In gravitational microlensing, astronomers often begin with a convenient simplification: they treat the distant source star as a single point of light. Usually, that approximation works remarkably well. But when the lens passes extremely close to the source, the star’s actual angular size can no longer be ignored.
The resulting change in the observed light curve is called the finite-source effect. Instead of magnifying every part of the background star equally, the gravitational lens magnifies different regions of the stellar disk by different amounts. The telescope records the combined light from all those regions.
This can smooth or reshape the peak of a microlensing event, alter the dramatic features produced near caustics, and even allow astronomers to measure quantities that cannot normally be extracted from a simple point-source light curve.
Why Microlensing Usually Treats a Star as a Point
Gravitational microlensing occurs when a foreground object passes close to the line of sight between Earth and a more distant star. The gravity of the foreground object bends the background star’s light, temporarily increasing its apparent brightness.
NASA’s overview of gravitational microlensing describes this basic geometry: sufficiently close alignment allows the foreground object to act as a natural gravitational lens, while planets orbiting that lens can produce additional disturbances in the magnification pattern.
For most events, the apparent angular diameter of the source star is tiny compared with the angular scale over which the lensing magnification changes. Treating the source as a mathematical point therefore produces an excellent approximation.
For a point-mass lens and a point source, the magnification is commonly written as:
A(u) = (u² + 2) / [u√(u² + 4)]
Here, u is the angular separation between the lens and the center of the source, expressed in units of the angular Einstein radius.
The catch appears when u becomes extremely small. In the ideal point-source model, perfect alignment with a point lens would make the mathematical magnification diverge. A real star, however, is not infinitely small. Its finite disk spreads the lensing signal over a measurable angular area, preventing that point-source singularity from describing the physical observation.
What the Finite-Source Effect Actually Means
Imagine that the gravitational lens is moving across the apparent disk of a distant star. Light coming from the portion of the star closest to the lens may be magnified much more strongly than light coming from the opposite side.
The telescope cannot normally resolve those individual patches of the distant star. It simply measures their combined brightness. Astronomers therefore calculate the observed magnification by integrating the lensing magnification across the projected surface of the source star, weighted by the brightness of each region.
That integration is the heart of finite-source microlensing.
In other words, the lens no longer magnifies a geometric point. It magnifies an extended stellar disk.
The Key Parameter: ρ = θ* / θE
The importance of the finite-source effect is commonly described using the dimensionless parameter ρ:
ρ = θ* / θE
Here, θ* is the angular radius of the source star and θE is the angular Einstein radius of the lensing system.
This ratio provides an intuitive way to think about the phenomenon. If the source star looks extremely small compared with the Einstein radius, ρ is small and the point-source approximation is usually adequate except during exceptionally close approaches to high-magnification structures.
If the angular source radius becomes significant relative to the Einstein radius, finite-source effects become much harder to ignore.
The lens-source separation also matters. Even a relatively small source can show conspicuous finite-source effects if the lens passes close enough to its disk. In binary or planetary microlensing, the effect can become especially important when the source approaches or crosses a caustic, a boundary in the lensing map where point-source magnification changes extremely rapidly.
What Happens to the Microlensing Light Curve?
A point source samples the magnification pattern at essentially one location at a time. An extended star samples a whole patch of that pattern simultaneously.
The result is a kind of spatial averaging.
For a very close single-lens event, the finite size of the source can make the peak broader and less sharply amplified than the ideal point-source prediction. Instead of the lensing signal becoming arbitrarily large as the separation approaches zero, the observed magnification remains finite because the measured light comes from the entire stellar disk.
Near a binary-lens caustic, the effect can be even more striking. Different portions of the star cross the high-magnification region at different times. The resulting light curve contains information about the angular size and surface-brightness structure of the star.
So finite-source effects do not simply make an event “weaker.” They change the shape of the light curve in a way that depends on the geometry of the event, the source radius, and the brightness distribution across the stellar surface.
Why Finite-Source Effects Are Scientifically Valuable
A standard microlensing light curve often provides an Einstein crossing time, tE. Unfortunately, that time alone does not uniquely reveal the mass of the lens because it depends on several quantities, including lens mass, lens and source distances, and their relative motion.
A measurable finite-source effect provides an additional piece of the puzzle: ρ.
If astronomers can independently estimate the angular radius of the source star, θ*, from its color, brightness, and stellar properties, they can calculate the angular Einstein radius:
θE = θ* / ρ
This is one of the major scientific rewards of detecting a finite-source signature.
