Stellar-Field Microlensing

• Stellar-field microlensing is a gravitational lensing phenomenon in dense stellar regions where transient flux variations reveal the presence of compact lenses.
• It employs detailed photometric and astrometric techniques to resolve sub-milliarcsecond angular scales and break degeneracies in lens mass and distance.
• Modern applications extend to mapping Galactic structure, probing dark remnants like isolated black holes, and diagnosing stellar atmospheres through finite-source effects.

Stellar-field microlensing is the transient form of gravitational lensing that occurs when a compact foreground mass passes sufficiently close to the line of sight to a background star in a dense stellar field. In the Galactic bulge, spiral-arm and disk sightlines, and the Magellanic Clouds, the lens-generated images are ordinarily separated by only sub-milliarcsecond angles, so the defining observable is not resolved image splitting but a time-dependent change in source flux, sometimes accompanied by centroid motion or polarization. Its distinctive strength is that it operates precisely where conventional stellar astrophysics is observationally most difficult: crowded fields containing enormous numbers of unresolved or barely resolved stars, where rare alignments can be harvested statistically and, in favorable events, turned into precision measurements of compact lenses, Galactic structure, and source-star atmospheres (Rahvar, 2015).

1. Definition, geometry, and natural scales

The standard geometry is the observer–lens–source configuration. If $\hat\beta$ is the angular position of the unlensed source, $\hat\theta$ the angular image position, $D_s$ the observer–source distance, $D_d$ the observer–lens distance, and $D_{ds}$ the lens–source distance, the lens equation is

$\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$

with point-lens deflection

$$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$

For a point lens, this gives the scalar equation

$\theta^2=\beta\theta+\theta_E^2,$

where the angular Einstein radius is

$\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$

The two image positions are

$$\theta=\frac{\beta\pm\sqrt{\beta^2+4\theta_E^2}}{2}.$$

In stellar-field applications, $\hat\theta$0 sets the fundamental scale. The Galactic form emphasized in the literature is

$\hat\theta$1

For typical bulge or LMC geometries this is sub-milliarcsecond, which is why ordinary Galactic stellar-field microlensing is usually a photometric and astrometric phenomenon rather than a direct image-resolving one. The most important sightlines are the Galactic bulge, the Galactic disk and spiral arms, and the Large and Small Magellanic Clouds, the last of which were historically used to probe the Galactic halo (Rahvar, 2015).

2. Photometric events in crowded stellar fields

For a point source lensed by a point mass, the total magnification is

$\hat\theta$2

Using the normalized separation $\hat\theta$3, the standard Paczyński light curve is

$\hat\theta$4

with $\hat\theta$5 the minimum impact parameter, $\hat\theta$6 the epoch of peak magnification, and

$\hat\theta$7

the Einstein crossing time for relative proper motion $\hat\theta$8. The canonical single-lens stellar-field event is therefore a smooth, symmetric, achromatic brightening.

Dense stellar fields are essential because the instantaneous probability that any one star is strongly lensed is tiny. The same crowding that frustrates conventional photometry becomes an advantage: one can monitor millions of stars simultaneously and search for the rare transient amplification signature. In practice, however, crowding introduces blending. If the lensed source contributes flux $\hat\theta$9 and unresolved neighbors contribute constant background $D_s$0, then

$D_s$1

and the observed magnification becomes

$D_s$2

Blending suppresses the apparent amplitude, biases $D_s$3, and corrupts source-star counts used in optical-depth work, which is why image-difference photometry became central in crowded bulge and Magellanic Cloud fields (Rahvar, 2015).

The survey history of stellar-field microlensing is dominated by EROS, MACHO, MOA, and OGLE, with a later expansion into Galactic-plane time-domain surveys. In ZTF DR5, a Galactic-plane search using 2018–2021 data over 194 fields with $D_s$4 found 60 candidate events, including 3 with strong microlensing parallax signatures, and an efficiency-corrected mean timescale $D_s$5 days; the event counts decreased with Galactic longitude with scale length $D_s$6 (Rodriguez et al., 2021). In ZTF DR17, the same general region yielded 124 high-confidence events and 54 possible events, with a detection-efficiency-corrected mean $D_s$7 days, confirming that Galactic-plane stellar-field microlensing is observationally viable and systematically longer-lived than typical bulge samples (Zhai et al., 2023).

