ANALYTIC RESULTS FOR THE GRAVITATIONAL LENS STATISTICS OF SINGULAR ISOTHERMAL SPHERES IN GENERAL COSMOLOGIES

The natural cosmological distance for the analysis of gravitational lens statistics with a singular isothermal sphere model for the lens potential is the proper motion distance. By using the proper motion distance we derive analytic expressions for the cross-sections, averages of the image separations, and averages of the distances to the lens for both point-like and resolved sources that depend only on the total density of the cosmological model and the proper motion distance to the source. The ratio of the square of the mean image separation to the mean square image separation for the lenses in a survey is a robust measure of survey completeness. The test shows that the sample of lenses in the Snapshot lens survey of bright quasars is complete within the limits of statistical errors, while the sample of lenses in the MIT-Greenbank radio survey for gravitational lenses is only 50 per cent complete. In a complete sample, the mean and mean square image separations can be used to estimate the velocity dispersion scale of galaxies. The Snapshot survey sample shows that the velocity dispersion scale of L* galaxies is in the range 240 km s-1 less than or similar sigma* less than or similar 310 km s- 1 and the uncertainty is comparable to the uncertainties from dynamical studies. The probability of lensing extended radio sources discriminates more strongly between low-density universes and the Einstein-de Sitter model than does the probability of lensing unresolved sources such as quasars.