# What is the theoretical lower mass limit for a gravitationally stable neutron star?

I ask here intentionally not for the size of the smallest possible observed size of neutron stars, which corresponds approximately to the well-known Chandrasekhar-limit for the upper limit of the white dwarfs. This is defined by the minimal size of a stellar core to collapse into a neutron star, instead of white dwarf.

But, I think this is not the smallest possible neutron star mass - it is only the smallest mass that can be produced by stellar evolution processes.

For example, black holes have also a lower limit: the Tolman-Oppenheimer-Volkoff limit, which is around 1.5-3.0 Solar masses. Corresponding to that, the known smallest black hole is observed to be around 4 Solar masses. But this doesn't define the smallest possible size of a black hole, it only defines the smallest black hole size which can be formed. Theoretically, even Earth-sized or much more smaller black holes could exist, but there is no known process which could create them. Despite the extensive searches for micro black holes, nothing was found.

By analogy, I am asking is this a similar situation for neutron stars? What is the minimal mass of a neutron star, which could remain stable? Is this mass smaller than the Chandrasekhar-limit?

• Could you check I have not changed the meaning of your question? – Rob Jeffries Oct 26 '14 at 13:48
• I wonder if smaller neutron stars could be formed by collisions of neutron stars that fragmented them. The cross-section is extremely low, but such collisions might occur in the far future. – user4552 Oct 26 '14 at 15:30
• @RobJeffries: As I understand the state of Cosmology, there are several competing theories on the long-term fate of the universe, but the serious contenders agree that with time, the intensity of background radiation approaches zero. No black hole can have become small yet, but every black hole will become small some day. – Beta Apr 28 '15 at 17:19
• @Beta Yep, perhaps. Have you worked the numbers to check that the absorption of the CMB decreases more quickly than the Hawking radiation? – Rob Jeffries Apr 28 '15 at 22:28
• @Beta? Did you ever solve that? I'd be interested to see the result. – TLW Jul 25 '16 at 1:46

We think that most neutron stars are produced in the cores of massive stars and result from the collapse of a core that is already at a mass of $\sim 1.1-1.2 M_{\odot}$ and so as a result there is a minimum observed mass for neutron stars of about $1.2M_{\odot}$ (see for example Ozel et al. 2012). Update - the smallest, precisely measured mass for a neutron star is now $1.174 \pm 0.004 M_{\odot}$ - Martinez et al. (2015).

The same paper also shows that there appears to be a gap between the maximum masses of neutron stars and the minimum mass of black holes.

You are correct that current thinking is that the lower limit on observed neutron star and black hole masses is as a result of the formation process rather than any physical limit (e.g. Belczynski et al. 2012 [thanks Kyle]).

Theoretically a stable neutron star could exist with a much lower mass, if one could work out a way of forming it (perhaps in a close binary neutron star where one component loses mass to the other prior to a merger?). If one just assumes that you could somehow evolve material at a gradually increasing density in some quasi-static way so that it reaches a nuclear statistical equilibrium at each point, then one can use the equation of state of such material to find the range of densities where $\partial M/\partial \rho$ is positive. This is a necessary (though not entirely sufficient) condition for stability and would be complicated by rotation, so let's ignore that.

The zero-temperature "Harrison-Wheeler" equation of state (ideal electron/neutron degeneracy pressure, plus nuclear statistical equilibrium) gives a minimum stable mass of 0.19$M_{\odot}$, a minimum central density of $2.5\times10^{16}$ kg/m$^3$ and a radius of 250 km. (Colpi et al. 1993). However, the same paper shows that this is dependent on the details of the adopted equation of state. The Baym-Pethick-Sutherland EOS gives them a minimum mass of 0.09$M_{\odot}$ and central density of $1.5\times10^{17}$ kg/m$^3$. Both of these calculations ignore General Relativity.

More modern calculations (incorporating GR, e.g. Bordbar & Hayti 2006) get a minimum mass of 0.1$M_{\odot}$ and claim this is insensitive to the particular EOS. This is supported by Potekhin et al. (2013), who find $0.087 < M_{\rm min}/M_{\odot} < 0.093$ for EOSs with a range of "hardness". On the other hand Belvedere et al. (2014) find $M_{\rm min}=0.18M_{\odot}$ with an even harder EOS.

A paper by Burgio & Schulze (2010) shows that the corresponding minimum mass for hot material with trapped neutrinos in the centre of a supernova is more like 1$M_{\odot}$. So this is the key point - although low mass neutron stars could exist, it is impossible to produce them in the cores of supernovae.

Edit: I thought I'd add a brief qualitative reason why lower mass neutron stars can't exist. The root cause is that for a star supported by a polytropic equation of state $P \propto \rho^{\alpha}$, it is well known that the binding energy is only negative, $\partial M/\partial \rho>0$ and the star stable, if $\alpha>4/3$. This is modified a bit for GR - very roughly $\alpha > 4/3 + 2.25GM/Rc^2$. At densities of $\sim 10^{17}$ kg/m$^3$ the star can be supported by non-relativistic neutron degeneracy pressure with $\alpha \sim 5/3$. Lower mass neutron stars will have larger radii ($R \propto M^{-1/3}$), but if densities drop too low, then it is energetically favorable for protons and neutrons to combine into neutron-rich nuclei; removing free neutrons, reducing $\alpha$ and producing relativistic free electrons through beta-decay. Eventually the equation of state becomes dominated by the free electrons with $\alpha=4/3$, further softened by inverse beta-decay, and stability becomes impossible.

• This astrobites review discusses the NS-BH gap in reference to this paper. This suggests the gap is caused by delayed vs rapid bounces in the collapse of CCSNe. – Kyle Kanos Oct 26 '14 at 13:57
• Has anyone calculated what it would look like when a hypothetical neutron star that loses mass "popped back" into simple degenerate matter? Could we detect such an event at all? – biziclop Oct 26 '14 at 23:21
• There are a number of papers that explore the possibility of mass transfer in a binary neutron star shortly after birth. Passing below the minimum mass ought to result in an explosion of some sort. adsabs.harvard.edu/abs/2002ApJ...581.1271C – Rob Jeffries Oct 26 '14 at 23:37
• @biziclop Considering that the per-particle energy difference of free neutrons and hidrogen is in the order of the deuterium fusion, it would be essentially a supernova. Lsp? – peterh - Reinstate Monica Apr 28 '15 at 18:24
• @peterh Yes. :) Would this supernova have a recognisable radiation signature? – biziclop Apr 28 '15 at 18:31