# Tolman–Oppenheimer–Volkoff limit

The Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. Observations of GW170817, the first gravitational wave event due to merging neutron stars (which are thought to have collapsed into a black hole[1] within a few seconds after merging[2]), suggest that the limit is close to 2.17 solar masses.[3][4][5][6] A neutron star in a binary pair (PSR J2215+5135) has been measured to have a mass close to or slightly above this limit, 2.27+0.17
−0.15
M.[7] A more secure measurement of PSR J0740+6620, a pulsar being eclipsed by a white dwarf, yields a mass of 2.14+0.10
−0.09
M.[8][9] Earlier theoretical work placed the limit at approximately 1.5 to 3.0 solar masses,[10] corresponding to an original stellar mass of 15 to 20 solar masses. In the case of a rigidly spinning neutron star, the mass limit is thought to increase by up to 18–20%.[2][6]

## 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 work of Lev Landau. In 1932, he reasoned 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

${\displaystyle P={\frac {1}{\lambda ^{4}}}}$ .

That pressure must be balanced by 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

${\displaystyle P^{3}=M^{2}\rho ^{4}}$ ,

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

${\displaystyle M={\frac {1}{m^{2}}}}$ .

From this, 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, [11][12] 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.[10] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known. The mass of the pulsar PSR J0348+0432, at 2.01±0.04 solar masses, puts an empirical lower bound on the TOV limit.

## Applications

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.[13]: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.[14][15] 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 most massive neutron stars

Below is a list of neutron stars which approach the TOV limit from below.

Name Neutron star mass
(solar masses)
Distance from Earth
(light years)
Companion Class Notes Refs.
PSR J1748−2021B 2.74+0.21
−0.21
27,700 D In globular cluster NGC 6440. Mass inferred from rate of advance of periastron. [16]
4U 1700-37 2.44+0.27
−0.27
6,910±1,120 O6.5Iaf+ HMXB system. Mass calculated from Monte Carlo simulations of thermal comptonization process. [17][18]
PSR J1311–3430 2.15–2.7 6,500–12,700 Substellar object Black widow pulsar. Mass determined by spectroscopic and photometric observation. [19][20]
PSR B1957+20 2.4+0.12
−0.12
6,500 Substellar object Prototype star of black widow pulsars. Mass inferred from rate of advance of periastron. [21]
PSR J2215+5135 2.27+0.17
−0.15
10,000 G5V Redback pulsar. Mass determined from accurate measurement of radial velocity of companion. [7]
XMMU J013236.7+303228 2.2+0.8
−0.6
2,730,000 B1.5IV In M33, HMXB system. Mass determined by spectroscopic observation. [22]
PSR J0740+6620 2.14+0.10
−0.11
4,600 D Mass measured by range and shape parameter of Shapiro delay. [8]
PSR J0751+1807 2.10+0.2
−0.2
6,500±1,300 D Mass measured by precision pulse timing measurements of relativistic orbital decay. [23]
PSR J0348+0432 2.01+0.04
−0.04
2,100 D Mass measured by spectroscopic observation and gravity wave induced orbital decay of companion. [24]
PSR J1614−2230 1.97+0.04
−0.04
3,900 D Mass measured by range and shape parameter of Shapiro delay. [25]
PSR B1516+02B 1.94+0.17
−0.19
24,500 D In globular cluster M5. Mass inferred from rate of advance of periastron. [26]
Vela X-1 1.88+0.13
−0.13
6,200±650 B0.5Ib Prototypical detached HMXB system. Mass inferred from rate of advance of periastron. [27]

## List of least massive black holes

Below is a list of black holes which approach the TOV limit from above.

Name Black hole mass
(solar masses)
Distance from Earth
(light years)
Companion Class Notes Refs.
2MASS J05215658+4359220 3.3+2.8
−0.7
10,000 K-type (?) giant In Milky Way outskirts. Mass inferred from properties of noninteracting companion. [28][29]
GRO J0422+32/V518 Per 3.97 ± 0.95 8500 M4.5V SXT system. [30]
GRO J1719-24/
GRS 1716−249
≥4.9 8500 K0–5 V LMXB system. [31]

