3. Flare modelling
Both models that we will apply to the X-ray light curve of the 1994 flare of HU Virginis were initially developed to describe solar flares. Since they approach the problem from two different sides it is very interesting to see how consistent the results will be. Earlier applications of these models to ROSAT observations of long duration flares were discussed by Schmitt (1994) and Kürster & Schmitt (1996).
3.1. Rebinning the light curve
Since both flare models just describe the long-term trend in the light curve and cannot account for the fine structure in the onset and at the beginning of the maximum, we rebinned the data into one point per ROSAT-observation "slot" (which typically lasts for 2000 seconds) by taking the mean of all original points weighted with the inverse error. As a conservative approach we adopted the standard deviation of all (original) data points within a slot from the slot mean value as the error of the new points. These rebinned values are shown in Figs. 4 and 5.
3.2. Model 1: a quasi-static cooling flare
The quasi-static cooling loop model of van den Oord & Mewe (1989) assumes that a single coronal loop cools via X-ray emission whitout any further heating (conductive cooling is negligible). This model provides information only for the decay phase of the flare light curve and does not consider the flare onset and thus the heating mechanism. A more detailed description of the model can be found in the paper by van den Oord & Mewe (1989). The radiative energy per unit time released during the flare is given by:
where is the radiated energy at the peak of the flare, t is the time counted from flare peak, and is the radiative cooling time:
is the temperature at the peak of the flare, the particle density and () the emissivity or radiative cooling function. From a -fit to the decay phase, using and as free parameters, we obtain the best value for the radiative cooling time of = sec and = cts sec-1. The errors indicate the 68% (" ") confidence region for two parameters using the method described by Lampton et al. (1976).
This best-fit model is plotted along with the observed decay phase of the flare in Fig. 4. Since we have no spectral information with the HRI detector we must estimate the peak temperature and the emission measure by using values from PSPC observations of similar X-ray flares on other RS CVn-type stars. We adopt = 5.8 K and = 7.3 cm-3 from the CF Tuc flare (Kürster & Schmitt 1996). The value of has to be adjusted to the lower sensitivity of the HRI detector (by a factor of 3.2, Kürster et al. 1997), the greater distance of HU Viriginis (125 pc compared to 54 pc for CF Tuc) and the lower countrate. We thus derive = 1.6 cm-3 for the flare on HU Virginis. For the plasma density at the peak of the flare we obtain = cm-3 and for the flare volume we find V = cm3 using the relation . Having an estimate for the total flare volume we may further estimate the flare extension based upon a single-loop model. We may use where is the total length of the loop and is the ratio of the loop diameter to the total length of the loop. From solar analogy, where a typical value of is 0.1, we derive cm 23 4 ( 5.6 for the K-star). Approximating the height H of the loop as / (which is not exact of course, since the loop is not a half circle), we find H = cm 7.4 1.3 . Since we do not know if the flare on CF Tuc was hotter or cooler than the one on HU Virginis, we varied by a factor of 2 and adjusted the corresponding emission measures from Raymond-Smith plasma models (Raymond & Smith 1977) reproducing the observed peak count rate and redetermined the loop size. For a temperature = 1.2 K and a corresponding emission measure = 2.27 cm-3 we obtain a loop height of 5.8 cm 8.3 1.5 . For a cooler temperature, = 2.9 K, and the smaller corresponding emission measure, = 1.29 cm-3, we obtain H = 4.8 cm 6.9 1.2 . In all cases the resulting loop height is comparable to the stellar radius.
3.3. Model 2: a two-ribbon flare
The two-ribbon flare model (Kopp & Poletto 1984) describes a quite different astrophysical scenario. Most importantly, further heating is implicitely accounted for. The model describes the two-ribbon type flares observed on the Sun where reconnection of an open magnetic field structure delivers the energy from which a certain fraction goes into X-rays. The open field structure was created before the flare by a disruptive event and the emission occurs in an arcade of "post-flare loops". Actually, the energy reservoir of the flare is the difference between the non-potential magnetic field before and the potential magnetic field after the reconnection. The excess magnetic energy gained by the reconnection of open magnetic field lines shows up as thermal energy of the bright X-ray loops. The model is a 2-D field representation of the magnetic field geometry of the flaring region. Again, for more details of the model we refer to the papers of Kopp & Poletto (1984) and Poletto et al. (1989). For our purpose it is sufficient to know that the energy-release rate is given by:
where , and is the Legendre polynomial of the order n that describes the morphology of the magnetic field. is the upward-motion function of the reconnection point in the corona. Once more it is the solar analogy where this motion was determined as a simple exponential function:
where is the time it takes the neutral point to reach its maximum height ( is the stellar radius). The parameter n describes the angular width of the flaring region (higher n values correspond to smaller regions).
If one could fit the model to the observational data to distinguish between different values of n one would obtain some information on the size of the active region responsible for the flare. Schmitt (1994) already fitted curves of n =2, 6, and 40 to a long duration flare on EV Lac but found that there is no substantial difference between these curves and all of them gave a reasonable fit to the data. Kürster & Schmitt (1996) also compared theoretical two-ribbon light curves with their highly structured light curve of the flare on CF Tuc. Again, no decision in favour of one model could be made, primarily because of the lack of good coverage of the onset phase of the flare. Fortunately, our observation of HU Virginis covered the onset and the maximum of the flare quite well. During the fit process the upward motion time , the order of the Legendre polynomial n, and the overall normalization were used as free parameters. Table 1 compares the best-fit (smallest ) results for different model parameters. Degree of freedom is 8 (11 data points minus 3 free parameters) hence .
Table 1. Results of two-ribbon flare model fits. is given in sec and n denotes the order of the Legendre polynomial. Degree of freedom is 8 (11 data points minus 3 free parameters). For the n -values within the 68 % confidence region () the corresponding -range is given.
Four values for n (3, 4, 5 and 6) and -values from 151000 to 178500 sec are located within the combined 68 % confidence region for 3 parameters ( according to Lampton et al. 1976). These values correspond to latitudinal widths of the flaring region from to and maximum heights of the flaring loops between 0.5 and 1.05 . Fig. 5 shows the two-ribbon model light curve with the smallest -value (n =3 and = 171500 sec) compared to the observed flare light curve of HU Virginis. The maximum height of the neutral point for such a large flare is 4.2 km 5.9 1.05 .
© European Southern Observatory (ESO) 1997
Online publication: March 26, 1998