Tolman–Oppenheimer–Volkoff limit

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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, [1] 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. [2]

Observations of GW170817, the first gravitational wave event due to merging neutron stars (which are thought to have collapsed into a black hole [3] within a few seconds after merging [4] ), placed the limit in the range of 2.01 to 2.17  M (solar masses). [5]

In the case of a rigidly spinning neutron star, [n 1] the mass limit is thought to increase by up to 18–20%. [4] [5]


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, [6] [7] 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. [1] 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. [8] :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. [9] [10] 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.

List of the most massive neutron stars

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 methodNotesRefs.
PSR J1748−2021B2.74±0.2127,700 D Rate of advance of periastron.In globular cluster NGC 6440. [11]
4U 1700-37 2.44±0.276,910 ± 1,120 O6.5Iaf+ Monte Carlo simulations of thermal comptonization process. HMXB system. [12] [13]
EXODIA J0952–0607 2.35±0.173,200-5,700Fastest and heaviest known galactic neutron star [14]
PSR J1311–3430 2.15–2.76,500–12,700 Substellar object Spectroscopic and photometric observation.Black widow pulsar. [15] [16]
PSR J1600−30532.3+0.7
6,500 ± 1,000 D Fourier analysis of Shapiro delay's orthometric ratio. [17] [18]
PSR J2215+51352.27+0.17
10,000 G5V Innovative measurement of companion's radial velocity.Redback pulsar. [19]
XMMU J013236.7+3032282.2+0.8
2,730,000 B1.5IV Detailed spectroscopic modelling.In M33, HMXB system. [20]
PSR J0751+18072.10±0.26,500 ± 1,300 D Precision pulse timing measurements of relativistic orbital decay. [21]
PSR J0740+6620 2.08±0.074,600 D Range and shape parameter of Shapiro delay.Most massive neutron star with a well-constrained mass [22] [23] [24]
PSR J0348+0432 2.01±0.042,100 D Spectroscopic observation and orbital decay due to radiation of gravitational waves. [17] [25]
PSR B1516+02B1.94+0.17
24,500 D Rate of advance of periastron.In globular cluster M5. [17] [26]
PSR J1614−2230 1.908±0.0163,900 D Range and shape parameter of Shapiro delay.In Milky Way's galactic disk. [17] [18] [27]
Vela X-1 1.88±0.136,200 ± 650 B0.5Ib Rate of advance of periastron.Prototypical detached HMXB system. [28]
PSR B1957+20 1.81±0.176,500 Substellar object Rate of advance of periastron.Prototype star of black widow pulsars. [29] [30] [31]

List of least massive black holes

Below is a list of black holes.

Companion class Mass determination methodNotesRefs.
V723 Monocerotis 3.04±0.061,500 K0/K1III Spectroscopic radial velocity measurements of companion.Mass may be underestimated due not accurate measurement distance to the companion star.[ clarify ] [32]
2MASS J05215658+43592203.3+2.8
10,000 K-type (?) giant Spectroscopic radial velocity measurements of noninteracting companion.In Milky Way outskirts. [17] [33] [34]
GW190425's remnant3.4+0.3
518,600,000N/AGravitational wave data of neutron star merger from LIGO and Virgo interferometers.97% chance of prompt collapse into a black hole immediately after merger. [17] [35] [36]
NGC 3201-14.36±0.4115,600(see Notes)Spectroscopic radial velocity measurements of noninteracting companion.In globular cluster NGC 3201. Companion is 0.8M main sequence turn-off. [17] [37]
HR 6819 (QV Tel)5.0±0.41,120 Be/B3III Spectroscopic radial velocity measurements of companion.Unconfirmed black hole. [38]
GRO J1719-24/
GRS 1716−249
≥4.98,500 K0-5 V Near-infrared photometry of companion and Eddington flux. LMXB system. [17] [39]
4U 1543-475.0+2.5
30,000 ± 3,500 A2 (V?) Spectroscopic radial velocity measurements of companion.SXT system. [17] [40]
XTE J1650-500 ≥5.18,500 ± 2,300 K4V Orbital resonance modeling from QPOs Transient binary X-ray source [41]
GRO J1655-40 5.31±0.07<5,500 F6IV Precision X-ray timing observations from RossiXTE.LMXB system. [42] [43]
GX 339-4 5.9±3.626,000N/A [17]

List of objects in mass gap

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 methodNotesRefs.
GW170817's remnant2.74+0.04
144,000,000N/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. [44]
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. [45] [46] [47]
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. [48] [49]
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. [50] [51]
LS I +61 303 1.0 - 4.07,000 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. [52] [53]
LS 5039 3.7+1.3
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. [54]
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. [17] [55]

See also


  1. Meaning that different levels in the interior of the star all rotate at the same rate.

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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).

<span class="mw-page-title-main">Neutron star</span> Collapsed core of a massive star

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.

<span class="mw-page-title-main">X-ray binary</span> Class of binary stars

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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.

<span class="mw-page-title-main">Intermediate-mass black hole</span> Class of black holes with a mass range of 100 to 100000 solar masses

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.

<span class="mw-page-title-main">Stellar black hole</span> Black hole formed by a collapsed star

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.

<span class="mw-page-title-main">47 Tucanae</span> Globular cluster in the constellation Tucana

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.

<span class="mw-page-title-main">Astrophysical jet</span> Beam of ionized matter flowing along the axis of a rotating astronomical object

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<span class="mw-page-title-main">Millisecond pulsar</span> Pulsar with a rotational period less than about 10 milliseconds

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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.

<span class="mw-page-title-main">Ultraluminous X-ray source</span>

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.

<span class="mw-page-title-main">Gravitational-wave astronomy</span> Branch of astronomy using gravitational waves

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<span class="mw-page-title-main">Stellar rotation</span> Angular motion of a star about its axis

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<span class="mw-page-title-main">Hypernova</span> Supernova that ejects a large mass at unusually high velocity

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<span class="mw-page-title-main">UY Volantis</span> Low mass X-ray binary in the constellation Volans

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<span class="mw-page-title-main">PSR J0952–0607</span> Massive millisecond pulsar in the Milky Way

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.


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