Elliptic Mass Distributions versus Elliptic Potentials in Gravitational Lenses

We present analytic expressions for deflections by gravitational lenses made of homoeoidal elliptic mass distributions (EMDs) with a variety of radial profiles. We study in detail pseudoisothermal elliptic mass distributions (PIEMDs) and give analytic formulae for their time delay, deflection, magnification matrix, sizes of caustics and critical lines, and parameters of umbilici. The analytic tractability makes elliptic mass lenses almost as convenient as lenses with elliptic potentials (EPs), both for computer simulations and for qualitative analysis. We compare the topography, critical behavior and imaging properties of PIEMDs and of lenses with a pseudoisothermal elliptic potential (PIEP). The two lenses are similar in many aspects, however: PIEMD can represent masses of any elongation, while physically meaningful PIEP lenses are limited to mass distributions with axis ratios b/a ≳ 0.5 and therefore cannot represent almost half of the galaxies and clusters in the universe. A notable difference between the imaging properties of the two lens models is that PIEMDs have much more freedom to form caustics with naked cusps. Naked cusps are involved in the formation of some of the giant arcs and in statistics of multiple QSO images. To study statistics of multiple imaging, we simulated altogether ∼2 × 106 multiple QSO cases imaged by isolated galaxies, identified seven types of image configurations, and obtained predictions for relative frequencies of these types for a variety of sampling conditions. The sampling conditions include thresholds for resolution, for next flux, and for flux ratio between the brightest and the faintest resolved images. Next, we adopted a requirement to be satisfied by our predictions for relative frequencies of two- and three-image cases, to fit observations. This requirement leads to constants on the central surface mass density of galaxies, which are stronger for PIEMDs than for PIEP models, largely because of the triple images of sources inside naked cusps. Finally, we demonstrate that galaxies responsible for QSO lens cases with four or five images are biased to larger ellipticities than galaxies in general. In addition to the simplest pseudoisothermal example, combinations of elementary analytic lens models with elliptic mass distributions have a considerable freedom to approximate gravitational lenses with variable radial profiles and ellipticities.