The extended nonlinear model has been applied to construct neutron star matter equation of state. In the case of neutron star matter with nonzero strangeness the extension of the vector meson sector by the inclusion
of nonlinear mixed terms results in the stiffening of the equation of state and accordingly in the higher value of the maximum neutron star mass

Bednarek, Ilona

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Hyperon Star  Author: Ilona Bednarek  Citation style: Bednarek Ilona. (2009). Hyperon Star. "Acta Physica Polonica B" (Vol. 40, no. 11 (2009), s. 3071-3077).   Vol. 40 (2009) ACTA PHYSICA POLONICA B No 11HYPERON STAR∗Ilona BednarekInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007, Katowice, Poland(Received October 29, 2009)The extended nonlinear model has been applied to construct neutronstar matter equation of state. In the case of neutron star matter with non-zero strangeness the extension of the vector meson sector by the inclusionof nonlinear mixed terms results in the stiffening of the equation of stateand accordingly in the higher value of the maximum neutron star mass.PACS numbers: 97.60.Jd, 26.60.Dd, 26.60Kp, 26.60.–c1. IntroductionAt the core of a neutron star the matter density ranges from a few timesthe density of normal nuclear matter (ρ0) to about an order of a magni-tude higher. At such densities exotic forms of matter such as hyperons areexpected to emerge. The appearance of these additional degrees of free-dom and their impact on a neutron star structure have been the subject ofextensive studies [1–4].The analysis of the role of strangeness in nuclear structure in the aspectof multi-strange system is of great importance for both nuclear physics andfor astrophysics. In the latter case understanding the properties of hyperonstar is still a relevant item. In the core of a neutron star, in the very highdensity environment, there is also the possibility for strange gravitationalphenomena [5].Recent results of neutron star mass measurements with the most spec-tacular and promising one, obtained for neutron star–white dwarf binarysystem, with the Arecibo radio telescope, suggest that the maximum neu-tron star mass is rather high [6]. According to original observations, thementioned above binary system, includes a neutron star with the largestmass ever reported M(PSR J0751+1807) = 2.1± 0.2 M(1σ) [7]. However,∗ Presented at the XXXIII International Conference of Theoretical Physics, “Matter tothe Deepest”, Ustroń, Poland, September 11–16, 2009.(3071)3072 I. Bednarekthis result has been corrected down to the new value 1.26 M with the esti-mated errors 1.12–1.30 M(1σ) and 0.98–1.53 M(2σ) [8, 9]. This reversedvalue of the mass has been obtained on the basis of the improved value ofthe orbital decay and the detection of the Shapiro delay.The most accurately measured neutron star masses come from observa-tions of the radio binary pulsars. These reliably determined neutron starmasses are in the range 1.5–1.65 M. The largest mass obtained recentlyfrom the analysis of timing observations of the pulsar PSR J1903+0327 hasbeen estimated at the value of 1.7 ± 0.4 M. This is the largest preciselyknown neutron star mass. There are other reported results of measure-ments of neutron star masses which indicate their high values. Among oth-ers there are observations of pulsars in globular clusters: the pulsar PSRB1516+02B located in the globular cluster M5 with estimated value of themass 1.96+0.09−0.12 M, pulsar PSR J1748-2021B in the globular cluster NGC6440 with the median mass of 2.74 ± 0.21 M. However, these results arerather uncertain. Another examples of the high neutron star masses in-clude compact star in the low mass X-ray binary (LMXB) 4U 1636-536with the mass estimated at the value of 2.0 ± 0.1 M [10] or the X-raysource EXO 0748-676 which has been constrained by the detection of grav-itational redshift of certain absorption lines. In the latest case the obtainedresults combined with other observational data lead to individually