Problem How do relativistic models, used to build EoS of compact stars, behave at subsaturation densities? EoS at subsaturation densities/crust of compact stars: how do relativistic and Skyrme nuclear models compare? Can we constrain the temperature and isospin dependence of the nuclear EoS studying the subsaturation density behaviour of the EoS of ANM?
Compact star crust: relativistic versus Skyrme nuclear models Camille Ducoin, Philippe Chomaz, Alexandre Santos, Constança Providência, Universidade de Coimbra, Portugal The Complex Physics of Compact Stars Ladek Zdroj, 18-29 February 28
Outline Energy functionals Skyrme interactions Relativistic Mean-Field Nuclear Models Equilibrium properties Thermodynamical instabilities Dynamical instabilities Formalism Instability Region Stellar Matter Summary Conclusions
Skyrme Functional The Skyrme energy-density functional homogeneous, spin-saturated matter, no Coulomb interaction H = K + H + H 3 + H eff K = 2 2m τ H = C ρ 2 + D ρ 2 3 H 3 = C 3 ρ σ+2 + D 3 ρ σ ρ 2 3 H eff = C eff ρτ + D eff ρ 3 τ 3
Skyrme Functional: parametrizations chosen SIII - fitted to double magic nuclei(beiner et al A238 76(1974)) SGII - with spin and pairing effects and constraints on K (Van Giai et al NPA371(1982)) Sly23a - improves the isospin description (Chabanat et al NPA627 (1997)) LNS -fitted to a Brueckner-Hartree-Fock calculation extended to spin and isospin polarized homogeneous matter (Cao et al PRC 76(26)) NRAPR- fitted to the APR EoS obtained from a variational microscopic calculation (Steiner et al PhysRep 411(25))
Skyrme Functional: parametrizations chosen Model B/A ρ K E sym M /M (MeV) (fm ) (MeV) (MeV) (MeV) SIII 15.85.145 355.5 28.16.76 SGII 15.59.159 215.4 26.85.79 SLy23a 15.99.16 229.9 31.97.697 NRAPR 15.86.16 225.7 32.79.7 LNS 15.32.175 21.9 33.4.825
RMF Lagrangian for npe matter Lagrangian density L NLWM = L i + L mesons + L γ + L e, i=p,n Nucleon contribution: Li = ψ i [γ µ D µ M ] ψ i, D µ = i µ g v V µ gρ 2 τ bµ ea µ 1+τ 3 2 M = M g s φ g δ τ δ. Meson contribution L mesons = L σ + L ω + L ρ + L δ + L non linear... Electromagnetic contribution: Lγ = 1 4 F µνf µν Electron contribution: Le = ψ e [γ µ (i µ + ea µ ) m e ]ψ e
Relativistic Nuclear Models Parametrizations tested: NL3: includes non-linear σ terms (Lalazissis et al PRC55(1997)) Density Dependent Hadron Models: TW, DDME2 (Typel & Wolter, NPA656, Niksic PRC66 (22)) The coupling meson-nucleon parameters are density dependent Thermodynamical consistency: requires the inclusion of a rearrangement contribution Σ R in the self-energies NLδ, DDHδ: include the δ meson (Greco et al PRC67 (23), Gaitanos et al NPA732 (24)) Isovector channel has a saturation mechanism similar to isoscalar channel The symmetry energy at larger densities increases faster
Relativistic Nuclear Models Model B/A ρ K E sym M /M (MeV) (fm ) (MeV) (MeV) (MeV) NL3 16.3.148 272 37.4.6 NLδ 16..16 24 3.5.6 TW 16.3.153 24 32..56 DD-ME2 16.14.152 251 32.3.57 DDHδ 16.3.153 24 25.1.56
