Gravitational Lensing
Point-mass lens: Einstein angle, image positions and point-source magnification. A finite source is imaged through the lens equation.
About this tool
Select an example, change the offset or radius and Calculate. The lensed image comes first, with the unlensed circular source beside or below it. Both use the same angular scale in θ_E; the orange dashed curve marks the Einstein angle. An exactly centred source makes a finite annulus; off-axis sources make arcs or distorted images. Reset restores the two-arcs example and image view, and clears results without calculating again.
Distances: Dₗ, Dₛ and Dₗₛ are angular-diameter distances observer–lens, observer–source and lens–source in Mpc. Cosmological distances do not obey Dₗₛ=Dₛ−Dₗ, so all three are entered separately. They must come from the same cosmology. Positive ranges are checked, not cosmological consistency. Examples with additive distances are Euclidean teaching cases; 0.004 Mpc = 4 kpc.
Model: θ_E²=4GM Dₗₛ/(c²DₗDₛ). Table positions θ₊, θ₋ and μ=(u²+2)/(u√(u²+4)) describe an ideal point source at the source centre, u=β/θ_E. μ is not the total magnification of the finite disk. Only the point-source model diverges at u=0. The uniformly bright finite-source image follows the inverse lens equation β⃗=θ⃗−θ⃗/|θ⃗|² in Einstein units, conserving surface brightness. Raster resolution limits thin arcs.
Single point mass, thin lens and small deflection angles: no realistic galaxy profile, shear, time delay, wave optics, black-hole shadow or observed image. 3D uses exaggerated angles and arbitrary plane spacing; it is not physical ray integration. Without WebGL, calculations and source images remain available.