Symmetry energy and the neutron star core-crust transition with Gogny forces

Symmetry energy and the neutron star core-crust transition with Gogny forces Claudia Gonzalez-Boquera, 1 M. Centelles, 1 X. Viñas 1 and A. Rios 2 1 Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB), Facultat de Física, Universitat de Barcelona, Spain 2 Department of Physics, Faculty of Engineering and Physical Sciences, University of Surrey, United Kingdom ECT* Workshop 5-9 June 2017

Introduction Introduction Aim of this work: Study the Core Crust Transition in NSs and their crustal properties. Previous results of the core-crust transition using Skyrme interactions and RMF models, but not with Gogny interactions. Core-crust transition studied from the core to the crust. Different approaches: dynamical method, thermodynamical method. The essential ingredient is the EoS of Asymmetric Nuclear Matter. Sometimes, it is approximated by a Taylor expansion in terms of the isospin asymmetry δ = (ρ n -ρ p )/ρ.

The Gogny Interaction The Gogny Interaction The Gogny two-body effective nuclear interaction in a homogeneous system reads as 2 V (r1, r2 ) = X (Wi + Bi Pσ Hi Pτ Mi Pσ Pτ ) e r2 µ i i=1,2 + t3 (1 + x3 Pσ ) ρα (R)δ(r) We have used several representative Gogny forces: D1, D1S, D1M, D1N, D250, D260, D280 and D300.

Asymmetric Nuclear Matter The Asymmetric Nuclear Matter (ANM): Theory I From V (r1, r2 ), we can obtain the Equation of State (EoS), Eb (ρ, δ), in terms of the total nuclear density ρ = ρn + ρp and the isospin asymmetry parameter, δ = (ρn ρp )/ρ. I We have used the exact Eb (ρ, δ) to perform the calculations. Also, the EoS can be expressed as a Taylor expansion around δ = 0: Eb (ρ, δ) = Eb (ρ, δ = 0) + Esym,2 (ρ) δ 2 +Esym,4 (ρ) δ 4 +... + Esym,2k (ρ) δ 2k +O(δ 2k +2 ),

Asymmetric Nuclear Matter The Asymmetric Nuclear Matter (ANM): Theory I Energy per baryon in symmetric nuclear matter. Eb (ρ, δ) = Eb (ρ, δ = 0) + Esym,2 (ρ) δ 2 +Esym,4 (ρ) δ 4 +... + Esym,2k (ρ) δ 2k +O(δ 2k +2 ),

Asymmetric Nuclear Matter The Asymmetric Nuclear Matter (ANM): Theory I I Energy per baryon in symmetric nuclear matter. Esym,2 (ρ) = 1 2 Eb (ρ, δ) 2! δ 2 Eb (ρ, δ) = Eb (ρ, δ = 0) + Esym,2 (ρ) δ 2 +Esym,4 (ρ) δ 4 +... + Esym,2k (ρ) δ 2k +O(δ 2k +2 ), δ=0

Asymmetric Nuclear Matter The Asymmetric Nuclear Matter (ANM): Theory I I Energy per baryon in symmetric nuclear matter. Esym,2 (ρ) = 1 2 Eb (ρ, δ) 2! δ 2 Eb (ρ, δ) = Eb (ρ, δ = 0) + Esym,2 (ρ) δ 2 +Esym,4 (ρ) δ 4 +... + Esym,2k (ρ) δ 2k +O(δ 2k +2 ), I Esym,2k (ρ) = 1 2k Eb (ρ, δ) (2k )! δ 2k δ=0 δ=0

Asymmetric Nuclear Matter The Asymmetric Nuclear Matter (ANM): Theory Eb (ρ, δ) = Eb (ρ, δ = 0) + Esym,2 (ρ) δ 2 +Esym,4 (ρ) δ 4 +... + Esym,2k (ρ) δ 2k +O(δ 2k +2 ), The slope of the symmetry energy is defined as L = 3ρ0 Esym,2 (ρ) ρ ρ=ρ0 with ρ0 ' 0.16 fm 3 the saturation density.

