Transcript EoS - BAS

Equation of state for dense
supernova matter
Ch.C. Moustakidis
Department of Theoretical Physics
Aristotle University of Thessaloniki
Greece
28o International Workshop on Nuclear Theory
Rila Mountains, Bulgaria 21-27 June 2009
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 Msun) .
Evaluation of supernova
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).
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.
 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
 We concentrate our study at this stage.
SN 1987A
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.
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.
 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
 Various theoretical models have been applied for the study of hot nuclear matter
based on realistic or phenomenological interactions.
 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.
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.
Main points of the present work
1) Only few calculations of the EOS of SNM at high densities
2) The effect of the nuclear symmetry energy dependence on the EOS
3) The simultaneously study of thermal effects on the kinetic and interaction
part of the nuclear symmetry energy
4) Thermal and interaction effects on the chemical composition of the SNM
5) Comparison of the EOS of hot and cold neutron stars
6) Construction of adiabatic EOS
Momentum-dependent Yukawa interaction (MDYI)
The most general two-body interaction is a sum of a momentum-independent
part and a momentum-dependent part:
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 model
The energy density of the asymmetric nuclear matter (ANM)
The potential contribution
The function g (k,Λ) suitably chosen to simulate finite range effects is of the form
The contribution of the various terms
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
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
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:
Leptons density-energy –pressure
 Density:
 Energy density:
 Pressure:
 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
Equation of state of hot nuclear matter
 The total energy density is given by
 The total pressure is given by
 The baryon contribution:
 The entropy density (both
for baryons and leptons):
 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
Results for supernova matter
Energy per baryon of supernova matter ESM and
cold neutron star matter ENS
Conclusions-Comments
 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.
 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.
 The most striking feature, in adiabatic case (S=1), is the slight dependence of the
fractions Yi from the baryon density.
 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 above EOS can be applied to the evaluation of the gross properties of hot neutron
stars i.e. mass and radius.
 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).
Supernova matter
 Supernova matter (SNM) exist in a collapsing supernova
core and eventually forms a hot neutron star (protoneutron 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