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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).

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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


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


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


http://www.physics.montana.edu

EoS:the challenge

Wide range of

r,T,Ye in the core

during core collapse

and NS formation :

In NS: T = 0 approximation,

but: very high density

 composition uncertain!

different states

of matter

(inhomogeneous,

homogeneous,

exotic particles?)


Construction of effective functionals

 see N. Chamel’s talk!

  • 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

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

softer

stiffer

see also: Chamel et al., PRC 80, 065804 (2009)


Eos the model
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 b 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 b equilibrium

  •  same nuclear model to treat the interacting nucleons


Eos at finite t the method 1
EoS at finite T : the method (1)

  • Inhomogeneousphase: ETF: Extended (4th order) Thomas-Fermi

  •  high-speedapproximationto HF

  •  Wigner-Seitzcell (spherical) containingAnucleons

  •  T dependent minimizationof the free energy per nucleon

  • (integratonover the WS cell)

 tq, Jq, sq : expansion up to the 4th order

expressedas a functionofanassumed density distributionrq

 minimizationwrtgeometricalparametersof the cell, and wrt N,Z

 onegetsapproximationto the HF values

Skyrme type (BSk functionals)

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)

7


Eos at finite t the method 2
EoS at finite T : the method (2)

+ protonshellcorrectionsadded viaStrutinsky-Integral (SI) tocorrectfTETF

SI correction perturbative

from first minimization (previous slide)

shell corrections

  • Homogeneousphase:

  •  n, p, e, muons

  •  sameSkyrmefunctionaltotreat the interactingnucleons

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)

8


EoS: results

Outer crust

Pearson et al., PRC83, 065810 (2011)

Inner crust + Core

Pearson et al., PRC 85, 065803 (2012)

We construct the NS structure with these EoSs, solving TOV equations

Use of LORENE (http://www.lorene.obspm.fr) library for rotational configuration

9


NS properties: P vs energy relation

10

BSk19, BSk20, BSk21 compatible withobservations of X-ray bursts


NS properties: moment of inertia

from Crab: P, vexp, Mneb, Rneb estimation of lower limit on moment of inertia

11

BSk19, BSk20, BSk21 compatible for lowest limit of I


NS properties: gravitational redshift

BSk19, BSk20, BSk21 compatible

withvalues extracted from observations

12


NS properties: M vs R relation (1)

Non-rotating configurations

Chamel, Fantina, Pearson, Goriely, PRC 84, 062802(R) (2011)

13


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…

14


NS with phase transition (1)

  • We assume that nucleonic matter undergoes a 1st order phase transition to some

  • “exotic” matter at baryon densities above nN , so that:

  • n < nN : matter is in the nucleonic phase

  • nN ≤ n ≤ nX : phase coexistence 

  • n > nX : matter is in the exotic phase (the energy is lowered).

  • The stiffest possible EoS satisfying causality is:

  • n = nC: the two phases have the same energy.

  • For n > nC the ground state of matter would be again nucleonic.

Chamel et al.,

arXiv:1205.0983

15


NS with phase transition (2)

Chamel et al., arXiv:1205.0983

Only imposed constraints: 1. causality;

2. thermodynamical consistency

16


NS with phase transition (3)

Fantina et al., Proceedings ERPM (2012)

17

BSk19 + phase transition compatible with observations!


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 p+/p- 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

18


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

19



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