Influence of Entropy on Composition and Structure of Massive Protoneutron Stars

Transcription

1 Commun. Theor. Phys. 66 (216) Vol. 66, No. 2, August 1, 216 Influence of Entropy on Composition and Structure of Massive Protoneutron Stars Bin Hong ( Ê), Huan-Yu Jia ( ), Xue-Ling Mu (½ ), and Xia Zhou ( ) Institute for Modern Physics, Southwest Jiaotong University, Chengdu 6131, China (Received April 11, 216; revised manuscript received June 2, 216) Abstract Adjusting the suitable coupling constants in relativistic mean field (RMF) theory and focusing on thermal effect of an entropy per baryon (S) from to 3, we investigate the composition and structure of massive protoneutron stars corresponding PSR J and PSR J It is found that massive protoneutron stars (PNSs) have more hyperons than cold neutron stars. The entropy per baryon will stiffen the equation of state, and the influence on the pressure is more obvious at low density than high density, while the influence on the energy density is more obvious at high density than low density. It is found that higher entropy will give higher maximum mass, higher central temperature and lower central density. The entropy per baryon changes from to 3, the radius of a PNS corresponding PSR J will increase from km to km and PSR J will increase from 13.3 km to km. The entropy per baryon will raise the central temperature of massive PNSs in higher entropy per baryon, but the central temperature of massive PNSs maybe keep unchanged in lower entropy per baryon. The entropy per baryon will increase the moment of inertia of a massive protoneutron star, while decrease gravitational redshift of a massive neutron star. PACS numbers: c, 26.6.Kp, Jk, 24.1.Pa Key words: massive protoneutron stars, entropy, PSR J , PSR J Introduction Neutron stars are the smallest and densest stars known. They may exhibit conditions and phenomena not observed elsewhere, so are attracting the interest of many researchers not only in astrophysics, but also in nuclear and particle physics. The typical neutron star in generally refers to a star with the mass M on the order of 1.5M and the radius R of 12 km. [1] The composition and corresponding equation of state (EOS) of this typical neutron star matter had drawn much attention. [2 4] However this has been changed since the massive neutron stars were found, such as PSR J , which is measured to have a mass of 1.97±.4M using the method of shapiro delay, [5] and PSR J , which has a mass of 2.1±.4M, measured by a combination of radio timing and precise spectroscopy of the white dwarf companion by Antoniadis et al. [6] These observations support the stiff equation of state. A lot of studies have been performed on the composition and global structure of these massive neutron stars at zero temperature. For example, Tsuyoshi Miyatsu et al. reconstructed the EOS for neutron star matter at zero temperature, including nuclei in the crust and hyperons in the core, and obtained the resultant maximum mass is 1.95M, which is consistent with the PSR J [7] Zhao and Jia attempted to find a possible model in relativistic mean field (RMF) theory to describe the neutron star of PSR J through adjusting different hyperon coupling parameters. [8] As we know, a massive cold neutron star comes from a protoneutron star which is formed after enormous supernova explosions that occur in the last few moments in the evolution of a massive star. The properties of the massive protoneutron star and how the mass of a PNS changes among different evolution stages (with different entropy per baryon) are of practical importance, but little attention has been on this subject. In this paper, considering the dense matter with strangeness-rich hyperons, we apply the relativistic mean field theory at finite entropy to investigate the composition and structure of massive protoneutron stars. Here we do not consider the neutrino concentrations and focus on the thermal effect of the entropy per baryon from to 3. [9] The paper is organized as follows. In Sec. 2, we give the complete form of relativistic mean field (RMF) theory at finite entropy. In Sec. 3, the details in selecting coupling constants are presented. In Sec. 4, some calculation results of the massive PNS are given. In Sec. 5, the summary will be presented. 