Astron. Astrophys. 344, 154-162 (1999)

4. Application of the model

We fit the model of Eq. (3) to that part of the lightcurves from the observations introduced in Sect. 2 that are by visual inspection identified with the outburst due to their enhanced count rates. Except for V773 Tau (see Fig. 7) none of these lightcurves (Fig. 3c, Fig. 5, and Fig. 6) can be explained by simple sine-like variations due to rotational modulation of the quiescent emission. There is always an additional feature present, namely a flare. The lightcurves discussed here are characterised by a concave shape of the (e-folding) decay phase typical for flares (whether or not rotationally modulated), while a simple sine-like rotational modulation of quiescent emission always produces a convex shape in the decay part.

In all cases we examined, the quiescent count rate is held fixed on its average pre-flare value. Thus, three free parameters have to be adjusted to the data: the strength of the flare, in cts/s, the decay timescale of the lightcurve, , and the radius r of the flaring volume relative to the stellar radius, R. An additional freedom allowing for an offset between rotational phase of the flare and the apparent outburst of the flare, which is observed as a rise in count rate, is used for the modeling of the Ginga (Algol) and ASCA (V773 Tau) lightcurves (see the explanation in the previous section).

The rotational periods of our sample stars were known from optical photometry, except for the case of SR 13. In principle, could be included as a further free parameter in the fit. But with this additional freedom the fit does not result in a unique solution, as we will show in the example of SR 13, and thus may not be uniquely determined from our model. For Algol we assume synchronous rotation.

Our best fit results will be discussed in detail in the following subsections. The best fit parameters for all flares are listed in Table 2 together with the rotation periods and measured quiescent count rates. Note, however, that the model depends to some degree on the initial parameters, and the parameters are not well determined due to correlations, such that similar solutions are obtained for different combinations of parameter values. For the flares on Algol and V773 Tau we computed 90 % confidence levels for the best fit parameters according to the method described by Lampton et al. (1976). The low statistics in the data of the lightcurves of SR 13 and P 1724 do not allow to apply this method. We, therefore, do not give uncertainties for the best fit parameters of these events.

Table 2. Best fit parameters of the rotating-flare model and 1 uncertainties. No uncertainties are given for the ROSAT observations (see text). Dots indicate that no uncertainties could be determined within the limits of the parameters given by model restrictions (ie. ).The rotational periods were fixed on the values given in column 2 except for SR 13 whose rotational period is unknown: ^a Hill et al. 1971 (assuming synchronous rotation of Algol B), ^b Cutispoto et al. 1996, and ^c Welty 1995. The quiescent count rate, , was determined from the pre-flare data except for V773 Tau.

4.1. Algol

A two day long continuous Ginga observation of Algol in January 1989 (first presented by Stern et al. 1990, 1992) includes a large flare event. Secondary eclipse begins during the decay of that flare, but it seems to affect the count rate only marginally. Preceding the large flare, primary eclipse and a small flare are observed (see discussion in Sect. 3). We therefore base our estimate for the quiescent emission on the time between the two flare events, i.e. immediately before the rise phase of the large outburst which marks the onset of the time interval to which we apply the `rotating-flare model'. From fitting Eq. (3) to the data after in Fig. 3c we obtain a best fit of 5.34 for 108 degrees of freedom. The fit can be significantly improved when the critical phase is allowed to vary around the start of observation as explained in Sect. 3 ( for 107 dofs), and all stages of the flare are well represented by the model. Although this value of is still far from representing an excellent fit, the ability of the model to reproduce the overall shape of the X-ray lightcurve is convincing.

A detailed spectral analysis of the flare event on Algol was undertaken by Stern et al. (1992). The emission measure EM they obtained from a thermal bremsstrahlung spectrum + Fe line emission for 11 time-sliced spectra covering all phases of the flare is displayed in Fig. 4. According to the best fit of our `rotating flare model' to the lightcurve, the flare volume has become almost completely visible () around , i.e. about 10 hours after the rise in count rate was observed to set in. The exponential decay of the emission measure for the last three values of Fig. 4 can thus be extrapolated to the previous part of the observation to find the values for the emission measure intrinsic to this flare event. The observed emission measure during the rotationally dominated beginning of the flare is then fairly well reproduced by correcting the extrapolated values for the time dependence of the volume (), where we neglect possible variations of the plasma density . Thus, in contrast to the observation, the actual emission measure of the flare event is highest at the onset of the flare at and it decays simultaneously with the count rate () due to a decrease of or shrinking loop volume. The good agreement between the emission measure observed by Stern et al. (1992) and the values expected from our model (see Fig. 4) provide convincing evidence that the application of the `rotating-flare model' is justified for this flare. The development of the temperature during the flare (see Stern et al. 1992) does not show the characteristic hump shape, but is close to a pure exponential decay as expected for volume unrelated parameters.

