View source for Tolman–Oppenheimer–Volkoff limit

{{Short description|Upper bound to the mass of cold, nonrotating neutron stars}}

The '''Tolman–Oppenheimer–Volkoff limit''' (or '''TOV limit''') is an upper bound to the [[Star#Mass|mass]] of cold, non-rotating [[neutron star]]s, analogous to the [[Chandrasekhar limit]] for [[white dwarf]] stars. Stars more massive than the TOV limit collapse into a [[black hole]]. The original calculation in 1939, which neglected complications such as nuclear forces between neutrons, placed this limit at approximately 0.7 [[solar mass]]es ({{Solar mass|link=y}}). Later, more refined analyses have resulted in larger values.

Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0 {{Solar mass}},<ref name=Bombaci>{{cite journal |first=I. |last=Bombaci |date=1996 |title=The Maximum Mass of a Neutron Star |journal=[[Astronomy and Astrophysics]] |volume=305 | pages=871–877 |bibcode=1996A&A...305..871B}}</ref> corresponding to an original stellar mass of 15 to 20 {{Solar mass}}; additional work in the same year gave a more precise range of 2.2 to 2.9 {{Solar mass}}.<ref>{{cite journal |title=The Maximum Mass of a Neutron Star
|date=11 August 1996|arxiv=astro-ph/9608059v1|last1=Kalogera|first1=V|last2=Baym|first2=G|journal=The Astrophysical Journal|volume=470|pages=L61–L64|doi=10.1086/310296|bibcode=1996ApJ...470L..61K|s2cid=119085893}}</ref>

Data from [[GW170817]], the first gravitational wave observation attributed to [[Neutron star merger|merging neutron stars]] (thought to have collapsed into a black hole<ref name="Pooley2018">{{cite journal|last1= Pooley|first1= D.|last2= Kumar|first2= P.|last3= Wheeler|first3=J. C.|last4= Grossan|first4= B.|title= GW170817 Most Likely Made a Black Hole|journal= The Astrophysical Journal|volume= 859|issue= 2|date= 2018-05-31|pages= L23|doi= 10.3847/2041-8213/aac3d6|arxiv= 1712.03240|bibcode= 2018ApJ...859L..23P|s2cid= 53379493|doi-access= free}}</ref> within a few seconds after merging<ref name = "Cho2018">{{cite journal | last= Cho | first= A. | title= A weight limit emerges for neutron stars | journal= Science | volume= 359 | issue= 6377 | pages= 724–725 | date= 16 February 2018 | doi= 10.1126/science.359.6377.724 | pmid= 29449468 | bibcode= 2018Sci...359..724C }}</ref>) placed the limit in the range of 2.01 to 2.17 {{Solar mass}}.<ref name="Rezzolla2018">{{cite journal|last1= Rezzolla|first1= L.|last2= Most|first2=E. R.|last3= Weih|first3=L. R.|title= Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars|journal= Astrophysical Journal|volume= 852|issue= 2|date= 2018-01-09|pages= L25|doi= 10.3847/2041-8213/aaa401|arxiv= 1711.00314|bibcode= 2018ApJ...852L..25R|s2cid= 119359694|doi-access= free}}</ref>

In the case of a rigidly spinning neutron star, meaning that different levels in the interior of the star all rotate at the same rate, the mass limit is thought to increase by up to 18–20%.<ref name = "Cho2018" /><ref name="Rezzolla2018" />