A well-known observational example is OGLE-2003-BLG-262. In that event, the lens crossed the face of a K giant source. Yoo and colleagues used the finite-source signal to measure an angular Einstein radius of about 195 microarcseconds and thereby place substantially stronger constraints on the lens than the event timescale alone could provide.
Once θE is known, astronomers can also infer the lens-source relative proper motion using approximately:
μrel = θE / tE
If a microlensing parallax measurement is also available, combining parallax information with θE can go much further, potentially allowing the lens mass and distance to be determined rather than merely statistically estimated.
Why Low-Mass Lenses Can Show Strong Finite-Source Effects
Finite-source effects are particularly interesting in searches for very low-mass lenses, including free-floating planet candidates.
The reason comes from the Einstein radius. Its angular size decreases as lens mass decreases, all else being equal. Because ρ is the source angular radius divided by the Einstein radius, a smaller θE can produce a larger value of ρ.
A normal background star can therefore occupy a surprisingly large fraction of the lensing scale when the lens has planetary mass.
This creates an interesting observational trade-off. A large finite source can smooth the magnification pattern and reduce a sharp peak, potentially making some events harder to detect. At the same time, detecting the finite-source signature gives researchers access to θE, which is particularly valuable when trying to determine whether an unusually short microlensing event was caused by a planetary-mass object.
Finite-Source Effects Can Reveal Limb Darkening
A star is not a uniformly illuminated disk. In visible wavelengths, the center generally appears brighter than the outer edge, or limb. This phenomenon is known as limb darkening.
When a gravitational lens passes across or extremely close to a stellar disk, it magnifies different radial zones at different times. In favorable events with sufficiently precise and frequent observations, the microlens effectively scans the star’s surface-brightness profile.
Finite-source modeling therefore may need to include a limb-darkening law rather than assuming uniform brightness. Research on microlensing transit events has shown that such events can provide information about the atmospheric intensity profiles of distant stars that normally cannot be spatially resolved directly.
This is an elegant side benefit of microlensing: an event originally detected because of the foreground lens can become a probe of the background star itself.
Finite-Source Effect vs. Finite-Lens Effect
The terms finite-source effect and finite-lens effect should not be confused.
A finite-source effect concerns the nonzero angular size and surface-brightness distribution of the background source. Instead of treating the source as a point, the calculation integrates magnification across its disk.
A finite-lens effect concerns the physical size or opacity of the foreground lens. In suitable geometries, the lens itself may block part of a lensed image or otherwise introduce effects that would not occur if the lens were treated as an infinitesimal point mass.
Both depart from idealized point-like models, but they describe physically different parts of the lensing system.
A Simple Way to Picture the Effect
Think of a microlensing magnification map as terrain containing steep hills. A point source behaves like the tip of a pencil moving across that terrain: at every moment it samples one precise location.
A finite star behaves more like a coin sliding across the same terrain. Different parts of the coin cover different elevations at the same time, so the measured result represents an average over an area rather than the value at one point.
If the terrain is nearly flat across the coin, the distinction hardly matters. If the coin passes over a sharp ridge, however, averaging across its surface changes the signal dramatically.
In a microlensing event, caustics and extremely close lens-source alignments provide those steep ridges.
Does Every Microlensing Event Show a Finite-Source Effect?
No. Most ordinary microlensing events can be modeled very accurately with the point-source approximation because the apparent stellar disk is tiny relative to the scale over which magnification changes.
A detectable finite-source signal generally requires favorable geometry, sufficient photometric precision, and observations taken frequently enough to resolve the altered part of the light curve.
This last requirement matters because the strongest finite-source features can occur over only a small fraction of the total event. An event lasting weeks might contain a scientifically crucial source-crossing feature lasting hours.
The Bottom Line
The finite-source effect in gravitational microlensing occurs when the angular size of the background source star becomes large enough relative to the lensing magnification pattern that the star can no longer be treated as a single point of light.
Different parts of the stellar disk experience different magnifications, so astronomers must integrate the lensing effect across the surface of the source. This modifies the observed light curve, particularly during extremely close lens passages and caustic crossings.
Far from being merely an inconvenience in the model, the effect can unlock extra information. Measuring the normalized source radius ρ and estimating the source’s angular radius θ* yields the angular Einstein radius θE. That measurement can constrain the relative motion of the lens and source and, when combined with additional information such as microlens parallax, help reveal the lens’s physical mass and distance.
So the moment when the point-source approximation breaks is often exactly when a microlensing event becomes more informative.