An empirical Equation of State for nuclear physics and astrophysics

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1 An empirical Equation of State for nuclear physics and astrophysics Collaborators: Debarati Chatterjee LPC/ENSICAEN, caen, france Francesca Gulminelli Jerome Margueron Adriana Raduta Sofija Antic Debora P. Menezes XIV HADRON PHYSICS march 18-23, 218 florianopolis, brazil

2 OBSERVABLES SENSITIVE TO CRUST-CORE PROPERTIES Glitches in spin period Image: W. Ho NS cooling Image: C. Miller Gravitational wave detectors (LIGO/VIRGO) NS oscillations Image: L. Rezzolla

3 magnetars Images: NASA Image: Handbook of Pulsar Astronomy, ( Lorimer and Kramer)

4 NUCLEAR EQUATION OF STATE Nuclear EoS Image: Physics of NS crusts Haensel & Chamel

5 NON-UNIFIED VS UNIFIED EOS Crust-Core Matching for non-unified EoSs Lattimer and Prakash, (21) Matching is done so that pressure is an increasing function of energy density different models leads to arbitrary results uncertainty in crust thickness upto 3% and for radius 4% Fortin et al., (216)

6 e.g. Non-relativistic (Skyrme, BHF), relativistic (RMF, RHF)

7 e.g. Non-relativistic (Skyrme, BHF), relativistic (RMF, RHF) Fortin et al. (216) Constraints in J-L plane From n skin thickness of 28 Pb From HIC From electric dipole polarizability αd From giant dipole resonance (GDR) of 28 Pb From measured nuclear masses From isobaric analog states (IAS)

8 WHY WE NEED A MODEL INDEPENDENT EOS Spurious correlations! Margueron, Casali, Gulminelli, PRC 97, 2585 (218) Skyrme RMF BHF Ducoin et al. PRC 211 Reinhard and Nazarewicz PRC 216

9 EMPIRICAL META-MODEL

10 EMPIRICAL COEFFICIENTS: REFERENCE PARAMETERS Margueron, Casali, Gulminelli, PRC 97, 2585 (218) N = 2 Prior distribution of Bayesian analysis N = 3 N = 4

11 APPLICATION TO NUCLEAR PHYSICS : NUCLEI.4 Aymard, Gulminelli, Margueron (216) (E tot -E exp )/A [MeV] EoS-C fin -C so EoS-C fin full ETF H-F.2.15 (b) Z=28 Energy residuals vs A for symmetric and asymmetric nuclei AME (212) Mass tables A (E tot -E exp )/A [MeV] EoS emp -.15 full ETF H-F I DC, Gulminelli, Raduta and Margueron, PRC 96, 6585 (217)

12 PREDICTION OF NUCLEAR OBSERVABLES EoS emp expt full ETF !4)5!6!7 123!4)5!6! 4859 (:;;!1<= <r ch > (fm) !"# $% &!'()*! (a) I = (N-Z)/A Z=2 Charge Radii expt: ADNDT, Marinova and Angeli (213) Z= !+!,!'-.*/ charge radii.25.2 neutron skin! " #$!%&'(! Trzcinska et al. (21).15 DC, Gulminelli, Raduta and Margueron, PRC 96, 6585 (217).1.2 /1!2'$!3!4 /1!2'$!3! 25$6 &788!/9: !)!*!%+,(-.

13 CORRELATIONS: EMPIRICAL PARAMETERS & NUCLEAR OBSERVABLES n sat E sat K sat E sym L sym K sym m*/m <r n > <r ch > R np C fin a 1 a C fin R np <rch > <r n > m*/m K sym L sym E sym Ksat E sat n sat DC, Gulminelli, Raduta and Margueron, PRC 96, 6585 (217)

14 CRUST-CORE PHASE TRANSITION NM 25 2 Thermodynamic Spinodals: Sly23a = =.2 =.4 =.6 =.8 = Thermodynamic Spinodals: EoS SLy23a Exact (MeV.fm 3 ) 5 p mer. mars 14 15:3: n Thermodynamical spinodals

15 CRUST-CORE PHASE TRANSITION NM NMe Thermodynamic Spinodals: Sly23a = =.2 =.4 =.6 =.8 = Thermodynamic Spinodals: Sly23a - (MeV.fm 3 ) (MeV.fm 3 ) mer. mars 14 15:3: = =.2 =.4 =.6 =.8 = 1 Thermodynamical spinodals

16 CRUST-CORE PHASE TRANSITION Finite size fluctuations: Dynamical spinodals.12.1 TD spinodal 2 MeV/c 4 MeV/c 8 MeV/c 16 MeV/c Dyn spinodal envelope.8 p n NMe Ref: Thesis, Camille Ducoin bulk density-gradient Coulomb

17 APPLICATION OF META-MODEL IN ASTROPHYSICS: CCPT Thermodynamic Spinodals: EoS SLy23a.12 Exact.1 NM.8 p n Thermodynamical spinodals

18 APPLICATION OF META-MODEL IN ASTROPHYSICS: CCPT Thermodynamic Spinodals: EoS SLy23a.12 Exact N=2 N=4.1 NM.8 p n Thermodynamical spinodals: effect of HO terms of MM

19 SENSITIVITY TO EMPIRICAL PARAMETERS N = 2 N = 3 N = 4

20 THE NEXT STEP : EFFECT OF MAGNETIC FIELD 2 18 B= B=1e15G B=1e16G B=1e17G B=1e18G B=2e18G Energy density (MeV/fm3) B=1e18G B=2e18G B=5e18G B=7e18G B=1e19G Baryon density (1/fm3) Energy density (MeV/fm3) Baryon density (1/fm3)

21 EFFECT OF MAGNETIC FIELD ON THERMODYNAMIC SPINODALS.12.1 B * =1 2 Thermodynamic Spinodals: empirical EoS B * =1 2, N=4 B=, N=4 B=, N=4 Preliminary!.8 B * =B/B e c = 4.4x1 13 G p B * =1 3 B= B * = n p Ref: Thesis, Jianjun Fang n

22 EFFECT OF MAGNETIC FIELD ON THERMODYNAMIC SPINODALS.12.1 B * =1 4 B= B * = 1 4 Preliminary!.8 B * =B/B e c = 4.4x1 13 G p B * =5x B= B * = 5x1 5 B * = n.15 p.1.5 Ref: Thesis, Jianjun Fang n

23 SUMMARY We propose an empirical MetaModel to describe homogeneous nuclear matter as well as asymmetric nuclei within the same formalism DFT in the ETF approximation to construct an energy functional for HNM and clusterized matter In HNM, the coefficients of the energy functional directly related to experimentally determined empirical parameters {ρsat, λsat, Ksat, Jsym, Lsym, Ksym} and m* In clusterized matter, a single extra parameter required (Cfin ) to reproduce the experimental measurements of nuclear masses We apply this MetaModel to calculate the thermodynamical instability (spinodal) region that determines the crust-core phase transition in neutron stars We study the influence of the uncertainty in empirical parameters on the crust-core phase transition We investigate the influence of strong magnetic fields on the crust-core phase transition This may have important consequences on the crust thickness, radii, moment of inertia and other astrophysical observables