2 Relativistic Mean Field Theory at Finite Entropy Relativistic mean field (RMF) theory is an effective field theory of hadron interaction. [1 11] The degrees of freedom relevant to this theory are baryons interacting through the exchange of σ, ω, ρ mesons, of which the scalar meson σ provides the medium-range attraction, the vector meson ω provides short-range repulsion, and the vectorisospin vector meson ρ describes the difference between neutrons and protons. We study the properties of thermal neutron stars in RMF theory. The partition function of system is the start- Supported by National Natural Science Foundation of China under Grant No hyjia@home.swjtu.edu.cn c 216 Chinese Physical Society and IOP Publishing Ltd

2 No. 2 Communications in Theoretical Physics 225 ing point. From the partition function we could get various thermodynamic quantities at equilibrium. For the grand canonical ensemble, the partition function can be written as: Z = Tr{exp[ (Ĥ µ ˆN)/T]}, (1) where Ĥ and ˆN are the Hamiltonian operator and the particle operator respectively, µ is the chemical potential, T is the temperature. From the partition function we can get particle population density, energy density and pressure: n = T lnz V µ, (2) ε = T 2 V lnz T + µn, (3) P = T lnz, (4) V here, V is the volume. Considering the baryons B and leptons l as fermions, we can get: lnz B,l = 2J B,l +1 k 2 dk{ln[1+ e (ε B,l(k) µ B,l )/T ] B,l + V L, (5) T where J B,l is the spin quantum number and µ B,l is the chemical potential of baryon and lepton. L is the Lagrangian density. ε B,l (k) = k 2 + m 2 B,l is thermal excitation energy of baryon and lepton. The total partition function Z total = Z B Z l, where Z B and Z l are the partition function of baryons and the standard noninteracting partition function of leptons respectively. The additional condition of charge neutrality equilibrium is listed as the following: 2J B,l + 1 q B,l k 2 n B,l (k)dk =, B,l where n B (k) and n l (k) are Fermi distribution function of baryons and leptons respectively. They are given by 1 n i (k) = (i = B, l). (6) 1 + exp[(ε i (k) µ i )/T] When neutrinos are not trapped, the set of equilibrium chemical potential relations required by the general condition: µ i = b i µ n q i µ e, (7) where b i is the baryon number of particle i and q i is its charge. The properties of a neutron star at finite temperature can be described by the entropy per baryon, the total entropy per baryon is calculated using S = (S B +S l )/(Tρ B ), where S B = P B + ε B i=b µ iρ i and S l = P l + ε l i=l µ iρ i. [12] The Lagrangian density of hadron matter is given by: [11] L = B Ψ B (iγ µ µ m B + g σb σ g ωb γ µ ω µ 1 2 g ρbγ µ τ ρ µ) Ψ B ( µσ µ σ m 2 σ σ2 ) 1 4 ω µνω µν m2 ω ω µω µ 1 4 ρ µν ρ µν m2 ρ ρ µ ρ µ 1 3 g 2σ g 3σ 4 + l=e,µ Ψ l (iγ µ µ m l )Ψ l, (8) where the sum on B runs over the octet baryons (n, p, Λ, Σ, Σ, Σ +, Ξ, Ξ ), and Ψ B is the baryon field operator. The last term represents the free lepton Lagrangian. The relativistic mean field theory gives the formula of energy density and pressure of a neutron star at finite temperature as follows: ε = 1 3 g 2σ g 3σ m2 σ σ m2 ω ω m2 ρ ρ2 3 + B + l 2J B + 1 2J l + 1 k2 + (m ) 2 (exp[(ε B (k) µ B )/T] + 1) 1 k 2 dk k 2 + m 2 l (exp[(ε l(k) µ l )/T] + 1) 1 k 2 dk, (9) P = 1 3 g 2σ g 3σ m2 σσ m2 ωω m2 ρρ B l 2J B + 1 2J l + 1 k 2 k2 + (m ) 2 (exp[(ε B(k) µ B )/T] + 1) 1 k 2 dk k 2 (exp[(ε k2 + m 2 l (k) µ l )/T] + 1) 1 k 2 dk, (1) l where, m = m B g σb σ is the effective mass of baryon. B and l denote baryons and leptons respectively. Once the equation of state is specified, the mass and radius of the neutron star can be obtained by solving the well-known hydrostatic equilibrium equations of Tolman Oppenheimer Volkoff. [13] dp dr = (p + ε)(m + 4πr3 p), (11) r(r 2M) M(r) = 4π r εr 2 dr. (12)

