(3.21) |

where

(3.23) |

(3.24) |

If we set , we obtain our final result for the potential due to the surface density perturbation

(3.26) |

(3.27) |

and

Using these equations [(3.28), (3.29), and (3.30)], , and , we obtain the dispersion relation for the self-gravitating instability of the rotating gaseous thin disk

Generally speaking, the epicyclic frequency depends on the rotation law but is in the range of (see Table 3.1 for for typical rotation laws). It is shown that the system is stabilized due to the epicyclic frequency compared with a nonrotating thin disk [eq.(2.65)].