Gravitational Microlensing Magnification Calculator

Microlensing Magnification Introduction

Gravitational microlensing magnification is easiest to understand when a compact foreground lens drifts almost directly across the line of sight to a distant star. The lens may be far too faint to detect on its own, but its gravity can still bend the source light enough to create a temporary rise in brightness. In the point-lens picture, the split images are usually too close together to resolve, so observers record a single light curve that brightens and then fades as the alignment changes. That brief flash is what makes microlensing such a valuable way to find dim stars, brown dwarfs, free-floating planets, and other compact objects.

This calculator focuses on the three quantities that matter most in the simplest microlensing event: the Einstein radius, the peak magnification, and the Einstein crossing time. The Einstein radius gives the size of the lensing zone in physical units. The peak magnification tells you how much brighter the source becomes at closest approach. The crossing time estimates how long the lens needs to move across one Einstein radius at the chosen transverse speed. Read together, those outputs give a compact description of the strength, scale, and duration of the event.

For students, observers, and anyone sketching a microlensing scenario, the calculator provides a fast way to see how mass, distance, alignment, and velocity interact. A heavier lens usually enlarges the Einstein radius. A smaller impact parameter usually raises the peak magnification. A slower relative velocity usually stretches the event out over more days. Seeing those trends numerically is often the quickest way to connect the equations to the shape of a real light curve.

Gravitational Microlensing and the Einstein Ring

In a gravitational microlensing magnification problem, the foreground lens and background source are separated by a tiny angle on the sky, but that tiny offset is enough for the lens to act like a natural telescope. The source does not need to line up perfectly for the brightness to change; even a near miss can produce a measurable signal. Surveys toward the Galactic bulge, the Magellanic Clouds, and other crowded fields use this effect to identify objects that would otherwise be invisible because they emit little or no light.

Central to a microlensing magnification calculation is the Einstein radius, the scale at which the lensing geometry becomes most effective. For a point mass lens of mass $M$, a lens distance $D\_l$, and a source distance $D\_s$, the angular Einstein radius is given by

${\mathrm{\backslash theta}}_E=\sqrt{\frac{4GMD\_s-D\_l}{c^2D\_lD\_s}}$.

In practice, astronomers often work with the physical Einstein radius $R_E=D\_l{\mathrm{\backslash theta}}_E$, which usually spans a few astronomical units for stellar-mass lenses in the Milky Way. The apparent magnification depends on the projected lens-source separation. Let $u$ denote the separation between lens and source in units of the Einstein radius. The magnification for a single point lens is

$A\left(u\right)=\frac{u^2+2}{u\sqrt{u^2+4}}$.

When the lens passes directly in front of the star ($u=0$), the point-source formula predicts an infinite magnification, although real stars have finite size and the peak stays finite. For larger impact parameters, the amplification drops quickly. Events are usually summarized by the minimum impact parameter $u\_0$, which occurs at closest approach. The brightness curve is symmetric in the simplest model and follows the Paczyński profile as the lens moves in and out of alignment.

The Einstein radius also sets the characteristic duration of the event. If the relative transverse velocity between lens and source is $v$, the Einstein timescale is $t\_E=\frac{R_E}{v}$. For typical Galactic lenses with $M$ on the order of one solar mass and velocities around 200 km/s, $t\_E$ runs from days to months. Observing programs use this timescale to plan cadence and to separate microlensing from ordinary stellar variability, which often evolves on different timescales.

How to Use the Microlensing Magnification Calculator

To use the gravitational microlensing magnification calculator, enter the lens mass in solar masses, the lens distance and source distance in kiloparsecs, the minimum impact parameter u₀ as a dimensionless number, and the relative transverse velocity in kilometers per second. Then press the compute button. The calculator returns the Einstein radius in astronomical units, the peak magnification at closest approach, and the Einstein timescale in days.

Each field controls a different part of the microlensing geometry. The lens mass sets how strongly gravity bends the light. The lens distance is the observer-to-lens distance. The source distance is the observer-to-source distance. The impact parameter u₀ is the closest projected separation in Einstein-radius units, so smaller values mean a tighter alignment and a brighter peak. The relative velocity is the effective transverse speed of the event across the sky. The calculator handles the unit conversions internally, so you can work in the displayed astronomy units without converting them by hand.

For the formulas to make physical sense, the source should be farther away than the lens. If the lens is placed beyond the source, the standard microlensing geometry no longer applies and the Einstein radius expression is not meaningful. It is also best not to enter exactly zero for the impact parameter, because the ideal point-source model predicts an infinite peak there. Real observations are always softened by finite source size, blending from nearby stars, and instrument limits.

