We assume the wave driven by the self-gravity has a form of tightly-wound spiral
[Fig.3.7(left)].
When we move radially, the density varies rapidly.
While, it changes its amplitude slowly in the azimuthal direction.
In a mathematical expression, if we write the density perturbation as

(3.22) |

where

(3.24) |

(3.25) |

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

(3.27) |

(3.28) |

Neglecting the terms compared to the terms containing
,
equations (3.19), (3.20), and (3.21) are rewritten as

and

Using these equations [(3.29), (3.30), and (3.31)], , 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)].

Kohji Tomisaka 2012-10-03