Asymmetric Nuclear Matter The Asymmetric Nuclear Matter (ANM): Theory It is also common to use the empirical parabolic approximation (PA) in order to estimate the symmetry energy. PA Eb (ρ, δ) ' Eb (ρ, δ = 0) + Esym (ρ)δ 2 PA This Esym (ρ) corresponds to the difference between the energy in pure neutron matter and the energy in symmetric nuclear matter: PA Esym (ρ) = Eb (ρ, δ = 1) Eb (ρ, δ = 0)

Asymmetric Nuclear Matter Taylor expansion coefficients up to 6 th order (ρ) (MeV) E PA sym E sym,4 (ρ) (MeV) 45 40 35 30 25 20 15 10 5 E PA sym D1 D1S D1M D1N D250 D260 D280 D300 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ρ (fm 3 ) 1.0 D1 D1S 0.8 D1M D1N D250 D260 0.6 D280 D300 0.4 0.2 E sym,4 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ρ (fm 3 ) E sym,2 (ρ) (MeV) E sym,6 (ρ) (MeV) 45 40 35 30 25 20 15 10 5 E sym,2 D1 D1S D1M D1N D250 D260 D280 D300 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ρ (fm 3 ) 0.5 D1 D1S 0.4 D1M D1N D250 D260 0.3 D280 D300 0.2 0.1 E sym,6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ρ (fm 3 )

Asymmetric Nuclear Matter β-stable Nuclear Matter The homogeneous npe matter in the core is in β-equilibrium, and therefore the chemical potentials obey the condition: µnp µn µp = µe with µq = H ρq We have performed the calculations with the full EoS. Moreover, if the Taylor expansion of Eb (ρ, δ) is used, we get Eb (ρ, δ) = 4δEsym,2 (ρ) + 8δ 3 Esym,4 (ρ) δ + 12δ 5 Esym,6 (ρ) + O(δ 7 ) = µe. µ n µp = 2

Asymmetric Nuclear Matter The Thermodynamical Method The thermodynamical method is driven by the mechanical and chemical stability conditions in the core P v >0 and µnp µnp q > 0. v For nuclear matter in β-equilibrium, and treating the electrons as a free Fermi gas, the stability condition implies 2ρ 2 2 2 1 Eb (ρ, xp ) 2 Eb (ρ, xp ) Eb (ρ, xp ) Eb (ρ, xp ) + ρ2 ρ >0 ρ ρ2 ρ xp xp2 where xp = (1 δ)/2 is the proton fraction.

Neutron Star Core-Crust Transition Neutron Star Core-Crust Transition: Density 0.13 0.094 ρ t 0.118 ρ t (fm 3 ) 0.12 0.11 D1 D260 D300 D250 D1S r = -0.8800 up to δ 2 up to δ 4 up to δ 6 EXACT 0.10 D1M D1N D280 15 20 25 30 35 40 45 50 L (MeV)

Neutron Star Core-Crust Transition Neutron Star Core-Crust Transition: Asymmetry and Pressure δ t 0.945 0.940 0.935 0.930 0.925 0.920 0.915 0.910 0.905 D260 D1 up to δ 2 up to δ 4 up to δ 6 EXACT D1S D1M D250 D300 15 20 25 30 35 40 45 50 L (MeV) D1N 0.909 δ t 0.931 r= 0.2478 D280 P t (MeV fm 3 ) 0.9 0.8 0.7 0.6 0.5 0.4 D260 D1 D1S D250 D300 D1M 15 20 25 30 35 40 45 50 L (MeV) D1N up to δ 2 up to δ 4 up to δ 6 EXACT D280 r= -0.0315 0.399 P t 0.665

Neutron Stars Structure Bulk properties of neutron stars: Mass and Radius Integration of the TOV equations, using the exact β-equilibrium EoS. Use of a polytropic EoS for the inner crust P=a+bɛ 4/3. SLy unified EoS of DH. All Gogny functionals provide maximum NS masses below the observational limit (M 2M ): D1M: 1.76M, D280: 1.66M. Only D1M and D280 can generate masses above the canonical M 1.4M. Gogny forces have not been fit to reproduce highdensity matter.

Neutron Stars Structure Bulk properties of neutron stars: Moment of Inertia I computed from the static mass distribution and gravitational potentials in the TOV equations. Moments of inertia are relatively small (below SLy). Case of D1M and D280, I max 1.3-1.4 10 45 g cm 2, below the typical value of I max 2 10 45 g cm 2. Dimensionless quantity I/MR 2 vs. compactness χ = GM/R. D1M results agree with the SLy results.

Neutron Stars Structure Crustal Properties Gogny interactions give good estimates for finite nuclei. Study of the extrapolation to neutron rich matter. Crust could be well described by them. D280 provides a thicker and heavier crust than D1M and SLy. D280 has higher I crust /I (%) than the other models. D1M overall comparable to SLy results.

