This exploratory model was developed by Battaner et al. (1992) following the magnetic hypothesis previously proposed by Nelson (1988). The two models are not equivalent. For instance, Nelson's magnetic field must have a non-vanishing pitch angle (i.e. the angle between the direction of the field -assumed to be contained in the galactic plane- and the azimuthal direction). Battaner et al. (1992) consider pure azimuthal (toroidal) fields or, rather, the azimuthal component of the field producing the required force.

In the radial component of the equation of motion, it is necessary to
include magnetic forces, which are of the form
^{ . } - *B*^{2}
(e.g. Battaner (1996)).
The first term in cylindrical coordinates
(R, , z) will be

(115) |

if the field is azimuthal, (
*B*_{R} = *B*_{z} = 0), under azimuthal symmetry,
(
/ = 0), the radial component of this force is
simply
- *B*_{}^{2}/*R*.

The magnetic pressure gradient force
- *B*^{2}
would have a radial component simply given by
- (1/8)*B*_{}^{2}/*R*, and therefore the radial component of the
magnetic force would be

(116) |

By including this force in the radial component of the equation of motion, we obtain

(117) |

in which steady-state conditions and vanishing viscosity are assumed. is the gravitational potential. Here, the pressure gradient force in the radial direction is usually ignored.

From this equation, with current estimates for the different terms,
but excluding any dark matter halo, it is easy to integrate numerically
and obtain
*B*_{}(*R*). This was done by
Battaner et al. (1992).

For didactic purposes only, it could be interesting to consider an ideal analytical calculation, assuming, for such large radii, that gravitation itself could be considered negligible. In such a case

(118) |

where
*B*_{0} = *B*_{}(*R* = *R*_{0})
and *R*_{0} is the radius where the
integration begins. As decreases exponentially (or faster, due
to flaring, truncation, etc.) the last integral converges. For
large enough *R* the first term would become dominant and we obtain
that
*B*_{} should become asymptotically

(119) |

This profile,
*B*_{}^{*}(*R*), does not
produce any force, neither
inwards nor outwards, and will be called critical. The real profile
should have a slope lower than the critical one, to produce the fast
rotation, as the magnetic pressure gradient force is probably an
outward force. On the other hand, the magnetic tension
*B*^{2}_{}/*R*
is always inward and does not depend on the gradient,
.

An intuitive reasoning underlying these equations, about how a magnetic
tension produces an inward force, is that the term
1/4^{ . } pushes the gas along the field
lines. In a ring where the magnetic field lines are circular and
contained in the ring, a radial inward force will be produced. This
force will also be present in the disk composed of many rings and if
it is higher than that produced by the magnetic pressure, a net inward
force would be added to gravity, which can only be compensated with
an enhanced centrifugal force. Therefore the disk must rotate more
rapidly.

In the exploratory model by
Battaner et al. (1992),
the calculated strengths are close to the critical (or asymptotic)
profile
*B*_{}^{*}(*R*) *R*^{-1} for very large radii. In this
basic model a strength of about 6 G at
*R* = 25*kpc* was obtained,
which is indeed very high. In more recent models, which will commented
later, much lower values of *B*_{} are obtained, even less than
1G . The authors considered that the predicted synchrotron
radiation was not in conflict with observations reported by
Beck (1982)
and that stability problems in the disk
could arise if the strength reached such high values.

To explore this problem,
Cuddeford and Binney (1993)
developed a
single model, with which they demonstrated that a disk with such a large
magnetic field would produce excessive flaring. They assumed in this
model that the magnetic pressure was times the density, and
this constant, , was considered independent of *z*, but it was
allowed to have a dependence on *R*. There exist several observations
that prove that *B*^{2}/8 decreases with *z* much more slowly than
does the
density, mainly in galaxies with a radio-halo
(Ruzmaikin, Shukurov &
Sokoloff, 1988;
Hummel et al. 1989;
Hummel, Beck & Dahlem,
1991;
Breitschwerdt, McKenzie &
Völk, 1991;
Wielebinski, 1993;
Han & Qiao, 1994 ...)
and so the large flaring calculated by
Cuddeford and Binney was clearly overestimated. Nevertheless, this
paper was very illustrative in showing that the vertical component of
the motion equation must unavoidably be integrated with the
radial one, to assess the problem properly. If the magnetic field
strength capable of driving the rotation curve is too high the disk may
become thicker and flare. Moreover,
Vallée (1997)
considers that
the strength required would unacceptably expand the HI disk.

But even when adopting
*B*^{2}/8 = with (*r*) being
independent of *z* (i.e. the condition assumed by
Cuddeford and Binney, 1993),
the flaring of the disk would not be so large as
estimated by these authors. If the disk were too thick the gas far
from the plane would escape if it were slightly perturbed by
very small vertical winds. As such winds are probably present in
spiral galaxies, gravitation at such a high *z* would be too weak to
retain gas moving outward.

An expanded disk cannot be retained. Clouds very far from the plane would be blown away by instabilities producing vertical winds and strong disk-corona interaction.

To demonstrate this and solve the problem raised by
Cuddeford and Binney (1993)
and Vallée (1997),
Battaner and Florido (1995)
developed a second
model in which they adopted the most disadvantageous magnetic vertical
profile, according to the
*B*^{2} condition, but
considered vertical winds and escape of gas, in the z-direction, to a
galactic corona. Let us include a summary of this model.