3. Degeneracy breaking: parallax, finite sources, astrometry, and polarimetry

Ordinary photometric monitoring of a point-lens event measures $D_s$8, but only $D_s$9 carries direct physical information, and even that is degenerate:

$D_d$0

Thus the light curve alone does not uniquely determine the lens mass $D_d$1, lens distance $D_d$2, or transverse speed $D_d$3. The higher-order observables used in stellar fields are therefore all variants of degeneracy breaking.

Annual parallax distorts the symmetry of long-duration light curves through Earth’s orbital acceleration; Earth-plus-satellite parallax measures the microlens parallax vector through differences in peak time and impact parameter between two observatories; terrestrial parallax is possible only in extreme high-magnification cases. Finite-source effects arise when the source angular size is not negligible relative to the impact parameter. In that case the point-source approximation fails, the source-crossing light curve resolves the stellar disk, and the normalized source size $D_d$4 can be measured. At exact central crossing, the maximum finite-source magnification is

$D_d$5

When $D_d$6 is combined with an independent angular radius estimate for the source, it yields $D_d$7.

Astrometric microlensing measures the centroid shift of the unresolved images rather than only the flux change. The centroid shift relative to the unlensed source is

$D_d$8

with maximum

$D_d$9

This is conceptually decisive because astrometry gives $D_{ds}$0 directly. Polarimetry is complementary: microlensing breaks the cancellation of the source’s local polarization pattern, so the lensed Stokes parameters encode both event geometry and atmospheric physics. In dense fields, these measurements are rare but disproportionately informative because long events, bright sources, and favorable transits occur often enough across very large survey samples to be exploited (Rahvar, 2015).

A later astrometric-parallax study sharpened this distinction between photometric and astrometric information for isolated stellar-mass black-hole events. For Galactic-bulge observations, the efficiencies for discerning parallax amplitudes with relative error $D_{ds}$1 were reported as $D_{ds}$2 through astrometric observations and $D_{ds}$3 through photometric observations, whereas toward the LMC the corresponding efficiencies were $D_{ds}$4 and $D_{ds}$5. The underlying reason is that the parallax amplitude in astrometric deflections is proportional to the relative parallax $D_{ds}$6, which does not strongly depend on lens mass in the same way as the normalized photometric parallax amplitude $D_{ds}$7 (Sajadian et al., 2023).

4. Stellar-surface and atmospheric diagnostics

One of the most distinctive astrophysical applications of stellar-field microlensing is that high-magnification and source-crossing events turn the lens into an effective scanning probe of the source surface. Because stellar disks are limb-darkened rather than uniform, finite-source distortions encode the surface-brightness profile. A simple limb-darkening law used in this context is

$D_{ds}$8

Well-covered bulge events can therefore measure limb-darkening coefficients in multiple passbands for stars at kiloparsec distances. The same logic extends to stellar spots: a cool or otherwise anomalous region on the source surface produces localized perturbations when the lens scans across it, allowing indirect study of stellar activity and atmospheric inhomogeneity (Rahvar, 2015).

Polarimetric follow-up formalizes this atmospheric leverage. A Galactic-bulge simulation study using VLT/FORS2 found that, assuming detection of about 3000 microlensing events per year by OGLE, one should expect almost $D_{ds}$9, $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$0, $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$1, and $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$2 detectable polarization microlensing events per year for the criteria of three consecutive polarimetry points above baseline at $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$3, $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$4, $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$5, and $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$6, respectively. In that framework, joint photometry and polarimetry can measure the scattering optical depth of the atmosphere and the inner radius of the stellar envelope of cool RGB stars, thereby constraining dust opacity and dust-formation radii in remote bulge giants (Khalouei et al., 2020).

