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Astron. Astrophys. 324, 155-160 (1997)

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3. Statistical significance of the data and observational results

3.1. The optical data

The high-speed photometer measures the object, a comparison star and the sky simultaneously in all colors. Therefore the observational errors are dominated by photon noise, unless weather conditions are very unstable. Since we did not record any significant variability on time-scales of less than 1 minute, all the optical observations analysed in the following sections have been binned from 2 seconds to 50 seconds time resolution, with corresponding 1 [FORMULA] errors of [FORMULA] in B, V and R and [FORMULA] 10.008 in U and I.

During the observations BP Tau showed night-to-night variations in its brightness level in all colors. The total amplitude in the V-band of these variations was [FORMULA] 0.25 mag (see Fig 1, upper panel). On one night the star did also show short-term variations on time-scales less than hours. The details of these variations and their relation to the variations in X-ray are discussed in Sect. 3.3.

[FIGURE] Fig. 1. Optical UBVRI light-curves (upper panels) and X-ray light-curves (lower panel) for the whole observing run. During and after the end of the second night the U-band channel of the instrument malfunctioned. The X-ray data are binned at 400 sec intervals and the error bars are set at the 68% confidence level.

3.2. The X-ray data

In order to assess any variability in the recorded X-ray count rates we calculated the Poisson probability to record the observed counts in a given time bin for a single trial, given an assumed count rate level. This calculation was first performed on the data summed over each observing night in order to search for night-to-night variability. In a second step we repeated the calculation to search for variability on shorter time scales.

Let us first consider night-to-night variations. If we use the overall mean count rate [FORMULA] as reference level, we find the count rates too low during the first three nights, and count rates too high during nights four and five. A more reasonable hypothesis therefore appears to assume that the true quiescent level is defined by the mean rate of the first three nights [FORMULA], while nights four and five represent highly significant upward excursions from this mean level. Of course, the alternate hypothesis that night four and five represent the true quiescent level and that nights one to three represent downward X-ray flux excursions is statistically also possible but physically less plausible. The results of our statistical analyses are summarized in Table 1; for each night we quote the recorded number of counts, the adopted count rate reference level and the probability to record at least the observed number of counts both for [FORMULA] and [FORMULA]. Obviously, if the overall mean rate is used as a reference level, one expects almost always to record more counts than observed during the first three nights, while for nights four and five the observed counts are extremely unlikely. If the mean rate for the first three nights is adopted instead as a reference level, nights 1 - 3 show no significant variability, while nights 4 and 5 showed enhanced count rates. At any rate we are forced to conclude that the observed count rate of BP Tau did exhibit nightly variations during our observations.


Table 1. Statistical significances for the variations in the nightly X-ray count rates. See text for details and definition of variables. Note that the different probabilities for nights 2 and 3 comes from the different distribution of counts in each night.

Next we investigated the individual observations separately and used as a reference level the observed mean count rate during each night. To this end we binned the X-ray data into contiguous 400 sec bins and compared the number of recorded counts with the expected number of counts. Since the ROSAT observations of BP Tauri were split up into individual observation intervals of about 1 - 1.5 ksec, i.e., non-multiples of 400 sec, we decided to retain any "left over" bins in excess of 300 seconds for analysis. Thus in total we have 63 such bins. We then looked for bins such that the probability to record at least the observed number of counts was no more than 0.02, and found two such events, one during the first night, and another one during the second night. Since this probability refers to a single trial, we have to fold in the total number of trials (i.e., bins). To find at least two events with a single trial existence probability of 0.02 in 63 trials, has itself a probability of 0.46. Thus - statistically speaking - we are forced to conclude that we have no hard evidence for short term (i.e, few hundred seconds) variability in the X-ray flux of BP Tauri. However, the event during the first night of our ROSAT observations, displayed in Fig. 4, does look very suggestive of a stellar flare; in this particular case two consecutive time bins show elevated count rates (16 recorded counts vs. 8.77 expected counts, and 12 counts recorded vs. 6.70 expected) and we therefore consider this event as real although it does not formally satisfy our criteria. A physical interpretation of this event will be presented in Sect. 5.

3.3. Relation between the optical and X-ray variability

We showed above that BP Tauri exhibited brightness variability in both X-ray and in the optical. Now we investigate if there exist any correlations between the variations in these two spectral regions. The maximum level in the X-ray count rates was observed during the fourth night of our observations. Unfortunately this night was clouded out so no simultaneous optical data was obtained. For the remaining four nights we binned the optical data with the same time resolution and observational window as the X-ray measurements (cf., Fig. 2). As can be seen in both Fig. 1 and Fig. 2, there is no correlation between the optical and X-ray level of the star at all. A formal correlation analysis resulted in a linear correlation coefficient for the data in Fig. 2 of 0.19 which occurs with a probability of 0.3 in uncorrelated data.

[FIGURE] Fig. 2. Correlation diagram over the optical (B-band) and the X-ray brightness.

In Fig. 3 we plot the data of the second night of observation, where BP Tau revealed two events in the optical, clearly detectable in all photometric bands. The amplitude of the second bump in Fig. 3 was [FORMULA] [FORMULA] in V and [FORMULA] [FORMULA] in B with a time duration of [FORMULA] 1.2 hours. Data from the U-channel is not shown in Fig. 3 since the channel started to malfunction during this night. It was, however, possible to recover a U-band amplitude of the event of [FORMULA] [FORMULA]. The energy of the excess flux of the optical flare was a few times 10 [FORMULA] ergs. From Fig. 3 it is also apparent that the simultaneous X-ray observations showed no significant increase in the count rates neither during the decay phase nor after the time of the optical event and therefore we conclude that the optical event had no counterpart in soft X-ray emission. Thus, within the limits of our observations, the brightness changes of BP Tauri in the soft X-ray and in the optical of time duration spanning from days to [FORMULA] 1 hour occurred without any correlation.

[FIGURE] Fig. 3. An optically active night (JD 2449243) shown in the B and V-bands (solid lines) together with the corresponding X-ray count rates (dots). The optical data is smoothed by 100 sec. The U-band is not shown since this channel started to malfunction during this night.
[FIGURE] Fig. 4. A possible X-ray flare during the first night (JD 2449241) at a level just about significant.
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998