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Accretion Rate
Using equation (2.26), the necessary time for a mass-shell at to reach the center
(free-fall time) is expressed as
|
(4.79) |
(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 (4.79) as
|
(4.80) |
Mass in the shell between and ,
, accretes
on the central object in .
Thus, mass accretion rate for a pressure-free cloud is expressed
as
.
This leads to the expression as
|
(4.81) |
Figure 4.8:
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
|
(4.82) |
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 (4.81) gives a constant accretion rate in time.
In Figure 4.8 we plot the mass accretion rate against the
cloud density.
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
|
|
|
(4.83) |
as shown in Figure 4.9.
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
as
.
If ,
for
.
Since and
,
equation(4.81) predicts
.
A constant accretion rate is expected for this power-law and
the accretion rate is converged to a constant value after
.
If ,
for
.
Since for this power and
,
equation(4.81) predicts
.
They gave
for .
Figure 4.9:
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).
|
Subsections
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Kohji Tomisaka
2007-07-08