Modeling Binary Lenses and Sources with the BAGLE Python Package

TL;DR

This work extends the BAGLE Python package to robustly model binary microlensing systems by incorporating complete Keplerian orbital motion and efficient linear/accelerated approximations. It develops comprehensive BSPL, PSBL, and BSBL formalisms, using Thiele–Innes constants and cordoned Keplerian elements $ (oldsymbol{ earrow ext{eight elements}}) $ to compute sky-plane trajectories, including center-of-mass motion and parallactic effects. Validation against established microlensing tools shows good agreement with small discrepancies arising from parallax implementations and root-solving vs contour methods, while demonstrating BAGLE’s competitive runtimes for many scenarios. The results emphasize the value of joint photometric and astrometric fitting in breaking degeneracies and enabling precise mass and orbital determinations for binary lenses and sources, with direct relevance to Rubin Observatory and Roman Telescope datasets and dark-lens searches. BAGLE thus provides a versatile framework for simulating and fitting complex binary microlensing events, paving the way for improved measurements of compact objects like black holes and exoplanets in forthcoming surveys.

Abstract

Gravitational microlensing is a powerful tool that can be used to find and measure the mass of isolated and dark compact objects. In many microlensing events, the lens, the source, or both may be a binary system. Therefore, in this study we present lensing equations for binary source and lens models in the Bayesian Analysis of Gravitational Lensing Events (BAGLE) Python microlensing package. The new binary source and lens models in BAGLE account for the complete Keplerian orbit. BAGLE also includes binary models that approximate the orbital motion as linear or accelerating motion of the secondary companion; these are useful when the orbit has a very low eccentricity or the orbital period is much longer than the microlensing timescale. The model parameterizations based on these binary lensing equations will enable joint fitting of photometric and astrometric data sets. Consequently, binary microlensing events with complex astrometric trajectories can be used to break several microlensing degeneracies that plague photometry-only microlensing modeling. These binary models will be used to fit microlensing event data from the Vera C. Rubin Observatory, the Nancy Grace Roman Telescope, and other surveys.

Figures (19)

• Figure 1: Binary geometries available in BAGLE. Source trajectories on the sky plane are represented in pink and lens trajectories are represented in black. The change in Right Ascension is plotted on the x-axis and the change in Declination is plotted on the y-axis. Models with binary lenses ( left column), binary sources ( middle column), and binary lens and source ( right column) are supported. Secondary companions can have fixed separation and angle relative to the primary ( top row), linear motion ( 2nd row), accelerating motion ( 3rd row), or full Keplerian orbital motion ( 4th row).
• Figure 2: Trajectory of a binary orbit at $\textit{i} = 0 \degree$ (binary disk is face-on) simulated using BAGLE for stationary proper motion. The orbit has an eccentricity of 0.8. The primary and secondary objects at the time of periastron passage have been highlighted.
• Figure 3: Source and lens trajectories for linear (Top) and accelerated (Bottom) approximations of orbital motion in binary sources. We present the unlensed sources (solid lines) and the lensed images (dashed + dotted lines). For each source, there is a major and a minor image. The minor image is seen around the lens, and the major image is seen around the source. The green line is the flux-weighted average of lensed source positions, as observed on the sky. Note that $mag_{S,pri}$ = 16 and $mag_{S,sec}$ = 17. In both panels, the primary's proper source motion $\boldsymbol{\mu}_{\boldsymbol{S},\sun}$ is [6 mas yr$^{-1}$, 3 mas yr$^{-1}$]; the secondary's proper source motion relative to the primary source $\boldsymbol{\mu}_{\boldsymbol{S_s},\sun}$ is [9 mas yr$^{-1}$, 7 mas yr$^{-1}$]. The acceleration of the secondary source $\bold{a}_{\boldsymbol{S\mathrm{rel}},\sun}$ is [0.5 mas yr$^{-2}$, -2 mas yr$^{-2}$] in the lower panel. In both cases, the Einstein time ($t_{E,\sun}$)=269 days and $u_{0,\sun}$ =1.01. The lens mass is $10 M_\odot$, and it is held stationary.
• Figure 4: Source and lens trajectories for a simulated binary source point lens microlensing event involving Keplerian orbits. We present the unlensed sources (solid lines) and the lensed images (dashed + dotted lines). The green line is the flux-weighted average of lensed source positions, as observed on the sky. Note that $mag_{S,pri}$ = 16 and $mag_{S,sec}$ = 17. This event has the following Keplerian elements: $\omega_{pri} = 30 \degree$, $\Omega_{sec} = 10 \degree$, $\textit{i} = 90 \degree$, $e=0.6$, $\textit{P} = 1000$ days, $\aleph_{pri} = 3$ mas and $\aleph_{sec} = 8$ mas. We simulate this event over $t_E=208.47$ days. The lens (8 $M_\odot$) is held stationary.
• Figure 5: Source and lens trajectories for linear (upper panel) and accelerated (lower panel) approximations of orbital motion of binary lenses. The solid black line is the primary lens, the solid purple line is the secondary lens, and the solid yellow lines are the image positions. The green line is the flux-weighted average of the lensed source position, as observed on the sky. Note that $mag_{S}$ = 16. In both panels, the source is stationary; the primary lens has a proper motion of $\boldsymbol{\mu}_{\boldsymbol{L},\sun}$ = [-3.76 mas yr$^{-1}$, -3.76 mas yr$^{-1}$]; the secondary lens has a proper motion of $\boldsymbol{\mu}_{\boldsymbol{L_s} ,\sun}$ = [-2.76 mas yr$^{-1}$, -2.76 mas yr$^{-1}$]. For our model with acceleration, we provide the following input for $\bold{a}_{\boldsymbol{L\mathrm{rel}},\sun}$ = [1 mas yr$^{-1}$, -1 mas yr$^{-1}$]. In both cases, the Einstein time is $t_{E,\sun}$=412 days and $u_{0,\sun}$ = 0.5. Note that this scenario does not involve a caustic crossing and thus produces only 3 images.