Gravitational Microlensing

Gravitational microlensing is an astronomical phenomenon predicted by Albert Einstein's general theory of relativity in which the gravitational field of a compact foreground mass (the lens, such as a star, brown dwarf, planet, white dwarf, neutron star, or stellar-mass black hole) bends and focuses light rays arriving from a more distant background star (the source). Because the angular separation between the split relativistic micro-images for a solar-mass lens at kiloparsec distances inside the Milky Way is of order ~0.5 to 1 milliarcsecond (mas)—far smaller than the ~0.5–1 arcsecond seeing limit of ground-based optical telescopes—the individual images cannot be resolved spatially as in galaxy-scale strong gravitational lensing. Instead, as the foreground lens and background source move relative to one another across the sky with transverse proper motion μ_rel, the unresolved combined flux of the micro-images exhibits a smooth, achromatic, time-symmetric brightening and fading known as a microlensing event. Because gravitational light deflection depends strictly on mass rather than lens luminosity, microlensing provides a unique observational channel in astronomy capable of weighing dark compact remnants, free-floating "rogue" planets, and cold exoplanets orbiting near and beyond the circumstellar water snowline.

In the weak-field, thin-lens limit of general relativity, a light ray passing a point mass M_L at physical impact parameter b is deflected toward the mass by the Einstein deflection angle α_E = 4*GM*_L / (c²b). Consider an observer at distance D_L from the lens and distance D_S from a background source star separated from the lens by an unperturbed angular angle β on the sky. The geometric lens equation relating the true source angle β to the observed image angle θ is:

β = θ − (θ_E² / θ)

When the foreground lens lies directly along the line of sight to the source (β = 0), azimuthal symmetry deforms the background star into a luminous circular annulus known as the Einstein ring, whose characteristic angular radius θ_E is given by:

θ_E = [(4 G M_L / c²) ((D_S − D_L) / (D_L D_S))]^1/2 ≈ 1.0 mas × (M_L / M_☉)^1/2 (D_rel / 8 kpc)^−1/2

where D_rel⁻¹ = D_L⁻¹ − D_S⁻¹ is the relative lens–source parallax distance. At the distance of the lens plane, the physical Einstein radius is R_E = D_L θ_E ≈ 2.5 to 4.0 astronomical units (au) for a typical stellar lens in the Galactic disk or bulge (D_L ≈ 4 kpc, D_S ≈ 8 kpc). It is this fortunate coincidence—that R_E for Galactic bulge microlensing matches the physical orbital radius of the circumstellar water snowline (a ≈ 2–5 au) in protoplanetary disks—that gives gravitational microlensing unmatched sensitivity to cold Jupiter-, Neptune-, and Earth-mass exoplanets.

Single-Lens Image Splitting and the Paczyński Light Curve

For any nonzero angular source–lens separation β > 0, let u = β / θ_E denote the dimensionless impact parameter normalized to the Einstein ring radius. Solving the quadratic lens equation u = y − y⁻¹ (where y = θ / θ_E) yields two distinct micro-images positioned on opposite sides of the lens: a bright major image outside the Einstein ring at y₊ = ½(u + (u² + 4)^1/2) and a fainter, inverted minor image inside the Einstein ring at y₋ = ½(u − (u² + 4)^1/2). Because gravitational lensing conserves surface brightness (I_ν) in accordance with Liouville's theorem, the flux magnification A_± of each image equals the Jacobian determinant of the solid-angle transformation |(y_± / u) (dy_± / du)|. Summing the two image magnifications yields the total point-source, point-lens magnification formula derived by Bohdan Paczyński in 1986:

A(u) = A₊ + A₋ = (u² + 2) / [u (u² + 4)^1/2]

At the boundary of the Einstein ring (u = 1), the source flux is magnified by A(1) = 3 / √5 ≈ 1.34 (a brightening of 0.32 magnitudes), whereas for small impact parameters (u ≪ 1), magnification scales inversely with impact parameter as A(u) ≈ 1 / u.

For uniform rectilinear relative proper motion μ_rel between lens and source, the dimensionless separation evolves in time t as:

u(t) = [u₀² + ((t − t₀) / t_E)²]^1/2

where u₀ is the minimum impact parameter attained at peak magnification time t₀, and t_E = θ_E / μ_rel is the Einstein radius crossing time (or Paczyński timescale). For a stellar lens (M_L ≈ 0.3–1.0 M_☉) with typical Galactic transverse velocity v_⊥ ≈ 100–200 km s⁻¹, t_E ≈ 15 to 40 days, whereas a Jupiter-mass lens (M_L ≈ 10⁻³ M_☉) produces a brief event lasting t_E ≈ 1 to 2 days, and an Earth-mass lens (M_L ≈ 3 × 10⁻⁶ M_☉) yields an ultrashort event of t_E ≈ 1 to 4 hours.

As proposed by Shude Mao and Bohdan Paczyński in 1991 and refined by Andrew Gould and Abraham Loeb in 1992, if the foreground lens star is orbited by an exoplanet of mass m_p (mass ratio q = m_p / M_L ≪ 1) at projected separation s = r_⊥ / R_E near the Einstein ring (s ≈ 0.6 to 1.6, the "lensing zone"), the planet perturbs the gravitational potential of the host star and breaks the circular symmetry of the Einstein ring. Instead of a single central degenerate point, the binary lens produces closed curves of formally infinite point-source magnification in the source plane known as caustics (comprising a central astroid-shaped caustic near the host star and one or two planetary caustics).

