# Tolman–Oppenheimer–Volkoff limit

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The Tolman–Oppenheimer–Volkoff limit (or TOV limit) (also referred to as the Landau–Oppenheimer–Volkoff limit (or LOV limit)) is an upper bound to the mass of stars composed of neutron-degenerate matter (i.e. neutron stars). The TOV limit is analogous to the Chandrasekhar limit for white dwarf stars. It is approximately 1.5 to 3.0 solar masses,[1] corresponding to an original stellar mass of 15 to 20 solar masses.

## 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, whose reasoning was based on the Pauli exclusion principle according to which the Fermionic particles in sufficiently compressed matter would be forced into energy states so high that their rest mass contribution would become negligible compared with the relativistic kinetic contribution determined just by the relevant quantum wavelength ${\displaystyle \lambda }$ which would be of the order of the mean interparticle separation. In terms of Planck units with his constant ${\displaystyle \hbar }$ and the speed of light ${\displaystyle c}$ and Newton's constant ${\displaystyle G}$ all set equal to one, there will be a corresponding pressure given roughly by ${\displaystyle P=1/\lambda ^{4}}$, that must be balanced by the pressure needed to resist gravity, which for a body of mass ${\displaystyle M}$ will be given according to the virial theorem roughly by ${\displaystyle P^{3}=M^{2}\rho ^{4}}$, where ${\displaystyle \rho }$ is the density, which will be given by ${\displaystyle \rho =m/\lambda ^{3}}$ where ${\displaystyle m}$ 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

${\displaystyle M=1/m^{2},}$

in which ${\displaystyle m}$ can be taken to be given roughly by the proton mass, even in the white dwarf case (that of the Chandrasekhar limit) for which the Fermionic particles providing the pressure are electrons, because the mass density is provided by the nuclei in which the neutrons are at most about as numerous as the protons while the latter, 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, [2][3] 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.[1] 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.[4]: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.[5][6]