deletions | additions       diff --git a/sectionAnalysis.tex b/sectionAnalysis.tex  index 9c1e18c..39774e3 100644  --- a/sectionAnalysis.tex  +++ b/sectionAnalysis.tex 
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\section{Weak Lensing Analysis:}  \label{sec:method}  %  %  We perform a pipeline based on python language to make the  
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%  We used SExtractor (Bertin \& Arnouts, 1996) for the detection and fotometry of the   sources, in a two-pass mode. A first run is made to detect bright objects, with a detection level of 5$ \sigma $ above the background, in order to estimate the seeing and the saturation level of each image. The seeing is estimated using the average of the FWHM parameter of the point-like objects, selected from the FWHM/MAG\_BEST diagram since for these objects the FWHM is independent of the magnitude (\textbf{se entiende, o les parece mejor agregar el grafico?)}). Determining the seeing is important for SExtractor to perform the star-galaxy classification. The saturation level is estimated as 0.8 times the maximum value of the FLUX\_MAX parameter. These parameters, \textit{seeing} and \textit{saturation level}, are taking into account in the SExtractor configuration file for the second run, with a lower threshold detection limit of 1.5$ \sigma $. Second run is made in dual mode, detecting objects on the \textit{r'} image, while astrometric and photometric parameters are measured on all individual images.\\  %  \begin{figure}  \centering  \includegraphics[width=.5\textwidth]{./../plots/new-ones/mu_mag.eps}~\\ 
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\caption{Objects detected in the galaxy cluster [VMF98]102 \textit{r'-}band image, stars (\textit{blue points}), galaxies (\textit{green points}), and artifacts (\textit{black points}). In the MU\_MAX/MAG\_BEST plane (\textit{left}), stars are in the region marked by the red solid line $\pm$ 0.4 magnitudes. To check the selection, we plot the sources in the MAG\_BEST/FWHM plane (\textit{left}), where point like object selected as stars have almost the same FWHM despite the magnitude.}  \label{sources}  \end{figure}  %  For the object classification in stars, galaxies and false detections,  we considered a similar analysis as Bardeau\,et\,al.\,2005, taking into account the position of the source in the magnitude/central flux diagram, the FWHM respect to the seeing and the stellarity index, according to the CLASS\_STAR parameter. Objects that are more sharply peaked than the PSF (FWHM replace_contentlt;$ $<$  \textit{seeing} - 0.5 pixel) and with FLAG parameter replace_contentgt;$ $>$  4, are considered as false detections. As the light distribution of a point source scales with magnitude, objects on the line magnitude/central flux, $\pm$ 0.4 magnitudes (Figure\,\ref{sources}), FWHM replace_contentlt;$ $<$  \textit{seeing} + 1 pixel and CLASS\_STAR replace_contentgt;$ $>$  0.8 are considered as stars. The rest of the objects are considered as galaxies. \\ %  \subsection{Shape measurements}  % 
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\subsection{Background Galaxies selection and redshift distribution}  %  Background galaxies for the shear estimation, were selected as the galaxies with magnitude in the filter \textit{r'}, $m_{r}$, higher than $m_{L}$, and lower than $m_{max} + 1.0$, where $m_{L}$ is defined as the lowest magnitude $r'$ such that the probability that the galaxy is behind the cluster is higher than 0.7 and $m_{max}$ correspond to the peak of the magnitude distribution of the image in the filter \textit{r'} (both magnitudes for each cluster are listed in Table\,\ref{table:2}). The later cut in magnitude ensures that we are not taking into account galaxies that are too faint, given that they could have great errors in the shape measurements. Also, as we did for the simulated data, we discard galaxies with FWHM replace_contentlt;$ $<$  5, with companions closer than 16 pixels and with $\sigma_{e} > 0.2$.\\ To compute $m_{L}$ we used the catalogue of photometric redshifts computed by Coupon et al. 2009, based on the public release Deep Field 1 of the Canada-France-Hawaii Telescope Legacy Survey. We estimated the fraction of galaxies with $z > z_{cluster}$ in magnitude bins for the \textit{r'} filter, and then we chose $m_{L}$ as the lowest magnitude for which the fraction of galaxies was greater than 0.7. Background galaxy density after the selection are listed for each cluster in Table 2.\\  Once we obtained a catalogue for the background galaxies, we average the two components of the ellipticities (E-mode and B-mode) in nonoverlapping logarithmic annuli, to conserve the S/N ratio in each bin. In order to take into account the contamination of foreground galaxies in the catalogue, we weighted the average value of the components of the galaxie's ellipticity with the probability that the galaxy was behind the cluster. We compute this probability using Coupon's catalogue, from the fraction of galaxies with $z > z_{cluster}$ for each bin in magnitude, \textit{r'}, and color (\textit{g' - r'} and \textit{r' - i'}) - Figure\,\ref{weigh}. Hence, given the magnitude and the color of each galaxy, we assigned to it a weigh, \textit{w}, as the fraction of galaxies with $z > z_{cluster}$ in that bin. \\  \begin{figure} 
