What can a microlensing light curve tell us about its lens?

Predict the light each telescope collects

For telescope $k$, let $F_{s,k}$ be the unlensed source flux and $F_{b,k}$ the constant light from other unresolved objects. The predicted measurement uses the exposure-averaged magnification from the preceding chapter:

Divide by the baseline $F_{s,k}+F_{b,k}$ and define the source fraction $f_k=F_{s,k}/(F_{s,k}+F_{b,k})$. Then

Worked example: the source supplies 70 units of baseline flux and another star supplies 30. If lensing brightens the source fourfold, the aperture receives $70\times4+30=310$ units. The measured brightening is $3.10$, even though the source itself brightens fourfold. Fitting the aperture curve as though all baseline light were lensed would change the inferred event geometry.

Reading: Beaulieu et al. (2006), OGLE-2005-BLG-390 discovery, astro-ph/0601563v1

Use the measured error model

The published five-file exercise supplies instrumental magnitudes and their reported errors. Magnitude is a logarithmic measure of flux. For each telescope, an unknown baseline magnitude sets its zero point; its source fraction sets how strongly the event appears against constant light.

Assume the reported magnitude errors $\sigma_i$ are independent Gaussian standard deviations. For a measured magnitude $m_i$ and predicted value $\widehat m_i(\vartheta)$, the normalized likelihood is

With the same data and fixed errors, maximizing this likelihood minimizes $\chi^2=\sum_i[(m_i-\widehat m_i)/\sigma_i]^2$. Here the signed residual is predicted magnitude minus measured magnitude, divided by the stated error. Plotting residuals by telescope can expose a local failure hidden by the total score.

Correlated atmospheric or instrumental errors would require a covariance model. In the reproducible exercise the unknown covariance is taken as zero and the supplied errors are kept unchanged. Rescaling every competing fit to the same reduced chi-square would remove part of the comparison we are trying to make.

Reading: Beaulieu et al. (2006), OGLE-2005-BLG-390 discovery, astro-ph/0601563v1 · Bozza et al. (2018), VBBinaryLensing finite-source methods

A fitted minimum has neighbors and alternatives

A numerical optimizer follows local improvements. A narrow planetary feature can send different starts toward different valleys in parameter space. Search several separations, mass ratios and trajectory directions, retain the failed or inferior solutions, and compare all models on exactly the same measurements.

A profile asks what happens when one quantity is fixed and the remaining parameters are fitted again. For mass ratio $q$, write

Near and wide binary configurations can sometimes fit similarly. The words describe candidate geometries, not a guarantee of two acceptable solutions for every event. Finite source size, source blending and a changing observer trajectory may shift their relative fit quality.

A Bayesian posterior additionally needs a specified prior: $p(\vartheta\mid d)\propto\mathcal L(\vartheta)\,\pi(\vartheta)$. A uniform prior in a logarithm differs from one uniform in the original parameter. Finite, proper priors and their volume factors matter when comparing whole models. An improvement in chi-square compares likelihoods at the selected fits. An integrated model probability also needs integration over the allowed parameters and their prior weights.

Reading: Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025) · Beaulieu et al. (2006), OGLE-2005-BLG-390 discovery, astro-ph/0601563v1

Extract the source-crossing clock

The finite disk supplies a dimensionless radius $\rho=\theta_*/\theta_E$, where $\theta_*$ is the source’s true angular radius. With the relative angular speed $\mu_{\mathrm{rel}}$,

For a synthetic event with $t_E=10.0\,\mathrm d$ and $\rho=0.0400$, the source-radius time is $0.400\,\mathrm d$. A larger radius lengthens this clock at fixed motion. Crossing a caustic at a slant also changes the observed feature width, so that width alone is not a direct ruler for the stellar radius.

If independent stellar information supplies $\theta_*=1.00\,\mu\mathrm{as}$ and the fit gives $\rho=0.0100$, then $\theta_E=0.100\,\mathrm{mas}$. The angular size estimate has its own error, which must travel into the Einstein-radius uncertainty.

Reading: Beaulieu et al. (2006), OGLE-2005-BLG-390 discovery, astro-ph/0601563v1 · Dong et al., A Saturn-mass microlensing object, arXiv:2601.00057v1; Science

Use the observer’s motion to obtain another scale

Earth moves during the event. Subtract its position and straight-line motion at a reference epoch, then project the remaining displacement into north and east components $\Delta s_N,\Delta s_E$ in astronomical units divided by one astronomical unit. These are dimensionless numbers.

Define a parallax vector $(p_N,p_E)$ along the source’s motion relative to the lens. In the trajectory frame its effect is

The vector magnitude $\pi_E=\sqrt{p_N^2+p_E^2}$ equals the relative trigonometric parallax divided by the Einstein radius. Two well-separated observers can measure a difference in the event trajectories. An annual effect from one short ground-based event may be weak because Earth moves nearly in a straight line during it.

Using the synthetic $\theta_E=0.100\,\mathrm{mas}$ above and a separately measured $\pi_E=0.200$ gives $M\simeq0.0614\,M_\odot$. Without that parallax measurement there is a family of mass-distance solutions. Two lenses with different Einstein radii can also share the same duration if their proper motions change in proportion.

Reading: Dong et al., A Saturn-mass microlensing object, arXiv:2601.00057v1; Science · JPL Horizons vector ephemeris documentation

Decide what the event actually supports

Look for repeated information: the same anomaly in independent telescopes, a finite-source shape that survives source-profile and exposure checks, and a fit that remains stable when its starting point changes. Each contribution answers a different part of the inference.

The original OGLE-2005-BLG-390 analysis used a Galaxy model to turn its geometric measurements into a distribution of possible physical masses. The later ground-plus-Gaia case in the paper guide uses finite-source and space-parallax information to obtain an additional angular and distance scale. Reading those two procedures side by side shows what a short brightening invites us to do next: investigate a low-mass lens and measure the extra quantities needed to determine its mass.

Reading: Beaulieu et al. (2006), OGLE-2005-BLG-390 discovery, astro-ph/0601563v1 · Dong et al., A Saturn-mass microlensing object, arXiv:2601.00057v1; Science