== History ==
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 [[fermion]]ic 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 [[matter wave|quantum wavelength]] {{mvar|λ}}, which would be of the order of the mean interparticle separation. In terms of [[Planck units]], with the [[reduced Planck constant]] {{mvar|ħ}}, the [[speed of light]] {{mvar|c}}, and the [[gravitational constant]] {{mvar|G}} all set equal to one, there will be a corresponding [[pressure]] given roughly by
{{block indent|<math>P=\frac{1}{\lambda^4}</math>.}}
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 {{mvar|M}} will be given according to the [[virial theorem]] roughly by
{{block indent|<math>P^3=M^2\rho^4</math>,}}
where {{mvar|ρ}} is the density. This will be given by {{math|''ρ'' {{=}} {{sfrac|''m''|''λ''<sup>3</sup>}}}}, where {{mvar|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
{{block indent|<math>M=\frac{1}{m^2}</math>.}}
In this relationship, {{mvar|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 [[neutron]]s in a neutron star formed a [[Degenerate matter|degenerate]] cold [[Fermi gas]]. They thereby obtained a limiting mass of approximately 0.7&nbsp;[[solar mass]]es,<ref>{{cite journal
 |first=R. C. |last=Tolman
 |date=1939
 |title=Static Solutions of Einstein's Field Equations for Spheres of Fluid
 |journal=[[Physical Review]]
 |volume=55 |issue=4 |pages=364–373
 |doi=10.1103/PhysRev.55.364
 |bibcode=1939PhRv...55..364T
 |url=https://resolver.caltech.edu/CaltechAUTHORS:TOLpr39
 }}</ref><ref name=OppenheimerVolkoff>
{{cite journal
 |first1=J. R. |last1=Oppenheimer
 |first2=G. M. |last2=Volkoff
 |date=1939
 |title=On Massive Neutron Cores
 |journal=[[Physical Review]]
 |volume=55 |issue=4 |pages=374–381
 |doi=10.1103/PhysRev.55.374
 |bibcode=1939PhRv...55..374O
}}</ref> which was less than the [[Chandrasekhar limit]] for white dwarfs. 

Oppenheimer and Volkoff's paper notes that "the effect of repulsive forces, i.e., of raising the pressure for a given density above the value given by the Fermi equation of state ... could tend to prevent the collapse."<ref name="OppenheimerVolkoff"/> And indeed, the most massive neutron star detected so far, [[PSR J0952–0607]], is estimated to be much heavier than Oppenheimer and Volkoff's TOV limit at {{val|2.35|0.17}} ''M''<sub>☉</sub>.<ref>{{Cite journal |last1=Romani |first1=Roger W. |last2=Kandel |first2=D. |last3=Filippenko |first3=Alexei V. |last4=Brink |first4=Thomas G. |last5=Zheng |first5=WeiKang |date=2022-08-01 |title=PSR J0952−0607: The Fastest and Heaviest Known Galactic Neutron Star |journal=The Astrophysical Journal Letters |volume=934 |issue=2 |pages=L17 |doi=10.3847/2041-8213/ac8007 |doi-access=free |bibcode=2022ApJ...934L..17R |issn=2041-8205|arxiv=2207.05124 }}</ref><ref>{{Cite web |date=2022-07-22 |title=The heaviest neutron star on record is 2.35 times the mass of the sun |url=https://www.sciencenews.org/article/heaviest-neutron-star-mass-sun-record-black-holes |access-date=2024-01-04 |language=en-US}}</ref> More realistic models of neutron stars that include [[baryon]] [[Strong interaction|strong force]] repulsion predict a neutron star mass limit of 2.2 to 2.9 <var>M</var><sub>☉</sub>.<ref>{{Cite web |last=Siegel |first=Ethan |title=The Surprising Reason Why Neutron Stars Don't All Collapse To Form Black Holes |url=https://www.forbes.com/sites/startswithabang/2018/06/13/the-surprising-reason-why-neutron-stars-dont-all-collapse-to-form-black-holes/ |access-date=2024-01-04 |website=Forbes |language=en}}</ref><ref>{{Cite journal |last1=Burkert |first1=V. D. |last2=Elouadrhiri |first2=L. |last3=Girod |first3=F. X. |date=2019-05-05 |title=The pressure distribution inside the proton |url=https://www.nature.com/articles/s41586-018-0060-z |journal=Nature |language=en |volume=557 |issue=7705 |pages=396–399 |doi=10.1038/s41586-018-0060-z |pmid=29769668 |s2cid=21724781 |issn=1476-4687}}</ref> The uncertainty in the value reflects the fact that the [[equation of state (cosmology)|equations of state]] for [[QCD matter|extremely dense matter]] are not well known.

