1. Institut d’Astronomie et d’Astrophysique Université Libre de Bruxelles Structure of neutron stars with unified equations of state Anthea F. FANTINA Nicolas CHAMEL, Stéphane GORIELY (IAA, ULB) Michael J. PEARSON (University of Montreal) From nucleon structure to nuclear structure and compact astrophysical objects 19th June 2012, Beijing, China

2. Outline  Motivation  Introduction - Construction of the functionals  EoS: the model  EoS: results at T = 0 - EoS in the NS - NS properties and astrophysical observations  Conclusions & Outlook

3. 3 Motivations & Aims  Unified EoS  based on energy-density functional theory  valid in all regions of NS interior  outer / inner crust and crust / core transition described consistently  obtained with the same functional  EoS both at T = 0  cold non-accreting NS and at finite T  SN cores, accreting NS  EoS has to satisfy:  Astrophysical constraints  Nuclear experimental data

4. 4 EoS: the challenge Wide range of ,T,Y e in the core during core collapse and NS formation : different states of matter (inhomogeneous, homogeneous, exotic particles?) In NS: T = 0 approximation, but: very high density  composition uncertain! http://www.physics.montana.edu

5. 5  Underlying forces: BSk19-20-21 Goriely et al., PRC 82, 035804 (2010)  microscopic mass models based on HFB method with semi-local functionals of Skyrme type and microscopically deduced pairing force  fit available experimental mass data (2149 masses, rms = 0.581 MeV)  reflect current lack of knowledge of high-density behaviour of nuclear matter  constrained to neutron matter EoS at T = 0 softer stiffer BSk19 constrained to fit Friedman & Pandharipande n matter BSk20 constrained to fit Akmal, Pandharipande & Ravenhall n matter BSk21 constrained to fit Li & Schulze n matter see also: Chamel et al., PRC 80, 065804 (2009) Construction of effective functionals  see N. Chamel’s talk!

6. 6 EoS: the model  OUTER CRUST (up to neutron drip) (Pearson et al., PRC 83, 065810 (2011))  one nucleus (bcc lattice) + electrons, in charge neutrality and  equilibrium  experimental nuclear masses + microscopic mass models (HFB)  minimization of the Gibbs energy per nucleon (BPS model)  INNER CRUST (Onsi et al., PRC 77, 065805 (2008), Pearson et al., PRC 85, 065803 (2012) )  one cluster (Wigner-Seitz cell) + n, p, e  semi-classical model: Extended Thomas Fermi (4th order) + proton shell corrections (  see next slide)  CORE  homogeneous matter: n, p, e, muons in  equilibrium  same nuclear model to treat the interacting nucleons

7. 7 EoS at finite T : the method (1) Onsi et al., PRC 77,065805 (2008); PRC 55, 3139 (1997); PRC 50, 460 (1994), and Refs. Therein Pearson et al., PRC 85, 065803 (2012)  Inhomogeneous phase: ETF: Extended (4th order) Thomas-Fermi  high-speed approximation to HF  Wigner-Seitz cell (spherical) containing A nucleons  T dependent  minimization of the free energy per nucleon (integraton over the WS cell) Skyrme type (BSk functionals)   q, J q, s q : expansion up to the 4th order expressed as a function of an assumed density distribution  q  minimization wrt geometrical parameters of the cell, and wrt N,Z  one gets  approximation to the HF values

8. 8 EoS at finite T : the method (2) + proton shell corrections added via Strutinsky-Integral (SI) to correct f TETF from first minimization (previous slide)  Homogeneous phase:  n, p, e, muons  same Skyrme functional to treat the interacting nucleons shell corrections SI correction perturbative Onsi et al., PRC 77,065805 (2008); PRC 55, 3139 (1997); PRC 50, 460 (1994), and Refs. Therein Pearson et al., PRC 85, 065803 (2012)

9. 9 EoS: results Inner crust + Core Pearson et al., PRC 85, 065803 (2012) Outer crust Pearson et al., PRC83, 065810 (2011) We construct the NS structure with these EoSs, solving TOV equations Use of LORENE (http://www.lorene.obspm.fr) library for rotational configuration

10. 10 NS properties: P vs energy relation BSk19, BSk20, BSk21 compatible with observations of X-ray bursts

11. 11 NS properties: moment of inertia from Crab: P, v exp, M neb, R neb  estimation of lower limit on moment of inertia BSk19, BSk20, BSk21 compatible for lowest limit of I

12. 12 NS properties: gravitational redshift BSk19, BSk20, BSk21 compatible with values extracted from observations

13. 13 Chamel, Fantina, Pearson, Goriely, PRC 84, 062802(R) (2011) NS properties: M vs R relation (1) Non-rotating configurations

14. 14 NS properties: M vs R relation (2) BSk20, BSk21 compatible with observations, BSk19 too soft, but if we consider a possible phase transition to exotic phase…

15. We assume that nucleonic matter undergoes a 1st order phase transition to some “exotic” matter at baryon densities above n N, so that: n < n N : matter is in the nucleonic phase n N ≤ n ≤ n X : phase coexistence  n > n X : matter is in the exotic phase (the energy is lowered). The stiffest possible EoS satisfying causality is:  n = n C : the two phases have the same energy. For n > n C the ground state of matter would be again nucleonic. 15 NS with phase transition (1) Chamel et al., arXiv:1205.0983

16. 16 NS with phase transition (2) Only imposed constraints: 1. causality; 2. thermodynamical consistency Chamel et al., arXiv:1205.0983

17. 17 BSk19 + phase transition compatible with observations! NS with phase transition (3) Fantina et al., Proceedings ERPM (2012)

18. 18 Conclusions  Unified EoSs both for NS matter (and SN matter)  same nuclear model to describe all regions of NS interior  but: only one cluster (ok for thermodynamical properties)  Nuclear models fitted on - experimental nuclear data - nuclear matter properties  EoSs BSk20, BSk21 consistent with astrophysical observations! BSk19 favoured by  + /  - experiments, but seems too soft for astro… but: ok if we include a possible phase transition in the core!  EoSs available as table / analytical fit

19. 19 Outlooks  EoS for NS (T=0) and SN cores (finite T)  T = 0: EoS : table analytical fit (easy to implement!)  T ≠ 0: work in progress  generate tables for SN cores  implement in hydro codes  possibility to treat non-spherical cluster (in progress)  application to accreting NS

20. Thank you