Gravitational Microlensing and the Einstein Ring

Gravitational microlensing occurs when a massive object passes nearly in front of a distant background star. The gravitational field of the foreground lens bends the light from the source, producing multiple images whose magnified light briefly brightens the star. Unlike strong lensing by galaxies, microlensing images are so close together that they cannot be resolved with current telescopes; instead, observers see a time-dependent change in brightness. Microlensing surveys toward the Galactic bulge and the Magellanic Clouds have revealed thousands of such events, providing insights into the distribution of stellar and sub-stellar objects and offering a means to detect dark matter candidates.

Central to microlensing is the concept of the Einstein radius, the angular scale at which light from the source is significantly deflected. 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 typically spans a few astronomical units for stellar-mass lenses within the Milky Way. The apparent magnification of the source depends on how closely the lens and source align. 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 magnification formally diverges, though in reality the finite size of the star limits the peak brightness. For larger impact parameters, the magnification decreases rapidly. Events are typically characterized by their minimum impact parameter $u\_0$, which occurs at the time of closest approach. The brightness as a function of time follows a symmetric curve described by the Paczyński profile, rising as the alignment improves and falling as the lens moves away.

The Einstein radius also sets the characteristic timescale 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, $t\_E$ ranges from days to months. Monitoring campaigns use this timescale to schedule observations and to distinguish microlensing from intrinsic stellar variability, which often occurs on different timescales.

The calculator above takes as input the lens mass, distances to the lens and source, the minimum impact parameter in units of the Einstein radius, and the relative velocity. From these it computes the physical Einstein radius, the expected peak magnification, and the event duration. Distances are expressed in kiloparsecs (kpc), where one kpc equals 3,262 light-years, a convenient unit for Galactic-scale phenomena. The lens mass is specified in solar masses, and the velocity in kilometers per second. Internally, the script converts all quantities to SI units, evaluates the Einstein radius using the formula above, computes the magnification using the Paczyński relation, and derives the timescale.

Gravitational microlensing provides a unique probe of compact objects regardless of the light they emit. The technique has been used to discover exoplanets, including some of the lowest-mass planets ever detected, by analyzing small deviations in the light curve caused by a planet orbiting the lens. It has also placed limits on the abundance of dark compact objects in the Milky Way halo, constraining models of dark matter composed of MACHOs—massive compact halo objects. Future surveys such as the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) and the Nancy Grace Roman Space Telescope will monitor millions of stars and detect microlensing events across the sky, potentially revealing isolated black holes and free-floating planets.

To illustrate how the inputs affect the results, consider the example combinations listed in the table. Each row assumes a relative velocity of 200 km/s, a common value for stellar motions in the Galactic bulge. The magnification column shows the peak brightening of the source star, while the timescale indicates how long the lens takes to traverse one Einstein radius.

Lens Mass (M☉)	Dl (kpc)	Ds (kpc)	u0	Peak A	tE (days)
0.3	4	8	0.1	10.0	20
1.0	6	8	0.3	3.5	40
5.0	3	10	0.5	1.7	90
10.0	2	9	0.2	5.0	150

The table demonstrates several trends. Increasing the lens mass enlarges the Einstein radius, which lengthens the event and raises the magnification for a fixed impact parameter. 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 exerts a strong influence: even a modest increase in $\mathrm{u\_0}$ dramatically lowers the magnification, highlighting the rarity of high-magnification events. Survey strategies often prioritize dense star fields to maximize the chance of near-perfect alignments.

Microlensing light curves are typically symmetric and achromatic—independent of wavelength—unless additional physics is involved. Planetary companions to the lens introduce short-lived anomalies, binary lens systems create caustic crossings with sharp spikes in brightness, and finite-source effects round off the peak when $\mathrm{u\_0}$ becomes comparable to the angular radius of the source star. Studying these deviations allows astronomers to extract rich information about planetary systems, binary star fractions, and even the sizes of distant stars.

For students and enthusiasts, experimenting with this calculator offers insight into how microlensing observables relate to the underlying physical parameters. By adjusting the lens mass or distances, one can see how the Einstein radius responds, building intuition about the scale of gravitational lensing in our Galaxy. Understanding the dependence on impact parameter and velocity sheds light on why some events last only days while others unfold over several months. With these tools, observers can plan monitoring campaigns, and theoreticians can explore the conditions under which exotic objects—like rogue planets or primordial black holes—might be detected through their fleeting gravitational fingerprints.