Hydrostatic Balance

Consider a hydrostatic balance of isothermal cloud. By the gas density, $\rho$, the isothermal sound speed, $c_{\rm is}$, and the gravitational potential, $\Phi$, the force balance is written as

$$-\frac{c_{\rm is}^2}{\rho}\frac{d \rho}{d r}-\frac{d \Phi}{d r}=0,$$ (4.1)

and the gravity is calculated from a density distribution as

$$-\frac{d \Phi}{d r}=-\frac{GM_r}{r^2}=-\frac{4\pi G}{r^2}\int_0^r \rho r^2 dr,$$ (4.2)

for a spherical symmetric cloud, where $M_r$ represents the mass contained inside the radius $r$. The expression for a cylindrical cloud is

$$-\frac{d \Phi}{d r}=-\frac{2G\lambda_r}{r}=-\frac{4\pi G}{r}\int_0^r \rho r dr,$$ (4.3)

where $\lambda_r$ represents the mass per unit length within a cylinder of radius being $r$.

For the spherical symmetric case, the equation becomes the Lane-Emden equation with the polytropic index of $\infty$ (see Appendix C.1). This has no analytic solutions. However, the numerical integration gives us a solution shown in Figure 4.1 (left). Only in a limiting case with the infinite central density, the solution is expressed as

$$\rho(r)=\frac{c_{\rm is}^2}{2\pi G}r^{-2}.$$ (4.4)

Increasing the central density, the solution reaches the above Singular Isothermal Sphere (SIS) solution.

On the other hand, a cylindrical cloud has an analytic solution (Ostriker 1964) as

$$\rho(r)=\rho_c \left( 1+ \frac{r^2}{8H^2} \right)^{-2},$$ (4.5)

where

$$H^2={c_{\rm is}^2}/{4\pi G \rho_c}.$$ (4.6)

Far from the cloud symmetric axis, the distribution of equation (4.5) gives

$$\rho(r)\propto r^{-4},$$ (4.7)

while the spherical symmetric cloud has

$$\rho(r)\propto r^{-2}$$ (4.8)

distribution.