A Mathematical Treatment of the Offset Microlensing Degeneracy

TL;DR

This paper provides a theoretical foundation for the offset microlensing degeneracy, demonstrating its mathematical nature, conditions of validity, and generalization to multiple lens components, enhancing understanding of planetary microlensing interpretations.

Contribution

It offers the first theoretical justification of the offset degeneracy, showing its exactness under certain conditions and extending it to N-body microlenses with a combinatorial degeneracy.

Findings

The offset degeneracy is exact to zeroth-order in mass ratio for specific crossing scenarios.

The degeneracy persists under certain conditions related to source trajectory and caustic crossing.

The formalism generalizes to N-body systems, resulting in a 2^{N-1}-fold degeneracy.

Abstract

The offset microlensing degeneracy, recently proposed by Zhang et al. (2022), has been shown to generalize the close-wide and inner-outer caustic degeneracies into a unified regime of magnification degeneracy in the interpretation of 2-body planetary microlensing observations. While the inner-outer degeneracy expects the source trajectory to pass equidistant to the planetary caustics of the degenerate lens configurations, the offset degeneracy states that the same mathematical expression applies to any combination of the close, wide, and resonant caustic topologies, where the projected star-planet separations differ by an offset ($s_{\rm A}\neq s_{\rm B}$) that depends on where the source trajectory crosses the star-planet axis. An important implication is that the $s_{\rm A}=1/s_{\rm B}$ solution of the close-wide degeneracy never strictly manifests in observations except when the source crosses a singular point near the primary. Nevertheless, the offset degeneracy was proposed upon numerical calculations, and no theoretical justification was given. Here, we provide a theoretical treatment of the offset degeneracy, which demonstrates its nature as a mathematical degeneracy. From first principles, we show that the offset degeneracy formalism is exact to zeroth-order in the mass ratio ($q$) for two cases: when the source crosses the lens-axis inside of caustics, and for $(s_{\rm A}-s_{\rm B})^6\ll1$ when crossing outside of caustics. The extent to which the offset degeneracy persists in oblique source trajectories is explored numerically. Lastly, it is shown that the superposition principle allows for a straightforward generalization to $N$-body microlenses with $N-1$ planetary lens components ($q\ll1$), which results in a $2^{N-1}$-fold degeneracy.