Binary Source & Lens Models

• Binary source and lens models describe gravitational microlensing scenarios where either the source or lens is binary, producing complex light curve anomalies.
• They employ rigorous mathematical formalism, numerical root-finding, and Bayesian inference to extract key parameters such as mass ratios, separations, and caustic structures.
• Advanced modelling techniques, including high-resolution imaging and CMD analysis, help break degeneracies and refine physical interpretations in microlensing surveys.

Binary source and lens models comprise a class of gravitational microlensing configurations in which either the lens (foreground mass) or the source (background light-emitting body) is a binary system, or both. These models are integral to the analysis of complex microlensing events exhibiting multi-featured anomalies, caustic crossings, or deviations from simple single-lens single-source (1L1S) light curves. The mathematical structure, parameterization, fitting methodologies, and degeneracy-breaking techniques employed in such modeling ensure robust physical interpretation of microlensing data, supporting constraints on stellar, planetary, and remnant populations across the Galaxy.

1. Formalism of Binary Source and Lens Equations

The generalized microlensing lens equation in complex notation for $N_L$ lenses and $N_S$ sources is given by

$$\zeta = z - \sum_{i=1}^{N_L} \frac{m_i}{\overline{z} - \overline{z_{L,i}}},$$

where $\zeta$ represents the complex source position, $z$ the complex image position, $z_{L,i}$ the lens component positions, and $m_i$ their fractional masses normalized so that $\sum_i m_i = 1$ (Han et al., 2021, Han et al., 2022). For binary lens scenarios ($N_L=2$) this quintic equation yields up to five images for each source. Each source component $S_j$ ($j = 1,2$) independently follows its own trajectory, characterized by closest approach time $t_{0,j}$, impact parameter $u_{0,j}$, normalized angular radius $\rho_j = \theta_{*,j}/\theta_E$, and unlensed flux $F_{S,j}$.

For triple-lens configurations ($N_L=3$), the equation generalizes with an additional mass, commonly representing a planetary companion to a binary stellar lens (Han et al., 2021). Solving the lens equation numerically via root-finding, contour integration, or ray-shooting yields all image positions and instantaneous magnifications.

The combined magnification for unresolved binary sources is the flux-weighted average: $$A(t) = \frac{F_{S,1} A_1(t) + F_{S,2} A_2(t)}{F_{S,1} + F_{S,2}} \equiv \frac{A_1(t) + \Gamma A_2(t)}{1+\Gamma},$$ where $\Gamma = F_{S,2}/F_{S,1}$ denotes the flux ratio (Han et al., 2021, Chung et al., 26 Jun 2025).

2. Parameter Space and Physical Interpretation

Binary-lens and binary-source models are defined by a suite of geometric and physical parameters:

• Binary Lens: projected separation $s$ (in units of $\theta_E$), mass ratio $q = m_2/m_1$, trajectory angle $\alpha$, normalized source radius $\rho$, impact parameter $u_0$, closest approach time $t_0$, Einstein timescale $t_E$, and source radius $\theta_*$ (Shin et al., 2011, Hwang et al., 2010, Han et al., 2021).
• Binary Source: per-star $t_{0,j}$, $u_{0,j}$, $\rho_j$, and baseline flux $F_{S,j}$, with flux ratio $q_F = F_{S,2}/F_{S,1}$ (Han et al., 2022, Han et al., 2024).
• Higher-order Effects: microlens parallax vector $\pi_E$, lens and source orbital motion, and blend flux terms (Shin et al., 2011, Bhadra et al., 3 Dec 2025).

Physical lens properties, such as mass ($M$), distance ($D_L$), and projected separation ($a_\perp$), are inferred via

$$M = \frac{\theta_E}{\kappa \pi_E}, \quad D_L = \frac{\mathrm{AU}}{\pi_E \theta_E + \pi_S},$$

where $\kappa = 4G/(c^2 \mathrm{AU})$, and $\pi_S = \mathrm{AU}/D_S$ (Shin et al., 2011).

In events with strong finite-source effects, angular Einstein radii are estimated independently for each source component, often using de-reddened color-magnitude diagrams and empirical color–surface-brightness relations (Han et al., 2022). Consistency between $\theta_{E,1}$ and $\theta_{E,2}$ corroborates the binary-source interpretation (Han et al., 2022, Han et al., 2024).

3. Caustics, Anomaly Morphologies, and Degeneracies

Binary lens systems generate extended caustic structures—regions in the source plane where the Jacobian determinant vanishes and which correspond to divergence in point-source magnification: $$\det J(z) = 0 \Longleftrightarrow \text{critical curve in lens plane} \to \text{caustic in source plane}$$ (Han et al., 2022, Han et al., 2024, Han et al., 2021). The topology of caustics varies by $s$ and $q$:

• Resonant (central) caustics: large, multi-cusped caustics for $s \sim 1$.
• Close/Wide degeneracy ($s \leftrightarrow 1/s$): solutions with similar caustic structures and nearly identical light curves (Hwang et al., 2010, Han et al., 2024).
• Triple-lens caustics: addition of a planet generates a small secondary caustic near its projected position, overlaid on a central binary caustic (Han et al., 2021).

Light curve anomalies—multiple spikes, bumps, or excursions—arise when source(s) traverse caustic limbs. Binary-source models reproduce complex multi-peak events through independent caustic crossings or approaches by each source star (Han et al., 2022, Han et al., 15 Sep 2025).

