1 Introduction

A protoneutron star is a very hot ($T \le 50-60$ MeV) and lepton rich object in quasi-hydrostatic equilibrium (Burrows & Lattimer 1986). These stars are formed in the gravitational collapse of iron cores in massive stars (for alternative scenarios on the origin of neutron stars see, e.g., the review by Canal et al. 1990). During the first 25-30 s. in the life of the protoneutron star - the Kelvin epoch for neutrino cooling or the core deleptonization era - the temperature in the core decreases to a typical central value of $\sim 8$ MeV, with a profile (in radius) like that given by the relativistic thermal equilibrium condition (see Burrows 1990 for a review on this subject).

Recent two-dimensional hydrodynamical calculations show that convection in the protoneutron star can encompass the whole star within $\approx\!1$ s, and in this epoch a near constant profile in entropy is observed, with a typical value of s =3 (Janka & Müller 1996; Keil et al. 1996). The extension of the convective region in a protoneutron star is a matter of debate. Many independent models must prove that protoneutron stars have long-lasting convection, before this can be considered as an established fact.

On the other hand, detailed studies of the evolution of massive stars carried out by Woosley and collaborators lead to the conclusion that massive iron cores up to $\approx 2.15\ M_{\odot}$ (Model 2.0S of $40\ M_{\odot}$ of the the review of evolutionary calculations, without mass loss, by Weaver & Woosley 1993), can be formed during the late stages of their evolution. In Woosley et al. (1993) was reported for the first time the complete evolution of a set of massive stars, from the main sequence to the presupernova stage, including mass loss, and the authors found models which develop iron cores of up to $\approx 1.9\ M_{\odot}$(presupernova model 85WRA of $85\ M_{\odot}$).

Consequently, it would be interesting to take these facts into account in order to clarify the conditions of the interior of a protoneutron star: isentropic or isothermal (see, Goussard et al. 1997, for a discussion) and its evolutionary history.

The study of the properties of such protoneutron stars is important to understand supernova neutrinos emitted during the cooling epoch (Burrows & Lattimer 1986; Suzuki 1994). The characteristics of a neutrino burst reflect the thermal condition inside the star and its cooling history. The profile of a neutrino burst may also affect the mechanism of the supernova explosion itself (See Janka & Müller 1996, for example). Axisymmetric neutrino emission from rotating protoneutron stars have been of particular interest. The effect this may have on the observation of supernova neutrinos has been pointed out (Janka & Mönchmeyer 1989) and a mechanism has been proposed for supernova explosion through anisotropic neutrino heating (Shimizu et al. 1994). To discuss these effects it is essential to know the size of the deformation and the density profile (density gradient, for example), which determine the flux of neutrinos, of rotating protoneutron stars. Therefore, it is of great interest to investigate the structure of rotating protoneutron stars taking into account the thermal profile and history.

After the seminal paper by Burrows & Lattimer (1986) several authors have studied the properties of static non-rotating protoneutron stars. Let us cite some of them: Sumiyoshi & Toki (1994); Takatsuka (1995); Bombaci et al. (1995); Sumiyoshi et al. (1995a,c); Bombaci (1996); Bombaci et al. (1996) and the ones contained in Glendenning's book (Glendenning 1997).

A few papers have appeared recently which focus on the analysis of the stability of rapidly rotating protoneutron stars (Hashimoto et al. 1994; Goussard et al. 1997) using a full general-relativistic description of axisymmetric configurations. The main differences between these calculations refer to the assumptions concerning: i) the thermal state of the hot interior of the protoneutron star - isothermal (according to the Newtonian definition in Hashimoto et al. 1995, or the Einsteinian definition in Goussard et al. 1997), or isentropic (in the case of Goussard et al. 1997) -, and ii) the lepton composition, $Y_{\rm L}$, is zero in Hashimoto et al. (1994), but Goussard et al. (1997) consider two extreme values $Y_{\rm L} = 0, 0.4$ in their calculations. The EOS adopted in these studies are different and several sets of EOSs which differ in stiffness are used in Hashimoto et al. (1994) whereas Goussard et al. (1997) use a set of EOS by Lattimer & Swesty (1991).

It should be stressed that the computations carried out by Goussard et al. (1997; see also Goussard 1997 for details) are careful as they combine the following: i) high accuracy of spectral methods, ii) checking their results with the general-relativistic virial theorem, and iii) a thermodynamically consistent interpolation in the EOS tables. A main conclusion coming from the calculations made by Goussard et al. (1997) is the existence of a minimum rotation period of rapidly rotating protoneutron stars which is significantly larger than the corresponding limit for cold neutron stars. For the range of gravitational masses $1.3-1.8\ M_{\odot}$, Goussard et al. (1997) obtain minimum rotation periods of 1.25-1.1 ms (1.6-1.1 ms), respectively, for their isothermal and deleptonized (trapped lepton, $Y\rm _L=0.4$, and isentropic) models. Hence, a fundamental lower limit of 1.1 ms seems to arise from these calculations (Hashimoto et al. 1994, obtain a lower limit of 0.9 ms).

