## 3. Equation of state for protoneutron starsThe EOS of PNS matter is the basic input quantity whose knowledge
over a wide range of densities, ranging from the density of iron at
the star's surface up to about eight times the density of normal
nuclear matter reached in the cores of the most massive stars of a
sequence, is necessary to solve the structure equations. Due to the
high lepton number, the EOS of PNS's is different from the EOS's for
cold and hot NS's with low lepton numbers. The nuclear EOS used in
this paper for the description of a PNS is a Thomas-Fermi model of
average nuclear properties, with a momentum- and density-dependent,
effective nucleon-nucleon interaction developed by Myers &
Swiatecki (1990, 1991). The parameters of the nuclear EOS were
adjusted to reproduce a wide range of properties of normal nuclear
matter and nuclei (Myers & Swiatecki 1995, 1996, 1998; Wang et al.
1997). Strobel et al. (1999) extended this approach to the case of
finite
temperature Figs. 1-3 show the pressure, , as
function of the baryon number density,
Fig. 2 shows the general trend that the leptons dominate the
pressure in the density region around and above the neutrino sphere,
fm With increasing density the temperature dependence of the nuclear
EOS increases. At the highest densities possible in PNS's
( fm The EOS of Lattimer & Swesty (1991) shows a smaller temperature
dependence at high densities (see Goussard et al. 1997) in comparison
with our EOS. The reason for this is that the temperature dependence
of the EOS of Lattimer & Swesty (1991) lies entirely in the
kinetic part of the enegy density, since they choose
in their
approach The number density and the mean energy of the neutrinos
,
,
,
,
,
are given by respectively. The trapped electron neutrinos and anti-neutrinos are
in chemical equilibrium, . The
chemical potentials of all other neutrino species vanish,
(with
,
,
,
) due to the lepton family number
conservation (the muon number density is small compared to the
electron number density in PNS's and is therefore neglected, for
simplicity). The factor and a linear temperature dependence of the mean neutrino energy: with , , , . Due to the high chemical potential of the electron neutrinos in the case of a high trapped lepton number (; see Fig. 4 and Fig. 5), the number density and the mean energy of the electron anti-neutrinos can be approximated by: This leads to the simple, linear temperature dependence of the mean energy of the electron anti-neutrinos: the energy of an ultra-relativistic Boltzmann gas. The mean neutrino energies of all neutrino species are shown in Fig. 6 for the case . Fig. 4 shows the chemical potential of the electron neutrinos and the electrons for different models of the EOS.
In Fig. 5 we show the density dependence of the temperature for
different EOS's. One can see the temperature drop in the
EOS at the interface between the hot
shocked envelope ( fm The fractions of electrons and electron-neutrinos are shown in
Fig. 7 for the and
cases with constant lepton number.
The lepton fractions of the unphysical
EOS are nearly identical to that of
the EOS and are not shown in Fig. 7
for that reason. The fractions are nearly constant
( for electrons and
for electron neutrinos) in a wide
range of densities. The electron fraction decreases for
fm
The effective masses, , of neutrons and protons for symmetric and pure neutron matter are shown in Fig. 8. The effective mass for symmetric nuclear matter at saturation density is found to be for neutrons and protons in our model. This is in good agreement with the generally accepted, experimental value (e.g. Bauer et al. 1982). In the case of pure neutron matter, the effective neutron mass increases up to , at nuclear matter density, whereas the effective proton mass in pure neutron matter decreases to .
is shown for different EOS's in Fig. 9. The adiabatic index
decreases with increasing temperature and lepton numbers for densities
around and above nuclear density. In contrast, it decreases with
increasing lepton number for densities
fm
The speed of sound in units of
the speed of light is tabulated in Table 2 for different EOS's. The speed of sound increases with density up to nearly the speed of light, in the most massive stars of a sequence. Nevertheless, is always smaller than the speed of light and all EOS's used in this paper are causal. The speed of sound increases with temperature and lepton number at fixed density in contrast to the results of Goussard et al. (1998). This is probably caused by the smaller temperature dependence of the EOS of Lattimer & Swesty (1991), which was discussed before in this section.
© European Southern Observatory (ESO) 1999 Online publication: October 4, 1999 |