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.