esti-mated mass and radius of the star at the value of M ≥ 2.10± 0.28 M andR ≥ 13.8± 1.8 km [11,12].Besides the measurements of neutron star masses also the estimated valueof the radius of the isolated neutron star RX J1856.5-3754 [13] indicates itshigh value of the mass. The analysis of thermal emission from the starsurface allows the determination of the ratio of the photospheric radius R∞to the distance d. This result together with the assumed model of theatmosphere support the large value of radius of the star R > 12 km.Nuclear matter in neutron star interiors represents environments withextremely high value of the isospin asymmetry fa which defines the neutronexcess in the system fa=(ρn−ρp)(ρn+ρp), ρn and ρp represent neutron andproton densities, respectively. The sum ρn+ρp=ρb defines the baryon dens-ity. The equation of state (EOS) of neutron-rich matter can be approximatedby the following parabolic formula (ρb, fa) = (ρb, 0) + S2(ρb)f2a , where(ρb, fa) is the energy per particle of asymmetric infinite nuclear matter.The EOS includes the symmetric matter contribution which does notdepend on the isospin asymmetry (ρb, 0) and the symmetry energy termS2(ρb). The last one totally determines the isospin dependence of the system.The most important characteristics of a neuron star, namely its maxi-mum mass and the typical neutron star radius are strictly connected withthe form of the equation of state involved, but as the maximum mass isHyperon Star 3073controlled by stiffness of the EOS, the radius is determined by the densitydependence of the symmetry energy S2(ρb). The presented above results ofastrophysical observations which suggest high value of neutron star massesshould be compared with the data obtained from terrestrial nuclear experi-ments [14].The reported results from the elliptic flow in the heavy ion collisionsare used to constrain the symmetric part of nuclear matter EOS. Basingon the analysis of the flow data and the results from transport calculationsthe values of pressure in the density range 2 < ρb/ρ0 < 4.6 have been es-timated [14]. On the other hand, the density dependence of the symmetryenergy is still poorly known. Data from intermediate energy heavy ion reac-tions, which include subthreshold kaon production, constrain the form of thesymmetry energy in the density range (2 < ρb/ρ0 < 3). The one-parameterfit to the low-density behavior of the symmetry energy has been introduced,where u = ρ/ρ0 and J is the symmetry energy coefficient equal to S2(ρ0).This dependence also allows one to determine the transition density ρt be-tween the crust and the core of a neutron star. The constraints on the valueof γt, for the transition density ρt, obtained from the intermediate-energyheavy-ion collisions provides the range γt ∼ 0.69–1.05 [15–17].In general, heavy ion experiments support the soft EOS, at moderatedensities. The considerable stiffening of the EOS is indispensable for higherdensities in order to reproduce the observed high value of neutron starmasses. Thus, the combined current astrophysical and heavy-ion constraintslead to a very specific form of the EOS which in the case of non-strange nu-clear matter is satisfactorily reproduced by many models. However, thereis a contradiction between the very high value of the observed neutron starmasses and the EOS of strangeness-rich nuclear matter. In general, the ap-pearance of hyperons in neutron star interiors is connected with the substan-tial softening of the EOS and this results in the low value of the maximummass.2. The modelVector densities, which are defined by the value of the vector mesonfields, are the decisive factors that contribute to the EOS of dense matterin neutron star interiors. The construction of the extended vector mesonsector has been done by the inclusion the fourth-order self-interaction termof the vector mesons. Different forms of the SU(3) invariants have beenincluded. This allows one to take into account divers nonlinear vector mesoninteraction terms. This results in the