The effective cross-sections of a lensing galaxy: singular isothermal sphere with external shear.

We present numerical studies of the imaging and caustic properties of the singular isothermal sphere (SIS) under a wide range of external shear (from 0.0 to 2.0). Using a direct inverse-mapping formula for this lensing system,we investigate various lensing properties for both low-shear (i.e. γ < 1.0) and high-shear (i.e. γ > 1.0) cases: the image separations, the total or individual magnifications, the flux ratios of two images, the maximum number of images and the lensing cross-sections. We systematically analyse the effective lensing cross-sections of double-lensing and quadruple-lensing systems, based on the radio luminosity function obtained by the Jodrell–VLA Astrometric Survey (JVAS) and the Cosmic Lens All-Sky Survey (CLASS). We find that the limit of a survey selection bias (i.e. between brighter and fainter images) preferentially reduces the effective lensing cross-sections of two-image lensing systems. By considering the effects of survey selection bias, we demonstrate that the long-standing anomaly over the high quads-to-doubles ratios (i.e. 50–70 per cent for JVAS and CLASS) can be explained by the moderate effective shear of 0.16–0.18, which is half that of previous estimates. The derived inverse-mapping formula could make the SIS + shear lensing model useful for galaxy-lensing simulations. [ABSTRACT FROM PUBLISHER]