Hydrodynamic equation of motion using the Lagrangean time derivative [eq.(A.3)] is
Multiplying the position vector r and integrate over a volume of a cloud, we obtain the Virial relation as
is an inertia of the cloud,
is a term corresponding to the thermal pressure plus turbulent pressure,
comes from a surface pressure, and
is a gravitational energy.
To derive the last expression in each equation, we have assumed the cloud is spherical and uniform.
Here we use a standard notation as the radius , the volume
the average pressure , and the mass .
To obtain a condition of mechanical equilibrium, we assume .
Equation (4.18) becomes
Assuming the gas is isothermal
, the average pressure is written as
Using equation (4.24) to eliminate from equation (4.18),
the external pressure is related to the mass and the radius as
Keeping constant and increasing from zero, increases first, but it takes a maximum,
, and finally declines.
This indicates that the surface pressure must be smaller than
for a cloud
to be in the equilibrium.
In other words, keeping and changing , it is shown that has a maximum value to
have a solution.
The maximum mass is equal to
The cloud massive than cannot be supported against the self-gravity.
This corresponds to the Bonnor-Ebert mass [eq.(4.9)],
although the numerical factors are slightly different.