Gravitational Lensing Calculator

The Einstein radius is the angular radius of the ring-shaped image that forms when a background source, lens, and observer are in perfect alignment. It depends on the lens mass and the geometry of the lens-source-observer system. Larger lens masses and more favorable distance configurations produce larger Einstein radii. It is typically measured in arcseconds for galaxy-scale lenses.

General relativity predicts a deflection angle α = 4GM / (c² × b), where G is Newton's gravitational constant, M is the lens mass, c is the speed of light, and b is the impact parameter (closest approach distance of the light ray). This is exactly twice the Newtonian prediction — a key confirmation of Einstein's theory made during the 1919 solar eclipse.

An Einstein ring is the perfect circular image of a background source that appears when the source, lens, and observer are exactly aligned along a straight line. In practice, exact alignment is extremely rare. Near-perfect alignments produce partial arcs. The angular radius of the ring is the Einstein radius, which this calculator computes from the lens mass and distances.

Magnification (μ) describes how much brighter a lensed source appears compared to its unlensed brightness. Lensing preserves surface brightness but distorts and stretches images, increasing their solid angle on the sky and thus the total flux. A magnification of 10 means the source appears 10 times brighter than it would without lensing. High magnification occurs when the source is very close to the Einstein ring.

Strong lensing produces dramatic, easily visible distortions — multiple images, arcs, or Einstein rings — and occurs when the source lies very close to the Einstein radius. Weak lensing causes subtle statistical shape distortions in background galaxy populations and requires averaging over many galaxies to detect. Microlensing is a third regime involving compact lens objects like stars, causing temporary brightness fluctuations.

Lensing efficiency depends on the effective angular diameter distance combination D_eff = D_L × D_LS / D_S, where D_L is the distance to the lens, D_S is the distance to the source, and D_LS is the distance between lens and source. A lens placed at roughly half the distance to the source produces the strongest lensing effect. Very nearby or very far lenses relative to the source reduce the lensing efficiency.

Yes — gravitational lensing is one of the most powerful probes of dark matter because it responds to total mass regardless of whether the mass emits light. By mapping lensing distortions of background galaxies, astronomers can reconstruct the full mass distribution of galaxy clusters and cosmic filaments, revealing dark matter halos that far outweigh the visible baryonic matter.

The Hubble constant determines the physical scale of the universe, which affects how angular diameter distances are computed from redshifts. Since the Einstein radius and lensing geometry depend on these distances, the inferred lens mass from a lensing observation depends on the assumed cosmology. Interestingly, time delays between multiple lensed images of quasars can independently constrain the Hubble constant — a method called H0LiCOW.