Steady state equilibrium condition of npe gas and its application to astrophysics

Research in Astron. Astrophys. 2011 Vol. 11 No. 1. 91-102Research inhttp://www.raa-joumalorghttp://www.ioporg/journals/raaAAstrophysicsSteady state equilibrium condition of npe* gas and itsapplication to astrophysicsMen-Quan LiuCenter for Astrophysics, University of Science and Technology of China, Hefei 230026, China;minquan@maiL.ustc.edu.cnInstitute of Theoretical Physics, China West Normal University, Nanchong 637002, ChinaReceived 2010 May 25: accepted 2010 July 30Abstract The steady equilibrium conditions for a mixed gas of neutrons, protons.electrons, positrons and radiation fields(abbreviated as npe gas)with or withoutexternal neutrino flux are investigated, and a general chemical potential equilibriumequation un=Ap Cue is obtained to describe the steady equilibrium at high tem-(T>109K)tic fitting formula of coefficient C isthe sake of simplicity, when neutrinos and antineutrinos are transparent. It is a sim-ple method to estimate the electron fraction for the steady equilibrium npe- gas thatadopts the corresponding equilibrium condition. As an example, we apply this methodto the grB accretion disk and confirm that the composition in the inner region is approximately in equilibrium when the accretion rate is low. For the case with extermalneutrino flux. we calculate the initial electron fraction of neutrino-driven wind fromthe proto-neutron star model MI5-11-rl. The results show that the improved equilib-rium condition makes the electron fraction decrease significantly more than the caseAn=Pp + He when the time is less than 5s post bounce, which may be useful forr-process nucleosynthesis modelsKey words: nuclear reactions nucleosynthesis, weak-interaction GRB accretiondisk- neutrino-driven wind1 INTRODUCTIONIt is a classic and simple approximation for the practical application in many astrophysical situationsthat matter compositions can be considered as a mixture of the neutrons, protons and electrons, theso called npe" system. If the temperature is very high(T> 10 K), lots of photons, positrons, andeven neutrinos and antineutrinos will appear in the system, i.e., the system becomes a mixture ofelectrons, positrons, nucleons and radiation fields(abbreviated as npe* gas ) Many astrophysicalsituations can be regarded as an npe* gas, such as(i)the hot fireball jetted from the central engineof a Gamma Ray Burst( GRB)(Pruet Dalal 2002), (i)the matter produced after the core-collapseof a supernova due to the photodisintegration of the iron nuclei( Marek Janka 2009),(iii)theneutrino-driven wind emanating from a proto-neutron中国煤化工:9K(Martinez-Pinedo 2008),(iv)the outer core of a young neutron staldo aucoinCNMHGSupported by the National Natural Science Foundation2009),(v)the accretion disk of a Grb Liu et al. 2007; Janiuk Yuan 2010)and(vi)the earlyuniverse before the decoupling of neutrinos( Dutta et al. 2004; Harwit 2006). In summary, npe andnpe gas is applied widely in present studies. The equilibrium steady state of npe or npt gasis an important stage for many cases. Many authors have addressed this issue for several decadesA typical approach to the steady state equilibrium of the npe system was proposed by ShapiroTeukolsky(1983). They gave an important result that Pn pp the for a steady equilibriumnpe system, where A's are the chemical potentials for neutrons, protons and electrons respectivelyThis result has been accepted by most authors. However, because Shapiro Teukolsky(1983)onlyconsidered the electron capture and its reverse interaction at 'low temperature, they ignored theappearance of positrons when the temperature of the system is high enough for this to be a factorRecently, Yuan(2005)argued that many positrons can exist at high temperatures, which leads to agreat increase of the positrons'capture rate. The positron capture significantly affects the conditionof steady state equilibrium. If the neutrinos can escape freely from the system with plenty of epairs, the equilibrium condition should be An =Ap +2e instead. However, for a more generalcondition when the temperature is moderate, the equilibrium condition has not be researched. Liuet al.