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Astron. Astrophys. 321, 123-128 (1997)

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3. Statistical indices of the variability

3.1. The structure function

To study the ensemble variability properties of the SA94 sample we have used the structure function (Simonetti et al. 1985, Paper I), defined as:


where mag [FORMULA] are the magnitudes at the time t, [FORMULA] is a time lag evaluated in the rest-frame of each QSO and the brackets " [FORMULA] " indicate a mean over the ensemble. The effect of the measurement errors has been subtracted according to the procedure described in detail in Paper I. Only objects detected in at least 5 epochs and with a median magnitude brighter than the completeness limit have been used to compute the SF. Only 6 QSOs have been measured on Plate R9477 brighter than the completeness limit. We have checked that the SF and the following statistical results do not depend on the inclusion/exclusion of these data.

In Fig. 1 the R-band SF in the quasar rest frame is shown. We observe a steady rise, steeper up to about 5 years and a flattening afterwards (within the errors), reaching values corresponding to about 0.1 mag2 of RMS variability.

[FIGURE] Fig. 1. QSO rest-frame structure function in the R-band

It should be noted that points in Fig. 1 with [FORMULA] are all derived from the comparison of the POSS plate with the remaining plates. Being 28 years apart from Plate R5413, Plate E1453 does not play any role in the estimates of the SF on the shorter timescales (5 yr or less).

In Fig. 2 a comparison between the SF s in the B-band (as derived from the data discussed in Paper I) and in the R-band is reported. The SF in the B-band shows a larger variability amplitude with respect to the one in the R-band. To quantify the statistical difference between the two SF s, we have calculated the weighted average of the ratio [FORMULA], for lags between 1 and 5 yr in the QSOs rest frame. The result is [FORMULA], a difference from the unity significant at a [FORMULA] level.

[FIGURE] Fig. 2. Structure function for QSOs in the R-band (continuous line) and in the B-band (dashed line)

In order avoid any possible bias deriving from different time samplings or time dilation effects, a direct comparison of the variability in the R and B band has been carried out. As shown in Fig. 3, for each object we have computed the quantities [FORMULA], as the difference between the magnitude at a given epoch and the average magnitude. Since the epochs of the R plates are in large part coincident with the epochs sampled with the B plates (epochs 2, 3, 4, 5, 6, 7 of the present paper correspond to epochs 2, 6, 8, 9, 10, 11+12 of Paper I, respectively) it has been possible to plot the [FORMULA] versus the [FORMULA] for all the objects with at least 5 epoch-magnitudes in common. Again, the POSS plate does not play any role in this comparison.

[FIGURE] Fig. 3. [FORMULA] vs. [FORMULA] (see text). Dashed line: straight line with a slope 1. Continuous line: least-squares fit between [FORMULA] and [FORMULA] (see text). The cross in the lower-right corner shows the typical measurement errors

We have then fitted a straight-line relationship between [FORMULA] and [FORMULA], with a least-squares technique that takes into account the errors of the data points on both axes (Fasano & Vio 1988). The result is [FORMULA]. The dispersion of the points around this relation is fully compatible with the measurement errors, with no intrinsic scatter. If we make the additional assumption that the R flux varies in phase with the B flux, i.e. that for [FORMULA] [FORMULA], then we obtain [FORMULA], again fully consistent with no intrinsic scatter. The coefficient between the R and the B variability obtained in this way is consistent within the errors with what has been estimated on the basis of the SF analysis.

In Fig. 4 the distribution of QSOs in the redshift-magnitude plane is illustrated. To further investigate and disentangle the dependences of the variability on the luminosity and redshift we have examined two subsamples. The first is defined within the redshift limits [FORMULA] (the area between the dashed lines in Fig. 4). The second is defined within the absolute magnitude limits [FORMULA] (the area between the continuous lines in Fig. 4). As in Paper I, the SF s indicate that the QSO variability (in magnitude) is, also in the R band, clearly anticorrelated with absolute luminosity, in the sense that more luminous objects have a smaller variability amplitude (Fig. 5). No correlation between redshift and variability - of the type discovered in Paper I for the B-band variability - has been detected within the errors (Fig. 6). However, such a failure is not surprising considering that in the present work we are dealing with only one sample (in place of the three of Paper I) and the still non-uniform coverage of the redshift-absolute magnitude plane of the subsample defined between [FORMULA] introduces a luminosity-redshift correlation in the data that may mask any redshift-variability correlation.

[FIGURE] Fig. 4. Distribution of the SA94 QSO sample in the [FORMULA] plane
[FIGURE] Fig. 5. The structure function for quasars of different luminosity in the redshift interval [FORMULA]. Continuous line: QSOs with [FORMULA]. Dashed line: QSOs with [FORMULA]
[FIGURE] Fig. 6. The structure function for quasars of different redshifts in the luminosity interval [FORMULA]. Dashed line: QSOs with [FORMULA]. Continuous line: QSOs with [FORMULA]

3.2. The variability index

In order to further verify the results obtained in the previous section, we have evaluated the variability index (IDX) for each QSO, defined as the variance of the normal process representing the intrinsic variability that, added to the photometric errors, reproduces the observed light variations on a given range of timescales (see Paper I). The index has been evaluated in one time bin (from 1 to 15 years in the QSOs rest frame) to avoid on the one hand time scales for which the structure function is still quickly rising, as observed in the previous section, and on the other hand time lags unaccessible with the present data for the higher-redshift QSOs. Then we have analysed the correlation matrix, reported in Table 3.


Table 3. Correlation coefficients

The correlation coefficients indicate a correlation of the variability index with the absolute magnitude and an almost equal anticorrelation with redshift (correlation coefficients respectively of [FORMULA] and -0.18, both with a significance greater than 99%). Due to the strong flux-redshift anticorrelation in the QSO sample it is not possible to disentangle if one of the two correlations is spurious. However, if, on the basis of the results of the previous subsections and of Paper I, we assume as fundamental the absolute magnitude - variability correlation, it is possible to subtract its influence from the variability-redshift anticorrelation via the method of partial correlation analysis (Spiegel 1991). Applying this recipe to the results of Table 3, the corrected correlation coefficient between variability index and redshift, turns out to be [FORMULA] (65%), not significant, in agreement with the result shown in Fig. 6.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998