The crust-core transition and the stellar matter equation of state

Helena Pais CFisUC, University of Coimbra, Portugal

Nuclear Physics, Compact Stars, and Compact Star Mergers YITP, Kyoto, Japan, October 17-28, 2016

In collaboration with C. Providência, D. P. Menezes, S. Antic, S.Typel, N. Alam, B. K. Agrawal

Acknowledgments:

Neutron stars • One of the most dense objects in Universe: R~10km and M~1.5

1.Outer crust 2.Inner crust 3.Core

M�

N. Cham

el and P

. Haens

el.

Liv. Re

v. Rel. 1

1, 10, 2

008

• The choice of inner crust EoS and the matching to the core EoS can be critical :

• Divided in 3 main layers:

Variations have been found of 0.5km for a M=1.4 star!M�

Crust-core transition important:

• plays crucial role in fraction of I in crust of star:Pt Icrust ⇠16⇡

3

R6tPtRs

which also depends on crust thickness, Rt

Describing neutron stars

1.EoS: for a system at given and

2.Compute TOV equations 3.Get star M(R) relation

⇢ TP (E)

Prescription:

Problem: Which EoS to choose?

•Phenomenological models (parameters are fitted to nuclei properties): RMF, Skyrme…

•Microscopic models (starts from n-body nucleon interaction): (D)BHF, APR…Solution: Need Constrains!!

Many EoS models in literature:

P.B. Demorest et al, Nature 467, 1081, 2010

1

10

100

1 2 3 4

P (M

eV fm

-3)

/ 0

T=0, yp=0.5

Experiments

flow exp.KaoS exp.

EoS Constrains•Experiments

P. Danielewicz et al, Science 298, 1592, 2002

W. G. Lynch et al, PPNP 62, 427 2009

•Microscopic calculations

S. Gandolfi et al, PRC 85, 032801, 2012

K. Hebeler et al, Astrophys. J. 773,11, 2013

•Observations

J. M. Lattimer and M. Prakash, arXiv: 1012.3208 [astro-ph.SR] 2010

0.1

0.2

0.4

0.6

1

23

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

P (M

eV fm

-3)

(fm-3)

T=0, neutron matterMicroscopic calculations

Chiral EFTMonte Carlo

J. M. Lattimer and A. W Steiner, EPJA 50, 40, 2014

Choosing the EoS(s)We need unified EoS, but if we don’t have it.. !Choose 1 EoS for each NS layer:

!•Outer crust EoS (BPS or HP or RHS) •Inner crust EoS (1) •Core EoS

arXiv:1604.01944 [astro-ph.SR] 2016

M(R) not affectedpasta phases ?, unified core EoS ?

homogeneous matter

•Match OC EoS at the neutron drip with IC EoS •Match IC EoS at crust-core transition (2) with Core EoS

and then

We are going to focus on (1) and (2)

to obtain the transition densities and pressures!

The pasta phases•Competition between Coulomb and nuclear forces leads to frustrated system

•Geometrical structures, the pasta phases, evolve with density until they melt crust-core transition

•Criterium: pasta free energy must be lower than the correspondent hm state

G. Watanabe et al, PRL 103, 121101, 2009

C. J. Horowitz et al, PRC 70, 065806, 2004

QMD calculations:

Pasta phases - calculation (I)• Thomas-Fermi (TF) approximation:

• Nonuniform npe matter system described inside Wigner-Seitz cell:!! Sphere, cilinder or slab in 3D (spherical symmetry), 2D (axial symmetry !! ! around z axis) and 1D (reflexion symmetry).!• Matter is assumed locally homogeneous and, at each point, its density is

determined by the corresponding local Fermi momenta. !• Fields are assumed to vary slowly so that baryons can be treated as

moving in locally constant fields at each point. !• Surface effects are treated self-consistently. !• Quantities such as the energy and entropy densities are averaged over the

cells. The free energy density and pressure are calculated from these two thermodynamical functions.

