Consider a hydrostatic balance of isothermal cloud.
By the gas density, , the isothermal sound speed, , and the gravitational potential,
, the force balance is written as

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

where represents the mass per unit length within a cylinder of radius being .

For the spherical symmetric case, the equation becomes the Lane-Emden equation with the polytropic index of
(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

(4.4) |

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

(4.6) |

(4.7) |

(4.8) |