Gravitational Lensing Formalism - Thin Lens Approximation

Thin Lens Approximation

In the limit of a "thin lens", where the distances between the source, lens, and observer are much larger than the size of the lens (this is almost always true for astronomical objects), we can define the projected mass density

where is a vector in the plane of the sky. The deflection angle is then

$\vec{\hat{\alpha}}= \frac{4 G}{c^2} \int \frac{(\vec{\xi}-\vec{\xi}^{\prime})\Sigma(\vec{\xi}^{\prime})}{|\vec{\xi}-\vec{\xi}^{\prime}|^2}d^2 \xi^{\prime 2}$

As shown in the diagram on the right, the difference between the unlensed angular position and the observed position is this deflection angle, reduced by a ratio of distances, described as the lens equation

where is the distance from the lens to the source is the distance from the observer to the source, and is the distance from the observer to the lens. For extragalactic lenses, these must be angular diameter distances.

In strong gravitational lensing, this equation can have multiple solutions, because a single source at can be lensed into multiple images.

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