BonnorEbert Mass
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 lowdensity 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
centraltosurface density ratio in Figure 4.2 (right).
The axis represents a normalized mass as
.
The maximum value of means

(4.9) 
This is the maximum mass which is supported against the selfgravity by the thermal pressure with an
isothermal sound speed of , when the cloud is immersed in the pressure .
This is called BonnorEbert 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
with
.
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

(4.10) 
In this case, a relation between the partial derivatives such as

(4.11) 
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

(4.12) 
equation (4.11) gives us

(4.13) 
For a cloud with
the mass is an increasing function of the central density as

(4.14) 
Thus, equation (4.13) leads to the relation

(4.15) 
for
.
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
(for
) behaves

(4.16) 
and this represents that an extra external pressure decreases the central density and thus the pressure.
Pressure encourages expansion.
This reaction is unstable.
Figure 4.2:
(Left) radial density distribution.
Each curve has different .
The xaxis denotes the radial distance normalized by a scalelength 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 xaxis represents the central density normalized by
.
The yaxis represents the cloud mass normalized by
.

Kohji Tomisaka
20121003