The structure, stability, and form of marginally stable axisymmetric perturbations of rotating gas clouds with simple conformal density distributions. I - General features and analytical solutions

A family of conformal solutions of the equilibrium equations for differentially rotating and selfgravitating gas clouds is considered. The clouds are infinitely extended and have infinite central densities. As the radial parts of the solutions are given by power laws, the problem of the equilibrium structure is reduced to a system of ordinary differential equations for the angular parts. A case of radiative equilibrium and cases of rotating polytropes are discussed. Axially symmetric, marginally stable, polytropic perturbations of these equilibrium states are considered, and by separation of the variables the perturbation equations are reduced to ordinary differential equations for the angular parts of the perturbations. The radial parts are simple oscillating functions with Titius-Bode law features. Special attention is called to polytropic perturbations of polytropic equilibrium states, in particular to the case of isothermal perturbations of isothermal undisturbed clouds.