Implication of Existence of Hybrid stars and Theoretical Expectation of Submillisecond Puls
a r X i v :a s t r o -p h /0609011v 2 8 J a n 2008
Implication of Existence of Hybrid stars and Theoretical Expectation of Submillisecond
Pulsars
Xiaoping Zheng a ,?,Nana Pan a ,Shuhua Yang a ,Xuewen Liu a ,
Miao Kang a and Jiarong Li b
a Institute of Astrophysics,Huazhong Normal University,Wuhan 430079,P.R.
China
b Institute
of Particle Physics,Huazhong Normal University,Wuhan 430079,P.R.
China
Probing the composition of matter in the interior of compact stars is of in-terest for understanding of pulsars.The available theories have not yet given a unique solution to this problem.Since the decon?nement transition from hadron matter into quark matter is possible at high density,one logically pro-poses that some pulsars could be strange stars.But how to distinguish strange stars from neutron stars is a di?cult and an urgent issue.First,one suggested that bulk properties of neutron stars such as radius and moment of inertia,may be marked o?due to the di?erence between self-bound and gravity-bound matter.Unfortunately,the bulk properties of these two kinds of stars are simi-lar in the observed mass range 1 frequency,since the discovery by Andersson,Friedman and Morsink of r-mode instabilities in the rotating compact stars put rather severe limits on the high-est rotation frequency of pulsars(Andersson,1998;Friedman and Morsink, 1998).A pioneer investigation found that the limiting rotation of strange stars is in millisecond period range but for normal neutron stars in the period range of15～20ms(Madsen,1998,2000).However,a strange star evolving to the limiting frequency corresponds to temperature in106??107K range instead of the inferred core temperature of millisecond pulsars of a few times108K. Obviously,this is a serious drawback if we attributed the uncorrespondence to the under-representing statistics due to a low number of objects as suggested by Madsen(Madsen,2000).Although some works also argued that neutron stars with solid crusts would be the candidates to explain the rapidly rotating accreting pulsar’s data through thermal runaway recycle of r-mode.Due to the presence of viscous boundary layer damping(Bildsten and Ushomirsky, 2000;Andersson et al.,2000),the resulting viscous heating is so intense that it can heat the crust-core interface to the melting temperature of the solid crust when the r-mode amplitude is larger than some critical value,which is crudely estimated to be～10?3by Owen(1999)and a more accurate value is given as5×10?3by(Lindblom,Owen and Ushomirsky,2000). Following the above train of thought,we will here investigate the r-mode instability window of hybrid stars,i.e.,the neutron stars containing quark matter core.We show that the instability window of hybrid stars explains the millisecond pulsar data very well,but the strange stars and normal neutron stars would not?t because of the weaknesses concerned above. In rotating relativistic stars,gravitational radiation drives the r-mode while various dissipation mechanisms counteract the?uid motion.In general,shear viscosity,or surface rubbing that could also be applied to the hybrid stars, which also have a solid crust similar to the normal neutron stars,suppresses the mode at low temperatures but bulk viscosity dominates at high temper-atures.The critical rotation frequency for a given stellar model as a function of temperature follows 1 τsv + 1 τsr =0,(1) whereτgr<0is the characteristic timescale for energy loss due to gravity wave emission,τsv andτbv are the damping times due to shear and bulk viscosities, andτsr is that for surface rubbing due to the presence of the viscous boundary layer. A polytropic equation of state with a low index n is a good approximation for compact stars,as discussed in many papers(Kokkotas and Stergioulas,1999; Lindblom,Mendell and Owen,1999).The timescale for n=1polytrope from gravity wave emission can be written as τgr=?47s× M10km ?4 1000 R108K 1000 1.4M⊙ κ2?2 m s (5) κ2?2 for hadron matter(Ipser and Lindblom,1991),where the constantκ=2/3,the angular velocity?=2πνandρis the density,m s is strange quark mass.To construct hybrid stars,we consider the decon?nement transition occurring at high density inside the compact stars and use BPS(Baym,Pethick and Sutherland, 1971)and GPS(Ghosh,Phatak and Sahu,1995)equations of state for hadrons and MIT model for quarks.As in most cases,the variation of volume per unit mass and the chemical potential imbalance due to the oscillation in the star is so small that it is the liner part that contributes most to the viscosity,so we have used the relaxation time approximation method(Lindblom and Owen, 2002;Drago,Lavagno and Pagliara,2005;Nayyar and Owen,2005)to com-pute the bulk viscosity of the mixed phase matter, P(γ∞?