4 Discussion

Attempts to measure correlations of amplitude (or some related variability parameter) with redshift and luminosity have a long history, which is summarised by Hawkins ([1996]). Among recent work, a useful place to start is with the paper by Hook et al. ([1994]). They analyse a sample of $\sim 300$ quasars in the SGP area from 12 UK 1.2 m Schmidt plates taken in 5 separate yearly epochs spanning 16 years. They find a convincing anti-correlation between their variability parameter $\sigma_{{\rm v}}$ (a measure of variation about the mean) and luminosity, but a much weaker anti-correlation with redshift. They attribute this to the degeneracy between redshift and luminosity in their sample. The same sample is re-analysed by Cid Fernandes et al. ([1996]) who use a variability index related to variance, and also one related to the structure function. They concur with Hook et al. ([1994]) that there is an anti-correlation between their variability indices and luminosity, but claim a positive correlation with redshift. This effect is not apparent to the eye, but is interpreted as a variability-wavelength dependence rather than an intrinsic variability-redshift dependence. The net result in an un-binned sample cancels out with the luminosity variability relationship. Cid Fernandes et al.'s claim for a positive correlation between amplitude and redshift appears to be motivated at least in part by expectations arising from a paper by Di Clemente et al. ([1996]). This interesting paper examines the relation between their variability parameter S₁ (an amplitude based on the structure function) and wavelength. Their sample is composed of PG quasars (Schmidt & Green [1983]), which are mostly low redshift and relatively low luminosity objects. With the help of archival IUE observations they find that S₁ decreases with wavelength. This effect can clearly be seen in the light curve from the intensive monitoring programme of the Seyfert galaxy NGC 5548 (Clavel et al. [1991]), which has a larger amplitude at shorter wavelength. Figure 8 shows the relation between amplitude and wavelength taken from the light curves, with a best-fit quadratic curve.

The relation between rest wavelength and amplitude is essentially equivalent to the relation between redshift and amplitude, where one is seeing progressively shorter wavelengths at higher redshift. This is illustrated in the top panel of Fig. 9 which shows the UVX sample with two curves superimposed. The solid line is converted from Fig. 8 and does not appear to follow the trend of the data, but the large scatter makes it hard to construct a convincing test. Nonetheless it suggests that any relation which holds for Seyfert galaxies might have to be modified for quasars. The dotted line is from Di Clemente et al. ([1996]), and shows a very small effect, which does nonetheless follow the flat distribution of the data.

The decrease of amplitude for the smallest redshift and luminosity seen in the top two panels of Fig. 4 has a number of possible explanations. It could be the effect of the underlying galaxy dominating any change in nuclear brightness for low luminosity objects, it could be a consequence of the small optical depth to microlensing at low redshift, or it could be a consequence of the wavelength dependence of variability (Cristiani et al. [1996]). The present dataset is not adequate to settle the question, which is best done by looking at luminous quasars at very low redshift and Seyfert galaxies at high redshift.

All the plots in Figs. 3 and 4 show a trend of decreasing amplitude towards higher redshift or more luminous objects. Although the trend as a function of luminosity is more marked, the old problem of degeneracy makes it hard to say for certain that it is a luminosity effect which is being observed. However, if we look at Fig. 6 where the data are binned in luminosity and redshift we see that while there is no significant trend of amplitude with redshift in either luminosity bin, there is a marked inverse correlation between amplitude and luminosity.

The relation between amplitude and luminosity is in agreement with that found in earlier work (Hook et al. [1994]; Hawkins [1996]; Cristiani et al. [1996]) and may well turn out to be a useful way of distinguishing between various schemes for quasar variability. The evidence for a constant amplitude with redshift is more debatable. It would appear to be consistent with the early claim of Hook et al. ([1994]) for a weak anti-correlation with redshift which they ascribed to degeneracy with luminosity. It is also in agreement with the results of Cristiani et al. in the observer's frame. When they correct their structure function to the quasar rest frame they inevitably imprint a positive correlation between their variability parameter and redshift.

