The crust-core transition and the stellar matter equation of state

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1 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 7-28, 26 In collaboration with C. Providência, D. P. Menezes, S. Antic, S.Typel, N. Alam, B. K. Agrawal Acknowledgments:

2 Neutron stars One of the most dense objects in Universe: R~km and M~.5 M Divided in 3 main layers: N. Chamel and P. Haensel. Liv. Rev. Rel.,, 28.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.5km for a M=.4 M P t plays crucial role in fraction of I in crust of star: which also depends on crust thickness, R t star! I crust 6 3 R 6 t P t R s

3 Describing neutron stars Prescription:.EoS: P (E) for a system at given and 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!! P.B. Demorest et al, Nature 467, 8, 2

4 EoS Constrains Observations Experiments J. M. Lattimer and A. W Steiner, EPJA 5, 4, 24 Experiments T=, y p =.5 P (MeV fm -3 ) P. Danielewicz et al, Science 298, 592, 22 W. G. Lynch et al, PPNP 62, flow exp. KaoS exp / Microscopic calculations P (MeV fm -3 ) Microscopic calculations T=, neutron matter S. Gandolfi et al, PRC 85, 328, 22 K. Hebeler et al,.2 Astrophys. J. 773,, 23 Chiral EFT Monte Carlo (fm -3 ) J. M. Lattimer and M. Prakash, arxiv: [astro-ph.sr] 2

5 Choosing the EoS(s) We need unified EoS, but if we don t have it..! Choose EoS for each NS layer:! arxiv: [astro-ph.sr] 26 Outer crust EoS (BPS or HP or RHS) M(R) not affected Inner crust EoS () pasta phases?, unified core EoS? 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 () and (2) to obtain the transition densities and pressures!

6 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 3, 2, 29 C. J. Horowitz et al, PRC 7, 6586, 24

7 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 D (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.

8 Pasta phases - calculation (II) Coexistence Phase (CP) approximation: check PRC 9, 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 p µ I n = µ II 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.! p Total F and total of the system:! F = ff I +( f)f II + F e + surf + coul! p = e = y p = f I p +( f) II p P I = P II

9 Pasta phases - calculation (III) Compressible Liquid Drop (CLD) approximation: check PRC 9, The total free energy density is minimized, including the surface and Coulomb terms. The equilibrium conditions become: µ I n = µ II n, µ I p = µ II p surf f( f)( I p II p ), P I = P II surf 2 II p f( f)( I p II p )

10 Non-linear Walecka Model L = X L i = i [ µ id µ nucleons M ] i electrons em L e = e [ µ (i@ µ + ea µ ) m e ] e L = 2 L = 4 F µ F µ i=p,n non-linear mixing couplings terms: responsible for density dependence of Esym! mesons: mediation of nuclear force L i + L e + L + L + L! + L + L! mesons non-linear mixing µ m 2 s 2 3 apple L! = 4 µ µ + 2 m2 vv µ V µ + 4! g4 v(v µ V µ ) 2 L = 4 B µ B µ + 2 m2 b µ b µ L! = 3 g s g 2 v V µ V µ + 2 g 2 sg 2 v 2 V µ V µ + g s g 2 b µ b µ + g 2 sg 2 2 b µ b µ + v g 2 vg 2 b µ b µ V µ V µ

11 How to calculate transition density? courtesy: C. Providência courtesy: C. Providência ) 3 (fm p coexistence spinodal binodal equil. Y =.4 L ) Get the instability region: Dynamical spinodal or Thermodynamical spinodal 2) Intersect EoS with that boundary to get t p (fm 3 ) thermodynamical dynamical Y = n (fm ) n (fm ).6.8 PRC 82, 5587, 2 PRC 85, 5994(E), 22 For -eq. matter and T=, dyn. spinodal very coincident with TF calculation

12 Thermodynamical spinodal check PRC 74, The (free) energy curvature matrix for asymmetric NM is defined 2 F C j Stability conditions: T r(c) >,Det(C) > The spinodal is given by (T, p, n ) for which Det(C) = i.e., one of eigenvalues is negative in the region of instability and goes to zero at border: = p Tr(C) Tr(C) 2 4Det(C) = 2