(2007)have previously proposed a method in which they assume that the coefficient of pevaries exponentially from un =Hp +He to An=up+ 2ue in the accretion disk of GRBs, but itis not a rigorous method. Therefore, a detailed and reliable database or fitting function to describethe steady state equilibrium of the npe gas at any temperature is necessary. Furthermore, the abovediscussions are limited to an isolated system ignoring the external neutrino fux. In this paper, weinvestigate the chemical equilibrium condition for npe* gas at any temperature from 10 to 10 K,and give a concrete application to the grB accretion disk. we also calculate the initial electronfraction of the neutrino-driven wind in a PNS, in which the external strong neutrino flux cannot beignored. This paper is organized as follows In Section 2, we present the equilibrium conditions whenneutrinos are transparent or opaque for an isolated system. Section 3 contains a detailed discussionof the initial electron fraction of a neutrino-driven wind for the pn model M15-1l-rl(Arcones etal. 2007, 2008). Finally, we analyze the results and draw our conclusions2 EQUILIBRIUM CONDITION OF npeGAS WITHOUT THE EXTERNALNEUTRINO FLUXFor a mixed gas of npe and a radiation field in different physical conditions, we divide it into twocases: neutrino transparency and opacity. To guarantee self-consistency, we give a simple estimatefor the opaque critical density of the npe- gas. The mean free path of the neutrino islwhere n and nn are the number density of baryons and neutrons respectively. HereP(1-Ye)d gabe ar10-44( Kippenhanhn Weigert 1990), where E, is the energy of the neutrino, mec is the mass energy ofan electron, and c is the speed of light. In addition中国煤化工ob≈Epe≈CNMHGSteady State Equilibrium Condition of npe*Gas(Qian& Woosley 1996; Lai& Qian 1998), whereA=πGcos2(+3C),GF=1436×10-4 erg cIn3is the Fermi weak interaction constant, and cos28c=0.95 refers to the Cabbibo angle Cy=1CA= 1.26, Ee and pe are the energy and momentum for the electron, respectively. Due to the energyconservation of the nuclear reactionEe=Ev +Q, Q=(mn-mp)e=1.29 MeV,where mn and mp are the mass of a neutron and proton respectively. At high density, the electronsare strongly degenerate and relativistic, so Ee R EF=[(372A3ne )2/3+1/2(in units of mec2)and Ae=mc is the reduced electron Compton wavelength. Substituting preNa for ne, we obtainE≈(3n23pYNA)1/3Therefore, the mean free path of the neutrino isLNA(3x23pyNA-Q)2/3×10-4+14(323YN4)2/3(1-)MA(1)If we assume ly 10 km is the criterion for neutrino opacity, pcr= 5.58, 4.50, 4.10, 3.963.96 x 10g cm for Ye =0.1, 0.2, 0.3, 0.4, and 0.5 respectively. Rigorously speaking, hereoverestimate the absorption cross section because we ignore a block factor (l-fe), so Pcri is theminimum critical density. When p< pcri, the neutrino is transparent, otherwise it is opaque. Bya similar method, one only needs to replace v, ee Ev +Q, and nn with v, Ee Ep-Qand np respectively to obtain the mean free path of an antineutrino. The critical density for anantineutrino pcs=1.43,0.86,0.62,0.48,and0.40×101gcm-3 for Ye=0.1,0.2,0304,and0.5, respectivelyAnother more precise way to judge the transparency of a neutrino is by defining a parameter,neutrino optical depth T, which is closely related to the object's composition and structure. FollowingArcones et al (2008), we obtain( drhere r is the neutrino transport distance,(kr)=√/(6ab)(ad)+(m)where Kabs and Ngac are the absorption opacity and scatter opacity respectively, Ksac = nosacMabs=2in.Oabs(, and aabs() and ni are the neutrino absorption cross section and number density of the target particle respectively. Usually authors define t 10gcm-3, Ye tends to zero, especially for lower temperaturesThis is consistent with the results in figure 5 of Reddy et al. (1998). At high density, the B decay isalmost forbidden and the positron capture rate is smaller than that of the electron. In order to sustainthe equilibrium, electron number density ne must be very low, which causes the Ye to