Baltic Astronomy, vol. 25, , 2016

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1 Baltic Astronomy, vol. 25, , 216 THE STUDY OF RADIO FLUX DENSITY VARIATIONS OF THE QUASAR OJ 287 BY THE WAVELET AND THE SINGULAR SPECTRUM METHODS Ganna Donskykh Department of Astronomy, Odessa I. I. Mechnikov National University, Dvoryanskaya Str. 2, Odessa, 6582, Ukraine; Received: 216 June 3; accepted: 216 August 14 Abstract. Flux density variations of the extragalactic radio source OJ 287 are studied by applying the wavelet and the singular spectrum methods to the long-term monitoring data at 14.5, 8. and 4.8 GHz acquired at the University of Michigan Radio Astronomy Observatory during 4 years. This monitoring significantly supplements the episodic VLBI data. The wavelet analysis at all three frequencies revealed the presence of quasiperiods within the intervals and years. The singular spectrum analysis revealed the presence of quasiperiods within the intervals 6 1 and years. For each quasiperiod the time interval of its existence was determined. Key words: quasars, BL Lacertae objects: individual (OJ 287) 1. INTRODUCTION QSO J or OJ 287 is a BL Lacertae object with the redshift z =.36 (Stickel et al. 1989) known as providing quasiperiodic optical and radio outbursts. This paper presents a study of the radio flux density variations of OJ 287 based on long-term monitoring observations obtained at the University of Michigan Radio Astronomy Observatory. We analyze the monitoring data at the three frequencies: 14.5 GHz for a period of 37 yr ( ), 8 GHz for a period of 4 yr ( ) and 4.8 GHz for a period of 32 years ( ). For the data analysis, the wavelet and the singular spectrum methods are applied. The object exhibits its variability through the full range of the electromagnetic spectrum. There is a suggestion that it is a binary system of two supermassive black holes (Sillanpää et al. 1988; Valtonen et al. 28). In this model, the smaller black hole perturbs the accretion disk of the bigger primary black hole (e.g., Valtonen & Lehto 1997). The mass of the central supermassive black hole is solar masses (Valtonen et al. 28). A precessing jet model that assumes the existence of helical Kelvin-Helmholtz instability in the jet flow may explain the variability of the object (Hardee 2; Mignone et al. 21; Valtonen & Pihajoki 213). For the data processing of flux density of OJ 287 Hughes et al. (1998) have applied the wavelet analysis method. They found the periods of 1.66 and 1.12 yr. To explain the existence of these periods the authors proposed a shock-in-jet model (Blandford & Königl 1979; Marscher & Gear 1985), in which a short-term

2 238 G. Donskykh 1 8 Flux density (Jy) GHz 8 GHz 4.8 GHz Fig. 1. Flux density of OJ 287 at the frequencies 14.5, 8. and 4.8 GHz. periodicity is associated with the passage of a shock. A long-lasting radio monitoring of OJ 287 at the 22, 37 and 9 GHz frequences has been conducted at the Metsähovi Radio Observatory. Hovatta et al. (28) by the wavelet analysis found a period of 1.4 yr at 22 and 9 GHz, and a period of 1.7 yr at 37 GHz. For study of the radio flux variations of OJ 287 we applied the UMRAO monitoring database. The methods for calibration and data reduction are described by Aller et al. (1985). The average flux densities for seven-day intervals were taken as the initial data. The extract of the short-period component of the signal was made by the Fourier transform (Donskykh et al. 215). The initial data plot for the flux density variations of OJ 287 at 14.5, 8. and 4.8 GHz are presented in Fig RESULTS OF THE WAVELET ANALYSIS Unlike the classical Fourier transform, the wavelet transform allows to obtain not only quasiperiods, but also the time range of their presence. The wavelet transform provides the time-frequency presentation of the signals. The following expression defines a continuous wavelet transform: W (a, b) = 1 a ( ) t b x(t)ψ dt. (1) a Here, a is the scale factor and b is the translational value (a, b R, a ). A continuous function ψ(t) is called the mother wavelet (Polikar 21). The initial function can be recovered by the inverse wavelet transform x (t) = C 1 ψ W (a, b) ψ ( t b a ) 1 a da db a 2, (2)

