5 Conclusion

This study has revealed that the striking results obtained by Antonov (1962) and Lynden-Bell & Wood (1968) in their investigations on the thermodynamics of self-gravitating systems are not restricted to isothermal configurations but also occur for polytropic configurations. It is possible that this formal analogy is the mark of a generalized thermodynamics (Tsallis entropy) but this point needs to be discussed in greater detail. In any case, the study presented in this paper and in our companion papers (Chavanis 2002a,b) provide a unified description of the structure and stability of isothermal and polytropic gas spheres. This is a useful complement to the existing literature on the subject (Chandrasekhar 1932). Quite remarkably, the stability analysis can be performed analytically or by using graphical constructions. Unfortunately, our study is limited in its applications by the requirement of a confining box in the case of isothermal configurations and polytropes of index $n\ge 5$. In future works (in preparation), we shall extend our analytical results to the case of composite configurations made of an isothermal core and a polytropic envelope with index n<5. This is an important generalization for a better description of self-gravitating systems from the viewpoint of statistical mechanics and thermodynamics.

Acknowledgements

I acknowledge interesting comments from C. Tsallis and J. Perez on the first draft of this paper. This work was initiated during my stay at the Institute for Theoretical Physics, Santa Barbara, during the program on Hydrodynamical and Astrophysical Turbulence (February-June 2000). This research was supported in part by the National Science Foundation under Grant No. PHY94-07194.