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

The virial analysis is also applicable to the magnetohydrostatic clouds.
The terms related to the magnetic fields are

(4.28) |

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

(4.30) |

This shows the effects of the magnetic fields:

- 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

is

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
Bonnor-Ebert mass.
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

Kohji Tomisaka 2012-10-03