# 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.

Theoretical work in 1996 placed the limit at approximately 1.5 to 3.0 solar masses, 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.

Observations of GW170817, the first gravitational wave event due to merging neutron stars (which are thought to have collapsed into a black hole within a few seconds after merging), placed the limit at close to 2.17M (solar masses). This value was inconsistent with short gamma-ray burst X-ray plateau data however, which suggested a value of MTOV = 2.37M. Reanalysis of the GW170817 event data in 2019 resulted in a higher value of MTOV = 2.3M. 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. 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.

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

## 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

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

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

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

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

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

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,  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. 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.: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. 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 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. 
4U 1700-37 2.44+0.27
−0.27
6,910 ± 1,120 O6.5Iaf+ Monte Carlo simulations of thermal comptonization process. HMXB system. 
PSR J1311–3430 2.15–2.7 6,500–12,700 Substellar object Spectroscopic and photometric observation. Black widow pulsar. 
PSR B1957+20 2.4+0.12
−0.12
6,500 Substellar object Rate of advance of periastron. Prototype star of black widow pulsars. 
PSR J1600−3053 2.3+0.7
−0.6
6,500 ± 1,000 D Fourier analysis of Shapiro delay’s orthometric ratio. 
PSR J2215+5135 2.27+0.17
−0.15
10,000 G5V Innovative measurement of companion's radial velocity. Redback pulsar. 
XMMU J013236.7+303228 2.2+0.8
−0.6
2,730,000 B1.5IV Detailed spectroscopic modelling. In M33, HMXB system. 
PSR J0740+6620 2.14+0.10
−0.11
4,600 D Range and shape parameter of Shapiro delay. Most massive neutron star with a well-constrained mass 
PSR J0751+1807 2.10+0.2
−0.2
6,500 ± 1,300 D Precision pulse timing measurements of relativistic orbital decay. 
GW190425-A 2.03+0.15
−0.14
518,600,000 NS Gravitational wave data of neutron star merger from LIGO and Virgo interferometers. Merged with companion to form 3.4M black hole 
PSR J0348+0432 2.01+0.04
−0.04
2,100 D Spectroscopic observation and gravity wave induced orbital decay of companion. 
PSR B1516+02B 1.94+0.17
−0.19
24,500 D Rate of advance of periastron. In globular cluster M5. 
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. 
Vela X-1 1.88+0.13
−0.13
6,200 ± 650 B0.5Ib Rate of advance of periastron. Prototypical detached HMXB system. 

## List of least massive black holes

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.
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. 
GW190425’s remnant 3.4+0.3
−0.1
518,600,000 N/A Gravitational wave data of neutron star merger from LIGO and Virgo interferometers. 97% chance of prompt collapse into a black hole immediately after merger. 
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. 
GRO J0422+32/V518 Per 3.97+0.95
−0.95
8,500 M4.5V Photometric light curve modelling. SXT system. 
LS I +61 303 ≤4.0 7,000 B0Ve Spectroscopic radial velocity measurements of companion. Microquasar system. 
NGC 3201-1 4.36+0.41
−0.41
15,600 (see Notes) Spectroscopic radial velocity measurements of noninteracting companion. In globular cluster NGC 3201. Companion is 0.8M main sequence turn-off. 
GRO J1719-24/
GRS 1716−249
≥4.9 8,500 K0-5 V Near-infrared photometry of companion and Eddington flux. LMXB system. 
4U 1543-47 5.0+2.5
−2.3
30,000 ± 3,500 A2 (V?) Spectroscopic radial velocity measurements of companion. SXT system. 
XTE J1650-500 ≥5.1 8,500 ± 2,300 K4V Orbital resonance modeling from QPOs Transient binary X-ray source 
GRO J1655-40 5.31+0.07
−0.07
<5,500 F6IV Precision X-ray timing observations from RossiXTE. LMXB system. 

## List of objects in mass gap

This objects may contain as neutron stars, as black holes, as quark stars, as either exotic objects; separated from list of least massive black holes due to the unclear nature of these objects.

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. Possibly collapsed into a black hole 5–10 seconds after merger. 
SS 433 3.0–30.0 18,000 ± 700 A7Ib First discovered microquasar system. 
LB-1 2.0–70.0 approx. 7,000 Be star/stripped helium star Initialy thought to be first black hole in pair-instability mass gap. 
Cygnus X-3 2.0–5.0 24,100 ± 3,600 WN4-6 Near-infrared spectroscopy and atmosphere model fitting of companion. Microquasar system.
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