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Teoria a molti-corpi della materia nucleare. Lezione IV Implicazioni per le stelle di neutroni 2. Cenni sulla fase superfluida 3. Indicazioni sulla EoS da dati osservativi e da collisioni fra ioni pesanti 4. Confronto con EoS fenomenologiche - PowerPoint PPT Presentation
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Teoria a molti-corpi della materia nucleare
Lezione IV
1. Implicazioni per le stelle di neutroni
2. Cenni sulla fase superfluida
3. Indicazioni sulla EoS da dati osservativi e da collisioni fra ioni pesanti
4. Confronto con EoS fenomenologiche
5. Formulazione relativistica, l’ approssimazione Dirac-Brueckner
6. Transizione alla fase di quark, modelli per la fase deconfinata
Rappresentazione schematica di una stella massivain condizioni pre-collasso
SN 1987a
Exploding Before explosion
La “nuvola” espulsa e il rimanente oggetto compatto
Abbondanza di oggetti compatti !
Visione schematica di una pulsar e del suo “faro”
“faro” in direzione della terra “faro” fuori direzione
Distribuzione delle pulsars in cielo rispetto al piano galattico
A section (schematic)
of a neutron star
La parte piu’ internadi una Stella di neutroni“convenzionale”e’ dominata da materianucleare omogenea efortemente asimmetrica
Piu’ avanti ci occuperemodella “crosta”
HHJ : Astrophys. J. 525, L45 (1999
BBG : PRC 69 , 018801 (2004)AP : PRC 58, 1804 (1998)
The baryonic Equations of State
Kh. Gad Nucl. Phys. 747 (2005) 655
Phenomenolocical area from Danielewicz et al.,Science 298 (2002) 1592
Nonostante le incertezzedell’ analisi sembra esserci unaben definita discriminazionetra le diverse EOS
Composition of asymmetric and beta-stable matterComposition of asymmetric and beta-stable matter
•Parabolic approximation
),0,(),1,(),(
),(),0,(),,(
2x-1parameter Asymmetry
2
p
YYYsym
YsymYY
pn
xA
Bx
A
BxE
xExA
Bx
A
B
•Composition of stellar matter
i) Chemical equilibrium among the
different baryonic species
ii) Charge neutrality
iii) Baryon number conservation
np
ep
e
epn
Symmetry energyas a function of density
Proton fraction as afunction of density inneutron stars
AP becomes superluminal at high density and has no DU
Hyperon influence on hadronic EOS
Composition of asymmetric and beta-stable matterComposition of asymmetric and beta-stable matterincludingincluding hyperonshyperons
•Parabolic approximation
),0,(),1,(),(
),(),0,(),,(
2x-1parameter Asymmetry
2
p
YYYsym
YsymYY
pn
xA
Bx
A
BxE
xExA
Bx
A
B
•Composition of stellar matter
i) Chemical equilibrium among the
different baryonic species
ii) Charge neutrality
iii) Baryon number conservation
np
ep
n
pn
e
epn
2
extended to hyperons
•Shift of the hyperon onset points
down to 2-3 times saturation density
•At high densities N and Y present almost in the same percentage.
Including hyperons inside the neutron stars
Mass-Radius relationMass-Radius relationMass-Radius relationMass-Radius relation
• Inclusion of Y decreases the maximum mass value
H.J. Schulze et al., PRC 73, 058801 (2006)
Including Quark matter
Since we have no theory which describes both confined and deconfined phases, we uses two separate EOS for baryon and quark matter and assumes a first order phase transition. a) Baryon EOS. BBG AP HHJ
b) Quark matter EOS. MIT bag model Nambu-Jona Lasinio Coloror dielectric model
The three baryon EOS for beta-stable neutron star matterin the pressure-chemical potential plane.
MIT bag model. “Naive version”
PRC , 025802 (2002)
Al decrescere del valore della bag constant la massa massimadelle NS tende a crescere. Tuttavia B non puo’ essere troppopiccolo altrimenti lo stato fondamentale della materia nucleareall densita’ di saturazione e’ nella fase deconfinata !
