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Astron. Astrophys. 329, 863-872 (1998)

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4. Definition of the variability parameters

From the PCA results, it is possible to derive line and continuum variability parameters. The mean continuum [FORMULA] and line [FORMULA] fluxes are derived from the component k normalized by its mean flux [FORMULA] (Fig. 1a). The continuum [FORMULA] and line [FORMULA] flux dispersions are estimated from the principal component normalized by its flux dispersion [FORMULA]. This derivation is motivated by the theoretical argument exposed in Sect. 4.1. The determination of the continuum and the method of line integration is described in Sect. 4.2.

4.1. Theoretical basis

In the following, the temporal average [FORMULA] and the temporal dispersion [FORMULA] of the n spectra [FORMULA] measured at time [FORMULA] ([FORMULA]) are defined as


where [FORMULA]. At each [FORMULA], the total spectrum is the sum of a continuum and a line contribution [FORMULA]. This relation combined with Eqs. (1) and (2) gives


Eq. (4) shows that contrary to the mean spectrum, the variance spectrum can not be simply decomposed into a continuum and a line contribution.

It would however be possible to decompose the dispersion spectrum (i.e. the flux dispersion in each [FORMULA]) into a continuum and a line contribution if


because this would imply that [FORMULA] via Eq. (2) and thus that Eq. (4) reduces to


Equation (5) would correspond to substitute a time independent response function [FORMULA] for actual convolution with the transfer function. Since there is a significant lag between the line and the continuum variations, Eq. (5) is incorrect and thus we can not derive the line variability [FORMULA] from the dispersion spectrum via Eq. (6).

However, we have shown in Sect. 3.2that the principal component is correlated with the continuum at zero lag. This is not only verified in NGC 5548, but is a direct consequence of the way we applied the PCA. Therefore, Eq. (5) is correct for the line-part in the principal component and thus, according to Eq. (6), we can estimate the line variability [FORMULA] by integrating the line-part in the principal component normalized by its flux dispersion [FORMULA]. This estimation gives a lower limit to the actual line variability, since it does not consider the delayed line variations, but has the advantage of excluding spurious variability due to noisy spectra.

4.2. Continuum and emission-line determination

On the basis of Eqs. (3) and (6), we can derive both the mean and the dispersion of the line flux by integrating the emission-line above a defined continuum in the appropriate component. To do this, we first defined five 30 Å bands that are usually free of emission or absorption features in the rest frame of the source at [FORMULA] 1120-1150 Å, 1320-1350 Å, 1430-1460 Å, 1700-1730 Å, 1810-1840 Å. The two last bands are both outside of the component domain [FORMULA] 1229-1948 Å in 3C 273 and GQ Comae, due to their higher redshift. For these two objects, we had to define a new band from [FORMULA] 1665 Å to the end of the component, despite the possible He II [FORMULA] 1640 and O III ] [FORMULA] 1663 contamination at those wavelengths. Doing so, there are always four bands that constrain the continuum, whatever the redshift of the object is.

The continuum is estimated by fitting a straight line through the points in these four bands. We used the ordinary least-squares regression "OLS(X [FORMULA] Y)" of Isobe et al. (1990) with its uncertainties on the slope and on the intercept. To take into account systematic errors, the uncertainty on the continuum is assumed to be twice the uncertainty on the fit that was calculated, as all uncertainties in this paper, according to the general formula


where f is a function of the n variables [FORMULA] with uncertainties [FORMULA].

The Ly [FORMULA] and C IV emission-lines are integrated above the continuum in the velocity ranges [ -15 000 ; [FORMULA] 15 000 ] and [ -15 000 ; [FORMULA] 12 000 ] respectively, as far as enabled by the redshift of the object. We choose these ranges in order to include the N V [FORMULA] 1240 line in the Ly [FORMULA] [FORMULA] 1216 integration and to exclude the He II [FORMULA] 1640 and O III ] [FORMULA] 1663 lines from the C IV [FORMULA] 1549 integration. In the following, the Ly [FORMULA] line actually refers to the Ly [FORMULA] [FORMULA] N V line. The uncertainty on the line contribution is determined by integrating the line above the fitted continuum plus and minus its uncertainty. The continuum parameters are arbitrary defined at [FORMULA] 1350 Å.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997