appearance of various nonlinear vectormeson couplings, among which there are terms which relate the strangeand non-strange mesons. As a consequence strong connections between theasymmetry and strangeness fraction of the model have emerged. For detailsconcerning the model considered see the paper [18].3074 I. BednarekThe Lagrangian function of the system consists of a baryonic part whichincludes the full octet of baryons together with terms describing interactionof baryons with scalar and vector mesons, and a mesonic part. The mesonicpart contains also additional interactions between mesons. The most generalform of the Lagrangian function can be written as followsL =∑Bψ¯BiγµDµψB −∑BmB(σ, σ∗)ψBψB + LM , (1)where baryon fields ΨTB = (ψN , ψΛ, ψΣ , ψΞ) are composed of the followingisomultiplets [1]:ΨN =(ψpψn), ΨΛ = ψΛ ,ΨΣ = ψΣ+ψΣ0ψΣ− , ΨΞ = (ψΞ0ψΞ−),Dµ is the covariant derivative of baryons which in terms of ωµ, ρaµ and φµfields is given byDµ = ∂µ + igωBωµ + igφBφµ + igρBI3Bτaρaµ . (2)The meson part of the Lagrangian functionLM = 12∂µσ∂µσ + 12∂µσ∗∂µσ∗ − Ueff(σ, σ∗, ωµ, ρaµ, φµ)−14ΩµνΩµν − 14RaµνRaµν − 14ΦµνΦµν (3)includes the field tensors Ωµν , Φµν and Raµν defined as Ωµν = ∂µων − ∂νωµ,Φµν = ∂µφν−∂νφµ and Raµν = ∂µρaν−∂νρaµ. All meson interaction terms arecollected in the potential function which can be written as a sum of linearand nonlinear parts, respectivelyUeff(σ, σ∗, ωµ, ρaµ, φµ)= Ulin(σ, σ∗, ωµ, ρaµ, φµ)+Unl(σ, σ∗, ωµ, ρaµ, φµ). (4)The linear scalar and vector meson part of the potential takes the formUlin(σ, σ∗, ωµ, ρaµ, φµ)= 12m2σσ2 + 12m2σ∗σ∗2 − 12m2ω (ωµωµ)−12m2ρ(ρaµρµa)− 12m2φ (φµφµ) , (5)whereas its nonlinear part is given byUV = 14c3(ρaµρµa)2 + 34c3(ρaµρµa)(φνφν) + 18c3(φµφν)2+34c3(φµφµ)(ωνων) + 14(gρgω)2ΛV (φµφµ)2+14c3(ωµωµ)2 − 12(gρgω)2ΛV (ρaµρµa)(φνφν)+(gρgω)2ΛV (ρaµρµa)(ωνων)− 12(gρgω)2ΛV (φµφµ)(ωνων) . (6)Hyperon Star 3075Calculations performed in this paper are based on the standard TM1 pa-rameter set [19]. The inclusion of the mixed nonlinear isoscalar–isovectorcoupling ΛV provides the possibility of modifying the high density depen-dence of the symmetry energy and requires the adjustment of the gρN cou-pling constant to keep the same value of the symmetry energy at saturation.Thus, in the isovector sector the parameters gρN and ΛV have been fittedto reproduce the symmetry energy coefficient at the value J = 36.89 MeV.The remaining ground state properties are left unchanged.Vector mesons–hyperon coupling constants are taken from the quarkmodel. Whereas the scalar couplings gσB of the Λ, Σ and Ξ hyperons requireconstraining in order to reproduce the estimated values of the potential feltby a single Λ, Σ and Ξ in the saturated nuclear matter. The knowledgeabout the hyperon–hyperon interaction is also indispensable. The couplingof hyperons to the strange meson σ∗ has been limited by the estimatedvalue of hyperon potential depths in hyperon matter and this has directconsequences for neutron star parameters. Recent experimental data [20]indicate a much weaker strength of hyperon–hyperon (Y –Y ) interaction.The strong Y –Y interaction is related to the value of the potential U (Λ)Λ =−20 MeV, whereas the weak one corresponds to U (Λ)Λ = −5 MeV [20,21].3. Results and discussionThe integration of the Tolman–Oppenheimer–Volkov (TOV) equationswith a specific equation of state leads to the mass-radius relation and allowsone to determine the value of the maximum mass which in a sense can givea measure of the impact of particular nonlinear couplings between vectormesons.In Fig. 1 the equations of state obtained for different cases of nonlinearpotentials presented in this paper have been shown. Extreme, dashed curvesrepresent results obtained for the standard TM1 parameterization, for thenon-strange and strangeness rich matter, respectively. The case when thematter includes only nucleons and leptons gives the stiffest EOS whereas thedirectly opposed result namely the softest EOS can be obtained for the stan-dard TM1 parameterization extended by the inclusion of