obviouslydecrease. Note that it is quite different from the direct Urca process for strongly degenerate baryons(Shapiro Teukolsky 1983), in which np/nn>1/8. The baryons here are nondegenerate sincetheir chemical potentials(minus their rest mass)are very low, even negative. After p, T and Ye arefound, chemical potentials An and up can be calculated as below(energy is in units of mec andmomentum in units of me c),(11)En -AnU kTdp(12)where conservation of baryon number and charge density are also included.In order to describe the numerical relationship of Ae, An and ap, we define a factor C: Anp+Cue. Tables I and 2 are the results at T=10 K and 5 x 10 K(ap, pe and pn are chemicalpotentials without rest mass), respectively. It can be seen from Table I that Ae"p=An Ae+n,i.ethe positron capture rate at this time can be ignored. C A l means that An=Pp +He is valid, whilefrom Table 2 one can find Ae-p A Ae+n >An, i. e, beta decay becomes negligible because manypositrons take part in the reactions at high temperature. Correspondingly, the equilibrium conditionbecomes n=Pp+Cue, withC N 2. It is quite different from the well known result An=Pp+.This result was first observed by Yuan(2005), and a detailed explanation can be found in Yuan(2005). Here we give a simple explanation that中国煤化工Ae-p anent a ffp, Ae+CNMHGAe-p-Ae+n a fe/p-fnfe+=fpf eConsideringfe- exp("KT/'yes exp(Ee+Hef≈expEp-lfn≈expwe find un =Pp 2e is valid when Ae-p= Ae+n For a more universal case, none of the termsAe-p, Ae+n or An can be ignored, and the coefficient C will vary with the physical conditionsTable 1 Steady state chemical equilibrium condition when the neutrino is transparent for T=10°K(cm-3s-1)0101.50E+08856E+260.84043062l1.090.206.83E+070800.510631.00.304.27E+070780.56-0.631.060403.02E+07279E+26076060-0641.050.502.29E+07214E+260.740640.64103Table 2 Steady state chemical equilibrium condition when the neutrino is trans-parent for T=5×10k(Mev)0.102.32E+081.57E+29428E+28|064-300-394|190.208.38E+07773E+28179E+28045-3.494081.96030443E+0745E+28961E+270.33-38241819714E+28564E+270.234.104271.980501.78E+072L3E+283.38E+270.144354.35199Figure 2 shows the coefficient C at different T and Ye values. It can be found that C mainlydepends on temperature T. When T< 10 K, C a 1; when T increases from 10 K to 5 x 10K, Cincreases significantly from I to 2: when T>5x10 K, C a 2. However, when T>3 andYe >0.4, C is obviously larger than 2. The reason is that the fiducial analysis in Yuan(2005)ignoresthe Fermi function. If we set the Fermi function equal to l, C A 2 is still valid For convenience inpractical application, we give a practical analytic fitting formula,C=2-1+expT9-A1Bwhere A-{28643,29249,29785,2.9902,30094]andB=[0.79138,0.72181,06331,0.61813,0.57999]corresponds to Ye =[0.1, 0.2, 0.3, 0.4, 0.5 ). Tg is the temperature in units of 10K(Tg e1-6). The accuracy of the fitting is generally better thaAs an example, we introduce an application to analyzedisk. A grb is one of the most violent events in the univH中国煤化工RB accretionCNMHSteady State Equilibrium Condition of npe Gas03念之20Fig 2 Coefficient C of chemical potential equilibrium condition An =Ap+ Cue as afunction of T and Yenot clear. Many authors support the view that a grb originates from the accretion disk of a stellars black hole. Various accretion rates( from 0.01 Mos-I to 10 Mos-)cause quite a significantdifference in the disk structure and composition. Since the temperature of the accretion disk is gen-erally larger than 100K, all nuclei are dissociated to become free nucleons, so the npe- gas candescribe the composition well. For lower accretion rates(M<0.1 Mos-), the disk is transparentto neutrinos and antineutrinos, and neutrino and antineutrino absorption are not important (SurmanMclaughlin 2004). Adopting the steady equilibrium condition, Ye of the disk model PWF99(Popham et al. 1999)(accretion rate M=0. 1 Mos, alpha viscosity a= 0. 