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

2.3 Distortion analysis of subaru images
galaxies with Pg s < 0. After smoothing s in the object parameter space,
all of the objects yield positive values of 〈 Pg〉 s , with the minimum of ≈ 0.04.
The median value of 〈 Pg〉 s over the weak lensing galaxy sample, including
galaxies with Pg s < 0, is calculated as ≈ 0.32. For a reference, the subsample
of galaxies with Pg s > 0 gives the median average of ≈ 0.33, mostly
weighted by galaxies with r g = 2 − 2.5 pixels. Finally, we use the following
estimator for the reduced shear:
g α = e ′ α/ 〈 P s g〉
. (2.4)
The quadratic shape distortion of an object is described by the complex
reduced-shear, g = g 1 + ig 2 . The tangential component g T is used to obtain
the azimuthally averaged distortion due to lensing, and computed from the
distortion coefficients g 1 , g 2 :
g T = −(g 1 cos 2θ + g 2 sin 2θ), (2.5)
where θ is the position angle of an object with respect to the cluster center,
and the uncertainty in the g T measurement is σ ≡ σ g / √ 2 in terms of the
rms error σ g for the complex shear measurement. The cluster center is well
determined from symmetry of the strong lensing pattern (Broadhurst et al.
2005a). The estimation of g T only has significance when evaluated statistically
over large number of galaxies, since galaxies themselves are not round
objects but have a wide spread in intrinsic shapes and orientations. In radial
bins we calculate the weighted average of the g T s and the weighted error:
〈g T (r n )〉 =
∑
gT /σ 2
∑ 1/σ
2
(2.6)
σ T (r n ) =
(∑ ) −1/2
1/σ
2 . (2.7)
It has been shown that such weights depend on the size of the objects
but mostly on their magnitudes (see e.g. Hoekstra et al. (2006)). Therefore,
as apparent magnitude increases with redshift the redshift distribution of
sources will be modified to some extent by this weighting scheme. We have
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