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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
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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
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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*
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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 /
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UV14+TNI Model
320
2342
2340
,
15333517838
33871330959
fmnMeVVV
nnnnnV
nnnnnV
UV
UV
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AV14+UVII Model
nnn
nnnnV
nnn
nnnnV
AV
AV
1888601844
1930961187
36312652560
31801786378
23
4562
23
4560
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UV14+UVII Model
nnnn
nnnnV
nnn
nnnnV
UVU
UVU
152357140935
1528873175
34710651602
1749848152
234
5672
23
4560
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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
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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,,
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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
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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
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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
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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
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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
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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
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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
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Comparisonof simple variational method
and self-consistent variational method
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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
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Mass and radii of neutron stars:• Tolman-Oppenheimer-Volkoff
equation • Equation of state
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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
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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
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[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: