3.2 Einstein radius

For a point lens of mass M, the deflection angle is given by Equation (4View Equation). Plugging into Equation (6View Equation) and using the relation ξ = DL 𝜃 (cf. Figure 3View Image), one obtains:
DLS---4GM--- β (𝜃) = 𝜃 − DLDS c2𝜃 . (8 )
For the special case in which the source lies exactly behind the lens (β = 0), due to the symmetry a ring-like image occurs whose angular radius is called Einstein radius 𝜃E:
∘ ------------- 4GM---DLS-- 𝜃E = c2 D D . (9 ) L S
The Einstein radius defines the angular scale for a lens situation. For a massive galaxy with a mass of M = 1012M ⊙ at a redshift of zL = 0.5 and a source at redshift zS = 2.0, (we used here H = 50 km sec–1 Mpc–1 as the value of the Hubble constant and an Einstein–deSitter universe), the Einstein radius is
∘ ---M---- 𝜃E ≈ 1.8 --12----arcsec (10 ) 10 M ⊙
(note that for cosmological distances in general DLS ⁄= DS − DL!). For a galactic microlensing scenario in which stars in the disk of the Milky Way act as lenses for bulge stars close to the center of the Milky Way, the scale defined by the Einstein radius is
∘ ----- 𝜃E ≈ 0.5 -M-- milliarcsec. (11 ) M ⊙
An application and some illustrations of the point lens case can be found in Section 4.7 on galactic microlensing.
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