Astron. Astrophys. 330, 569-577 (1998)

1. Introduction

According to the current models of Type II supernova explosions (Bethe 1990), the core collapse of a massive star between 8 and 30 due to electron capture and photodisintegration proceeds until the nuclear matter density is reached, and a shock wave is produced as a consequence of the sudden stiffening of the equation of state (EOS). However, as the shock propagates through the iron core, it weakens and then perhaps stalls because of nuclear dissociation and neutrino losses. A complete analysis of the core collapse, bounce and shock movement requires not only detailed modeling of neutrino transport, general relativity and convection, but also careful investigation of EOS at densities higher than the nuclear matter density.

The EOS for dense matter in a supernova core plays important roles in the whole supernova-explosion process. First, a number of studies have attempted to delineate the exact role of the EOS in the collapse and the bounce phases (Baron,et al. 1985a,b; Baron et al. 1987; Myra & Bludman 1990; Bruenn 1989a,b; Cooperstein & Baron 1990; Swesty, et al. 1994), while the other studies (e.g., Takahara & Sato 1985; Gentile et al. 1993; Dai,et al. 1995) found that the evolution of the EOS, such as phase transitions, is likely to be in favor of explosions. Many of these investigations have been largely motivated by the observation that softening the EOS above the nuclear matter density in some cases leads to stronger bounces and shocks. Baron et al. (1985a,b, 1987) used the Cooperstein & Baron (1990) EOS, in which the bulk incompressibility and adiabatic index of supernuclear matter were varied, found that shock strength was correlated with EOS stiffness, and claimed that sufficiently soft EOSs could result in explosions before collapse to a black hole. Second, the structure of the post-bounce core depends upon the extent to which electron capture reactions deleptonize the core during the collapse epoch. A small electron-capture rate, which is determined by the bulk symmetry energy of nuclear matter (Bruenn 1989a; Swesty et al. 1994), results in a larger trapped lepton concentration at the core bounce. Third, in order to reenergize the shock if it stalls, one studied the so-called delayed mechanism, in which neutrino heating revives the shock (Bethe & Wilson 1985; Wilson 1985), and convection rapidly transports energy into the region behind it (e.g., Burrows & Fryxell 1992, 1993; Janka 1993; Janka & Müller 1993a,b; Wilson & Mayle 1993; Bruenn & Mezzacappa 1994; Herant et al. 1994; Burrows, et al. 1995). Moreover, the strength of the initial bounce and the structure of the post-bounce core are crucial to the operation of these two mechanisms.

Even after the supernova explosion, the EOS for supernuclear matter also plays important roles in determining the evolution of the matter at the birth stage (Burrows & Lattimer 1986). Such a birth event always companies with a neutrino burst of which the energy spectrum can be detected on the terrestrial experiments. In addition, theoretically, the properties of neutron stars such as the maximum mass, the maximum rotation frequency, the moment of inertia, and the proton concentration during the cooling, which affects the reaction rates of neutrino processes inside the stars, on one hand, are determined by the EOS together with the Tolman-Oppenheimer-Volkoff equation. On the other hand, many observations have put constraints on these properties (for a brief review see Cheng, Dai et al. 1996). Therefore, the determination of the EOS for dense matter is also crucial to the study of the birth, evolution and physics of neutron stars.

To determine an EOS for dense matter through the many-body theory of interacting hadrons, one has studied many approaches, among which the relativistic many-body approach on nuclear systems is of growing interest during recent years. The relativistic Brueckner-Hartree-Fock theory reproduces the saturation property of nuclear matter, which is not possible in the nonrelativistic approach unless one introduces three-body force by hand (Brockmann & Machleidt 1990). Moreover, rather promising results within the framework of this theory have been obtained by M ther, et al. (1990) and Li, et al. (1992), and the application of this theory to neutron stars has been investigated by Engvik et al. (1994) and Bao et al. (1994). On the other hand, the relativistic mean field (RMF) theory is successful both for elastic scattering and for nuclear ground-state property (Walecka 1974; Chin 1977; Serot & Walecka 1986). Hence, the RMF theory has been suggested to calculate the EOS for neutron-star matter (Walecka 1974). This approach contains both nucleonic and mesonic degrees of freedom and can be considered as phenomenological. The coupling constants and meson masses of the effective meson-nucleon Lagrangian are taken as free parameters which are adjusted to fit the properties of nuclear matter and finite nuclei.

In the standard model of Walecka (1974) the incompressibility of nuclear matter is overestimated. There are two ways to solve this question. First, Boguta & Bodmer (1977, hereafter BB) introduced cubic and quartic terms for the scalar field into the Lagrangian. This shifts the incompressibility to reasonable values in comparison with empirical data. Along this direction, many authors (Glendenning 1982, 1985, 1987a,b; Weber & Weigel 1989a,b; Kapusta & Olive 1990; Ellis, et al. 1991; Sumiyoshi, et al. 1992; Sumiyoshi & Toki 1994; Sumiyoshi, et al. 1995a; Cheng et al. 1996; Schaffner & Mishustin 1996) have studied the EOS for dense matter and the properties of neutron stars. Zimanyi & Moszkowski (1990, hereafter ZM) proposed an alternative nonlinear model, in which the non-linearity is contained in the connection between the effective nucleon mass and the scalar field. Thus the Lagrangian of this model has no extra terms, and consequently deals with fewer parameters as compared with the BB model. The ZM model also yields reasonable values of incompressibility and an effective nucleon mass for nuclear matter. This model was recently used to study the properties of neutron stars by Cheng et al. (1996), who suggested that observations on surface radiation of neutron stars may discriminate between these two models.

The application of the RMF theory to studies of supernova matter is also of much interest. Sumiyoshi & Toki (1994) and Chiapparini, Rodrigues & Duarte (1996) have studied the EOS for nonstrange dense matter in a supernova core by using the BB model. The scope of our work is to investigate the properties for asymmetric nuclear matter and to study the EOS of supernova matter, by using the ZM model. We arrange this paper as follows. In Sect. 2, we describe the framework based on the RMF theory. In Sect. 3, we calculate the properties of asymmetric nuclear matter and supernova matter. Astrophysical implications of our results are discussed in Sect. 4, and conclusions are given in the final section.

© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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