13 The crust-core transition - thermodynamical spinodal approach p (fm -3 ).8 a) density-dependent models with S. Antic, S. Typel and C. Providência.6.4 D D2 T=2 MeV T=6 MeV preliminary! p (fm -3 ) b) non-linear mixing meson couplings models preliminary! F 2 F T=2 MeV T=6 MeV with N. Alam, B. K. Agrawal and C. Providência n (fm -3 ) D* models: scalar and vector self-energies depend on E: the couplings are adjusted to the optical potential in nuclear matter. Nucl. Phys. A 938, n (fm -3 ) Different mixing couplings: different L: L(F )=7 MeV, L(F2 )=46 MeV. e.g. PRC 8,

14 Dynamical spinodal check PRC 94, 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, ; PRC 85, 5994(E) 22 These small deviations are described by linearized equations of motion. Perturbed fields: F i = F i + F i Perturbed distribution function: f i = f i + f i

15 Dynamical spinodal (cont) The time evolution of the distribution functions is described by the Vlasov + {f i,h i } =, 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 +F pp L p F pn L p C pe A L p F np L n +F nn L n C ep A L e CA eel e The dynamical spinodal surface is defined by the region in space, for a given wave vector k and temperature T, limited by the surface! =. In the k= MeV limit, the thermodynamic spinodal is obtained. A 2P F p 3k 2P F n 3k 2P F e 3k p p n n e e C A = ( p, n )

16 The crust-core transition - dynamical spinodal approach PRC 94, p (fm -3 ) L=8, NL3 L=68, 4 4 L=55, 6 6 t (fm -3 ) TM NL3 Z27 TM NL3 Z27.2 T= MeV, k=75 MeV The larger L, the smaller the spinodal section. 2.The term ωρ makes the spinodal section larger compared to σρ. n (fm -3 ) but 3.Crossing of the ωρ and σρ spinodals, for a given L, occurs close to the crossing of the β-eq EoS with the spinodals P t (MeV.fm -3 ).55 t for ωρ or the σρ models about the same L (MeV).55 ωρ terms σρ terms.3.25 L<8MeV, Pt is larger for ωρ:! direct implication in I L (MeV)

17 a) effect of pasta: bps+pasta+hm bps+hm M(R) relations 2 Effect on Mmax is negligible, not true for the radius!! M (M sun ).5 L=8, NL3 L=55, 6 6 Stars with unified inner crust-core EoS (black lines) have larger (smaller) radii than configurations without inner crust (pink lines) for the NL3ωρ (NL3σρ) models.!.5 σρ give slightly larger radii than ωρ models, the differences being larger for M >~.4M. For.4M star, difference is of m.! R (km) We get the transition density from a dyn. spin. calculation. b) effect of different inner crust EoS with L close to core EoS: bps+pasta+hm bps+pasta(fsu)+hm the error on the determination of the radius is negligible for all masses! exceptions: NL3σρ6, difference of ~5 m ( 4 m) for a M (.4M ) star; NL3ωρ6, 2 m for a M star M (M sun ) 2.5 PSR J PSR J tested for other models (TM and Z27): same result!! R (km)

18 P (MeV fm -3 ) (P NM -P micro )/ (P NM -P micro )/ T=, neutron matter Chiral EFT (a) (b) NL3 6 6 TM 6 (fm -3 ) T=, y p =.5 (fm -3 ) Monte Carlo 6 Z flow exp. KaoS exp. NL3 TM Z27 Z27, cut / If we combine the 3 constrains, we get the following models: M (M sun ) exceptions: Z27 TM Z27* NL3 did not pass exp. constrain R (km) NL3 NL3 6 6 TM 6 6 Z * 6* For.4M stars, these models predict R=3.6 ±.3 km and a crust thickness of.36 ±.6km. and Z27 did not pass obs. constrain but Z27*: extra potential dependent on σ meson, that makes M* to stop decreasing above saturation density, as suggested in K. A. Maslov, E. E. Kolomeitsev, and D. N. Voskresensky, Phys. Rev. C 92, 528 (25).

