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Neutron Stars and Strange Matter Bnl—39402 De37 006746 J NEUTRON STARS AND STRANGE MATTER BNL—39402 DE37 006746 J. Cooperstein* Brookhaven National Laboratory, Upton NY 11973 and SUNY at Stony Brook NY 11794 We investigate the likelihood that quark matter with strangeness of order unity resides in neutron stars. In the strong coupling regime near Po this is found to be unlikely. Considering higher densities where perturbative expansions are used, we find a lower bound to be at 7p0 for the transition density. This is higher than the inferred density of observed neutron stars, and thus the transition to quark matter is precluded. I. Introduction We will deal with the question: does bulk strange matter exist in nature? For this we will draw upon our recent investigation of this subject done in collaboration with Hans Bethe and Gerry Brown1. Strange matter is usually taken to mean deconfined i .lark matter with strangeness of order unity. If such material exists in the universe today, the core of neutron stars is the oi?ly reasonable place to search for it in the bulk. Observationally the problem is difficult. We know that all well measured neutron stars have masses in the range 1.4 ~ M/MQ ~ 1.6, so we have at least a lower limit to the maximum mass. As far as an upper limit is concerned one might be tempted to believe it is not much higher than the above range for two reasons. First there is the fact that Type II supernovae calculations always tend to leave behind a mass at the lower end of the above range, since this is about the iron core mass of the presupernova star, and the explosion should sweep out almost all of the overlying burning shells. Afterwards, however, matter may continue to accrete onto the neutron star, especially if it is in a binary system (and the measured masses have only been accurately determined for such systems) and the companion overflows its Roche lobes. The second reason is the lack of any strong indication of a neutron star with a large mass, say ~ 2MQ. But at any rate, the upper limit is not very precise, there having been only a handful of measured neutron star masses. Radii of neutron stars are not as well known as their masses and other static properties such as surface temperatures may not tell us anything directly about the interior. The most hope for discriminating amongst equations of state may come from evaluation of dynamic properties of neutron stars. For instance cooling rates could give us clues as to the existence of pion condensates or quark matter. Examination of core quakes and glitches can also yield * Supported in part by the U. S. Department of Energy under Contracts No. DE-AC02- 76CH00016 and No. DE-AC02-76ER13001. m B , '£•;* / DISTRIBUTION OF THIS [iOCUMFjf some (parameterization dependent) constraints. From such an analysis of the free precession of the Hercules X-l pulsar, Pandharipande, Pines and Smith2 posited, for example, that soft equations of state such as that constructed from the Reid soft core potential, could be excluded as requiring uncomfortably high values of oblateness and initial angular velocity. However such dynamical studies are not fully developed, and so unfortunately, at the present time, our only "good" constraint is on the maximum neutron star mass; many equations of state can satisfy this criteria. On the theoretical plane evaluating the probability of the existence of strange quark matter in neutron stars involves two steps. First one must calculate (estimate) the density at which such a phase transition would take place, based on the properties of the confined and deconfined phases. Second one must ask what density is achieved at the center of neutron stars. If this density lies below that of the phase transition, strange matter will not find a home in neutron stars. II. Strange Matter The possibility of quark stars is not a new subject. In the late 1970's Baym and Chin3, Chapline and Nauenberg4, Freedman and McLerran0 and others considered this question. A general consensus emerged tljaS the density of the transition would be too high to occur in neutron stars, about 10 po °r greater. Recently Witten6 has revived interest in this question and it has been further explored by Farhi and Jaffe7 and others8. This work floats upon the observation that one can find a set of the relevant parameters (a3, m3, B), such that strange matter could be stable (either in • «i\\ or for small nuggets). Why nature should opt to dial in such values is far from obvious, *'.' ••• ough it does keep some theorists amused. One recent investigation8 even claims to find a phase transition from neutron matter to strange quark matter