Gravitational Lens (2D)

Deflection of light

General relativity deflects a light ray passing a mass M at impact parameter b by α = 4 G M / (c^2 b), twice the Newtonian value. For the Sun's limb this is 1.75 arcseconds, the 1919 eclipse result. The graph compares the full and the Newtonian half value.

Point-mass lens

For a source at angle β behind a point lens the lens equation is β = θ - θ_E^2 / θ with the Einstein angle θ_E = sqrt(4 G M / c^2 · D_ls / (D_l D_s)). It has two solutions, θ± = (β ± sqrt(β^2 + 4 θ_E^2)) / 2, one on each side of the lens. At β = 0 they merge into the Einstein ring. Each image is magnified by μ± = [u^2 + 2 ± ...]; the sum is A(u) = (u^2 + 2) / (u sqrt(u^2 + 4)) with u = β / θ_E.

Microlensing light curve

When a lens drifts in front of a distant star, u changes with time, u(t) = sqrt(u0^2 + ((t - t0)/t_E)^2), and the star's brightness follows the symmetric Paczynski curve A(u(t)); the images are too close to resolve, only the total brightness is seen.

How the picture is made

Every pixel of the image plane is traced back through the lens equation to the source plane and takes the brightness of the source there (surface brightness is conserved), so the arcs, ring and the demagnified inner image are produced by the same equation that gives the numbers. An optional external shear γ breaks the symmetry and splits the ring into arcs.

Simplifications

Thin lens, point mass (plus optional shear), small angles, no source evolution, no time delay, static lens in the picture. Real galaxy lenses have extended mass distributions.

• Drag the source close to the lens axis: the two images stretch into the Einstein ring.
• Increase the shear to see the ring split into four images.
• Choose "Sun's limb" in the deflection graph and read the 1.75 arcsecond value.
• Press the microlensing button and watch the brightness curve while the lens passes.