Behaviour of Proton Componentin Neutron Star Matter

for Realistic Nuclear Models

1. Realistic Nuclear Models Used in Calculations.

2. Symmetry Energy and Proton Localization.3. Model of Proton Component in Neutron

Matter.4. Astrophysical Consequences.5. Conclusions.

Adam Szmagliński, Marek Kutschera, Włodzimierz Wójcik

Neutron Star Structure

• Atmosphere

• Envelope

• Crust

• Outer Core

• Inner Core

3610 cmg3116 103.410 cmg31411 10)4.22(103.4 cmg

core 1410)4.22(

core

Realistic Nuclear Models:

1. Myers-Świątecki (MS)

2. Skyrme (Sk)

3. Friedman-Pandharipande-Ravenhall (FPR)

4. UV14+TNI (UV)

5. AV14+UVII (AV)

6. UV14+UVII (UVU)

7. A18

8. A18+δv

9. A18+UIX

10. A18+δv+UIX*

Energy per particle of asymmetric nuclear matter:

nVxnVxnTxnE F 22

0 21,,

3

53

5

3

22 13

10

3, xxn

mxnT

NF

Kinetic Energy of Fermi gas:

Energy density: xnnExn ,, where: nnx P /

UV14+TNI Model

320

2342

2340

,

15333517838

33871330959

fmnMeVVV

nnnnnV

nnnnnV

UV

UV

AV14+UVII Model

nnn

nnnnV

nnn

nnnnV

AV

AV

1888601844

1930961187

36312652560

31801786378

23

4562

23

4560

UV14+UVII Model

nnnn

nnnnV

nnn

nnnnV

UVU

UVU

152357140935

1528873175

34710651602

1749848152

234

5672

23

4560

Equilibrium

MeVmc

MeVcm

MeVcm

P

N

29.1

26.938

55.939

2

2

2

enep

NeP

PNi

n

nnnn

i

PNPNi ,,

,,

nnn ep ee

e

Symmetry Energy of Nuclear Matter

nVnTx

xnEnE

xnEnExnE

F

x

S

S

2

2

12

2

2

2

1,

9

5,

8

1

122

1,,

Coexistence Conditions are Implemented by Constructing Isobars

CCCNN xnPxnP ,0,

CCNNN xnxn ,0,

CCPNP xnxn ,0, Protons do not diffuse into neutron matter

The neutron matter coexists with proton clusters of nucleon density and proton fraction in a pressure range .Cn Cx 21;PP

n

nnP

/2

The most important izobar parameters

Potential Skyrme UV14+TNI AV14+UVII UV14+UVII

71 398 3904 2464

0.202 0.445 0.371 0.078

0.477 1.061 1.668 1.587

0.422 0.957 1.617 1.537

97 405 4037 2579

0.5 0.5 0.5 0.183

0.526 1.069 1.677 1.652

0.438 0.962 1.622 1.529

MeVP1

MeVP2

Cx

Cx

3fmnN

3fmnN

3fmnC

3fmnC

Model of Proton Impurities in Neutron Star Matter (M. Kutschera, W. Wójcik)

We divide the system into spherical Wigner-Seitz cells,each of them enclosing a single proton:

PnV

1The volume of the cell:

The energy of the cell of uniform phase: PN nnVE ,0 PNPNPN nnnnn ,

NNP nVnE 0

2

24

3

2

4

3exp

3

2

PPP R

rRr

V

NPPP

PL rdrnBrnrrnm

rE 322

*

2

0

22

20

4

8

9

drdr

rdnBnrnnrnrpr

RmEEE

NNNPP

PPL

rrrp PP *where

Variational Method for single protonin neutron matter

rpnrn N

25

2

3

0

2

0

22

1

3

4

2

94

48

9

NP

NNN

NPPPP

BR

ndrnrnr

drnrnrprRm

E

EWe minimize the energy difference

PR,00

- neutron density enhancement around the proton

- neutron density reduction in the proton vicinity

with variational parameters

PR - the mean square radius of the localized proton probability distribution

The self-consistent variational method

V V

PPNP rdrrErdnrnErrnf 1, 3*3

boundary conditions:

V

PP

V

N rdrrrdnrn 010 3*3

We look for such functions and , that minimize functional: r rn

E, - Lagrange multipliers

By differentiation with respect to and we obtain rP* rn

following Euler-Lagrange equations:

rErnrnrm PPPNPPPP

2

2

1

022

2*

dr

rndBrnrr

rn

rnNNPP

P

NN nr - variational parameterPR

Comparisonof simple variational method

and self-consistent variational method

Potential1 Method 2 Method

MS 1.070 1.047 1.030 0.905

Sk 0.364 1.949 0.351 1.519

FPR 0.743 1.483 0.721 1.262

UV14+TNI 0.749 1.340 0.731 1.209

AV14+UVII 0.816 1.181 0.789 0.971

UV14+UVII 0.796 1.117 0.766 0.913

A18 1.515 1.250 1.493 1.136

A18+δv 1.657 1.018 1.627 0.915

A18+UIX 0.675 1.124 0.645 0.911

A18+δv+UIX* 0.853 1.080 0.819 0.878

locn locPR locn loc

PR

Mass and radii of neutron stars:• Tolman-Oppenheimer-Volkoff

equation • Equation of state

Potential

MS 2.019 2.038 2

Skyrme 0.876 2.242 32

FPR 1.447 1.797 17

UV14+TNI 1.489 1.827 14

AV14+UVII 1.793 2.123 3

UV14+UVII 1.986 2.207 2

A18 1.647 1.673 7

A18+δv 1.805 1.805 3

A18+UIX 2.095 2.386 2

A18+δv+UIX* 2.052 2.213 2

MINM MAXM MINMM

J.M. Lattimer and M. Prakash, astro-ph/0405262

Conclusions1. Symmetry energy – inhomogeneity of dense nuclear matter

in neutron stars.2. Proposal of self-consistent variational method – neutron

background profile as a solution of variational equation with parameter (mean square radius of proton wave function).

3. Localization of protons as an universal state of dense nuclear matter in neutron stars.

4. Finding of neutron star regions with localized proton phase.5. Astrophysical consequences of proton localization:

• spontaneous polarization of localized protons as a source of strong magnetic fields of pulsars;

• influence on neutron star cooling rate.

PR

[1] M. Kutschera, Phys. Lett. B340, 1 (1994).[2] M. Kutschera, W. Wójcik, Phys. Lett. B223, 11 (1989).[3] M. Kutschera, W. Wójcik, Phys. Rev. C47, 1077 (1993).[4] M. Kutschera, S. Stachniewicz, A. Szmagliński, W. Wójcik, Acta Phys. Pol. B33, 743 (2002).

References: