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Astron. Astrophys. 358, 451-461 (2000)

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3. Time scales of the radio variability

3.1. Visual inspection

We analyse the radio light curves of 0202+149 at 37 and 22 GHz (Fig. 1 top) obtained in 1984-1998 with the 13.7 m telescope at Metsähovi and at 14.5, 8.0 and 4.8 GHz (Fig. 1 bottom) from the 26 m telescope at UMRAO during the same period. The total flux density of the source shows complex variability over a wide range of time scales from as short as [FORMULA] 1 year to longer than 10 years; short timescale variations dominate at mm wavelengths, while longer timescales are prominent at cm wavelengths.

[FIGURE] Fig. 1. The total flux density of 0202+149 as a function of time

Close inspection of the mm-wave total flux density curves suggests the possibility of a periodicity on a timescale of about 4 years. Fig. 2 presents three 4-year cycles of the flux variability at 22 and 37 GHz when the long-term trend [FORMULA] (see below) is subtracted from the flux density S. The top panel gives the first cycle (C1), which starts at [FORMULA]=1985.3 (the data are very sparse). The middle and bottom panels show the second (C2, [FORMULA]=1990.1) and the third (C3, [FORMULA]= 1994.2) cycles.

[FIGURE] Fig. 2. Light curves separated into three 4-year cycles at 22 (open circles) and 37 (solid circles) GHz after subtraction of the long-term trend.

All three cycles display a similar structure of variability, with global maxima near the middle of the period, despite the fact that the 1996 outburst is nearly 5 times fainter than two previous outbursts. Fig. 2 shows that the 4-year cycle probably consists of two parts: a bright first half ("outburst") with two maxima ([FORMULA] 1991 and [FORMULA] 1992, in cycle C2, labeled below in Fig. 3 as I and II, respectively, and [FORMULA] 1995 and [FORMULA] 1996, in cycle C3, labeled as 1 and 2) and more quiescent second half ("post-outburst") with very faint maximum ([FORMULA] 1988.7, in cycle C1; [FORMULA] 1993.7, in cycle C2, labeled as III; [FORMULA] 1997.7, in cycle C3, labeled as 3). The time between the outbursts is 4.4[FORMULA]0.4 yr. Taking into account this estimate, we can expect a new maximum at mm wavelengths near [FORMULA] 1999.0; the 22 GHz total flux density curve (Fig. 1) does show a maximum at this epoch quite clearly. Some indications of the maximum are seen also at 14.5 and 8 GHz.

[FIGURE] Fig. 3. Flux densities at frequencies 37 (asterisks), 22(crosses), 14.5 (squares), 8 (circles) and 4.8 (triangles) GHz after removing the long-term trend for cycles C2 (top) and C3 (bottom).

Fig. 3 shows cycles C2 and C3 at all frequences from 37 to 4.8 GHz after substraction of the long-term trend (see below under time series analysis). The main maximum of a cycle (denoted as II in cycle C2 and as 2 in cycle C3) takes place at all the frequencies almost simultaneously and has a rather flat spectrum (see Sect. 3.2). On the contrary, the preceding maximum (denoted as I and 1, respectively, in cycles C2 and C3) is delayed by approximately 0.7[FORMULA]0.1 yr at 4.8 and 8 GHz relative to 14.5, 22 and 37 GHz. Whereas in cycle C2 this maximum is most prominent at 22 and 37 GHz, in cycle C3 it is much higher at 8 GHz.

3.2. Evolution of the total flux density spectrum

We construct the spectral energy distribution in the range from 4.8 to 37 GHz for several similar phases of cycles C1, C2, and C3 (Fig. 4), where the phase is a duration in years from the start point of the cycle.

[FIGURE] Fig. 4. Spectra of the total flux density over the range 4.8 to 37 GHz in cycles C1(bottom), C2(middle), and C3 (top).

The figure shows that the overall spectrum steepened from cycle C1 to cycle C3 and that the turnover frequency evolved from 37 to 8 GHz, respectively. Fig. 4 demonstrates a difference between the spectra at the two maxima of cycle C2 (I - corresponding to [FORMULA] 1.0 and II - corresponding to [FORMULA] 2.0). The spectrum at maximum I peaked at 37 GHz and steeply declined to 8 GHz, while the spectrum at maximum II was rather flat and peaked at 22 GHz.

In order to separate the spectra of outbursts from the spectra of the long-term components, we substracted from the flux densities at maxima I and II the average values of flux densities just before and after the corresponding outbursts. The spectra obtained are presented in Fig. 5;

[FIGURE] Fig. 5. Spectra of radio emission for outburst components during maxima I (solid circles) and II (open circles). Solid curves show second order polynomial approximation.

these show that the turnover frequency at maximum I is [FORMULA] 37 GHz while at maximum II it is close to  22 GHz. The spectral indexes ([FORMULA]) for the frequency range of the spectra from 8 to 22 GHz are [FORMULA] -2.1 to -1.1 for the I and II maxima correspondingly. The spectral index for the optically thin part of the spectra at maximum II can be estimated as [FORMULA] 0.43. The spectral index of optically thin part of spectrum at maximum I can't be estimated due to lack of data above 37 GHz. Unfortunately, analogous estimates for maxima 1 and 2 in cycle C3 are rather uncertain due to the low flux densities of the outbursts at high frequencies. It is interesting to point out that the spectrum at epoch T=1991.7 (the intermediate between the two maxima) has two humps, at frequencies 8 and 22 GHz.

3.3. Time series analysis

To confirm our visual estimate of the characteristic time scales of the radio variability of 0202+149 we performed a time series analysis of the radio light curves and array of spectral indices in the range of 4.8 - 37 GHz, using two methods to search for a periodicity - "whitening" (Hagen-Thorn et al. 1997) and CLEAN (Roberts et al. 1987).

The method "whitening" is based on the construction of the periodograms


where [FORMULA] is the initial time series with the mean flux density subtracted, N is the total number of points, and [FORMULA] is the array of frequencies. The periodograms are constructed for residual time series formed by the successive subtraction of a sinusoidal component corresponding to the maximum periodogram value of the previous series. Subtraction of the long-term harmonics (the period of the harmonic is longer than half of the total duration of observations) removes the underlying long-term trend. Fig. 6a presents the first periodogram of the initial light curve at 4.8 GHz, which has a primary maximum at a frequency corresponding to 13.7[FORMULA]1.1 years. The first periodograms for other light curves have the same form and show the presence of long-term trends with characteristic time scales in the range 11-14 years (The CLEAN method gives similar results, see Table 1). Fig. 6b shows a fit between the data at 4.8 GHz and the long-term trend.

[FIGURE] Fig. 6. Long-term trend of the flux variability at 4.8 GHz: a) the first periodogram of the original light curve at 4.8 GHz; b) comparison of the flux densities at 4.8 GHz (circles; the average value of the flux density is subtracted) with the long-term trend (solid curve), obtained as a sum of the 13.7- and 11.1-year harmonics that give the main maxima in the first and second periodograms.


Table 1. Parameters of harmonics found by CLEAN method

For all light curves, the functions of the spectral window were constructed. As an example, Fig. 7a presents the function of the spectral window at 22 GHz, where labels A, B and C mark the maxima at frequencies [FORMULA]=0.00055[FORMULA]0.00002 (5-year harmonic), [FORMULA]=0.00115[FORMULA]0.00001 (2-year harmonic), and [FORMULA]=0.00280[FORMULA]0.00001 (1-year harmonic) caused by unequally spaced data that generate aliases in the power spectra. The aliases, which usually appear in pairs on either side of "real" peaks in the power spectra, are labeled by a letter with a number, where the letter corresponds to a peak in a spectral window (see Fig. 7a) and the number is the modulus of the corresponding period found in the data.

