4 Discussion

The availability of both microwave spectra and spatial information at the same time gives us the opportunity to model the coronae of AR Lac. As shown in previous papers (Umana et al. 1993; Umana et al. 1999), the observed flat radio spectra of Algols and RS CVn type binary systems cannot be reproduced by an homogeneous source model. VLBI observations of the close binary systems Algol and UX Arietis pointed out the existence of a two component structure in the coronal layers (Mutel et al. 1985; Lestrade et al. 1998): a compact core, coinciding with the active star, and a larger halo, having approximatively the size of the entire system.

We used the core-halo model developed by Umana et al. (1993) to fit the observed spectra of AR Lac, in order to check if the flux variability can be attributed to the variation of one of the parameters. Although it is not possible to derive a unique solution, we have limited the sizes of core and halo on the basis of the VLBA observation results. In particular, for the halo we assumed the size measured from the VLBA data, and for the core a size smaller than the resolution limit of the interferometer.

Figure 5: Comparison between the observed radio spectra of AR Lac shown in Fig. 2 and the computed spectra obtained by assuming a core-halo structure for the radio source (thick line). The contribution of the halo (dot-dashed line) and core (dashed line) to the composite spectrum are also shown.

For the magnetic field strength we assumed $B\sim B_{\rm phot}(\frac{R}{R_\ast})^{-3}$, where $B_{\rm phot}\sim 600-1000$ Gauss, as derived for other RS CVn systems (Gondoin et al. 1985; Donati et al. 1992).

According to these constraints, we derived the best core-halo model fit labeled a in Fig. 2. We then used the derived values of the average magnetic field strength Band energetic electron number density ( $N_{\rm rel}$), in fitting the other 5 spectra, under the hypothesis that these physical properties of the coronal emitting regions are stationary. Then we tried to fit the other spectra by varying only the structure size. The results of our analysis are summarised in Table 3 and are shown in Fig. 5. For the spectra from a to d, we can get a good agreement between the observed data and a core-halo structure by assuming that the flux variations are due to structure size changes in the range of 1.1 and 1.4 times the stellar radius, corresponding a variation of 0.3 mas, which is below the errors of our measurements.

The low s/n ratio of the VLBA data does not permit us to investigate the behaviour of the emitting region size for each single scan, and so as a function of the orbital phase. The source size is very close to the beam width, so it was not possible to obtain an estimate of the source size for the time ranges during which the flux density was lower then $\approx$2 mJy. Nevertheless, the results obtained for the scans with sufficient s/n ratio, that are shown in Fig. 6 by diamond symbols, suggest that the source size remains almost constant, as the measured changes fall within errors.

Figure 6: Lower panel: source size estimates as a function of orbital phase. The diamonds represent the value obtained from UVFIT, the triangles represent the value of source size we used to fit the observed spectra with the core-halo model. Upper panel: the corresponding total flux density at 8.4 GHz.

On the other hand, it should be noted that the slow modulation of the modelled halo size is in good agreement with the results obtained from an independent analysis of the VLBA data and plotted in Fig. 6. To fit the spectra e and h, we had to assume an "ageing'' of the relativistic electron population and a variation of its number density from $8.0\times 10^{5}$ to $7.0\times 10^{5}$ cm^-3, that is needed to explain the faster decay at the higher frequency.

If the core-halo model is able to account for the radio corona of AR Lac, the visibilities of the VLBA data should fit by a two Gaussian model corresponding to the core and halo. For the Nov. 2-3 data, the average size of the core from the analysis of the spectra a, b, c, d is 0.15 mas, and the average flux density at 8.4 GHz is 1.21 mJy; for the halo, 1.25 mas and 1.74 mJy. For the Nov. 3-4 data, the same parameters from the analysis of the spectra e and f are 0.145 mas, 1.05 mJy for the core and 1.20 mas, 1.17 mJy for the halo. We then model the normalized visibility function at 8.4 GHz with that corresponding to the core-halo model (Fig. 4). It is evident that the VLBA data are consistent with the core-halo scenario derived from the analysis of the radio spectra.

VLBA data indicate a source size close to the separation of the binary components, suggesting the possibility of an emitting region located between the system components. UV emission from plasma close to the Lagrangian point in between the system components has been suggested for AR Lac (Pagano et al. 2001) and other RS CVn-type systems (Busà et al. 1999).

