Nowadays, studying the evolution of a protoneutron star during its Kelvin epoch is a very active line of research, during which energy loss through neutrino emission dominates the star's outcome, and its consequence on the supernova neutrinos and the mechanism of supernova explosion. Sophisticated numerical codes have been built up in order to understand the physics involved in convection, neutrino transport and so on (see e.g., Burrows & Lattimer 1986; Suzuki 1994; Janka & Müller 1996; Keil et al. 1996). Realistic numerical simulations of the cooling of rotating protoneutron stars, including those relevant physical processes, are still in their infancy (see Janka & Keil 1997). Therefore, in order to understand the generic features - structural and thermal changes - of the evolutionary history of rotating protoneutron stars, it might be of interest to follow the approach of just linking their stationary macroscopic properties corresponding to equilibrium configurations with different values of entropy per baryon and lepton fraction.

By collecting the data generated by our computations, discussed
in the previous section, and reordering them we have generated a
set of figures (Fig. 8), in which we have plotted
- in a *entropy versus central rest-mass density* ()diagram - the geometrical loci of those equilibrium configurations
with the same baryonic mass (labels stand for this mass in solar units).

In Fig. 8, upper diagrams refer to non-rotating (i.e., zero angular
momentum) protoneutron stars with (left) and (right). Lower diagrams in Fig. 8 display (for the same
values of ) two families of protoneutron stars, parametrized by
the baryonic mass: those non-rotating ones (continuous lines) and those
having constant angular momentum (dashed lines). In this set of diagrams we
have selected the values of the baryonic mass of a few ones. For
the rotating configurations, and for the sake of simplicity (notice
the existence of scaling laws), for each value
of the baryonic mass, we have plotted those with maximum angular momentum.
This maximum of the angular momentum is, in units of 10^{48} cgs:
i) 4.0, 9.5, 16.9, 25.0 and 30.5 for, respectively, the baryonic mass
(in solar units) 0.9, 1.4, 1.9, 2.4, 2.63 of the models shown in lower left
diagram of Fig. 8 (), and ii) 3.5, 8.5, 15.5, 24.0
and 25.7 for, respectively, the baryonic mass (in solar units) 0.9, 1.4,
1.9, 2.4, 2.5 of the models shown in lower right diagram of Fig. 8
().

The analysis of the diagrams in Fig. 8 can help one to understand the fundamental trends of the cooling of a protoneutron star, regarding to basic properties of the equilibrium configurations. First, we are going to make the assumption - which is quite obvious - that once the protoneutron star has formed, it starts to cool down - keeping both its baryonic mass and its angular momentum constant- and, hence, the entropy (and the temperature at its inner core) decreases. Secondly, we are going to ignore the short interval of time in which the unbounced inner core heats due to thermal waves (propagating inwards from the region where the shock was formed) as well as the outward diffusion of electron neutrinos produced by electron captures onto protons in the inner core. Guided by these two basic principles, the information contained in diagrams of Fig. 8 illustrates the following issues.

i) For a given value of , a critical value of the baryonic mass exists, (the maximum for the cold configurations for the same ) such that stars with a mass will become unstable and will evolve towards a black hole, increasing in central density and entropy. From Fig. 8 (see also Tables 1 and 2) we can find some of the values of : they are, roughly, () for non-rotating (rotating) configurations, with negligible differences due to .

ii) The value of of the rotating configurations (for a given ) is an upper bound for the non-rotating ones. This statement seems obvious, but let us draw the reader's attention towards the curve labelled by 2.5 in lower right diagram of Fig. 8. There are no stable protoneutron stars with (for ). If they were born with that baryonic mass it would not be possible for them to cool down into a stable neutron star for both rotating and non-rotating cases.

iii) Only allowing the value of to decrease during cooling (that is what should happen when neutrinos leak out of the star, according to standard theory) it is possible to relax the upper bound quoted in ii) a bit. A rotating protoneutron star of - according to this picture - will cool down to lower and lower values of entropy eventually reaching a zero value for or the bifurcation point to the unstable branch for .

iv) For a given value of the baryonic mass and the entropy, the central rest-mass density of equilibrium configurations decreases with the angular momentum.

v) Finally, let us point out that during cooling, rotating protoneutron stars undergo a significant increase of their rest-mass density at the center. This feature displayed in Fig. 8 together with the above issues can be of potential interest if phase transitions in dense matter exist (as already pointed out in Martí et al. 1988).

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