Gravitational lensing

How it works

Mass bends the path of light. When a galaxy lies almost exactly between us and a distant source, the source’s light reaches us from several directions around it. Perfect alignment produces an Einstein ring with angular radius θE; a small offset breaks the ring into arcs, and further from the axis two images remain. The lens maps every direction θ we look in to the true source position β.

β = θ − θE² / θθE ∝ √(M · Dₗₛ / (Dₗ · Dₛ))

Try it yourself

1. Choose “Einstein ring”. With the default relative mass of 1.00× and the lens at 50 % Dₛ, the ring has radius θE = 0.90 u.
2. Raise “Relative mass M” to 2.00×. The ring grows only √2 times, to about 1.27 u: the radius grows with the square root of the mass.
3. Change “Position between us and the source”. At 20 % Dₛ, θE grows to 1.80 u; at 80 % Dₛ it shrinks to 0.45 u.
4. Click “Luminous arcs” and then “Double image”. The further the source is from the axis, the further one image moves outwards while the other fades near the centre of the lens. Turn off “Lensing” to see where the source really is.

Model limitations

The galaxy is replaced by a point mass, and angles and mass are in normalised units. Distances are simplified to Euclidean ones; the model ignores cosmology, dark matter, time delays and redshift. The galaxy glow and foreground stars are illustrative.

Related experiments

• Black hole
• Special relativity
• Ray optics