Einstein Ring Angular Radius Calculator
Calculate the Einstein ring radius for gravitational lensing.
Enter lens mass and distances to find the angular radius in arcseconds.
Gravitational lensing occurs when a massive object bends the light from a more distant source. When the source, lens, and observer are perfectly aligned, the source appears as a ring — an Einstein ring.
Einstein ring angular radius:
θ_E = √(4GM/c² × D_LS / (D_L × D_S))
In arcseconds (using distances in kiloparsecs and mass in solar masses):
θ_E ≈ √(M/M☉ × D_LS / (D_L × D_S × kpc)) × 0.9026" (approximate)
Where:
- D_L = distance from observer to lens (lensing object)
- D_S = distance from observer to source
- D_LS = distance from lens to source
What it tells us: The Einstein radius defines the region of maximum magnification. Light sources within θ_E of the lens-observer line are strongly magnified. Objects further than θ_E are weakly lensed.
Physical Einstein radius: The actual physical size of the Einstein ring on the lens plane:
R_E = θ_E × D_L (in AU or km)
Famous Einstein rings:
- B1938+666: one of the first complete Einstein rings discovered (1998)
- SDSS J0946+1006: a “double Einstein ring” with two lensed galaxies
- Gravitational lensing has been observed for everything from stars (microlensing) to entire galaxy clusters
Microlensing: When a compact object (star, brown dwarf, or planet) passes in front of a background star, the Einstein radius determines the magnification event duration. Used to detect dark matter candidates and extrasolar planets.