LI~TTERE AL NUOVO ClMENTO VOL. 13, ~. 2 10 Maggio 1975

Superttuid Neutron Star Cores.

U. DE ANGELIS

Osservatorio As tronomico - N apo l i

L. DE CESA~E

Is t i tu to di F i s ica dell' Univers i t5 - Salerno

(ricevuto il 19 Febbraio 1975)

The possibility that the neutron gas, at temperatures and densities typical of neutron stars, becomes a superfiuid has been pointed out by several authors (1-3).

The following discussion is based upon results from the above-cited references. There are two ways for degenerate neutrons near the top of the Fermi sea to become

superfiuid: at lower densities ((1013--10 I4) g/cm 3) by forming couples in the s-state and at higher densities (e ~> 1014 g/cm3) by forming couples in the P-state. The critical temperature of transition to the superfluid state (T~) is estimated to be of the order of 1013 ~ in both eases, although for higher densities this is still uncertain.

These results are obtained in the framework of the BCS theory of superconductivity where the mechanism for the formation of couples (Cooper pairs) is due to weak attrac- tive forces between two charged fermions near their Fermi surface (superconductivity of electrons and protons).

In order to apply the same mechanism to the neutrons one has to go up to densities such that not only the neutron gas is degenerate bu t also strong interactions begin to play their role.

Then before two neutrons become strongly coupled and give way to hyperon-forming reactions one can think of a situation where the coupling is still weak so that Cooper pairs of neutrons can be formed with consequent transition to the superfluid state.

In this picture the weakly bound neutrons do not form bineutrons (bosons) and the fluid has then to be treated according to Fermi statistics though the energy spectrum is somehow modified.

Since the central densities of neutron stars in equilibrium are higher than the transi- tion densities to the superfluid state, it is generally concluded that superfluidity may be present in the neutron gas below the crus~ of a neutron star bu t not in the star 's core.

(') V. L. GINZBU'RG and D. A. B:IRZm~ITS: SoY. Phys. JETP , 20, 1346 (1965). (~) V. L. GINZBIm~: Joum. Stat. Phys., 1, 3 (1969); i u Proceedings ol the InternationaZ Conlerence on the Sclcnce of Superconductivity, Vol. 55 (Amsterdam) , p. 207. (8) G. B&YM, C. PETHICK and D. PINES: Nature, 224, 674 (1969).

79

~ 0 U. DE ANGELI8 a n d L, DE CESARE

In the present note we wish to make some comments on the possible existence of superfluidity in neutron star cores and investigate some immediate consequences due to the new properties of the core.

Consider the formation of a neutron star as the result of the implosion of a supernova remnant in the currently accepted picture of neutron star formation as the density in the core increases, the material (originally consisting of nuclei and electrons) undergoes (~ neutronization ~) (formation of neutron-rich nuclei) and then formation of degenerate gases of neutrons, protons and electrons. If the mass of the remnant is below some limiting value, an equilibrium configuration (neutron star) will be reached at some central density where the pressure of the degenerate neutrons balances the gravitat ional pull. The equilibrium central density depends on the mass but it is always Q~ ~ 1014 g/cm a.

Then for internal temperatures T KT~ the neutron fluid, soon after i t has been formed, should become a superfluid in the sense stated above. As the density keeps increasing to eventually reach the equilibrium value for the given mass, the conpling of the Cooper pairs of neutrons should become stronger, so strong in fact as to give birth to bineutrons just before strong reactions take place (reaction thresholds for hyperon formation begin a little above l0 ~ g/cm3: the first one to be formed is the E- at e -~ 1-12"1015 g/eroS) (4).

We therefore suggest that there should be a point, just before the star reaches its equilibrium density, when neutron pairs can actually be treated as individual bineutrons, i.e. bosons of spin 1 (neutronions).

At this very moment one should switch from Fermi statistics to Bose statistics and, if this is the case, then one main consequence should be investigated: the Bose- Einstein condensation of the neutronion gas and its effect on the equation of state.

Next we show that condensation does indeed occur in such a system and that the drop in pressure is so large that the quasi-equilibrium initial configuration has to collapse.

Some possible consequences of such a collapse are then discussed t.ogether with some further comments on our hypothesis and results.

Consider the core of the would-be neutron star just at the moment when the degenerate neutron gas can be treated as a Bose gas of neutronions.

If there are n , neutrons with mass m n per unit volume, there will be n~--~n/2 neutronions with mass m = 2m n per unit volume.