Fig. 4. Emission measure during the large flare on Algol. Small, filled circles represent the spectral results of Stern et al. (1992). Large, open circles are our extrapolation of the exponential decay fitted to the last three data points of Stern et al. (1992). The lower curve denotes the extrapolated emission measure after correction for the time dependence of the volume.

Fig. 5a-d. ROSAT PSPC observation of a flare on SR 13: a  X-ray lightcurve (400 s bins, 1 uncertainties) and best fit model. Data used in the fitting process are represented by filled circles. Open circles are pre-flare data. b  Time evolution of the hardness ratio HR2 during the flare on SR 13. Note, that the first value (open circle) was determined from the data of a 1991 March observation and is plotted here at an arbitrary time prior to the flare. Time dependency of the temperature c and emission measure d of a Raymond-Smith model spectrum display large uncertainties (shown are 90 % confidence levels).

Fig. 6. ROSAT HRI lightcurve of the flare on P1724 (400 s bins, 1 uncertainties) and best fit model. The meaning of the plotting symbols is the same as in Fig. 5

The values of the flare parameters (, r) resulting from the fit of our model to the lightcurve are similar to those derived from normal (ie. neither occulted nor rotationally modulated) Algol flares observed by various instruments. Ottmann et al. (1996) summarize the characteristic parameters of three Algol flares (see their Table 5): the decay timescale seems to vary by one order of magnitude between and hours, while the loop length found from standard loop modeling extends from stellar radii. We conclude that modelling the January 1989 X-ray flare on Algol in terms of rotational modulation yields flare properties which are perfectly consistent with those of other X-ray flares.

4.2. SR 13

Casanova (1994) discusses the similarity between a flare of SR 13 observed by the ROSAT PSPC and the Algol flare analysed in the previous subsection. Besides the absolute values of the count rate which is by a factor of 500 higher for Algol (note, that the observations were performed by different instruments and, therefore, the differences in count rate are no direct measure for the differences in flux), the shape of the SR 13 flare is very similar to that of the flare on Algol.

The rotational period of the CTTS SR 13 is unknown to the present. We determined the quiescent emission of SR 13 from the pre-flare data of the first satellite orbit. Our attempt to find the rotational period from the modeling of the flare according to Eq. (3) with a free parameter failed, since the uncertainties in the data do not allow to distinguish between different fit solutions. In Fig. 5a we overlay the data points by two solutions of the `rotating-flare model', one was found assuming a period of 3 d, the other one corresponds to twice that period.

A detailed spectral analysis of this specific flare event similar to the one carried out for the Algol flare (see Stern et al. 1992 and Sect. 4.1) is not practicable due to the low number of counts. To underline the difficulty in evaluating the spectral information for the flare on SR 13, we briefly discuss the results from our attempts to fit a Raymond-Smith model (Raymond & Smith 1977) to the spectra during four stages of the flare that were defined in the following way: phase 1 is given by the quiescent stage, phase 2 is the observed, apparent flare rise, and phases 3 and 4 correspond to the observed decay. The three flare time intervals are marked in Fig. 5a. The quiescent spectrum was computed from an earlier observation obtained in 1991 March 05-10 by the ROSAT PSPC due to the scarcity of non-flare data in the September observation.

A two-temperature Raymond-Smith model was needed to obtain acceptable fits with for each of the four phases, where we held the temperature of the softer component fixed at . The graphs in Fig. 5c and d display the best fit values for the temperature and emission measure of the hotter component. The large uncertainties of the best fit values shown in Table 2 prohibit a spectral study with better time resolution, but having only four time bins to define the spectral evolution, the decay of the emission measure after the flaring volume became visible could not be pinned down, and thus a check of the `rotating flare model' by modeling of the time development of the emission measure is not possible for this flare on SR 13. A slight indication for cooling is present in the evolution of kT during the flare suggesting that the actual outburst might in fact have occurred as early as during the second phase.

In cases of insufficient data quality hardness ratios may be used to give a clue to spectral properties. Neuh"auser et al. (1995) showed that the ROSAT hardness ratio HR2 (see Neuh"auser et al. 1995 for a definition) is related to the temperature of the plasma (see their Fig. 4). We computed HR2 for the four different time intervals defined above. The time evolution of the hardness ratio HR2 is displayed in Fig. 5b. The decreasing HR2 during the last three intervals supports the decline in temperature measured in the spectra and presents further evidence for cooling.

To conclude, the results on the SR 13 lightcurve, while having an admittedly reduced statistical significance, are fully consistent with an interpretation in terms of flare cooling combined with rotational modulation.

