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Consider here the effect of the magnetic field.
In the magnetized medium, the Lorentz force

(4.27) 
works in the ionized medium.
The first term of equation (4.27), which is called the magnetic pressure,
has an effect to support the cloud against the selfgravity.
The virial analysis is also applicable to the magnetohydrostatic clouds.
The terms related to the magnetic fields are
where represents a magnetic flux and it is assumed to be conserved if we change the radius,
, that is
.
Equation (4.23) becomes

(4.29) 
where we ignored the term
.
The last two terms are rewritten as

(4.30) 
where is defined as
.
This shows the effects of the magnetic fields:
 Bfields effectively reduce the gravitational mass as
.
This plays a part to support a cloud.
 However, even a cloud contracts (decreasing its radius from to ), the ratio of the
gravitational to the magnetic terms keeps constant since these two terms are proportional to
.
Thus, if the magnetic term does not work initially, the gravitational term continues to predominate over
the magnetic term.
If , a sum of last two terms in equation (4.29) is positive.
Since the second term of rhs of equation (4.25) is positive,
there is one which satisfies equation (4.29) irrespective of the external pressure .
While, if , there is a maximum allowable external pressure .
Therefore, gives a criterion whether the magnetic fields work to support the cloud
or not.
More realistic calculation [Mouschovias (1976a,1976b), Tomisaka et al (1988)] gives us a criterion

(4.31) 
where and means the column density and the magnetic flux density.
A cloud with a mass

(4.32) 
is sometimes called magnetically supercritical, while that with

(4.33) 
is magnetically subcritical.
More precisely speaking, the criterion showed in equations (4.32) and
(4.33) should be applied for a cloud which has a much larger mass than the
BonnorEbert mass.
That is, even without magnetic fields, the cloud lessmassive than the BonnorEbert mass has a hydrostatic
configuration shown in Figure 4.2 (left).
The cloud with central density of has a stable density distribution.
To fit the numerical results, Tomisaka et al (1988) obtained an expression for the critical mass
when the cloud has a masstoflux ratio , the isothermal sound speed ,
and the external pressure as

(4.34) 
This shows that the critical mass is a decreasing function of the masstoflux ratio or increasing function
of the magnetic flux.
And the critical mass becomes much larger than the BonnorEbert mass
only when the masstoflux ratio at the center of the cloud is reaching
at which the term in the curry bracket goes to zero.
Hereafter, we call here the cloud/cloud core with mass larger than the critical mass
a supercritical cloud/cloud core.
The cloud/cloud core lessmassive than the critical mass is subcritical.
Next: Subcritical Cloud vs Supercritical
Up: Virial Analysis
Previous: Virial Analysis
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Kohji Tomisaka
20070708