For a collapsing star, at what mass is the formation of a black hole inevitable?

Stellar mass black holes are formed from the collapse of stars. I have read figures, normally around 2-3 solar masses, that correspond to the mass at which it becomes inevitable that a star will collapse to form a singularity. Firstly, is this the correct figure - and if so, how is it derived?

Also, this limit does seem to be particularly low. Surely this should lead to a number of black holes greater than what we can observe today?

Not sure how comprehensive this will be, but it'll point to what is known, and various references.

The TOV limit for neutron stars was first worked out around 1939 and was 0.7 solar masses. More recent estimates lead to 1.5-3 solar masses, in 1996, accounting for the strong force (see Ref. 1 in the wiki article with url above, I. Bombaci (1996). "The Maximum Mass of a Neutron Star". Astronomy and Astrophysics. 305: 871–877. Bibcode:1996A&A...305..871B).

There's also been some calculations that it could become more massive by turning into a quark star, and in fact this could happen in the cores area of a neutron star -- it is not likely the quarks could be the outer area as it is conjectured that they would not be stable in contact with empty space, so they would either be strange quark matter (i.e., from strange quarks) or just possible in the core areas so they would be overdense neutron stars.

The mechanism is always the collapse, but then the possibility of an equilibrium configuration for neutron degenerate matter. The equations of state for those are not well known. The understanding is that the limit might be in the 3-5 solar masses, but it is not a firm conclusion. See the plot in that wiki article for the possible equilibrium states, without the quark star possibilities included, and the LOV limit -- Landau instead of Tolman).

The Wikipedia article cites Carroll as a reference for the conclusion that no known mechanism can stop the collapse, though the exact mass limit is not known. Note that in supernova explosions masses even greater than about 20 solar masses can leave a leftover object not much more than 5 solar masses, so it's the end state that is limited by the degeneracy or strong force pressures, the collapse itself can vary.

It depends what you mean by a "star". There is indeed a maximum mass for a neutron star. This must be at least 2 solar masses since there are now two examples with measured masses at around this value - PSR 1614-2230 at $1.97\pm0.04\ M_{\odot}$ (Demorest et al. 2010) and PSR J0348+0342 at $2.01\pm 0.04\ M_{\odot}$ (Antoniadis et al. 2013).

The exact value depends on the very uncertain equation of state of nucleonic matter at high densities.

"Harder" equations of state are able to support more massive neutron stars. This might be the case if neutrons maintain their identity at densities much higher than nuclear matter, where the strong force becomes highly repulsive. Alternatively, the neutrons might undergo a phase change to hyperonic or even quark matter. This would soften the equation of state leading to a lower maximum mass.

The extreme upper limit is found by extrapolating a well known equation of state (e.g. an n, p, e fluid at subnuclear densities) up to $P=\rho c^2$ at high densities ( a limit set by causality, where the speed of sound is $c$), and solving the Tolman-Oppenheimer-Volkhoff hydrostatic equilibrium equation under GR conditions. The star becomes unstable at finite density, with a maximum mass of around 3.5 solar masses, which could be increased a tiny bit by rapid rotation.

Physically what is happening here is that the addition of more and more momentum to the particles in order to provide the required higher and higher central pressures is ultimately counterproductive, because in GR this additional pressure and momentum simply add to the gravitational field that is crushing the star inwards.

So the limit is somewhere between 2 and 3.5 solar masses. It is notable however that there are no convincing black hole candidates with masses below 4 solar masses (see Ozel et al. 2012). It is not yet clear whether this is because they do not form or they have not been seen, but there is a gap in measured compact object masses between 2 and 4 solar masses.

There are of course lots of "normal" stars with masses above these limits. These are stars supported by normal gas pressure, with nuclear reactions heating their interiors. The end phases of a massive star's life involve it shedding large quantities of mass in winds and a final supernova. The mass limits referred to above, pertain to the inert core of the star after nuclear reactions have ceased and the envelope has been blown away in a supernova. Frequently, that would preclude the formation of a black hole (estimates vary, but it could require a star of initial mass $>25$ solar masses to produce a black hole) , so they are relatively rare (though there could be $\sim 10^8$ in our Galaxy and they are also nearly impossible to detect unless in close binary systems).

• So there are no known neutron stars above 2 solar masses, and no black holes below 4 solar masses. Clearly, collapsing objects of these masses must have some end state, so presumably there are some to be discovered. – Peter Shor Aug 8 '16 at 19:08
• @PeterShor or the final stages of stellar evolutuon and mass loss arrange it so that the intermediate objects are never formed (in binary systems). Topical research. – Rob Jeffries Aug 8 '16 at 19:11
• I could believe rarely. I think never is rather unlikely. Wouldn't a neutron star amassing gas from a companion eventually go over the limit and collapse from one of the largest neutron stars possible to one of the smallest black holes possible? – Peter Shor Aug 8 '16 at 19:28
• There are neutron stars more massive then 2 solar masses. In 2010 and 2013 two neutron stars (PSR J1614-2230 and PSR J0348+0432) with 1.97(4) and 2.01(4) solar masses where measured. There also is a candiate for a 2.5 solar mass pulsar (Black Widow Pulsar). The theoretical mass range of static neutron stars is between 1.5 and 3.2 solar masses as Rob Jeffries pointed out. The evolution of neutron stars and low mass black holes is still in active research and in general not an easy task. A lot of unknowns are involved: equation of state, GR+MHD,... – N0va Aug 8 '16 at 21:33
• @M.J Steil Those are the 2 solar mass neutron stars I was talking about. No other neutron stars have a higher, precisely determined mass. – Rob Jeffries Aug 9 '16 at 6:04