Isoscalar channel Binding energy: E/A Pressure: P = ρ 2 (E/A)/ ρ Incompressibility: K = 9 P/ ρ
Isoscalar channel E/A (MeV) ) P (MeV fm 2 15 1 5 5 1 15 SIII SGII Sly23a NRAPR LNS 2.5.1.15.2.25.3.35.4.45 1.8.6.4.2.2.4.6 ρ (fm ) P (MeV fm ) E/A (MeV) 2 15 1 5 5 1 15 2 1.5.5 NL3 TW DDME2.5.1.15.2.25.3.35.4.45 NL δ TW δ ρ (fm ).8 1.2.4.6.8.1.12.14.16.18.2 ρ (fm ) 1.5.1.15.2 ρ (fm ) 4 4 3 3 K (MeV) 2 1 K (MeV) 2 1 1 1.2.4.6.8.1.12.14.16.18.2 ρ (fm ).5.1.15.2 ρ (fm )
Density of upper spinodal border ρ s ρ (fm ) (fm ) Sly23a.12.16 SGII.11.159 SIII.98.145 RATP.13.16 NR-APR.13.161 NLS.111.175 NL3.96.148 NLδ.12.16 TW.96.153 DDHδ.95.153 ME2.99.152
Isovector channel Symmetry energy: a s = 1 2 2 F y 2, Slope of symmetry energy: L = 3ρ a s Symmetry incompressibility: K sym = 9ρ 2 a s, y = ρ 3 /ρ
Symmetry energy 5 SIII SGII Sly23a NRAPR 4 LNS (MeV) a s 3 2 1.5.1.15.2.25.3.35.4.45 ρ (fm )
Symmetry energy (MeV) a s 5 SIII SGII Sly23a NRAPR 4 LNS NL3 ME2 3 TW 2 1 NL δ TWδ.5.1.15.2.25.3.35.4.45 ρ (fm )
Slope of the symmetry energy L = 3ρ a s L (MeV) 5 45 4 35 3 25 2 SIII SGII Sly23a NRAPR LNS NL3 ME2 TW TWδ NL δ 15 1 5.5.1.15.2.25.3.35.4.45 ρ (fm )
Symmetry incompressibility K sym = 9ρ 2 a s, (MeV) 5 1 15 SIII SGII Sly23a NRAPR LNS K sym 2 25 NL3 ME2 TW TW δ NL δ.5.1.15.2.25.3.35.4.45
Isovector chemical potential µ 3 = µ n µ p = 2 F/ ρ 3 parabolic approximation: µ par 3 = 4 a s ρ 3 /rho para 3 /µ 3 µ 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 SIII SGII Sly23a LNS NRAPR NL3 TW DDME2 ρ =.5 fm para 3 /µ 3 µ 1.25 1.2 1.15 1.1 1.5 ρ =.3 fm 1.1 1.1.2.3.4.5.6.7.8.9 1 ρ /ρ 3 1.1.2.3.4.5.6.7.8.9 1 ρ /ρ 3
Thermodynamical Stability Conditions Stability condition: the free energy curvature matrix C is positive. ( 2 ) F C ij = F ij = ρ i ρ j The free energy curvature matrix: C = ( µn ρ n µ p ρ n µ n ρ p µ p ρ p Stability conditions for asymmetric nuclear matter (ANM) : T ) Tr(C) >, Det(C) >.,
Thermodynamical instability: RMF ρ (fm ) ρ (fm ) ρ (fm ).12.1.8.6.4.2.1.8.6.4.2.1.8.6.4.2 NL3 TM1 TW NL δ DDHρδ δ T= MeV T=1 MeV T=14 MeV.2.4.6.8 1 Largest differences: at finite T and large δ Avancini et al PRC7 (24)
Thermodynamical Spinodal (ρ p, ρ n ) Curvature at ρ s : 2 det C(ρ s,) ρ 2 3 = 4 ρ 4 [ a s (ρ 2 a s + 2ρa s) 2(ρa s) 2 ].12.1.8 SIII SGII Sly23a NRAPR LNS T=.6 ρ p.4.2.12.2.4.6.8.1.12 ρ (fm ) n
Thermodynamical Spinodal (ρ p, ρ n ).12.1.8 SIII SGII Sly23a NRAPR LNS NL3 TW DD ME2 T= p ρ.6.4.2.12.2.4.6.8.1.12 ρ n (fm )
Thermodynamical Spinodal (ρ p, ρ n ): meson δ.12 ) (fm p ρ.1.8.6.4 NL3 TW DD ME2 NL TW δ δ.2.2.4.6.8.1.12 ρ (fm ) n
Thermodynamical instability direction the crossing of the spinodal with the axis of symmetric matter: ρ s corresponds to the zero of the incompressibility function K(ρ) Skyrme interaction predict larger instability regions. At low densities all spinodals reach high asymmetry and cannot be distinguished.
