5 Summary and conclusions

We have studied the properties of general relativistic slowly rotating protoneutron stars. We have surveyed the structure of rotating protoneutron stars with a wide range of the entropy per baryon, the lepton fraction and the baryonic mass of the stars in order to shed light on the evolutionary history of protoneutron stars during cooling after birth. We have adopted the relativistic equation of state of dense matter derived by the relativistic mean field theory, which is based on the microscopic nuclear many-body framework and checked by the experimental data of many nuclei.

We have presented the macroscopic properties of warm rotating protoneutron stars in the tables and the figures, so as to provide basic information on the effect of rotation and general relativity on protoneutron stars with various baryonic masses for a wide variety of thermal conditions. The information enables us to construct the evolutionary history for cooling and deleptonizing protoneutron stars by following the change of thermal states (s and $Y_{\rm L}$)depending on baryonic masses.

The masses and radii of protoneutron stars are affected by rotation in up to 10$\%$.The effects of rotation become weaker for stars with larger entropy per baryon. This is due to the fact that larger radii of the stars with larger entropy per baryon lead to smaller critical angular velocities. Macroscopic properties of stars depend rather weakly on the lepton fraction but depend more significantly on the entropy per baryon.

The shapes of rotating protoneutron stars are deformed with a value of about 0.7 for $e\rm _s$. The gradients of density at equator and pole are found to be different. This leads to the anisotropy of neutrino flux and may have an influence on the observational neutrino events and the mechanism of delayed supernova explosion. During the cooling epoch, both the shape of stars and the density gradient change according to the corresponding entropy per baryon and lepton fraction and the neutrino flux may vary accordingly.

Following the history of protoneutron stars having the same baryonic mass, the central density is found to increase during the cooling and this may lead to a possible phase transition of dense matter inside during the cooling of protoneutron stars. The analysis reveals that, for a given value of $Y\rm _{L}$, a critical value of the baryonic mass exists, which is the maximum of the cold configurations for the same $Y\rm _{L}$. For rotating protoneutron stars, the critical mass is about $2.5\ M_{\odot}$, above which protoneutron stars will evolve towards a black hole.

Acknowledgements

This work has been supported by the Spanish DGICYT (grant PB94-0973). K.S. is grateful for fruitful discussions with members of Universidad de Valencia, where the most of this work has been done, and their warm hospitality. K.S. thanks also for the Alexander von Humboldt Stiftung, which partly supports his visit to Valencia during his research stay in Max-Planck-Institute für Astrophysik.