==Applications==
In a star less massive than the limit, the gravitational compression 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.<ref name=Illari2019>{{Cite journal |last=Illari |first=Phyllis |date=2019 |title=Mechanisms, Models and Laws in Understanding Supernovae |journal=Journal for General Philosophy of Science |language=en |volume=50 |issue=1 |pages=63–84 |doi=10.1007/s10838-018-9435-y |issn=0925-4560|doi-access=free }}</ref>{{rp|74}} 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 [[Degenerate matter#Quark degeneracy|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 [[stellar black hole|black hole formed by the collapse of an individual star]] must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because of [[Stellar mass loss|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 mass]]es.<ref name="evo2">
{{cite journal
 |first1=S. E. |last1=Woosley
 |first2=A. |last2=Heger
 |first3=T. A. |last3=Weaver
 |date=2002
 |title=The Evolution and Explosion of Massive Stars
 |journal=[[Reviews of Modern Physics]]
 |volume=74 |issue=4 |pages=1015–1071
 |doi=10.1103/RevModPhys.74.1015
 |bibcode=2002RvMP...74.1015W
 |s2cid=55932331
 }}</ref><sup>:Fig. 16</sup> 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 mass]]es.<ref>
{{cite arXiv
 |first1=J. E. |last1=McClintock
 |first2=R. A. |last2=Remillard
 |date=2003
 |title=Black Hole Binaries
 |eprint=astro-ph/0306213
}}</ref><ref>
{{cite journal
 |first=J. |last=Casares
 |date=2006
 |title=Observational Evidence for Stellar-Mass Black Holes
 |arxiv=astro-ph/0612312
 |doi=10.1017/S1743921307004590
 |volume=2
 |journal=Proceedings of the International Astronomical Union
 |page=3
|s2cid=119474341
 }}</ref> [[LIGO]] has [[List of gravitational wave observations|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.

Oppenheimer and Volkoff discounted the influence of heat, stating in reference to work by Landau (1932), 'even [at] 10<sup>7</sup> degrees... the pressure is determined essentially by the density only and not by the temperature'<ref name=OppenheimerVolkoff/> – yet it has been estimated<ref name=":3">{{Cite journal |last1=Kaminker |first1=A. D. |last2=Kaurov |first2=A. A. |last3=Potekhin |first3=A. Y. |last4=Yakovlev |first4=D. G. |date=2014-08-21 |title=Thermal emission of neutron stars with internal heaters |url=http://academic.oup.com/mnras/article/442/4/3484/1357581/Thermal-emission-of-neutron-stars-with-internal |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=442 |issue=4 |pages=3484–3494 |doi=10.1093/mnras/stu1102 |doi-access=free |issn=1365-2966|arxiv=1406.0723 }}</ref> that temperatures can reach up to approximately >10<sup>9</sup> K during formation of a neutron star, mergers and binary accretion. Another source of heat and therefore collapse-resisting pressure in neutron stars is 'viscous friction in the presence of differential rotation.'<ref name=":3" />

Oppenheimer and Volkoff's calculation of the mass limit of neutron stars also neglected to consider the rotation of neutron stars, however we now know that neutron stars are capable of spinning at much faster rates than were known in Oppenheimer and Volkoff's time. The fastest-spinning neutron star known is PSR J1748-2446ad, rotating at a rate of 716 times per second<ref>{{Cite journal |last1=Hessels |first1=Jason W. T. |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 |date=2006-03-31 |title=A Radio Pulsar Spinning at 716 Hz |journal=Science |volume=311 |issue=5769 |pages=1901–1904 |doi=10.1126/science.1123430 |pmid=16410486 |arxiv=astro-ph/0601337 |bibcode=2006Sci...311.1901H |issn=0036-8075}}</ref><ref>{{Cite web |date=2007-12-29 |title=SkyandTelescope.com – News from Sky & Telescope – Spinning Pulsar Smashes Record |url=http://www.skyandtelescope.com/news/3311021.html?page=1&c=y |access-date=2024-01-05 |archive-url=https://web.archive.org/web/20071229113749/http://www.skyandtelescope.com/news/3311021.html?page=1&c=y |archive-date=2007-12-29 }}</ref> 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). Star rotation interferes with convective heat loss during supernova collapse, so rotating stars are more likely to collapse directly to form a black hole <ref>{{Cite journal |last1=Fryer |first1=Chris L. |last2=Heger |first2=Alexander |date=Oct 2000 |title=Core-Collapse Simulations of Rotating Stars |url=https://iopscience.iop.org/article/10.1086/309446 |journal=The Astrophysical Journal |language=en |volume=541 |issue=2 |pages=1033–1050 |doi=10.1086/309446 |arxiv=astro-ph/9907433 |bibcode=2000ApJ...541.1033F |issn=0004-637X}}</ref>{{rp|1044}}

