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Astron. Astrophys. 363, 837-842 (2000)

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2. Lensing properties of the DLA absorbing galaxy/quasar pairs

The observed characteristics of the seven quasar/galaxy pairs are presented in Table 1. We present three different methods to evaluate the lensing properties of these seven configurations: (i) a model-independent formalism; (ii) a formalism which assumes a Singular Isothermal Sphere (SIS) lens model, only based on the geometry of the systems (i.e. redshifts of the quasar and the galaxy in each association and their angular separation); (iii) a formalism which assumes a SIS model whose velocity dispersion is determined from the observed luminosity of the DLA galaxies and the Tully-Fisher relation.


Table 1. Characteristics of the seven galaxy/quasar pairs: quasar emission redshift [FORMULA], apparent magnitude in the F450 filter (similar to the B band), DLA line redshift [FORMULA], neutral hydrogen column density [FORMULA], angular impact parameter [FORMULA], linear projected impact parameter [FORMULA], k -corrected absolute magnitude [FORMULA] and its estimated 1[FORMULA] error of the absorbing galaxy (see Boissé et al. 1998 for details). The last column provides the galaxy morphological type, derived from the WFPC2 images as precisely as possible.

The values of the angular impact parameter [FORMULA] and absolute luminosity [FORMULA] of the absorbing galaxy image were obtained by Le Brun et al. (1997) and Boissé et al. (1998).

No secondary lensed QSO image is detected in any of the systems presented here at more than 0.3" from the quasar, down to a limiting magnitude of [FORMULA], i.e. 6.8 to 9 magnitudes fainter than the main observed QSO image. The absence of any secondary image either means that the lensing configuration is indeed not capable to produce multiple imaging, or that the apparent luminosity of the secondary image is too faint to be detected, possibly due to extinction by dust.

2.1. (Nearly) model-independent constraints

Subramanian & Cowling (1986) showed that, for a spherical mass distribution, with surface density decreasing from the center to the outer parts, as expected for individual galaxy halos, a sufficient and necessary condition to have multiple images is that the central value [FORMULA] is larger than a critical value [FORMULA] defined by:


and thus independently of the model chosen for the halo. In this relation, c is the speed of light, G the gravitational constant, [FORMULA], [FORMULA] and [FORMULA] the angular-diameter distances between the observer and the source (QSO), the observer and the lens, the lens and the source, respectively. Furthermore, they show that the absence of a secondary lensed QSO image ensures that [FORMULA] still for a spherical mass distribution. They conjecture that this result is also valid for centrally peaked elliptical lenses. We therefore conclude that the absence of a secondary QSO image at a separation larger than [FORMULA] in our sample ensures that [FORMULA] over such an angular scale.

On the other hand, for a lens with circular symmetry, the mean surface density within the Einstein radius [FORMULA] is equal to [FORMULA] (cf. Schneider et al. 1992); in addition, the location of the main image is always such that its angular separation from the lensing galaxy [FORMULA]. In particular, this relation is true even in the case of multiple images with any secondary image being hidden due to extinction by dust. Consequently, we can derive an upper limit on the projected mass [FORMULA] enclosed in a disk of radius [FORMULA] centered on the galaxy, as


and thus also on the average M/L ratio within [FORMULA].

No constraint can be set on the amplification A of the QSO image.

2.2. SIS lens model with only geometrical constraints

We here assume that the distribution of matter within each galaxy can be described as a singular isothermal sphere (SIS), whose volume mass density [FORMULA] is given as a function of the distance r to the galaxy center by:


where [FORMULA] is the 1-dimensional velocity dispersion of the SIS. As a consequence, the total projected mass enclosed in a disk of radius r is


We then compute the angular Einstein radius for each quasar/galaxy configuration, by


In the case of an SIS, the observed images are located at:


where [FORMULA] is the true (unobserved) position of the source.

Multiple (double) imaging only occurs for [FORMULA], in which case the separation between the two images is [FORMULA], and their amplification is [FORMULA]. If [FORMULA] only one image is formed and its amplification is [FORMULA].

