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

## TL;DR

This paper introduces a comprehensive method to identify and analyze all possible critical curves in triple-lens gravitational microlensing systems with arbitrary mass configurations, enhancing understanding of their diverse caustic structures.

## Contribution

It provides a novel approach to map critical curves and topologies for any triple-lens configuration, including new nested critical-curve topologies, and assesses their occurrence probabilities.

## Key findings

- Identified 39 parameter space regions for equal-mass triples with nine topologies.
- Found 11 critical-curve topologies in star-planet-moon models, including new nested loops.
- Demonstrated the method on three models, revealing mass-dependent lensing regimes.

## Abstract

Since the first observation of triple-lens gravitational microlensing in 2006, analyses of six more events have been published by the end of 2018. In three events the lens was a star with two planets; four involved a binary star with a planet. Other possible triple lenses, such as triple stars or stars with a planet with a moon, are yet to be detected. The analysis of triple-lens events is hindered by the lack of understanding of the diversity of their caustics and critical curves. We present a method for identifying the full range of critical curves for a triple lens with a given combination of masses in an arbitrary spatial configuration. We compute their boundaries in parameter space, identify the critical-curve topologies in the partitioned regions, and evaluate their probabilities of occurrence. We demonstrate the analysis on three triple-lens models. For three equal masses the computed boundaries divide the parameter space into 39 regions yielding nine different critical-curve topologies. The other models include a binary star with a planet, and a hierarchical star--planet--moon combination of masses. Both have the same set of 11 topologies, including new ones with doubly nested critical-curve loops. The number of lensing regimes thus depends on the combination of masses -- unlike in the double lens, which has the same three regimes for any mass ratio. The presented approach is suitable for further investigations, such as studies of the changes occurring in nonstatic lens configurations due to orbital motion of the components or other parallax-type effects.