==List of the least massive black holes==
{{See also|Stellar black hole}}
{{Hatnote|Content here is [[help:transclusion|transcluded]] from [[List of least massive black holes]].}}
{{:List of least massive black holes}}

== 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.
{| class="wikitable sortable sticky-header"
! scope="col" class="unsortable" | Name
! scope="col" | Mass<br>({{Solar mass|link=y}})
! scope="col" | Distance<br>([[light year|ly]])
! scope="col" | Companion [[Stellar classification|class]]
! scope="col" class="unsortable" | Mass determination method
! scope="col" class="unsortable" | Notes
! scope="col" class="unsortable" | Refs.
|-
|[[GW170817]]'s remnant || {{sort|274|{{val|2.74|0.04|0.01}}}} || 144,000,000 || style="text-align: center;" | {{TableTBA|N/A}} || [[Gravitational wave]] data of [[neutron star merger]] from [[LIGO]] and [[Virgo interferometer|Virgo]] interferometers. || In [[NGC 4993]]. Possibly collapsed into a black hole 5–10 seconds after merger. || <ref>{{cite journal |first1=Maurice H P M |last1=van Putten |first2=Massimo |last2=Della Valle |title=Observational evidence for extended emission to GW&nbsp;170817 |journal=Monthly Notices of the Royal Astronomical Society: Letters |volume=482 |issue=1 |date=January 2019 |pages=L46–L49 |doi=10.1093/mnrasl/sly166 |doi-access=free |arxiv=1806.02165 |bibcode=2019MNRAS.482L..46V |quote=we report on a possible detection of extended emission (EE) in gravitational radiation during GRB170817A: a descending chirp with characteristic time-scale ''&tau;<sub>s</sub>'' = {{val|3.01|0.2|u=s}} in a (H1,L1)-spectrogram up to 700&nbsp;Hz with Gaussian equivalent level of confidence greater than 3.3&nbsp;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&nbsp;kHz indicate a hypermassive magnetar rather than a black hole, spinning down by magnetic winds and interactions with dynamical mass ejecta.}}</ref>
|-
|[[SS 433]] || {{sort|330| 3.0–30.0 }} || 18,000 ± 700 || style="background: {{star-color|A7}}; text-align: center;" | [[main sequence|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.|| <ref>{{Cite journal|doi=10.1023/A:1021356630889|year=2002|last1=Cherepashchuk|first1=Anatol|title=Observational Manifestations of Precession of Accretion Disk in the SS 433 Binary System|journal=Space Science Reviews|volume=102|issue=1|pages=23–35|bibcode=2002SSRv..102...23C|s2cid=115604949}}</ref><ref>{{cite journal|doi=10.1038/s41586-018-0565-5|pmid=30283106|title=Very-high-energy particle acceleration powered by the jets of the microquasar SS 433|journal=Nature|volume=562|issue=7725|pages=82–85|year=2018|last1=Abeysekara|first1=A. U.|last2=Albert|first2=A.|last3=Alfaro|first3=R.|last4=Alvarez|first4=C.|last5=Álvarez|first5=J. D.|last6=Arceo|first6=R.|last7=Arteaga-Velázquez|first7=J. C.|last8=Avila Rojas|first8=D.|last9=Ayala Solares|first9=H. A.|last10=Belmont-Moreno|first10=E.|last11=Benzvi|first11=S. Y.|last12=Brisbois|first12=C.|last13=Caballero-Mora|first13=K. S.|last14=Capistrán|first14=T.|last15=Carramiñana|first15=A.|last16=Casanova|first16=S.|last17=Castillo|first17=M.