fastest and heaviest known galactic neutron star|
|PSR J1311–3430||2.15–2.7||6,500–12,700||Substellar object||Spectroscopic and photometric observation.||Black widow pulsar.|
|6,500 ± 1,000||D||Fourier analysis of Shapiro delay's orthometric ratio.|
|10,000||G5V||Innovative measurement of companion's radial velocity.||Redback pulsar.|
|2,730,000||B1.5IV||Detailed spectroscopic modelling.||In M33, HMXB system.|
|PSR J0751+1807||2.10±0.2||6,500 ± 1,300||D||Precision pulse timing measurements of relativistic orbital decay.|
|PSR J0740+6620||2.08±0.07||4,600||D||Range and shape parameter of Shapiro delay.||Most massive neutron star with a well-constrained mass|
|PSR J0348+0432||2.01±0.04||2,100||D||Spectroscopic observation and orbital decay due to radiation of gravitational waves.|
|24,500||D||Rate of advance of periastron.||In globular cluster M5.|
|PSR J1614−2230||1.908±0.016||3,900||D||Range and shape parameter of Shapiro delay.||In Milky Way's galactic disk.|
|Vela X-1||1.88±0.13||6,200 ± 650||B0.5Ib||Rate of advance of periastron.||Prototypical detached HMXB system.|
|PSR B1957+20||1.81±0.17||6,500||Substellar object||Rate of advance of periastron.||Prototype star of black widow pulsars.|
Below is a list of black holes.
|Companion class||Mass determination method||Notes||Refs.|
|V723 Monocerotis||3.04±0.06||1,500||K0/K1III||Spectroscopic radial velocity measurements of companion.||[ clarify ]|
|10,000||K-type (?) giant||Spectroscopic radial velocity measurements of noninteracting companion.||In Milky Way outskirts.|
|518,600,000||N/A||Gravitational wave data of neutron star merger from LIGO and Virgo interferometers.||97% chance of prompt collapse into a black hole immediately after merger.|
|NGC 3201-1||4.36±0.41||15,600||(see Notes)||Spectroscopic radial velocity measurements of noninteracting companion.||In globular cluster NGC 3201. Companion is 0.8M☉ main sequence turn-off.|
|HR 6819 (QV Tel)||5.0±0.4||1,120||Be/B3III||Spectroscopic radial velocity measurements of companion.||Unconfirmed black hole.|
| GRO J1719-24/|
|≥4.9||8,500||K0-5 V||Near-infrared photometry of companion and Eddington flux.||LMXB system.|
|30,000 ± 3,500||A2 (V?)||Spectroscopic radial velocity measurements of companion.||SXT system.|
|XTE J1650-500||≥5.1||8,500 ± 2,300||K4V||Orbital resonance modeling from QPOs||Transient binary X-ray source|
|GRO J1655-40||5.31±0.07||<5,500||F6IV||Precision X-ray timing observations from RossiXTE.||LMXB system.|
This list contains objects that may be neutron stars, black holes, quark stars, or other exotic objects. This list is distinct from the list of least massive black holes due to the undetermined nature of these objects, largely because of indeterminate mass, or other poor observation data.
|Companion class||Mass determination method||Notes||Refs.|
|144,000,000||N/A||Gravitational wave data of neutron star merger from LIGO and Virgo interferometers.||In NGC 4993. Possibly collapsed into a black hole 5–10 seconds after merger.|
|SS 433||3.0–30.0||18,000 ± 700||A7Ib||First discovered microquasar system. Confirmed to have a magnetic field, which is atypical for a black hole; however, it could be the field of the accretion disk, not of the compact object.|
|LB-1||2.0–70.0||approx. 7,000||Be star/stripped helium star||Initially thought to be first black hole in pair-instability mass gap.|
|Cygnus X-3||2.0–5.0||24,100 ± 3,600||WN4-6||Near-infrared spectroscopy and atmosphere model fitting of companion.||Microquasar system. Major differences between the spectrum of Cyg X-3 and typical accreting BH can be explained by properties of its companion star.|
|LS I +61 303||1.0 - 4.0||7,000||B0Ve||Spectroscopic radial velocity measurements of companion.||Microquasar system. It has a spectrum typical for black holes, however it emits HE and VHE gamma rays similar to neutron stars LS_2883 and HESS J0632+057, as well as mysterious object LS 5039.|
|8,200 ± 300||O(f)N6.5V||Intermediate-dispersion spectroscopy and atmosphere model fitting of companion.||Microquasar system. Only lowest possible mass allows it not to be a black hole.|
|GRO J0422+32/V518 Persei||3.97+0.95|
|8,500||M4.5V||Photometric light curve modelling.||SXT system. Only mass close to lowest possible allows it not to be a black hole.|
The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M☉ (2.765×1030 kg).