[FIGURE] Fig. 7. Time series analysis: a) the function of the spectral window for the observational grid at 22 GHz; b-f) the periodograms of the 0202+149 light curves after subtraction of the long-term trend at 22, 37, 14.5, 8.0, and 4.8 GHz, respectively.

We subtract the long-term trends approximated by the sum of (usually two) sinusoids (see Fig. 6) from the light curves. Fig. 7 (b-f) present the main periodograms for all frequencies after subtraction of the long-term trend, which, however, is not removed completely (especially at 8.0 GHz, where the CLEAN method indicates the presence of the most complicated long-term trend; see Table 1), but whose influence is decreased sufficiently to study shorter characteristic time scales of variability.

The analysis of the periodograms shows that the light curves have a typical time scale of variability of P[FORMULA](4.2[FORMULA]0.8) yr  (at 8 GHz the most prominent harmonic, 3.2[FORMULA]0.6 yr, is within the 2[FORMULA]-uncertainties of the 4-year harmonic). The 4-year cycle has substructures with typical timescales of 2.3[FORMULA]0.2 yr and 1.3[FORMULA]0.1 yr. The 1.3[FORMULA]0.1 yr scale is more pronounced in the 37 and 22 GHz periodograms, while the 4.8, 8.0, and 14.5 GHz periodograms show the 2.3[FORMULA]0.2 yr timescale.

Parameters of harmonics found by the CLEAN method are given in Table 1 (the first column gives the frequency of the analysed light curve; the second column, the period of each harmonic P; the third column, the amplitude of each harmonic A; and the fourth column, the phase of each harmonic [FORMULA]). The table contains the harmonics that exceed the level of noise, defined at every frequency by the formula: [FORMULA], where [FORMULA] is the mean value of the power spectrum, sn is the ratio of the standard deviation of a single observation to the error of a single measurement, [FORMULA], where Pr is the level of significance. A comparison of Fig. 7 and Table 1 shows that the results of analysis by the CLEAN and "whitening" methods are in good agreement. Existence of close pairs of harmonics (13.7 & 11.0, 4.6 & 3.0 & 5.5, 2.3 & 1.7 years) can be easily explained by deviations from strict periodicity, namely, by amplitude modulations and phase delays at different frequencies.

It is noteworthly that the light curves of 0202+149 at 318, 430, 660, 880, and 1400 MHz obtained during a 5-year program of multifrequency monitoring by Mitchell et al. (1994) over the period 1980-1985 reveal the possible presence of a similar 4-year cycle starting between the end of 1980 and beginning of 1981. The recently published long-term light curves at 318 and 430 MHz based on the Arecibo measurements (Salgado et al.1999) seem to show a variability on the timescale [FORMULA] 13 yr.

3.4. Evolution of the spectral index

We construct the array of spectral indices of 0202+149 in the radio region from 4.8 GHz to 37 GHz, considering that observations at different frequencies are simultaneous if the difference between dates of observations does not exceed 0.05 yr. The values of spectral indices [FORMULA] ([FORMULA]) estimated by the least-squares method are given in Fig. 8a. The figure shows a gradual steepening of the spectrum of the source from 1990 to 1998 that corresponds to a long-term trend of variability with time scale [FORMULA]11 yr and amplitude 0.35[FORMULA]0.06.

[FIGURE] Fig. 8. a Spectral index of the total flux density in the range from 4.8 to 37 GHz versus time; b the periodogram of the spectral index array after subtraction of the long-term trend.

The time series analysis of the array after subtraction of the long-term trend reveals the presence of a 4-year cycle of spectral index variability with amplitude 0.08[FORMULA]0.02 (Fig. 8b). This is further confirmation that a 4-year cycle is the most characteristic time scale of the flux density variations at mm- and cm- wavelengths. The periodogram in Fig. 8b also shows a 1-year time scale of variability. The amplitude of the variations reaches [FORMULA]0.4 near 1991 and significantly decreases after 1993.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000