3 Flux density variations of the quasar OJ Fig. 2. The wavelet spectrum for OJ 287 at 14.5 GHz. The text boxes indicate the periods in years. C ψ = ˆψ 2 ω 1 dω, (3) where C ψ is the admissible constant, which must satisfy the criterion of admissibility C ψ <. In this paper, we used a well-localized in time and frequency Morlet wavelet. The description of the method is given by Van den Berg (24). At the first stage, the long-period component of the radio source flux variability was analyzed. Having the trend component eliminated by the (O C) digital filtering method, the short-period component of flux variability was then analyzed. The wavelet spectrum of OJ 287 in the logarithmic scale at 14.5 GHz is shown in Fig. 2. When calculating the continuous wavelet transform, fast Fourier transform (Cooley et al. 1969) for convolution as well as B-spline interpolating surface for visualization were used. The spectral power was calculated by integrating of the interpolating surface. To obtain the peaks of the wavelet spectrum, which correspond to the real quasi-harmonic components in the studied time series, the critical levels (an autoregressive (AR) model) and the cones of influence (Torrence & Compo 1998) were applied. The peaks (the values of quasiperiods), which exceeded the critical level of 95%, were chosen. The results of the wavelet analysis at the frequencies 14.5, 8. and 4.8 GHz are given in Table 1. The periods obtained are shown in Figs RESULTS OF THE SINGULAR SPECTRUM ANALYSIS To specify the periods of the OJ 287 flux variations, obtained by the wavelet transform, application of alternative methods is required. For this purpose, the singular spectrum analysis (SSA) technique implemented in the Caterpillar-SSA software package was applied. A more detailed description of this method is given by its authors (Golyandina 24; Golyandina & Shlemov 215). Application of the

4 24 G. Donskykh Table 1. The results of wavelet analysis for the data on OJ 287 at the frequencies 14.5, 8. and 4.8 GHz. Quasiperiod Error Lifetime Year of the maximum (yr) (yr) of quasiperiod spectral power 14.5 GHz ; GHz GHz Ghz 8 Spectral power years 1.6 years 1.2 years Fig. 3. The periods obtained by the wavelet analysis at 14.5 GHz. singular spectrum analysis implies decomposition of the initial signal into a set of narrow-band filters. They contain trend and periodic components, as well as noise. The special feature of the singular spectrum analysis is that the analyzing function is not used in the calculations; hence, these calculations enable to extract different components of the investigated time series to a high accuracy. Computation of main components is one of the SSA stages, and it had been previously used by other authors to study the AGN X-ray flux variability (e.g. Parker et al. 214).

5 Flux density variations of the quasar OJ GHz Spectral power years 4.5 years 2 years 1.6 years 1.2 years Fig. 4. The periods obtained by the wavelet analysis at 8. GHz GHz Spectral power years 4 years 2.1 years 1.6 years 1.1 years Fig. 5. The periods obtained by the wavelet analysis at 4.8 GHz. In order to analyze the time series F N = (f,..., f N 1 ) of a length N, the parameter L (called the window length) was used being chosen arbitrarily (Golyandina 24). Firstly, the trajectory matrix of the series which is L K matrix (and K = N L + 1) is formed: X = [X 1,..., X K ]. (4)

