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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, 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]), placed the limit at close to 2.17M☉ (solar masses).[3][4][5][6] This value is inconsistent with short gamma-ray burst X-ray plateau data however, which suggests a value of MTOV = 2.37M☉.[7] Reanalysis of the GW170817 event data in 2019 resulted in a higher value of MTOV = 2.3M☉.[8] A neutron star in a binary pair (PSR J2215+5135) has been measured to have a mass close to this limit, 2.27+0.17
−0.15 M☉.[9] 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☉.[10][11]

Earlier theoretical work placed the limit at approximately 1.5 to 3.0 solar masses,[12] corresponding to an original stellar mass of 15 to 20 solar masses.

In the case of a rigidly spinning neutron star,[n 1] 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 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, [13][14] 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.[12] 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.[15]: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.[16][17] 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 Mass
(M☉) 		Distance
(ly) 		Companion class Mass determination method Notes Refs.
PSR J1748−2021B 2.74+0.21
−0.21		 27,700 D Rate of advance of periastron. In globular cluster NGC 6440. [18]
4U 1700-37 2.44+0.27
−0.27		 6,910±1,120 O6.5Iaf+ Monte Carlo simulations of thermal comptonization process. HMXB system. [19][20]
PSR J1311–3430 2.15–2.7 6,500–12,700 Substellar object Spectroscopic and photometric observation. Black widow pulsar. [21][22]
PSR B1957+20 2.4+0.12
−0.12		 6,500 Substellar object Rate of advance of periastron. Prototype star of black widow pulsars. [23]
PSR J1600−3053 2.3+0.7
−0.6		 6,500±1,000 D Fourier analysis of Shapiro delay’s orthometric ratio. [24]
PSR J2215+5135 2.27+0.17
−0.15		 10,000 G5V Innovative measurement of companion's radial velocity. Redback pulsar. [9]
XMMU J013236.7+303228 2.2+0.8
−0.6		 2,730,000 B1.5IV Detailed spectroscopic modelling. In M33, HMXB system. [25]
PSR J0740+6620 2.14+0.10
−0.11		 4,600 D Range and shape parameter of Shapiro delay. [10]
PSR J0751+1807 2.10+0.2
−0.2		 6,500±1,300 D Precision pulse timing measurements of relativistic orbital decay. [26]
PSR J0348+0432 2.01+0.04
−0.04		 2,100 D Spectroscopic observation and gravity wave induced orbital decay of companion. [27]
PSR B1516+02B 1.94+0.17
−0.19		 24,500 D Rate of advance of periastron. In globular cluster M5. [28]
PSR J1614−2230 1.908+0.016
−0.016		 3,900 D Range and shape parameter of Shapiro delay. In Milky Way’s galactic disk. [24][29]
Vela X-1 1.88+0.13
−0.13		 6,200±650 B0.5Ib Rate of advance of periastron. Prototypical detached HMXB system. [30]

List of least massive black holes

See also: Stellar black hole

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

Name Mass
(M☉) 		Distance
(ly) 		Companion class Mass determination method Notes Refs.
GW170817’s remnant 2.74+0.04
−0.01		 144,000,000 N/A Gravitational wave data of neutron star merger from LIGO and Virgo interferometers. In NGC 4993. Delayed collapse into a black hole 0.98+0.31
−0.26 seconds after merger.		 [3][31]
2MASS J05215658+4359220 3.3+2.8
−0.7		 10,000 K-type (?) giant Spectroscopic radial velocity measurements of noninteracting companion. In Milky Way outskirts. [32][33]
LS 5039 3.7+1.3
−1.0		 8,200±300 O(f)N6.5V Intermediate-dispersion spectroscopy and atmosphere model fitting of companion. Microquasar system. [34]
GRO J0422+32/V518 Per 3.97 ± 0.95 8,500 M4.5V Photometric light curve modelling. SXT system. [35]
LS I +61 303 ≤4.0 ~7,000 B0Ve Spectroscopic radial velocity measurements of companion. Microquasar system. [36][37]
GRO J1719-24/
GRS 1716−249		 ≥4.9 8,500 K0-5 V Near-infrared photometry of companion and Eddington flux. LMXB system. [38]
4U 1543-47 5.0+2.5
−2.3		 ~30,000±3,500 A2 (V?) Spectroscopic radial velocity measurements of companion. SXT system. [39]