3 226 Communications in Theoretical Physics Vol Coupling Parameters Among the coupling constants for the RMF models, the nucleon coupling constants can be determined from the saturation properties of nuclear matter, such as nuclear saturation density, binding energy per baryon number, effective mass of the nucleon, nuclear compression modulus and asymmetry energy coefficient. [14] For this study, we choose the parameter set GL85 listed in Table 1, [11] which may well describe cold massive neutron stars. [8] Table 1 GL85 nucleon coupling constants. m m σ m ω m ρ g σ g ω g ρ g 2 Mev MeV MeV MeV fm g 3 ρ B/A K a sym m /m fm 3 MeV MeV MeV When hyperons are included, their coupling constants are needed. For the coupling constants related with hyperons, we define the ratios: x σh = g σh g σ = x σ, (13) x ωh = g ωh g ω = x ω, (14) x ρh = g ρh g ρ = x ρ, (15) where H denotes hyperons (Λ, Σ and Ξ). The ratios of hyperon coupling constant to nucleon coupling constant exist considerable uncertainty. It cannot be decided by the saturation properties of nuclear matter, but could be extrapolated through the hypernuclear experimental data. The hypernuclear potential depth in nuclear matter U N H, which is known in accordance with available hypernuclear data, serves to strictly correlate the value of x σh and x ωh : [15] U N H = x ωhv x σh S, (16) where S = m m, V = (g ω /m ω ) 2 ρ are the values of the scalar and vector field strengths for symmetric nuclear matter at saturation respectively. With UH N, if we give the value of x ωh, we can get the value of x σh. The experimental data of hypernuclear potential depth of UΛ N, UN Σ, and UΞ N are:[16 21] U N Λ = 3 MeV, U N Ξ U N Σ = +3 MeV, = 15 MeV. (17) In studying the properties of a neutron star with RMF theory, due to the considerable uncertainty in the value of x ωh. Reference [22] points that its value should be restricted at 1/3 to 1. We are presumably familiar with the truth that the higher value of x ωh will give the higher mass of a neutron star. [8] In order to get the massive neutron star, such as 2.M, we select x ωλ = x ωσ = x ωξ = 1 without considering the difference between hyperon coupling with ω and nucleon coupling with ω. Then the coupling constants x σλ, x σσ and x σξ can be calculated by formulas (16) (17): x σλ =.85, x σσ =.57, x σξ =.78. (18) The hyperon coupling constants x ρλ, x ρσ and x σξ are determined by using SU(6) symmetry: [23] x ρλ =, x ρσ = 2, x σξ = 1. (19) Using these coupling constants, we calculate the mass of a zero temperature neutron star. The resultant maximum mass is as high as 2.1M, the corresponding radius is 11.8 km, which is consistent with observation results [5 6] and other works. [24 25] So this result shows that above coupling constants are suitable for describing massive cold neutron stars like PSR J and PSR J observed recently. Sequentially, these coupling constants can be extrapolated to study PNSs. 4 Results and Discussion 4.1 Composition Now above sets of nucleon and hyperon coupling constants, which have given massive cold neutron stars, may be used to describe the massive PNSs. For these newborn stars, the thermal effect should be considered in an approximately uniform entropy per baryon from to 1. [26] In this work, we do not consider the neutrino concentrations and select entropy per baryon from to 3. In neutron star interior, some of the nucleons can be converted to hyperons which carry strangeness. The octet baryons comprise some of the least massive baryons which include the Λ, Σ, Ξ. These hyperons form a significant population of massive protoneutron stars and indeed are dominant in the high density. As a result, in Fig. 1 and Fig. 2, we give the particle populations of a massive neutron star among different evolutional stages (with different entropy per baryon). Figure 1, we can see that the hyperon number of S = 2 are more than that of S = 1. The density at where Λ appears is about.12 fm 3 of S = 2,.26 fm 3 of S = 1, while the Σ is about.23 fm 3 of S = 2,.42 fm 3 of S = 1. For S = 2 all the hyperons including Λ, Σ, Ξ appear at.1 fm 3 < ρ <.8 fm 3. But for S = 1, the hyperons appear at ρ >.2 fm 3 and Ξ, Ξ does not appear. This can account for that it is massive and there is not compensated by the presence of the electron chemical potential in the equilibrium condition. The total nucleon fraction and hyperon fraction versus the total baryon number density are shown in Fig. 2. The upper part denotes the total nucleon fraction (n, p), the lower part denotes the total hyperon fraction (Λ, Σ, Ξ). The higher entropy per baryon will impel more nucleons to convert to hyperons. S =, the nucleons convert to hyperons at ρ =.44 fm 3 and S = 3, the nucleons convert to hyperons at ρ =.1 fm 3.