1, and black hole spinarametera=0.95)are obtained in figure 3. The dashed line and solid line are the results from thesteady equilibrium condition and the full calculation by Surman mclaughlin(2004) respectivelyIt shows that in the inner region of the disk(from 20 km to 120 km), the electron fraction calculatedfrom different methods are principally confirmed, which indicates that the composition in the diskis approximately in an equilibrium state, but our result is generally smaller than that of SurmanMcLaughlin(2004). This is caused by the Fermi function correction in our calculation. Although inthe outer region of the disk Ye deviates from equilibrium, this deviation increases with the accretiondisk's radiusSurman& McLaughlin(2004)did not bother with specifying the radial profile of the temperature and the density of the accretion disk when calculating the electron fraction as a function of theradius for the model introduced by Popham et al. (1999). Here we rewrite the temperature and thedensity formulae of Popham et al. (1999)'s analytical model asT=1.3×101la0.M102R03K,(14)p=1.2×1014a-13M17r-25Mgcm(15)where Mi is the mass of the accreting black hole in Mo, and R is the radius given in terms of grav-stational radius rg(Tg E GMi/e, which is equal to 1. km for Mi=1 Mo). Since the explicitformulae are given, we adopt the equilibrium condition of npe gas to obtain some representativeralues of Ye in Figure 4 at a radius larger than the inner edge(six gravitational radii)of the accre-tion disk. One can find from Figure 4 that Ye values have a rapid increase with radius because bothdensities and temperatures decrease rapidly when radius increases and the variation of Ye is verysensitive to density and temperature as shown in Figure中国煤化工 cretion ratesthe accretion rate is larger, and the Ye value is larger. Thithe radius strongly depends on the structure equations ofCNMHGM.Q. Liu40B080100120140160180R(km)Fig 3 Ye as a function of accretion disk radius for model M =0.1, alpha viscosity a=0. 1, andblack hole spin parameter a= 0.95. The dashed line shows Ye from the steady state equilibriumcondition, while the solid line is the full calculation by Surman McLaughlin(2004)0.6R()60100Fig, 4 Y. as a function of accretion disk radius for the thin disk analytical model (a=0. 1, a=0,MI 3). The long-dashed line, dotted line and dot-dashed line show Ye as the accretion rateM=0.01, 0.05, and 0. 1, respectively. The vertical solid line denotes the inner boundary of thedisk(six gra2.2 Case 2. Neutrinos are OpaqueIn neutrino-opaque and antineutrino-opaque matter, neutrinos and antineutrinos will be absorbed byprotons and neutrons, except for the reactions(2) (4)as follows,+7→e+p,中国煤化工(16)e+p→e++nCNMHGSteady State Equilibrium Condition of npe GasBy using cross sectionsc=(E+Q(En+Q)2-12(4-),(E-Q)(E-Q)2-1/2(1-f+),e obtain their rates (in the system of natural units)hDn/(E,+Q)(E+Q)2-1]F(z+1,E+Q1-A)EfdE,(18Mn2(.---p2+1-9-h)Ewhere fv, and f are the Fermi-Dirac distribution function of neutrinos and antineutrinos,fu.=1+exp(kTdf+exp(The number densities of neutrinos and antineutrinos arem=#/c1+P)/p1+6xp(r"When nve =nve, i. e, number density of neutrinos is equal to that of antineutrinos, Av.=Pp. =0In this case, the equilibrium condition of Equation(5)becomepOne can find from Table 3 that even at T=5x 100K(up, pe and un'are chemical potentiawithout rest mass), C is still approximately equal to 1. In other words, when systems with neutrinosand antineutrinos are opaque and their chemical potentials are zero, An=Pp+Ae is always effectiveno matter what the temperature is, just as expectedTable 3 Steady state chemical equilibrium condition when neutrinos are opaque for T=5x10 KY010232E+ll+37789E+377.88E+367.56E+36123E+310.18-15.14-2465|1.010.206.17E+10+37201E+37417E+36401E+36587E+306692140-27.38|1.010.30246E+1036726E+36248E+36239E+36310E+30437-2595-29601010.401.05E+10362.71E+361.40E+361.35E52E+3024R-3029-3031010.503,40E+0935740E+35559E+35543E+中国煤化工93|102CNMHGM. Q. Liu3 EQUILIBRIUM CONDITION OF npe GAS WITH EXTERNAL NEUTRINOFLUXAs discussed in Section 2, we only consider that npe gas is isolated, but for many astrophysical en-vironments, the extermal strong neutrino and antineutrino fluxes cannot be ignored. These processesinvolve some complex and difficult problems that concern both the neutrino