Conclusions Conclusions If we look at the results for the transition properties using different orders of approximation, we observe that even when adding terms of higher orders the results may still be rather far from the ones calculated with the exact EoS. We find that Gogny interactions predict an anticorrelation in the transition density values with L, whereas the asymmetry and pressure do not present correlations with the slope parameter L. Only the D1M, D1N, D280 and D300 forces provide numerically stable solutions of the TOV equations. Some of the bulk stellar properties predicted by the Gogny forces are incompatible with observations. In comparison with SLy results, D280 and D1M interactions seem to provide overall realistic results for the crust.

Conclusions Thank you for your time. References 1. J. Xu, L.-W. Chen, B.-A. Li and H.-R. Ma, Astrophys. J., 697, 1549 (2009) 2. R. Sellahewa and A. Rios, Phys. Rev. C, 90, 054327 (2014) 3. C. C. Moustakidis, Phys. Rev. C, 86, 015801 (2012) 4. T. R. Routray, X. Viñas, D. N. Basu, S. P. Pattnaik, M. Centelles, L. Robledo and B. Behera, J. Phys. G, 43, 105101 (2016) 5. C. C. Moustakidis, T. Nikšić, G. A. Lalazissis, D. Vretenar and P. Ring, Phys. Rev. C, 81, 065803 (2010) 6. F. Douchin and P. Haensel, Astron. Astrophys. 380, 151 (2001)

Back up slides

Saturation properties of nuclear matter studied using Gogny interactions. The saturation density ρ 0 has units of fm 3, and all other properties have units of MeV. Force D1 D1S D1M D1N D250 D260 D280 D300 ρ 0 0.167 0.163 0.165 0.161 0.158 0.160 0.153 0.156 E 0 16.31 16.01 16.03 15.96 15.80 16.26 16.33 16.22 K 0 229.37 202.88 224.98 225.65 249.41 259.49 285.20 299.14 E sym,2 (ρ 0 ) 30.70 31.13 28.55 29.60 31.54 30.11 33.14 31.23 E sym,4 (ρ 0 ) 0.76 0.45 0.69 0.21 0.43 1.20 1.18 0.80 E sym,6 (ρ 0 ) 0.20 0.16 0.24 0.15 0.16 0.27 0.29 0.20 L 18.36 22.43 24.83 33.58 24.90 17.57 46.53 25.84 L 4 1.75 0.52 1.04 1.96 0.33 4.73 4.36 2.62 L 6 0.46 0.08 0.42 0.08 0.09 0.99 1.19 0.63 Esym PA (ρ 0 ) 31.91 31.95 29.73 30.14 32.34 31.85 35.89 32.44 L PA 21.16 22.28 24.67 31.95 24.94 24.33 53.25 29.80

Asymmetry and Pressure in β-stable Nuclear Matter δ (ρ) 1.00 0.98 0.96 0.94 0.92 0.90 0.88 up to δ 2 up to δ 4 up to δ 6 EXACT Parabolic D1S D280 P(ρ) (MeV fm 3 ) 100 10 1 0.1 0.01 D280 up to δ 2 up to δ 4 up to δ 6 EXACT Parabolic D1S 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ρ (fm -3 ) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ρ (fm 3 ) D1S: L = 22.43 MeV. D280: L = 46.53 MeV.

Comparison with Skyrme models ρ t (fm -3 ) 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 r = -0.9405 SKYRME GOGNY 0.05 0 20 40 60 80 100 120 140 L (MeV) P t (MeV fm -3 ) 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 SKYRME GOGNY r= -0.6494 0.20 0 20 40 60 80 100 120 140 L (MeV)

DERIVATION OF THE THERMODYNAMICAL METHOD In order to obtain the transition between the core and the crust of the Neutron Star, we use the thermodynamical method, which presents the following mechanical and chemical stability conditions: ( ) P > 0 (1) v µ np and ( µnp q ) v > 0, (2) This method is the long-wavelength limit of the dynamical method, which requires the convexity of the energy per particle in the single phase, when neglecting the Coulomb interaction.

DERIVATION OF THE THERMODYNAMICAL METHOD First we study the mechanical stability condition. Considering the electrons as a free Fermi gas, the pressure given by them is P e = µ 4 e/(12π 2 ). Therefore, when calculating the derivative of the total pressure with respect to v it gives no contribution, and Eq. (1) can be rewritten as ( ) Pb > 0. (3) v µ np

DERIVATION OF THE THERMODYNAMICAL METHOD Knowing that in β-stable matter the proton fraction is a function of the density, x p (ρ), and that µ np = 2 E b / δ = E b / x p, we can rewrite the mechanical stability condition as ( ) [ Pb = ρ 2 2ρ E b(ρ, x p ) v µ np ρ ( ρ 2 E b (ρ,x p) ρ x p ) 2 2 E b (ρ,x p) x 2 p + ρ 2 2 E b (ρ, x p ) ρ 2 > 0. (4)

