Upper bound to the mass of cold, nonrotating neutron stars
TheTolman–Oppenheimer–Volkoff limit (orTOV limit) is an upper bound to themass of cold, non-rotatingneutron stars, analogous to theChandrasekhar limit forwhite dwarf stars. Stars more massive than the TOV limit collapse into ablack hole. The original calculation in 1939, which neglected complications such as nuclear forces between neutrons, placed this limit at approximately 0.7solar masses (M☉). Later, more refined analyses have resulted in larger values.
Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0M☉,[1] corresponding to an original stellar mass of 15 to 20M☉; additional work in the same year gave a more precise range of 2.2 to 2.9M☉.[2]
Data fromGW170817, the first gravitational wave observation attributed tomerging neutron stars (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 2.17M☉.[5]
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%.[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 ofLev Landau, based on thePauli exclusion principle. Pauli's principle shows that thefermionic particles in sufficiently compressed matter would be forced into energy states so high that theirrest mass contribution would become negligible when compared with the relativistic kinetic contribution (RKC). RKC is determined just by the relevantquantum wavelengthλ, which would be of the order of the mean interparticle separation. In terms ofPlanck units, with thereduced Planck constantħ, thespeed of lightc, and thegravitational constantG all set equal to one, there will be a correspondingpressure 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 massM will be given according to thevirial theorem roughly by
,
whereρ is the density. This will be given byρ =m/λ3, wherem 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 theproton mass. This even applies in thewhite dwarf case (that of theChandrasekhar 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.
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."[7] 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 at2.35±0.17M☉.[8][9] More realistic models of neutron stars that includebaryonstrong force repulsion predict a neutron star mass limit of 2.2 to 2.9M☉.[10][11] The uncertainty in the value reflects the fact that theequations of state forextremely dense matter are not well known.
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.[12]: 74 If its mass is above the limit, the star will collapse to some denser form. It could form ablack hole, or change composition and be supported in some other way (for example, byquark degeneracy pressure if it becomes aquark star). Because the properties of hypothetical, more exotic forms ofdegenerate 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.
Ablack hole formed by the collapse of an individual star must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because ofmass loss duringstellar evolution, a black hole formed from an isolated star of solarmetallicity can have a mass of no more than approximately 10solar masses.[13]:Fig. 16 Observationally, because of their large mass, relative faintness, and X-ray spectra, a number of massive objects inX-ray binaries are thought to be stellar black holes. These black hole candidates are estimated to have masses between 3 and 20solar masses.[14][15]LIGO hasdetected 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] 107 degrees... the pressure is determined essentially by the density only and not by the temperature'[7] – yet it has been estimated[16] that temperatures can reach up to approximately >109 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.'[16]
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[17][18] 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[19]: 1044
Spectroscopic radial velocity measurements of noninteracting companion.
In Milky Way outskirts. Recent Hubble ultraviolet data has suggested the companion to this object is actually a warm subgiant, and ruling out smaller compact companions such as awhite dwarf.[22]
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.
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.
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 starsLS_2883 and HESS J0632+057, as well as mysterious objectLS 5039.
^Orosz, Jerome A.; Jain, Raj K.; Bailyn, Charles D.; McClintock, Jeffrey E.; Remillard, Ronald A. (2002). "Orbital Parameters for the Soft X-Ray Transient 4U 1543-47: Evidence for a Black Hole".The Astrophysical Journal.499 (1):375–384.arXiv:astro-ph/9712018.Bibcode:1998ApJ...499..375O.doi:10.1086/305620.S2CID16991861.
^van Putten, Maurice H P M; Della Valle, Massimo (January 2019)."Observational evidence for extended emission to GW 170817".Monthly Notices of the Royal Astronomical Society: Letters.482 (1):L46 –L49.arXiv:1806.02165.Bibcode:2019MNRAS.482L..46V.doi:10.1093/mnrasl/sly166.we report on a possible detection of extended emission (EE) in gravitational radiation during GRB170817A: a descending chirp with characteristic time-scaleτs =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.
^Abeysekara, A. U.; Albert, A.; Alfaro, R.; Alvarez, C.; Álvarez, J. D.; Arceo, R.; Arteaga-Velázquez, J. C.; Avila Rojas, D.; Ayala Solares, H. A.; Belmont-Moreno, E.; Benzvi, S. Y.; Brisbois, C.; Caballero-Mora, K. S.; Capistrán, T.; Carramiñana, A.; Casanova, S.; Castillo, M.; Cotti, U.; Cotzomi, J.; Coutiño De León, S.; De León, C.; de la Fuente, E.; Díaz-Vélez, J. C.; Dichiara, S.;Dingus, B. L.; Duvernois, M. A.; Ellsworth, R. W.; Engel, K.; Espinoza, C.; et al. (2018). "Very-high-energy particle acceleration powered by the jets of the microquasar SS 433".Nature.562 (7725):82–85.arXiv:1810.01892.Bibcode:2018Natur.562...82A.doi:10.1038/s41586-018-0565-5.PMID30283106.S2CID52918329.
