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

1.2 Gravitational Lensing
1.2.3.1 Measuring Galaxy Shapes and Sizes
With the surface brightness I( ⃗ θ) measured for a galaxy at ⃗ θ, we define the
tensor of second brightness moments,
Q ij =
∫
d 2 θW [I( θ)](θ ⃗ α − ¯θ i )(θ j − ¯θ j )
∫
d2 θW [I( θ)] ⃗ , i, j ∈ 1, 2, (1.33)
where the center ¯⃗ θ of the shape is defined as
¯⃗θ =
∫
d 2 θW [I( ⃗ θ)] ⃗ θ
∫
d2 θW [I( ⃗ θ)]
(1.34)
and W [I( θ)] ⃗ is a properly chosen weight function, e.g., in our case a Gaussian
window function.
The size of the object can be found from the determinant of the tensor
Q,
ω = (Q 11 Q 22 − Q 2 12) 1/2 . (1.35)
We define the shape of the image by the complex ellipticity
e ≡ Q 11 − Q 22 + 2iQ 12
Q 11 + Q 22
. (1.36)
For the case of an elliptical galaxy, this simplifies to
e = 1 − r2
1 + r 2 e2iϑ (1.37)
where r ≤ 1 is the axis ratio of the elliptical isophote, and the phase is twice
the position angle of the major axis, ϑ.
Thus, the ellipticity is the same
if the galaxy image is rotated by π, since a rotation of an ellipse leaves it
unchanged.
Defining the complex ellipticity of the intrinsic source in analogy to
eq. 1.36 with in terms of the tensor Q (s)
ij of the source, leads to
e (s) =
e − 2g + g2 e ∗
1 + |g| 2 − 2R[ge ∗ ]
(1.38)
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