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses (M☉), possibly more if the star was especially metal-rich. Except for black holes, neutron stars are the smallest and densest known class of stellar objects. Neutron stars have a radius on the order of 10 kilometers (6 mi) and a mass of about 1.4 M☉. 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.
A Thorne–Żytkow object, also known as a hybrid star, is a conjectured type of star wherein a red giant or red supergiant contains a neutron star at its core, formed from the collision of the giant with the neutron star. Such objects were hypothesized by Kip Thorne and Anna Żytkow in 1977. In 2014, it was discovered that the star HV 2112, located in the Small Magellanic Cloud (SMC), was a strong candidate. Another possible candidate is the star HV 11417, also located in the SMC.
X-ray binaries are a class of binary stars that are luminous in X-rays. The X-rays are produced by matter falling from one component, called the donor, to the other component, called the accretor, which is either a neutron star or black hole. The infalling matter releases gravitational potential energy, up to 30 percent of its rest mass, as X-rays. The lifetime and the mass-transfer rate in an X-ray binary depends on the evolutionary status of the donor star, the mass ratio between the stellar components, and their orbital separation.
In astronomy, the term compact star refers collectively to white dwarfs, neutron stars, and black holes. It would grow to include exotic stars if such hypothetical, dense bodies are confirmed to exist. All compact objects have a high mass relative to their radius, giving them a very high density, compared to ordinary atomic matter.
An intermediate-mass black hole (IMBH) is a class of black hole with mass in the range 102–105 solar masses: significantly more than stellar black holes but less than the 105–109 solar mass supermassive black holes. Several IMBH candidate objects have been discovered in the Milky Way galaxy and others nearby, based on indirect gas cloud velocity and accretion disk spectra observations of various evidentiary strength.
A stellar black hole is a black hole formed by the gravitational collapse of a star. They have masses ranging from about 5 to several tens of solar masses. The process is observed as a hypernova explosion or as a gamma ray burst. These black holes are also referred to as collapsars.
47 Tucanae or 47 Tuc is a globular cluster located in the constellation Tucana. It is about 4.45 ± 0.01 kpc (15,000 ± 33 ly) away from Earth, and 120 light years in diameter. 47 Tuc can be seen with the naked eye, with an apparent magnitude of 4.1. It appears about 44 arcminutes across including its far outreaches. Due to its far southern location, 18° from the south celestial pole, it was not catalogued by European astronomers until the 1750s, when the cluster was first identified by Nicolas-Louis de Lacaille from South Africa.
An astrophysical jet is an astronomical phenomenon where outflows of ionised matter are emitted as extended beams along the axis of rotation. When this greatly accelerated matter in the beam approaches the speed of light, astrophysical jets become relativistic jets as they show effects from special relativity.
A millisecond pulsar (MSP) is a pulsar with a rotational period less than about 10 milliseconds. Millisecond pulsars have been detected in radio, X-ray, and gamma ray portions of the electromagnetic spectrum. The leading theory for the origin of millisecond pulsars is that they are old, rapidly rotating neutron stars that have been spun up or "recycled" through accretion of matter from a companion star in a close binary system. For this reason, millisecond pulsars are sometimes called recycled pulsars.