To investigate this dependence of amplitude on redshift for the present sample, Fig. 10 shows the epoch at which quasars achieve their maximum amplitude as a function of redshift. Apart from very low redshifts (z < 0.3) this relation is flat, implying that at least over the 21 years of the present dataset, time dilation effects will not bias the measurement of amplitude. Since the conclusions of Cid Fernandes et al. ([1996]) are largely based on a sample for which a time dilation correction has been applied, it is not feasible to make a direct comparison with the present work. However, it appears that the main difference between their results and those of Hook et al. is in the definition of a variability parameter and the method of analysis (both papers are based on the SGP sample).

There are perhaps three currently discussed schemes for quasar variability. The least well constrained is the accretion disk model, where instabilities are propagated across the disk leading to variation in light. The details of this approach have proved hard to work out, especially in the context of the constraints imposed by existing observations, but it does not seem to lead to an inverse correlation between amplitude and luminosity. An interesting recent attempt to model variation on the basis of accretion disk instabilities by Kawaguchi et al. ([1998]) may provide a means for producing the observed variations. It does however appear to predict variations which are either too asymmetric or of too small an amplitude to be consistent with the current observations. The timescales which they predict are also rather short, around 200 days for reasonable input parameters, and much shorter than the observed timescale of a few years.

An alternative approach, developed by Terlevich and his collaborators, accounts for the variation by postulating that the quasar is powered by a series of supernova explosions. Qualitatively, this model can account for the observed relation between luminosity and amplitude, and works quite well for Seyfert galaxies (Aretxaga & Terlevich [1994]). However, for quasars (Aretxaga et al. [1997]) large numbers of supernovae are required to achieve the luminosity, which results in smaller variations. For example, even a relatively modest quasar with absolute magnitude $M_{B} \sim -26$ would need some 300 type II supernovae per year to power it, which given typical decay times would lead to very little variation at all. This clearly conflicts with observations in this paper which show that most quasars vary by around 0.5 to 1 magnitude on a timescale of a few years.

The third way of explaining quasar variability is to invoke microlensing. This approach has been explored in several recent papers (Hawkins [1993], [1996]; Hawkins & Taylor [1997]), and seems to account well for a number of statistical properties of quasar light curves. It can also explain the inverse correlationi between luminosity and amplitude in a natural way. It is well known that when a point source is microlensed by a population of compact bodies of significant optical depth, the lenses combine non-linearly to form a caustic pattern which produces sharp spikes in the resulting light curves (Schneider & Weiss [1987]). As the source becomes comparable in size with, or larger than, the Einstein radius of the lenses the amplitude decreases (Refsdal & Stabell [1991]). Thus if one assumes a uniform temperature for quasar disks, and that the luminosity is determined by the disk area, the larger more luminous disks will be amplified less. Using a relation between source size (in terms of Einstein radius) and amplitude given by Refsdal & Stabell ([1991]) (Eq. 1), one can thus derive a relation between amplitude $\delta m$ and absolute magnitude M of the form:

$$\delta m = 10^{0.2(M+c)}$$

where c is a constant. The bottom panel of Fig. 9 shows a plot of amplitude versus absolute magnitude for the UVX sample with this relation superimposed. The constant c was adjusted to allow the curve to track the upper envelope of the points, which has the effect of defining the quasar disk size. This ranges from 1.1 Einstein radii for M_B = -23 to 11 for M_B = -28. The points scatter downwards from the upper envelope because the quasars do not necessarily attain their maximum possible amplitude. Although one cannot say that the curve provides a fit to the data, the trend is certainly well represented.

The possibility of a distinct population of large amplitude, low luminosity quasars suggested by Fig. 7 may also be used to test the models of quasar variability. Again, an accretion disk provides no obvious mechanism for such an effect. The Christmas Tree model certainly does imply large amplitude variations for low luminosity objects, where each event is of comparable brightness to the nucleus itself. Perhaps the most natural explanation comes from microlensing, where the nucleus of low luminosity quasars would plausibly become very small compared with the Einstein radii of the lenses, resulting in large amplifications from caustic crossing events.

Figure 8: Relation between amplitude and wavelength for the Seyfert galaxy NGC 5548 from IUE data published by Clavel et al. (1991)

Figure 9: Plots of amplitude versus redshift and luminosity for UVX selected quasars. In the top panel the solid line represents the relation from Fig. 8, with wavelength converted to redshift; the dotted line is the model from Di Clemente et al. (1996)

Figure 10: Plot of the epoch at which the largest magnitude difference is achieved in quasar light curves at different redshifts. Poisson error bars are shown based on the number of objects in each bin