---
title: "What can a microlensing light curve tell us about its lens?"
description: "Build a blended-light likelihood, distinguish fitted geometry from physical mass, and see how source size and parallax supply missing scales."
canonical_url: "https://sandboxphysics.com/stories/microlensing-planets/read-a-light-curve-without-weighing-too-soon/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/microlensing-planets/read-a-light-curve-without-weighing-too-soon/"
last_modified: "2026-11-03"
---

[A dark planet bends starlight](https://sandboxphysics.com/stories/microlensing-planets/)

AC23 · L3

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

Sandbox Physics · 2026-11-03

A well-sampled light curve can reveal a companion’s mass ratio, a projected separation and the source’s passage across a caustic. The absolute mass needs more information. Start by predicting what each telescope measures, including light that stays nearly constant throughout the event.

## Before you begin

Read the two-image and finite-disk chapters. You need their source trajectory and image-area integral. This chapter introduces a likelihood as a probability model for measured errors and a profile as a fit repeated with one quantity held fixed.

## 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:

$$
F_k(t)=F_{s,k}\langle A\rangle(t)+F_{b,k}.
$$

The lensing calculation acts on the background source. Constant blended light is added in the telescope’s own flux scale.

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

$$
A_{\mathrm{ap},k}(t)=1+f_k[\langle A\rangle(t)-1].
$$

This is the brightening of the whole photometric aperture.

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.

$$
m_k(t)=m_{0,k}-2.5\log_{10}\{1+f_k[\langle A\rangle(t)-1]\}.
$$

The baseline magnitude and source fraction belong to the telescope. The lens geometry is shared.

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

$$
\mathcal L(\vartheta)=\prod_i\frac1{\sqrt{2\pi}\sigma_i}\exp\!\left[-\frac{(m_i-\widehat m_i(\vartheta))^2}{2\sigma_i^2}\right].
$$

The parameter vector contains the common trajectory and source geometry, plus each telescope’s baseline and blend.

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

$$
\chi^2_{\mathrm{prof}}(q)=\min_{\text{other parameters}}\chi^2(q,\ldots).
$$

The profile follows a possible compensation among parameters; a slice with the others fixed would answer a narrower question.

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 $\mu_{\mathrm{rel}}$,

$$
t_E=\frac{\theta_E}{\mu_{\mathrm{rel}}},\qquad t_*=\rho t_E=\frac{\theta_*}{\mu_{\mathrm{rel}}}.
$$

The source-radius crossing time is a geometric clock. The full duration of an anomaly also depends on the caustic and the angle of crossing.

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

$$
\delta\tau=p_N\Delta s_N+p_E\Delta s_E,\qquad\delta\beta=p_N\Delta s_E-p_E\Delta s_N.
$$

The first component shifts progress along the track; the second shifts the transverse offset. This states the sign convention explicitly.

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.

$$
\theta_E^2=\kappa M\pi_{\mathrm{rel}},\qquad\pi_E=\frac{\pi_{\mathrm{rel}}}{\theta_E},\qquad M=\frac{\theta_E}{\kappa\pi_E}.
$$

The source-size scale and measured microlens parallax together can supply a physical mass, with $\kappa\simeq8.144\,\mathrm{mas}\,M_\odot^{-1}$.

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

## Exercises and explanations

1. A lensed source brightens fivefold but supplies only one quarter of the baseline aperture light. What brightening is measured?

   Read the explanation

   The aperture brightening is $1+0.25(5-1)=2$. The other three quarters stay constant.

2. A narrow uncertainty in event duration is advertised as an equally narrow uncertainty in mass. Which unmeasured quantity appears before mass can be inferred?

   Read the explanation

   The relative proper motion enters $t_E=\theta_E/\mu_{\mathrm{rel}}$, and the angular scale still mixes mass with relative parallax. A precise duration does not remove those missing scales.

3. A second fit improves chi-square by a small amount when two parallax components are added. What must be examined before interpreting their nonzero best values?

   Read the explanation

   Compare the fit with zero parallax on the same data, inspect the likelihood profile and its boundaries, check observer-coordinate and timing conventions, and test residual behavior. An optimizer returns a best value even when a broad range, including zero, remains plausible.

## Try the prediction

### [Separate image magnification from aperture brightening](https://sandboxphysics.com/labs/gravitational-lens-telescope/?mode=binary)

1. Pause a fixed geometry and decrease the source fraction. Predict the aperture brightening before reading it.
2. Change the source radius at fixed Einstein time. Compare the source-radius crossing time with the full width of the anomaly, and explain their different meanings.

[Previous: What happens when a stellar disk crosses a caustic?](https://sandboxphysics.com/stories/microlensing-planets/a-stellar-disk-crosses-a-caustic/) [Next: Read a dark planet’s mass from Earth and space](https://sandboxphysics.com/stories/microlensing-planets/read-ground-space-and-roman-evidence/)

## Sources and calculation scope

1. [Beaulieu et al. (2006), OGLE-2005-BLG-390 discovery, astro-ph/0601563v1](https://arxiv.org/abs/astro-ph/0601563v1)

   Read the main text, Table 1 and Figure 1: finite-source planetary anomaly, source limb coefficients, 650 original points and population-conditioned physical mass. Our independently rerun five-file subset contains 308 points.

2. [Bozza et al. (2018), VBBinaryLensing finite-source methods](https://doi.org/10.1093/mnras/sty1791)

   Sections 2–4 and equations 14–20, 25–26 explain contour area, error refinement and radial source profiles. The pinned LGPL code remains a separately downloaded offline reference.

3. [Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025)](https://arxiv.org/abs/2410.13660v1)

   Read equations 1–16 and section 6 before interpreting binary roots, parity, critical curves or a missing image pair. This is a methods reading, not a replication of its benchmark suite.

4. [Dong et al., A Saturn-mass microlensing object, arXiv:2601.00057v1; Science](https://arxiv.org/abs/2601.00057v1)

   Fixed author version, Figure 2 and supplement S5–S11: finite-source scale, ground/Gaia parallax, alternative blending and solution weights. arXiv identifies the paper as published in Science. This guide reads the evidence; it does not rerun that event.

5. [JPL Horizons vector ephemeris documentation](https://ssd.jpl.nasa.gov/horizons/manual.html)

   The exercise retains a fixed 161-node DE441 geometric ICRF Earth vector response, positions in AU, velocities in AU/day and TDB epochs. The headerless photometry keeps its separate timing qualification.