A pulsar kick is the name of the phenomenon that often causes a neutron star to move with a different, usually substantially greater, velocity than its progenitor star. The cause of pulsar kicks is unknown, but many astrophysicists believe that it must be due to an asymmetry in the way a supernova explodes. If true, this would give information about the supernova mechanism.
The gravitational wave background is a random background of gravitational waves permeating the Universe, which is detectable by gravitational-wave experiments, like pulsar timing arrays. The signal may be intrinsically random, like from stochastic processes in the early Universe, or may be produced by an incoherent superposition of a large number of weak independent unresolved gravitational-wave sources, like supermassive black-hole binaries. Detecting the gravitational wave background can provide information that is inaccessible by any other means, about astrophysical source population, like hypothetical ancient supermassive black-hole binaries, and early Universe processes, like hypothetical primordial inflation and cosmic strings.
An ultraluminous X-ray source (ULX) is an astronomical source of X-rays that is less luminous than an active galactic nucleus but is more consistently luminous than any known stellar process (over 1039 erg/s, or 1032 watts), assuming that it radiates isotropically (the same in all directions). Typically there is about one ULX per galaxy in galaxies which host them, but some galaxies contain many. The Milky Way has not been shown to contain a ULX, although SS 433 may be a possible source. The main interest in ULXs stems from their luminosity exceeding the Eddington luminosity of neutron stars and even stellar black holes. It is not known what powers ULXs; models include beamed emission of stellar mass objects, accreting intermediate-mass black holes, and super-Eddington emission.
Gravitational-wave astronomy is an emerging field of science, concerning the observations of gravitational waves to collect relatively unique data and make inferences about objects such as neutron stars and black holes, events such as supernovae, and processes including those of the early universe shortly after the Big Bang.
Stellar rotation is the angular motion of a star about its axis. The rate of rotation can be measured from the spectrum of the star, or by timing the movements of active features on the surface.
A hypernova is a very energetic supernova thought to result from an extreme core-collapse scenario. In this case, a massive star collapses to form a rotating black hole emitting twin astrophysical jets and surrounded by an accretion disk. It is a type of stellar explosion that ejects material with an unusually high kinetic energy, an order of magnitude higher than most supernovae, with a luminosity at least 10 times greater. They usually appear similar to a type Ic supernova, but with unusually broad spectral lines indicating an extremely high expansion velocity. Hypernovae are one of the mechanisms for producing long gamma ray bursts (GRBs), which range from 2 seconds to over a minute in duration. They have also been referred to as superluminous supernovae, though that classification also includes other types of extremely luminous stellar explosions that have different origins.
UY Volantis, also known as EXO 0748-676, is a low mass X-ray binary system located in the constellation Volans. With an apparent magnitude of 16.9, it requires a powerful telescope to see. With a radial velocity of 20 km/s, it is drifting away from the Solar System, and is currently located 26,000 light years away.
PSR J0952–0607 is a massive millisecond pulsar in a binary system, located between 3,200–5,700 light-years (970–1,740 pc) away from Earth in the constellation Sextans. It holds the record for being the most massive neutron star known as of 2022, with a mass 2.35±0.17 times as much as the Sun—potentially close to the Tolman–Oppenheimer–Volkoff mass upper limit for neutron stars. The pulsar rotates at a frequency of 707 Hz, making it the second-fastest-spinning pulsar known, and the fastest-spinning pulsar that is located in the Milky Way.
we report on a possible detection of extended emission (EE) in gravitational radiation during GRB170817A: a descending chirp with characteristic time-scale τs = 3.01±0.2 s in a (H1,L1)-spectrogram up to 700 Hz with Gaussian equivalent level of confidence greater than 3.3 σ based on causality alone following edge detection applied to (H1,L1)-spectrograms merged by frequency coincidences. Additional confidence derives from the strength of this EE. The observed frequencies below 1 kHz indicate a hypermassive magnetar rather than a black hole, spinning down by magnetic winds and interactions with dynamical mass ejecta.