Pasta phases - calculation (II)• Coexistence Phase (CP) approximation:

• Separated regions of higher and lower density: pasta phases, and a background nucleon gas.!

• Gibbs equilibrium conditions: for : !!

• Finite size effects are taken into account by a surface and a Coulomb terms in the energy density, after the coexisting phases are achieved.!

• Total and total of the system:
 !

!

µIp = µIIp µ

In = µ

IIn P

I = P IIT = T I = T II

F = fFI + (1� f)FII + Fe

+ ✏surf

+ ✏coul

F ⇢p

⇢p = ⇢e = yp⇢ = f⇢Ip + (1� f)⇢IIp

check PRC 91, 055801 2015

Pasta phases - calculation (III)• Compressible Liquid Drop (CLD) approximation:

The total free energy density is minimized, including the surface and Coulomb terms.

The equilibrium conditions become:

µIn = µIIn ,

P I = P II � ✏surf⇣ 12↵

+1

2�

@�

@f�

⇢IIpf(1� f)(⇢Ip � ⇢IIp )

⌘

µIp = µIIp �

✏surff(1� f)(⇢Ip � ⇢IIp )

,

check PRC 91, 055801 2015

Non-linear Walecka Model

nucleons electronsmesonsem

non-linear mixing couplings

Li = ̄i [�µiDµ �M⇤] iLe = ̄e [�µ (i@µ + eAµ)�me] e

L� =1

2

✓@µ�@

µ��m2s�2 �1

3�3 � 1

12��4

◆

L! = �1

4⌦µ⌫⌦

µ⌫ +1

2m2vVµV

µ +1

4!⇠g4v(VµV

µ)2

L⇢ = �1

4Bµ⌫ ·Bµ⌫ +

1

2m2⇢bµ · b

µ

L� = �1

4Fµ⌫F

µ⌫

L�!⇢ = ⇤3�gsg2v�VµV µ + ⇤2�g2sg2v�2VµV µ + ⇤1�gsg2⇢�bµ · bµ

+⇤�g2sg

2⇢�

2bµ · bµ + ⇤vg2vg2⇢bµ · bµVµV µ

L =X

i=p,n

Li + Le + L� + L� + L! + L⇢ + L�!⇢mesons: mediation of nuclear force

non-linear mixing couplings terms: responsible for density dependence of

Esym!

)−3 (fmn0 0.02 0.04 0.06 0.08

)−3

(fm

p

0

0.02

0.04

0.06

0.08

coexistencespinodal

binodal

Y =0

Y =0.4L

− equil.

How to calculate transition density?

• Thermodynamical spinodal• Dynamical spinodal1) Get the instability region:

2) Intersect EoS with that boundary to get ⇢t

PRC 82, 055807, 2010 PRC 85, 059904(E), 2012

For -eq. matter and T=0, dyn. spinodal very coincident with TF calculation

�

or

.n

thermodynamical

dynamical

p(fm

)

−3

(fm ) −3 0

0.02

0.04

0.06

0.08

0.08 0.06 0.04 0.02 0

courtesy: C. Providênciacourtesy: C. Providência

Thermodynamical spinodal

•The (free) energy curvature matrix for asymmetric NM is defined by:

check PRC 74, 024317 2006

C =⇣ @2F@⇢i@⇢j

⌘

•Stability conditions: Tr(C) > 0, Det(C) > 0•The spinodal is given by for which Det(C) = 0i.e., one of eigenvalues is negative in the region of instability and goes to zero at border:

�� =1

2

⇣Tr(C)�

pTr(C)2 � 4Det(C)

⌘= 0

(T, ⇢p, ⇢n)

The crust-core transition - thermodynamical spinodal approach

D* models: scalar and vector self-energies depend on E: the couplings are adjusted to the optical potential in nuclear matter.

Nucl. Phys. A 938

, 92 2015

preliminary!

with S. Antic, S. Typ

el and C. Providên

cia

a) density-dependent models

0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

p (fm

-3)

n (fm-3)

T=12 MeVT=6 MeV

D1D2

b) non-linear mixing meson couplings models

e.g. PRC 81

, 034323 201

0

with N. Alam, B. K. Agrawal

and C. Providência

preliminary!

