Sound wave seems to be modified in the medium where the self-gravity is important.
Beside the continuity equation (2.35)
and the equation of motion (2.37)
the linearized Poisson equation
should be included.
where we used equation (2.47) to eliminate .
where we used
This is the equation which characterizes the growth of density perturbation owing to the self-gravity.
Here we consider the perturbation are expressed by the linear combination of plane waves as
where and represent the wavenumber and the angular frequency of the wave.
Picking up a plane wave and putting into equation (2.49), we obtain the dispersion relation
for the gravitational instability as
Reducing the density to zero, the equation gives us the same dispersion relation as that of the sound wave as
For short waves (
), since the wave is ordinary oscillatory wave.
Increasing the wavelength (decreasing the wavenumber), becomes negative for
For negative , can be written
using a positive real .
In this case, since
, the wave which has
its amplitude exponentially.
This means that even if there are density inhomogeneities only with small amplitudes,
they grow in a time scale of and form density inhomogeneities with large amplitudes.
The critical wavenumber
corresponds to the wavelength
which is called the Jeans wavelength.
Ignoring a numerical factor of the order of unity, it is shown that the Jeans wavelength is approximately equal to
the free-fall time scale (eq.[2.26]) times the sound speed.
The short wave with
does not suffer from the self-gravity.
For such a scale, the analysis we did in the preceding section is valid.
Typical values in molecular clouds,
, give us the Jeans wavelength as
The mass contained in a sphere with a radius is often called Jeans mass,
which gives a typical mass scale above which the gas collapses.
Typical value of the Jeans mass is as follows
Using again the above typical values in the molecular clouds,
the Jeans mass of this gas is equal to