Tolman–Oppenheimer–Volkoff limit

Tolman–Oppenheimer–Volkoff limit

The Tolman–Oppenheimer–Volkoff limit (or TOV 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.


The limit was computed 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 cold, degenerate Fermi gas. This leads to a limiting mass of approximately 0.7 solar masses.[1][2] Modern estimates range from approximately 1.5 to 3.0 solar masses.[3] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known.

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 mass no more than approximately 10 solar masses.[4]:Fig. 21 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]

See also


  1. ^ R.C. Tolman (1939). "Static Solutions of Einstein's Field Equations for Spheres of Fluid". Physical Review 55 (4): 364–373. Bibcode 1939PhRv...55..364T. doi:10.1103/PhysRev.55.364. 
  2. ^ J.R. Oppenheimer and G.M. Volkoff (1939). "On Massive Neutron Cores". Physical Review 55 (4): 374–381. Bibcode 1939PhRv...55..374O. doi:10.1103/PhysRev.55.374. 
  3. ^ I. Bombaci (1996). "The Maximum Mass of a Neutron Star". Astronomy and Astrophysics 305: 871–877. Bibcode 1996A&A...305..871B. 
  4. ^ S.E. Woosley, A. Heger, and T.A. Weaver (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. 
  5. ^ J.E. McClintock and R.A. Remillard (2003). "Black Hole Binaries". arXiv:astro-ph/0306213 [astro-ph]. Bibcode 
  6. ^ J. Casares (2006). "Observational Evidence for Stellar-Mass Black Holes". arXiv:astro-ph/0612312 [astro-ph]. Bibcode 

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