4 Flickering properties

To study flickering we used both the power spectra and autocorrelation function (ACF). As it was mentioned above, PS of MV Lyr show a red noise. Although the presence of red noise in PS does not specify the underlying source variability, investigation of PS properties is useful from the point of view of their comparison with the ones predicted from the theoretical and numerical models. An important characteristic in this aspect is the power law index $\gamma$, as well as the common shape of the PS.

In order to determine $\gamma$ we plotted PS in log-log scale and fitted their linear parts by least-squares linear fit. The interval in which the fits were performed varied from $5-10\ [c/h]$ to $70-100\ [c/h]$ (with one exception on Jul. 18, 1993 where the upper end of the linear part was $50 \ [c/h]$). At frequencies higher than $70-100\ [c/h]$ the spectra become flat because the white noise dominates over all other sources of variability. The values of $\gamma$ are listed in Table 3 together with the errors determined from fitting program. The value of $\gamma$ determined from the mean power spectrum and the mean $\gamma$ from the individual values are also given in the table.

The data contain gaps with common length $10{-}30\%$ of the run length. They cause power leakage effects and may reduce the power spectrum slope. To investigate the influence of these gaps on the accuracy of $\gamma$ determination, we used the three series containing 10% gaps only (Jul. 06, 1992, Jul. 17 and 19, 1993). The gaps were filled using the method developed by Fahlman & Ulrych (1982). After that all original gaps were enlarged randomly, but keeping their common length 15, 20, 25 and 30% respectively. For every run and common length of the gaps, 100 different gaps distributions were introduced and the mean values of $\gamma$ were compared with those determined from the filled and original runs. The mean values of $\gamma$ showed slight tendency to decrease with the growth of the gaps length, but all values remained in the interval $\pm \sigma_{\gamma}\simeq 0.1$ determined from the 100 simulated gaps distributions. Because the PS are very noisy, the determination of $\gamma$ also depends strongly on the interval in which the fit is performed. Small changes in this interval caused changes in the values of $\gamma$reaching 0.2-0.3. So the errors listed in Table 3 should be regarded as a lower limit.

The time scale of the flickering is not an easily definable quantity. An objective way to define it is by autocorrelation function (ACF). The typical time scale of the flickering $\tau$ may be defined as the time shift at which the ACF $r(\tau)$ first accepts the value r₀=1/e. Thus determined correlation times are strongly biased from the presence of periodic brightness variations or some trends in the data. In order to flee the influence of these factors the following procedure was applied. The runs were divided into non-overlapping sections of $\sim 15$ min each and the average points in these bins were interpolated by cubic spline. This roughly corresponds to output from a filter cutting modulations with periods approximately longer than 40 min. Finally the residuals between the original data and the smooth curve were analysed. Because of gaps in the observations the ACFs were calculated according to Edelson & Krolik (1988) and are shown in Fig. 4. The bin sizes were chosen to be equal to integration time. Determined correlation times listed in Table 3 are of the order of 1.5 min. The errors are determined from shift times at which the functions $r(\tau) \pm \Delta r(\tau)$ first reach the value r₀=1/e, where $\Delta r(\tau)$ are the errors of the autocorrelation coefficients. In the table is also given the mean from the individual values. As Robinson & Nather (1971) and Panek (1980) note, these correlation times can be additionally biased by the presence of weakly correlated noise and the way of trend removal.

  

Table 3: Flickering properties

When periodic components are detrended the real shape of the signal cannot be found exactly and this leads to so called residual noise. In the procedure described above this noise is caused by the influence of the flickering on the determination of the mean values. However in case of averaging over long enough bins the amplitudes of the residuals will be small compared with those of the flickering and the results will be weakly affected by the residual noise.

  

Figure 4: Autocorrelation functions of MV Lyr light curves. Bars show the errors of the autocorrelation coefficients

