Mass and Light distributions in Clusters of Galaxies - Henry A ...

2.5 Photometric redshifts
Weak Lensing Dilution in A1689
We need to estimate the respective depths of our color-magnitude selected
samples when estimating the cluster mass profile, because the lensing signal
increases with source distance, and therefore must differ between the samples.
The effect of this difference in distance on the weak lensing signal is simply
linear as we can see from the relation between the dimensionless surface mass
density,
where
and the tangential distortion:
κ(r) = Σ(r)/Σ crit , (2.10)
Σ crit =
c2
4πGd l
D s
D ds
(2.11)
〈g T (r)〉 = (¯κ(r) − κ(r))/(1 − κ(r)) (2.12)
so that in the weak limit where κ is small,
〈g T (r)〉 ∝ D ds
D s
(¯Σ(r) − Σ(r)) (2.13)
and hence for an individual cluster, with a fixed redshift and a given mass
profile, the observed level of the weak distortion simply scales with the lensing
distance ratio. Further details are presented in the appendix. The mean ratio
of D ds /D s , which is weighted by the redshift distribution of the background
population corresponding to our magnitude and color cuts, is calculated using
the expression
〈D〉 ≡ 〈 D ds
D s
〉 =
∫ D ds
D s
(z)N(z)dz
∫
N(z)dz
. (2.14)
Since we cannot derive complete samples of reliable photometric redshifts
from our limited 2-color V, i ′ images of A1689, we instead make use of other
deep field photometry covering a wider range of passbands, sufficient for photometric
redshift estimation of the faint field redshift distribution appropriate
for samples with the same color and magnitude limits as our red, green and
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