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Astronomical Data Analysis Software and Systems VI ASP Conference Series, Vol. 125, 1997 Gareth Hunt and H. E. Payne, eds.

Determination of Variable Time Delay in Uneven Data Sets
V. L. Oknyanskij Sternberg Astronomical Institute, Universitetskij Prospekt 13, Moscow, 119899, Russia Abstract. Time delay determinations in astrophysics are used most often to find time delays b etween flux density variations in different sp ectral bands and/or lines in AGNs, and different images of gravitationally lensed QSOs. Here we consider a new algorithm for a complex case, when the time delay is itself a linear function of time and the intensity of echo resp onse is exp onential function of the delay. We apply this method to the optical-to-radio delay in the lensed double quasar Q0957+561. Radio-optical variability correlation in Q0957+561 was first rep orted by Oknyanskij & Beskin (1993, hereafter OB) on the basis of radio observations made in the years 1979 to 1990. OB used an idea to take into account the known gravitational lensing time delay to get combined radio and optical light curves and then to use them for determination of the p ossible radio-from-optical time delay. It was found this way that radio variations (5 MHz) followed optical ones by ab out 6.4 years with high level of correlation ( 0.87). Using new radio data (Haarsma et al. 1997), for the interval 1979­1994, we find nearly the same value for the optical-to-radio delay as had b een found b efore. Additionally, we susp ect that the time delay value is linearly increasing at ab out 110 days p er year while the p ortion of reradiated flux in the radio resp onse is decreasing. We conclude that the variable radio source is ejected from the central part of the QSO compact comp onent.

1.

Introduction

Time delay determinations in astrophysics are used most often to find time shifts b etween variations in different sp ectral bands and/or lines in AGNs, as well as time delays b etween different images of gravitationally lensed QSOs. In most cases, the task is complicated by uneven spacing of data, so that standard crosscorrelation methods b ecome useless. Two different methods are most often used: CCF (Gaskell & Spark 1986) and DCF (Edelson & Krolik 1988), which are based on line interp olation of data sets or binning of correlation coefficients, resp ectively. We have introduced several simple improvements to CCF (Oknyanskij 1994) and this modernized MCCF combines the b est prop erties of CCF and DCF methods. With MMCF we calculated regression coefficients as functions of time shift. Here, this calculation is generalized for the more complex case where the time delay is a linear function of time, and a p ortion of the flux density is itself a p ower-law function of the delay. We apply this method to the optical-to-radio time delay in the gravitationally lensed double quasar Q0957+561. The data 162

© Copyright 1997 Astronomical Society of the Pacific. All rights reserved.


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sets used here were obtained to determine the gravitational lensing time delay . Our results are nearly identical for values of in the interval of 410­550 days. In the discussion b elow, we take = 425 days. 2. Method and Results

Our method includes several steps, which are briefly explained b elow: Combined light curves. We take the radio (Haarsma et al. 1997) and optical (Vanderriest et al. 1979; Schild & Thompson 1995) data sets for A and B images and determine (using MCCF) the line regression coefficients k( ) and m( ). Then we transform A(ti ) values into the B image scale system for the known value of : B (ti - ) = k( ) · A(ti )+ m( ). (1)

We combine these values B with the usual B ones, sorting by time. The resulting optical light curve was then smoothed by averaging in 200 day intervals with steps of 30 days. This accounts for the physical argument that radio sources should b e bigger than optical ones. The value of 200 days for smoothing was taken as ab out optimal from the autocorrelation analysis of light curves. Correction for change of time delay and radio flux. Taking the opticalto-radio time delay or to b e a linear function of time, let V b e the change of optical-to-radio time delay or p er year. We fix some moment of time as t0 . It is attractive to choose t0 so that it falls near a strong maximum in the optical light curve (here J.D. 2445350), which obviously correlates with the high maximum in the radio light curve if take or (t0 ) = 2370 days. So we can calculate the needed correction: S (t) = V · (t - t0 ) 365d (2)

to b e added to dates in the optical light curves: ti = ti + S (ti ) (3)

Assuming that a p ortion of radio flux decreases as a p ower-law function of time with exp onent . We should also correct the optical flux for that fading b efore computing the cross-correlation function: Iop (ti ) = Iop (ti ) · (1 + S (ti )/2370d )-


(4)

Computing MCCFs. We compute an array of MCCFs for combined radio and optical light curves, varying V and . Map cross-correlation as a function of V and . For p oints (V, )we map the MCCF values (see Figure 1). The b est correlation occurs for V 110 days/year, and 0.7.


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Oknyanskij

Figure 1.

Two-dimensional cross-correlation function (see text).

Comparison of optical and radio light curves. We correct the optical combined light curve using (3) and (4) with the parameters V = 110d and = 0.70, shift ahead by or (t0 ) = 2370d , and then fit to the radio data by analogy with (1). The corrected optical light curve is shown with with the radio light curves in Figure 2. Most features in b oth light curves coincide quite well. So the investigation supp orts our assumption on the lengthening of the optical-to-radio time delay. As a result we can give an expression for the optical-to-radio time delay as a linear function of time: = 2370d + 110d · (t - t0 ) 365d



or

(5)

3.

Conclusion

We have calculated the time delay b etween radio and optical flux variations using a new method. In addition, we have investigated the p ossibilities that (1) there is a change of the time delay that is a linear function of time, and (2) the radio resp onse has p ower-law dep endence on the time delay value. Finally, let us stress some additional consequences from our results: 1. For some ob jects, optical-to-radio time delays were probably not found b ecause they were too long compared to the duration of monitoring program. 2. Optical-radio correlations may not have b een recognized in some ob jects since the time delays as well as resp onse functions probably were variable. This p ossibility has never b een entertained b efore.


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Figure 2. Radio and optical combined light curves. (The optical light curve is corrected as describ ed in the text.) 3. The variable radio flux in Q0957+561 may originate in a very compact jet comp onent moving away from the optical source. Only after another jet comp onent app ears (whose time delay value will of course b e different) will the QSO again show some optical-radio correlation. 4. If several compact jet may have no chance jet comp onents move is probable for Blazar found, since the time b e very close to zero. comp onents exist simultaneously in a QSO then we to find any radio-optical correlation. Only if these toward the observer closely to the line of sight (as it s), will radio-optical correlation have a chance to b e delays for all these variable radio comp onents would

Acknowledgments. In conclusion, we thank Debb orah Haarsma for sending us the preprint with the new radio data for Q0957+561 b efore publication. References Edelson, R. A., & Krolik, J. H. 1988, ApJ, 333, 646 Gaskell, C. M., & Spark, L. S. 1986, ApJ, 305, 175 Haarsma, D. B., Hewitt, J. N., Lehar, J., & Burke, B. F. 1997, ApJ, 479, 102 Oknyanskij, V. L., & Beskin, G. M. 1993, in Gravitational Lenses in the Universe: Proceedings of the 31st Liege International Astrophysical Colloquium, eds. J. Surdej et al. (Liege, Belgium: Universite de Liege, Institut d'Astrophysique), 65 Oknyanskij, V. L. 1994, Ap&SS, 222, 157 Schild, R. E., & Thomson, D. J. 1995, AJ, 109, 1970 Vanderriest, C., et al. 1989, A&A, 215, 1