Materia nucleare simmetrica
1.1Q GeV3fm
Density dependent bag “constant”
Density profiles of different phasesMIT bag model
Evidence for “large” mass ?
Nice et al. ApJ 634, 1242 (2005) PSR J0751+1807 M = 2.1 +/- 0.2
Ozel, astro-ph /0605106 EXO 0748 – 676 M > 1.8
Quaintrell et al. A&A 401, 313 (2003) NS in VelaX-1 1.8 < M < 2
2
4
222
442 4
3
4
3
a
a
Baa effQM
Non-perturbative corrections ; Strange quark mass
14 a corresponds to the usual MIT bag model
Alford et al. , ApJ 629 (2005) 969
Freedman & McLerran 1978
Maximum mass depends mainly on the parametrizationand not on the transition point
HHJ
BBG
The problem of nuclear matter ground state is solved.
But, in any case one needs an additional repulsion in
quark matter at high density
NJL Model
The model is questionable at high density where the cutoffcan be comparable with the Fermi momentum
Including Color Superconductivity in NJLSteiner,Reddy and Prakash 2002Buballa & Oertel 2002.
Application to NSCT + GSI , PLB 562,,153 (2003)
Mass radius relationshipMaximum mass
NJL , the quark current masses as a function of density
Equivalence between NJL and MIT bag model above chiraltransition (two flavours). For NJL B = 170 MeV
The pressure is zero at zero density ! (no confinement)
The CDM model : the equation of state for symmetric matterC. Maieron et al., PRD 70, 043010 (2004)
The model is confining
The CDM model : maximum mass of neutron star
The effective Bag constsnt in the CDM model
Some (tentative) conclusions
1. The transition to quark matter in NS looks likely, but the amount of quark matter depends on the quak matter model.
2. If the “observed” high NS masses (about 2 solar mass) have to be reproduced, additional repulsion is needed with respect to “naive” quark models .
The situation resembles the one at the beginning of NS physics with the TOV solution for the free neutron gas
The confirmation of a mass definitely larger than 2would be a major breakthrough
3. Further constraints can come from other observational
data (cooling, glitches …….)
Comparison between phenomenological forces andmicroscopic calculations (BBG) at sub-saturationdensities.
M.Baldo et al.. Nucl. Phys. A736, 241 (2004)
Asymmetry (isospin) dependence of EOS
Symmetry energy as a function of density. A comparisonat low density.
Microscopic results approximately fitted by 6.00 )/(3.31
Trying connection with phenomenology : the case.Density functional from microscopic calculations
microscopic functional
The value of r_n - r_p from mic. fun. is consistent with data
rel. mean field
Skyrme and Gogny
Pb208
A section (schematic)
of a neutron star
Negele & Vautherin classical paper. Simple functional,and no pairing.
The structure of nuclei and Z/N ratio are dictated by betaequilibrium epn
No drip region Drip region
Outer Crust Inner Crust
Position of the neutron chemical potential
Looking for the energyminimum at a fixedbaryon density
Density = 1/30 saturationdensity
Wigner-Seitzapproximation
The neutron matter EOS
Solid line : Fayans functional ; Dashes : SLy4Dotted line : microscopic (Av-18)
Including pairing in crust structure calculations
M.B., E. Saperstein et al. , Nucl. Phys. A750, 409 (2005)
Dependence on the functionals
In search of theenergy minimum as a function ofthe Z value insidethe WS cell
Neutron density profile at different Fermi momenta
..
..
. ..
.
...
Proton density profile at different Fermi momenta
1 2
1 Negele & Vautherin
2 Uniform nuclear matter (M.B.,Maieron,Schuck,Vinas NPA 736, 241 (2004))
11
Comparing different Equations of State for low densityDespite the quite different lattice structure, the EoS appears stable.