4 No. 2 Communications in Theoretical Physics 227 of.1 g/cm 3 (.6%). At ρ =.5 fm 3, to 3 of S gives g/cm 3 to 15.3 g/cm 3 of lgε, an increase of.5 g/cm 3 (.33%). The effect of S on the energy density are more obvious at high density than low density. Fig. 1 Populations of various particles in neutron star matter with entropy per baryon S = 1 and S = 2 relative to the baryon number density as a function of density. Fig. 3 The EOS of a PNS for different entropy per baryon. Fig. 2 Total nucleon fraction (upper part) and hyperon fraction (lower part) in neutron star matter with different entropy per baryon as a function of density. These results show that the entropy is in favor of the production of hyperons, and it means that a massive protoneutron stars have more hyperons than a cold neutron star. 4.2 Equation of State The equation of state of PNS matter is shown in Fig. 3 and Fig. 4. Figure 3 gives the relation between pressure and energy density, while Fig. 4 gives the relation of the pressure and energy density with baryon number density. In Fig. 3 the pressure all increase with the energy density increases in different entropy cases. The influence of entropy per baryon on EOS can be read in Fig. 4, in upper panel, at ρ =.145 fm 3 (saturation density), the value of S changes from to 3, and the value of lgp increases from 33.8 dyne/cm 2 to dyne/cm 2, an increase of.43 dyne/cm 2 (1.2%). At ρ =.5 fm 3 (around central density), to 3 of S gives 35.2 dyne/cm 2 to dyne/cm 2 of lgp, an increase of.12 dyne/cm 2 (.3%). The effect of S on the pressure are more obvious at low density than high density. In comparison, in the lower panel, at ρ =.145 fm 3, as the value of S changes from to 3, the value of lgǫ increases from 14.4 g/cm 3 to g/cm 3, an increase Fig. 4 Pressure (upper panel) and energy density (lower panel) of neutron star matter as a function of baryon density for different values of entropy per baryon. At low density, this is explained that the nucleons convert to hyperons little, entropy obviously increase the pressure and stiffen the EOS. At high density, we can get the information from the Fig. 2, in core of a PNS, more hyperons appear, their populations are driven by the pauli principle in dense matter so as to reduce more energy of the baryon Fermi Seas (softening the EOS). 4.3 Mass, Radius, Temperature, Moment of Inertia and Gravitational Redshift We can get the properties of a massive PNS by substituting the equations of state into the TOV equation. The results are shown in Fig.5 Fig. 9. In Fig. 5, the mass as a function of the central density is given for different entropy per baryon. It is found that the mass will increase significantly with the entropy per baryon. When S =, corresponding to the cold neutron star, gives the maximum mass 2.1M at ρ =.94 fm 3.

