4. Comparison with observations
Eqs. (6) and (7) derived in the previous section have been used to compute the gyrosynchrotron emission from a flaring loop. To calculate the plasma emissivity, , and the absorption coefficient, , the results by Klein (1987) have been used. The only difference is that in our case the integration over the pitch angle has been performed numerically from to , since the reflection at the mirror points makes the electron density n independent on in the interval and zero outside. In this calculation, the effect of the ambient plasma through free-free emission and absorption and the Razin effect have been also taken into account. The intensity is obtained integrating along the line of sight, s, as
where s is perpendicular to the loop projection as appears in Fig. 2. For simplicity every quantity has been assumed constant along s and the loop thickness is assumed proportional to r but independent on (as in Franciosini & Chiuderi-Drago 1995). The successive integration of over the whole source, i.e. with r varying between the dipole lines of force ( and ) shown in Fig. 2, yields the flux density. This last quantity can be compared with the observed radio spectra.
The resulting flux density is determined by the following parameters:
1) The spectral index of the initial and injected electron distributions. This parameter determines the slope in the optically thin region but is not crucial for the observed optically thick spectra; for this reason it has been arbitrarily fixed at reasonable values .
2) The thermal plasma density, . Analyses of its effect on the emission by Massi & Chiuderi Drago (1992) and by Franciosini & Chiuderi Drago (1995) have determined an upper limit cm-3 for this quantity. We have assumed cm-3 for all the computed spectra.
3) The time at which the injection and hence the flare activity starts. It is not possible to infer from our data, since the flare onset can have occurred at any time between the observation of the first rising spectrum and a preceding observation of quiescent emission.
4) The magnetic field at the base of the loop. As shown by Eq. (4), increasing the magnetic field makes radiative losses increase, and this results in a cutoff in the spectrum at lower and lower frequencies due to the suppression of the emission from higher energy electrons. The influence of the choice of is clearly tied to the time interval during which synchrotron losses are important. The value of has been chosen in order that, given , it is possible to reproduce the observed position of the spectral peak.
5) The spot separation , which determines the extension of the loop (in particular a big makes the diffuse halo very large). In fact, a variation of changes the relative importance of the regions at the footpoint and at the top, and hence it is determinant for changing the optically thick part of the spectrum. In particular, decreasing gives a steeper spectrum at lower frequencies because it reduces the contribution from regions far from the star where the magnetic field is lower. The choice of is however limited from the choice of to a range of values that give reasonable loop sizes.
6) The radius of the two starspots on the stellar surface on which the bipolar magnetic loop is anchored. Optical observations give typical spot radii (e.g. Elias et al. 1995); we have therefore considered values in this range.
7) The relativistic electron density at the flare onset (). Its value defines a ground level of emission present before the start of the injection.
8) The injection rate which, together with , determines the intensity of the emission at any time after the start of the injection.
The method used to fit the observed spectra is the following. A ground level emission is reproduced assuming a given value for and deriving . The evolution of the spectrum is then determined by choosing , and according to the observed characteristics of the following spectrum. The comparison of the theoretically predicted spectra at later times with the observed ones constitutes a test for the model.
The results are shown in Figs. 3 - 6 and the parameters used for the fits are given in Table 1. The values of in Table 1 imply very elongated loops, which in some cases extend to dimensions comparable to the binary system separation. The derived is of the order or slightly lower than the photospheric values ( G) derived from optical observations (Giampapa et al. 1983; Gondoin et al. 1985; Donati et al. 1990). These lower values are due to the fact that we do not "see" the emission originating near the photosphere (since it fades in a few seconds), but only the one coming from upper regions. In fact, the lower values are found in the case of flares 1 and 3, where we are observing a later phase, when the injection mechanism and hence the growth are almost over, as explained hereafter.
Table 1. Parameters used for the fits of the observed flare spectra
In Figs. 3 - 6 each spectrum is labeled with the time used in the calculation. This time has been derived from the exact time intervals between the observations, which is generally different from the one deduced from the approximated orbital phases shown in the figures. For flares 2 and 4 (Figs. 4 and 6, respectively) the ground level of emission at corresponds to the quiescent phase of the star. However, the assumption of a quiescent-like level of emission for the ground level, which seems the most obvious and elegant choice, is not always the best for our model. In fact, the evolution of the electron distribution has been derived for a constant injection rate and therefore the model must be applied in this limit. For example in the case of flare 1 the emission remains high but almost at the same level for a time hour, implying that the flare is near its maximum, i.e. in a phase where the electron injection is almost over. To reproduce such a case the only possibility is to assume a very low in this time interval, but the same low value of cannot reproduce the whole rising phase, starting from a quiescent flux of about 10 mJy. For this reason we have assumed as starting time and the corresponding emission as ground level. The same happens for flare 3 where we have assumed that phase corresponds to hours. Flares 2 and 4, for which the evolution can be followed from the beginning, clearly show the longest time evolution. In particular, in this framework it becomes clear that flare 4 is the only one to have a non-positive spectral index between 5 and 10 GHz because many hours have passed since the start of its rising phase.
It is apparent that the model reproduces remarkably well the time evolution of the spectra for the four flares considered in this paper. The physical implications are further discussed in the next section.
© European Southern Observatory (ESO) 1998
Online publication: April 28, 1998