The 'ensemble' method

The 'ensemble' method was introduced by [Horne & Stiening (1985)] who applied it to the nova-like variable RW Tri. In a quite straight forward way they define the rms-light curve as the root-mean-squared deviation between the individual light curves and their mean light curve as a function of phase. [Horne & Stiening (1985)] found the scatter to drop sharply in a small phase interval around eclipse centre and concluded ``that the flickering light source is more compact than the steady disc light'' and ``that the flickering distribution is approximately centred on the disc''.

The simple approach of [Horne & Stiening (1985)] furnished sensible results because they made sure that all of their light curves were observed within a short interval of just four days and that the object of their study, RW Tri, is a UX UMa type nova-like variable and thus a member of the most stable class of CVs. Both points together minimize the importance of long term variations. Thus, the mean light curve is stable at all orbital phases.

This was different in the case of the study of HT Cas by [Welsh & Wood (1995)]. They used the 'ensemble' method to calculate the scatter curve based on light curves which were collected over a time span of several years at different telescopes. Moreover, as a SU UMa type dwarf nova HT Cas is by far not as stable as RW Tri. These systems do not only exhibit flickering but also aften much stronger long term variations and variations on orbital time scales than nova-like variables. Furthermore, there is no guarantee that those variations which are not attributable to flickering influence the entire light curve in the same way. On the contrary, there is ample evidence for variations in such stars which affect only a part of the orbit, e.g. variable hump amplitudes, appearance and disappearance of intermediate humps, or variations of the eclipse depth. Such effects have to be accounted for before the scatter curve of the flickering can be calculated from the differences between the individual and the mean light curves.

Moreover, [Welsh & Wood (1995)] introduced a bias into their analysis of the HT Cas light curves. They find that the mid-eclipse flux is variable and attribute it to long term secular trends. They remove this effect by applying ``a small additive constant to each curve so that the mean flux at mid-eclipse remained constant''. This procedure is problematic because by definition it reduces the scatter of the individual light curves around the mean during eclipse minimum (and even minimizes it if the flickering during eclipse is not stronger than at other phases).

In order to assess if this bias is serious or insignificant the method of [Welsh & Wood (1995)] was applied to the data of HT Cas used in this paper (see Sect. 3), assuming that the long-term behaviour of the system is not altogether different in the data set of [Welsh & Wood (1995)] and the present one. Since the light curves available here are only expressed in count rates (as opposed to fluxes) they were normalized to the mean out-of-eclipse brightness in order to put them onto a comparable scale. Thus, long-term brightness variations are assumed not to exist, leading to a lower limit of the scatter. Due to strongly variable eclipse depths (with respect to the out-of-eclipse light level) of HT Cas the resulting scatter curve has a maximum during eclipse phases if no correction is applied to the light curves. It turns into a very deep minimum if the data are treated in the same way as [Welsh & Wood (1995)] did.

Thus, the bias is potentially quite serious. The minimum in the scatter curve of [Welsh & Wood (1995)] at the eclipse phase is therefore in the first place caused by the method. Even if the flickering really is eclipsed at these phases - as shown in Sect. 4.1 - this is not the primary cause for the minimum found by [Welsh & Wood (1995)] in their scatter curve. This conclusion can only be avoided if the actual flux at eclipse minimum does not change much, leading to only small additive corrections. But then, the strongly varying eclipse depths seen at least in the present data suggests that [Welsh & Wood (1995)] would have measured the long-term variations instead of the flickering. Therefore, without knowledge of further details of their work the results of [Welsh & Wood (1995)] must be regarded with suspicion.

This shows how important - and dangerous - long term variations in the light curves are when applying the 'ensemble' method. [Bennie et al. (1996)] devised a way to overcome this problem and applied it to RW Tri. Unfortunately they published only a very concise and qualitative description of their method. Here, I will try to reconstruct it.

Due to the briefness of the presentation of [Bennie et al. (1996)] details of the reconstruction may differ from the original method. But this should not affect the basic idea. However, I will introduce one major difference: [Bennie et al. (1996)] (just as Horne & Stiening 1985 and Welsh & Wood 1995) expressed their light curves in fluxes, implying that they had calibrated them. The archival light curves for the present study (see Sect. 3) - all of them obtained with photon counting devices - are only available in count rates. They were observed at very different epochs with a variety of instruments. Thus, an otherwise desirable transformation into fluxes is unfortunately not possible. At least for objects (and instruments) furnishing a weak signal, Poisson noise is not negligible in the present context. Whereas it is not obvious how to handle this effect easily once the light curves have been transformed into fluxes, the above mentioned disadvantage is - however only slightly - made up for by the easy application of a Poisson noise correction if the light curves are given in count rates. For these reasons the formulation presented here is expressed in terms of count rates and includes a correction for Poisson noise.

In the spirit of the 'ensemble' method the scatter as a function of orbital phase phi can be expressed as:

  

Let us regard the individual terms of this equation. The index i stands for an individual light curve. C²_i,ref is a ``reference count rate''. It can be taken as the mean count rate in parts of the light curve i which are not disturbed by features such as an eclipse or orbital hump. In order to account for long term variations [Bennie et al. (1996)] assumed that all variations not due to flickering scale linearly with the reference count rate, i.e. that a relation exists such that

  

Here is the count rate which one would observe in the absence of flickering. Note that this is the basic assumption of the method! It is important to realize that the slope b of this relation is permitted to be a function of phase. Thus, each part of the light curve may scale in a different (but linear) way with the reference count rate.

The difference between the observed count rate and the count rate predicted in the absence of flickering is then interpreted as being due to flickering. The square of this difference [i.e. of the expression enclosed in square brackets in Eq. (1)] is the contribution of light curve i to the variance around the mean light curve at phase phi. It is biased by Poisson noise.

The variance due to the latter effect is equal to the mean count rate at phase phi. If the variations in the light curve in a small interval around phi are dominated by Poisson noise can be taken as the mean count rate within this interval. If it is dominated by flickering (i.e. by real variations of the system brightness), is the best estimate for In praxis, this does not make a significant difference, and I assume . However, this holds only true if the light curve has not been binned in phase. Otherwise, is decreased by where n is the number of original data points (if the light curve is expressed in counts per integration time) or time units (if it is expressed in counts per time unit) in a phase bin. Thus, the second term in the curved brackets in Eq. (1) constitutes the correction for Poisson noise.

The total variance at phase phi is then the sum of the individual terms divided by , where is the number of light curves available at phase phi. However, it would be misleading to take the straight mean because this would give exaggerated weight to those light curves which were observed at high count rates (e.g. with a larger telescope) because then the absolute numbers of and would be large. An equal weight can be assigned to each light curve by dividing its contribution to the total variance by the square of the reference count rate.

The square root of the total variance is finally the scatter at the phase phi which is now independent of the actual order of magnitude of the count rates. Somewhat loosely spoken it can be regarded as the mean amplitude of the flickering in terms of the reference count rate. Its absolute value therefore depends on the strength of the flickering relative to the other light sources in the system.