Astron. Astrophys. 350, 497-512 (1999)

3. Equation of state for protoneutron stars

The 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 ⁴, where they use exact numerical solutions for the integration over the Fermi-Dirac distribution functions. Appendix A contains a brief description of this approach. We extend the nuclear EOS to subnuclear densities and to different compositions of PNS and HNS matter (i.e. trapped neutrinos, constant entropy per baryon, ). In the subnuclear regime, the EOS was also obtained by means of the homogeneous Thomas-Fermi model. The electron number is derived by fitting the pressure to the subnuclear EOS's of Baym et al. (1971) and Negele & Vautherin (1973) for densities below the density of the neutrino sphere in the case of EPNS and LPNS models and below the nuclear density in the case of HNS models. Our results for subnuclear densities are comparable to the EOS's derived by Lattimer & Swesty (1991) and used in the investigations of Goussard et al. (1997, 1998) and Gondek et al. (1997).

Figs. 1-3 show the pressure, , as function of the baryon number density, n, for different physical scenarios. The envelope region with densities,  fm^-3 is depicted in Fig. 1 for the isothermal part with of the EOS and the EOS, the isentropic EOS, and the EOS for a CNS. The neutrinos do not contribute to the pressure for densities  fm^-3, since they are not trapped in this region. In the EOS with  MeV, the pressure is dominated by the contribution of the photons, MeV^-3 fm^-3 = MeV fm^-3, for low densities. Nevertheless, the influence of this low density region on gross properties of LPNS's and HNS's is only small (see Sect. 4). In contrast to the isothermal EOS, the isentropic EOS is almost identical with the cold EOS in this density region.

Fig. 1. Pressure versus baryon number density for densities fm-3 of hot dense matter. The curve CNS corresponds to cold matter (see Sect. 2.4). The and curve is the isothermal envelope part of an isentropic core with entropy per baryon (see Sects. 2.2 and 2.3 for explanation). Finally the curve corresponds to the envelope part of this EOS with and no trapped lepton number in this density region.

Fig. 2. Pressure versus baryon number density in the density region fm-3 for different EOS's of hot dense matter. The abbreviations are described in Table 1.

Fig. 3. Pressure versus baryon number density for densities fm-3 for different EOS's of our model of hot dense matter. The abbreviations are described in Table 1. The pressure of the EOS is nearly identical to the case in this density region and is not shown for that reason.

Fig. 2 shows the general trend that the leptons dominate the pressure in the density region around and above the neutrino sphere,   fm^-3 for LPNS's. Both thermal effects and the trapped lepton number contribute significantly to the pressure and increase it by a factor of each at  fm^-3. The thermal effects are even larger for the high entropy model labeled . The impact of the shape of the neutrino sphere can be inferred by comparison of the curves labeled and .

With increasing density the temperature dependence of the nuclear EOS increases. At the highest densities possible in PNS's ( fm^-3) the pressure increase due to thermal effects and due to high lepton numbers become comparable (see Fig. 3). This feature is clearly expressed by the nearly identical pressure of the and the EOS's.

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 ⁵. At this point it should be mentioned, that the behaviour of at high densities is highly uncertain. At  fm^-3 we obtain, for instance, a pressure increase due to thermal effects of (from the to the case), whereas Lattimer & Swesty (1991) obtain an increase of only (see Fig. 1b in Goussard et al. 1997). The impact on the structure of PNS's and HNS's are discussed in Sect. 4, where we compare our results with the results of Goussard et al. (1997, 1998) and Gondek et al. (1997) who used the EOS of Lattimer & Swesty (1991).

The number density and the mean energy of the neutrinos , , , , , are given by ⁶:

and

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 g denotes the spin-degeneracy factor and is related to the spin, , of the particles by . Since only positive helicity neutrinos and negative helicity anti-neutrinos exist ⁷, the degeneracy factor is equal 1 for the neutrinos. In the case of vanishing chemical potential Eqs. (2) and (3) lead to a temperature dependence of the number density:

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:

and

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.

Fig. 4. Chemical potential versus baryon number density for electrons and electron-neutrinos of the and cases. The dotted lines correspond to the CNS, the and the EOS's.

Fig. 5. Temperature versus baryon number density for different EOS's.

Fig. 6. Mean neutrino energies versus baryon number density for all neutrino types in the case for densities larger than fm-3 ( and ).

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^-3) and the unshocked core ( fm^-3) (see Burrows & Lattimer 1986; Burrows et al. 1995). The temperature increases with increasing entropy per baryon (see discussion in Sect. 4.2) and decreases with increasing lepton number, since the neutron fraction and the proton fraction become more equal (see Prakash et al. 1997). The maximum temperature in the most massive PNS's and HNS's reaches values between 80 and 120 MeV (see Fig. 5 and Table 5). The temperature in PNS's and HNS's with a typical baryonic mass of has values between 20 and 40 MeV (see Table 3).

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^-3 since the symmetry energy decreases for densities  fm^-3 (Strobel et al. 1997). Non-relativistic EOS's derived within variational approaches behave similar in this respect, whereas the symmetry energy derived in relativistic and non-relativistic Brückner-Bethe calculations monotonically increases with density (see Strobel et al. 1997). Higher temperatures cause a small decrease (increase) of the electron (electron neutrino) fraction (see Takatsuka et al. 1994). Without trapped neutrinos the sum of the electron and muon fraction increases, in first approximation, quadratically with increasing temperature (see Fig. 7 and Keil & Janka 1995).

Fig. 7. Lepton fractions versus baryon number density for different EOS's. The lepton fractions of the unphysical EOS are nearly identical to that of the EOS and is not shown for that reason. The lower dotted line corresponds to the CNS EOS, the middle dotted line to the EOS and the upper dotted line to the EOS. The dotted lines show the sum of the electron and muon fraction. The stars correspond to the positron fraction of the case. The positron fraction of all other EOS's lies below the resolution of this figure.

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 .

Fig. 8. Effective mass of neutrons and protons versus baryon number density. The solid line shows the effective mass of neutrons and protons for symmetric nuclear matter (the difference in the values for neutrons and protons are negligibly small). The long dashed line shows the effective mass of neutrons in pure neutron matter. The dot-dashed line shows the effective mass of protons brought into pure neutron matter.

The adiabatic index :

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^-3 (see Gondek et al. 1997). The steep behaviour of the adiabatic index around nuclear matter density is the reason for the core bounce of the collapsing iron core of the progenitor star (Shapiro & Teukolsky 1983).

Fig. 9. Adiabatic index versus baryon number density for different EOS's.

The speed of sound in units of the speed of light c:

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.

Table 2. The speed of sound in the density region around and above nuclear matter density for different EOS's. The abbreviations are described in Table 1. The maximum value is reached in the CNS EOS: (in units of c).

© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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