t: just less than 2p0. We have reinvestigated these possibilities in a recent article1 and find agreement with the negative results of the 1970's. Perturbative QCD calculations cannot be used reliably to determine strange matter properties near p0- Here the leading log expression for the running strong coupling constant, a. = (33 - 2n/) In (fc/A) is useless, the momenta corresponding to distances r ~ A"1 ~ 1 fm. The precise behavior of a.a in this regime is not known; strong coupling must be used in the confinement regime. Consider a system of A particles confined on a lattice. Each A has one up, down and strange quark, and thus has no charge, like the neutron. Chiral lag model calculations indicate the A particle is significantly larger than the nucleon due to the absence of a pion cloud providing a ram pressure9. These find R^ ~ 1.15 fm10, just equal to the radius of a sphere containing one nucleon at p0. Since this is the equilibrium radius of the A, the pressure vanishes at the edge of the bags which can be considered as Wigner-Seitz spheres, and since the A energy is minimized at this radius, small shifts about it give only negligible energy changes. Inclusion of interactions between the A particles are neglected but should also be relatively small and not change our arguments. The energy per A would be equal to its mass, m^ = 1116 MeV, substantially above the energy per nucleon of neutron matter, ~ rn^ ~ 940 MeV. (In fact, constraining the center of mass motion of the A in a crystal or liquid leads to another 100 ~ 200 MeV additional energy, as pointed out to us by Leonardo Castillejo.) Taking the A-matter as the model of equilibrium strange matter with vanishing pressure near pQ, we see there is no possibility near this density of joining neutron matter to strange matter. This would require equality of chemical potential and pressure for the two phases. A general thermodynamic prescription for locating the phase transition is to require continuity of the intensive variables across the phase boundary. Specifically we require Pt = PN = Pq (2) where Pt is the pressure at the transition point and "N" and "q" refer to nucleons and quarks respectively; and r M: = /"A = 3Ai? (3) for the chemical potentials, three quarks being needed to achieve baryon number of unity. The temperatures of the two phases should also be equal of course, but here we assume T = 0. The phase transition (assumed to be first order) can be constructed graphically as shown schematically in figure 1. Plotting Pr* vs. fj.^ and Pq vs. 3/i? on the same graph, we find the curves cross a: point "t" where eqs. (2) and (3) are satisfied. Since we have the Maxwell relation the baryon number density discontinuity at tHe transition is given by the change in the slopes at point "t". We must have the quark curve cross from below of course, in order to have our normal world made up of baryonic matter and not quark matter. The discontinuity in the energy density Ae is given by Ae = Eq - e.v = MtAp = m (pg/3 - pN) (5) For the neutron matter equation of state we employ a simple power law for illustration12: KNp0 V?N —g — (6a) PN _ -, Neutrons \ I i i i I i ( i I i i i I ill or 3 FIGURE 1 The dotted line displays Pg + B vs 3^? for the quarks; the solid line P^ vs fx^ for the neutron matter. The quark curve must be displaced downward by the value B required to get the first order phase transition. where ujq — p;v/po, KN is the compression modulus of neutron matter at the symmetric matter saturation density po = .16 fm3, 7 is the high density adiabatic exponent, and fh^ is fhN = mN + ^ + £8ym - —T- -r (7) ?he chemical potential is then given by 7 = (eN + PN) lpN = mN+ -zn (3) We can eliminate p^ in (8) by using (6a) to obtain 7 (9) giving us the nuclear curve in figure 1. Equation (9) indicates that stiffening the neutron matter equation of state tends to move the transition downward. If this stiffening is done by increasing iCjv, then at fixed the pressure is reduced, so the nucleon jlurve in figure 1 is displaced downward, crossing the quark curve sooner; it is also easy to s/ee that the neutron density, pjy, is also lowered from equation (4). If stiffening is accomplished by raising 7 exemption of eqs. (6-8) shows the effect to be the same as long as UJV > (7 - 1) /7, which is no problem. Thus using a stiff neutron matter equation of state always promotes the transition to occur at lower densities. However, as we shall see later, stiffer equations of state lower the neutron star central density at a fixed star mass, and so there is a pincers effect which gives some stability to conclusions about the occurrence of the transition.
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