# Magnetohydrostatic Configuration

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

 (C.13)

In the axisymmetric case, the poloidal magnetic fields is obtained by the magnetic flux function, , or the -component of the vector potential as
 (C.14)

 (C.15) (C.16) (C.17)

with
 (C.18)

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
 (C.19)

This means is also constant along one magnetic field line, . Since the forces are expressed by the defrivative of function
 (C.20)

where equation (C.19) is used, equation (C.15) and (C.16) are rewritten as
 (C.21) (C.22)

Finally, using the fact that , , and are functions of , these two equations are reduced to
 (C.23)

Another equation to be coupled is the Poisson equation as
 (C.24)

The source terms of equations (C.23) and (C.24) are given by determining the mass and the angular momentum contained in a flux tube - . Mass and angular momentum distribution of
 (C.25) (C.26)

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. Since
 (C.27) (C.28)

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.

Kohji Tomisaka 2012-10-03