Binary-lens microlensing events produce caustic crossings when a background star’s apparent path intersects a caustic created by two foreground masses. At an ordinary fold crossing, a pair of lensed images appears or disappears, producing a rapid change in the star’s observed brightness.
The star does not physically pass through either lens. The crossing describes changing alignment on the sky. Nor does the star become infinitely bright: that prediction belongs to an idealized point-source model. A real star has a finite disk, which turns the mathematical singularity into a finite feature in its light curve.
How two masses create a caustic
A binary lens can consist of two stars or a star and a planet. Both masses deflect the background star’s light. Their combined lens mapping relates the source’s true angular position to the positions of its lensed images.
A critical curve lies in the image plane, where that mapping becomes singular. Mapping it into the source plane produces a caustic. Its smooth sections are called folds; its pointed junctions are cusps. These are geometric features of the lens mapping, not glowing structures surrounding the foreground objects.
The caustic pattern depends on the mass ratio and projected separation. Martin Dominik’s analysis of binary-lens geometry describes three configurations, away from their transition boundaries:
| Configuration | Caustic pattern |
|---|---|
| Close | One four-cusp caustic and two three-cusp caustics |
| Intermediate | One six-cusp caustic |
| Wide | Two four-cusp caustics |
“Close” and “wide” refer to separation relative to the lens’s Einstein scale, a characteristic angular scale set by mass and distance. The boundaries between configurations depend on mass ratio; they are not one universal physical distance.
What changes during a fold crossing
For an isolated binary point-mass lens, a point source away from a caustic has three images outside a caustic region and five inside. At entry through a fold, two images emerge together. At exit, a pair merges and disappears. Ordinary microlensing photometry records their combined light rather than separating the individual images.
Near a generic fold, the extra pair’s magnification follows a simple local rule:
Apair ≈ K / √d
Here, d is the perpendicular source-to-fold distance on the inside, measured in consistent normalized units, and K describes the fold’s strength. This is a local point-source approximation, not a formula for the entire event. The other images also contribute light.
The fold-caustic analysis by B. Scott Gaudi and Arlie Petters derives this image-pair behavior. As the distance approaches zero from inside, the idealized pair’s magnification diverges. That geometric amplification creates the sharp crossing feature.
Why a real star produces a finite peak
A star’s leading edge reaches a fold before its center does, and its trailing edge crosses later. During that interval, different parts of the stellar disk experience different magnifications. The observed flux sums those contributions, weighted by the star’s surface brightness.
This finite-source averaging removes the infinite peak. Limb darkening—the tendency for a stellar disk to look dimmer toward its edge—also changes the profile. A caustic therefore acts like a moving, highly selective magnifier across the star’s face. It does not deliver an ordinary resolved photograph.
Why crossings often appear as two peaks
In a simple passage through a caustic region, entry produces a rapid rise and subsequent decline; approach to the exit produces another rise followed by a rapid decline. The two features mark different boundary crossings, not separate brightenings caused independently by the two lens masses.
However, two clean spikes are not compulsory. A complicated path can cross several folds. Nearby entry and exit features can overlap after averaging over the stellar disk. A trajectory passing near a cusp can also produce a bright peak without crossing the caustic at all. Microlensing Source’s illustrated caustic explanations show the distinction between a crossing and a cusp approach.
Consequently, counting peaks alone cannot identify the lens configuration. The shape and timing of the wider light curve matter too.
What controls the crossing duration?
For a locally straight fold and approximately constant relative motion, the time for the entire stellar diameter to cross is:
Δt ≈ 2θ* / μ⊥
Here, θ* is the star’s angular radius and μ⊥ is its angular speed relative to the caustic, perpendicular to the fold. A shallow crossing angle reduces that perpendicular speed and lengthens the crossing. This duration differs from the time between entering and leaving the whole caustic region.
As a hypothetical example, take an angular radius of 0.5 microarcseconds and perpendicular motion of 5 milliarcseconds per year. The diameter-crossing time is 1 microarcsecond divided by 5,000 microarcseconds per year: approximately 1.75 hours. Halving the perpendicular speed doubles the duration. These values illustrate the calculation; they do not describe a measured event.
What astronomers can learn—and what remains ambiguous
Well-sampled crossings help constrain finite-source effects and stellar brightness profiles. But a crossing alone does not uniquely determine the binary. Dominik’s study of fold-crossing light curves explains why observations outside the crossing are necessary to connect its local behavior to the full lens model.
Even broader observations can admit competing solutions. Research on degeneracies in two-body microlensing shows how distinct configurations can produce similar signals. A dramatic spike is therefore not, by itself, proof of a planet or a unique measurement of its mass.
The essential physical story is geometric: two masses create a structured magnification pattern, relative motion carries the background star across it, and an image pair appears or disappears at a fold. The star’s finite size shapes that transition into the brightness feature astronomers measure.