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 massiva
in 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’ interna
di una Stella di neutroni
“convenzionale”
e’ dominata da materia
nucleare omogenea e
fortemente asimmetrica
Piu’ avanti ci occuperemo
della “crosta”
The baryonic Equations of State
HHJ : Astrophys. J. 525, L45 (1999
BBG : PRC 69 , 018801 (2004)
AP : PRC 58, 1804 (1998)
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
Composition of asymmetric and beta-stable matter
•Parabolic approximation
Asymmetry parameter  
n   p
 1 - 2x p

B
B
(  ,  , xY )  (  ,   0, xY )   2 E sym (  , xY )
A
A
B
B
E sym (  , xY )  (  ,   1, xY )  (  ,   0, xY )
A
A
•Composition of stellar matter
 n   p  e
i) Chemical equilibrium among the
different baryonic species
e   
ii) Charge neutrality
 p  e   
iii) Baryon number conservation
   p  n
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
Hyperon influence on hadronic EOS
Composition of asymmetric and beta-stable matter
including hyperons
•Parabolic approximation
Asymmetry parameter  
n   p
 1 - 2x p

B
B
(  ,  , xY )  (  ,   0, xY )   2 E sym (  , xY )
A
A
B
B
E sym (  , xY )  (  ,   1, xY )  (  ,   0, xY )
A
A
•Composition of stellar matter
n   p  e
i) Chemical equilibrium among the
e  
different baryonic species
2 n   p    
n  
 p  e    
extended to hyperons
ii) Charge neutrality
iii) Baryon number conservation

   p  n    
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.
Mass-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 matter
in the pressure-chemical potential plane.
MIT bag model. “Naive version”
PRC , 025802 (2002)
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 !
Density dependent bag “constant”
 Q  1 .1
GeV fm
3
Density profiles of different phases
MIT 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
Alford et al. , ApJ 629 (2005) 969
QM  
a4
a2
3
4 2
a4  
4
3
4 2
a2  2  Beff
Non-perturbative corrections ;
Strange quark mass
a4  1 corresponds to the usual MIT bag model
Freedman & McLerran 1978
Maximum mass depends mainly on the parametrization
and not on the transition point
BBG
HHJ
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 cutoff
can be comparable with the Fermi momentum
Including Color Superconductivity in NJL
Steiner,Reddy and Prakash 2002
Buballa & Oertel 2002.
Application to NS
CT + GSI ,
PLB 562,,153 (2003)
Mass radius relationship
Maximum mass
NJL , the quark current masses as a function of density
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)
The CDM model : the equation of state for symmetric matter
C. 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 2
would be a major breakthrough
3. Further constraints can come from other observational
data (cooling, glitches …….)
Comparison between phenomenological forces and
microscopic calculations (BBG) at sub-saturation
densities.
M.Baldo et al.. Nucl. Phys. A736, 241 (2004)
Asymmetry (isospin) dependence of EOS
Symmetry energy as a function of density. A comparison
at low density.
Microscopic results approximately fitted by 31.3 (  / 0 )0.6
208
Trying connection with phenomenology : the
Pb 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
A section (schematic)
of a neutron star
The structure of nuclei and Z/N ratio are dictated by beta
n   p  e
equilibrium
Negele & Vautherin classical paper. Simple functional,
and no pairing.
Outer Crust
No drip region
Inner Crust
Drip region
Position of the neutron chemical potential
Looking for the energy
minimum at a fixed
baryon density
Density = 1/30 saturation
density
Wigner-Seitz
approximation
The neutron matter EOS
Solid line : Fayans functional ; Dashes : SLy4
Dotted 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 the
energy minimum
as a function of
the Z value inside
the WS cell
..
.
..
. .
.
.
.
Neutron density profile at different Fermi momenta
Proton density profile at different Fermi momenta
1
11
2
1 Negele & Vautherin
2 Uniform nuclear matter (M.B.,Maieron,Schuck,Vinas NPA 736, 241 (2004))
Comparing different Equations of State for low density
Despite the quite different lattice structure, the EoS
appears stable.