Inversely, even in the case of an isothermal sphere with a core-radius, Narayan & Schneider (1990) showed that the condition [FORMULA] is sufficient to avoid the formation of multiple images. Using only the geometry of the system, we can set the following constraint on the mass of the DLA galaxy. We successively consider the cases of single and double image systems:

2.2.1. Single image system

If only one image is present, [FORMULA]. Using Eq. 4 and inverting Eq. 5 imply that:


The only constraint on the amplification affecting the observed image is [FORMULA].

2.2.2. Double image system

Suppose now that two images are actually produced by the lens. In order to explain the observations, we have to assume that only the brightest image is detected, while the faintest one is affected by dust extinction by an amount [FORMULA] so that its apparent magnitude is fainter than the limiting magnitude of the corresponding WFPC2 observations.

The angular separation between the observed image and the center of the lensing galaxy [FORMULA] is such that [FORMULA]. Consequently, the total projected mass located within a disk of radius [FORMULA] is constrained by:


Let us define [FORMULA] as the difference between the limiting magnitude of the corresponding WFPC2 frame and the magnitude of the main observed QSO image: [FORMULA]. We then have the following relations between the amplifications of the two images I and B of the quasar:


where [FORMULA] and [FORMULA] are the extinction affecting the main and secondary images, respectively.

Since the secondary image can be very close to the center of the deflecting galaxy, it might be extremely extinguished, and thus undectable on our data.

2.2.3. Summary

Only using geometrical quantities with a SIS lens model does not greatly improve the limits based on a model-independent formalism: the value [FORMULA] is within a factor of two the upper limit of the mass of a SIS lens only constrained by the angular separation between the lensing galaxy and the quasar, independently of the fact that multiple images exist or not.

2.3. SIS lens model constrained by the Tully-Fisher relation

In this section, we consider one additional piece of information brought by the HST observations: the luminosity of the galaxies considered to be responsible for the DLA absorption.

If we assume that the 1-dimensional velocity dispersion of the SIS [FORMULA] is equal to the velocity dispersion of matter in the galaxy [FORMULA], which is related to the maximal value [FORMULA] of the rotational velocity of a galaxy by [FORMULA], we can indeed use the Tully-Fisher relation (Tully & Fisher 1977) to derive its value from the galaxy luminosity:


where [FORMULA]  km s-1, [FORMULA] (h is the Hubble constant in units of 100 km s-1 Mpc-1), and [FORMULA]; these values are in accordance with the work by Fukugita & Turner (1991) and SCS. This relation is derived from local galaxies, but most recent studies (see Bershady 1996 for a review) indicate that there is no evolution of the parameters in this relation at intermediate redshifts. Then, even if the value of [FORMULA] is known to evolve with redshift (see e.g. Lilly et al. 1995), this only indicates an evolution of the density of galaxies of a given luminosity, but not of the relation between the luminosity and the velocity dispersion of a given galaxy.

We can then set an upper limit to the projected mass enclosed in a disk of radius [FORMULA] centered on the galaxy, as Eq. 4 becomes


We can also estimate the amplification factor [FORMULA] of the quasar apparent luminosity due to gravitational lensing by the galaxy responsible for the DLA absorption, as [FORMULA] is derived by inserting [FORMULA] in Eq. 5:


The values of [FORMULA] and [FORMULA] derived for each of the QSO-galaxy association are presented in Table 2. As expected, the inferred mass-to-light ratios [FORMULA], also listed in Table 2, are close to the mass-to-light ratios for spiral and elliptical galaxies at the present epoch estimated by various, classical methods (see Bahcall et al. 1997 for a review). This result, discussed further in Sect. 3.3, ensures us that the values obtained for the amplification factor [FORMULA] are probably good estimates.


Table 2. Lensing properties of the seven quasar/galaxy pairs. Symbols are defined in the text. The TF upper-script indicates that the corresponding quantity has been evaluated with the Tully-Fisher relation. Calculations have been done using [FORMULA] km/s/Mpc, [FORMULA], [FORMULA], and no correction for extinction.

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Online publication: December 5, 2000