Thermodynamical Spinodal (µ 3, ρ) (MeV) 1 5 SIII SGII Sly23a NRAPR LNS (MeV) µ 3 3 1 5 NL3 TW DD ME2 TWδ NLδ µ 5 5 1 1.2.4.6.8.1.12.2.4.6.8.1.12 ρ (fm ) ρ (fm ) (ρ,µ 3 ) spinodal: all models are distinct at all densities,
Spinodal Surface The eigenvalues of the stability matrix are given by λ ± = 1 ( ) Tr(F) ± Tr(F) 2 2 4Det(F), with eigenvectors δρ ± δρ ± i δρ ± j = λ ± F jj F ji, i, j = p, n The largest eigenvalue λ + is always positive The smallest λ becomes negative at subsaturation densities
Distillation Effect T = MeV, y p =.2, ρ p /ρ n =.25.9.8 δρ p /δρ n.7 NL3 TM1.6 NL δ TW DDH δ.5.2.4.6.8.1 ρ (fm ) Avancini et al PRC74 (26)
Distillation Effect T = MeV, y p =.2,.4 1.9 δρ p /δρ n.8.7 NL3 TW Λ=.25 no rearr Avancini et al PRC74 (26).6.2.4.6.8.1 ρ (fm )
Thermodynamical instability direction /δρ 3 δρ Distillation effect if δρ 3 /δρ <.4 for Z/A=.3.5.45.4.35.3.25.2.15.1.5 Z/A=.3, T= SIII SGII Sly23a LNS NRAPR.2.4.6.8.1.12 ρ (fm ) NL δ TW δ DDME2 NL3 TW Avancini et al PRC74 (26)
Thermodynamical instability direction Common trend: reduction of the distillation effect with density. DD-ME2 follows quite closely the Skyrme parametrization Sly23a with the smallest distillation effect TW follows the general trend of all the other models. NL3 and NLδ: distillation effect with density increases with density Multifragmentation heavy-ion collisions: isospin content of large fragments sensitive to δρ 3 /δρ
Dynamical instabilities The Relativistic Vlasov Equation The time evolution of f i is described by the Vlasov equation df i dt =, i = p, n, e f i t + {f i, h i } =, i = p, n, e
Small perturbation of the system Equilibrium state characterized by: P Fn, P Fp, P Fe Charge neutrality: P Fe = P Fp Perturbed fields: F i = F i + δf i, Perturbed distribution function: f = f + δf, δf i = {S i, f i } Generating function: S(r, p, t) = diag(s p, S n, S e ),
Linearized Equations of Motion The linearized relativistic Vlasov equation ds i dt + {S i, h i } = δh i Longitudinal fluctuations: ( Si δf j δρ i δh i ) = ( Sω,i (x) δf ω,j δρ ω,i δh ω,i ) e i(q r ωt) x = cos(p q), i = e, p, n, j = σ,ω,ρ,γ
Linearized Equations of Motion One-body hamiltonian variation Electrons [ δh e = e δa p δa ], ǫ e nucleons: relativistic models δh i = g s δφ M ǫ + δv i p δv i ǫ, i = p, n nucleons: Skyrme interactions δh i = [ ] U (1) ij + xu (2) ij δρ j j
Finite size instabilities: npe matter The instability growth ratio and direction of the unstable mode 1 1/τ ((fm/c).7.6.5.4.3.2 SIII SGII Sly23a LNS NRAPR NL3 TW ME2 ρ=.5 fm, Z/A=.3, T= Vlasov /δρ 3 δρ.5.45.4.35.3.25.2.15 NL3 TW DD ME2.1 5 1 15 2 25 3 q (MeV/c).1.5 5 1 15 2 25 3 q (MeV/c)
Instability growth rates/ Direction of instability Sly23a: the smallest growth rates of the Skyrme interactions Relativistic models: smaller growth rates, and q ranges Low q: all models similar behavior defined by electromagnetic interaction anti-distillation is model dependent ( δρ3 /δρ>.4) no anti-distillation: NL3 and DDME2 Large q behavior: defined by the finite range of the nuclear force Distillation behavior: NL3 shows the strongest distillation TW and DDME2 behave like Skyrme interactions
