# Fermi energy from gravitational collapse

When a cloud of hydrogen collapses to form a star, the particles gain energy; potential energy is converted into heat. This eventually causes the star to ignite; the thermal energy becomes high enough to allow fusion to take place.

Eventually the nuclear fuel of this star is all consumed and it collapses again. Part of it would blow away in a supernova if it was big enough, but some part may remain to form a neutron star of perhaps a black hole.

My question is concerned with this final collapse. Surely, the material must again heat up. One can even imagine that the heat must become quite extreme. Moreover, even if it collapses to form a black hole, the matter would be a ball of almost pure neutrons. So one can imagine that the Fermi energy of this ball must be extremely high. To make things more interesting, this ball is being compressed into a smaller volume. This should also increase the Fermi energy. Is it possible that the Fermi-energy in this final state of collapse (before the event horizon forms) could be so high (perhaps above the QCD scale or even the electro-weak scale) that it could actually allow some interesting physics to take place?

Yes of course! That is what motivates much of the study of neutron stars.

Tackling your first point, yes the interior temperatures of recently collapsed neutron stars are very high - perhaps $10^{11}$ K, but cool rapidly (in seconds) through the emission of thermal neutrinos to "cold" configurations where the Fermi energies are much greater than $kT$. There is plenty of theoretical interest in how matter behaves in these circumstances and how the neutrinos interact with other matter, since it controls the physics of supernovae.

The physics of neutron stars is reasonably well understood up to the nuclear saturation density at $3\times 10^{17}$ kg/m$^3$, but a typical neutron star probably has a density several times to an order of magnitude larger than this. The higher Fermi energies at these increased densities could well lead to exotic possibilities such as the creation of massive hadrons like hyperons or to the production of pions or kaons which then form a boson condensate. Alternatively, quarks may attain asymptotic freedom at high densities leading to a quark-gluon plasma or neutron stars that are entirely made of (strange) quark matter. Another alternative is that neutrons form some sort of solid core, held in a lattice by the strong nuclear force.

I suppose what you are really asking is when a neutron star exceeds the Tolman-Oppenheimer-Volkhoff limit and collapses, does the density increase even more to the extent that even more exotic (non-equilibrium) physics becomes possible? One thing to note here is that the radius of neutron star when it collapses is probably only a factor $\sim 1.5$ larger than its Schwarzschild radius, so the average density is only going to increase by a factor of a few before it departs the practically observable universe (I am not going to speculate on what cannot be observed). Given the current uncertainties in the behaviour of matter beyond the nuclear saturation density, I suspect that all the candidate physics considered for neutron cores is also relevant for collapsing neutron stars.

EDIT: Thus to answer your edit: At the highest densities inside neutron stars and at the highest densities achieved by material collapsing to a black hole (perhaps a few $10^{18}$ kg/m$^3$, the Fermi (kinetic) energy of the neutrons becomes large enough (a few hundred MeV $\sim$ the QCD scale) (or equivalently, the separation between neutrons becomes small enough $<10^{-15}$ m) that quarks may attain asymptotic freedom and quark matter is formed. The Fermi energies never get anywhere near the electroweak scale of 246 GeV.

• Interesting. Didn't know that the size of a neutron star is so close to its Schwarzschild radius. Is this always the case? How does the size of the neutron star depends on its mass? I'd expect something like: mass ~ radius$^3$. Nov 22 '16 at 4:29
• If it takes seconds for the super dense matter to cool down then that gives plenty of time for exotic physics to take place, since these reactions usually take orders of magnitude smaller amounts of time to take place. Nov 22 '16 at 4:34
• @flippiefanus Temperatures of $10^{11}$K correspond to thermal energies of only 10 MeV, so not exotic Physics in the sense you are thinking of perhaps. Neutron star radii are almost independent of mass for the favoured equations of state, with radii of 10 km compared with Schwarzschild radii of >6km for a NS beyond the TOV limit. Nov 22 '16 at 7:46