Microlensing Magnification Formula

The gravitational microlensing magnification calculator uses the standard point-lens, point-source relations. The physical Einstein radius is computed from the lens mass and the observer-lens-source geometry. In the JavaScript, the mass is converted from solar masses to kilograms, distances are converted from kiloparsecs to meters, and the result is reported in astronomical units for readability. The peak magnification is then evaluated from the minimum impact parameter, and the event timescale is found by dividing the Einstein radius by the transverse speed.

Put another way, stronger gravity and a lens placed in a favorable position produce a larger Einstein radius. A larger Einstein radius means the lens influences light over a broader region, which usually stretches the event over a longer time. The magnification expression depends only on the dimensionless separation u. That is why two microlensing events can share the same peak magnification even if their lens masses are very different, while their durations still diverge because the physical Einstein radius changes.

It is often more useful to read the outputs together than separately. A very bright event with a short timescale may require frequent monitoring to catch the maximum. A milder brightening with a long timescale can be easier to follow but less dramatic. The Einstein radius also gives a sense of the physical scale of the lensing region, which matters when thinking about planetary perturbations, binary lenses, or other departures from the simplest model.

Gravitational Microlensing Magnification Example

Suppose you enter a lens mass of 0.3 solar masses, a lens distance of 4 kpc, a source distance of 8 kpc, an impact parameter of 0.1, and a relative velocity of 200 km/s. That combination is a reasonable toy model for a Galactic bulge microlensing event. With those inputs, the Einstein radius comes out to a few astronomical units, the peak magnification is about 10, and the event timescale is on the order of a few weeks. In practical terms, the source would brighten by roughly a factor of ten at maximum and remain observably magnified long enough for repeated follow-up.

This example makes the role of alignment very clear. If you keep the mass and distances fixed but increase u₀ from 0.1 to 0.5, the peak magnification falls quickly. The event still happens, but it is far less striking. If you keep the geometry fixed and increase the lens mass, the peak magnification does not change directly in the point-lens formula, but the Einstein radius and the timescale both increase. That distinction is important: in the simplest model, mass controls the scale and duration, while the closest approach controls the brightness peak.

The table below shows the same idea across a few sample microlensing setups. Each row assumes a relative velocity of 200 km/s, a common order-of-magnitude for stellar motions in the Galactic bulge. The magnification column gives the peak brightening of the source star, while the timescale column shows how long the lens needs to traverse one Einstein radius.

The table highlights the trends that matter most when reading a microlensing curve. Increasing the lens mass enlarges the Einstein radius, which lengthens the event. Moving the lens closer to the observer also increases the radius because the geometry places the lens more centrally along the line of sight. The impact parameter has an especially strong effect: even a modest increase in $u_0$ can cut the magnification sharply, which is why the most dramatic events are comparatively rare. Survey teams therefore monitor dense star fields to maximize the odds of catching close alignments.

Gravitational Microlensing Magnification Limitations and Assumptions

This calculator uses the simplest microlensing magnification model: a single point-mass lens and a point-like source. That approximation is excellent for building intuition, but real events can be more complicated. Binary lenses, planetary companions, Earth’s orbital motion, finite-source effects, limb darkening, and blended light from nearby stars can all reshape the observed light curve. In those cases, the peak magnification and timescale from this tool should be treated as baseline estimates rather than full observational predictions.

Another limitation is that the reported timescale is the Einstein crossing time, not necessarily the full interval during which a survey would flag the event. Detectability depends on cadence, photometric precision, source brightness, extinction, and on how much brightening is needed to rise above the noise. The magnification formula also assumes a point source. When the source star has a meaningful angular size compared with the lensing geometry, the formal divergence at very small u₀ is softened and the peak becomes rounded.

The calculator also assumes that the user supplies sensible inputs. If the source distance is less than or equal to the lens distance, or if the velocity is zero or negative, the output will not correspond to a realistic microlensing event. The script has no extra astrophysical validation beyond the formulas shown here, so it is best used for quick theoretical estimates rather than as a substitute for a full light-curve fit to actual data.

Microlensing light curves are usually symmetric and achromatic—independent of wavelength—unless additional physics is present. Planetary companions can create short anomalies, binary lenses can produce caustic crossings with sharp spikes in brightness, and finite-source effects can round off the peak when $u_0$ becomes comparable to the source star's angular radius. Studying those deviations lets astronomers infer planet masses, binary fractions, and even the sizes of distant stars.

For students and enthusiasts, experimenting with this calculator is a practical way to see how microlensing observables connect to the underlying parameters. By changing the lens mass or distances, you can see how the Einstein radius responds and build intuition for the scale of gravitational lensing in our Galaxy. Exploring the dependence on impact parameter and velocity shows why some events fade after only a few days while others remain bright for months. With that intuition, observers can plan monitoring campaigns, and theorists can explore the conditions under which compact objects—such as rogue planets or primordial black holes—might announce themselves through a brief gravitational signature.