Sound wave seems to be modified in the medium where the self-gravity is important.
Beside the continuity equation (2.35)
![\begin{displaymath}
\frac{\partial \delta \rho}{\partial t}+\rho_0 \frac{\partial \delta u}{\partial x}=0,
\end{displaymath}](img455.png) |
(2.45) |
and the equation of motion (2.37)
![\begin{displaymath}
\rho_0 \frac{\partial \delta u}{\partial t}= -c_{is}^2\frac...
...o}{\partial x}-\rho_0\frac{\partial \delta \phi}{\partial x},
\end{displaymath}](img471.png) |
(2.46) |
the linearized Poisson equation
![\begin{displaymath}
\frac{\partial^2 \delta \phi}{\partial x^2}=4\pi G \delta \rho,
\end{displaymath}](img472.png) |
(2.47) |
should be included.
(2.46) gives
![\begin{displaymath}
\rho_0\left(\frac{\partial^2 \delta u}{\partial x \partial t...
...artial^2 \delta \rho}{\partial x^2}-4\pi G \rho_0 \delta \rho.
\end{displaymath}](img473.png) |
(2.48) |
where we used equation (2.47) to eliminate
.
This yields
![\begin{displaymath}
\frac{\partial^2 \delta \rho}{\partial t^2}=c_{is}^2 \frac{\partial^2 \delta \rho}{\partial x^2}+4\pi G \rho_0 \delta \rho.
\end{displaymath}](img475.png) |
(2.49) |
where we used
(2.45).
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
![\begin{displaymath}
\delta \rho (x,t)=\sum A(\omega,k)\exp(i\omega t-ikx),
\end{displaymath}](img476.png) |
(2.50) |
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
![\begin{displaymath}
\omega^2=c_{is}^2k^2 -4\pi G \rho_0.
\end{displaymath}](img478.png) |
(2.51) |
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
increases
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.
Figure 2.2:
Dispersion Relation
![\begin{figure}\centering\leavevmode
\epsfxsize =.45\columnwidth \epsfbox{eps/dispersion.ps}\end{figure}](img488.png) |
The critical wavenumber
![\begin{displaymath}
k_J=(4\pi G \rho_0)^{1/2}/c_{is}
\end{displaymath}](img489.png) |
(2.52) |
corresponds to the wavelength
![\begin{displaymath}
\lambda_J=\frac{2\pi}{k_J}=\left(\frac{\pi c_{is}^2}{G \rho_0}\right)^{1/2},
\end{displaymath}](img490.png) |
(2.53) |
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,
such as
,
, 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
![\begin{displaymath}
M_J\simeq \frac{4\pi}{3}\rho_0\left(\frac{\lambda_J}{2}\righ...
...{\pi}{6}\left(\frac{\pi}{G \rho_0}\right)^{3/2}c_{is}^3\rho_0.
\end{displaymath}](img496.png) |
(2.54) |
Using again the above typical values in the molecular clouds,
,
,
the Jeans mass of this gas is equal to
.
Kohji Tomisaka
2012-10-03