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

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4. Discussion

The decrease of the variability amplitude with increasing wavelength observed in the present QSO sample is consistent with the results obtained for individual objects at low redshift by Edelson et al. (1990), Kinney et al. (1991), Paltani & Courvoisier (1994). Is this variability-frequency correlation sufficient to account for the positive correlation between redshift and variability found in Paper I?

Let us assume a simple power-law representation of a typical QSO spectrum (Giallongo et al. 1991, Di Clemente et al. 1996):


where [FORMULA] is the flux at an observed wavelength [FORMULA], [FORMULA] an arbitrary wavelength and [FORMULA] the spectral index. The variability can then be modeled as a change in the spectral index:


where [FORMULA] is a function of time. In this way the flux variations are idealized as changes in the slope of the power-law "hinged" on the point of wavelength [FORMULA], that we imagine somewhere redwards of the B and R bands.

Converting in magnitudes,


we obtain the following expression for the magnitude variations:


Applying the same equation to the B-band, we can write:


From Eq. 7 we would expect a decreasing ratio [FORMULA] with increasing redshift. In the present data we observe, if any, a very small decrement of [FORMULA] with the redshift (from [FORMULA] to [FORMULA] for two sub-samples with redshift [FORMULA] and [FORMULA] respectively).

Introducing the value of [FORMULA] at a [FORMULA] reported in Sect. 3 in Eq. 7, allows us to put a lower limit to the wavelength [FORMULA] Å.

If we turn back to Eq. 6, we can see how this lower limit to [FORMULA], together with the typical observed RMS for the variability [FORMULA] and [FORMULA] of 0.35 and 0.28 mag, respectively (see Fig. 2), puts an upper limit to the RMS variation of the spectral index [FORMULA].

This value can be compared with the typical dispersion in the spectral indices observed in various QSO samples: Pei et al. (1991) have obtained [FORMULA] in a control sample of 15 QSOs, Francis (1993) observed [FORMULA] in a much larger sample, showing also a systematic hardening of the spectrum with increasing redshift. In this way the dispersion in the observed spectral indices of the QSO population can be only partially accounted for by variability and significant intrinsic differences between QSO and QSO have to be invoked (but see also the dispersion [FORMULA] estimated by Elvis et al., 1994).

Putting in Eq. 6 our best estimates for [FORMULA] and [FORMULA] Å, we obtain for the typical RMS variability in the B band 0.26 and 0.33 mag at [FORMULA] and [FORMULA], respectively and 0.22 and 0.30 in the R band at the same redshifts, consistent with the observations of Paper I.

The dependence of the variability time scale on the wavelength is a standard prediction of the accretion disk model. The temperature of the disk decreases with the radius, and the dynamical, thermal and viscous time scales all increase as [FORMULA] (see Baganoff & Malkan 1995 and references therein). In particular, in the standard model, involving instabilities in a thin accretion disk around a supermassive black hole, the intrinsic characteristic time scale should vary as [FORMULA]. More in general, if the source is thermal and if the flux variability is due to temperature changes, the observed wavelength dependence can be naturally produced (Paltani & Courvoisier 1994). The only requirement concerns the turnover in the spectral energy distribution that should not be located at such a high frequency that the spectral slope in the observed band becomes independent of the temperature (e.g. the Rayleigh-Jeans domain for a black body spectrum). As remarked by Di Clemente et al. (1996), if brighter objects are on average hotter and the variability is due to temperature changes, one would expect a negative correlation between amplitude of the variability and luminosity, as observed. Assuming for example a spectrum of black body or thermal bremsstrahlung, the spectral turnover of brighter objects is progressively shifted at higher frequencies producing progressively smaller flux changes, from [FORMULA] for [FORMULA], to [FORMULA] for [FORMULA].

The broad-band variability in the starburst model (Aretxaga et al. 1996 and references therein) is defined by the superposition of a variable component, supernova explosions (SNe) generating rapidly evolving compact supernova remnants (cSNR), and a non-variable component, a young stellar cluster and the other stars of the galaxy. The problem of predicting the wavelength dependence of the variability can be reduced to computing the spectra of the variable and non-variable components, and their relative luminosities. The spectrum of the non-variable component has been predicted to show a [FORMULA] with [FORMULA] (Cid Fernandes & Terlevich 1995). The variable/non-variable relative luminosities can also be estimated on the basis of stellar evolution (Aretxaga & Terlevich 1994). Much more difficult is to predict the optical/UV spectrum of the variable component. It is expected to be harder than the non-variable component with a [FORMULA] and [FORMULA]. We can model the total QSO flux according to a Simple Poissonian model (Cid Fernandes et al. 1996) as


where [FORMULA] is the non-variable fraction of the total flux and [FORMULA], [FORMULA] are the slopes of the non-variable and variable components. Indicating with [FORMULA] and [FORMULA] the effective wavelengths of the B and R-band and putting for simplicity [FORMULA], we obtain for the ratio between the B-band and R-band variability:


with [FORMULA].
For example, with [FORMULA] and [FORMULA] respectively (Cid Fernandes et al. 1996), the ratio between the B-band and R-band variability is expected to be about 1.4 at [FORMULA], tending to 1 for increasing redshift, independent on the intrinsic luminosity of the QSO. The above reported slight tendency for a decreasing [FORMULA] ratio with increasing redshift is compatible with this model. However, an increase of the [FORMULA] with the absolute luminosity is also observed. We find the values [FORMULA] and [FORMULA] for two subsamples with [FORMULA] ([FORMULA]) and [FORMULA] ([FORMULA]), respectively. This suggests that more general situations have to be envisaged in which the fraction on the non-variable component and/or the pulse properties (e.g. the spectral distribution) depend on the global luminosity of the object.

Recently Hawkins (1996) has proposed that the variability of nearly all the QSOs (except the very low redshift ones) is produced by microlensing. A non-achromatic behaviour can be accommodated in this kind of model. For a source comparable in size to the Einstein ring of the lens, with a quasar disk redder at larger radii, the bluer compact core would produce a larger amplitude of the variations in the blue passband whereas the larger extent of the red image would cause a small amplitude variation, albeit with a longer duration, in the red. However it cannot be denied that achromatic light variations would have been one of the most significant characteristics of the microlensing model and the present observation of the wavelength dependence make it less appealing/compelling. In particular the superposition of the two competing effects of time dilation and correlation of the variability with frequency, i.e. redshift, accounts naturally, also in the accretion disk and in the starburst scenarios, for the observed lack of correlation in the observer's rest frame between the variability time scale and the redshift, otherwise claimed as an evidence in favor of the microlensing model.

Future detailed comparisons between simulated and observed light curves will be fundamental to disentangle among these three scenarios and to start exploring the parameter space of each model.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998