|last18=Cotti|first18=U.|last19=Cotzomi|first19=J.|last20=Coutiño De León|first20=S.|last21=De León|first21=C.|last22=de la Fuente|first22=E.|last23=Díaz-Vélez|first23=J. C.|last24=Dichiara|first24=S.|last25=Dingus|first25=B. L.|author25-link= Brenda Dingus |last26=Duvernois|first26=M. A.|last27=Ellsworth|first27=R. W.|last28=Engel|first28=K.|last29=Espinoza|first29=C.|last30=Fang|first30=K.|display-authors=29|arxiv=1810.01892|bibcode=2018Natur.562...82A|s2cid=52918329}}</ref><ref name="2018-10-04_SD">{{cite web |url=http://www.spacedaily.com/reports/Scientists_discover_new_nursery_for_superpowered_photons_999.html |title=Scientists discover new nursery for superpowered photons |author=Staff Writers |website=[[Space Daily]] |date=2018-10-04}}</ref>
|-
|[[LB-1]] || {{sort|720| 2.0–70.0 }} || approx. 7,000 || style="background: {{star-color|A7}}; text-align: center;" | [[Be star]]/[[Extreme helium star|stripped helium star]] ||  || Initially thought to be first black hole in pair-instability mass gap. || <ref>{{cite journal |first=Jifeng |last=Liu |display-authors=et al. |title=A wide star–black-hole binary system from radial-velocity measurements |journal=[[Nature (journal)|Nature]] |volume=575 |issue=7784 |pages=618–621 |date=27 November 2019 |doi=10.1038/s41586-019-1766-2|pmid=31776491 |bibcode=2019Natur.575..618L |arxiv=1911.11989 |s2cid=208310287 }}</ref><ref>{{cite journal |doi=10.1051/0004-6361/201937343 |journal=Astronomy and Astrophysics |title=A stripped helium star in the potential black hole binary LB-1 |type=Letter to the Editor |volume=633 |pages=L5 |date=January 2020 |author1=Irrgang, A. |author2=Geier, S. |author3=Kreuzer, S. |author4=Pelisoli, I. |author5=Heber, U.|arxiv=1912.08338 |bibcode=2020A&A...633L...5I |doi-access=free }}</ref>
|-
| [[Cygnus X-3]] || {{sort|250| 2.0–5.0 }} || 24,100 ± 3,600 || style="background: {{star-color|WN}}; text-align: center;" | [[Wolf-Rayet star|<span style="color:white;">WN4-6</span>]] || [[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. || <ref>{{cite journal|bibcode=2017MNRAS.472.2181K|title=Gemini/GNIRS infrared spectroscopy of the Wolf-Rayet stellar wind in Cygnus X-3|journal=Monthly Notices of the Royal Astronomical Society|volume=472|issue=2|pages=2181|last1=Koljonen|first1=K. I. I.|last2=MacCarone|first2=T. J.|year=2017|doi=10.1093/mnras/stx2106|doi-access=free |arxiv=1708.04050|s2cid=54028568}}</ref><ref>{{cite journal|bibcode=2013MNRAS.429L.104Z|title=Cyg X-3: A low-mass black hole or a neutron star|journal=Monthly Notices of the Royal Astronomical Society|volume=429|pages=L104–L108|last1=Zdziarski|first1=A. A.|last2=Mikolajewska|first2=J.|last3=Belczynski|first3=K.|year=2013|doi=10.1093/mnrasl/sls035|doi-access=free |arxiv=1208.5455|s2cid=119185839}}</ref>
|-