Binary-lens caustic crossings sharpen this stellar-surface sensitivity further. A spot-polarimetry analysis found that photometric observations are more efficient than polarimetry for spot detection, but polarimetry can specify the magnetic field of source spots. In the reported simulations, about $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$7 of spotted caustic-crossing events had detectable polarimetric signatures, whereas $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$8 had detectable photometric signatures; for $\hat\theta D_s=\hat\beta D_s+\hat\alpha D_{ds},$9 million monitored stars over $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$0 years, the estimated yields were about $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$1 polarimetric spot detections and $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$2 photometric ones (Sajadian, 2015). Rotation introduces a related class of source effects: stellar ellipticity and gravity darkening shift the timing of polarimetric peaks and generate asymmetric photometric and polarimetric perturbations. With the characterizations adopted for VLT/FORS2, the probability of measuring the ellipticity-induced time shift was reported as only about $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$3, implying that the effect is real but instrumentation-limited in current datasets (Sajadian, 2016).

5. Galactic structure, halo compact objects, and population statistics

In Galactic-structure work, the central statistical observable is the optical depth,

$$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$4

the instantaneous probability that a source lies within one Einstein radius of some lens. It depends on the mass density $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$5 along the line of sight, not on the individual lens masses. The event rate is then related to the timescale distribution by

$$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$6

Toward the Magellanic Clouds, these statistics were historically used to test whether the Galactic halo could be dominated by MACHOs; toward the bulge and disk, they now constrain the stellar and remnant populations of the inner Galaxy (Rahvar, 2015).

The historical Magellanic-Cloud result summarized in the review literature is that, after long-term monitoring by EROS, MACHO, MOA, and OGLE, MACHOs in the mass range $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$7 to a few $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$8 account for less than about $$\hat\alpha=\frac{4GM}{c^2D_d}\frac{\hat\theta}{\theta^2}.$$9 of the halo (Rahvar, 2015). Later work stressed that this conclusion is model-dependent at the level of the exact excluded mass scale. In a dedicated LMC PBH analysis, the smallest PBH mass for which $\theta^2=\beta\theta+\theta_E^2,$0 remained allowed by EROS-2 shifted from $\theta^2=\beta\theta+\theta_E^2,$1 in the fiducial standard halo to $\theta^2=\beta\theta+\theta_E^2,$2 in a “massive” Evans power-law halo and to $\theta^2=\beta\theta+\theta_E^2,$3 in a “light” halo, showing that the largest mass constrained by stellar microlensing can vary by about an order of magnitude under plausible halo assumptions; the same study concluded that these uncertainties weaken but do not eliminate the tension with dynamical and accretion constraints, and that finite-width PBH mass functions generally worsen rather than relieve the tension (Green, 2017).

A later overview synthesized the modern status more broadly: under standard assumptions, current observations exclude compact objects making up all of the dark matter over the mass range

$\theta^2=\beta\theta+\theta_E^2,$4

In much of the planetary-to-stellar mass regime, the limits reach the sub-percent level. This statement folds together the stellar-field microlensing results from Magellanic-Cloud surveys, Galactic-bulge and Galactic-plane programs, Kepler, Subaru/HSC, and long-timescale reanalyses, while also emphasizing that cadence, finite-source effects, and halo modeling remain integral to the interpretation (Green, 17 Feb 2026).

6. Dark remnants and higher-order lens architectures

Long-timescale stellar-field microlensing has long been recognized as a route to dark remnants, but for many years the central difficulty was that photometric timescales alone could not separate a massive dark lens from a slower or more distant ordinary star. A near-infrared VVV event projected 3.5 arcmin from the center of NGC 6553 illustrated this earlier regime: if the lens were a globular-cluster member, the combination of known cluster distance and relative proper motion implied a lens mass of about $\theta^2=\beta\theta+\theta_E^2,$5–$\theta^2=\beta\theta+\theta_E^2,$6, suggestive of a massive stellar remnant and, for low blending, an isolated black-hole candidate, but still model-dependent because neither microlens parallax nor a finite-source angular scale was measured directly (Minniti et al., 2015).

The decisive breakthrough came from joint photometric-plus-astrometric analysis. For MOA-2011-BLG-191 / OGLE-2011-BLG-0462 toward the Galactic bulge, HST astrometry over eight epochs and six years, combined with long-baseline ground-based photometry, measured

$\theta^2=\beta\theta+\theta_E^2,$7

with

$\theta^2=\beta\theta+\theta_E^2,$8

The lens emitted no detectable light, and its mass was too high for a white dwarf or neutron star, yielding the first unambiguous detection and mass measurement of an isolated stellar-mass black hole by microlensing and, more generally, the first direct mass measurement of an isolated stellar-mass black hole by any technique (Sahu et al., 2022).