## References

1. ^ Pooley, D.; Kumar, P.; Wheeler, J. C.; Grossan, B. (2018-05-31). "GW170817 Most Likely Made a Black Hole". The Astrophysical Journal. 859 (2): L23. arXiv:1712.03240. Bibcode:2018ApJ...859L..23P. doi:10.3847/2041-8213/aac3d6.
2. ^ a b Cho, A. (16 February 2018). "A weight limit emerges for neutron stars". Science. 359 (6377): 724–725. Bibcode:2018Sci...359..724C. doi:10.1126/science.359.6377.724. PMID 29449468.
3. ^ Margalit, B.; Metzger, B. D. (2017-12-01). "Constraining the Maximum Mass of Neutron Stars from Multi-messenger Observations of GW170817". The Astrophysical Journal. 850 (2): L19. arXiv:1710.05938. Bibcode:2017ApJ...850L..19M. doi:10.3847/2041-8213/aa991c.
4. ^ Shibata, M.; Fujibayashi, S.; Hotokezaka, K.; Kiuchi, K.; Kyutoku, K.; Sekiguchi, Y.; Tanaka, M. (2017-12-22). "Modeling GW170817 based on numerical relativity and its implications". Physical Review D. 96 (12): 123012. arXiv:1710.07579. Bibcode:2017PhRvD..96l3012S. doi:10.1103/PhysRevD.96.123012.
5. ^ Ruiz, M.; Shapiro, S. L.; Tsokaros, A. (2018-01-11). "GW170817, general relativistic magnetohydrodynamic simulations, and the neutron star maximum mass". Physical Review D. 97 (2): 021501. arXiv:1711.00473. Bibcode:2018PhRvD..97b1501R. doi:10.1103/PhysRevD.97.021501. PMC 6036631. PMID 30003183.
6. ^ a b Rezzolla, L.; Most, E. R.; Weih, L. R. (2018-01-09). "Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars". Astrophysical Journal. 852 (2): L25. arXiv:1711.00314. Bibcode:2018ApJ...852L..25R. doi:10.3847/2041-8213/aaa401.
7. ^ a b Linares, M.; Shahbaz, T.; Casares, J.; Grossan, Bruce (2018). "Peering into the Dark Side: Magnesium Lines Establish a Massive Neutron Star in PSR J2215+5135". The Astrophysical Journal. 859 (1): 54. arXiv:1805.08799. Bibcode:2018ApJ...859...54L. doi:10.3847/1538-4357/aabde6.
8. ^ a b Cromartie, H. T.; Fonseca, E.; Ransom, S. M.; et al. (2019). "Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar". Nature Astronomy. doi:10.1038/s41550-019-0880-2.
9. ^ Plait, Phil (2019-09-17). "Record breaker: Astronomers find the most massive neutron star known". Bad Astronomy. Retrieved 2019-09-19.
10. ^ a b Bombaci, I. (1996). "The Maximum Mass of a Neutron Star". Astronomy and Astrophysics. 305: 871–877. Bibcode:1996A&A...305..871B.
11. ^ Tolman, R. C. (1939). "Static Solutions of Einstein's Field Equations for Spheres of Fluid" (PDF). Physical Review. 55 (4): 364–373. Bibcode:1939PhRv...55..364T. doi:10.1103/PhysRev.55.364.
12. ^ Oppenheimer, J. R.; Volkoff, G. M. (1939). "On Massive Neutron Cores". Physical Review. 55 (4): 374–381. Bibcode:1939PhRv...55..374O. doi:10.1103/PhysRev.55.374.
13. ^ Woosley, S. E.; Heger, A.; Weaver, T. A. (2002). "The Evolution and Explosion of Massive Stars". Reviews of Modern Physics. 74 (4): 1015–1071. Bibcode:2002RvMP...74.1015W. doi:10.1103/RevModPhys.74.1015.
14. ^ McClintock, J. E.; Remillard, R. A. (2003). "Black Hole Binaries". arXiv:astro-ph/0306213.
15. ^ Casares, J. (2006). "Observational Evidence for Stellar-Mass Black Holes". Proceedings of the International Astronomical Union. 2: 3. arXiv:astro-ph/0612312. doi:10.1017/S1743921307004590.
16. ^ Lattimer, James M. (2015-02-25). "Introduction to Neutron Stars". American Institute of Physics. AIP Conference Proceedings 1645, 61.
17. ^ Clark, J. S.; Goodwin, S. P.; Crowther, P. A.; Kaper, L.; Fairbairn, M.; Langer, N.; Brocksopp, C. (2002). "Physical parameters of the high-mass X-ray binary 4U1700-37". Astronomy & Astrophysics. 392 (3): 909–920. arXiv:astro-ph/0207334. Bibcode:2002A&A...392..909C. doi:10.1051/0004-6361:20021184.