The rotation of a star is limited by the Kepler frequency, $\rm \Omega_{k}$, at which the equatorial surface velocity equals the orbital velocity of a particle at the equator and, hence, mass shedding at the stellar equator makes the star unstable. If $M\rm _{max}$ and $R\rm _{max}$ are the gravitational mass and the radius, respectively, of the non-rotating equilibrium model at the maximum of the corresponding curve for a given equation of state (EOS), the following empirical formula has been proved to be a reasonable approximation of the limiting frequency  

$$\Omega_{\rm k} = C (M_{\rm max}/M_{\odot})^{1/2} (R_{\rm max}\rm /10~km)^{-3/2} ,$$ (1)

where parameter C can have some of the following values: i) $C\rm _N \approx 1.15 \ 10^4\ s^{-1}$ in the Newtonian case, ii) $C\rm _{HZ} \approx 0.775 \ 10^4\ s^{-1}$ according to the fitting, derived by Haensel & Zdunik (1989), to the full (axisymmetric) general relativistic results of Friedman et al. (1986, 1989) and iii) $C\rm _G \approx (0.750 - 0.830) \ 10^4\ s^{-1}$ according to the calculations carried out by Goussard et al. (1997; see also Goussard 1997), the general trend being that $C\rm _G$ increases with the central temperature (isothermal models), decreases with entropy (isentropic models) and increases with the incompressibility modulus.

If we define the adimensional parameter $\rm \eta \equiv \Omega /\Omega_k$where $\Omega$ is the angular velocity of the star as measured by a distant observer (we will assume that $\Omega$ is constant inside the star, i.e., the star rotates uniformly) and consider the Newtonian limit of $\rm \Omega_{k}$, then two values of $\eta$ are of particular relevance: i) $\eta \approx 0.7$, for a star having the limit frequency given by the above Haensel & Zdunik fitting, and ii) $\eta \approx 0.3$ which is the approximate value for a millisecond pulsar (like, e.g., PSR 1937+21).

Consequently, with the above estimate the assumption of slow rotation following Hartle's approach (1967) seems to be a good way to look more deeply into the properties of rotating relativistic equilibrium configurations. This approach combines: i) simplicity, ii) it is sufficient to understand the main macroscopic properties of most pulsars (with exception , perhaps, of the millisecond ones), and iii) it enables one to know the influence of general relativistic effects on the pulse profile (Kapoor & Datta 1985, 1986; Datta 1988). Therefore, consistent with Hartle's approach, we will assume that the star rotates like a solid rigid object, in spite of the fact that, in a more realistic scenario, rotation is differential (Janka & Mönchmeyer 1989).

Unlike the paper by Goussard et al. (1997) we have made an exhaustive analysis of the properties of slowly rotating protoneutron stars, and focussed on the dependence of these properties on the two thermodynamical parameters which we consider constant inside the star: $Y_{\rm L}$ (the lepton fraction) and s (the entropy per baryon). The assumption of a constant profile for $Y_{\rm L}$ is realistic, once $\beta$-equilibrium conditions have been reached inside the inner core, during the infall epoch. We have also paid particular attention to the evolutionary track of cooling protoneutron stars by exploring a wide range of values for the thermodynamical parameters as well as the baryonic mass of the star. The assumption of a constant profile for s is based on the considerations above regarding convection in protoneutron stars.

Another aim of the current study is to explore the properties of protoneutron stars adopting the relativistic EOS, which is qualitatively different from conventional EOSs (Sumiyoshi et al. 1995). The recent success of relativistic nuclear many-body frameworks (see Serot & Walecka 1986, for a review) is of great importance for astrophysical applications. In particular, the relativistic mean field (RMF) theory has been quite successful in describing the EOS as well as the properties of the nuclei. It should be stressed that the RMF theory has been extensively checked by experimental data of nuclei in the region away from stability, where the environment is closer to neutron stars, and thus the EOS is well constrained by the current knowledge of nuclear physics (Hirata et al. 1991; Sugahara & Toki 1994; Hirata et al. 1997).

In Sect. 2 we describe the relativistic EOS used in the present paper. Section 3 discusses the properties of the slowly rotating protoneutron stars coming from our calculation. Section 4 shows some evolutionary sequences at fixed baryon mass and angular momentum in a diagram entropy vs. central rest-mass density. Conclusions are summarized in Sect. 5.