DERIVATION OF THE THERMODYNAMICAL METHOD For the chemical stability condition in Eq. (2), we know that the chemical charge is q = x p ρ e /ρ, where ρ e = µ 3 e/(3π 2 ) is the density of electrons. Then, we can express the inequality in Eq. (2) as ( ) [ q = µ np v 2 E b (ρ, x p ) x 2 p ] 1 + µ2 e π 2 > 0. (5) ρ Given that Eq. (5) is usually valid, the stability condition for β-stable matter becomes V ther = 2ρ E b(ρ, x p ) + ρ 2 2 E b (ρ, x p ) ρ ρ 2 ( ) 2 ( ) 1 ρ 2 E b (ρ, x p ) 2 E b (ρ, x p ) > 0. (6) ρ x p x 2 p

DERIVATION OF THE THERMODYNAMICAL METHOD Rewriting the condition V ther in terms of the Taylor expansion in δ one obtains V ther = ρ 2 2 E b (ρ, δ = 0) ρ 2 + 2ρ E b(ρ, δ = 0) ρ + ( δ 2k ρ 2 2 E sym,2k (ρ) ρ 2 + 2ρ E ) sym,2k(ρ) ρ 2k ( ρ 2 2kδ 2k 1 E ) 2 sym,2k(ρ) ρ 2k [ 1 2k(2k 1)δ 2k 2 E sym,2k (ρ)] > 0. 2k (7)

DERIVATION OF THE THERMODYNAMICAL METHOD V ther (ρ) (MeV) 8 6 4 2 0-2 -4-6 -8-10 D280 D1S up to δ 2 up to δ 4 up to δ 6 EXACT Parabolic 0.04 0.06 0.08 0.10 0.12 0.14 0.16 ρ (fm 3 )

TRANSITION VALUES. Force D1 D1S D1M D1N D250 D260 D280 D300 δt δ2 0.9215 0.9199 0.9366 0.9373 0.9167 0.9227 0.9202 0.9190 δt δ4 0.9148 0.9148 0.9290 0.9336 0.9119 0.9136 0.9127 0.9128 δt δ6 0.9127 0.9129 0.9265 0.9321 0.9101 0.9112 0.9110 0.9110 δt exact 0.9106 0.9111 0.9241 0.9310 0.9086 0.9092 0.9110 0.9096 δt PA 0.9152 0.9142 0.9296 0.9327 0.9111 0.9153 0.9136 0.9134 ρ δ2 t 0.1243 0.1141 0.1061 0.1008 0.1156 0.1228 0.1001 0.1161 ρ δ4 t 0.1222 0.1129 0.1061 0.0996 0.1143 0.1198 0.0984 0.1145 ρ δ6 t 0.1211 0.1117 0.1053 0.0984 0.1131 0.1188 0.0973 0.1136 ρ exact t 0.1176 0.1077 0.1027 0.0942 0.1097 0.1159 0.0938 0.1109 ρ PA t 0.1222 0.1160 0.1078 0.1027 0.1168 0.1168 0.0986 0.1142 Pt δ2 0.6279 0.6316 0.3325 0.4880 0.7032 0.5891 0.6984 0.6775 Pt δ4 0.6479 0.6239 0.3531 0.4676 0.6906 0.6482 0.7169 0.6998 Pt δ6 0.6452 0.6156 0.3553 0.4581 0.6810 0.6508 0.7051 0.6955 Pt exact 0.6184 0.5817 0.3391 0.4164 0.6464 0.6273 0.6492 0.6647 Pt PA 0.6853 0.6725 0.3985 0.5172 0.7367 0.6808 0.7667 0.6777

Neutron Star Core-Crust Transition: Density 0.13 0.094 ρ t 0.118 ρ t (fm 3 ) 0.12 0.11 D1 D260 D300 D250 D1S r = -0.8800 up to δ 2 up to δ 4 up to δ 6 EXACT Parabolic 0.10 D1M D1N D280 15 20 25 30 35 40 45 50 L (MeV)

Neutron Star Core-Crust Transition: Asymmetry and Pressure δ t 0.945 0.940 0.935 0.930 0.925 0.920 0.915 0.910 0.905 D260 D1 up to δ 2 up to δ 4 up to δ 6 EXACT Parabolic D1S D1M D250 D300 15 20 25 30 35 40 45 50 L (MeV) D1N 0.909 δ t 0.931 r= 0.2478 D280 P t (MeV fm 3 ) 0.9 0.8 0.7 0.6 0.5 0.4 D260 D1 D1S D250 D300 D1M 15 20 25 30 35 40 45 50 L (MeV) D1N up to δ 2 up to δ 4 up to δ 6 EXACT Parabolic D280 r= -0.0315 0.399 P t 0.665