18. ^ Martinez-Chicharro, M.; Torrej ́on, J. M.; Oskinova, L.; F ̈urst, F.; Postnov, K.; Rodes-Roca, J. J.; Hainich, R.; Bodaghee, A. (2018). "Evidence of Compton cooling during an X-ray flare supports a neutron star nature of the compact object in 4U1700−37". Royal Astronomical Society: Letters. 473 (1): L74–L78. arXiv:1710.01907.
19. ^ Romani, Roger W.; Filippenko, Alexei V.; Silverman, Jeffery M.; Cenko, S. Bradley; Greiner, Jochen; Rau, Arne; Elliott, Jonathan; Pletsch, Holger J. (2012-10-25). "PSR J1311-3430: A Heavyweight Neutron Star with a Flyweight Helium Companion" (PDF). The Astrophysical Journal Letters.
20. ^ Romani, Roger W. (2012-10-01). "2FGL J1311.7−3429 Joins the Black Widow club". The Astrophysical Journal Letters.
21. ^ Van Kerkwijk, M. H.; Breton, R. P.; Kulkarni, S. R. (2011). "Evidence for a Massive Neutron Star from a Radial-Velocity Study of the Companion to the Black-Widow Pulsar Psr B1957+20". The Astrophysical Journal. 728 (2): 95. arXiv:1009.5427. Bibcode:2011ApJ...728...95V. doi:10.1088/0004-637X/728/2/95.
22. ^ Varun, B. Bhalerao; van Kerkwijk, Marten H; Harrison, Fiona A. (2018-06-08). "Constraints on the Compact Object Mass in the Eclipsing HMXB XMMU J013236.7+303228 in M 33" (PDF). arXiv.
23. ^ Nice, David J.; Splaver, Eric M.; Stairs, Ingrid H.; Loehmer, Oliver; Jessner, Axel; Kramer, Michael; Cordes, James M. (2005). "A 2.1 Solar Mass Pulsar Measured by Relativistic Orbital Decay". The Astrophysical Journal. 634. arXiv:astro-ph/0508050. doi:10.1086/497109.
24. ^ Demorest, P. B.; Pennucci, T.; Ransom, S. M.; Roberts, M. S. E.; Hessels, J. W. T. (2010). "A two-solar-mass neutron star measured using Shapiro delay". Nature. 467 (7319): 1081–1083. arXiv:1010.5788. Bibcode:2010Natur.467.1081D. doi:10.1038/nature09466. PMID 20981094.
25. ^ Crawford, F.; Roberts, M. S. E.; Hessels, J. W. T.; Ransom, S. M.; Livingstone, M.; Tam, C. R.; Kaspi, V. M. (2006). "A Survey of 56 Midlatitude EGRET Error Boxes for Radio Pulsars". The Astrophysical Journal. 652 (2): 1499. arXiv:astro-ph/0608225. Bibcode:2006ApJ...652.1499C. doi:10.1086/508403.
26. ^ Freire, Paulo C. C. (2007-11-30). "Super-Massive Neutron Stars" (PDF). arXiv. National Astronomy and Ionosphere Center, Cornell University.
27. ^ Quaintrell, H.; et al. (2003). "The mass of the neutron star in Vela X-1 and tidally induced non-radial oscillations in GP Vel". Astronomy and Astrophysics. 401: 313–324. arXiv:astro-ph/0301243. Bibcode:2003A&A...401..313Q. doi:10.1051/0004-6361:20030120.
28. ^ Thompson, T. A.; Kochanek, C. S.; Stanek, K. Z.; et al. (2019). "A noninteracting low-mass black hole–giant star binary system". Science. 366 (6465): 637–640. doi:10.1126/science.aau4005.
29. ^ Kumar, V. (2019-11-03). "Astronomers Spot A New Class Of Low-Mass Black Holes". RankRed. Retrieved 2019-11-05.
30. ^ Gelino, D. M.; Harrison, T. E. (2003). "GRO J0422+32: The Lowest Mass Black Hole?". The Astrophysical Journal. 599 (2): 1254–1259. arXiv:astro-ph/0308490. Bibcode:2003ApJ...599.1254G. doi:10.1086/379311.
31. ^ Chaty, S.; Mirabel, I. F.; Goldoni, P.; Mereghetti, S.; Duc, P.-A.; Martí, J.; Mignani, R. P. (2002). "Near-infrared observations of Galactic black hole candidates". MNRAS. 331 (4): 1065–1071. arXiv:astro-ph/0112329. Bibcode:2002MNRAS.331.1065C. doi:10.1046/j.1365-8711.2002.05267.x.