The one-dimension model

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 ${1 \over {4 \pi}}$$\vec{B}\,$ ^. $\nabla$$\vec{B}\,$ - ${1 \over {8\pi}}$$\nabla$B² (e.g. Battaner (1996)). The first term in cylindrical coordinates (R, $\varphi$, z) will be

(115)

if the field is azimuthal, ( B_R = B_z = 0), under azimuthal symmetry, ( $\partial$/$\partial$$\varphi$ = 0), the radial component of this force is simply - B_$\varphi$²/R.

The magnetic pressure gradient force - ${1 \over {8\pi}}$$\nabla$B² would have a radial component simply given by - (1/8$\pi$)$\partial$B_$\varphi$²/$\partial$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. $\cal {F}$ 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_$\varphi$(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 B₀ = B_$\varphi$(R = R₀) and R₀ is the radius where the integration begins. As $\rho$ 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_$\varphi$ should become asymptotically

(119)

This profile, B_$\varphi$^*(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 B²_$\varphi$/R is always inward and does not depend on the gradient, ${{\partial B_\varphi} \over {\partial R}}$.

An intuitive reasoning underlying these equations, about how a magnetic tension produces an inward force, is that the term 1/4$\pi$$\vec{B}\,$ ^. $\nabla$$\vec{B}\,$ 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_$\varphi$^*(R) $\propto$ R^-1 for very large radii. In this basic model a strength of about 6 $\mu$G at R = 25kpc was obtained, which is indeed very high. In more recent models, which will commented later, much lower values of B_$\varphi$ are obtained, even less than 1$\mu$G . 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 $\alpha$ times the density, and this constant, $\alpha$, was considered independent of z, but it was allowed to have a dependence on R. There exist several observations that prove that B²/8$\pi$ 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 B²/8$\pi$ = $\alpha$$\rho$ with $\alpha$(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 B² $\propto$ $\rho$ 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.