The crust-core transition and the stellar matter equation of state Helena Pais CFisUC, University of Coimbra, Portugal Nuclear Physics, Compact Stars, and Compact Star Mergers YITP, Kyoto, Japan, October 17-28, 2016 In collaboration with C. Providência, D. P. Menezes, S. Antic, S.Typel, N. Alam, B. K. Agrawal Acknowledgments:
• One of the most dense objects in Universe: Neutron stars R~10km and M~1.5 M � . l e s n e 8 a 0 H 0 2 . P , 0 d 1 n a , 1 l 1 e m . l e a R h C . v e . N R . v i L • Divided in 3 main layers: 1.Outer crust 2.Inner crust 3.Core Crust-core transition important : • The choice of inner crust EoS and the matching to the core EoS can be critical : Variations have been found of 0.5km for a M=1.4 star! M � R 6 I crust ∼ 16 π t P t • plays crucial role in fraction of I in crust of star: P t R s 3 which also depends on crust thickness , R t
Describing neutron stars P . B . D e m o r e s t e t a l , N a t u r e 4 6 7 , 1 0 8 1 , 2 0 1 0 Prescription: 1.EoS: for a system at given P ( E ) and ρ T 2.Compute TOV equations 3.Get star M(R) relation Problem: Which EoS to choose? Many EoS models in literature: •Phenomenological models (parameters are fitted to nuclei properties): RMF, Skyrme… •Microscopic models (starts from n-body nucleon interaction): (D)BHF, APR… Solution: Need Constrains!!
EoS Constrains •Observations J. M. Lattimer and A. W Steiner, •Experiments EPJA 50, 40, 2014 Experiments T=0, y p =0.5 100 P (MeV fm -3 ) P. Danielewicz et al , 10 Science 298, 1592, 2002 W. G. Lynch et al , PPNP 62, 427 2009 flow exp. KaoS exp. 1 1 2 3 4 � / � 0 •Microscopic calculations 3 Microscopic calculations 2 T=0, neutron matter P (MeV fm -3 ) 1 S. Gandolfi et al , J. M. Lattimer and M. Prakash, 0.6 PRC 85, 032801, 2012 arXiv: 1012.3208 [astro-ph.SR] 2010 0.4 K. Hebeler et al , Astrophys. J. 773,11, 2013 0.2 Chiral EFT Monte Carlo 0.1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 � (fm -3 )
Choosing the EoS(s) We need unified EoS, but if we don’t have it.. ! Choose 1 EoS for each NS layer: arXiv:1604.01944 [astro-ph.SR] 2016 ! •Outer crust EoS (BPS or HP or RHS) M(R) not affected pasta phases ?, unified core EoS ? •Inner crust EoS (1) •Core EoS homogeneous matter and then •Match OC EoS at the neutron drip with IC EoS •Match IC EoS at crust-core transition (2) with Core EoS We are going to focus on (1) and (2) to obtain the transition densities and pressures!
The pasta phases •Competition between Coulomb and nuclear forces leads to frustrated system •Geometrical structures, the pasta phases , evolve with density crust-core transition until they melt •Criterium: pasta free energy must be lower than the correspondent hm state QMD calculations: G. Watanabe et al , PRL 103, 121101, 2009 C. J. Horowitz et al , PRC 70, 065806, 2004
Pasta phases - calculation (I) • Thomas-Fermi (TF) approximation: • Nonuniform npe matter system described inside Wigner-Seitz cell : ! ! Sphere, cilinder or slab in 3D (spherical symmetry), 2D (axial symmetry ! ! ! around z axis) and 1D (reflexion symmetry). ! • Matter is assumed locally homogeneous and, at each point, its density is determined by the corresponding local Fermi momenta. ! • Fields are assumed to vary slowly so that baryons can be treated as moving in locally constant fields at each point. ! • Surface effects are treated self-consistently . ! • Quantities such as the energy and entropy densities are averaged over the cells . The free energy density and pressure are calculated from these two thermodynamical functions.
