In the preceding section [Fig.4.1 (left)], we have seen the radial density distribution of
a hydrostatic configuration of an isothermal gas.
Consider a circumstance that such kind of cloud is immersed in a low-density medium with a pressure .
To establish a pressure equilibrium, the pressure at the surface
must equal to .
This means that the density at the surface is constant
Figure 4.2 (left) shows three models of density distribution,
, and .
Comparing these three models,
it should be noticed that the cloud size (radius) decreases
with increasing the central density .
The mass of the cloud is obtained by integrating the distribution, which is illustrated against the
central-to-surface density ratio in Figure 4.2 (right).
The -axis represents a normalized mass as
The maximum value of means
This is the maximum mass which is supported against the self-gravity by the thermal pressure with an
isothermal sound speed of , when the cloud is immersed in the pressure .
This is called Bonnor-Ebert mass [Bonnor (1956), Ebert (1955)].
It is to be noticed that the critical state
is achieved when the density contrast
is rather low
Another important result from Figure 4.2 (right) is the stability of an isothermal cloud.
Even for a cloud with
any clouds on the part of
are unstable, whose clouds are distributed on the branch
This is understood as follows:
For a hydrostatic cloud the mass should be expressed with the external pressure and
the central density [Fig.4.2 (right)] as
In this case, a relation between the partial derivatives such as
is satisfied, unless each term is equal to zero.
Figure 4.1 (left) shows that
the cloud mass is a decreasing function of the external pressure
if the central density is fixed.
Since this means
equation (4.11) gives us
For a cloud with
the mass is an increasing function of the central density as
Thus, equation (4.13) leads to the relation
This means that gas cloud contracts (the central density and pressure increase),
when the external pressure increases.
This is an ordinary reaction of a stable gas.
On the other hand, the cloud on the part of
and this represents that an extra external pressure decreases the central density and thus the pressure.
Pressure encourages expansion.
This reaction is unstable.
(Left) radial density distribution.
Each curve has different .
The x-axis denotes the radial distance normalized by a scale-length as
It is shown that the radius increases with decreasing in this range
(Right) The relation between mass () and the central density () is plotted,
under the condition of constant external pressure.
The x-axis represents the central density normalized by
The y-axis represents the cloud mass normalized by