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Current Status of Equation of State in Nuclear Matter and Neutron Stars

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Current Status of Equation of State in Nuclear Matter and Neutron Stars Open Issues in Understanding Core Collapse Supernovae June 22-24, 2004 Current Status of Equation of State in Nuclear Matter and Neutron Stars J.R.Stone1,2, J.C. Miller1,3 and W.G.Newton1 1 Oxford University, Oxford, UK 2Physics Division, ORNL, Oak Ridge, TN 3 SISSA, Trieste, Italy Outline 1. General properties of EOS in nuclear matter 2. Classification according to models of N-N interaction 3. Examples of EOS – sensitivity to the choice of N-N interaction 4. Consequences for supernova simulations 5. Constraints on EOS 6. High density nuclear matter (HDNM) 7. New developments Equation of State is derived from a known dependence of energy per particle of a system on particle number density: EA/(==En) or F/AF(n) I. E ( or Boltzman free energy F = E-TS for system at finite temperature) is constructed in a form of effective energy functional (Hamiltonian, Lagrangian, DFT/EFT functional or an empirical form) II. An equilibrium state of matter is found at each density n by minimization of E (n) or F (n) III. All other related quantities like e.g. pressure P, incompressibility K or entropy s are calculated as derivatives of E or F at equilibrium: 2 ∂E ()n ∂F ()n Pn()= n sn()=− | ∂n ∂T nY, p ∂∂P()nnEE() ∂2 ()n Kn()==9 18n +9n2 ∂∂nn∂n2 IV. Use as input for model simulations (Very) schematic sequence of equilibrium phases of nuclear matter as a function of density: <~2x10-4fm-3 ~2x10-4 fm-3 ~0.06 fm-3 Nuclei in Nuclei in Neutron electron gas + ‘Pasta phase’ Electron gas ~0.1 fm-3 0.3-0.5 fm-3 >0.5 fm-3 Nucleons + n,p,e,µ heavy baryons Quarks ??? (β−equilibrium) + leptons Ground state: Symmetric (SNM) - saturation density n0 0.16 fm-3 np = nn = 1/2 energy per particle E/A (n0) - 16 MeV symmetry energy as E/A(SNM)-E/A(PNM) 28-32 MeV Incompressibility K(n0) 210 – 240 MeV Pressure P(n0) 0 Excited states: Asymmetric matter – n+p+e+µ (in β-equilibrium) density dependence of : - proton fraction yp = np/n - symmetry energy S (coefficient as) - chemical potentials µn, µp, µe, µµ Pure neutron (PNM) - does not saturate (E/A > 0) To construct the energy functional we need nuclear and particle physics models: Expectation value of the total energy: ET=<+φ,( V)φ > T kinetic energy, V total potential energy of the system Φ is a Slater determinant of single particle states φi Theories Relativistic Non-relativistic Potentials Realistic Phenomenological Reid 93 } Local non-rel Skyrme Nijmegen II } Gogny } Argon v18 (A18) Nijmegen I Non-local non-rel SMO } CD Bonn Local rel NL + Realistic Phenomenological bare nucleon (OBE) density dependence Hjorth-Jensen, Phys.Rep,261,125(1995) ~20-60 adjustable parameters 10-15 adjustable parameters Several thousand data points Several hundred data points Free nucleon-nucleon scattering Finite nuclei (g.s.) and properties and properties of deuteron of nuclear matter at saturation Renormalization needed density for use in dense medium ------------ Brueckner-Hartree-Fock Hartree-Fock, Dirac-Hartree, Dirac-Brueckner-Hartree-Fock Dirac-Hartree-Fock etc to Variational methods, etc to calculate selfconsistently calculate G-matrix equilibrium potential energy EOS for SNM and PNM (usually) EOS for SNM, PNM and + inhomogeneous NM Further parameterization needed to obtain inhomogeneous NM ------------ Examples Non-relativistic Hartree method - phenomenological Skyrme potential Vautherin and Brink, PRC 5, 626 (1972) (2) (3) Vv=+∑∑ij,,vij,k ij<<ij<k p2 11 E =<∑∑i ||i >+ <ij |v12 |ij >ASA+∑<ijk ||v123| ijk >S ii22m ,,j6ij,k V12 (v123) parameterized two (three)-body potential Relativistic Dirac-Brueckner method – realistic Bonn potential Li et al., PRC 45, 2782 (1992) 2 2 mm%%+ pi 1 m% Ei=+∑∑<j|(G% z%)|ij−ji>−m ik≤<ffEE%%ii2 i, jk E%j For notation see the above reference G-matrix Equation of State of Akmal et al. PRC 58, 1804 (1998) thought of as the most complete study to date of HDNM Potentials: A18 (two-body): UIX (three-body) static, long range one-pion exchange static, long-range two-pion exchange + + PHENOMENOLOGICAL medium and PHENOMENOLOGICAL medium short range part dependent on 18 range repulsive term two-body operators + Relativistic boost correction to the two body N-N interaction + 2 -γ ρ Ad hoc density dependent term γ2ρ e 3 which provides 25% (~4 MeV) correction to the E/A of symmetric nuclear matter to reach the empirical value 16 MeV For densities <0.1 fm-3 EOS by Lorenz et al., PRL 70,379 (1993) is used. Akmal et al., E/A as a function of baryon number density At saturation density 0.16 fm-3 the expected value of E/A=16 MeV Akmal et al., Sensitivity of density dependence of E/A to the form of a realistic potential Symmetric nuclear matter Pure