5 228 Communications in Theoretical Physics Vol. 66 With the entropy per baryon increasing, the maximum mass of S = 3 is 2.18M at ρ =.74 fm 3. S = ). This means that the evolution of massive PNS is a stellar contract process. Fig. 5 The masses of a massive neutron star for different cases. Fig. 7 The interior temperature of the neutron star as a function of the baryon number density for different cases. Fig. 6 Mass-Radius relation for different cases. The entropy is higher, the corresponding central density of the maximum mass PNS is lower. The higher entropy suggests the violent thermonuclear reaction in interior structure during beginning of evolutional period and leads to stiffen the EOS, so an interesting phenomenon is presented that the mass of PNS gives an uneven increasing span with every.5 step rising from to 3. Here, in Fig. 5, we also give the central density of PNS of PSR J and PSR J without considering the stellar accretion, the density tendency illustrates that the central density will increase during the evolution of massive PNS(S = 3 to S = ). The mass-radius relation is shown in Fig. 6. In Fig. 6, the mass as a function of the radius is given for different entropy per baryon. We give the radius of the PNS corresponding to PSR J and PSR J respectively. When S changes from to 3, the radius of PSR J increases from km to km, and the radius of PSR J increases from 13.3 km to km. It substantially demonstrates that radius will decrease from initial stage of evolution (we can hypothesize this stage S = 3) to stable stage (here we hypothesize Fig. 8 The relation between the moment of inertia and mass for different cases. Fig. 9 The relation between gravitational redshift and mass for different cases. The profile of temperature in massive neutron star for different entropy is shown in Fig. 7. It can be seen that the temperature will increase with the entropy per baryon. For S = 3, the temperature will increase from MeV (at ρ =.1 fm 3 ) to MeV (at ρ = 1. fm 3 ). For S =.5, from 4.2 MeV to MeV and then the temperature will keep unchanged in core.

6 No. 2 Communications in Theoretical Physics 229 The entropy is higher, the temperature is higher. From above discussion, entropy will be in favor of the production of hyperons. When the hyperons are included in the protoneutron star, the temperature can not increase so high, that nucleons convert to hyperons will exhaust majority of temperature. The bottom of Fig. 7 shows that the increasing speed of temperature with the density of the protoneutron star will become slow when the hyperons appear, and the temperature gradually gets to the maximum value with the relative fraction of the hyperons increase and extends slowly to a stable temperature platform. It means that the temperature at the center of massive protoneutron star may be a constant when including the hyperons in lower entropy per baryon. However, when the entropy per baryon increases continuously, the increased effect of entropy per baryon on temperature will exceed the decreased effect of nucleon convert to hyperon on temperature, so the temperature will increase with the density till very high at the upper panel of Fig. 7. The central density and temperature corresponding to the maximum mass PNS from to 3 are also pointed in Fig. 7. It is found that higher entropy will give higher maximum mass, higher central temperature and lower central density. Table 2 The PNS properties of PSR J whose mass is 1.97M denoting by and PSR J whose mass is 2.1M. R is radius, ρ c is central density, T c is central temperature, I is moment of inertia and Z indicates gravitational redshift. R/km ρ c/fm 3 T c/mev I/(g cm 3 ) Z S = S = S = S = S = S = S = Table 3 The properties of the maximum mass PNS. R is radius, ρ c is central density, T c is central temperature, I is moment of inertia and Z indicates gravitational redshift. M max (M ) R/km ρ c/fm 3 T c/mev I/(g cm 3 ) Z S = S = S = S = S = S = S = The behavior of the moment of inertia and gravitational redshift vs. mass are illustrated in Fig. 8 and Fig. 9. It is found that the higher is the entropy per baryon, the larger is the moment of inertia and the smaller is the gravitational redshift. Because the bigger entropy per baryon will give the larger radius which can be read from the Fig. 6, this obviously increases the moment of inertia and decrease the gravitational redshift of a PNS. From above discussion we list the specific properties of the massive protoneutron star corresponding to PSR J and PSR J , which include radius, central density, central temperature, moment of inertia and gravitational redshift. We also list the properties of the maximum mass PNS corresponding to above calculations. The results are listed in Table 2 and Table 3 respectively. 