4.3. P1724

The ROSAT HRI observation of P1724 comprises 13 satellite orbits (see Fig. 6). Similar to the flare on SR 13, constant count rate is observed only during the very first orbit. We, therefore, base our value for the quiescent emission, , on this time interval and find that it is consistent with most of the observations of P1724 presented by Neuh"auser et al. (1998). However, in March 1991 the count rate was higher by a factor 4, possibly indicating long-term variations in the quiescent emission.

The lightcurve during the second orbit resembles a small flare event and is omitted from our analysis. The maximum of the large flare that dominates this observation extends over almost 4 hours. During the decline of the count rate irregular variations are observed that might be due to short timescale activity superposed on the large flare event. We ignore these fluctuations and model the lightcurve beginning after the second data gap by Eq. (3).

The total number of source counts measured in this observation is smaller than 1000, and thus far too low for a timesliced hardness ratio analysis. Having in view the similarity between the X-ray lightcurve of this flare and the previously discussed flares, and the good description of the data by our best fit, we trust that the `rotating flare model' applies also to this observation.

4.4. V773 Tau

An intense X-ray flare on V773 Tau has been reported by Skinner et al. (1997) and interpreted as a sinusoidal variation whose period is approximately equal to the known optical period of V773 Tau, i.e. 71.2 h.

The ASCA lightcurve of this event (see Fig. 7) is characterized by constant count rate at maximum emission which lasts over more than 2 h making the event a candidate for a rotationally modulated flare. No data is available prior to the peak emission, but observations resumed about 10 h after the maximum and display a steady decrease in count rate. Since the pre-flare stage and the rise of the flare are completely missing in the data, the flare volume must have emerged from the backside of the star well before the start of the observation, and an additional time offset parameter has to be included in the fit (analogous to the modeling of the flare on Algol), to determine the time that elapsed between phase (= emergence of the flare volume) and the first measurement.

Fig. 7a and b. a  ASCA lightcurve of the observation of V773 Tau (128 s bins, 1 uncertainties) and best fit. Modeling of a rotating flare alone (dotted line) does not lead to an acceptable fit, while adding an additional feature producing enhanced X-rays until JD 2449977.85 and then gradually disappearing describes the bending of the lightcurve and improves the fit perceptibly (solid line). b  Time evolution of the visible flare volume and the secondary X-ray emitter during the flare on V773 Tau as found from the best fit of the `rotating-flare model' to the lightcurve.

Since the flare covers the complete observation a value for the quiescent count rate, cps, was adopted from a later ASCA SIS0 observation in February 1996, also presented by Skinner et al. (1997). Despite the fact that the broad maximum of the September 1995 lightcurve can be explained by the loop rotating into the line of sight, no satisfying fit could be obtained for the flare on V773 Tau by the model of Eq. (3) even after a time offset was added ( for 149 degrees of freedom): The decay of the observed lightcurve seems to be faster than our model predictions (see Fig. 7 dotted curve). We note that the data is slightly bended towards the time axis around the 6th data interval after the start of the observation. This behavior, producing an overall `convex' shape of the X-ray lightcurve, could be due to an additional feature on the surface of the star. We suggest that a localized region with enhanced X-ray emission can be responsible for this break if this region disappears due to the star's rotation at . For comparison we show a fit of our `rotating flare model' where such a feature has been included (solid line in Fig. 7, for 149 degrees of freedom). Since we are interested in a qualitative description of the shape of the lightcurve only we assumed that this region makes up for 0.2 cps during its visibility and begins to disappear gradually at JD 2449977.85. Representing this X-ray emitter by another set of free parameters would certainly further improve the already good agreement between data and model.

Skinner et al. (1997) also present the time behavior of the emission measure derived from a two-temperature fit to the ASCA spectrum. If our interpretation of adding a soft X-ray spot, which gradually rotates away, is correct, then the emission measure of the soft component should stay constant for most of the time, but decrease towards the end of the observation. However, the S/N of the time-sliced spectral fits (Skinner et al. 1997, his Fig. 10, middle panel) is not sufficient to judge whether this is indeed the case.

To conclude, other interpretations such as a different kind of anomalous flaring cannot be excluded from the data of this observation. Tsuboi et al. (1998) have presented another ASCA flare observation of V773 Tau. In that observation V773 Tau shows the typical flare behavior in the sense of a sharp rise and a subsequent longer decay of count rate, temperature, and emission measure. However, their attempt to fit an e-folding decay to the lightcurve of the hard X-ray count rate was not successful the count rate remaining too high towards the end of the observation. Hence, unusually long decays seem to be characteristic for V773 Tau.

© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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