Astron. Astrophys. 332, 809-813 (1998)

2. The shape of magnetically induced warps

According to Battaner et al. (1990), the vertical magnetic force which induces the warp is of the order , where is the component of the extragalactic field in the equatorial plane of the galaxy perpendicular to the line of nodes, is the component in the direction of the rotation axis of the galaxy and L is an undetermined characteristic length. To deduce this expression, we assume a constant mean value of the magnetic field strength in a large characteristic length connecting the inner disc and the intergalactic medium. This seems to be an extremely simple assumption, but it will be shown later that it does not introduce a quantitatively important error. Let us assume in this simplified model that these quantities , , and L are constants in the region of interest.

The characteristic length L would depend on the degree of ionization which in turn depends on the galactocentric radius. Let us however assume that at any radius the degree of ionization is large enough as to assure infinite conductivity and frozen-in magnetic field lines. In the inner part of the disc this force is negligible compared with the gravitational force, but it becomes increasingly important towards the outer parts of the disc.

The expression is valid for a point in the equatorial plane of the galaxy. For a point above this plane the vertical force becomes

where and is the slope of the direction of the magnetic field in a diagram. It is easily derived that this expression is equivalent to

where is the radial coordinate in the plane

In this simple model we assume that the gravitational potential is that of a point mass in the centre of the galaxy, as a simplifying assumption for the outer part of a disk not embedded in a massive halo. This potential has already been used before, for instance by Cuddeford and Binney (1993). Equilibrium in the vertical direction gives

A full solution for the distribution of the gas in the combined magnetic-gravity force field would reproduce the whole geometry of the warp. This full solution is beyond the scope of this paper. In any case, it would be interesting to, simply but precisely define the warp curve. In the inner unwarped region, at , in the galactic plane. Let us therefore define the warp curve as the locus of points where . We adopt an exponential law for the disc density with length scale R, i.e. . Using R as length unit for x and z, we obtain for the warp curve

where

is one of the adjustable parameters which compares the extragalactic magnetic energy density with the gravitational energy density. The other adjustable parameter is which specifies the direction of the extragalactic magnetic field. The warp curve is therefore defined with just two free parameters: k and .

For small values of x a series expansion gives

which is a very simple expression for small warps. This simple formula illustrates the fact that the maximum efficiency in producing warps is obtained for , as in Battaner et al. (1990)

For very large values of x, we obtain

i.e. the slope of the warp curve matches the direction of the magnetic field at large radii.

In Fig. 1 we plot the obtained curves for , for and . The curves for seem to be unrealistic at first glance. Note, however, that in practice values at are unobservable (or the galaxy simply does not exist at these radii). For instance if the curve is truncated at , the obtained warp curve becomes quite familiar. We reproduce the curve for larger values of x and z in order to see the region where the slope becomes equal to the direction of the extragalactic magnetic field, which probably takes place for radii far from observational capacities or galaxy limits.

Fig. 1. Warp curves for magnetically induced warps. Values of k are indicated beside each curve and the value of is indicated at the top of the plot

It is worth noting that, for , there is a change in slope. It is higher at intermediate regions before reaching its asymptotic value . This is a common feature of real warps, and can even be directly observed in the early contour maps of NGC 5907 and NGC 4565 by Sancisi (1976).

Fig. 2 reproduces the observational curve for one of the best known prominent warps, in NGC 5907, adopted from Sancisi (1976). We cannot exclude that the warp of this galaxy is due to other mechanisms, but we choose this warp because it is one of the most representative and is studied very often. This figure also reproduces a fitting to the model, with parameters and . It is not straightforward to deduce the magnetic field strength from the value of k, mainly because L is an equivalent undetermined quantity. For , , and , it is obtained that values of k in the range to correspond to field strengths between and , in agreement with reported measurements by Kronberg (1994). Probably better results would be obtained with more realistic calculations, but given the present, still exploratory, character of the magnetic model it is preferable to deal with idealized systems. The noticeable fitting of the NGC 5907 warp curve suggest that the magnetic model and the approximations considered here are not unreasonable.

Fig. 2. Warp curve for NGC 5907. Open symbols are experimental data adopted from Sancisi (1976) and the solid line corresponds to the model for and .

© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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