E b (ρ, δ) of ANM, obtained using the Gogny interaction. E b (ρ, δ) = Eb kin (ρ, δ) + Eb zr (ρ, δ) + E dir exch (ρ, δ) + E (ρ, δ), where each one of them reads as follows E kin b (ρ, δ) = 3 2 20m b ( ) 3π 2 2/3 [ ρ 2/3 (1 + δ) 5/3 + (1 δ) 5/3] 2 E zr b (ρ, δ) = 1 8 t 3ρ α+1 [ 2(x 3 + 2) (2x 3 + 1)(1 + δ 2 ) ] E dir b (ρ, δ) = 1 2 µ 3 [ i π3/2 A i ρ + B i ρδ 2] i=1,2 b

{ Ci Eb exch (ρ, δ) = 1 µ 3 i=1,2 i k F 3 2 [F (µ ik Fn ) + F(µ i k Fp )] [ π D i s 4 µ3 i (k Fn + sk Fp ) s=±1 ( (kfn 2 + kfp 2 µi ) sk Fn k Fp )erf 2 (k Fn + sk Fp ) ( ) ]} µ 2 + i ( k 2 2 Fn + kfp 2 ) sk Fn k Fp 1 e µ2 i (k Fn+sk Fp ) 2 /4, with F(η) = and the error function defined as ( ) π η 2 2 η3 erf(η) + 2 1 e η2 3η2 2 + 1 erf(x) = 2 x dte t 2. π 0

The constants A i and B i define, respectively, the isoscalar and isovector part of the direct term, and in the exchange part of the energy E b (ρ, δ), C i relates to the interaction between particles with the same isospin (neutron-neutron or proton-proton) whereas the constant D i considers interactions between particles with different isospin (neutron-proton, proton-neutron). A i = 1 4 (4W i + 2B i 2H i M i ) B i = 1 4 (2H i + M i ) C i = 1 π (W i + 2B i H i 2M i ) D i = 1 π (H i + 2M i ).

SYMMETRY ENERGIES UP TO SIXTH ORDER E sym,2 (ρ) = 2 6m E sym,4 (ρ) = E sym,6 (ρ) = + 1 6 ( 3π 2 2 ) 2/3 ρ 2/3 1 8 t 3ρ α+1 (2x 3 + 1) + 1 2 [ C i G 1 (µ i k F ) + D i G 2 (µ i k F )] i=1,2 2 162m + i=1,2 ( ) 3π 2 2/3 ρ 2/3 2 1 324 [C ig 3 (µ i k F ) + D i G 4 (µ i k F )] 7 2 ( ) 3π 2 2/3 ρ 2/3 4374m 2 + 1 43740 [C ig 5 (µ i k F ) D i G 6 (µ i k F )] i=1,2 µ 3 i π3/2 B i ρ i=1,2

SYMMETRY ENERGIES UP TO SIXTH ORDER Splitting the contributions given by the like or unlike isospin interactions, we have defined the following functions G j (η): G 1 (η) = 1 ( η η + 1 ) e η2 η G 2 (η) = 1 η 1 η e η2 η G 3 (η) = 14 ( ) η + 2 14 e η η + 14η + 7η3 + 2η 5 G 4 (η) = 14 ( ) η 8η + η3 2e η 2 7 η + 3η G 5 (η) = 910 ( ) η + 2 910 e η η + 910η + 455η3 + 147η 5 + 32η 7 + 4η 9 G 6 (η) = 910 ( ) η + 460η 65η3 + 3η 5 + e η 2 910 + 450η + 60η3. η

Approach of Zdunik, Fortin and Haensel R crust (km) 3.5 3 2.5 2 1.5 Dashed: Polytropic inner crust + core EoS Solid: Zdunik et al. approach D1M D1N D280 D300 SLy4 M crust (M sun ) 0.06 0.04 Dashed: Polytropic inner crust + core EoS Solid: Zdunik et al. approach D1M D1N D280 D300 SLy4 1 0.02 0.5 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 M total (M sun ) 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 M total (M sun ) 1 2GM/R core c 2 R crust = φr core 1 φ (1 2GM/R core c 2 ) M crust = 4πP trcore 4 ( 1 2DM ) core GM core R core c 2 with φ = ( ) (µ t /µ 0 ) 2 1 R corec 2 2GM arxiv 1611.01357 (2016)