0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08p

(fm-3

)

n (fm-3)

T=12 MeVT=6 MeV

F2F

Different mixing couplings: different L: L(F )=70 MeV, L(F2 )=46 MeV.

Dynamical spinodal

•Dynamical instabilities are given by collective modes that correspond to small oscillations around equilibrium state.

•These small deviations are described by linearized equations of motion.

•Perturbed fields:

•Perturbed distribution function:

Fi = Fi0 + �Fi

fi = fi0 + �fi

check PRC 94, 015808 2016

•Very good tool to estimate crust-core transition in cold neutrino-free neutron stars. check PRC 82, 055807 2010; PRC 85, 059904(E) 2012

Dynamical spinodal (cont)

•We get a set of equations for the fields and particles, whose solutions form a complete set of eigenmodes, that lead to the following matrix:

•The dynamical spinodal surface is defined by the region in space, for a given wave vector k and temperature T, limited by the surface •In the k=0 MeV limit, the thermodynamic spinodal is obtained.

(⇢p, ⇢n)

! = 0.

0

@1 + F ppLp F pnLp C

peA Lp

FnpLn 1 + FnnLn 0CepA Le 0 1� CeeA Le

1

A

0

B@

2PFp3k

�⇢p⇢p

2PFn3k

�⇢n⇢n

2PFe3k

�⇢e⇢e

1

CA = 0

semiclassical approach, that is a good approximation to t-dependent Hartree-Fock eqs at low energies

•The time evolution of the distribution functions is described by the Vlasov equation: @fi

@t+ {fi, hi} = 0, i = p, n, e

The crust-core transition - dynamical spinodal approach

1.The larger L, the smaller the spinodal section.

PRC 94, 015808 2016

3.Crossing of the ωρ and σρ spinodals, for a given L, occurs close to the crossing of the β-eq EoS with the spinodals

ωρ terms

⇢t

σρ terms

L

M(R) relations

Stars with unified inner crust-core EoS (black lines) have larger (smaller) radii than configurations without inner crust (pink lines) for the NL3ωρ (NL3σρ) models.!

Effect on Mmax is negligible, not true for the radius!!

a) effect of pasta:

b) effect of different inner crust EoS

with L close to core EoS:

σρ give slightly larger radii than ωρ models, the differences being larger for M >~ 1.4M⊙. For 1.4M⊙ star, difference is of ∼ 100 m.!

the error on the determination of the radius is negligible for all masses!

tested for other models (TM1 and Z271): same result!!

0

0.5

1

1.5

2

2.5

3

13 14 15 16

M (M

sun)

R (km)

bps+pasta+hmbps+hm

L=118, NL3L=55, 6

6

0

0.5

1

1.5

2

2.5

3

13 14 15 16

M (M

sun)

R (km)

bps+pasta+hmbps+pasta(FSU)+hm

PSR J0348+0432PSR J1614-2230

exceptions: NL3σρ6, difference of ~50 m (∼ 40 m) for a 1M⊙ ( 1.4M⊙) star; NL3ωρ6, ∼ 20 m for a 1M⊙ star

We get the transition density from a dyn. spin. calculation.

-10

-5

0

5

10

0.04 0.06 0.08 0.1 0.12 0.14 0.16

(PN

M-P

mic

ro)/

(fm-3)

(b)

Monte Carlo

-10

-5

0

5

10

0.04 0.06 0.08 0.1 0.12 0.14 0.16

(PN

M-P

mic

ro)/

(fm-3)

(a)

T=0, neutron matterChiral EFT

NL366

TM16

6Z271

856

1

10

100

1 2 3 4

P (M

eV fm

-3)

/ 0

T=0, yp=0.5

flow exp.KaoS exp.