Stellar Matter β-equilibrium matter: neutrino free.12.1.8.6 DU SIII SGII Sly23a NRAPR LNS Y i (p, µ ).25.2.15 NL3 TW DDME2 DU DU p p, µ= Y i.4.2.1.5 p µ p, µ=.5.1.15.2.25 ρ (fm ).3.35.4.45.5.1.15.2.25.3.35.4.45 ρ (fm )
β-equilibrium matter: neutrino trapped Y i.4.35.3.25.2.15.1.5 Y l =.4 SIII SGII Sly23a LNS NRAPR.5.1.15.2.25.3.35.4.45 ρ (fm ) Y i (ν e e p).4.35.3.25.2.15.1.5 p ν NL3.5.1.15.2.25.3.35.4.45 ρ (fm ) TW, DDME2
Barionic density at µ onset model ρ (fm ) SIII.19 SGII.137 Sly23a.112 LNS.118 NRAPR.111 NL3.112 TW.115 DDME2.114 For SIII the muons disappear at ρ >.239123fm
β-equilibrium npeν e matter/ Instability region β-equilibrium EoS with/without neutrinos and the dynamical spinodal surface por npe matter 12 115 T=1 MeV (MeV) e +µ p µ t 11 15 1 95 SIII SGII Sly23a LNS NRAPR NL3 9 85 86 88 9 92 94 96 µ t (MeV) n β-equilibrium EoS without neutrinos: no crossing.
Edge of outer/inner crust Crossing of the β-equilibrium EoS with/without neutrinos with the dynamical spinodal surface por npe matter.12.1.8 = ν ρ Y l =.4 SIII SGII Sly223a LNS NRAPR NL3 cros b ρ.6.4.2 2 4 6 8 1 12 14 16 18 T (MeV)
Eos/dynamical spinodal crossing The behavior of different models follows the trend of the thermodynamical spinodals: at the larger densities and very asymmetric matter the Sl23a spinodal extends to larger asymmetries and the NL3 to smaller ones. 1 5 (MeV) 3 µ 5 1.2.4.6.8.1.12 ρ (fm )
Edge of inner crust: T = MeV Densities at the inner edge of the crust (fm ) from spinodal of pne matter for k = 8 MeV/c Y ν = Y L =.4 model ρ cross,sup ρ cross,sup SIII.1978.9296 SGII.7643.8821 SLy23a.7883.8758 LNS.7724.9579 NRAPR.7228.8828 NL3.5345.8146 TW.752.8387 DD-ME2.723.8258
Size of clusters Half-wave-length of the most unstable mode inside the instability region, along a path of β-equilibrium ) λ /2 (fm 2 18 16 14 12 1 8 6 ρ =, T= ν NL3 TW ME2 SIII SGII Sly23a LNS NRAPR ) λ /2 (fm 2 18 16 14 12 1 8 6 Y l =.4, T=1 MeV NL3 SIII SGII Sly23a LNS NRAPR 4 2 Vlasov 4 2 Vlasov.2.4.6.8.1.12 ρ (fm b ).2.4.6.8.1.12 ρ (fm b )
Most Unstable Modes Neutrino free: relativistic models predict larger clusters > 6 fm, Skyrme forces clusters with 4 fm may appear high density border: all models predict similar sizes. Neutrino trapped matter at T=1 MeV: The size of the clusters does not change much with density Mean value depends on the model, ranging from 5 fm for SIII to 8 fm for NL3 and SLy23a
Conclusions RMF/DDH versus Skyrme Equilibrium properties: larger differences in the isovector channel RMF/DDH: smaller thermodynamical spinodal regions RMF: distillation effect increases with density Skyrme/DDH: distillation effect decreases with density ρ >.1fm RMF/DDH: larger cluster in the crust Dynamic properties differ: RMF/DDH larger clusters, smaller growth rates, instabilities extend to smaller q. What do other approachs starting from NN interaction, including correlations, etc.. predict for the quantities discussed? Could transport calculations for isospin diffusion currents, isospin content of fragments, etc.. distinguish models?