γ0)τ ζ= 0.00.10.20.30.40.50.60.7 (g /c m s ) B (fm -3 ) Fig. 1.The bulk viscosity of the mixed phase matter in the hybrid star for m s =150Mev and B 1/4=165Mev in terms of GPS model(Ghosh,Phatak and Sahu,1995). Here P is the pressure of the mixed phase,γ∞and γ0are the ”in?nite”and ”zero”frequency adiabatic index,ωis the angular velocity of the perturbation and τis the relaxation time associated to the weak processes.As an exam-ple,we numerically calculate the bulk viscosity coe?cient of the mixed phase matter which is showed in ?gure 1for bag constant B 1/4=165Mev.From numerical estimate,we easily ?nd the fact that ζQP ≈ζMP ?ζHP . Now,we can use the bulk viscosities listed above to evaluate the bulk viscous damping times for the hybrid stars.For the given energy of modes E and its variation rate ˙E ≡d E/d t ,the timescale is de?ned as 1 2E . (7) The r-mode energy in a rotating relativistic star with radius R has been ex-pressed as(Lindblom,Mendell and Owen ,1999) E = α2π R 2m +2 d r +O (?4), (8) whereαis the normalization parameter,m is integer andρ(r)is the density function(Andersson and Kokkotas,2001).The energy dissipation rate due to bulk viscosity can be estimated with the following formula ˙E bv =? ζδσδσ?d3x,(9) whereδσis expansion tensor of Eulerian perturbation of velocity?eld,ex-pressed approximately as(Lindblom,Mendell and Owen,1999) δσ=?i(ω+m?) δρ R )m+1.In a stratiform star,we can decompose the energy E into four terms,i.e.,the quark core,mixed phase,nuclei envelope and solid crust denoted respectively by QP,MP,HP and CR E= R QP 2m (m+1)3(2m+1)!R4QP?2 R QP ρQP(r) r 2m (m+1)3(2m+1)!R4QP?2 R MP R QP ρMP(r) r 2m (m+1)3(2m+1)!R4QP?2 R HP R MP ρHP(r) r 2m (m+1)3(2m+1)!R4QP?2 R R HP ρCR(r) r R )2m?2(˙E QP+˙E MP),(16) Fig.2.The stratiform structure of the hybrid star with parameters M=1.4M , R=9.87km,m s=150Mev and B1/4=165Mev in terms of GPS model(Ghosh,Phatak and Sahu,1995).Here QP and MP represent the quark phase and mixed phase in the hybrid star respectively. with ˙E = dE QP 2m?2 dE dt R QP,ρMP.(18) R QP Thus,the timescale from Eq(4)is immediately expressed as (R),(19)τbv=AτQP bv where A is a constant related to the stellar structure that is showed in table 1,τQP (R)can be given by(Andersson and Kokkotas,2001) bv =7.9s× M10km ?4 T100MeV ?4 1000τQP bv Table1 The Values of A for di?erent Structures of a Hybrid Star with M=1.4M⊙ 165.09.870.6886 175.010.410.6929 185.012.67 6.5488 Zheng et al.,2003).It is accepted that the neutron stars with crust could also?t the pulsar data through r-mode runaway recycle mechanism(Levin, 1999;Bildsten and Ushomirsky,2000;Andersson et al.,2000).An important point is that,the fate of the crust of neutron stars is di?erent from that of hybrid stars:the former is melted probably because of r-mode dissipation due to viscous boundary layer.We can imagine that the existence of the higher rotation pulsars is possible if there would exist larger quark matter core in the interior of a compact star.This may imply that submillisecond pulsars,if they exist,should be hybrid stars instead of neutron stars or strange stars. In conclusion,the bulk viscous damping timescale for hybrid stars is estimated for a range of parameters through which the r-mode instability window is obtained.The rapid rotation pulsar data is a signi?cant indication of the existence of hybrid stars instead of neutron stars or strange stars.Naturally, this is a possible evidence of quark matter in the interior of the compact stars. We would like to thank the support by National Natural Science Foundation of China under Grant No.10373007,90303007and the Ministry of Education of China with project No.704035.We are especially grateful to Professor D.F. Hou for reading and checking the text. References Andersson,N.,1998,ApJ,502,708. Andersson,N.,Jones,D.I.,Kokkotas,K.D.&Stergioulas,N.,2000,ApJ,534, L75. Andersson,N.&Kokkotas,K.D.2001,Int.J.Mod.Phys.,D10,381. Bildsten,L.&Ushomirsky,G.,2000,ApJ,529,L33. Friedman,J.L.&Morsink,S.M.,1998,ApJ,502,714. Baym,G.,Pethick,C.&Sutherland,P.,1971,ApJ,170,299. Ghosh,S.K.,Phatak,S.C.&Sahu,K.,1995,Z.Physik A,352,457.(Proc. Suppl.),24,110. Lindblom,L.&Owen,B.J.,2002,Phys.Rev.D,65,063006. Drago,A.,Lavagno,A.&Pagliara,G.,2005,Phys.Rev.D,71,103004. Nayyar,M.&Owen,B.J.,2005,astro-ph/0512041. Haensel,P.,Zdunik,J.L.&Schae?er,R.1986,A&A,160,121. Ipser,J.R.&Lindblom,L.,1991,ApJ,373,213. Kokkotas,K.D.&Stergioulas,N.,1999,A&A,341,110. Levin,Y.,1999,ApJ,517,328. Lindblom,L.,Mendell,G.&Owen,B.J.,1999,Phys.Rev.D,60,064006. Lindblom,L.,Owen,B.J.&Ushomirsky,G.,2000,Phys.Rev.D,62,084030. Madsen,J.,1992,Phys.Rev.D,46,3290. Madsen,J.,1998,Phys.Rev.Lett,81,3311. Madsen,J.,2000,Phys.Rev.Lett,85,10 Owen, B.J.,3rd Edoardo Amalai conference on gravitational waves, gr-qc/9912065. Zheng,X.P.,Yang,S.H.&Li,J.R.,2003,ApJ,585,L135.