Francesca Gulminelli - LPC Caen, France

Extended Nuclear Statistical Equilibrium and

applications to (proto)neutron stars

Collaboration: LPC Caen, LNS Catania, LUTH Meudon, IFIN Bucarest

Dense matter in the universe

F.S.Kitaura et al, A&A 450 (06) 345

• Supernova explosion occurs via core-collapse in very massive stars (M>8Msun)

Dense matter in the universe• Supernova explosion

occurs via core-collapse in very massive stars (M>8Msun)

• 106 < <r 1015 g/cm3 0.01<T<50 MeV in the core

T.Fischer et al, 2011 ApJS 194 39

40 Msun progenitor

Dense matter in the universe• Supernova explosion

occurs via core-collapse in very massive stars (M>8Msun)

• 106 < <r 1015 g/cm3 0.01<T<50 MeV in the core

• The density in the residual pulsar (neutron star) is of the same order <r> ~r0~1014 g/cm3

=> Matter with nucleonic or sub-nucleonic dof !

Matter below saturation• Standard nuclear physics: theory very (?) well

known• Most of the supernova dynamics + NS crust• SN: T>0, yp given by weak rates

• NS: T=0, b-equilibrium • Same physical system but different thermo conditions => very different formalisms

Cluster dofNSE

Nucleon dofDFT

• HF(B) or (E)TF in the WS cell• Extended to finite temperature

6/27

I- DFT for neutron star matter

J. W. Negele and D. Vautherin, NPA 207, 298 (1973)

• HF(B) or (E)TF in the WS cell • Extended to finite temperature• Many applications: crust

composition, pasta, cooling….

7/27

I- DFT for neutron star matter

, R.Wolf et al, PRL 110, 041101 (2013)

M.Fortin et al., PRC 82 (2010), 065804

S.S.Avancini et al., PRC 79 (2009),035804

droplets

rods

slabs

homogeneous

II- NSE for supernova matterNon-interacting ideal-gas of nuclei + (interacting) nucleon gas A. R. Raduta and F. Gulminelli, Phys. Rev. C 82, 065801 (2010) A. S. Botvina and I. N. Mishustin, Nucl. Phys. A 843, 98 (2010)M. Hempel and J. Schaffner-Bielich, Nucl. Phys.A 837, 210 (2010) S. Furusawa, K. Sumiyoshi, S. Yamada, and H. Suzuki, A&A 772, 95 (2013)

• No WS cell: energy (and entropy) calculated in the space • No nucleus-gas interactions• Excluded volume

Þ No match with microscopic calculations in the WS cell

R.I.Epstein W.D.Arnett APJ 201 (1975) 202

Finite temperature stellar matter Cluster distribution + consistency with

DFT&correct T=0 limit

1. Elementary WS cell at T>0: volume associated to each (dressed) cluster

2. mapping microscopic WS cell <=> cluster+gas (Skyrme functionals for both)

3. distribution of WS cells

F.Gulminelli, A.Raduta, ArXiV:1504.04493

Free energy of the

in-medium cluster

Self-interactingnucleons Cluster distribution

𝑝𝑘=1𝑍 𝛽

(𝑧¿¿𝑔𝑎𝑧 )𝑉∏𝐴 , 𝐼

1

𝑛𝐴𝐼(𝑘) !

𝑒𝑥𝑝− 𝛽𝑛𝐴𝐼(𝑘) 𝑭𝒎(𝑨 , 𝑰)¿

Extended NSE: T=0F.Gulminelli, A.Raduta, ArXiV:1504.04493

° NV: Negele-Vautherin (HF)- BPS+BBP: Baym (cluster model) This work (Sly4) This work (SKM*)

M.Fortin, C.Providencia, F.Gulminelli, et al., in preparation

• Can be applied at very low density like cluster models

• Gives the correct melting behavior of clusters at high density like microscopic calculations

• Unified EoS below and above r0

SkI4 mean fieldPichon-Haensel (cluster model)

This work (SkI4)

r(fm-3)

Application: EoS dependence of the NS

crust width M.Fortin, C.Providencia, F.Gulminelli, et al., in preparation

• EoS dependence of NS properties needs a unified EoS!

Extended NSE: T>0

• The T>0 distribution cannot be reduced to the most probable cluster • The presence of a distribution changes the relation -m r => Even the

average quantites !

rB=10-3 fm-3 T=1.5 MeV

F.Gulminelli, A.Raduta, ArXiV:1504.04493

WS cell free-energy

surfaces

Cluster distribution and zoom on light clusters

Single ClusterApproximation

Core collapse 25 Mo - CoConuT

Application: electron capture during

core collapse F.Gulminelli, A.Raduta, M.Oertel, in preparation

tim

e

Inclusion of pairing in the T>0 BCS

approximation

Cao L.G., Lombardo U. and Schuck P., PRC74(2006)

BHF

in-medium surface tension modification

in the LDA

S.Burrello, F.Gulminelli, M.Colonna, in preparation

Excluded volume

T=0 results

Neutron drip

Crust-core transition

r B

Good agreement with full HFB

Temperature dependence of the proton

fraction in b-equilibrium

• The temperature evolution of the proton fraction is important at low density

• Pairing is important close to the crust-core transition

Effect of the cluster distribution

• The cluster distribution becomes wider with increasing temperature • At high temperature the clusters do not dissolve into a homogeneous gas, but

in a gas of neutron-rich resonances• Sizeable effects on the energy density close to the crust-core transition

Heat capacity

• Results consistent with HFB when heavy clusters dominate• Temperature dependence of the proton fraction cannot be neglected• Great sensitivity to the mass model• Extra peaks corresponding to the emergence and dissolution of light resonances

Conclusions• Unified theoretical modelling of T=0 (NS) and T>0

(SN) matter o T=0: a single WS cell variationally determinedo T>0: a statistical distribution of WS cells=dressed

clusters

• T=0: a unified EoS below and above saturation o Melting of clusters in the dense medium o No artificial discontinuity in the P(r) for TOVo => effect of the symmetry energy on the radius and

crust thickness

• T>0: cluster distribution in hot NSo Inclusion of pairing in the local BCS approximation o Pairing gap from BHFo Different peaks in CV due to the presence of light resonances