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

4. Discussion

We have used the ZM model in the relativistic mean-field theory to study the properties of asymmetric nuclear matter and the equation of state of supernova matter. We have also investigated the structure of neutron stars.

We now discuss astrophysical implications of our EOS for neutron star matter. First, we make an estimate of the maximum rotation frequency () of neutron stars, which is determined by the mass shedding condition. Haensel & Zdunik (1989) first noticed that for realistic EOSs of dense matter, the numerically calculated values of can be fitted by an empirical formula

where is the maximum mass of the nonrotating neutron stars with the same EOS, is the radius corresponding to , and C is a dimensionless phenomenological constant. Haensel and collaborators (Haensel & Zdunik 1989; Haensel, et al. 1995; Lasota, et al. 1996) further found that the best fit is for , where . Therefore, for our EOS, , which leads to the minimum rotation period of 0.69 ms. Second, for a neutron star with the gravitational mass of , our EOS reproduces the ratio of the inner crust to total moments of inertia to be about 8.6%, which is consistent with the result from the analysis of observational data of the glitches of four pulsars in Link, et al. (1992). Third, we found that the proton concentration at the density of in neutron stars is about 0.085. This value is smaller than 15%, which is required for the operation of direct Urca process (Lattimer et al. 1991). Thus, the modified Urca process is a main neutrino reaction in neutron stars. This conclusion is opposite to that of many authors (e.g., Boguta 1981; Glendenning 1985; Sumiyoshi & Toki 1984; Sumiyoshi et al. 1995a; Cheng et al. 1996), who found that the proton concentration in neutron stars based on BB model in the RMF theory is so large that the direct Urca process occurs. This may provide an observational signature for the ZM model.

We next discuss the effects of our EOS for supernova matter on Type II supernovae. First, we explore the role of the bulk incompressibility on the "prompt" phase of Type II supernovae. Baron et al. (1985b) studied the supernova explosions systematically by using the following parameterized EOS:

where the saturation density and the incompressibility determine the behavior of the saturation properties of asymmetric nuclear matter at the proton concentration, , and the adiabatic index expresses the stiffness of the EOS. They proposed that the value of the incompressibility at is a key quantity of prompt explosions, and further found that should be less than 180 MeV for the successful prompt explosions of massive stars with . This value is too small as compared to the experimental value extracted from the nuclear physics, and the softness of the EOS expressed by Eq. (27) with MeV and violates the constraint from the mass of PSR 1913+16 (Swesty et al. 1994). In our work, the value of K at is 210 MeV and the value at is 227.6 MeV. Thus, we can conclude that if the dense matter in a supernova core is described by the ZM model in the RMF theory, the prompt explosion mechanism cannot work. Furthermore, Swesty et al. (1994) used the Lattimer & Swesty's (1991) EOS and numerically showed that there is no discernible difference in the shock stall radius for the different incompressibility.

Second, at one time it was thought that the strength of supervova shocks also strongly depends on the bulk symmetry energy of nuclear matter. For example, Bruenn (1989a) investigated the role of the symmetry energy using the Cooperstein & Baron EOS, and found that decreasing the symmetry energy leads to weaker shocks. However, the work of Swesty et al. (1994) indicates that the shock stall radius is practically independent of the symmetry energy. In addition, Swesty et al. (1994) also found that the rate of electron capture which deleptonizes the core during the collapse epoch increases with increasing the symmetry energy. Our EOS in the RMF theory gives a smaller value of the symmetry energy (29.35 MeV), which may result in a smaller electron-capture rate during the collapse and a larger trapped lepton concentration () at the core bounce. This further leads to a stiffer EOS of the post-bounce core than for a smaller (see Fig. 9), and a larger radius of the protoneutron star for a given mass.

Third, the delayed explosion mechanism has been of particular interest in recent years. In this mechanism the stall shock is reheated through neutrino energy deposition behind it and is revived over hundreds of milliseconds. Clearly, a high neutrino luminosity is important for a successful shock. Using Sumiyoshi, Suzuki & Toki's (1995b) results based on the BB model, we can make an estimate of shock energy. Since our value of the symmetry energy is close to that for their parameter set TMS, at the core bounce in our work is larger than for their parameter set TM1. According to Burrows & Goshy's (1993) analytic theory of one-dimensional neutrino-heated supernovae, the total energy () which the shock obtains during the reheating stage is nearly proportional to . Thus, in our work may be larger than that for the parameter set TM1 of Sumiyoshi et al. (1995b). Therefore, our EOS in the RMF theory is likely to be favourable to the operation of the delayed explosion mechanism. Of course, convection in the region between the shock front and the neutrinosphere can rapidly transport neutrino energy into the region behind the shock and thus the shock will obtain more energy.

Fourth, Fig. 8 implies that during the evolution of a protoneutron star into a neutron star the adiabatic index near the baryon number density of strongly increase. This behavior might result in the Rayleigh-Taylor instability in a layer of the inner crust. This is because during the evolution of a protoneutron star neutrinos in the region close to diffuse more rapidly than neutrinos at higher densities do so that under the effect of gravity and pressure the mass density at a baryon number density near can be larger than that at a higher baryon number density at some evolution stage. This Rayleigh-Taylor instability might have an important impact on the evolution of the protoneutron star and the origin of a magnetic field of the neutron star (Thompson & Duncan 1993).

Finally, we note that Goussard, et al.(1997) recently studied the maximum rotation frequency of uniformly rotating protoneutron stars in general relativity and further gave an empirical formula for this frequency. This formula actually coincides with used in cold neutron stars (e.g., Eq. [26]). Thus, our EOS for dense matter in supernova cores can also be applied to a study of the maximum rotation frequency of uniformly rotating protoneutron stars.

© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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