Population-synthesis interpretation then connected stellar-field microlensing masses to formation channels. In one such analysis, most isolated black holes in the Milky Way, $\theta^2=\beta\theta+\theta_E^2,$9, were predicted to be of binary origin, with low-mass isolated black holes preferentially originating from disrupted binaries and high-mass isolated black holes tending to trace single-star evolution. In that framework, an event like OB110462, if truly free-floating, is more likely to be a binary-origin isolated black hole than a single-star-origin one (Vigna-Gómez et al., 2022).

Stellar-field microlensing has also moved well beyond the single-point-lens approximation. A 2024 simulation study of binary microlensing by high-eccentricity stellar-mass black-hole binaries, including unbound orbits, found that binary-lens models including orbital motion could recover the true parameters of high-eccentricity Galactic black-hole binaries within the $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$0 uncertainty of the inferred values, even for $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$1 (Kim et al., 2024). At the level of higher-order stellar multiplicity, KMT-2021-BLG-1122 required a triple-lens single-source interpretation, with fitted separations and mass ratios $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$2 and $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$3, and a Bayesian physical interpretation $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$4. The body of that paper identifies KMT-2021-BLG-1122L as the first triple stellar system detected via microlensing, expanding the domain of stellar-field microlensing from dark remnants to genuinely higher-order stellar architectures (Han et al., 2023).

7. Expanding observational regimes and future directions

Future stellar-field microlensing work is organized around space baselines, high-precision astrometry, and expansion into new source classes. The long-standing programmatic aims already identified in review work include routine parallax from space-based baselines, astrometric microlensing from missions such as GAIA, and dense bulge monitoring that can ultimately produce a stellar and remnant mass function together with line-of-sight maps of Galactic structure. Earth-plus-satellite parallax with baselines such as Spitzer and dedicated space-based microlensing telescopes was identified early as one of the cleanest ways to break the standard single-event degeneracy (Rahvar, 2015).

That logic now converges on Roman and Rubin. For isolated-remnant searches, Roman was explicitly described as transformative because it should combine HST-like astrometric precision with dense, space-based, high-cadence microlensing photometry over bulge fields, while Rubin should provide wide-area, homogeneous, long-baseline photometric discovery across many Galactic sightlines and thereby identify high-value astrometric follow-up targets (Sahu et al., 2022). The same expansion is visible in extragalactic applications. A forecasting study of gravitationally lensed Type Ia supernovae found that a sample of 50 well-modeled glSNe Ia with single-epoch observations at peak intrinsic luminosity should constrain an average stellar mass-to-light ratio to within $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$5, and that caustic-crossing light-curve information should constrain the mass of the microlenses to within $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$6 (Weisenbach et al., 3 Feb 2025).

An even newer direction is extragalactic stellar-field microlensing through ultra-diffuse galaxies. Using NGC1052-DF2 as a case study, one analysis estimated a total UDG microlensing event rate of $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$7 over its five background galaxies for typical JWST $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$8 mag visits, but only $\theta_E^2=\frac{4GM}{c^2}\frac{D_{ds}}{D_dD_s}.$9 for LSST. The same paper argued that Euclid is better suited for identifying large samples of low-redshift star-forming galaxies seen through local galaxies, for which a zeroth-order calculation gives $$\theta=\frac{\beta\pm\sqrt{\beta^2+4\theta_E^2}}{2}.$$0–$$\theta=\frac{\beta\pm\sqrt{\beta^2+4\theta_E^2}}{2}.$$1 events per year over the whole sky under LSST monitoring. In that regime, stellar-field microlensing is proposed as a route to the IMF and low-mass stellar multiplicity of UDG field stars themselves (Li et al., 13 Apr 2026).

Taken together, these developments preserve the defining logic of stellar-field microlensing while enlarging its domain. The phenomenon remains intrinsically a crowded-field experiment based on unresolved images and transient amplification. What has changed is the scale of the questions it can now address: from MACHO searches and bulge event harvesting to isolated black-hole mass measurement, stellar-surface tomography, higher-order stellar multiples, extragalactic stellar mass calibration, and eventually joint photometric–astrometric surveys in which lens masses, distances, and kinematics are recovered routinely rather than statistically.