Using equation (2.26), the necessary time for a mass-shell at to reach the center
(free-fall time) is expressed as
(for detail of this section see Ogino, Tomisaka, & Nakamura 1999).
Consider two shells whose initial radii are and .
The time difference for these two shells to reach the center
can be written down using equation (4.93) as
Mass in the shell between and ,
on the central object in .
Thus, mass accretion rate for a pressure-free cloud is expressed
This leads to the expression as
Mass accretion rate against the typical density of the cloud.
This gives time variation of the accretion rate.
Consider two clouds with the same density distribution
but different absolute value.
Since these two clouds have the same
the mass accretion rate depends only on , and is expressed as
This indicates that the accretion rate is proportional to ,
while the time scale is to .
This is confirmed by hydrodynamical simulations of spherical symmetric isothermal clouds (Ogino et al.1999).
When the initial density distribution is the SIS as
the mass included inside is proportional to radius
In this case, equation (4.95) gives a constant accretion rate in time.
In Figure 4.11 we plot the mass accretion rate against the
represents the cloud density relative to that of a hydrostatic Bonnor-Ebert sphere.
This shows clearly that the mass accretion rate is proportional to for massive clouds
This is natural since the assumption of pressure-less is valid only for a massive cloud in which
the gravity force is predominant against the pressure force.
Similar discussion has been done by Henriksen, André, & Bontemps (1997) to explain
a decline in the accretion rate from Class 0 to Class I IR objects.
They assumed initial density distribution of
as shown in Figure 4.12.
Since the free-fall-time of the gas contained in the inner core is the same,
such gas reaches the center once.
It makes a very large accretion rate at
equation (4.95) predicts
A constant accretion rate is expected for this power-law and
the accretion rate is converged to a constant value after
the stellar mass is much larger than than that was containd in ,
Since for this power and
A model proposed to explain time variation in accretion rate
by Henriksen, André, & Bontemps (1997).
The density distribution at (left) and expected accretion rate (right).