5 Summary Considering the octet baryons in relativistic mean field (RMF) theory and focusing on thermal effect of an entropy per baryon from to 3, we investigate the composition and structure of massive protoneutron stars corresponding PSR J and PSR J One set of coupling constant of nucleon and hyperon in RMF theory are selected to reproduce the properties of PSR J and PSR J and then are extended to describe these massive protoneutron stars. It is found that the massive protoneutron stars (PNSs) have more hyperons than cold neutron stars as well as the populations will increase with the entropy per baryon. The entropy per baryon will stiffen the EOS, and the influence on the pressure is more obvious at low density than high density, at ρ =.145 fm 3 (saturation density), the value of S changes from to 3, and the value of lgp increases from 33.8 dyne/cm 2 to dyne/cm 2. At ρ =.5 fm 3 (around central density), to 3 of S gives 35.2 dyne/cm 2 to dyne/cm 2 of lgp. While the influence on the energy density is more obvious at high density than low density, at ρ =.145 fm 3, as the value of S changes

7 23 Communications in Theoretical Physics Vol. 66 from to 3, the value of lgǫ increases from 14.4 g/cm 3 to g/cm 3. At ρ =.5 fm 3, to 3 of S gives g/cm 3 to 15.3 g/cm 3 of lgε. The entropy per baryon will increase the maximum mass of a PNS. When the S changes from to 3, the maximum mass of a PNS will increase from 2.1M to 2.18M and the entropy per baryon is higher, the corresponding central density is lower. The radius of a massive protoneutron star is bigger than a cold neutron star. The higher is the entropy per baryon, the bigger is the radius of a protoneutron star. When the S changes from to 3, the radius of a PNS corresponding PSR J will increase from km to km and PSR J will increase from 13.3 km to km. The entropy per baryon will raise the central temperature of a massive protoneutron star in higher entropy per baryon, the temperature of S = 3 will increase from MeV (ρ =.1 fm 3 ) to MeV (ρ = 1. fm 3 ). In the lower entropy per baryon, the temperature of S =.5 from 4.2 MeV (ρ =.1 fm 3 ) to MeV (ρ = 1. fm 3 ) and the central temperature may keep unchange. The entropy per baryon will increase the moment of inertia of a massive protoneutron star, contrarily decrease the effect of gravitational redshift of a massive neutron star. References [1] J.M. Lattimer and M. Prakash, Science 34 (24) 536. [2] Clifford E. Rhoades Jr. and Remo Ruffini, Phys. Rev. Lett. 32 (1974) 324. [3] J. Cooperstein, Phys. Rev. C 37 (1988) 786. [4] H.-J. Schulze, A. Polls, A. Ramos, and I. Vidaña, Phys. Rev. C 73 (26) [5] P.B. Demorest, T. Pennucci, S.M. Ransom, et al., Nature (London) 467 (21) 181. [6] John Antoniadis, et al., Science 34 (213) 448. [7] Tsuyoshi Miyatsu, Sachiko Yamamuro, and Ken ichiro Nakazato, Astrophys. J. 777 (213) 4. [8] Xian-Feng Zhao and Huan-Yu Jia, Phys. Rev. C 85 (212) [9] A. Burrows and J.M. Lattimer, Astrophys. J. 37 (1986) 178. [1] Zhong-Zhou Ren, Phys. Rev. C 65 (22) [11] N.K. Glendenning, Compact Star: Nuclear Physics, Particle Physics, and General Relativity, Springer-Verlag, New York (1997). [12] Madappa Prakash, Ignazio Bombaci, Manju Prakash, Paul J. Ellis, James M. Lattimer, and Roland Knorren, Phys. Reps. 28 (1997) 1. [13] J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55 (1939) 374. [14] N.K. Glendenning, Astrophys. J. 293 (1985) 47. [15] S. Weissenborn, D. Chatterjee, and B. J. Schaffner, Nucl. Phys. A 881 (212) 62. [16] D.J. Millener, C.B. Dover, and A. Gal, Phys. Rev. C 38 (1988) 27. [17] J. Schaffner, H. Stöecker, and C. Greiner, Phys. Rev. C 46 (1992) 322. [18] C.J. Batty, E. Friedman, and A. Gal, Phys. Reps. 287 (1997) 385. [19] B.J. Schaffner and A. Gal, Phys. Rev. C 62 (2) [2] S. Aoki, S.Y. Bahk, and K.S. Chung, Phys. Lett. B 355 (1995) 45. [21] P. Khaustov, et al., Phys. Rev. C 61 (2) [22] N.K. Glendenning and S.A. Moszkowski, Phys. Rev. Lett. 67 (1991) [23] S. Pal, M. Hanauske, I. Zakout, H. Stöcker, and W. Greiner, Phys. Rev. C 6 (1999) [24] A. Drago, A. Lavabno, and G. Pagloara, Phys. Rev. D 89 (214) [25] F. Ozel, et al., Astrophys. J. 724 (21) L199. [26] J.A. Pons, et al., Astrophys. J. 513 (1999) 78.