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

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3. Discussion

3.1. Absence of a secondary image

As can be seen in Table 2, application of the Tully-Fisher relation to the absolute luminosity of the DLA galaxies implies that the impact parameter of the line-of-sight to the QSO never falls within twice the value of the galaxy Einstein radius. Hence, we do not a priori expect any secondary lensed QSO images, which is confirmed by the HST/WFPC2 observations.

However, the total luminosity of the galaxies might be underestimated, because of extinction due to the presence of diffuse dust in the galaxies themselves (self-extinction), especially in the rest-frame B band in which our galaxies are observed. Since the Tully-Fisher relation given above in Eq. 10 was determined from a sample of local galaxies whose B magnitudes were corrected for self-extinction, even for inclined systems, we are led to use a global extinction.

In particular, a significant self-extinction correction might be applied to a DLA galaxy absolute B magnitude if it presents a large inclination. Thus, from the observed values, we can calculate the self-extinction which is necessary to give a real Einstein radius large enough to lead to [FORMULA] and the formation of a double image. For four absorbing galaxies, this value of the self-extinction is larger than 3 magnitudes, which we consider to be implausible, given that such values have not been observed in samples dedicated to the study of self-extinction in galaxies (Xu et al. 1997).

For the three remaining galaxies (toward 3C 196, Q 1209+107 and MC 1331+170), the self-extinction necessary to lead to a value of [FORMULA] compatible with multiple imaging is smaller, ranging from 1 magnitude for 3C 196 to 1.6 and 1.8 magnitude for the last two. The last two values are comparable to the highest ones detected in the sample of Xu et al. (1997). On the other hand, we consider unlikely that the galaxy responsible for the DLA in the spectrum of 3C 196 is affected by 1 magnitude of self-extinction: the galaxy is seen face-on and its redshift is [FORMULA], so that the F702W band is actually centered at 4900Å in the galaxy rest-frame. Furthermore, the galaxy extent as determined in the F450W image, which roughly corresponds to near UV in the galaxy rest frame, is nearly equal to the one seen in the F702W filter. The self-extinction correction is thus probably small in this object.

In summary, the self-extinction required to produce multiple imaging in these 3 systems are unlikely to be significant. However, they are not unrealistic. Let us assume, therefore, that multiple imaging is taking place. Consequently, the absence of detected secondary images allows us to provide some constraints on the extinction in these galaxies along the particular lines-of-sight to the QSO images.

Indeed, if multiple imaging is taking place for 3C 196, Q1209+107 and MC1331+170 and if the SIS model is an adequate representation of the matter distribution within the lens, then the observations could reveal two images with a magnitude difference [FORMULA] smaller than [FORMULA], where [FORMULA] represents each of the three quasars. From Eq. 9, such a situation occurs if




The probability [FORMULA] of not detecting a secondary image in the system i is thus given by


This last relation assumes that the sources are uniformly distributed behind the lenses: we neglect the amplification bias, which tends to select systems with small [FORMULA] and, consequently, to strengthen the following conclusion.

The probability [FORMULA] of not detecting a secondary image in any of the 3 systems is therefore


which is represented as a solid line in Fig. 1.

[FIGURE] Fig. 1. Probability P of not detecting any secondary image for the three selected quasars (see text) as a function of the differential extinction [FORMULA] toward the three sight-lines. The horizontal dashed line represents the 3[FORMULA] (or 0.25%) level, which we choose as a threshold

We can see that the non-detection hypothesis, i.e. the observations, is ruled out with a confidence level larger than 3 sigma if the differential extinction is smaller than 3.9 magnitudes on each sightline. But an extinction larger than 3.9 mag is only expected in very dense clouds, whose covering factor is very small.

Furthermore, Table 2 shows that the limits on the mass-to-light ratios inferred from lensing and geometrical constraints alone are usually an order of magnitude larger than the ones derived from an application of the Tully-Fisher relation. If a secondary image was hidden due to extinction, it would imply that the mean [FORMULA] ratio of galaxies is much higher than previously thought.

