LensCalc
Einstein radius, enclosed mass, image positions, magnification and time delay for a singular isothermal sphere or a point-mass lens, in flat ΛCDM.
Lens plane
Source (unlensed position)
Image, positive parity
Image, negative parity
Total magnification vs source offset
Dot: the current β. Hover the curve to read μ at other offsets.
Images
Formulas used
Distances are angular-diameter distances in a flat universe with matter and Λ only (radiation neglected; differences from astropy's Planck18 are below 0.3% at z < 10). Dls = (DC(zs) − DC(zl)) / (1 + zs). M(<θE) is the projected mass inside the Einstein radius, valid for any circularly symmetric lens. Real cluster lenses need a full mass model; the isothermal sphere gives order-of-magnitude numbers.