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

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2. Inside a protoneutron star

A PNS differs in several respects from a CNS: At the beginning of its lifetime the PNS contains a high lepton number, [FORMULA], since the core is opaque with respect to neutrinos. A further difference is the high temperature which cannot be neglected with respect to the Fermi temperature throughout the whole PNS. We define, as usual, the lepton number, [FORMULA], as the sum of the net electron fraction, [FORMULA], (where n, [FORMULA], and [FORMULA] are the baryon number density, the electron number density, and the positron number density, respectively) and the net electron neutrino fraction, [FORMULA], (where [FORMULA] and [FORMULA] are the electron neutrino number density and the electron anti-neutrino number density, respectively). Since the muon number density is small in a PNS, they are neglected here. The reason for the small muon number is that the muon lepton family number is conserved, [FORMULA], while the neutrinos are trapped 2. Relativistic calculations lead to the conclusion that NS's and PNS's are composed not only of nucleons and leptons but also of hyperons and, possibly, of nucleon isobars (see, e.g. Pandharipande 1971; Schaffner & Mishustin 1996; Balberg & Gal 1997; Huber et al. 1998). Nevertheless, we shall not take these particle species into account. In view of the rather large uncertainties of the hyperon couplings, we shall, as a first approach to this problem, neglect these additional degrees of freedom. In the following we select four different stages in the evolution of PNS's and NS's, namely at times [FORMULA] ms, [FORMULA] s, [FORMULA] s, and [FORMULA] some minutes after core bounce.

2.1. Protoneutron stars about 50-100 ms after core bounce

This early type protoneutron star (EPNS) is characterized by a hot shocked envelope with an entropy per baryon of [FORMULA] 4-5 for densities [FORMULA] fm-3, an unshocked core with [FORMULA] for densities [FORMULA] fm-3, and a transition region between these densities (Burrows et al. 1995). The entropy per baryon in the very outer layers of an EPNS is larger than [FORMULA]. However, these layers have only a small influence on the EPNS structure and are therefore neglected. We investigate EPNS models with constant lepton number, [FORMULA] for densities above [FORMULA] fm-3 where the neutrinos are trapped (Burrows et al. 1995). Below this density, the neutrinos can freely escape and the chemical potential of the neutrinos vanishes, [FORMULA] (Cooperstein 1988). We refer to Table 1 for the detailed parameters of the EPNS models studied here.


Table 1. Entropies, temperatures, densities, and lepton numbers used in this paper. The entries are: entropy per baryon or temperature in the envelope, [FORMULA], [FORMULA]; entropy per baryon or temperature in the core, [FORMULA], [FORMULA]; maximum baryon number density of the envelope correlated with the entropy per baryon or temperature in the envelope, [FORMULA], [FORMULA]); minimum baryon number density of the core correlated with the entropy per baryon or temperature in the core, [FORMULA], [FORMULA]); baryon number density below which the neutrinos are not trapped, [FORMULA]; baryon number density above which the neutrinos are totally trapped, [FORMULA]; lepton fraction inside the core, [FORMULA].
* This notation means that the first number in parentheses classifies the lower density boundary of the neutrino sphere, i.e. [FORMULA], the second number gives the upper density boundary, i.e. [FORMULA].

2.2. Protoneutron stars at [FORMULA] s after core bounce

At this later stage, the entropy per baryon is approximately constant throughout the star, [FORMULA], except in some outer regions (Burrows & Lattimer 1986; Keil & Janka 1995; Keil 1996). The lepton number is approximately constant since the neutrino diffusion time, [FORMULA] s, is by an order of magnitude larger than the PNS's age. We model this late type protoneutron star (LPNS) with a neutrino transparent envelope with densities [FORMULA] fm-3 and a neutrino opaque core with densities [FORMULA] and [FORMULA] (see Table 1). The transition region between [FORMULA] and [FORMULA] is called neutrino sphere (Janka 1993). We choose four different values for [FORMULA] to simulate the influence of the shape of the neutrino sphere on the structure of LPNS's. We show also a LPNS with [FORMULA] for densities [FORMULA] fm-3, for the sake of comparison, since different evolution calculations show different lepton numbers (Burrows & Lattimer 1986; Keil & Janka 1995; Pons et al. 1999).

These different choices are motivated by recent theoretical calculations of the neutrino-nucleon cross-section that include modifications due to the nucleon-nucleon interaction and spin-spin correlations. The problem was treated, for instance, by Raffelt (1996), Reddy et al. (1998), Burrows & Sawyer (1998), and Pons et al. (1999) using static correlation functions and by Raffelt (1996), Janka et al. (1996), Raffelt & Strobel (1997), and T. Strobel (work in preparation) by using dynamical correlation functions. Because of the high complexity of the problem, the behaviour of the cross section and thus of the location and shape of the neutrino sphere is rather uncertain. Another reason for the different choices is the uncertainty due to convection, that might considerably influence the cooling of PNS's (e.g. Burrows & Lattimer 1988; Keil et al. 1996; Mezzacappa et al. 1998).

For the sake of comparison, we investigate also a model with an isothermal envelope with temperature 3 [FORMULA] MeV for densities below [FORMULA] fm-3. This temperature value is motivated by the fact that the temperature in the central parts of the progenitor star approximately raises to this value before the onset of the core collapse (e.g. Shapiro & Teukolsky 1983; Bethe 1990). It is certainly an upper limit of the true temperature since it corresponds to an increase of the entropy per baryon by three or four orders of magnitude for densities [FORMULA] fm-3 (outer most layer of the star). This high entropy seems to be possible only in hot bubbles (see Mathews et al. 1993).

For comparison we investigate also an unphysical, cold EOS, [FORMULA], with a trapped lepton number of [FORMULA] for densities [FORMULA] fm-3.

2.3. Deleptonized hot neutron star at [FORMULA] s after core bounce

After [FORMULA] s the neutrinos can freely escape and the HNS is nearly deleptonized. This also means that the lepton family number is not conserved anymore. The [FORMULA]-equilibrium is thus given by [FORMULA], [FORMULA], and [FORMULA] for all neutrino species. At this stage muons have to be taken into account since the muon number density is comparable to the electron number density above nuclear matter density. The entropy per baryon is nearly constant, [FORMULA], during the evolution from the LPNS to the HNS (e.g. Burrows & Lattimer 1986; Keil & Janka 1995; Sumiyoshi et al. 1995; Pons et al. 1999).

We again compare the models with isentropic envelopes with models with isothermal envelopes, [FORMULA] MeV or [FORMULA] MeV for densities below [FORMULA] fm-3. The models for the deleptonized HNS's are summarized in Table 1.

2.4. Cold neutron star some minutes after core bounce

After some minutes the NS has a temperature of [FORMULA] MeV throughout the star and the EOS for cold NS matter can be used to describe the CNS, because the thermal effects are negligibly small (see Shapiro & Teukolsky 1983). We shall adopt the model derived by Baym et al. (1971) for densities below neutron drip density, [FORMULA] fm-3, and Negele & Vautherin (1973) for densities between neutron drip density and the transition density, [FORMULA] fm-3. Above this transition density, we use the model for CNS matter in [FORMULA]-equilibrium without neutrinos derived by Strobel et al. (1997).

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Online publication: October 4, 1999