b - stable matter of protoneutron star

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1 Mean-field study of the hot b - stable matter of protoneutron star Dao Tien Khoa INST Hanoi, VINATOM EOS of hot nuclear matter with a high neutron-proton asymmetry. EOS of hot b - stable baryon-lepton matter of PNS: n - free and n - trapped scenarios for PNS at entropy S/A=1,2 and 4; impact of the symmetry energy and nucleon effective mass. Configuration of the n free PNS matter at S/A=4 and comparison of the hydrodynamic simulation of the core-collapse (failed) supernova.

2 A. Burrows and J.M. Lattimer, Astrophys. J. 307, 178 (1986); J.M. Lattimer and M. Prakash, Phys. Rep. 442, 109 (2007) Proto-neutron star (PNS) is a unique test ground for the mean-field models of hot NM

3 Equation of state of hot asymmetric nuclear matter N.H. Tan, D.T. Loan, D.T. Khoa, J. Margueron, Phys. Rev. C 93, (2016). HF energy density of hot NM in-medium NN interaction Neutron-proton asymmetry Nucleon momentum distribution Single-particle energy Density - and momentum dependent single-particle potential at finite temperature (based on HvH theorem)

4 In-medium (density dependent) NN interaction D.T. Khoa and W. von Oertzen, Phys. Lett. B 304 (1993) 8; B 342 (1995) 6 E = Direct Exchange Direct Exchange = + CDM3Yn density dependence D.T. Khoa, G.R. Satchler and W. von Oertzen, Phys. Rev. C 56, 954 (1997); D.T. Loan, B.M. Loc, and D.T. Khoa, Phys. Rev. C 92, (2015). G-matrix based M3Y interaction N. Anantaraman, H. Toki, G.F. Bertsch Nucl. Phys. A 398 (1983) 269.

5 HF results given by some mean-field interaction CDM3Yn: D.T. Khoa, G.R. Satchler, and W. von Oertzen, Phys. Rev. C 56, 954 (1997); D.T. Khoa, H.S. Than, and D.C. Cuong, Phys. Rev. C 76, (2007). M3Y-Pn: H. Nakada, Phys. Rev. C 78, (2008); Phys. Rev. C 87, (2013) D1S: J.F. Berger, M. Girod, and D. Gogny, Comp. Phys. Comm. 63, 365 (1991). D1N: F. Chappert, M. Girod, and S. Hilaire, Phys. Lett. B 668, 420 (2008). SLy4: E. Chabanat et al., Nucl. Phys. A 635, 231 (1998) Ab-initio variational calculation using Argon V18 NN + NNN inter. APR: A. Akmal, V.R. Pandharipande, and D.G. Ravenhall, Phys. Rev. C 58, 1804 (1998)

6 Basic thermodynamic properties of hot temperature T Different nuclear EOS s affect the nucleon entropy density via the s/p potential entering the nucleon momentum distribution Different nuclear EOS s affect the Helmholtz free energy (per baryon) and pressure of hot NM via both the total HF energy and s/p potential. Free symmetry energy per baryon Parabolic law: Quadratic dependence on neutron-proton asymmetry d?

7 Helmholtz free energy K=252 MeV K=218 MeV Microscopic BHF calculation using Argon V18 NN + Urbana NNN term G.F. Burgio and H.J. Schulze, Astronomy & Astrophysics 518, A17 (2010).

8 K=221 MeV K=230 MeV D1N version of Gogny int. F. Chappert, M. Girod, S. Hilaire, Phys. Lett. B 668, 420 (2008); Sly4 version of Skyrme int. E. Chabanat et al., Nucl. Phys. A 635, 231 (1998).

9 Free symmetry energy stiff F 1 (n b ) based on the BHF results by Jeukenne, Lejeune, Mahaux, Phys. Rev. C 16, 80 (1977). Fine tuned to the CC results for charge exc. reactions to IAS. soft F 1 (n b ) has the same functional form as F 0 (n b ) suggested in D.T. Khoa, G.R. Satchler, W. von Oertzen, Phys. Rev. C 56, 954 (1997).

10 M3Y-Pn interactions: H. Nakada, Phys. Rev. C 78, (2008); H. Nakada, Phys. Rev. C 87, (2013). Soft sym. energy D1N version of Gogny int. F. Chappert, M. Girod, S. Hilaire, Phys. Lett. B 668, 420 (2008); Sly4 version of Skyrme int. E. Chabanat et al., Nucl. Phys. A 635, 231 (1998).

11 Parabolic law is not accurate at high temperature because of the finite entropy!

12 EOS of b - stable baryon-lepton matter of hot PNS N.H. Tan, D.T. Loan, D.T. Khoa, J.Margueron, Phys. Rev. C 93, (2016). Hartree-Fock energy density Relativistic Fermi gases EOS of baryon-lepton matter Total free energy and entropy per baryon

13 Initial and final conditions of hot PNS Neutrinos are trapped at the onset of PNS Electron lepton fraction Y e ~ 0.4 and Y m ~ 0. A. Burrows and J.M. Lattimer, Astrophys. J. 307, 178 (1986). Most of neutrinos escaped before PNS cools down to NS or collapses to form BH

14 b-equilibrium condition Charge neutrality Conservation of total lepton fractions Neutron-proton asymmetry becomes a dynamic variable in the b-stable PNS matter. Impact of the nuclear symmetry energy is weaker in the presence of trapped neutrinos

15 Density profiles of the entropy per baryon and temperature of the b-stable baryon-lepton matter of hot PNS Weaker impact of the symmetry energy at the onset of PNS (n-trapped case)

16 Particle fractions in the b-stable and n-free PNS matter at S/A=1,2,4 Diminishing impact of the symmetry energy with increasing entropy

17 Nucleon effective mass in the b - stable PNS matter Determined at each point of the (S,T) grid Neutron effective mass in n-free PNS matter at S/A=1,2 and 4 Proton effective mass in n-free PNS matter at S/A=1,2 and 4

18 Strong impact of the nucleon effective mass on temperature of PNS

19 Mass ~ 40 M o PNS collapses directly to black hole in a failed supernova!

20 Simulated PNS matter in a failed supernova small v the onset of collapse to BH v-free matter Entropy per baryon S/A ~ 4 Maximum temperature of the PNS matter T ~ MeV Only results given by CDM3Yn int. agree with simulation!

21 Gravitational mass of the b-stable PNS at different entropies PNS NS

22 Gravitational mass of the b-stable, n-free PNS at S/A=4 Delay time from the onset of collapse to the BH formation Open symbols based on the simulation by Hempel et al.

23 Thank you