3 Structure function analysis of variable UBVRI fluxes

3.1 Properties of structure function and analysis realization

Press (1978) suggested that the variability in quasars is caused by a process known as "noise''. The power spectrum of the simplest of them has a form $g(\nu) \sim\nu^{-\gamma}$. Compact variable AGNs exhibit three types of noise: white-noise ( $\gamma = 0$), flicker-noise ( $\gamma = 1$), and shot-noise ( $\gamma = 2$) (see Terebizh 1993). Such processes are easily revealed by the technique of SF analysis. In application to AGNs this has been discussed by Hufnagel & Bregman (1992), Hughes et al. (1992), Lainela & Valtaoja (1993) and others. The first-order SF is defined as:

$$SF = \\ < [F(t) - F(t+{\rm d}t)]^{2} > ,$$

F(t) being the flux at time t, and dt being the time delay (lag) between observations of fluxes F(t) and $F(t+{\rm d}t)$, the angular brackets denote an ensemble average. The SF of an "ideal'' stationary random process on a logarithmic scale consists of three components: a straight line with a slope $b = {\rm d} \log(SF)/ {\rm d} \log {\rm d}t$, which is located between two plateaus. For short time scales, the plateau is just twice the variance of the measurement noise, because it has a zero correlation time scale. The longest correlation time scale $-T_{\max}$ gives the time lag when the SF reaches the upper plateau with an amplitude equal to twice the variance of the fluctuation. The logarithmic slope "b'' characterizes the nature of the process: b = 0 corresponds to flicker-noise, b = 1 to shot-noise, $T_{\max}$ characterizes the duration of the flares.

The SF method is related to the autocorrelation function and Fourier power spectrum technique which are more commonly used in the literature. Only methods of SF and a Fourier power spectrum permit one to determine the type of process causing the flux variability. However, the results of analysis using the Fourier power spectrum depend on windowing the sampling whereas the results of analysis by SF do not. The SF realization for the variable flux of the NGC 4151 nucleus was carried out using a special program package by S. G. Sergeev.

Figure 2 shows the SFs of variable fluxes of NGC 4151 in the UBVRI spectral bands, calculated for the whole sample of observations performed from 1989-1996. One can see that SFs do not have a simple "ideal'' single-process form. The slopes of SFs of intranight variations are essentially higher than those of extranight variations.

 

 

Table 2: Parameters of the SFs in the UBURI bands for all observational samples on different time-scales.

spectral	intranight		yearly
band	$b_1\pm\sigma$	$k_1\pm\sigma$	$b_2\pm\sigma$	$k_2\pm\sigma$
1	2	3	4	5
U	$0.959 \pm 0.081$	$0.960 \pm 0.021$	$0.599 \pm 0.057$	$0.844 \pm 0.043$
B	$0.950 \pm 0.071$	$0.968 \pm 0.017$	$0.583 \pm 0.051$	$0.863 \pm 0.037$
V	$0.930 \pm 0.065$	$0.972 \pm 0.015$	$0.526 \pm 0.053$	$0.827 \pm 0.046$
R	$0.812 \pm 0.080$	$0.946 \pm 0.027$	$0.438 \pm 0.054$	$0.725 \pm 0.069$
I	$0.881 \pm 0.052$	$0.980 \pm 0.011$	$0.502 \pm 0.057$	$0.795 \pm 0.054$

Figure 2: The Structure Functions for obtained UBV samples. Solid lines are the regression lines (see text). Bars are error bars for each value of the Structure Function, if errors are more than the dimension of signs.

Figure 2:  continued for the RI bands.

The behavior of the SF on time intervals 0.1-1 days is defined only roughly because of the lack of observations during daylight. We did not consider these intervals in our analysis.

Parameters of the SFs are presented in the columns of Table 2: (1) spectral band; (2), (4) logarithmic slopes "b₁'' and "b₂'' of SF for intranight and for yearly variations, respectively; (3), (5) coefficients of correlation "k₁" and "k₂" between the log SF and log dt; regression lines are shown in Fig. 2. The logarithmic slopes "b'' of SF on time-scales of hours are equal to 0.81-0.96 and on time-scales of years they are 0.44-0.60. The confidence levels of the (log SF-log dt) correlations are equal to 1. $T_{\max}$ for intranight process is equal to 1 hour and for yearly variations - 4.2 years.

The values of the slopes indicate that the process causing the variations on a time-scale of hours was near to that of shot-noise, but on a time-scale of years it was mixed shot-noise and flicker-noise. This fact permits one to suppose that the sources causing intranight and extranight variations are different. We examined SFs of intranight variations and those on a time-scale of months and years separately.

The high confidence level for the SF parameters was obtained only for the period of 4.2 years (log dt = 3.2). Confidence levels of the SF parameters for the whole observational period 7.2 years (log dt = 3.4) were rather low. We suppose that characteristics of weekly and monthly variability over the periods with the lack of observations would be the same as those over our observational periods.

3.2 Structure function of intranight variability

Structure function parameters of the intranight variability of the NGC 4151 nucleus are presented in Table 3 separately for I-IV periods of observations. The columns contain: 1) period of observations; 2) spectral band; 3) slope "b''; 4) $T_{\max}$; 5) coefficient of correlation (k) between the values of log dt and log SF, obtained for regression lines as shown in Fig. 3. The slopes of SFs are in the range $-0.03\leq b \leq 1.16$ and $T_{\max}\sim(1.5{-}5.7$) hours. The confidence levels of correlations are in the range 0.917-1.000.

