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Application to artificial light curves
In order to test the validity of the concepts described in Sects.
2.1 and 2.2 they will first
be applied here to artificial light curves before real data are subjected
to them. 100 light curves with artificial flickering based on a shot noise
model were generated. Their mean count rates varied randomly between 1000 and
2000 (i.e. in a realistic range but not too large for Poisson noise to
become negligible). Each light curve contains an eclipse in the phase interval
with a constant residual count rate of 10% of the
mean out-of-eclipse count rate. During eclipse ingress
the mean count rate drops linearly and the weight of the flickering
drops from 1 at
phi=-0.15
to 0 at
phi=-0.05
during this interval.
Eclipse egress is simulated in an analogous manner at
.
An orbital hump was introduced as the positive half of a sine curve in the
interval
.
For each of the hundred light curves
its amplitude is a random number between 0 and the mean count rate in the
interval
.
Finally, Poisson noise was simulated by
adding to each data point a random number drawn
from a Gaussian distribution (good enough an approximation for a Poissonian
distribution at the assumed count rates) with a mean of 0 and a standard
deviation equal to the value of the data point itself. An example of such an
artificial light curve is shown in Fig. 1a.
Before applying the 'ensemble' method to these data they were binned in intervals of
= 0.005
.
The reference flux was taken to be the
mean count rate in the interval 0.15 < phi < 0.5
(i.e. disregarding
eclipse and hot spot). The resulting scatter curve is shown in
Fig. 1b. As expected, during eclipse
the scatter is reduced to 0. During egress it rises constantly; the giggles
are due to the progressively visible flickering light source. After egress,
the scatter remains on a more or less constant level. However, what happened
at phases before the eclipse? There is a hump reflecting the orbital hump
although the simulated flickering strength at these phases is the same as
after the eclipse. The answer is simple: During the construction of the light
curves the hump amplitude (relative to the mean count rate at phase
0.15 < phi < 0.5)
was varied in a manner completely independent of
the mean count rate. Thus, the basic assumption of the 'ensemble' method
is violated, namely that all variations not due to flickering scale linearly
with the reference flux. If, on the other hand, the variations of the hump
amplitude are considered as flickering, the hump in the scatter curve is
perfectly as expected: It then reflects the enhanced scatter due to this
additional flickering component.
In order to study the effect of an unsuitable choice of the reference flux
the calculations were repeated using as reference flux the mean count rate
in the interval -0.35 < phi < -0.2,
i.e. right on top or the (variable)
hump. The results, shown in Fig. 1c,
could have been foreseen: The scatter assumes a minimum at the phases where
was defined while it is augmented at other (out-of-eclipse)
phases. The eclipse bottom is somewhat elevated, but not enough to
significantly alter the properties of the scatter eclipse.
Finally, Fig. 1d shows the scatter curve of the same sample of artificial light curves, calculated with the 'single' method. Obviously, the scatter conforms perfectly well with the expectations with the exception of a slightly too small eclipse width. This is an artifact explained by the discontinuous transition between the ingress/egress slopes and the constant eclipse bottom which cannot be followed by the smoothed light curve. The orbital hump is not seen in the scatter curve because it is connected with variations well above the time scale for which the 'single' method is sensible. Whether this is desirable or not depends on the point of view: Should orbital variations of the hump amplitude be regarded as flickering or not?