3 Correlation analysis of variations

3.1 DCF method

OJ 287 is violently variable in the optical and infrared bands. We want to investigate whether the variations in these two bands are correlated or not. To do so, we use the DCF method described by Edelson & Krolik (1988) and generalized by Hufnagel & Bregman (1992) which is suitable for data sets which are not sampled evenly or at the same density.

Following Edelson & Krolik (1988) and Hufnagel & Bregman (1992) (see also Tornikoski et al. 1994), we have calculated the DCF. Firstly, we have calculated the set of unbinned correlation (UDCF) between data points in the two data streams a and b

$${UDCF_{ij}}={\frac{ (a_{i}- \bar{a}) \times (b_{j}- \bar{b})}{\sqrt{\sigma_{a}^2 \times \sigma_{b}^2}}}$$ (1)

where a_i and b_j are points in the data sets, $\bar{a}$ and $\bar{b}$ are the means in the data sets, and $\sigma_{a}$ and $\sigma_{b}$ are the standard deviations of each data sets. Secondly, we have averaged the points sharing the same time lag by binning the UDCF_ij in suitably sized time-bins to get the DCF for each time lag $\tau$

$${DCF(\tau)}=\frac{1}{M}\Sigma UDCF_{ij}(\tau)$$ (2)

where M is the number of pairs in the bin. The standard error for each bin is

$$\sigma (\tau) =\frac{1}{M-1} \{ \Sigma [ UDCF_{ij}-DCF(\tau) ]^{2} \}^{0.5}.$$ (3)

3.2 Correlation between optical and infrared bands

From the light curve, we know that there are more observations in the K band for the infrared data and in the V band for the optical data. So, we use the V and K bands to investigate the correlation between optical and infrared emissions. The results are shown in Fig. 4 (a: for bin = 30 days and b: for bin = 50 days). It is clear that the peak in DCF corresponds to "zero'' point suggesting that the two bands are correlated with almost no time delay.

  

Figure 4: a) DCF between B and K bands with bin = 30 days. b) DCF between B and K bands with bin = 50 days