 

 

Table 3: The Structure Function parameters of intranight variability of the NGC 4151 nucleus for I-IV periods separately.

Period	Spectral	b	$T_{\max}$,	k
of observ.	region		hours
1	2	3	4	5
I	U	$0.463 \pm 0.145$	2.0	0.590
I	B	$0.206 \pm 0.121$	2.0	0.364
I	V	$0.195 \pm 0.135$	2.0	0.314
I	R	$0.214 \pm 0.083$	2.0	0.508
I	I	$-0.03 \pm 0.40$:	1.14	0:
II	U	$0.540 \pm 0.064$	5.7	0.834
II	B	$0.505 \pm 0.063$	5.7	0.823
II	V	$0.525 \pm 0.063$	5.7	0.832
II	R	$0.320 \pm 0.099$	5.7	0.501
II	I	$0.401 \pm 0.069$	5.7	0.733
III	U	$0.652 \pm 0.039$	2.5	0.957
III	B	$0.603 \pm 0.027$	2.5	0.975
III	V	$0.742 \pm 0.039$	2.5	0.967
III	R	$0.826 \pm 0.052$	2.5	0.953
III	I	$0.656 \pm 0.077$	2.5	0.864
IV	U	$1.112 \pm 0.077$	1.5	0.963
IV	B	$1.112 \pm 0.079$	1.5	0.962
IV	V	$1.092 \pm 0.081$	1.5	0.958
IV	R	$1.120 \pm 0.080$	1.5	0.962
IV	I	$1.162 \pm 0.079$	1.5	0.965

Figure 3: The Structure Functions for the U and the I bands for different observational periods. Solid lines are regression lines for intranight and extranight flux variations, broken lines are regressions for 10-150 day flares. Error bars for Structure Function values are shown if errors were more than the sign dimension. a) for period I.

Figure 3:   b) continued for period II.

Figure 3:   c) continued for the period III.

Figure 3:   d) continued for period IV.

Figure 4: Variation of the Structure Function slope "b'' with time. Top - for the variability on time-scale of hours, bottom - for yearly variations.

The slopes of the SFs for each period change with wavelength. Table 3 shows that the highest change was in the first period: the slopes decreased from 0.46 for the U band to 0 for the I band. In the second period they are equal to 0.3-0.5. In the third period all slopes increased to 0.6-0.8, and in the fourth period they increased to 1.1. These data show that during the first period the processes causing the flux variation depend on wavelength: mixed shot-noise and flicker-noise for the U band and flicker-noise for the BVRI bands. The dependence of the SF slopes on wavebands was not so clearly expressed during the three following periods. We mentioned the evolution of the process with time in all spectral bands: being near to flicker-noise in the I period, mixed shot-noise and flicker-noise in the II and III periods, and approaching strong shot-noise in the last period.

The obtained results can be interpreted as an indication of increasing intranight nucleus flare activity with increasing nucleus brightness. A clear dependence of the SF slope "b'' on waveband is exhibited in the first period but not obvious in the II-IV periods.

3.3 Structure Function of extranight flux variability

Figure 3 shows that SFs for time lags by more than one day curved around dt = 100 days
$(\log ~{\rm d}t = 2)$, being flatter for yearly variation compared to monthly variation. In the first approximation we neglected this SF curvature and calculated SF parameters for dayly-yearly variations together. Table 4 contains the parameters of the SFs for time lags more than one day for four periods of observation separately: b - logarithmic slope of SFs, k - coefficient of correlation between the values log SF and log dt, obtained from regression lines (Fig. 3). Table 4 and Fig. 3 show that the SFs exhibit slopes $0.25\leq b \leq 1.08$. The confidence level of the correlation equals 0.977-1.000.

Figure 4 shows the evolution of slopes with time. In the first period the slopes were in the range 0.7-1.1. In the second period they decline to 0.3-0.7. In the third period all the slopes increased to 0.6-0.8 and in the last period they decreased to 0.4-0.5. These data show that the processes causing the general brightness increase from 1989 to 1996 evolved with time. In the first period there was shot-noise in all spectral bands. Then it weakened and approached flicker-noise. In the third period the process was of shot-noise type (coefficient "b'' - 0.7-0.8), and at the end of the observational period the process again became mixed shot-noise and flicker-noise ( $b \sim 0.4{-}0.5$).

Figure 4 shows the dependence of the slopes on spectral bands for each period of time. The extreme values of slope are observed in the U band. This effect is clearest for the first period.

In the second approximation we tried to take into consideration the curvature of the SF around $\log ~{\rm d}t = 2$ (see Fig. 3). We calculated SF slopes on a time interval $10 \leq{\rm d}t \leq 150$ days ( $1.0 \leq \log~ {\rm d}t \leq 2.2$). Calculated logarithmic slopes b₁ for corresponding time-scales are presented in Col. 6 of Table 4; the coefficients of correlations (k₁) are in Col. 7. In all cases the b₁ were higher than the SF slopes "b'' on a time scale of (1.5-2.0) years. One can see that the strongest slopes for flares were observed during the first period of observations when the b₁ were as much as 2. We speculate that process of flux variation during the 10-150 day flares was stronger than during the moderate brightening of the nucleus.