1. Ch.C. Moustakidis Department of Theoretical Physics Aristotle University of Thessaloniki Greece Equation of state for dense supernova matter 28o International Workshop on Nuclear Theory Rila Mountains, Bulgaria 21-27 June 2009

2. Supernova theory   Supernova is spectacular event. The recent one, SN1987A, emitted light at a rate 100 million times that of the Sun.   Supernovae are very rare. The last one seen in our galaxy was Kepler’s in 1604.   Zwicky and Baade proposed that supernovae derive their energy from the gravitational collapse of the central core of a star (type II supernova).   Type II supernovae occur at the end of the evaluation of massive stars (M>8 M sun ).

3. A) The fusion reaction continues till the core of the star consists of heavy elements (like iron). B) The core begins to collapse. C) Once the densities of the central part of the core surpasses the normal nuclear matter density, the repulsive part of the nuclear force offers a powerful resistance for further compression. D) The shock waves produced lead to a spectacular explosion resulting also in the formation of a hot neutron star (protoneutron star). Evaluation of supernova

4. Supernova matter   Protoneutron star with radius of about 100 Km and T>50 MeV is short lived and then contracts rapidly (radius 10 Km, T<1 MeV).   This stage is identified as the birth of a neutron star, it is hot and composed of the so-called supernova matter.   We concentrate our study at this stage.   It is characterized by: 1) almost constant entropy per baryon (S=1-2 k B ) 2) high and almost constant lepton fraction Yl=0.3-0.4

5. In February 1987 a supernova appeared near the Tarantula nebula in our satellite galaxy the Large Magellanic Cloud, about 169,000 light years away. As the first supernova discovered in 1987, it was called SN1987A following astronomical convention. SN1987A was the first "nearby" supernova of the modern era and the closest supernova since Kepler's supernova in 1604.Large Magellanic Cloud SN1987A

7. Introductory remarks   The equation of state (EOS) of hot asymmetric nuclear matter determines the structure inside a supernova and hot neutron star, plays important roles for the study of the supernova explosion and the evolution of a neutron star at the birth stage.   Various theoretical models have been applied for the study of hot nuclear matter based on realistic or phenomenological interactions.   EOS is important for understanding the liquid-gas phase transition of asymmetric nuclear matter and for theoretical predictions of the properties of heavy-ion collisions   In the present work we apply a momentum-dependent effective interaction model. The model is able to reproduce the results of the microscopic calculations of both nuclear and neutron-rich matter T=0. The model can be extended to finite temperature.   The model is flexible to reproduce a variety of density dependent behaviors of the nuclear symmetry energy and symmetry free energy which are of importance for the study of nuclear equation of state and mainly the proton fraction and as a consequence the composition of hot β-stable nuclear matter.

8. Why we are interested about thermal effects on nuclear matter?   The main part of the calculations concerning cold nuclear matter (T=0).   There is an increasing interest for the study of the hot nuclear matter, the properties of neutron star and supernova, and the heavy-ion collisions properties at finite temperature.   We apply a momentum dependent effective interaction model. In that way, we are able to study simultaneously thermal effects not only on the kinetic part of the energy but also on the interaction part.   The temperature dependence of the proton fraction as well as of the electron and neutrino chemical potentials are related with the thermal evaluation of the supernova and the proton-neutron stars.

9. Main points of the present work 1) Only few calculations of the EOS of SNM at high densities 2) 2)The effect of the nuclear symmetry energy dependence on the EOS 3) 3)The simultaneously study of thermal effects on the kinetic and interaction part of the nuclear symmetry energy 4) 4)Thermal and interaction effects on the chemical composition of the SNM 5) 5)Comparison of the EOS of hot and cold neutron stars 6) Construction of adiabatic EOS

10. Momentum-dependent Yukawa interaction (MDYI) The momentum-independent part is approximated by a zero-range coordinate space interaction The momentum-dependent part is parametrized by MDYI which is also of zero range in coordinate space The most general two-body interaction is a sum of a momentum-independent part and a momentum-dependent part:

11. The model The energy density of the asymmetric nuclear matter (ANM)

12. The potential contribution The function g (k,Λ) suitably chosen to simulate finite range effects is of the form

13. The contribution of the various terms

17. Thermodynamic description of hot nuclear matter   The key quantities for the study of hot nuclear matter is the Helmholtz free energy F and internal energy E

18. Free energy and chemical potentials   The connection between free energy and chemical potentials is the basic ingredient of the present calculations   The free energy can be approximated by the parabolic relation   The key relation between free energy and chemical potentials

20. β-equilibrium-leptons contribution   Stable nuclear matter must be in chemical equilibrium for all types of reactions including the weak interaction   Chemical equilibrium can be expressed as:   Charge neutrality condition provides:   Total fraction:   Basic equation:

21. Leptons density-energy –pressure   Energy density:   Pressure:   Density:   One can solve self-consistently a system of equations in order to calculate the proton fraction Yp, the leptons fractions Yl and Yνε and the corresponding chemical potentials

22. Equation of state of hot nuclear matter   The total energy density is given by   The total pressure is given by   The baryon contribution:   From the above equations we can construct the isothermal and adiabatic curves for energy and pressure and finally to derive the isothermal (adiabatic) behavior of the EOS of hot nuclear matter under β-equilibrium   The entropy density (both for baryons and leptons):

23. Results for supernova matter

35. Energy per baryon of supernova matter E SM and cold neutron star matter E NS

37. Conclusions-Comments   Nuclear symmetry energy dependence on baryon density is less important in supernova matter than in cold neutron star matter.   Temperature affects appreciably both baryon and lepton contribution on the entropy.   The lepton energy dominates in the internal energy of the matter up to n~0.7 fm^(-3). The baryon contributions dominated only for n>0.7 fm^(-3). This is a characteristic of the supernova matter and is remarkable contrast with the situation of cold NSM.   The baryon pressure dominates on the total pressure especially for n>0.2 fm^(-3). The main part of lepton pressure originates from electrons.   We investigate thermal effects on equation of state of β-stable hot nuclear matter by applying a model with a momentum-dependent effective interaction. We study thermal effects both on the kinetic and the interaction part of the energy density.

38.   The most striking feature, in adiabatic case (S=1), is the slight dependence of the fractions Yi from the baryon density.   The above EOS can be applied to the evaluation of the gross properties of hot neutron stars i.e. mass and radius.   The internal energy of supernova matter is remarkably larger than that of neutron star matter. This is due mainly on the remarkably larger contribution of the leptons (large lepton fraction). High temperature also contributes but is less effective than the high lepton fraction.   The model can be applied for the study of formation of nonhomogeneities of neutron star (location of the inner edge of the crust, by considering neutrino-free matter) as well as on supernova core (by including finite temperature and effect of neutrino trapping).

39. Supernova matter   Supernova matter (SNM) exist in a collapsing supernova core and eventually forms a hot neutron star (proto- neutron star)   SNM is another form of nuclear matter distinguished in the participation of degenerate electrons and trapped neutrinos.   It is characterized by: 1) almost constant entropy per baryon (S=1-2 kB ) 2) high and almost constant lepton fraction Yl=0.3-0.4