Gravitational Lensing Calculator

The Einstein radius is the angular radius of the ring that would form if the source, lens, and observer were perfectly aligned. It sets the characteristic scale of lensing: for angular separations much less than the Einstein radius, lensing effects are strong (multiple images, high magnification). For separations much greater, lensing effects are weak.

An Einstein ring is the circular image of a distant source that forms when the source, lens, and observer are perfectly aligned along the same line of sight. Einstein rings are rare but have been observed by the Hubble Space Telescope in galaxy-lens systems. They provide extremely precise measurements of the lens mass through the Einstein ring radius.

Strong lensing produces visibly distinct multiple images, arcs, or rings around the lens, detectable in individual images. Weak lensing produces tiny coherent distortions (shear) of background galaxy shapes that are only detectable statistically by analyzing thousands of galaxies. Weak lensing maps the dark matter distribution on large scales without requiring detection of individual lensing events.

When a star and its planet pass in front of a more distant background star, the gravitational field of the foreground star (and planet) briefly magnifies the background star. The light curve shows a smooth magnification peak from the star, sometimes with a brief additional spike from the planet. This method can detect planets of any mass, including those in the outer reaches of their solar systems, and has found dozens of exoplanets.

When a quasar is lensed into multiple images, the different images traverse different paths around the lens and thus take different times to arrive at Earth. Measuring this time delay between image flux variations, combined with a model of the lens mass distribution, gives the Hubble constant H0 independently of other methods. This approach is called time delay cosmography or H0LiCOW.

The Hubble Frontier Fields is a program that used the Hubble Space Telescope to obtain ultra-deep images of six massive galaxy clusters. The clusters act as gravitational telescopes, magnifying background galaxies by factors of up to 50-100, allowing Hubble to see galaxies that would otherwise be too faint to detect — some of the most distant galaxies ever observed.

During the total solar eclipse of 1919, Arthur Eddington led expeditions to Sobral (Brazil) and Principe (Africa) to photograph stars near the eclipsed Sun. They measured the apparent positions of these stars and compared them to their positions when the Sun was not nearby. The shifts matched Einstein's prediction of 1.75 arcseconds (twice the Newtonian prediction), confirming general relativity and making Einstein an international celebrity.

Gravitational lensing conserves surface brightness — it magnifies both the area and the flux of a source proportionally, so extremely faint sources become bright enough to detect. This has allowed telescopes to observe galaxies at redshifts above z = 10 that would otherwise be undetectable, effectively extending the reach of our telescopes beyond what their aperture alone would allow.

A point mass lens produces simple symmetric Einstein rings or double image geometries. Extended mass distributions (like galaxies or clusters) produce more complex image configurations — quadruple images (quads), irregular arcs, and caustic patterns. Modeling these requires detailed mass models that reveal the internal dark matter distribution. The complexity of lensed image configurations is a direct probe of the lens mass structure.