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# Hydrostatic Balance

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 (4.1)

and the gravity is calculated from a density distribution as (4.2)

for a spherical symmetric cloud, where represents the mass contained inside the radius . The expression for a cylindrical cloud is (4.3)

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)

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 (4.5)

where (4.6)

Far from the cloud symmetric axis, the distribution of equation (4.5) gives (4.7)

while the spherical symmetric cloud has (4.8)

distribution.

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Kohji Tomisaka 2007-07-08