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
⃗ ∫ DS Σ(ξ) = 0 ρ(⃗r)dz. (12 )
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:
4G ∫ (⃗ξ − ⃗ξ′)Σ(⃗ξ′) ⃗&tidle;α(⃗ξ) = --2 -------------d2ξ′. (13 ) c |⃗ξ − ⃗ξ′|2
For a finite circle with constant surface mass density Σ the deflection angle can be written:
2 α(ξ) = DLS-4G- Σπ-ξ-. (14 ) DS c2 ξ
With ξ = D 𝜃 L this simplifies to
α(𝜃) = 4πG-Σ- DLDLS--𝜃. (15 ) c2 DS
With the definition of the critical surface mass density Σ crit as
2 Σ = --c----DS---, (16 ) crit 4πG DLDLS
the deflection angle for a such a mass distribution can be expressed as
Σ &tidle;α (𝜃) = ----𝜃. (17 ) Σcrit
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.
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