Critical surface mass density

3.3 Critical surface mass density

In the more general case of a three-dimensional mass distribution of an extended lens, the density $ρ(⃗r)$ can be projected along the line of sight onto the lens plane to obtain the two-dimensional surface mass density distribution $Σ (⃗ξ)$ , as

Here $⃗r$ is a three-dimensional vector in space, and $⃗ ξ$ is a two-dimensional vector in the lens plane. The two-dimensional deflection angle $⃗&tidle;α$ is then given as the sum over all mass elements in the lens plane:

For a finite circle with constant surface mass density $Σ$ the deflection angle can be written:

With $ξ = D 𝜃 L$ this simplifies to

With the definition of the critical surface mass density $Σ crit$ as

the deflection angle for a such a mass distribution can be expressed as

The critical surface mass density is given by the lens mass $M$ “smeared out” over the area of the Einstein ring: $Σcrit = M ∕(R2Eπ)$ , where $RE = 𝜃EDL$ . The value of the critical surface mass density is roughly $Σ ≈ 0.8 g cm −2 crit$ for lens and source redshifts of $z = 0.5 L$ and $z = 2.0 S$ , respectively. For an arbitrary mass distribution, the condition $Σ > Σcrit$ at any point is sufficient to produce multiple images.