Tightly wound (left) vs loosely wound (right) spirals.
We assume the wave driven by the self-gravity has a form of tightly-wound spiral
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
where the amplitude of spiral is a slowly varing function of ,
a tightly wound spiral means the shape function varies fast (the radial wavenumber
is large enough).
We consider the gravitational force from the vicinity of , since the oscillates
and cancels even if we integrate over large region.
Notice that the density perturbation [eq.(3.22)] is similar to that studied in
The potential should be expressed in a similar form to equation (2.64) as
which simply means
If we set , we obtain our final result for the potential due to the surface density perturbation
Differentiating this equation with and ignoring the term compared to that of
, we obtain
Neglecting the terms compared to the terms containing
equations (3.18), (3.19), and (3.20) are rewritten as
Using these equations [(3.28), (3.29), 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)].