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Modeling Binary Lenses and Sources with the BAGLE Python Package

This paper introduces new binary lens and source models, including Keplerian orbits and simplified acceleration-based approximations, into the BAGLE Python package to enable joint photometric and astrometric fitting of microlensing events for future surveys like the Vera C. Rubin Observatory and the Nancy Grace Roman Telescope.

Original authors: T. Dex Bhadra, J. R. Lu, Natasha S. Abrams, Andrew Scharf, Edward Broadberry, Casey Lam, Macy J. Huston

Published 2026-03-30

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CC BY 4.0

Original authors: T. Dex Bhadra, J. R. Lu, Natasha S. Abrams, Andrew Scharf, Edward Broadberry, Casey Lam, Macy J. Huston

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing in a dark room, trying to figure out what's floating in the air around you. You can't see the objects directly because they are either too dark, too small, or just invisible. However, if you shine a flashlight through the room, you might notice that the light bends or brightens in strange ways as it passes near these invisible objects. This is the basic idea behind gravitational microlensing.

In our universe, massive objects (like stars, black holes, or planets) act like giant, invisible magnifying glasses. When one of these objects passes in front of a distant star, its gravity bends the star's light, making the background star appear brighter and shift its position slightly. Astronomers use this trick to weigh invisible objects, like dark black holes or lonely planets drifting through space.

The Problem: It's Not Always Just One Thing

For a long time, scientists mostly assumed these events were simple: one invisible object (the lens) passing in front of one background star (the source). But the universe is messy. Often, both the lens and the source are actually binary systems—pairs of objects orbiting each other, like a double-star system or a star with a planet.

When you have two objects dancing around each other, the light doesn't just bend; it gets twisted, stretched, and amplified in complex, chaotic patterns. It's like trying to predict the ripples in a pond when you throw in two stones instead of one. If you try to model this with a "single stone" math formula, you get the wrong answer. You might think you found a black hole when it's actually a pair of stars, or you might miss a planet entirely.

The Solution: BAGLE

This paper introduces an upgrade to a software tool called BAGLE (Bayesian Analysis of Gravitational Lensing Events). Think of BAGLE as a super-smart detective's notebook. Before this update, the notebook was great at solving simple "one-on-one" cases. Now, the authors have taught BAGLE how to solve complex "two-on-two" cases.

Here is how the new BAGLE works, using some everyday analogies:

1. The "Dance Floor" Models

The authors added different ways to describe how the two objects in a binary system move relative to each other.

The Static Model: Imagine two dancers holding hands and spinning in place without moving across the floor. This is used when the event happens so fast that the dancers don't have time to move much.
The Linear/Accelerated Model: Imagine the dancers are walking in a straight line or speeding up/slowing down. This is a shortcut used when the dance is too long to track every step, but we know they are moving in a general direction.
The Keplerian Model (The Full Dance): This is the most detailed model. It tracks the dancers doing a full, complex waltz, including loops, speed changes, and elliptical orbits. This is crucial for long events where the binary system completes a full orbit while the astronomers are watching.

2. Mixing and Matching

The new BAGLE can handle four main scenarios:

One Lens, One Source: The old standard.
One Lens, Two Sources: A single invisible object passing in front of a double-star system.
Two Lenses, One Source: A double-star system passing in front of a single background star.
Two Lenses, Two Sources: The most chaotic scenario, where both the foreground and background are pairs of stars.

3. The "Map" Analogy

To understand how light bends, the software creates magnification maps.

Imagine the lens system is a landscape of hills and valleys.
If you roll a ball (the background star's light) across this landscape, some paths get amplified (bright spots), and others get distorted.
The software draws a map of these "bright spots" (called caustics). If the background star crosses one of these lines, the light flares up dramatically. The new BAGLE can draw these maps for complex, dancing pairs of lenses, helping astronomers know exactly what to look for.

Why Does This Matter?

The universe is full of binary systems. Recent simulations suggest that 55% of microlensing events actually involve binary systems, but many of them look like simple events if you don't have the right tools.

By upgrading BAGLE to handle these complex dances, astronomers can:

Stop getting fooled: They won't mistake a binary star for a single black hole.
Find the invisible: They can weigh dark objects (like black holes) more accurately by understanding how their binary partners tug on them.
Prepare for the future: New telescopes like the Vera C. Rubin Observatory and the Nancy Grace Roman Telescope will soon take millions of photos of the sky. They will find thousands of these events. BAGLE is being upgraded now so it can process this massive flood of data and tell us the true nature of the objects involved.

In a Nutshell

This paper is about giving astronomers a better pair of glasses. Before, they could only see simple, single-object gravity tricks. Now, with the new BAGLE software, they can see the complex, beautiful, and chaotic dances of binary systems, allowing them to map the invisible dark matter and lonely planets of our galaxy with much higher precision. It's like upgrading from a black-and-white sketch to a high-definition 3D movie of the universe's hidden mechanics.

1. Problem Statement

Gravitational microlensing is a primary method for detecting isolated, dark compact objects (e.g., black holes, free-floating planets) and measuring their masses. However, a significant challenge in interpreting microlensing data is that the lens, the source, or both are frequently binary systems.

