Changeset 40692 for trunk/doc/release.2015/ps1.analysis/analysis.tex

Timestamp:

Apr 13, 2019, 12:47:43 PM (7 years ago)

Author:

eugene

Message:

add lensing formulae

File:

• trunk/doc/release.2015/ps1.analysis/analysis.tex (modified) (2 diffs)

trunk/doc/release.2015/ps1.analysis/analysis.tex

@@ -2740 +2740 @@
 
 The observed ellipticity of an object observed in a real instrument
-will be affected by the instrumental signature of the instrument.  To
+will be affected by the point spread function of the instrument.  To
 first order, the effect on the polarization components can be
 described as a combination of ``smear'', in which the observed shape
@@ -2749 +2749 @@
 terms can be corrected.  
 
-
+\begin{eqnarray}
+X^{sh}_{1,1} = T^{-1} \sum f \left[ 2W(x^2 + y^2) + 2W^\prime (x^2 - y^2)^2 \\
+X^{sh}_{1,2} = T^{-1} \sum f \left[ 4W^\prime(x^2 - y^2) x y \\
+X^{sh}_{2,2} = T^{-1} \sum f \left[ 2W(x^2 + y^2) + 8W^\prime x^2 y^2 \\
+\end{eqnarray}
+
+\begin{eqnarray}
+e^{sh}_1 = 2 e_1 + 2 T^{-1} \sum f W^\prime (x^2 + y^2) (x^2 - y^2) \\
+e^{sh}_2 = 2 e_2 + 2 T^{-1} \sum f W^\prime (x^2 + y^2) 2 x y \\
+\end{eqnarray}
+
+\begin{eqnarray}
+X^{sm}_{1,1} = T^{-1} \sum f \left[ W + 2W^\prime (x^2 + y^2) + W^{\prime \prime} (x^2 - y^2)^2 \\
+X^{sm}_{1,2} = T^{-1} \sum f \left[ 2W^{\prime\prime} (x^2 - y^2) x y \\
+X^{sm}_{2,2} = T^{-1} \sum f \left[ W + 2W^\prime (x^2 + y^2) + 4W^{\prime \prime} x^2 y^2 \\
+\end{eqnarray}
+  
+\begin{eqnarray}
+e^{sm}_1 = T^{-1} \sum f \left[ 2W^\prime + W^{\prime \prime} (x^2 + y^2) \right] (x^2 - y^2) \\
+e^{sm}_2 = T^{-1} \sum f \left[ 2W^\prime + W^{\prime \prime} (x^2 + y^2) \right] 2 x y \\
+\end{eqnarray}
+  
 
 @ARTICLE{2017MNRAS.468.3499D,