19 ext. Nambu Jona-Lasinio Model PRC 93, Set of models with chiral symmetry included, unlike RMFs Since chiral symmetry is satisfied, EoS valid at higher densities! L = (i µ m) to make restoration of the chiral symmetry less abrupt +G s [( ) 2 +( i 5 ~ ) 2 ] G v ( µ ) 2 short range attraction short range repulsion G sv [( ) 2 +( i 5 ~ ) 2 ]( µ ) 2 G ( µ ~ ) 2 +( 5 µ ~ ) 2 G v ( µ ) 2 ( µ ~ ) 2 +( 5 µ ~ ) 2 density dependence of scalar coupling isospin asymmetric nuclear matter G s ( ) 2 +( i 5 ~ ) 2 ( µ ~ ) 2 +( 5 µ ~ ) 2 make the symmetry energy softer M = m 2G s s +2G sv s 2 +2G s s 2 3 nucleon effective mass

20 enjl models In this study, we used enjlx, enjlxωρy, and enjlxσρy type of models We also considered models with a current mass: enjlxm, and enjlxmσρy. To make Esym softer: *ωρ* and *σρ* models, where we fixed the Esym at ρ=. at the same value of enjlx (enjlxm), and we calculated the new Gρ, fixing the Gvρ (Gsρ) constant. Esym (MeV) enjl enjl enjl 2 enjl2 enjl2 enjl3 enjl3 Esym (MeV) enjlm enjlm enjl2m enjl2m (fm -3 ) (fm -3 ) without current mass with current mass

21 enjl models-constrains But the models need to fulfil the constrains only 2 models passed: enjl3σρ and enjl2mσρ Microscopic calculations Hebeler et al. Gandolfi et al. enjl enjl enjl2 enjl2 enjl3 enjl3 enjl2m enjl2m T=, y p =.5 Experiments P (MeV fm -3 ) 2.5 T=, neutron matter P (MeV fm -3 ) (a) Flow exp. Kaons exp. enjl enjl2 enjl3 enjlm enjl2m / (fm -3 )

22 M (M sun ) M(R) relations enjl3 psr J psr J BPS+hm BPS+pasta(CP)+hm BPS+pasta(CLD)+hm ) EoS: ) outer crust: BPS 2) inner crust: pasta from a CP or CLD calculation 3) core: hom. nucleonic matter, same model as inner crust 2) integrate TOV 3) get M(R) R (km) CP or CLD: no difference in M(R) M (M sun ) enjl3σρ: R(M=.4M )=3.22 km, with R(M=.4M )=.45km BPS+pasta+hm psr J psr J enjl3 enjl2m enjl2mσρ: R(M=.4M )=3.84 km, with R(M=.4M )=.48km R (km)

23 M(R) relations (cont) Considering hybrid stars: quark core described within SU(3) NJL model perform a Maxwell construction to get hadronic to quark transition 2.5 (a) M (M sun ) psr J psr J enjl3 +NJL enjl3 +NJL2 enjl3 +NJL3 enjl3 +NJL R (km) The deconfinement phase transition decreases maximum mass though We are still able to describe stable 2M stars with a quark core!!

24 Summary Inclusion of the inner crust EoS has strong effect on the radius of low and intermediate mass neutron stars! Unified EoS are needed! but Inner crust EoS with similar symmetry energy properties as the core EoS: effect on radii for stars with M > M is negligible! For RMF models, R(M=.4M )=3.6±.3km, with R(M=.4M )=.36±.6km. Inner-crust core unified EoS with chiral symmetry and pasta allows the description of 2 M stars, with R(M=.4M )=3.48±.64km.

25 Strong correlations of neutron star radii with the slopes of nuclear matter incompressibility and symmetry energy at saturation accep. PRC (R), arxiv: [nucl-th] set of 24 Skyrme-type effective forces and 8 RMF models, and 2 microscopic calculations, all describing 2M neutron stars. Unified EoSs for the inner-crust-core region have been built for all the phenomenological models, both relativistic and non-relativistic.. M NS (M O ) K (MeV) C(R., K ) =.655 C(R.4, K ) = R (km) PSR J PSR J ρ c (fm -3 ) 35 M (MeV) C(R., M ) =.646 C(R.4,M ) =.743 We started by calculating correlation of R with K, M and L L (MeV) C(R., L ) = R. (km) C(R.4, L ) = R.4 (km)

26 and then 5 4 K +αl (MeV) M +βl (MeV) α =.564 α =.883 C(R., K +αl ) =.92 C(R.4, K +αl ) =.85 β = C(R., M +βl ) =.945 C(R.4, M +βl ) = R. ( km) β = R.4 (km) K +αl (MeV) M +βl (MeV) C(R X, b) C(R X, b) and. b { b{ K L K +αl M L M +βl M NS (M O ). We found strong correlation of the neutron star R with linear combination of M and L, and almost independent of the neutron star mass in the range.6-.8m. This correlation can be linked to the empirical relation between R and P at a nucleonic density between -2 saturation density, and the dependence of P on K, M and L.

27 Thank you!