ONE FORMULA, ONE ANSWER
Einstein Ring Angular Radius Calculator
Calculate the Einstein ring angular radius for an aligned point lens in local Euclidean geometry. Choose input units and calculate one value within the stated conditions.
Inputs
Calculations run in this browser. Scientific notation such as 1.2e-3 is supported. Do not group thousands.
Result
Einstein ring angular radius θ_E
Enter values and select Calculate.
θ_E = √[(4GM/c²)(D_S−D_L)/(D_L D_S)]
Assumptions
Required: M>0 and D_S>D_L>0 with perfect alignment, a point mass and the thin-lens weak-field small-angle approximation. This local Euclidean model sets D_LS=D_S−D_L; this subtraction is invalid for general cosmological angular-diameter distances. The app requires θ_E≤0.1 rad and impact parameter b=D_Lθ_E≥100r_s. Rays must pass outside lens material without occultation. Excluded: extended mass distributions, strong fields, misalignment and cosmic expansion.
Displayed values are floating-point approximations. Design safety factors and unprovided error contributions are not calculated.
Formula references and credits
NaNa Lab created the code, descriptions, icons and preview images. Text, figures and code from the linked references have not been reproduced.
Physical constants: NIST / CODATA 2022. The values of μ₀ and ε₀ are measured recommendations, not exact defining constants. Values and precision notes are in the source record.
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