6 242 G. Donskykh 15 6 (1.95 %) 1 5 Amplitude Point number Fig. 6. Example of eigenvector (number 6) for OJ 287 at 14.5 GHz. The percentage value in brackets represents the contribution of the eigenvector in decomposition of the original series. Then comes the singular value decomposition of the matrix X, which results in X = X 1 + X X d, X i = λ i U i V T i, (5) where λ 1 λ 2 λ d > are the eigenvalues of XX T, U i are the eigenvectors, and V i are the factor vectors. The example of eigenvector, obtained using the SSA technique for the source OJ 287 at 14.5 GHz, is shown in Fig. 6. The set ( λ i U i Vi T ) is called the i-th eigentriple (Alexandrov & Golyandina 24; Chistyakova & Shamsha 211). Then, the grouping of decomposition components is carried out. Finally, the original series are restored from the grouped matrices. The result of reconstruction of the original series by eigenvectors 1 12 (at 14.5 GHz) is shown in Fig. 7. The application of singular spectrum analysis gave the following results. At the frequency 14.5 GHz, the quasiperiods 1±.2 yr, ( )±.2 yr, 1.9±.5 yr and 1.6±.3 yr were found. At the frequency 8 GHz, the quasiperiods 6.2±.1 yr, 3.1 ±.3 yr, 2.4 ±.2 yr and 1.6 ±.8 yr were found. At the frequency 4.8 GHz, the quasiperiods 6 ±.1 yr, 4 ±.5 yr, (2 2.4) ±.3 yr and 1.6 ±.2 yr were found. 4. CONCLUSIONS This paper demonstrates that a combination of the methods of wavelet analysis and the analysis of the singular spectrum allows us to study in detail the flux variations of the extragalactic radio source OJ 287. The wavelet analysis of the OJ 287 flux density changes at the frequency 14.5 GHz shows the presence of quasiperiods of 6.5, and 1.2 yr. At the frequency 8 GHz, the periods , 4.5 5, 2 2.2, and 1.2 yr were detected. At the frequency 4.8 GHz the periods , 4., , 1.6 and 1.2 yr were found. For each quasiperiod its lifetime was evaluated.

7 Flux density variations of the quasar OJ Original series Reconstruction Original series Fig. 7. Reconstruction of the original series at 14.5 GHz. The singular spectrum analysis at the frequency 14.5 GHz revealed the periods 1, , 1.9 and 1.6 yr. At the frequency 8. GHz, the periods 6.2, 3.1, 2.4 and 1.6 yr were detected. At the frequency 4.8 GHz, the periods 6, 4, and 1.6 yr were detected. ACKNOWLEDGMENTS. We are grateful to the referee for helpful comments on the first version of this article. We would like to thank Prof. S. M. Andrievsky for his help in preparing the manuscript. REFERENCES Alexandrov Th., Golyandina N. 24, Exponenta Pro, 3, 54 Aller H. D., Aller M. F., Latimer G. E., Hodge P. E. 1985, ApJS, 59, 513 Blandford R. D., Königl A. 1979, ApJ, 232, 34 Chistyakova A. A., Shamsha B. V. 211, Radioelectronic and Computer Systems, 4, 15 Cooley J. W., Lewis P., Welch P. 1969, IEEE Trans. on Education, 12, 1 Donskykh G. I., Ryabov M. I., Sukharev A. L., Aller M. F. 215, Odessa Astronomical Publications, 28(2), 217 Golyandina N. E. 24, The Caterpillar-SSA Method: Time Series Analysis, St. Petersburg State University, p. 74 Golyandina N., Shlemov A. 215, Statistics and its Interface, 8, 277 Hardee P. E. 2, ApJ, 533, 176 Hovatta T., Lehto H. J., Tornikoski M. 28, A&A, 488, 897 Hughes P. A., Aller H. D., Aller M. F. 1998, ApJ, 53, 662 Marscher A. P., Gear W. K. 1985, ApJ., 298, 114 Mignone A., Rossi P., Bodo G., Ferrari A., Massaglia S. 21, MNRAS, 42, 7 Parker M. L., Fabian A. C., Matt G. et al. 214, MNRAS, 447, 72 Polikar R. 21,

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