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Astron. Astrophys. 360, 627-632 (2000)

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2. The sources of the data and the analysis

Monitoring of many cataclysmic variables is almost entirely the domain of the associations of amateur observers due to the character of the long-term activity of these objects (unpredictable episodes of the low states in nova-like variables and, on the other hand, often relatively short outbursts separated by long intervals of quiescence in dwarf novae). The observations are mostly visual but they are quite numerous and come from a large number of observers; the objectivity of the features in the light curve can therefore be assessed. Visual data, if treated carefully, can be very useful for the analysis of long-term activity (Percy et al. 1985, Richman et al. 1994). Accuracy even better than 0.1 mag can be achieved by averaging the data. This is quite sufficient for analyses of these large-amplitude variable stars.

DO Dra has been extensively monitored by several associations of observers especially in the recent years. The presented analysis is based on the data from the AFOEV database, operated at CDS (France), VSNET database (Japan) and VSOLJ database (Japan). The combined data set was corrected for the observations repeated in several databases. The number of positive observations from the respective databases then amounts 346 (AFOEV), 419 (VSNET) and 211 (VSOLJ), respectively. The density of the coverage (including 4226 negative observations which yield the upper limits), important for detection of the rare outbursts in DO Dra, is unprecedented for this star.

The light curve was plotted and submitted to a visual inspection. The observations already marked as unreliable in the original files were rejected in most cases. Further, several observations, largely deviating from the neighbouring points in the light curve, were rejected. The curve was then smoothed by binning the positive observations (976) into one-day means (728). The resulting light curve, covering the years 1984-1999 (Fig. 1), was used for a large part of this study. Four outbursts, defined by multiple observations from several nights, are clearly visible . The two last outbursts also were partly covered by the CCD observations (Ouda station - Kyoto University, VSNET) which proved to be in good agreement with the visual data. It can be seen that although DO Dra is faint for most of the time (about 14.5 -16 [FORMULA]) and therefore beyond the reach of some amateur telescopes, the coverage by the negative observations (4226) is often quite dense, especially after JD[FORMULA] 2 450 000. The outburst reported by Hurst et al. (1985), marked by a vertical bar in Fig. 1, falls into a weakly covered interval of the light curve.

[FIGURE] Fig. 1. The light curve of DO Dra over the years 1984-1999. The points denote the one-day means of the positive observations and are connected by line for the densely covered intervals. The v symbols mark the upper limits of brightness and can be used to constrain the possibly missed outbursts. The outburst reported by Hurst et al. (1985) is marked by a vertical bar. See text for details.

2.1. Light curves of the outbursts

The light curves of four covered outbursts are displayed in detail in Fig. 2a-d. Due to the short duration of the outbursts and their rapid decline we preferred to plot the individual observations rather than the one-day means here to improve the coverage. It can be seen that the rise to the maximum is so fast that in three cases it is not covered by observations and in one case only the upper half is covered. However, the negative observations still enable us to put the upper limits on the duration of the respective events.

[FIGURE] Fig. 2a-d. The appearance of four outbursts in DO Dra. The individual (unbinned) observations are plotted to improve the coverage because the rises and decays are quite rapid. The scale of both axes in each plot is identical and allows one to compare the light curves of the respective events. See Sect. 2.1 for details.

In order to assess the common properties of the respective outbursts, it is instructive to superpose them into a common plot. We applied the method of alignment according to the decay branches, similar to that used for CH UMa in imon (2000, hereafter Paper I). The outburst having the maximum in JD = 2 451 443 was chosen as the template. The remaining outbursts were shifted along the time axis to match this template. The level of brightness [FORMULA] was chosen as the reference level in the vicinity of which the match was attempted. The result is shown in Fig. 3. The decay branches of the respective outbursts were then merged into a common file and smoothed by the program HEC13 (author Dr. P. Harmanec), based on the method of Vondrák (1969 and 1977). The input parameters of the fit [FORMULA] and the length of the bin [FORMULA][FORMULA] day satisfy the course of the decay. In our case the input [FORMULA] parameter was chosen so that the fit reproduces just the main course of the decay. The standard deviation of the residuals of this fit is [FORMULA]. The smoothed decay light curve is plotted as the thick solid line in Fig. 3.

[FIGURE] Fig. 3. The outbursts in DO Dra from Fig. 2, superposed into a common plot. The respective outbursts were shifted along the time axis to match the decay branch of the template - the time of crossing 12.0 [FORMULA] and the shifts with respect to the template are listed in the figure. The thick line represents the smoothed decay curve. See Sect. 2.1 for details.

