Equilibria of Cylindrical Cloud

In Figure 4.1 (right) we plotted the structure for a polytropic cloud. Inner structure is not dependent of $\Gamma$, it is clear the slope of the outer envelope is dependent on $\Gamma$.

1. In the case of the spherically symmetric, consider a polytrope ( $p\propto \rho^\Gamma$) with $\Gamma < 6/5$ (at least the envelope of $\Gamma = 6/5$ cloud extends to $\infty$.), in which gas extends to $\infty$. if $\rho\propto r^{-p}$, the mass inside of $r$ is proportional to $M_r\propto r^{3-p}$. Thus, the gravity per unit volume at a radius $r$, $GM\rho/r^2$, is proportional to $GM\rho/r^2\propto r^{1-2p}$. On the other hand the pressure force is $\vert(\partial p/\partial r)\vert= (\partial p/\partial \rho )\vert(\partial \rho/\partial r)\vert\propto (r^{-p})^{\Gamma-1}r^{-p-1}\propto r^{-p\Gamma-1}$. These two powers become the same, only if $p={2}/{(2-\Gamma)}$.
2. In the case of cylindrical cloud with $\Gamma \le 1$, the mass per unit length $\lambda \propto r^{2-p}$. The gravity at $r$, $G\lambda \rho /r \propto r^{1-2p}$. 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 $p$ should be the same $p={2}/{(2-\Gamma)}$.

The case of $\Gamma =0.9$, an envelope extending to a large radius indicates the power-law distribution much shallower than that of the isothermal $\Gamma=1$ one.