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.
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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