NL3TM1

Z271Z271, cut

0

0.5

1

1.5

2

2.5

3

10 11 12 13 14 15 16

M (M

sun)

R (km)

NL3

TM1

Z271*

Z271

NL3 66

TM1 66

Z271 56

5*6*

If we combine the 3 constrains, we get the following models:

NL3 did not pass exp. constrain

Z271*: extra potential dependent on σ meson, that makes M* to stop decreasing above saturation density, as suggested in K. A. Maslov, E. E. Kolomeitsev, and D. N. Voskresensky, Phys. Rev. C 92, 052801 (2015).

For 1.4M⊙ stars, these models predict R=13.6 ± 0.3 km and a crust thickness of 1.36 ± 0.06km.

Z271 did not pass obs. constrain

exceptions:

and but

ext. Nambu—Jona-Lasinio ModelPRC 93, 065805 2016

•Set of models with chiral symmetry included, unlike RMFs•Since chiral symmetry is satisfied, EoS valid at higher densities!

short range repulsionshort range attraction

density dependence of scalar coupling

isospin asymmetric nuclear matter

to make restoration of the chiral symmetry less abrupt

make the symmetry energy softer

L = ̄(i�µ@µ �m) +Gs[( ̄ )

2 + ( ̄i�5~⌧ )2] �Gv( ̄�µ )2

�Gsv[( ̄ )2 + ( ̄i�5~⌧ )2]( ̄�µ )2

�G⇢⇥( ̄�µ~⌧ )2 + ( ̄�5�

µ~⌧ )2⇤

�Gv⇢( ̄�µ )2⇥( ̄�µ~⌧ )2 + ( ̄�5�

µ~⌧ )2⇤

�Gs⇢⇥( ̄ )2 + ( ̄i�5~⌧ )

2⇤ ⇥( ̄�µ~⌧ )2 + ( ̄�5�

µ~⌧ )2⇤

M = m� 2Gs⇢s + 2Gsv⇢s⇢2 + 2Gs⇢⇢s⇢23nucleon effective mass

eNJL models

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5

Esym

(MeV

)

(fm-3)

eNJL1eNJL1 1eNJL1 2

eNJL2eNJL2 1

eNJL3eNJL3 1

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5

Esym

(MeV

)

(fm-3)

eNJL1meNJL1m 1

eNJL2meNJL2m 1

• In this study, we used eNJLx, eNJLxωρy, and eNJLxσρy type of models• We also considered models with a current mass: eNJLxm, and eNJLxmσρy.• To make Esym softer: *ωρ* and *σρ* models, where we fixed the Esym at ρ=0.1 at the

same value of eNJLx (eNJLxm), and we calculated the new Gρ, fixing the Gvρ (Gsρ) constant.

without current mass with current mass

eNJL models-Constrains

1

10

100

1 2 3 4 5

P (M

eV fm

-3)

/ 0

(a)

T=0, yp=0.5

Flow exp.Kaons exp.

eNJL1eNJL2eNJL3

eNJL1meNJL2m

0.5

1

1.5

2

2.5

3

3.5

0.04 0.06 0.08 0.1 0.12 0.14 0.16

P (M

eV fm

-3)

(fm-3)

T=0, neutron matter

Hebeler et al.Gandolfi et al.

eNJL1eNJL1 1

eNJL2eNJL2 1

eNJL3eNJL3 1

eNJL2meNJL2m 1

But the models need to fulfil the constrains…

Experiments

Microscopic calculations

only 2 models passed: eNJL3σρ1 and eNJL2mσρ1

M(R) relations

0

0.5

1

1.5

2

2.5

11 12 13 14 15 16 17 18

M (M

sun)

R (km)

eNJL3 1

psr J0348+0432psr J1614-2230

BPS+hmBPS+pasta(CP)+hm

BPS+pasta(CLD)+hm

1) outer crust: BPS 2) inner crust: pasta from a CP or CLD calculation 3) core: hom. nucleonic matter, same model as inner crust

1) EoS:

0

0.5

1

1.5

2

2.5

11 12 13 14 15 16 17 18

M (M

sun)

R (km)

BPS+pasta+hm

psr J0348+0432psr J1614-2230

eNJL3 1eNJL2m 1

CP or CLD: no difference in M(R)

eNJL3σρ1:

eNJL2mσρ1:

2) integrate TOV

3) get M(R)

R(M=1.4M⊙)=13.212 km, with ∆R(M=1.4M⊙)=1.405km.

