In section 4.2, we obtained the maximum mass which is supported
against the self-gravity using the virial analysis.
In this section, we will survey result of more realistic calculation.
Formalism was obtained by Mouschovias (1976a,6), which was extended by
Tomisaka, Ikeuchi, & Nakamura (1988) to include the effect of rotation.
Magnetohydrostatic equlibrium is achived on a balance between the Lorentz
force, gravity, thermal pressure force, and the centrifugal force as
In the axisymmetric case, the poloidal magnetic fields is obtained by
the magnetic flux function, ,
or the -component of the vector potential as
Equation (C.13) leads to
Equation (C.17) indicates is a function of as
, which is constant along one magnetic field line.
Ferraro's isorotation law demands,
that is, to satisfy the stead-state induction
equation is constant along a magnetic field.
This means is also constant along one magnetic field line,
From this, the density distribution in one flux tube is written
This means is also constant along one magnetic field line, .
Since the forces are expressed by the defrivative of function
where equation (C.19) is used,
equation (C.15) and (C.16) are rewritten as
Finally, using the fact that , , and are functions of ,
these two equations are reduced to
Another equation to be coupled is the Poisson equation as
The source terms of equations (C.23) and (C.24) are given by determining
and the angular momentum
contained in a flux tube -
Mass and angular momentum distribution of
is chosen artitrary in nature, where is the the height of the cloud surface where
the magnetic potential is equal to .
For example, and are chosen
as a uniformly rorating uniform-density spherical cloud threaded by uniform magnetic field.
The source terms of PDEs [eqs (C.23) and (C.24)] are given
from equations (C.27) and (C.28).
While the functons and are determined from the solution of these PDEs
after and are chosen.
This can be solved by a self-consistent field method.