Critical Curves and Caustics of Triple-Lens Models

Abstract

Among the 25 planetary systems detected up to now by gravitational microlensing, there are two cases of a star with two planets, and two cases of a binary star with a planet. Other, yet undetected types of triple lenses include triple stars or stars with a planet with a moon. The analysis and interpretation of such events is hindered by the lack of understanding of essential characteristics of triple lenses, such as their critical curves and caustics. We present here analytical and numerical methods for mapping the critical-curve topology and caustic cusp number in the parameter space of n-point-mass lenses. We apply the methods to the analysis of four symmetric triple-lens models, and obtain altogether 9 different critical-curve topologies and 32 caustic structures. While these results include various generic types, they represent just a subset of all possible triplelens critical curves and caustics. Using the analyzed models, we demonstrate interesting features of triple lenses that do not occur in two-point-mass lenses. We show an example of a lens that cannot be described by the Chang-Refsdal model in the wide limit. In the close limit we demonstrate unusual structures of primary and secondary caustic loops, and explain the conditions for their occurrence. In the planetary limit we find that the presence of a planet may lead to a whole sequence of additional caustic metamorphoses. We show that a pair of planets may change the structure of the primary caustic even when placed far from their resonant position at the Einstein radius.

Triple-lens Gravitational Microlensing: Critical Curves for Arbitrary Spatial Configuration

“…where φ ∈ [0, π) is a phase parameter (Witt 1990;Daněk & Heyrovský 2015a). The source positions corresponding to the critical curve form the caustic of the lens ζ c , obtained by setting z = z cc in Equation (1).…”

“…Several approaches to solve the problem were summarized in Daněk & Heyrovský (2015a,b). In this work we analyze critical-curve topologies by exploiting the Jacobian contour correspondence pointed out by Daněk & Heyrovský (2015a) for lenses consisting of n point masses. According to the correspondence, a positive-value contour of the Jacobian has the shape of the critical curve of a lens with the same components placed closer together.…”

“…The most intuitive way to characterize the shape of the triangle would be to use two of its angles. Unfortunately, such a parameterization fails to distinguish between different collinear configurations with the third component positioned along the line connecting the first two (Daněk & Heyrovský 2015b). All such configurations would correspond to the same set of angles {0, 0, π}.…”

“…In our previous work we developed techniques for the analysis of general n-point-mass lenses based on the properties of the lens-equation Jacobian (Daněk & Heyrovský 2015a). In Daněk & Heyrovský (2015b) we applied these methods to simple two-parameter triple-lens models and mapped the critical curves and caustics in their respective parameter spaces. Based on these results, we extend the approach further in this work and map the different lensing regimes for a given combination of three masses in an arbitrary spatial configuration.…”

OGLE-2018-BLG-1011Lb,c: Microlensing Planetary System with Two Giant Planets Orbiting a Low-mass Star

“…The lensing behavior of triple-lens systems is qualitatively different from that of binary-lens systems, resulting in a complex caustic structure, such as nested caustic and selfintersections. The range of the critical-curve topology and the caustic structure of the triple lens has not yet been fully explored, making it difficult to analyze triple-lens events (Daněk & Heyrovský 2015. As a result, there are some events suspected to be triple-lens events, but plausible models have yet not been proposed, e.g., OGLE-2008-BLG-270, OGLE-2012-BLG-0442/MOA-2012-BLG-245, OGLE-2012-BLG-0207/MOA-2012-BLG-105, and OGLE-2018-BLG-0043/MOA-2018-BLG-033.…”