If we assume the neutronions to be ideal bosons, then from the theory of Bose- Einstein condensation (5) the transition temperature to the condensed state is given by

2zeh2/m

where v = l / n is the specific volume, g~(1)- 2.612 ... and kn is the Boltzmann constant. In terms of mass density e eq. (1) gives

(2) Too=~ 102e ~ ~ �9

For densities in the range (10 la-1015) g/cm 3 where bineutrons are expected to form individual bosons, cq. (2) gives T c o ~ (1011-1012) ~

Thus we see that T~< Tee n and, since in neutron star interiors the expected tem- peratures are such that T ~ T ~ < T . . . . we conclude that the boson superfluid, if it exists, immediately upon formation undergoes a transition to the condensed state.

(4) S. TSURUTA ~ncl A. (~. l.V. CA~[ERO~: Can, Journ. Phys., ~4, 1895 [1966).

S U P E R F L U I D N E U T R O N S T A R C O R E S 81

This means that a fraction 5~o/N of the total number Y of bos ons occupies the single level of zero momentum: such a fraction is given (under conditions (5) which are largely verified in neutron star interiors) by

N

and for a typical T ~ 109 ~ this gives N 0~ N. So if condensation does occur it is almost complete. Since such a large fraction

of the neutronions drops at the lowest-energy level, it is clear that the pressure of the gas will suddenly fall to a much smaller value.

And indeed the pressure of an ideal Bose gas in the condensed state is given by

(3) P~ = 0.0851 mt h-3(k~ T)] ,

where in our ease m = 2m~. On the other hand, just before the double transition has occurred ( fermions~ super-

fluid bosons-+ condensation) the pressure was given by the Fermi pressure of the degen- erate neutron gas (nonrelativistic):

1 2 8 (4) PF 5- (3~2)~h2m[~ ~ ~ �9

The drop in prcssure is then

(5) PB 10 -6 T } 9 -~ /)F

5 5 and, since the range of typical values for neutron star interiors is T~6-~ ~1, it follows that P~ _~ 10-~Pr,.

As a result the imploding remnant, which was just adjusting itself to an equilibrium situation balancing the gravitational pressure with the Fermi pressure of the neutrons in the degenerate core, experiences a sudden (( vacuum ~) in its intcrior and can do nothing but collapse again.

One could think that the next major contribuent to the pressure after the neutron gas, the degenerate proton gas, could at this point assume the leading role in sustaining the star, but a quick, rough estimate shows that this is not the case.

In fact the density of the proton gas in neutron stars is about one order of magnitude below tlle density of the neutron gas: this means that the transition temperature to the superconducting state for the protons is of order 109 ~ and the conclusion is that, by the time the neutrons become superfiuid, thc protons are already superconducting and then, if the bineutrons condense, the biprotons are already in a condensed state and can be of no help in halting the collapse,

Next we come the degenerate electrons: they should not exhibit any superconduc- t iv i ty (2) and are therefore the only ones to keep their pressure at the corresponding Fermi value.

Since the electrons are relativistic at those values of density and temperature, their pressure is proportional to Q~ and for 9~ = 6, ~ 10-19. and it is found that the electron pressure is about one or two orders of magnitude below the neutron Fermi pressurc.

(~) K. HU~NG: Statistical Mechanics (New York, N.Y., 1963).

8 2 U. DE ANGELIS a n d L. DE CESARE

If such a pressure can indeed stop the collapse in a new equilibrium configuration depends essentially on the star's mass: if this is not only below the limiting value for neutron stars (as has been initially assumed) bu t also below the Chandraseckar limit, then a new equilibrium configuration could indeed be possible (a kind of while dwarf with central density typical of neutron stars).

In conclusion, the basic idea was that, as the density increases in the core of a supernova remnant approaching the neutron star configuration, the neutron gas becomes a superfiuid with weakly coupled pairs, that is a superfluid with strongly coupled pairs to be treated as individual bosons, and hence undergoes condensation (in momentum space).

The result is a collapse of the remnant that will never reach equilibrium as a neutron star.

The conjecture is based upon a kind of ~ composite nucleus )) theory of an n -kn strong reaction. In other words, just before the reaction n-kn-+hyperons occurs it should be possible to consider the state nA-n as a strongly coupled bineutron or neutronion and treat it properly according to Bose statistics. If this assumption can be made, ~hen neutron stars cannot be formed: they collapse just before formation.

The superdense configuration which could be formed in such a process is, as stated before, a kind of white dwarf, i .e. a configuration where the degenerate electrons, imbedded in a superfiuid neutron sea and a superconducting proton sea, support the gravitational pressure.

On the other hand, the collapse could also lead to a different situation: after B-E condensation has occurred the density reaches the value where bincutrons give birth to hyperons.

There is another sudden change in statistics (hyperons are fermions) and in pressure: this time the pressure increases enormously above Pn and this could lead to an explo- sion of the remnant .

Either way neutron stars cannot be formed.

This work has been supported by a contribution from the Consiglio Nazionale delle Ricerche through the Naples Department of the CNR National Group of Astronomy.