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

3. Rotational effects on flare lightcurves and spectra

The X-ray lightcurve during a flare event is commonly described by a steep, linear rise followed by an exponential decay. The e-folding time of the decay varies greatly between several minutes to hours depending on the nature of the flaring star. Disagreement prevails on the question whether the apparent quiescent emission might be attributed to continuous, unresolved short timescale activity. The detection of a high temperature spectral component in the quiescent spectrum might hint at such low-level flaring (Skinner et al. 1997).

Several flares have been observed that do not match the typical appearance: instead of displaying a sharp peak, the lightcurves of this type of events are characterized by smooth variations around maximum emission that sometimes goes along with a slower rise as compared to standard flare events. In these cases the shape of the lightcurve can be reproduced by taking account of the rotation of the star. Flares that erupt on the backside of the star become visible only gradually as the star rotates and drags the plasma loop around. The visible flare volume is thus a function of time which modulates the exponential decay. The scenario we have in mind, and that we will refer to as the `rotating flare model' henceforth, is sketched in Fig. 1.

Fig. 1. Top-view of the rotational plane of a flaring star. R is the radius of the star which rotates counterclockwise with angular velocity . The flaring volume is represented by the smaller circle of radius r. The angle is defined to be zero at the time when the plasma loop starts to disappear from the observer's view as a consequence of the rotation of the star.

For simplicity the emitting plasma loop is approximated by a sphere anchored on the star's surface. The fraction of the loop volume which is visible to the observer is given by

where R is the radius of the star and r the radius of the spherical plasma loop. The time dependency of V is hidden in , the angle between the current position of the flaring volume and the position where a flare just begins to become occulted by the star. Note that Eq. (1) does not hold for rotational phases during which the flaring volume is either completely behind () or completely in front of the star (). The time dependency of in Eq. (1) depends on whether the loop is disappearing or reappearing and is given by

where is the critical phase at which the plasma volume has just disappeared. Eq. (1) is not valid any more until the loop reaches phase and begins to move into the line of sight again. is a function of the relative size of the radius of the flaring volume and the radius of the star, . The visible fraction of the plasma volume as a function of time is plotted for different values of the radius ratio f in Fig. 2.

Fig. 2. Visible fraction of the plasma loop volume approximated by our rotating-flare model of Fig. 1 as a function of time for different values of the radius fraction .

Our model is based on several simplifying assumptions concerning the flare geometry. First, we imply that we look directly onto the rotational plane, i. e. , and that the flare takes place at low latitudes. Flares that erupt in polar regions, in contrast, in the configuration of Fig. 1 would remain partially visible during the whole rotation period. Furtheron, Eq. (1) does not take account of the curvature of the star. We content ourselves with these approximations because, given the present quality of the data, further sophistication of the model seems to be unnecessary.

Making use of the configuration described above, for a flare which is observed while the flaring region turns up from the backside of the star, the X-ray lightcurve can be modeled by

where is the quiescent X-ray count rate of the star, the strength of the outburst, the decay timescale of the count rate and the visible fraction of the volume of the plasma loop given by Eq. (1) for values of within the allowed range, and by 0 or 1 for angles outside the intervals of Eq. (2).

The hump-like shape of the lightcurves we will discuss in the next section can be reproduced if the visible volume V increases during the first observed part of the flare, ie. . Three critical moments determine the rotating flare event: the time of outburst, the time when the flare region passes phase and begins to move into the line of sight, and the time at which the observation started. In the next paragraphs the relation between these times will be examined.

First, an offset between flare outburst and the time when it becomes visible to an observer (at ) might be present, when the flare takes place on the occulted side of the star. In our model such a time offset contributes only to the normalization of the exponential and cannot be separated from the intrinsic brightness of the outburst. The upper limit for is given by

since for larger time offsets the flare would have been observed also at , that is before its occultation. Given the rotational periods of several days, exceeds the typical decay timescale for TTS flares (of a few hours). However, from an observational point of view it is impossible to exclude that the flares occurred already before they rotated away, because data extending over several hours before the reappearance of the flare are not available for the lightcurves analysed here, except in the case of Algol. Indeed, at first glance the combination of the two phases of enhanced count rate in the Algol observation (see Fig. 3a) looks similar to what is expected to be seen from one very long flare that disappeared behind the star shortly after outburst and reappeared half a rotational cycle later still displaying a strong count rate enhancement. Fig. 3b gives an example of a theoretical lightcurve for a flare which is occulted right after its outburst and whose duration is more than half the rotation period. However, our attempt to model the complete Algol lightcurve from Fig. 3a by such a single temporary occulted flare was not successful because the model restrictions concerning the relative strength of the pre- and post-occultation part of the flare are not met by the Algol lightcurve. Thus we can rule out offsets larger than for the Algol observation discussed in this paper, and the short rise in count rate observed before the large flare must be due to an independent event.

Fig. 3a-c. GINGA lightcurve of the Algol flare in comparison with the rotating-flare model: a  Algol lightcurve including the short event 13 hours before the large flare, b  theoretical shape of a lightcurve representing a strong flare which is occulted immediately after its outburst and reappears later, and c  large Algol flare overlaid by our best fit model curve. Binsize: 128 s (1 uncertainties)

Second, the start of the observation of a flare event, which is characterised by a beginning enhancement of the observed count rate, can differ from the time at which the outer edge of the plasma loop emerges from the back of the star due to gaps in the data stream. So, strictly speaking, another offset has to be included when fitting the `rotating flare model' to the data to take account of a possible delay of the observation with respect to flare phase . Such an additional parameter that allows to determine the rotational phase of the flare region at the beginning of the observed rise is needed to obtain acceptable fits for the flares on Algol and V773 Tau. For the ROSAT observations (of SR 13 and P1724), however, an offset does not improve the fit due to the low statistics of the data. We note here, that the observed flare rise is only apparent according to our model: The star is assumed to have flared (and thus exhibited its maximum emission) well before the observed maximum, and the count rate is low at that time only due to the fact that the flaring volume has not yet become visible.

The enhanced X-ray emission during flare events is produced by a hot plasma which has been heated to temperatures of and above. Optically thin plasma models show, when applied to spectra representing different stages of the flare, that after the outburst the temperature drops exponentially to the quiescent level. The temperature observed for a rotationally modulated flare should thus be highest during the phase where the flare emerges from the backside of the star when the observed lightcurve has not yet reached its maximum. The emission measure, on the other hand, being a volume related parameter is expected to show a time evolution similar to the lightcurve.

© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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