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In the case of a rotating disk with an angular rotation speed ,
basic equations are

(2.66) 

(2.67) 

(2.68) 
Equation (2.68) gives
a solution

(2.69) 
Choosing a direction in which the wavenumber vector can be expressed as
,
we can reduce equations (2.66)
and (2.67) to

(2.70) 

(2.71) 

(2.72) 
These three equations together with equation (2.69)
bring us a dispersion relation as

(2.73) 
Comparing with equation(2.65),
this indicates rotation works to stabilize the system.
Equation (2.73) is rewritten as

(2.74) 
Defining

(2.75) 
we can see that
for all wavenumbers if and
that if for some range of wavenumber becomes negative.
A rotating disk with is unstable for some range of wavenumber.
This number is called Toomre's .
This is useful to see whether a galactic disk is stable or not.
For the galactic disk, Toomre's must be modified as

(2.76) 
where represents the epicyclic frequency as

(2.77) 
See section 3.4 for the galactic disk.
Next: Convective Instability
Up: Gravitational Instability of Thin
Previous: Gravitational Instability of Thin
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Kohji Tomisaka
20070708