|[[LS I +61 303]] || {{sort|140|1.0–4.0}} || 7,000 || style="background: {{star-color|B}}; text-align: center;" | [[Be star|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 [[PSR B1259−63/LS 2883|LS_2883]] and HESS J0632+057, as well as mysterious object [[LS 5039]].|| <ref>{{cite journal|bibcode=2017MNRAS.468.3689M|arxiv=1704.01335|title=The black hole candidate LS I +61°0303|journal=Monthly Notices of the Royal Astronomical Society|volume=468|issue=3|pages=3689|last1=Massi|first1=M|last2=Migliari|first2=S|last3=Chernyakova|first3=M|year=2017|doi=10.1093/mnras/stx778|doi-access=free |s2cid=118894005}}</ref><ref>{{cite journal|bibcode=2006Sci...312.1771A|arxiv=astro-ph/0605549|title=Variable Very-High-Energy Gamma-Ray Emission from the Microquasar LS I +61 303|journal=Science|volume=312|issue=5781|pages=1771–3|last1=Albert|first1=J|last2=Aliu|first2=E|last3=Anderhub|first3=H|last4=Antoranz|first4=P|last5=Armada|first5=A|last6=Asensio|first6=M|last7=Baixeras|first7=C|last8=Barrio|first8=J. A|last9=Bartelt|first9=M|last10=Bartko|first10=H|last11=Bastieri|first11=D|last12=Bavikadi|first12=S. R|last13=Bednarek|first13=W|last14=Berger|first14=K|last15=Bigongiari|first15=C|last16=Biland|first16=A|last17=Bisesi|first17=E|last18=Bock|first18=R. K|last19=Bordas|first19=P|last20=Bosch-Ramon|first20=V|last21=Bretz|first21=T|last22=Britvitch|first22=I|last23=Camara|first23=M|last24=Carmona|first24=E|last25=Chilingarian|first25=A|last26=Ciprini|first26=S|last27=Coarasa|first27=J. A|last28=Commichau|first28=S|last29=Contreras|first29=J. L|last30=Cortina|first30=J|display-authors=1|year=2006|doi=10.1126/science.1128177|pmid=16709745|s2cid=20981239}}</ref>
|-
|[[LS 5039]] || {{sort|370|{{val|3.7|1.3|1.0}}}} || 8,200 ± 300 || style="background: {{star-color|O}}; text-align: center;" | [[O-type star|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. || <ref>{{cite journal|doi=10.1111/j.1365-2966.2005.09617.x|bibcode=2005MNRAS.364..899C|arxiv=astro-ph/0507549|title=A possible black hole in the γ-ray microquasar LS 5039|journal=Monthly Notices of the Royal Astronomical Society|volume=364|issue=3|pages=899–908|year=2005|last1=Casares|first1=J|last2=Ribo|first2=M|last3=Ribas|first3=I|last4=Paredes|first4=J. M|last5=Marti|first5=J|last6=Herrero|first6=A|doi-access=free |s2cid=8393701}}</ref>
|-
|[[GRO J0422+32]]/[[V518 Persei]] || {{sort|390|{{val|3.97|0.95|1.87}}}} || 8,500 || style="background: {{star-color|M}}; text-align: center;" | [[red dwarf|M4.5V]] || Photometric [[light curve]] modelling. || [[Soft X-ray transient|SXT]] system. Only mass close to lowest possible allows it not to be a black hole. || <ref name=stellargraveyard/><ref name= "Gelino2003">{{cite journal|last1= Gelino|first1=D. M.|last2= Harrison|first2=T. E.|title= GRO J0422+32: The Lowest Mass Black Hole?|journal= The Astrophysical Journal|volume= 599|issue= 2|year= 2003|pages= 1254–1259|doi= 10.1086/379311 |bibcode = 2003ApJ...599.1254G |arxiv= astro-ph/0308490|s2cid=17785067}}</ref>
|-
|}

==See also==
{{Portal|Astronomy|Physics}}
* [[Bekenstein bound]]
* [[Oppenheimer–Snyder model]]
* [[Quark star]]
* [[Tolman–Oppenheimer–Volkoff equation]]

==References==
{{Reflist}}

{{Neutron star}}
{{Black holes}}

{{DEFAULTSORT:Tolman-Oppenheimer-Volkoff limit}}
[[Category:Astrophysics]]
[[Category:Black holes]]
[[Category:J. Robert Oppenheimer]]
[[Category:Neutron stars]]