R(M=1.4M⊙)=13.084 km, with ∆R(M=1.4M⊙)=1.408km.

M(R) relations (cont)

Considering hybrid stars:• quark core described within SU(3) NJL model

0

0.5

1

1.5

2

2.5

11 12 13 14 15

M (M

sun)

R (km)

(a)

psr J0348+0432psr J1614-2230

eNJL3 1+NJL1eNJL3 1+NJL2eNJL3 1+NJL3eNJL3 1+NJL4

We are still able to describe stable 2M⊙ stars with a quark core!!

The deconfinement phase transition decreases maximum mass

though…

• perform a Maxwell construction to get hadronic to quark transition

Summary•Inclusion of the inner crust EoS has strong effect on the radius of low and intermediate mass neutron stars!

•Unified EoS are needed!

but…

•Inner crust EoS with similar symmetry energy properties as the core EoS: effect on radii for stars with M > 1 M⊙ is negligible!

•For RMF models, R(M=1.4M⊙)=13.6±0.3km, with ∆R(M=1.4M⊙)=1.36±0.06km.

•Inner-crust—core unified EoS with chiral symmetry and pasta allows the description of 2 M⊙ stars, with R(M=1.4M⊙)=13.148±0.064km.

Strong correlations of neutron star radii with the slopes of nuclear matter incompressibility and symmetry energy at saturation

set of 24 Skyrme-type effective forces and 18 RMF models, and 2 microscopic calculations, all describing 2M⊙ neutron stars.

Unified EoSs for the inner-crust-core region have been built for all the phenomenological models, both relativistic and non-relativistic.

accep. PRC (R), arXiv:1610.06344[nucl-th]

8 10 12 14 16R (km)

0

0.5

1

1.5

2

2.5

3

MN

S (M

O)

0 0.5 1 1.5ρc (fm

-3)

.

PSR J0348+0432PSR J1614-2230

240

270

300

K0 (

MeV

)

2000

2500

3000

3500

M0 (

MeV

)

12 13 14R1.0 (km)

40

80

120

L 0 (M

eV)

12 13 14 15R1.4 (km)

C(R1.0, K0) = 0.655

C(R1.0, M0) = 0.646

C(R1.0, L0) = 0.850

C(R1.4, K0) = 0.704

C(R1.4,M0) = 0.743

C(R1.4, L0) = 0.745

We started by calculating correlation of R with K, M and L…

We found strong correlation of the neutron star R with linear combination of M and L, and almost independent of the neutron star mass in the range 0.6-1.8M⊙.

300

350

400

450

500K

0+α

L 0 (M

eV)

250

300

350

400

K0+α

L 0 (M

eV)

12 13 14 15R1.0 ( km)

3000

4000

5000

6000

M0+βL

0 (M

eV)

11 12 13 14 15R1.4 (km)

3000

4000

5000

M0+βL

0 (M

eV)

C(R1.0, K0+αL0) = 0.902C(R1.4, K0+αL0) = 0.850

C(R1.0, M0+βL0) = 0.945 C(R1.4, M0+βL0) = 0.924

α = 1.564 α = 0.883

β = 17.089β = 28.370

and then…

0.6

0.8

1

C(R

X ,

b)

K0L0K0+αL0

0.6 0.8 1 1.2 1.4 1.6 1.8MNS (MO)

0.4

0.6

0.8

1

C(R

X ,

b)

M0L0M0+βL0

b {

{b.

and….

This correlation can be linked to the empirical relation between R and P at a nucleonic density between 1-2 saturation density, and the dependence of P on K, M and L.

Thank you!