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\label{weigh}  \end{figure}  The average E-mode components correspond to the shear value which depends on the geometrical factor $ \beta = D_{LS}/ D_{L} $, where $D_{LS}$ is the angular diameter distance from the lens to the background source galaxy, and $D_{L}$ is the distance from the observer to the background galaxy. A galaxy at the same radial distance from the center of the cluster but at a different background redshift is sheared differently. To remove this variation, we divide the estimated shear by $\beta$, which results in the infinite redshift limit (z $\rightarrow \infty$) shear value, that is only function of the radial distance from the cluster center. To estimate $\beta$ we used once more Coupon's catalogue, since we can consider that this catalogue is complete up to our magnitude limit. We applied the photometric selection criteria to the catalogue and then we computed $\beta$ for the whole distribution of field galaxies, taking into account the contamination by foreground galaxies given our selection criteria by setting $\beta(z_{phot} < z_{cluster}) = 0$. The averaged geometrical factors for each cluster are given in Table\,\ref{table:2}.  \subsection{Fiting the profiles}   We finally estimate the cluster mass fitting the shear data with the singular isothermal   sphere (SIS) and the NFW profile (Navarro et al. 1997). We will explain briefly the lensing formulae for these two profiles:  \subsubsection{SIS profile}  The SIS mass model is the simplest one for describing a relaxed massive sphere with a constant and isotropic velocity dispersion, $\sigma_{V}$. This is mainly described by the density distribution:  \begin{equation*}  \rho(r) = \dfrac{\sigma_{V}^{2}}{2 \pi G r^{2}}  \end{equation*}  where $c$ is the speed of light and \textit{G} is the gravitational constant. From this ecuation, we can get the critical Einstein radius for source at $z \rightarrow \infty $ ($\beta =1$) as:  \begin{equation*}  \theta_{E} = \dfrac{4 \pi \sigma_{V}^{2}}{c^{2}}  \end{equation*}  In terms of wich one obtains:  \begin{equation*}  \kappa_{\theta} = \gamma_{\theta} = \dfrac{\theta_{E}}{2 \theta}  \end{equation*}  where $\theta$ is the distance to the cluster center. Hence, fitting the shear for different radius, we can estimate the Einstein radius, and from that, we can obtain an estimation of the mass within a radius $\theta$ as:  \begin{equation*}  M(\theta) = \dfrac{\theta_{E} c }{4G} \theta  \end{equation*}  \subsubsection{NFW profile}  The NFW profile is derived from fitting the density profile  of numerical simulations of cold dark matter halos (Navarro  et al. 1995, 1997). This profiles depend on two parameters, the virial radius, $R_{200}$, and a dimensionless concentration parameter:  \begin{equation*}  \rho(r) = \dfrac{\rho_{c} \delta_{c}}{(r/r_{s})(1+r/r_{s})^{2}}   \end{equation*}  where $r_{s}$ is the scale radius, $r_{s} = R_{200}/c$ and $\delta_{c}$ is the characteristic overdensity of the halo,  \begin{equation*}  \delta_{c} = \frac{200}{3} \dfrac{c^{3}}{\ln(1+c)+c/(1+c)}   \end{equation*}  The properties of the NFW profile in the context of gravitational lensing have been discussed by many authors (Bartelmann 1996, Wright \& Brainerd 2000). The radial dependence of the shear as function of $x:=\theta / r_{s}$ is given by:  \begin{equation*}  \gamma(x) = \dfrac{2r_{s}\delta_{c}\rho_{c}}{\Sigma_{c}} j(x)   \end{equation*}  where $\Sigma_{c}$ is the \textit{critical surface mass density} and $j(x)$ is given by: \begin{eqnarray*}  $j(x < 1) & = & \dfrac{4 \arctanh \sqrt{\frac{1-x}{1+x}}}{x^{2} \sqrt{1-x^{2}}} + \dfrac{2 \ln(\frac{x}{2})}{x^{2}} - \dfrac{1}{x^{2}-1} \\  & &+ \dfrac{2 \arctanh \sqrt{\frac{1-x}{1+x}}}{(x^{2}-1) \sqrt{1-x^{2}}} \\ $  \end{eqnarray*}  \begin{eqnarray*}  $j(x = 1) = 2 \ln \left(\dfrac{1}{2} \right) + \dfrac{5}{3}$\\  $j(x < 1) = \dfrac{4 \arctan \sqrt{\frac{1-x}{1+x}}}{x^{2} \sqrt{1-x^{2}}} + \dfrac{2 \ln(\frac{x}{2})}{x^{2}} - \dfrac{1}{x^{2}-1} + \dfrac{2 \arctan \sqrt{\frac{1-x}{1+x}}}{(x^{2}-1)^{3/2}}} $  \end{eqnarray*}  If we fit the shear for different radius we can have an estimation of the parameters $c$ and $R_{200}$. Once we obtain these parameters we can compute the mass within a radius $\theta$ as:  \begin{equation*}  M(\theta) = \int^{\theta}_0 2\pi \theta \Sigma(\theta) d\theta = \int^{\theta}_0 4\pi \theta \rho_{c} \delta_{c} r_{s} F(x) d\theta  \end{equation*}  where,  \begin{eqnarray*}  $F(x < 1) & = & \dfrac{1}{x^{2}-1} (1 - \dfrac{1}{\sqrt{1-x^{2}}} \arcosh \dfrac{1}{x} )$  $F(x = 1) = \dfrac{1}{3}$ \\  $F(x > 1) & = & \dfrac{1}{x^{2}-1} (1 - \dfrac{1}{\sqrt{x^{2}-1}} \arccos \dfrac{1}{x} )$  \end{eqnarray*}  There is a well-known degeneracy between the parameters $R_{200}$ and $c$ when fitting the shear profile in the weak lensing regime. This is due to the lack of information on the mass distribution near the cluster center and only a combination of strong and weak lensing can raise it and provide useful constraints on the concentration parameter. Since we do not have strong lensing modeling of the clusters in the sample,   we decided to fix the concentration parameter, $c = 5$, and fit the mass profile with only one free parameter, $R_{200}$.