In Figure 4.1 (right) we plotted the structure for a polytropic cloud.
Inner structure is not dependent of , it is clear the slope of the outer envelope
is dependent on .
The case of , an envelope extending to a large radius indicates the power-law distribution
much shallower than that of the isothermal one.
- In the case of the spherically symmetric, consider a polytrope (
) with (at least the envelope of
cloud extends to .),
in which gas extends to .
, the mass inside of is proportional to
Thus, the gravity per unit volume at a radius , , is proportional to
On the other hand the pressure force is
These two powers become the same, only if
- In the case of cylindrical cloud with ,
the mass per unit length
The gravity at ,
Note that the power is the same as the spherical case.
Since the power of the pressure force should be the same as the spherical case, the resultant should be the same