Einstein Ring Angular Radius Calculator

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, with all three distances in kiloparsecs and the mass in solar masses:

θ_E ≈ 0.002854" × √(M/M☉ × D_LS / (D_L × D_S))

Sanity check that coefficient against a realistic case: a 10¹¹ M☉ galaxy a gigaparsec away (D_L = 10⁶ kpc) lensing a source at D_S = 2 × 10⁶ kpc, with D_LS = 10⁶ kpc, gives 0.002854 × √(10¹¹ × 5 × 10⁻⁷) = 0.002854 × 223.6 = 0.64 arcseconds. That is the scale real galaxy-galaxy lenses are found at, and it is why they need Hubble or adaptive optics to resolve. If you see this written with a coefficient near 0.9, that version is for mass in units of 10¹¹ M☉ and distances in gigaparsecs. Mixing the two up moves the answer by a factor of about 300.

Put the same galaxy a thousand times closer, at 1000 kpc, and the ring swells to 20 arcseconds. Nothing is wrong with the arithmetic; it is just that no galaxy that near is lined up behind anything, so rings that large in practice come from clusters rather than single galaxies.

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.