In partial overlap to our observations, X-ray observations of AR Lac were performed with the Beppo SAX satellite (Rodonò et al. 1999). This gave us the opportunity to determine whether the physical parameters of the radio emitting regions, derived from the comparisons between the observations and the core-halo models, are consistent with a co-spatial model for both the X-ray and radio emitting source.

Spectral analyses performed by several authors (Swank et al. 1981; Singh et al. 1995) showed that the X-ray emission from close binary systems requires at least two plasma components characterised by different temperature and volumetric emission measures (EM) to be modelled. On the basis of the first observation run, that started on Nov. 2 at 06:07 and ended on Nov. 4 at 17:50, Rodonò et al. (1999) derived for the first component $T_1=(0.93-1.02)\times 10^7$ K and $EM_1=2.3\times 10^{53}$ cm^-3 and for the second $T_2=(2.47-2.73)\times 10^7$ K and $EM_2=3.1\times 10^{53}$ cm^-3.

Assuming that the higher temperature component is associated with the halo and the the lower temperature component with the core, we can check if the magnetic field, as derived from the radio data, is strong enough to contain the X-ray source. This means that $\beta$, i.e. the ratio between the density of kinetic energy ( $P_{\rm c}\approx 2N_{\rm e}kT$) and the density of magnetic energy ( $P_{\rm M} \approx \frac{B^2}{8 \pi}$) has to be less then unity. If the plasma density $N_{\rm e}$ is constant over the emitting volume V, $EM=V\times N_{\rm e}^2$, and assuming the size (radius) from the analysis of the radio data of $4.7\times 10^{10}$ and $3.8\times 10^{11}$ cm for the core and the halo respectively, we get $N_{\rm e}\approx 2.3\times 10^{10}$ and $\approx$ $1.2\times 10^{9}\ {\rm cm}^{-3}$(see Table 3).

We obtain $\beta = 0.04$ for the core and $\beta =0.23$ for the halo. The physical parameters obtained from our analysis are therefore consistent with the hypothesis of a co-spatial X-ray and radio source.

We will furthermore test the possibility that the radio emission can be attributed to the same thermal electron population responsible for the observed X-ray emission. The brightness temperature of the resolved radio source at 3.6 cm, obtained from the relation

$$\begin{eqnarray*}T_{\rm B}=1.97\times 10^6 \frac{F_{\rm mJy}\lambda^2_{\rm cm}}{\theta_{\rm mas}^2} \end{eqnarray*}$$

is $T_{\rm B}\, \simeq \, 5.39\times 10^7$ K and $T_{\rm B}\, \simeq \, 3.70\times 10^7$ K for the first and the second session, respectively. These values are higher by a factor of two than the temperature derived from the Beppo SAX data, but the order of magnitude is the same. We simulated the spectra of a core-halo structure by adopting thermal gyrosynchrotron emission and using the expression given by Dulk (1985) for the emission and absorption coefficients. By combining the physical parameters derived from Rodonò et al. (1999) for the thermal coronal plasma with our VLBA results, we can fix the temperature, the thermal plasma density and the maximum size of the emitting regions. The only free parameter left is, thus, the magnetic field strength. The spectrum obtained for B=600 Gauss for the core and B=200 Gauss for the halo is plotted in Fig. 7 and compared with the spectra obtained from the VLA maps over the two sessions.

Figure 7: Simulated spectra from a core-halo structure for thermal gyrosynchrotron obtained with a magnetic field of 600 Gauss in the core and 200 Gauss in the halo (thick line) and $10\,000$ Gauss in the core and 300 Gauss (dashed line) for the halo. Overlaid are the average spectra for the two sessions.

It is evident that gyrosynchrotron emission from the same thermal population responsible for the X-ray emission is not able to account for the observed spectra, unless magnetic field strengths higher then 1000 Gauss are considered. Moreover, even assuming such an intense magnetic field, it is not possible to reproduce the quite flat observed spectra. Beasley & Guedel (2000) reached a similar result from simultaneous radio and X-ray observations of the RS CVn-type binary system UX Ari during quiescence.