When the background source star crosses or passes near a planetary caustic, two extra micro-images are created or annihilated, superimposing a sharp, asymmetric spike or dip lasting a few hours to a couple of days onto the smooth weeks-long stellar Paczyński light curve. By modeling the geometry of the caustic perturbation, astronomers directly extract the planet-to-star mass ratio q and the normalized projected separation s.

Breaking the Microlensing Mass–Distance Degeneracy

In a standard single-lens Paczyński light curve, the three fitted parameters (t₀, u₀, t_E) yield only one physical constraint—the Einstein timescale t_E = θ_E / μ_rel—which entangles three unknowns: the lens mass M_L, the lens distance D_L, and the relative transverse velocity v_⊥. To determine the unique physical mass M_L (in solar masses) and distance D_L (in kiloparsecs) of the lens, two higher-order observables are measured:

1. Finite-Source Effects (ρ_⋆ = θ_⋆ / θ_E): During caustic crossings or high-magnification events, the finite angular radius θ_⋆ of the background source star (calibrated from its de-reddened color and apparent magnitude) smooths the magnification peak, directly measuring θ_E = θ_⋆ / ρ_⋆.
2. Microlensing Parallax (π_E = π_rel / θ_E): Observed either via the annual orbital acceleration of Earth around the Sun or simultaneously from Earth and a space telescope separated by ~0.01 to 1 au (such as Spitzer or the Nancy Grace Roman Space Telescope), parallax shift yields π_E. Combining θ_E and π_E directly solves for the lens mass without any assumptions about Galactic kinematics:

M_L = θ_E / (κ π_E), where κ = 4G / (c² au) ≈ 8.144 mas M_☉⁻¹

High-cadence wide-field surveys by OGLE (Optical Gravitational Lensing Experiment), MOA (Microlensing Observations in Astrophysics), and KMTNet (Korea Microlensing Telescope Network) have also discovered dozens of ultrashort-timescale microlensing events (t_E ≈ 0.05 to 1 day) exhibiting prominent finite-source perturbations (ρ_⋆ ≳ 1) and no detectable host star within 10 au. These observations demonstrate an abundant Galactic population of free-floating (rogue) planets—ranging from super-Earths to Jupiter-mass bodies ejected from nascent planetary systems via planet–planet gravitational scattering or stellar flybys in open clusters.

In 1986, Bohdan Paczyński proposed monitoring millions of stars in the Large Magellanic Cloud (LMC, D_S ≈ 50 kpc) to test whether the dark matter halo of the Milky Way consists of Massive Astrophysical Compact Halo Objects (MACHOs)—such as brown dwarfs, faint white dwarfs, neutron stars, or primordial black holes. The microlensing optical depth τ—defined as the instantaneous probability that any background source star lies within one Einstein ring radius (πθ_E²) of a foreground lens—depends only on the average mass density along the line of sight and is independent of the individual lens masses:

τ = (4π G / c²) ∫₀^D_S ρ_lens(D_L) [D_L (D_S − D_L) / D_S] dD_L ≈ (v_rot / c)² ~ 5 × 10⁻⁷

Dedicated decades-long campaigns toward the Large and Small Magellanic Clouds and the Andromeda Galaxy by the MACHO, EROS, OGLE, and Subaru/HSC collaborations measured an optical depth toward the LMC of τ_LMC < 0.4 × 10⁻⁷, ruling out compact objects across the mass range 10⁻¹¹ M_☉ to 30 M_☉ as the dominant constituent (<10–15%) of Galactic halo dark matter. Conversely, toward the Galactic bulge (b ≈ −2° to −4°), the stellar microlensing optical depth reaches τ_bulge ≈ (1.5–2.5) × 10⁻⁶. By combining photometric microlensing light curves with milliarcsecond astrometric microlensing centroid shifts measured by the Hubble Space Telescope and Gaia (such as event OGLE-2011-BLG-0462 / MOA-2011-BLG-191, analysed by Kailash Sahu et al. and Casey Lam et al. in 2022), astronomers made the first unambiguous dynamical mass measurement of an isolated, X-ray-quiet stellar-mass black hole (M_L = 7.1 ± 1.3 M_☉ at D_L = 1.58 ± 0.18 kpc) wandering through the interstellar medium.

With the Galactic Bulge Time-Domain Survey aboard the Nancy Grace Roman Space Telescope, space-based infrared microlensing will monitor >10⁸ bulge stars at 15-minute cadence, discovering ~1,400 bound cold exoplanets down to the mass of Mars and hundreds of free-floating terrestrial planets.

1. Paczyński, B. (1986). "Gravitational microlensing by the galactic halo." The Astrophysical Journal, 304, 1–5. https://doi.org/10.1086/164140
2. Mao, S., & Paczyński, B. (1991). "Gravitational microlensing by double stars and planetary systems." The Astrophysical Journal Letters, 374, L37–L40. https://doi.org/10.1086/186066
3. Gould, A., & Loeb, A. (1992). "Discovering planetary systems through gravitational microlenses." The Astrophysical Journal, 396, 104–114. https://doi.org/10.1086/171700
4. Gaudi, B. S. (2012). "Microlensing surveys for exoplanets." Annual Review of Astronomy and Astrophysics, 50, 411–453. https://doi.org/10.1146/annurev-astro-081811-125518
5. Sahu, K. C., Anderson, J., Casertano, S., et al. (2022). "An isolated stellar-mass black hole detected through astrometric microlensing." The Astrophysical Journal, 933(1), 83. https://doi.org/10.3847/1538-4357/ac739e