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Astron. Astrophys. 326, L9-L11 (1997)

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3. Lensing cross-sections

We now estimate the total lensing cross-sections of galaxies with redshifts ranging from 0 to [FORMULA] for the distant sources like quasars. For simplicity, we still employ the singular isothermal sphere for the galaxy matter distribution and the evolutionary scenario of galaxy merging. The lensing cross-section for magnification greater than µ by a single galaxy at [FORMULA] is simply [FORMULA] where [FORMULA] is the Einstein radius with [FORMULA], [FORMULA] and [FORMULA] being the angular diameter distances to the galaxy, to the distant source and from the galaxy to the source, respectively. The total lensing cross-section by all the galaxies is

[EQUATION]

in which

[EQUATION]

and

[EQUATION]

where the symbols with a hat of tilde are the corresponding parameters in units of [FORMULA], and [FORMULA] is the proper distance within dz of z. Except for the factor of [FORMULA], eq.(4) identifies the eq.(6) of Rix et al. (1994) for [FORMULA], in which they concluded that the total optical depth to multiple images is quite insensitive to merging. This can be easily shown in terms of eq.(4) by noticing that [FORMULA]. Taking [FORMULA] and the galaxy morphological composition [FORMULA] (Postman & Geller 1984) and utilizing the numerical result of F found by Fukugita & Turner (1991) and the result of T found by Turner et al. (1984), we have

[EQUATION]

A straightforward computation yields [FORMULA] and [FORMULA], i.e., the total lensing cross-sections of [FORMULA] by all the galaxies even to [FORMULA] cannot cover the whole sky at all ! It is important to note that this conclusion is independent of the limiting magnitudes of the surveys which may reveal a remarkably high surface density of galaxies. Also, our computation has probably overestimated the lensing cross-sections of galaxies in the sense that a biasing factor of [FORMULA] between the velocity dispersion of stars and of dark matter particles is employed by Fukugita & Turner (1991) in obtaining T for E/S0 galaxies. If such a correction of velocity biasing is unnecessary (e.g. Kochanek 1994), the total lensing cross-section of galaxies [FORMULA] reduces to [FORMULA] for [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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