Point-Mass Gravitational Lensing

In astrophysics, a massive object can bend light from a background source. For a weak-field thin point-mass lens, the normalized lens equation is u = x - 1/x, where u is the source angular position and x is the image angular position, both normalized by the Einstein radius.

Worked example

Consider a source separated by u = 1 from the lens, with an Einstein radius of 1 arcsec. The image positions and total magnification can be derived as follows.

For u = 1 > 0, two images form at normalized positions x± = (u ± √(u² + 4)) / 2.
x+ = (1 + √5) / 2 ≈ 1.618, and x- = (1 - √5) / 2 ≈ -0.618.
Multiply by the Einstein radius (1 arcsec) to get angular positions: 1.618 arcsec and -0.618 arcsec.
Total magnification is μ = (u² + 2) / (u√(u² + 4)) = 3 / √5 ≈ 1.3416.

Native experiment

Open Point-Mass Gravitational Lensing to see how the images move relative to the source.

Set Source separation u to 1 and Einstein radius arcsec to 1. Note the positive image at 1.618 arcsec, the negative image at -0.618 arcsec, and total magnification of ~1.34.
Set Source separation u to 2. The images move to 2.414 arcsec and -0.414 arcsec, and magnification drops to ~1.06, closer to the unlensed case.
Set Source separation u to 0. The two images merge into a full Einstein ring of radius 1 arcsec, representing an infinite point-source magnification limit.

Model limitations

This is a weak-field thin-lens geometric model. It assumes the lens is a mathematical point mass and the deflection angles are small. It is not ray tracing near a black-hole horizon, and it does not account for extended sources, finite lens sizes, or wave optics effects. The negative-position image has negative parity (it is inverted), not negative flux.

References

The physics of gravitational lensing, including the point-lens equation and magnification, is covered in lecture notes such as University of Maryland ASTR 422 Lecture 13.

Point-Mass Gravitational Lensing