3. Lensing cross-sections
We now estimate the total lensing cross-sections of galaxies with redshifts ranging from 0 to 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 is simply where is the Einstein radius with , and 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
where the symbols with a hat of tilde are the corresponding parameters in units of , and is the proper distance within dz of z. Except for the factor of , eq.(4) identifies the eq.(6) of Rix et al. (1994) for , 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 . Taking and the galaxy morphological composition (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
A straightforward computation yields and , i.e., the total lensing cross-sections of by all the galaxies even to 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 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 reduces to for .
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998