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Astron. Astrophys. 351, 21-30 (1999)

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4. The structure function of the variable optical fluxes of the NGC 1275 nucleus

4.1. The structure function properties

Press (1978) suggested that the variability in quasars is caused by a process aliased as "noise". The power spectrum of the simplest of them has a form g([FORMULA]) [FORMULA] [FORMULA]. The compact variable AGNs exhibit three types of noise: white-noise ([FORMULA] = 0), flicker-noise ([FORMULA] = 1), and shot-noise ([FORMULA] = 2) (see Terebizh, 1993). Such processes are easily revealed by the technique of SF analysis. In application to AGNs it was discussed by Hufnagel & Bregman (1992); Hughes et al. (1992); Lainela & Valtaoja (1993) and others. The first-order SF is defined as:


F(t) being the flux at time t, and dt being the time delay (lag) between observations of fluxes F(t) and F(t+dt), the angular brackets denote an ensemble average. The SF of an "ideal" stationary random process on a logarithmic scale consists of three components: a slope b = d log(SF)/ d logdt, 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 - Tmax gives the time lag when 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. Tmax characterizes the duration of the flares.

There is a simple correspondence between the power spectrum of the Fourier analysis and the SF analysis: if SF(dt)[FORMULA] dtb, then g([FORMULA]) [FORMULA] [FORMULA]. SF has advantages compare to traditional power spectrum analysis, because there are no problems of windowing and aliasing. It provides more readily comprehensible information on the time scales, and is more successful in determining the noise process. It excludes the influence of stellar contamination on the observed flux. If we have several processes, or if the process is periodic, the interpretation of SF becomes more complicated.

4.2. Structure function analysis realization

The Structure Function analysis was done using a special program package by S.G.Sergeev.

The program package used was tested using the optical observations of the quasar 3C 273 during 1887-1967, reduced in a common system by Kunkel (1967). The light curve of 3C 273 in arbitrary units was given by Fahlmann & Ulrich (1975). Data are averaged over 100 days intervals.

This light curve was investigated with a correlation method and power spectra analysis by Kunkel (1967), Terrell & Olsen (1970), Terebizh (1993) and others. According to these investigations, the power spectrum of the variable flux of the quasar is g([FORMULA]) [FORMULA] [FORMULA] -2. The average duration of one flare equals 3.2 years according to Terrell & Olsen (1970), and 2.7 years according to Terebizh (1993). Using our program package we have calculated the SF at time lags from 100 to 1200 days. It is shown in Fig. 3. A regression line for this figure gives a logarithmic slope b = 0.98 with a coefficient of correlation k = 0.99[FORMULA]0.01, Tmax = 3.23 years. From this result follows that g([FORMULA]) [FORMULA] [FORMULA]. The maximum time of a correlated variability Tmax is 3.2 years. The evaluated characteristics of variability of the 3C 273 quasar are in good agreement with the data calculated by other authors using different methods. Results of this test allow us to conclude that our program package works rather well.

[FIGURE] Fig. 3. Structure function of variable flux of quasar 3C 273, obtained using data by Kunkel (1967). Straight line is a line of regression.

Fig. 4 shows SFs for the variable flux of NGC 1275 in 1982-1987 and 1987-1994. One can see that there is no SF of a simple "ideal" one - process form with one slope for all time lags. The slopes "b" of SFs for the intranight and years variations are essentially different. We examine these SFs on the intranight time-scale and on the months - years timescale separately.

[FIGURE] Fig. 4. Structure functions of the variable flux of the NGC 1275 nucleus: in the period 1982-1987 at [FORMULA] 5200 Å, in 1989-1994 for UBVRI system. Error bars of SF are absent in cases when their dimension are less than the dimension of circles. Each plot contains three straight lines being the regression lines, obtained separately for two types of intranight and one type of years variations of the NGC 1275 nucleus.

4.3. The structure function of the intranight variability of the nucleus of NGC 1275. Microvariability

Fig. 4 shows that the intranight part of all SFs does not contain a well-defined first plateau, caused by the observational errors. All logarithmic slopes are restricted only by Tmax and are not restricted by a plateau of the measurement errors. SF slopes are restricted only by the time of a single flux measurement (equal to 5 min). We cannot exclude that Tmin is less than 5 min. Shapes of SF for the intranight variations (microvariations) of the NGC 1275 nucleus show two parts with different slopes. These are shown by the two regression lines on each SF in Fig. 4 in the interval -2.8[FORMULA]logdt[FORMULA]-0.45 (2.5m[FORMULA]dt[FORMULA]8.6h). The corresponding Tmax of variations for the less steep part of SF in 1982-1987 and in 1989-1994 are about 4 hours and one hour respectively. The Tmax for the more steep part of SF is about 8.5 hours. We labelled these two types of variations as low and high level ones. Parameters of these two types of the microvariability of the NGC 1275 nucleus are represented in the columns of Tables 2 and 3: 4 - logarithmic slope - b; 3 - time interval (Tmax - Tmin); 5 - Tmax; 6 - coefficient of correlation - "k" between the values log dt and log SF, obtained for the regression lines of Fig. 4. The confidence levels of the correlations are given in Column 7.