Pasta phases - calculation (II) check PRC 91, 055801 2015 • Coexistence Phase (CP) approximation: • Separated regions of higher and lower density : pasta phases , and a background nucleon gas . ! • Gibbs equilibrium conditions : for : ! T = T I = T II µ I p = µ II µ I n = µ II P I = P II ! p n • Finite size effects are taken into account by a surface and a Coulomb terms in the energy density, after the coexisting phases are achieved . ! • Total and total of the system: F ρ p ! F = f F I + (1 − f ) F II + F e + ✏ surf + ✏ coul ! ρ p = ρ e = y p ρ = f ρ I p + (1 − f ) ρ II p
Pasta phases - calculation (III) check PRC 91, 055801 2015 • Compressible Liquid Drop (CLD) approximation: The total free energy density is minimized, including the surface and Coulomb terms. The equilibrium conditions become: µ I n = µ II n , ✏ surf µ I p = µ II p ) , p − f (1 − f )( ⇢ I p − ⇢ II ⇣ 1 ⇢ II 2 ↵ + 1 @� P I = P II − ✏ surf ⌘ p @ f − 2 � f (1 − f )( ⇢ I p − ⇢ II p )
Non-linear Walecka Model mesons: mediation of nuclear force X L = L i + L e + L γ + L σ + L ω + L ρ + L σωρ i = p,n mesons em electrons nucleons non-linear mixing couplings L i = ¯ ψ i [ γ µ iD µ − M ∗ ] ψ i ✓ ◆ L σ = 1 s φ 2 − 1 3 κφ 3 − 1 ∂ µ φ∂ µ φ − m 2 12 λφ 4 ψ e [ γ µ ( i ∂ µ + eA µ ) − m e ] ψ e L e = ¯ 2 L ω = − 1 4 Ω µ ν Ω µ ν + 1 v V µ V µ + 1 L γ = − 1 2 m 2 4! ξ g 4 v ( V µ V µ ) 2 4 F µ ν F µ ν L ρ = − 1 4 B µ ν · B µ ν + 1 2 m 2 ρ b µ · b µ non-linear mixing couplings terms: responsible for density dependence of Esym! v φ V µ V µ + Λ 2 σ g 2 v φ 2 V µ V µ + Λ 1 σ g s g 2 L σωρ = Λ 3 σ g s g 2 s g 2 ρ φ b µ · b µ ρ φ 2 b µ · b µ + Λ v g 2 + Λ σ g 2 s g 2 v g 2 ρ b µ · b µ V µ V µ
How to calculate transition density? courtesy: C. Providência courtesy: C. Providência 1) Get the instability region: 0.08 0.08 or binodal thermodynamical • Dynamical spinodal c o 0.06 e 0.06 � x − equil. i p (fm ) s ) t • Thermodynamical spinodal − 3 e − 3 n dynamical (fm c spinodal e 0.04 0.04 Y =0.4 2) Intersect EoS with L p � � that boundary to get ρ t 0.02 0.02 Y =0 � 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 − 3 � (fm ) � − 3 (fm ) n n PRC 82, 055807, 2010 PRC 85, 059904(E), 2012 For -eq. matter and β T=0, dyn. spinodal very coincident with TF calculation
Thermodynamical spinodal check PRC 74, 024317 2006 •The (free) energy curvature matrix for asymmetric NM is ⇣ ∂ 2 F defined by: ⌘ C = ∂ρ i ∂ρ j •Stability conditions: Tr ( C ) > 0 , Det ( C ) > 0 •The spinodal is given by ( T, ρ p , ρ n ) for which Det ( C ) = 0 i.e., one of eigenvalues is negative in the region of instability and goes to zero at border : λ − = 1 ⇣ ⌘ p Tr ( C ) − Tr ( C ) 2 − 4 Det ( C ) = 0 2
The crust-core transition - thermodynamical spinodal approach b) non-linear mixing meson a) density-dependent models couplings models p r e l i m i n 0.08 a a r c i y ! n ê d p i r e v l o i m r i F 2 � P n D1 a C . r y 0.08 ! d n a e l p y T S . F � , D2 c i n t A . S h t w i T=12 MeV 0.06 T=12 MeV T=6 MeV 0.06 T=6 MeV with N. Alam, B. K. Agrawal � p (fm -3 ) � p (fm -3 ) and C. Providência 0.04 0.04 0.02 0.02 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 � n (fm -3 ) � n (fm -3 ) D* models : scalar and vector Different mixing couplings: self-energies depend on E: the different L: L(F � )=70 MeV , couplings are adjusted to the L(F2 � )=46 MeV . optical potential in nuclear e.g. PRC 81, 034323 2010 Nucl. Phys. A 938, 92 2015 matter.
Dynamical spinodal c h e c k P R C 9 4 , 0 1 5 8 0 8 2 0 1 6 •Dynamical instabilities are given by collective modes that correspond to small oscillations around equilibrium state. •Very good tool to estimate crust-core transition in cold neutrino-free neutron stars. check PRC 82, 055807 2010; PRC 85, 059904(E) 2012 •These small deviations are described by linearized equations of motion. •Perturbed fields: F i = F i 0 + δ F i •Perturbed distribution function: f i = f i 0 + δ f i
Dynamical spinodal (cont) •The time evolution of the distribution functions is described by the Vlasov equation : ∂ f i ∂ t + { f i , h i } = 0 , i = p, n, e semiclassical approach, that is a good approximation to t-dependent Hartree-Fock eqs at low energies •We get a set of equations for the fields and particles, whose solutions form a complete set of eigenmodes, that lead to the following matrix: 2 P Fp C pe δρ p 1 + F pp L p F pn L p A L p 3 k ρ p F np L n 1 + F nn L n 2 P Fn δρ n 0 = 0 ρ n 3 k C ep 1 − C ee 2 P Fe A L e A L e δρ e 0 3 k ρ e •The dynamical spinodal surface is defined by the region in ( ρ p , ρ n ) space, for a given wave vector k and temperature T , limited by the ω = 0 . surface •In the k =0 MeV limit, the thermodynamic spinodal is obtained.
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