Prevalence: Recent simulations suggest that 55% of microlensing events involve binary systems, yet only ~5–11% are identified as obvious binaries. Many binary events masquerade as single-lens/single-source events, leading to incorrect mass and distance estimates.
Complexity: Binary systems introduce orbital dynamics (Keplerian motion) that can significantly alter lightcurves and astrometric trajectories, especially when the event timescale ( $t_E$ ) is comparable to or shorter than the orbital period ( $P$ ).
Software Limitations: Existing software packages (e.g., VBMicrolensing, MulensModel, pyLIMA) often lack the capability to perform joint fitting of photometric (lightcurve) and astrometric (centroid shift) data for complex binary configurations (Binary Source + Binary Lens, or BSBL). Furthermore, many lack efficient implementations for non-Keplerian approximations (linear/accelerated motion) needed for long-period binaries.

2. Methodology

The authors introduce a comprehensive set of binary models into BAGLE (Bayesian Analysis of Gravitational Lensing Events), a Python package designed for joint photometric and astrometric fitting. The methodology involves:

A. Geometric Frameworks

The paper defines models for three primary configurations:

BSPL: Binary Source, Point Lens.
PSBL: Point Source, Binary Lens.
BSBL: Binary Source, Binary Lens.

B. Orbital Motion Parameterizations

To handle binaries with varying orbital periods relative to the event duration, BAGLE implements four distinct motion models:

Static: The secondary companion has a fixed separation and angle relative to the primary. Valid when $P \gg t_E$ .
Linear: The secondary moves with a constant relative proper motion ( $\Delta\mu$ ) relative to the primary.
Accelerated: The secondary moves with constant relative acceleration ( $a_{rel}$ ) relative to the primary.
Full Keplerian: The system follows a full elliptical or circular orbit defined by eight Keplerian elements:
- $\omega_{pri}$ (argument of periastron), $i$ (inclination), $\Omega_{sec}$ (longitude of ascending node), $e$ (eccentricity), $P$ (period), $t_p$ (time of periastron), and projected semi-major axes ( $\aleph_{pri}, \aleph_{sec}$ ).
- The package uses Thiele-Innes constants to convert these orbital elements into rectangular coordinates on the sky plane.

C. Mathematical Formalism

Lensing Equations: For binary lenses, the paper utilizes the complex lens equation (Schneider & Weiss 1986) mapping the source plane to the image plane.
Root Solving: The lens equation is converted into a fifth-order polynomial ("lens polynomial"). BAGLE solves this using numpy.roots() and filters out "ghost roots" (non-physical solutions) using a Jacobian determinant criterion and a user-defined tolerance (root_tol).
Coordinate Transformations: The paper details non-trivial transformations between reference frames (e.g., geometric midpoint vs. center of mass) for parameters $u_0$ (impact parameter) and $t_0$ (time of closest approach), ensuring consistency across different model parameterizations.

D. Validation

The authors validated BAGLE against VBMicrolensing and MulensModel using simulated PSBL and BSPL events with parallax.

Accuracy: BAGLE and MulensModel showed residuals of $\sim 10^{-11}$ to $10^{-15}$ , while differences with VBMicrolensing were $\sim 10^{-4}$ (attributed to parallax implementation differences).
Performance: BAGLE was the fastest for full-scale simulations of BSPL events but slower than competitors for PSBL pre-instantiated runs due to the polynomial root solver.

3. Key Contributions

Unified Binary Modeling: BAGLE is now capable of modeling BSPL, PSBL, and BSBL systems with joint photometric and astrometric fitting. This is a unique capability among major microlensing packages (RTModel supports joint fitting but not for BSBL).
Flexible Orbital Approximations: The introduction of linear and accelerated motion models provides computationally efficient alternatives to full Keplerian solutions for long-period binaries where $P \gg t_E$ .
Robust Root Solving: Implementation of a rigorous ghost-image rejection algorithm using Jacobian determinants ensures physical validity in binary lens solutions.
Visualization Tools: New functions for generating magnification maps (using inverse ray shooting) and centroid shift maps (using forward ray shooting) to visualize caustic structures and astrometric deviations.
Coordinate Transformation Suite: Detailed mathematical derivations for transforming $u_0$ and $t_0$ between geometric midpoints, centers of mass, and primary objects, facilitating flexible model parameterization.

4. Results

Fitting Accuracy: In mock data tests simulating a binary-source event with Keplerian motion, models incorporating elliptical or circular orbital motion significantly outperformed linear approximations.
- Reduced Chi-Squared ( $\bar{\chi}^2$ ) for linear fit: 3.2.
- Reduced Chi-Squared for elliptical fit: 1.0 (consistent with error variance).
- This demonstrates that neglecting orbital dynamics in complex events leads to poor fits and biased parameters.
Caustic Analysis: Simulations showed that caustic structures become more symmetric as the mass ratio $q \to 1$ and elongate with increasing separation.
Computational Efficiency: While the polynomial root solver currently limits PSBL speed, the package is optimized for large-scale simulations of binary-source events. Future work plans to integrate JAX to accelerate the root solver.

5. Significance

This work significantly enhances the capability of the microlensing community to analyze data from upcoming major surveys, specifically the Vera C. Rubin Observatory (LSST) and the Nancy Grace Roman Telescope.

Dark Object Characterization: By accurately modeling binary dynamics, BAGLE will improve the mass measurements of isolated black holes and free-floating planets, reducing degeneracies caused by unmodeled binary companions.
Astrometric Precision: The ability to jointly fit astrometric data is crucial for the Roman Space Telescope, which will provide high-precision centroid shifts. This allows for the direct measurement of lens mass and distance without relying solely on photometric degeneracies.
Future-Proofing: The modular architecture and support for various orbital approximations ensure BAGLE remains a versatile tool for the next generation of high-cadence, high-precision microlensing surveys.