Consider here the effect of the magnetic field.
In the magnetized medium, the Lorentz force
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 self-gravity.
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
where we ignored the term
The last two terms are rewritten as
where is defined as
This shows the effects of the magnetic fields:
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
More realistic calculation [Mouschovias (1976a,1976b), Tomisaka et al (1988)] gives us a criterion
- B-fields 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.
where and means the column density and the magnetic flux density.
A cloud with a mass
is sometimes called magnetically supercritical, while that with
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
That is, even without magnetic fields, the cloud less-massive than the Bonnor-Ebert mass has a hydrostatic
configuration shown in Figure 4.2 (left).
The cloud with central density of has a stable density distribution.
Magnetohydrostatic clouds with different magnetic fluxes are calculated
by Mouschovias (1976a,1976b) and Tomisaka et al (1988)].
Mass of the cloud is obtained against the central density,
which is an increasing function of magnetic flux (Fig. 4.3).
The maximum allowable mass (critial mass) supported by some magnetic flux increases
with the magnetic flux.
To fit the numerical results, Tomisaka et al (1988) obtained an expression for the critical mass
when the cloud has a mass-to-flux ratio , the isothermal sound speed ,
and the external pressure as
This shows that the critical mass is a decreasing function of the mass-to-flux ratio or increasing function
of the magnetic flux.
And the critical mass becomes much larger than the Bonnor-Ebert mass
only when the mass-to-flux 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 less-massive than the critical mass is subcritical.
Cloud mass is plotted against the central density.
Respective curves correspond to clouds with different magnetic flux.
Plasma beta of the external medium ()
specifies the magnetic flux.