See also

Notes

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

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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  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. ^ a b 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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  7. ^ Gao, H.; Zhang, B.; Lü, H. (2016-02-24). "Constraints on binary neutron star merger product from short GRB observations". Physical Review D. arXiv:1511.00753. doi:10.1103/PhysRevD.93.044065.
  8. ^ Shibata, M.; Zhou, E.; Kiuchi, K.; Fujibayashi, S. (2019-07-26). "Constraint on the maximum mass of neutron stars using GW170817 event". Physical Review D. arXiv:1905.03656. doi:10.1103/PhysRevD.100.023015.
  9. ^ 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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  10. ^ 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: 439. arXiv:1904.06759. Bibcode:2019NatAs.tmp..439C. doi:10.1038/s41550-019-0880-2.{{cite journal}}: CS1 maint: bibcode (link)
  11. ^ Plait, Phil (2019-09-17). "Record breaker: Astronomers find the most massive neutron star known". Bad Astronomy. Retrieved 2019-09-19.
  12. ^ a b Bombaci, I. (1996). "The Maximum Mass of a Neutron Star". Astronomy and Astrophysics. 305: 871–877. Bibcode:1996A&A...305..871B.
  13. ^ 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.
  14. ^ 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.
  15. ^ 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.
  16. ^ McClintock, J. E.; Remillard, R. A. (2003). "Black Hole Binaries". arXiv:astro-ph/0306213.
  17. ^ 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.
  18. ^ Lattimer, James M. (2015-02-25). "Introduction to Neutron Stars". AIP Conference Proceedings. 1645 (1): 61–78. Bibcode:2015AIPC.1645...61L. doi:10.1063/1.4909560.
  19. ^ 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.
  20. ^ 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". Monthly Notices of the Royal Astronomical Society: Letters. 473 (1): L74–L78. arXiv:1710.01907. Bibcode:2018MNRAS.473L..74M. doi:10.1093/mnrasl/slx165.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  21. ^ 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". The Astrophysical Journal Letters. 760 (2): L36. arXiv:1210.6884. Bibcode:2012ApJ...760L..36R. doi:10.1088/2041-8205/760/2/L36.
  22. ^ Romani, Roger W. (2012-10-01). "2FGL J1311.7−3429 Joins the Black Widow club". The Astrophysical Journal Letters. 754 (2): L25. arXiv:1207.1736. Bibcode:2012ApJ...754L..25R. doi:10.1088/2041-8205/754/2/L25.
  23. ^ 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.
  24. ^ a b Arzoumanian, Zaven; Brazier, Adam; Burke-Spolaor, Sarah; Chamberlin, Sydney; Chatterjee, Shami; Christy, Brian; Cordes, James M.; Cornish, Neil J.; Crawford, Fronefield; Cromartie, H. Thankful (2018). "The NANOGrav 11-year Data Set: High-precision Timing of 45 Millisecond Pulsars". The Astrophysical Journal Supplement Series.
  25. ^ 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". arXiv:1207.0008. doi:10.1088/0004-637X/757/1/10. {{cite journal}}: Cite journal requires |journal= (help)
  26. ^ 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: 1242–1249. arXiv:astro-ph/0508050. doi:10.1086/497109.
  27. ^ 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.
  28. ^ Freire, Paulo C. C. (2008). "Super-Massive Neutron Stars". AIP Conference Proceedings. 983: 459–463. arXiv:0712.0024. doi:10.1063/1.2900274.
  29. ^ 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): 1499–1507. arXiv:astro-ph/0608225. Bibcode:2006ApJ...652.1499C. doi:10.1086/508403.
  30. ^ 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.
  31. ^ Gill, Ramandeep; Nathanail, Antonios; Rezzolla, Luciano (2019). "When Did the Remnant of GW170817 Collapse to a Black Hole?". The Astrophysical Journal. 876 (2). arXiv:1901.04138. doi:10.3847/1538-4357/ab16da. {{cite journal}}: Cite has empty unknown parameter: |1= (help)CS1 maint: unflagged free DOI (link)
  32. ^ Thompson, T. A.; Kochanek, C. S.; Stanek, K. Z.; Badenes, C.; Post, R. S.; Jayasinghe, T.; Latham, D. W.; Bieryla, A.; Esquerdo, G. A.; Berlind, P.; Calkins, M. L.; Tayar, J.; Lindegren, L.; Johnson, J. A.; Holoien, T. W.-S.; Auchettl, K.; Covey, K. (2019). "A noninteracting low-mass black hole–giant star binary system". Science. 366 (6465): 637–640. arXiv:1806.02751. doi:10.1126/science.aau4005. PMID 31672898. {{cite journal}}: Unknown parameter |displayauthors= ignored (|display-authors= suggested) (help)
  33. ^ Kumar, V. (2019-11-03). "Astronomers Spot A New Class Of Low-Mass Black Holes". RankRed. Retrieved 2019-11-05.
  34. ^ Casares, J; Ribo, M; Ribas, I; Paredes, J. M; Marti, J; Herrero, A (2005). "A possible black hole in the -ray microquasar LS 5039". Monthly Notices of the Royal Astronomical Society. 364 (3): 899–908. arXiv:astro-ph/0507549. Bibcode:2005MNRAS.364..899C. doi:10.1111/j.1365-2966.2005.09617.x.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  35. ^ 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.
  36. ^ Massi, M; Migliari, S; Chernyakova, M (2017). "The black hole candidate LS I +61°0303". Monthly Notices of the Royal Astronomical Society. 468 (3): 3689. arXiv:1704.01335. Bibcode:2017MNRAS.468.3689M. doi:10.1093/mnras/stx778.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  37. ^ Albert, J; et al. (2006). "Variable Very-High-Energy Gamma-Ray Emission from the Microquasar LS I +61 303". Science. 312 (5781): 1771–3. arXiv:astro-ph/0605549. Bibcode:2006Sci...312.1771A. doi:10.1126/science.1128177. PMID 16709745.
  38. ^ 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". Monthly Notices of the Royal Astronomical Society. 331 (4): 1065–1071. arXiv:astro-ph/0112329. Bibcode:2002MNRAS.331.1065C. doi:10.1046/j.1365-8711.2002.05267.x.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  39. ^ Orosz, Jerome A.; Jain, Raj K.; Bailyn, Charles D.; McClintock, Jeffrey E.; Remillard, Ronald A. (May 20, 1998). "Orbital Parameters for the Soft X-Ray Transient 4U 1543-47: Evidence for a Black Hole". The Astrophysical Journal. 499: 375–384. arXiv:astro-ph/0112329. doi:10.1086/305620.
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