In a diffuse cloud, the centrifugal force does not play an important role. Consider a gas parcel whose specific angular momentum is equal to . When the gas element contracts till the distance from the center, the centrifugal force per mass of

 (4.101)

works. Contraction stops when the gravitational acceleration is balanced by the centrifugal force. Its radius (centrifugal radius) is expressed as
 (4.102)

Assuming that the disk is near the hydrostatic balance in -direction, total column density is related to the volume density on the plane as
 (4.103)

We can show that a nondimensional ratio of specific angular momentum to the mass is equal to the ratio of the free-fall time to the rotation period as follow:
 (4.104)

Many numerical simulations confirm that the contracting disk in the runaway isothermal contraction phase has some `universality.' The ratio of the free-fall time to the rotation period is approximately equal to
 (4.105)

regardless of the initial conditions (Matsumoto, Nakamura, & Hanawa 1997). Thus, . Finally, we can see the centrifugal radius as
 (4.106)

increases with time in proportional to the mass , because gas element with large contracts later. As the centrifugal radius increases with time, the launching point of the outflow also expands with time.

Kohji Tomisaka 2009-12-10