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

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1. Introduction

It is generally believed that a neutron star (NS) is born as a result of the gravitational collapse of the iron core of a massive evolved progenitor star ([FORMULA]) in a Type-II supernova (e.g. Bethe 1990). The iron core of such a star collapses when its mass reaches the Chandrasekhar limit


where [FORMULA] denotes the number of electrons per baryon which depends on the mass of the progenitor star. Due to the Fermi pressure of the nucleons, the collapse stops when nuclear matter density is reached and the core bounces back. Shortly after core bounce (some 10 ms) a hot, lepton rich NS, called protoneutron star (PNS), is formed. This PNS consists of a shocked envelope with an entropy per baryon 1 s [FORMULA] 4-10 and an unshocked core with [FORMULA] (Burrows et al. 1995). The envelope and the core contain nearly the same mass of about 0.6 - [FORMULA] (slightly depending on the mass of the progenitor star, see Burrows et al. 1995; Keil et al. 1996). During the so-called Kelvin-Helmholtz cooling phase (e.g. Janka 1993) the lepton number decreases in the PNS due to the loss of neutrinos and consequently the PNS evolves in about 10 to 30 seconds into a hot, lepton poor neutron star (HNS) with an entropy per baryon, [FORMULA], depending on the model (e.g. Burrows & Lattimer 1986; Keil & Janka 1995; Sumiyoshi et al. 1995; Pons et al. 1999). After several minutes this HNS cools to a cold neutron star (CNS) with temperatures [FORMULA] MeV throughout the star (e.g. Keil & Janka 1995). Finally, it slowly cools via neutrino and photon emission until its thermal radiation is too weak to be observable after about [FORMULA] yr (e.g. Tsuruta & Cameron 1966; Schaab et al. 1996).

The PNS is in [FORMULA]-equilibrium during its lifetime, since the time scale of the weak-interaction is much smaller than the evolutionary time scale, i.e. the neutrino diffusion time scale or the neutrino cooling time scale, respectively. Hence the evolution of a PNS can be studied by considering quasi-stationary models at different times.

The properties of PNS's were investigated by different authors. For example, the case of non-rotating PNS's was studied by Takatsuka et al. (1994), Bombaci et al. (1995), and Prakash et al. (1997). The case of rotating PNS's was treated by Romero et al. (1992) and Takatsuka (1995, 1996) by means of an empirical formula for the Kepler frequency which was developed for CNS's (e.g. Haensel et al. 1995). Hashimoto et al. (1994) and Goussard et al. (1997) account for rapid rotation by using an exact, general relativistic approach. Finally, Goussard et al. (1998) have performed the case of differential rotation of PNS's. Most of these authors did not utilize an equation of state (EOS) of hot matter throughout the whole star (except Romero et al. 1992; Goussard et al. 1997, 1998), but used an EOS of cold matter for the envelope of the star instead. As we will show, this simplification leads to radii (Kepler frequencies) which are too small (large).

The aim of this work is to study the properties of non-rotating and rapidly, uniformly rotating PNS's. We use an exact, general relativistic approach to rapid rotation (Schaab 1998). The hot dense matter is described by a recently devolped EOS (Strobel et al. 1999), which is based on a modern parametrisation of the Thomas-Fermi approach for finite nuclei and cold nuclear matter performed by Myers & Swiatecki (1990, 1991, 1996). We generalize this approach to hot dense matter by taking the thermal effects on both the kinetic and the interaction energy into account. In this way, we construct a set of EOS's with different profiles of the entropy per baryon and different lepton numbers. We can follow the evolution of the PNS into a CNS, by means of this set, at different evolutionary stages. We also investigate the influence of the location and the shape of the neutrino sphere as well as the influence of the value of the temperature in the star's envelope.

The paper is organized as follows. Firstly, in Sect. 2 we will briefly review the physics in the interior of PNS's and describe the different evolutionary stages of PNS's and NS's. Furthermore, we discuss the location and the shape of the neutrino sphere. The EOS's of PNS and NS matter are described in Sect. 3, where we emphasize the influence of finite temperature and trapped lepton number. The properties of rotating and non-rotating PNS's and NS's are presented in Sect. 4. Finally, discussion of our results and conclusions are given in Sect. 5.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999