2 The magnetic model of CU Vir

We had preliminarily modelled the field of this star (Glagolevskij et al. 1998) under the assumption of a dipolar-quadrupolar configuration. However, it turned out later that such a concept describes rather the shape of the phase relation $B_{\rm e}(P)$ than the actual field structure. Besides of this, the distribution of the surface magnetic field strength in this case proves to be distorted (Gerth & Glagolevskij 2000). More correct results are provided by a technique that we term "magnetic charge method" (MCD), which selects the number of virtual charges, their coordinates and distances from the star's center and calculates the surface field strength repeatedly, using the procedure of sequential iterative approximations, so that the computed phase curves $B_{\rm e}(P)$ and $B_{\rm s}(P)$ would fit to the observed ones ($B_{\rm e}$ - effective magnetic field, $B_{\rm s}$ - average surface magnetic field). The final version of the relation $B_{\rm e}(P)$ is chosen by the least-squares method.

The MCD-method offers excellent advantages for the numerical computation. The potential of a point-like magnetic charge is spherically symmetric. All potentials and gradients are added up linearly. There are no constraints on the number and spatial distribution of sources. However, according to physics, the number of opposite (positive and negative) charges has to be equal and the total of all charges compensates to zero.

Let us return to the problem of modelling the CU Vir magnetic field to get a more reliable notion concerning the field structure than we had before. Unfortunately, in the case of CU Vir, we cannot investigate the fine structure of the field because of the small number and the inadequate accuracy of the observational data compiled up to now. But we are able to study the global structure.

By this way we obtained a new CU Vir magnetic field model. Figure 1A presents the data of measuring $B_{\rm e}$ (Borra & Landstreet 1980) versus the phase of the rotation period ( $P = 0\hbox{$.\!\!^{\rm d}$ }52$). The relationship is different from a sinusoid; the positive half-wave is wider than the negative one. The inclination angle of the star to the line of sight is determined accurately enough, provided that the relation $B_{\rm s}(P)$ is available. But there are no data for the average surface field $B_{\rm s}$, which should look like Fig. 1B. Therefore, the star inclination angle i is found from $v\,\sin \,i = 147 \pm 2$ kms^-1 (Hatzes 1997). We derive $i = 60^{\rm o}$ from the absolute bolometric magnitude $M_{\rm b}$ (see below), which is equal to the result of Hatzes (1997) based on the construction of the silicon map.

Figure 1: The result of modelling the magnetic field of CU Vir. A) dots - measuring data, solid line - model dependence B) calculated average surface magnetic field variation.

The best fit of the computed phase curve $B_{\rm e}(P)$ to the measurements we have obtained under the assumption of the following parameters:

No	Q	d	$\lambda$	$\delta$
1	+1	0.40	30	-3
2	-1	-0.20	210	3

The table gives the number of the virtual magnetic charge Q (relative units), its distance d from the star center as fraction of the radius, the longitude $\lambda$ and latitude $\delta$. The shape of the relation $B_{\rm e}(P)$ is well described by a dipole model with two equal "magnetic charges" of different sign - the sum of the magnetic charges being zero (Gerth & Glagolevskij 2000). The parameter values of d show that we deal with a decentered dipole displaced toward the negative charge by 0.3. The displacement is defined firstly of all by the ratio of the half-widths of the positive and the negative half-wave of the relation $B_{\rm e}(P)$. The value of $\delta$ is determined by the ratio of the maxima.

It has turned out that the dipole lies almost in the equatorial plane. The inclination angle of the dipole axis with respect to the rotational axes is $\beta = 87^{\rm o}$. As a result of the lack of data on the average surface field $B_{\rm s}$, the derived angles i and $\beta$ should be adopted as a first approximation. The modelling done with other inclination angles, differing from $60^{\rm o}$ by $\pm 10^{\rm o}$, has shown that the location of the magnetic poles in latitude changes by about the same value.

The model phase relation is shown in Fig. 1A with a solid line, while the distribution of the field strength over the surface is displayed in Fig. 2.

Figure 2: Surface distribution of the magnetic field strength Above - pseudo-Mercator map with iso-magnetic lines Below - spherical projections (globes) of the map.

The field of negative polarity is stronger: the values at the poles are $B_{\rm p}(-) = 7.9$ kG and $B_{\rm p}(+) = 1.2$ kG.
The average surface field computed from the obtained model varies from 1.2 kG to 3.2 kG (Fig. 1B). The dipole axis points away from the zero meridian by an angle of $+30^{\rm o}$. The surface region with the negative field is compact, whereas the region with the positive one is rather broad. Obviously, the average surface field with such a low intensity is nearly impossible to estimate from the splitted Zeeman components because of the fast star rotation. If we compare our model relation $B_{\rm e}(P)$ with the analogous relation in the paper of Hatzes (1997) for the central dipole model, then it can be noticed that our displaced dipole model agrees better with the observations.

Let us make some comments on the philosophy of our method to avoid misunderstandig. Since in the given case we deal with a dipolar field, the surface field distribution depends only on the value of the dipole displacement d, but does not depend of the separation l of the magnetic monopole charges Q, because the magnetic moment M = Ql is a constant of the dipole system.

The strange "separation" of a magnetic dipole into two magneticmonopoles of opposite polarity could easily lead to some confusion, because the well-known physics of closed lines of force seems to be violated. However, the lines of force converge generally to separate points in the space, which we regard as the virtual sources. The MCD-method proves to be a powerful heuristic approach for the determination of the virtualsources of a magnetic field. A definite theorem of the potential theory gives evidence, that any field configuration can be produced by the superposition of the fields of numerous point-like sources.