neutron matter Density dependence of E/A in nuclear matter for phenomenological Skyrme interactions (87 tested) 80 SkO (I) PNM BEM MSk7 (III) 60 SNM 40 SkX (II) 20 Energy per partlicle [MeV] 0 −20 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5 n [fm−3] n [fm−3] n [fm−3] 27 33 27 J.R.Stone et al., PRC 68, 034324 (2003) LATTIMER-SWESTY EOS Current standard in supernova modelling Nucl.Phys. A535, 331 (1991) and http://sbast.ess.sunysb.edu/dswesty/lseos.ftm Matter consisting from electrons, positrons, photons, Free nucleons, alpha particles (representing light nuclei) and single species for heavy nuclei. Compressible liquid drop model non-relativistic fermion kinetic energies, schematic two-body density dependent interaction (based on the Skyrme model) schematic model for multi-body interactions FINITE TEMPERATURE Temperature and effective mass: S = 1 , Yp = 0.3 m*/m ↓ T↑ Lattimer-Swesty: Three nuclear forces represented by K = 180, 220 and 375 MeV with m*/m = 1 Temperature is independent from the nuclear force for densities higher than ~0.1 fm-3, i.e for pure nuclear matter Onsi et al., PRC55,3139 (1997) Fully temperature dependent ETFSI with Skyrme interactions RATP (m*/m=0.67), SkSC6 (m*/m=1) Temperature is dependent on the force in the whole region of densities Lattimer-Swesty: Pressure/baryon as a function of ln n: Significant differences in nuclear matter phase in dependence on the nuclear force. Onsi et al: Pressure as a function of n n[fm-3] LS Onsi 0.1 13 16 0.15 16 23 Adiabatic index: ∂ ln P Γ= | ∂ ln n sY, p Very important in hydrodynamic calculations A measure of stiffness of the matter at phase boundaries Reflects nature and treatment of phase changes Examples I. Akmal et al : transition between normal low density matter (LDP) and high density matter with condensed pions (HDP) II. LS: transition from ‘pasta’ phase to pure nuclear matter phase ( proton rich nuclei in neutron vapor + α particles, + e neutron bubbles + α in dense proton rich matter) } III. Onsi et al: transition between droplet and bubble phase transition between bubble and homogeneous matter Adiabatic index as calculated by Akmal et al, PRC 58, 1804 (1998) Discontinuity at n~0.2 fm-3 represents a LDP -> HDP transition B - bag constant for quark admixtures Adiabatic index as a function of ln n (LS) and n (Onsi) for s=1 and Yp=0.3 LS Onsi et al. droplet-bubble bubble ↓ NM ‘pasta’ ↓ NM Note different x and y-axis scales Values of Γ differ from Akmal et al Bound nuclei in e + n gas beyond Wigner-Seitz model Magierski and Heenen, PRC 65, 045804 (2002) Shell effects in inhomogeneous asymmetric nuclear matter lead to a complicated pattern of density dependent phases and phase transitions not taken into account by previous models A STATE-OF-ART TREATMENT OF THE ‘PASTA’ PHASE "Core-Collapse Supernovae at the Threshold" H.-T. Janka, R. Buras, K. Kifonidis, A. Marek and M. Rampp (Proceedings IAU Coll.~192, Valencia, Spain, April 22--26, 2003, Eds. J.M.~Marcaide ad K.W.~Weiler, Springer Verlag) Shen et al, NP A637, 435 (1998) RMF Hillebrant and Wolff, 1985 – Skyrme T-dependent Hartree-Fock Shock radius Electron neutrino luminosity Sensitivity of predicted nuclear matter parameters to details of EOS Search for constraints to narrow down the variety of models. Example of an efficient constraint for a non-relativistic Skyrme potential in Hartree-Fock approximation Density dependence of the symmetry energy 1 n SE()nn==(,Y0)−E(n,Y=) where Y=p pp2 pn or 2 11d E 11/2 EE(,nYpp)==(nY, )+( 2 )(n)(Yp−) 22dYp 2 Bao-An Li, PRL 88, 192701 (2002) SNM ground state at all densities ρ //ρoo⇔ nn nn− δ = np nnnp+ PNM δ = 0 ground symmetric state δ = 1 at high pure neutron densities Density dependence of the symmetry energy is the main criterion for distinction between Skyrme parameterizations Group I Group II Group III 100 50 50 SkO (I) 80 25 25 60 0 0 [MeV] s a 40 SkX (II) MSk7 (III) −25 −25 20 0 −50 −50 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5 n [fm−3] n [fm−3] n [fm−3] SkO, SkX, MSk7 J.R.Stone at al., PRC68, 034324 (2003) examples of Skyrme potentials Gravitational mass versus radius for T=0 non-rotating neutron stars calculated using Skyrme EOS based on parameterizations I, II and III 2 SIII(II) * SkM (II) 1.6 SkX(II) Skz4(II) No reasonable neutron stars 1.2 ] are produced by sun Skyrme M [M 0.8 SLy4(I) interactions SkO(I) of group III SkI3(I) 0.4 SkI1(I) 0 8 101214168 10121416 R [km] R[km] Constraint from maximum mass of a neutron star Nice et al., Pulsar+white dwarf binary – Arecibo telescope – timing analysis orbital period 0.26 days –> 1.6 – 2.8 Msolar with 95% confidence If this observation is confirmed, a large number of EOS would be eliminated Structure of HDM would be questioned as the presence of hyperons and quarks is known to lower the maximum mass of a neutron star.
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