Fig. 3 clearly shows that the decay branches of the respective outbursts are remarkably similar to each other. The smoothed course reveals that the decay is curved downwards, it means that it is faster than exponential. If a linear approximation is made, we obtain the decay rate [FORMULA] days [FORMULA].

2.2. The outburst recurrence time and its variations

The searches of the archival plates covering several decades (Wenzel 1983a,b, Hazen 1986) already lead to the conclusion that the recurrence time [FORMULA] of the outbursts in DO Dra must be quite long, of the order of 1000 days. The densely covered light curve along with the upper limits in Fig. 1 confirms this conclusion. In total, we managed to accomplish timings of 11 outbursts, summarized in Table 1. They spread over 63 years. Of course, due to the very short duration of outbursts in DO Dra it is quite likely that some outbursts were missed. Determination of a reasonable value of [FORMULA] is then difficult. Using the residuals of some reference period can help to overcome this problem, assuming that the process is cyclic. In Sect. 3 it will be argued that the outbursts in DO Dra can be explained by the same mechanisms as those in dwarf novae. It is true that outbursts in dwarf novae are not strictly periodic but [FORMULA] often does not vary considerably between the neighbouring outbursts. The [FORMULA] diagram can then be constructed for some reference mean [FORMULA].


[TABLE]

Table 1. Outbursts in DO Dra. [FORMULA] refers to the moment of the maximum brightness in JD-2 400 000. The epoch number and [FORMULA] (days) are calculated according to Eq. (1). The last column gives the reference.


The method of determination of the recurrence time [FORMULA] of outbursts in dwarf novae using the [FORMULA] residuals from some reference period (e.g. Vogt 1980) removes the drawbacks of the widely used approach based on the measurements of separation of the neighbouring outbursts. The method of the [FORMULA] residuals is not sensitive to the exact length of the reference period and the [FORMULA] diagram can be constructed even if there are gaps in the data. A more detailed discussion of this method was given in Paper I.

A set of the [FORMULA] curves for slightly different reference periods was generated to obtain the mean slope of the [FORMULA] values as small as possible. The final [FORMULA] diagram is displayed in Fig. 4. The reference period of 868 days (Eq. (1)) keeps a large part of the [FORMULA] curve at a very small slope and shows the course of the [FORMULA] values of the outbursts with the best clarity. In most cases the error bars in Fig. 4 would be smaller than the symbols used. The overall course of the [FORMULA] values within the epochs [FORMULA] to 2 can be approximated by a straight line (i.e. the mean [FORMULA] is constant) with a standard deviation of 101 days. Only the last two outbursts have largely deviating [FORMULA] values and may suggest an increase of [FORMULA]. In conclusion, it can be seen from Fig. 4 that the method of residuals enabled us to determine the mean [FORMULA] in spite of several missing outbursts. It can also be inferred from Fig. 4 that an outburst might pass unobserved around JD = 2 449 250; an examination of the light curve in Fig. 1 revealed that the segment of the light curve around this date is not very densely covered and also contains several gaps in the coverage by the upper limits.

[EQUATION]

[FIGURE] Fig. 4. The [FORMULA] diagram for the outbursts in DO Dra. The [FORMULA] values were calculated according to Eq. (1). The vertical dotted lines at each epoch enable a better assessment of the missing outbursts. See Sect. 2.2 for details.

2.3. Variations of the quiescent level

The light curve in Fig. 1 displays the well visible fluctuations of the quiescent level. The inspection of the densely covered part within JD = 2 450 100 - 2 451 500 revealed that these fluctuations cannot be ascribed to the observational noise and that they display trends on the time scale of tens of days and longer. In order to diminish the noise of the quiescent light curve in this segment, only the data of four observers who covered long intervals of the curve were used. The light curves of the respective observers were interactively shifted to match each other. This procedure removed slight systematic shifts of some observers, not exceeding [FORMULA]. In the last step, the observations were binned into the means, mostly of 7 observations. We preferred to preserve the number of observations instead of the length of the bin to keep the noise as low as possible. This approach emphasized the slow component of variations whose full amplitude reaches almost [FORMULA] (Fig. 5).

[FIGURE] Fig. 5. Variations of the quiescent level of brightness in DO Dra. The points are bins mostly of seven observations, the bars represent their rms error. The empty circles denote the single observations in a weakly covered interval. Only the data for the quiescent level are plotted; the vertical lines denote the position of two outbursts, displayed in Fig. 2 cd. See Sect. 2.3 for details.

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© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000
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