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## Bonnor-Ebert 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 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

 (4.9)

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 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.

Next: Equilibria of Cylindrical Cloud Up: Hydrostatic Balance Previous: Problem 2   Contents
Kohji Tomisaka 2007-07-08