transport and the inter-actions with nucleons. Here we discuss the neutrino-driven wind(nDW)from a proto-neutron star(PNS)as a typical example NDw is regarded as a major site for r-process nucleosynthesis accordingto the observations of metal-poor stars in recent years(see e.g., Qian 2008, 2000; Martinez-Pinedo2008). Since the NDw was first proposed by Duncan et al. ( 1986), many detailed analyses of thisprocess have been done by many authors, including Newtonian and general relativity hydrodynamicsand other physical inputs: rotation, magnetic field, termination shock and so on(Qian Woosley1996; Thompson 2003: Metzger et al. 2007; Kuroda et al. 2008: Thompson et al. 2001; Fischer etal. 2009). A basic scenario of r-process nucleosynthesis in the NDw can be simply described as(seeMartinez-Pinedo 2008): soon after the birth of the PNS, many neutrinos are emitted from its surface;ecause of the photodisintegration of the shock wave, the main composition at the surface of a Pnsconsists of protons, neutrons, electrons and positrons (i. e, npe gas); in the circumambience of thePNS, the main reactions are the neutrino or antineutrinos absorption and emission by nucleons(theso called neutrino heat region); in a further region, the electron fraction Ye stays constant and aparticles are formed; above this region, other particles, such as 2C andBe, are produced until theseed nuclei are formed; abundant neutrinos are successively captured by seed nuclei. The previousresearches show that steady state is a good approximation to the nDw in the first 20s(Thompsonet al. 2001; Thompson 2003; Qian& Woosley 1996; Fischer et al. 2009)Usually, neutron-to-seed ratio, electron fraction, entropy and expansion timescale are four es-sential parameters for a successful r-element pattern. It is very difficult to fulfill all those conditionsself-consistently. Electron fraction Ye is one of the most important parameters. Recent research byWanajo et al. (2009)shows that the puzzle of the excess of the r-element of A= 90 may be solvedif Ye can increase 1%-2%(Wanajo et al. 2009). The evolution of Ye is usually obtained by solvingthe group of differential equations which is related to the Equation of State(EOS), neutrino reaction rates and hydrodynamic condition(Thompson et al. 2001). the initial value of Ye at the startof the wind is an important boundary condition. Considering that the neutrinos are emitted from aneutrino sphere Ye at the neutrino sphere can be regarded as the initial value of Ye for the windFor a given model, the initial value of Ye can be determined by making the assumption that the mat-ter in the neutrino sphere is in beta equilibrium(Arcones et al. 2008). To compare the results withthe previous work of Arcones et al.(2007, 2008), we employ the same PNS model M15-1l-rlThemodel has a baryonic mass of 1.4 Mo, which is obtained in a spherically symmetric simulation ofthe parameterized 15 Mo supernova explosion model. Detailed research shows that there are a fewa particles which will appear in the neutrino sphere, but the number density of the a particles ismuch smaller than that of the protons or neutrons, so it is reasonable to ignore the a particle effecton the electron fraction, i. e, the matter is regarded as an npe gas. Simultaneously, although manyneutrinos and antineutrinos are emitted from the pNS, their number densities are equal, which meansAv.=uD.=0. Since the neutrinos and antineutrinos are transparent to the matter in the neutrinophere, neutrinos produced by reactions(2) (3)cannot interact with nucleons, but for the neutrinosand antineutrinos coming from the core region of the PNS, absorption reactions(16)and(17)arepermitted. Their rates arep(1中国煤化工pYeCNMHG(23)Steady State Equilibrium Condition of npe Gaswhere Ln, y and Ln, v. are the number luminosity of the neutrino and antineutrino respectively, andRu is the neutrino sphere's radius Considering too many physical factors(EOS, transport equationand so on)will infuence the number luminosity and the neutrino energy, so we simply assume thnumber luminosity and the energy of neutrinos and antineutrinos are