The 'single' method

The 'single' method is described in detail by [Bruch (1996)] who applied it successfully to the dwarf nova Z Cha. The scatter as a function of orbital phase phi is defined in this case as:

  

In contrast to the 'ensemble' method no mean light curve is calculated here. Instead, the scatter of the data points around a smoothed version (in the sense that variations considered to be due to flickering are removed but others such as eclipses and variations on longer time scales are preserved) of each individual light curve is taken to define the scatter curve.

Let be the original count rate of light curve i at phase phi and the count rate of the smoothed light curve. Let there be n data points in a small interval around phase phi. The first term under the square root in Eq. (2) is then the variance of the data points of light curve i around the smoothed light curve in the interval .

It is biased by Poisson noise, the contribution of which to the total variance is just the mean of the count rates within (in contrast to the 'ensemble' method I assume here that the light curves are not binned; a corresponding correction would be trivial). Thus, the second term under the square root in Eq. (2) constitutes a correction for Poisson noise, and the square root itself as a function of phi is the scatter curve of light curve i.

Before taking the mean of the individual scatter curves it is necessary to normalize them in order not to lend higher weight to light curves which happen to have been observed at higher count rates for instrumental reasons. As normalization factor the average of the scatter of light curve i over all phases is taken. Finally, the normalized [thus the index n in Eq. (2)] mean scatter curve is the sum over all individual normalized curves, divided by the number of light curves contributing at phase phi.

The critical point of the method is the construction of the smoothed light curve. Is is important that the smoothing is homogeneous, i.e. to ensure that no part of the light curve (in particular the eclipse) is smoothed better or worse than the other parts because this would bias the scatter as a function of phase. A homogeneous smoothing can be achieved by binning the original light curve in suitable (fixed) phase bins and then performing a spline interpolation (better suited than a spline fit because it has fewer degrees of freedom and is thus less subject to arbitrariness) between the points of the binned light curve.

The method has two free parameters: (1) The interval used to calculate the variance between the original and the smoothed light curve defines the resolution of the scatter curve. Of course, a high phase resolution is always desirable but in praxis it is limited by the phase resolution of the original light curves and the acceptable statistical noise in the final scatter curve. (2) The phase interval over which the original light curves are binned before the smoothed light curve is calculated. Obviously the smoothed curve will follow variations on scales longer than the bin width which consequently will not contribute to the scatter. The choice of the appropriate bin width requires a balance between the maximum time scale of the flickering which is to remain detectable and the necessity for the spline to follow well the eclipse profile. The latter requirement sets an upper limit for the permissible bin width. The lack of sensitivity for flickering on time scales longer than the bin width is the main disadvantage of the 'single' method¹. Its main advantage, on the other hand, is its robustness against long term variations of the regarded system. Since the scatter of each individual light curve is calculated (before taking the mean) all long term variations are eliminated when the difference curve is constructed. This is in contrast to the 'ensemble' method where variations on long time scales (or even non-repetitive variations on orbital time scales) represent a major difficulty.