Astron. Astrophys. 344, 721-734 (1999)

## 4. Calculations

Following K96, we use the Hinshaw & Krauss (1987) softened isothermal sphere model for modeling the light deflection properties of the lens galaxies. For this model, the lens equation reads

where x is the angular position in the lens plane, y the angular position in the source plane, , denotes the one-dimensional velocity dispersion of the dark matter, s denotes the core radius, is the angular core radius and , and denote the angular size distances between the observer and the lens galaxy, the observer and the source and the lens galaxy and the source, respectively. Still following K96, we model the distribution of elliptical and lenticular lens galaxies using Schechter functions with constant comoving density

() and slope

The lens galaxy luminosities are converted to the dark matter velocity dispersions of the softened isothermal lens model by means of Faber-Jackson type relations,

where

and

The core radii of the softened isothermal lens model are varied with the dark matter velocity dispersions according to

where and . We consider elliptical and lenticular lens galaxies only. For the number-magnitude counts of quasars, we adopt the best-fit model from K96. We neglect here evolution, dust and other possible systematic effects and refer the reader to K96 for a discussion.

In our first calculations we apply Eq. (8) and compute the a priori likelihood

and the posterior probability density functions

and

in the limit where all nuisance parameters take precisely their mean values. To obtain an impression of the consequences of neglecting the uncerntainties of the nuisance parameters, in our second calculation we increase the value of the most uncertain nuisance parameter, , by two standard deviations.

For the computation of the innermost integral on the right side of Eq. (3), we consider the detectability of images in pairs: If the separation between the two closest images - these are always images 2 and 3, counting from the outside in - is more than the lower limit of the survey resolution limit , we define the image separation and flux ratio for the purpose of sample selection based on the two brightest images, usually 1 and 2. Otherwise we construct one image from the combined fluxes and flux-weighted positions of images 2 and 3 and define the image separation and flux ratio for the purpose of sample selection based on this combination image and image 1.

In general, if the separation between images 1 and 2 is too large for the survey and the separation between images 2 and 3 is large enough, then the image separation and flux ratio for the purpose of sample selection should be based on images 2 and 3. However, the present surveys are sensitive to the largest separations due to isolated galaxies, so this case doesn't need to be addressed in this paper (i.e. implementing it would lead to the same results in the present case).

For the calculation of the probabilities the function selects only those image configurations whose separation is per cent of the observed separation .

Each of the three integrals on the right side of Eq. (3) is approximated to an accuracy better than 0.004 by a family of recursive monotone stable formulae (Favati et al. 1991a,b).

© European Southern Observatory (ESO) 1999

Online publication: March 29, 1999