Table 2. Structure function parameters of low level microvariability of the NGC 1275 nucleus

The data in Table 2 show that the slopes of SFs of the low-level microvariations of the nucleus of NGC 1275 were in the range 0.25 [FORMULA] b [FORMULA] 0.66, and Tmax was about 4 hours in 1982-1987 and one hour in 1989-1994. The confidence level of the correlation in all cases was almost 1.0.

One can see that the slope "b" for [FORMULA] 5200 Å in 1982-1987 and for V band in 1989-1994 was 0.25-0.35 - just the same within the limits of errors. Obtained data show that the low-level microvariability of NGC 1275 in 1982-1994 was caused by a mixed process of flicker-noise and shot-noise. The character of the microvariability does not change with time, but Tmax decreases from 4 hours in 1982-1987 to one hour in 1989-1994.

Table 3 contains the SF parameters of the high-level flux microvariations. The logarithmic slopes of the SFs are in the range 0.66-1.19, and Tmax is about 9 hours. Tmax = 4 hours for variations in 1989-1994 is not real, it is restricted by observational time interval, as can be seen in Fig. 4. The confidence level of all correlations is almost 1.0.


Table 3. Structure function parameters of high level microvariability of the NGC 1275 nucleus

Comparison of the slope "b" for [FORMULA] 5200 Å in 1982-1987 and for the V band in 1989-1994 shows that it decreased from 1.14 to 0.66. The difference is insignificant, because it is equal to about 2.5[FORMULA].

The data of Fig. 4, Tables 2 and 3 show that the SF of the microvariability of the optical flux of the NGC 1275 nucleus in the period 1982-1994 shows the presence of two types of intranight process with different time scales. A mixed process of flicker-noise and shot-noise dominates on time scales less than 4 hours, and a shot-noise type process - on time scales more than 4 hours. The maximum time scale of the correlated flux variations is equal to 8.6 hours.

The slopes of the SFs obtained for the observations in the UBVRI system are not equal each to other. The highest slopes of the SF are calculated for the U and I bands: 0.42-0.66 against 0.35 in the V band for low level variations, and 1.19-1.04 against 0.66 in the V band for high level variations. The excess for the I band amounts to 0.66-0.35=0.31 (about 2.5[FORMULA]) and 1.04-0.66= 0.38 (more than 3[FORMULA]). The excess in the slopes of the SF in the U and I bands compared to the slope in the V band can be interpreted as an elevated activity of the nucleus in the ultraviolet and near-infrared regions of the galaxy spectrum compare to the visual region, but the differences are at a low confidence level.

4.4. Structure function of the optical flux of the NGC 1275 nucleus for time lags from 1 day to several years

Table 4 contains the SF parameters of the variability of the optical flux of the NGC 1275 on a time scale of years. The columns are the same as for Tables 2 and 3. Both Table 4 and Fig. 4 demonstrate that the SF for the 1982-1994 observations on time scales of more than one day exhibits the slope b [FORMULA] 0.06-0.14 with a confidence level of the correlation of up to 0.99. The second plateau of SFs is not clearly present. It is possible that in the NGC 1275 nucleus in 1982-1994 a weak flicker-noise type process was present on time lags more than 4 years, or entirely absent.


Table 4. Structure function parameters of years variability of the NGC 1275 nucleus

Fig. 4 exhibits several minima and maxima in SFs on time lags of days, months and years. Such peculiarities of SFs are usually interpreted as the evidence of cyclic processes or individual flares in active nuclei. Cyclic processes must be shown by SFs for both periods of observations. One can see that there is only one short interval of time lags with a common minimum for all SFs: -0.3[FORMULA] log dt [FORMULA]+0.4, or 0.9[FORMULA] dt [FORMULA]2.5 days. In this interval of time lags one can look for the cyclic variability of the optical flux of the NGC 1275 nucleus. All other maxima and minima are caused by noncorrelated individual flares.

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Online publication: November 2, 1999