Degeneracies in parameter space include:

• Close/wide ($s \leftrightarrow 1/s$), impact parameter sign ($u_0 \to -u_0$), ecliptic-trajectory symmetry, and caustic cycloid symmetry for extreme $s$ (Hwang et al., 2010, Bachelet et al., 2024).
• Model family ambiguity (2L2S vs. 3L1S): competing fits for multi-component events require $\Delta\chi^2$ criteria, proper motion consistency, and physical plausibility checks (Chung et al., 26 Jun 2025, Han et al., 2021).

4. Model Selection, Bayesian Inference, and Degeneracy Breaking

Model selection employs grid sampling, MCMC refinement, and explicit $\chi^2$ comparison:

• Fit standard 2L1S or 1L2S models to the bulk light curve, identify residuals near anomalies.
• Test extensions: add a source (2L2S), add a lens (3L1S), or invoke higher-order effects (parallax, orbital motion).
• Compare fit statistics, proper motion estimates, and physical parameter posteriors (Han et al., 2021, Chung et al., 26 Jun 2025, Han et al., 2021).

Bayesian inference leverages Galactic density, mass-function, and kinematic priors. Event observables ($t_E$, $\theta_E$, parallax $\pi_E$) constrain posterior distributions of lens and source masses and distances (Han et al., 2022, Han et al., 2024).

Degeneracy-breaking relies on independent observational diagnostics:

• Consistency of measured and predicted flux ratios via stellar isochrones for binary sources (Han et al., 2021).
• Color and magnitude constraints from CMD analysis, favoring source configurations with plausible spectral types and physical separations (Han et al., 2022, Han et al., 15 Sep 2025).
• Relative lens-source proper motion ($\mu_\mathrm{rel}$) derived from fit parameters and confirmed via high-resolution imaging; scenarios with implausibly large $\mu_\mathrm{rel}$, as for some 3L1S fits, are rejected (Chung et al., 26 Jun 2025, Han et al., 2024, Bachelet et al., 2024).
• Chromaticity and multi-band photometry validating source colors and blend fluxes (Dominik et al., 2018).

Astrometric microlensing (Gaia DR4), high-resolution AO imaging, and radial-velocity monitoring will resolve ambiguities in lens/source configurations, measure $\theta_E$ and $\mu_\mathrm{rel}$ directly, and enable unique physical characterization of many binary-lens and source systems (Bachelet et al., 2024).

5. Advanced Modeling Frameworks and Current Applications

Contemporary analysis employs packages such as BAGLE, which implement binary-lens and binary-source formalism, including:

• Full Keplerian orbital parameters for lens and source components: $a$, $e$, $i$, $\Omega$, $\omega$, $T_p$, $P$ (Bhadra et al., 3 Dec 2025, Shin et al., 2011).
• Approximations for linear and accelerating binary motion or static geometries when the orbital period far exceeds the microlensing timescale (Bhadra et al., 3 Dec 2025).
• Simultaneous photometric and astrometric Bayesian fitting, facilitating joint inference from multi-modal observational data (Bhadra et al., 3 Dec 2025).
• Computation of magnifications via root-finding, contour integrals, and ray-shooting; calculation of centroid shifts and caustic maps (Bhadra et al., 3 Dec 2025).

Application of these techniques to recent and ongoing microlensing campaigns (KMTNet, OGLE, Gaia, Rubin, Roman) has yielded robust characterization of binary lenses and sources, including systems composed of M dwarfs, brown dwarfs, white dwarfs, neutron stars, and planets (Han et al., 15 Sep 2025, Han et al., 2024, Bachelet et al., 2024).

6. Detection Biases, Physical Interpretation, and Event Population Statistics

Binary-source events preferentially feature secondaries with $q_F \gtrsim 0.5$, reflecting a selection bias toward systems where both stars are sufficiently bright to produce detectable anomalies (Han et al., 2024). This statistical tendency impacts the population of discovered binary-source events and should be accounted for in Galactic and extragalactic microlensing rate studies.

Physical interpretation is informed by color–surface–brightness calibrations, lens photometry, and proper motion estimates. Scenarios in which secondary source stars possess implausibly small separations or generate unobserved variability (eclipses, ellipsoidal distortion) are ruled out, refining the census of binary microlensing configurations (Dominik et al., 2018, Han et al., 2021).

Multi-instrument, multi-band, and multi-epoch follow-up is critical to the unambiguous identification and physical characterization of both binary source and binary lens events.

7. Outlook and Future Directions

Advances in survey cadence, photometric precision, and astrometric monitoring are making the routine detection and full orbital characterization of binary-lens and binary-source microlensing events feasible. High-resolution imaging, space-based astrometry, and spectroscopic diagnostics will enable direct measurement of lens masses, distances, and orbital parameters, removing degeneracies prevalent in photometry-only modeling. Statistical analysis of the growing sample will inform population synthesis and Galactic structure models, determining trends in binary occurrence, mass functions, and planetary frequency (Shin et al., 2011, Bhadra et al., 3 Dec 2025, Bachelet et al., 2024).

Recent events, such as KMT-2019-BLG-1715 (Han et al., 2021), OGLE-2005-BLG-018 (Shin et al., 2011), KMT-2022-BLG-0086 (Chung et al., 26 Jun 2025), and ongoing campaigns (Han et al., 15 Sep 2025), exemplify the sophistication of current analysis and the potential for further discoveries. Combined modeling approaches leveraging physical, statistical, and observational constraints will continue to enhance the precision and completeness of binary-source and binary-lens microlensing studies.