Institut d’Astronomie et d’AstrophysiqueUniversité Libre de Bruxelles

Structure of neutron starsStructure of neutron stars

with unified equations of statewith 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 objects19th 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

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

EoS: the challengeWide range of

,T,Ye in the core during core collapseand 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

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

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

EoS at finite T : the method (1)

Onsi et al., PRC 77,065805 (2008); PRC 55, 3139 (1997); PRC 50, 460 (1994), and Refs. ThereinPearson 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, Jq, sq : 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

EoS at finite T : the method (2) + proton shell corrections added via Strutinsky-Integral (SI) to correct fTETF

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

9

EoS: resultsInner crust + Core

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

Outer crustPearson et al., PRC83, 065810 (2011)

We construct the NS structure with these EoSs, solving TOV equationsUse of LORENE (http://www.lorene.obspm.fr) library for rotational configuration

10

NS properties: P vs energy relation

BSk19, BSk20, BSk21 compatible with observations of X-ray bursts

11

NS properties: moment of inertia

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

BSk19, BSk20, BSk21 compatible for lowest limit of I

12

NS properties: gravitational redshift

BSk19, BSk20, BSk21 compatible with values extracted from observations

13Chamel, Fantina, Pearson, Goriely, PRC 84, 062802(R) (2011)

NS properties: M vs R relation (1)

Non-rotating configurations

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…

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.

15

NS with phase transition (1)

Chamel et al., arXiv:1205.0983

16

NS with phase transition (2)

Only imposed constraints: 1. causality; 2. thermodynamical consistency

Chamel et al., arXiv:1205.0983

17BSk19 + phase transition compatible with observations!

NS with phase transition (3)

Fantina et al., Proceedings ERPM (2012)

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

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

Thank youThank you