hyperons. Otherequations of state presented in this figure aim to provide the analysis ofthe influence of additional vector meson nonlinear interaction. The casesΛV = 0.008 and ΛV = 0.01 have been taken into account. The obtainedresults indicate for the strong tendency for stiffening of the EOS for theincreasing value of the parameter ΛV . The inclusion of nonlinear vectormeson interactions has profound consequences for the structure of neutronstars and this can be deduced from Fig. 2 where the mass-radius relationsfor the obtained equations of state have been shown. Dotted curves depict3076 I. BednarekFig. 1. The EOS obtained for the nonlinear models. The EOSs for the nonlinearmodels with different values of the parameter ΛV are included. All parameters thatenter the considered models are taken from [18].Fig. 2. The mass-radius relations obtained for the chosen models.the results calculated for non-strange matter. The mass-radius relation forthe FSU Gold parameter set has been given, for comparison. The maximummass increases with increasing value of the parameter ΛV , giving in the re-sults neutron star models with masses exceeding 2 M and with reducedradii. Thus, one can expect that solutions with nonlinear vector meson cou-plings lead to neutron star models with substantially greater density. Themass-radius diagram includes the cases for non-strange and strange mat-ter. Changing the value of the parameter ΛV solutions with substantiallyreduced value of the transition density have been obtained [18]. This pointsHyperon Star 3077to the conclusion that the crust-core boundary moves to lower density regionleading to the models with reduced value of the non-homogenous phase. Ithas been shown that in the very nonlinear models the inclusion of hyperonsdoes not soften the EOS; on the contrary, it leads to its considerable stiffen-ing. The consequences for neutron star parameters are straightforward andappear as a considerable growth of neutron star masses. Thus one of theinevitable conclusions is that in the case of nonlinear models the inclusion ofhyperons does not result in the lowering of a neutron star mass. This is ofspecial interest when considering reported astrophysical data which indicatelarge neutron star masses.REFERENCES[1] N.K. Glendenning, Astrophys. J. 293, 470 (1985); Compact Stars, Springer–Verlag, New York 1997.[2] I. Bednarek, R. Manka, Int. J. Mod. Phys. D10, 607 (2001).[3] F. Weber, Pulsars as Astrophysical Laboratories for Nuclear and ParticlePhysics, IOP Publishing, Philadelphia 1999.[4] J. Schaffner-Bielich, A. Gal, Phys. Rev. C62, 034311 (2000).[5] J. Sladkowski, Int. J. Mod. Phys. D9, 311 (2001).[6] I. Sagert, M. Wietoska, J. Schaffner-Bielich, Ch. Sturm, J. Phys. G 35, 014053(2008).[7] D.J. Nice, E.M. Splaver, I.H. Stairs, O. Lohmer, A. Jessner, M. Kramer,J.M. Cordes, Astrophys. J. 634, 1242 (2005).[8] D. Blaschke, T. Klahn, F. Sandin, J. Phys. G 35, 014051 (2008).[9] D. Nice, 40 Years of Pulsars, Montrea 2007, conference contribution.[10] D. Barret, J.F. Olive, M.C. Miller, Mon. Not. R. Astron. Soc. 361, 855 (2005).[11] J. Cottam, F. Paerles, M. Mendez, Nature 420, 51 (2002).[12] F. Ozel, Nature 441, 1115 (2006).[13] J.E. Trümper, V. Burvitz, F. Haberl, V.E. Zavlin, Nucl. Phys. Proc. Suppl.B132, 560 (2004).[14] P. Danielewicz, R. Lacey, W.G. Lynch, Science 298, 1592 (2002).[15] Bao-An Li, L.W. Chen, C.M. Ko, Phys. Rep. 464, 113 (2008).[16] J. Piekarewicz, M. Centelles, Phys. Rev. C79, 054311 (2009).[17] M. Centelles, X. Roca-Maza, X. Vinas, M. Warda, Phys. Rev. Lett. 102,122502 (2009).[18] I. Bednarek, R. Manka, J. Phys. G 36, 095201 (2009).[19] Y. Sugahara, H. Toki, Prog. Theor. Phys. 92, 803 (1994).[20] H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001).[21] H.Q. Song, R.K. Su, D.H. Ku, W.L. Qian, Phys. Rev. C68, 055201 (2003).

Hyperon Star

https://rebus.us.edu.pl/bitstream/20.500.12128/7208/1/Bednarek_Hyperon_star.pdf

Hyperon Star

Abstract

Similar works

Full text

Available Versions

Repozytorium Uniwersytetu Śląskiego RE-BUŚ