The magnification invariant of circularly-symmetric lens models

In the context of strong gravitational lensing, the magnification of an image is crucially important for constraining various lens models. For several commonly used quadruple lens models, the magnification invariants, defined as the sum of the signed magnifications of images, have been analytically derived when the image multiplicity is a maximum. In this paper, we further study the magnification of several disk lens models, including (a) exponential disk lens, (b) Gaussian disk lens, (c) modified Hubble profile lens, and another two of the popular three-dimensional symmetrical lens models, (d) NFW lens and (e) Einasto lens. We find that magnification invariant exists for each lens model. Moreover, our results show that magnification invariants can be significantly changed by the characteristic surface mass density κc.