For all these reasons, we maintain our conclusion that the observed configurations of DLA absorbers are not likely to give rise to multiple images.

3.2. Biases due to gravitational lensing

A quasar located at an impact parameter slightly larger than the Einstein radius can still have its apparent luminosity increased by a factor A, due to gravitational lensing amplification. Column 10 of Table 2 presents this factor estimated for each of our systems: as one can see, it is always smaller than 0.3 mag., with an average value of 0.12 mag. The values for a [FORMULA] Universe are only slightly larger (by 3 to 6%).

As we now have an estimated value for the amplification, instead of estimating statistical lensing effects, we can actually compute lensing effects for each QSO individually. We used the approach of Narayan (1989) to evaluate the excess of quasars close to foreground galaxies. This method takes into account both the amplification bias and the by-pass effect. We point out, however, that it aims at estimating the excess number of quasars in the vicinity of galaxies, not the excess of quasars in the vicinity of galaxies giving rise to DLA systems in quasar absorption spectra. However, we prefer this method due to uncertainties in the determination of the inclination of the galaxies and other observational variables, and because it is good enough for our purpose.

The mean excess of quasars close to the DLA galaxies is found to be equal to a factor [FORMULA] for a [FORMULA] Universe. Therefore, if these quasars were drawn out of a magnitude limited sample, 14% of the quasars that present a DLA line in their spectrum would have been observed because of gravitational lensing (this calculation should be considered as a mere exercise, as the sample is not complete). This value is close to the 18% obtained by the method described in SCS (cf. Sect. 1).

On the other hand, if these quasars were drawn out of a volume limited sample, the magnification bias would be irrelevant: only the by-pass effect would be acting. In this case, the fact that the observed impact parameters [FORMULA] are larger than [FORMULA] leads to an underestimate of [FORMULA] by about 6%, based on the E+GH model of SCS.

3.3. Comparison of the [FORMULA] ratio constraints with other methods

We have plotted in Fig. 2 the upper limits to the model-independant mass-to-light ratios [FORMULA] we derive for the seven galaxies. The dotted and dashed lines show the power laws representing the overall mass-to-light ratio in spiral and elliptical galaxies, respectively, as estimated from Bahcall et al. (1995): this review presents recent results obtained on the mass-to-light ratio using the classical methods (H I emission, X-ray emission, motion of dwarf satellites...).

[FIGURE] Fig. 2. Upper limits on the mass-to-light ratio [FORMULA] derived from the lensing properties of damped Ly[FORMULA] absorbing galaxies. By comparison, we also show the results from classical methods (cf. Bahcall et al. 1995). They indicate that the mass-to-light ratio as a function of radius can be represented by power laws, plotted here as dotted and dashed lines for the spiral and elliptical galaxies, respectively.

As can be seen, the strongest limits set by gravitational lensing, which are obtained for the three spirals of the sample, are an order of magnitude above the values obtained from other methods. Thus, although they are compatible with other estimations, these observations do not allow to efficiently constrain the value of the hidden mass in the galaxies responsible for the DLAs.

3.4. Future work

We note that the published surveys to determine the cosmological density of neutral hydrogen [FORMULA] at [FORMULA] have been carried on using samples of quasars that are generally brighter than all the quasars on which the present study is based: only 15% of the quasars in the IUE survey (Lanzetta et al. 1995) have a B magnitude fainter than the quasars presented here; half of the Rao et al. (1995) quasars (followed up by Rao & Turnshek 2000) are brighter than the brightest quasar in our sample, and none are fainter than the faintest quasar in this same sample (some quasars are actually used both in this paper and in Rao et al.'s). These surveys may thus be more strongly affected by the magnification bias than the sample presented here; consequently, the results of this paper should not be interpreted as meaning that lensing effects have negligible effects on surveys of DLAs at [FORMULA].

In order to have a better constraint on the lensing effects in DLA surveys, we have carried a HST-NICMOS survey of 13 bright quasars whose spectra present a DLA system at low redshift, including the sample presented in this paper.

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Online publication: December 5, 2000