Teoria a molti-corpi della

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Teoria a molti-corpi della materia
nucleare
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• 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
• 5. Formulazione relativistica, l approssimazione
• Dirac-Brueckner
• 6. Transizione alla fase di quark, modelli per la
  fase
• deconfinata

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Rappresentazione schematica di una stella
massiva in condizioni pre-collasso

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SN 1987a
Exploding
Before explosion
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La nuvola espulsa e il rimanente oggetto
compatto
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Abbondanza di oggetti compatti !
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Visione schematica di una pulsar e del suo faro
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faro in direzione della terra
faro fuori direzione
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Distribuzione delle pulsars in cielo rispetto al
piano galattico
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A section (schematic)
of a neutron star
La parte piu interna di una Stella di
neutroni convenzionale e dominata da
materia nucleare omogenea e fortemente
asimmetrica Piu avanti ci occuperemo della
crosta
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The baryonic Equations of State
HHJ Astrophys. J. 525, L45 (1999
BBG PRC 69 , 018801 (2004) AP PRC
58, 1804 (1998)
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Phenomenolocical area from Danielewicz et
al., Science 298 (2002) 1592
Nonostante le incertezze dell analisi sembra
esserci una ben definita discriminazione tra le
diverse EOS
Kh. Gad Nucl. Phys. 747 (2005) 655
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Composition of asymmetric and beta-stable matter

• Parabolic approximation

• Composition of stellar matter

i) Chemical equilibrium among the different
baryonic species ii) Charge neutrality iii)
Baryon number conservation
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Symmetry energy as a function of
density Proton fraction as a function of
density in neutron stars
AP becomes superluminal at high density and has
no DU
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Hyperon influence on hadronic EOS
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Composition of asymmetric and beta-stable
matterincluding hyperons

• Parabolic approximation

extended to hyperons

• Composition of stellar matter

i) Chemical equilibrium among the different
baryonic species ii) Charge neutrality iii)
Baryon number conservation
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Including hyperons inside the neutron stars

• 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.

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Mass-Radius relation

• Inclusion of Y decreases the maximum mass value

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H.J. Schulze et al., PRC 73, 058801 (2006)
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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.
• Baryon EOS. BBG
• AP
• HHJ
  
• Quark matter EOS. MIT bag model
• 
  Nambu-Jona Lasinio
• 
  Coloror dielectric model

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The three baryon EOS for beta-stable neutron star
matter in the pressure-chemical potential plane.

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MIT bag model. Naive version

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PRC , 025802 (2002)
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Materia nucleare simmetrica
Al decrescere del valore della bag constant la
massa massima delle NS tende a crescere. Tuttavia
B non puo essere troppo piccolo altrimenti lo
stato fondamentale della materia nucleare all
densita di saturazione e nella fase
deconfinata !
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Density dependent bag constant
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Density profiles of different phases MIT bag
model
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Evidence for large mass ?
Nice et al. ApJ 634, 1242 (2005)
PSR J07511807 M 2.1
/- 0.2 Ozel, astro-ph /0605106
EXO 0748 676
M gt 1.8 Quaintrell et al.
AA 401, 313 (2003) NS in VelaX-1
1.8 lt M lt 2
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Alford et al. , ApJ 629 (2005) 969
Non-perturbative corrections Strange quark
mass
corresponds to the usual MIT bag model
Freedman McLerran 1978
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Maximum mass depends mainly on the
parametrization and not on the transition point

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BBG
HHJ
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The problem of nuclear matter ground state is
solved. But, in any case one needs an
additional repulsion in quark matter at high
density
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NJL Model
The model is questionable at high density where
the cutoff can be comparable with the Fermi
momentum
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Including Color Superconductivity in
NJL Steiner,Reddy and Prakash 2002 Buballa
Oertel 2002. Application to NS CT GSI ,
PLB 562,,153 (2003)
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Mass radius relationship Maximum mass
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NJL , the quark current masses as a function of
density
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Equivalence between NJL and MIT bag model above
chiral transition (two flavours). For NJL B
170 MeV
The pressure is zero at zero density ! (no
confinement)
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The CDM model the equation of state for
symmetric matter C. Maieron et al., PRD 70,
043010 (2004)
The model is confining
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The CDM model maximum mass of neutron star
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The effective Bag constsnt in the CDM model
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Some (tentative) conclusions

• The transition to quark matter in NS looks
  likely,
• but the amount of quark matter depends on the
  quak
• matter model.
  
• 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
  2
• would be a major breakthrough

3. Further constraints can come from other
observational data (cooling, glitches .)

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Comparison between phenomenological forces
and microscopic calculations (BBG) at
sub-saturation densities.
M.Baldo et al.. Nucl. Phys. A736, 241 (2004)
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Asymmetry (isospin) dependence of EOS
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Symmetry energy as a function of density. A
comparison at low density.
Microscopic results approximately fitted by
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Trying connection with phenomenology the
case. Density functional from microscopic
calculations
rel. mean field
Skyrme and Gogny
microscopic functional
The value of r_n - r_p from mic. fun. is
consistent with data
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A section (schematic)
of a neutron star
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The structure of nuclei and Z/N ratio are
dictated by beta equilibrium
Negele Vautherin classical paper. Simple
functional, and no pairing.
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Outer Crust
Inner Crust
No drip region
Drip region
Position of the neutron chemical potential
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Looking for the energy minimum at a fixed baryon
density
Density 1/30 saturation density
Wigner-Seitz approximation
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The neutron matter EOS
Solid line Fayans functional Dashes
SLy4 Dotted line microscopic (Av-18)
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Including pairing in crust structure calculations
M.B., E. Saperstein et al. , Nucl. Phys. A750,
409 (2005)
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Dependence on the functionals
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In search of the energy minimum as a function
of the Z value inside the WS cell
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Neutron density profile at different Fermi momenta
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Proton density profile at different Fermi momenta
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2
1
1
1
1 Negele Vautherin
2 Uniform nuclear matter (M.B.,Maieron,Schuck,Vi
nas NPA 736, 241 (2004))
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Comparing different Equations of State for low
density Despite the quite different lattice
structure, the EoS appears stable.