The determined values of $\gamma$ vary in the interval from 1.60 to 2.15. There are several mechanisms which may produce PS with power law shape and $\gamma$ in this interval (see Tremko et al. 1996). The low frequency flat part of PS allows us to suggest a model of the optical variability in terms of a "shot noise" process. According to this model the light curves are the result of overlapping, randomly distributed in time shots with some shape. In the classic "shot noise" model the shots are assumed as decaying exponents. Then PS shape is:

$$P(f)\propto \frac{1}{1+(2\pi\tau f)^2}$$ (2)

where $\tau$ is e-folding constant of the shots. Autocorrelation function of a shot noise curve has a shape $r(t) \propto \exp(-t/\tau)$,where t is the shift time. In practice, because of the finite length of the observations and the overlapping of the shots, the ACF does not follow this shape but crosses the zero level at some lag and after that oscillates around it (see for example Andronov 1994). To verify the influence of this effect on the determination of $\tau$,artificial shot noise series with time resolution 10 sec were simulated as a sum of randomly distributed in time one- and two-sided exponential shots. $\tau$ and the overlapping parameter $N=\tau \lambda$ ($\lambda$ is the mean shot rate) were varied between 1-5 min and 1-5 respectively. The correlation times determined by ACFs were compared with the accepted ones. In all cases the relative scatter of the values $\sigma_{\tau}/\tau$ was smaller than 0.25 and the mean value was close to the accepted one $\vert\tau_{\rm mean}-\tau\vert/ \tau<0.1$. Although the common shape of the ACF is strongly biased by the finite length of the data and the shots overlapping, for small lags the calculated ACFs are close to the theoretical ones and the mean value of $\tau$ may be an estimation of the correlation time. So we tried to fit the individual and the mean PS of non-detrended runs to function

$$P(f)=\frac{\alpha}{1+(2\pi\tau f)^\gamma}$$ (3)

as $\tau$ was fixed to be equal to the correlation times determined by ACFs and the mean value, respectively. Because of the strong influence of the power due to the periodic variations, the mean PS was fitted only for frequencies $\log(f) \geq 0.4$. These fits are shown in Fig. 5. It is seen that the shape of the individual spectra and the tendency of the mean PS to become flat between frequencies $\log(f) = 0.4{-}1.0$are fitted by the function (3). The power excess below $\log(f) = 0.4$ is due to the periodic variation. The values of $\gamma$ found by the fits are given in Fig. 5 and are a little greater than those determined by fitting of the linear parts. Additionally, we used these fits to estimate the significance of the peaks in the PS as was discussed in Sect. 3. In this case only the peaks corresponding to "50 min" QPOs remain statistically significant.

  

Figure 5: Individual and mean PS of non-detrended runs fitted to function $\frac{\alpha}{1+(2\pi\tau f)^\gamma}$. $\tau$ was fixed as was determined by ACFs and by mean from the individual values of $\tau$, respectively

The total amplitude of the flickering (i.e. the difference between the brightest and faintest points of the light curve) and standard deviation around the mean are listed in Table 3 also. There seems to be no difference between activity of the flickering in B and U bands if these quantities are taken as activity indication. Having an estimation of the total amplitude of the flickering we can calculate the contribution of the flickering light source to the total light of the star following the conception of Bruch (1992). The star brightness can be regarded as a sum of two sources - flickering light source and all other sources which are supposed to be constant on the flickering time scale. The magnitude of the constant light source can be defined as $m_{\rm c}=m+\Delta m/2$, where m is the mean magnitude of MV Lyr and $\Delta m$ is the total amplitude of the flickering. Then, if the amplitude of the flickering is assumed to be independent of the passband, the ratio of the flux of the flickering light source $F_{\rm f}$ to that of the constant ones $F_{\rm c}$ over the whole optical range is given by

$$\frac{F_{\rm f}}{F_{\rm c}}=10^{-0.4\Delta m'}-1$$ (4)

where $\Delta m'=m_{0}-m_{\rm c}$ and m₀ is the magnitude of some point of the light curve. As m_c is an upper limit for the constant light source, the calculated by Eq. (4) ratios have to be regarded as a lower limit. In Table 3 are given the ratios of the fluxes calculated for m₀ equal to the mean and maximal star brightness. It is seen that flickering light source emits at least 0.2-0.4 of the total radiation of the star.

Recently the most often discused mechanisms causing the flickering are turbulence in the accretions disc and/or unstable mass accretion on the white dwarf. These are stochastic processes and the resulting light curves would be described as a "shot noise". Although the flickering in cataclysmic variables is often discussed in terms of standard "shot noise" model (Williams & Hiltner 1984; Elsworth & James 1982; Panek 1980), it should be regarded only as a rough approximation. More complicated models should take into account the distribution of shot's durations which can change significantly the power spectra shape. Unfortunately, because of the overlapping of the shots, this distribution cannot be found from the light curves.