the same as those in the wind.First, we obtain the electron fraction by using a general equilibrium conditionAp-An=入+n-Aap+AIn other words, if the density and temperature are fixed for the equilibrium system, the electron fraction is unique. Then the coefficient C in the chemical potential equilibrium condition is determined(rightmost column in Table 4). The results for model M15-ll-rl are shown in Table 4In Table 4, t is the time post bounce, Ry is the neutrino sphere's radius, Ln is the numberluminosity for neutrino and antineutrino, and (Eve and(Eve are the average energy of the neutrinoand antineutrino respectively. All parameters above refer to Ref (Arcones et al. 2008). Ye is theelectron fraction for an extreme case, C=l, which is adopted in reference( arcones et al. 2008);Yis the result in which the steady equilibrium condition is valid and the extermal neutrino flux is alsoconsidered We can find ye is universally smaller than Ye, which means the extermal neutrino fluxstrongly infuences the composition of the equilibrium system. Comparing Ye with Ye, one can findthat the improved equilibrium condition makes the electron fraction significantly decrease when thetime is less than 5 s post bounce. After 5 s the electron fractions are similar to the case C= l. Notethat it is just a conclusion for the model M15-ll-rl. due to the huge differences between the differentmodels, the results may be quite different for the other models. More detailed consideration will bedone in our further work. Initial electron fraction is an important boundary condition to determine theelectron fraction of the wind. Since r-process nucleosynthesis is strongly dependent on the electronfraction, an accurate electron fraction is useful for the final r-process nucleosynthesisTable 4 Evolution of the initial electron fraction in chemical equilibrium conditions of differentsteady states(s)(km)OMev)(1056s-1)(Mev)g cn210.5563460520.7125645.50E+110.11300841.399825.143.5517.12261.30E+12005000391.229.683.0315921.691.40E+12004200351.15109594.37150521.862.00E+12002900281034 CONCLUSIONSIn this work, we derive the chemical potential equilibrium conditions Pn =p+ Cuo for npegasin two cases(with and without extermal neutrino Aux). Especially for neutrino-transparent matter,employing the fitting Equation(13)for the transition from low temperature to high temperatureis a more convenient method than the usual method for calculation of interaction rates sincechemical potentials are dependent on three parameters: density, electron fraction and temperature,any one of these three parameters can be determined if the other two parameters are given. Althoughthe variation of factor C is complicated when the extermal neutrino flux cannot be ignored, onecan obtain the extremum of those parameters assuming C= 1 or 2. Furthermore, our resultscan be regarded as the reference value for non-equilibrium states. Considering the simplicityfar-ranging aspects of the astrophysical environment, the中国煤化工 ected toused widely in the further related works.CNMHGM. Q LiuAcknowledgements The author would like to thank prof. Yuan. F for the many valuable con-versations and help with preparing this manuscript, and the referee for his/her constructive sug-gestions which have been helpful for improving this manuscript. This work is supported by theNational Natural Science Foundation of China(Grant Nos. 10733010, 10673010 and 10573016)the National Basic Research Program of China(2009CB824800), the Scientific Research Funds ofSichuan Provincial Education Department(10ZC014, 2009ZB087), China West Normal University(09A004), and Graduate Innovation Funds of USTOReferencesArcones, A, Janka, H -T,& Scheck, L. 2007, A&A, 467, 1227Arcones, A Martinez-Pinedo, G, O'Connor, E, et al. 2008, Phys. Rev. C, 78, 015806Baldo, M.,& ducoin, C. 2009, Phys. Rev. C, 79, 035801Cheng, K.S., Harko, T, Huang, Y. F, et al. 2009, Joumal of Cosmology and Astro-Particle Physics, 9, 7Duncan, R. C, Shapiro, S L,& Wasserman, I. 1986, ApJ, 309, 141Dutta, S L, Ratkovic, S,& Prakash, M. 2004, Phys. Rev. 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