SBCS+4.2+Spherical-Template+Mass

4.2.0. Generic Spherical-Template Mass
math M_{\Delta, {\rm lens}} math
 * __Notation:__** [[image:cue_mathinline.gif]]

math r math require some assumptions about the 3-D mass distribution in that region. The simplest way to proceed is by assuming a spherically-symmetric mass distribution described with a simple parametric model. Two of the most commonly used models are described below. Once a mass model has been chosen, one can fit it to the lensing data and derive an approximate spherical-overdensity mass for any desired scale radius.
 * __Introduction:__** Gravitational-lensing measurements of cluster mass within a spherical region of radius[[image:cue_mathinline.gif]]


 * __Algorithmic Description:__** Preparation of the lensing data for model fitting begins with the same basic steps for all spherical-template analyses. Here we outline those preparatory steps. Subsequent specific procedures unique to each spherical template are described below.

Weak-lensing analysis of a galaxy cluster begins with the detection of stars and galaxies in the same region of the sky, followed by unbiased measurement of the shapes of faint background galaxies that will be used to quantify the lensing signal. That signal can be expressed in terms of weighted quadrupole moments, math I_{\rm ij} = \int d^2 {\bf x} \, \, x_i x_j W({\bf x}) f(x) \, , math where math W({\bf x}) math is a Gaussian with a dispersion math r_g , math which is matched to the size of the object (see [|Kaiser et al. 1995]; [|Hoekstra et al. 1998], for more details). These quadrupole moments are then combined to form the two-component polarization math e_1 = \frac {I_{11} - I_{22}} {I_{11} + I_{22}} \; \;, \; \; e_2 = \frac {I_{12}} {I_{11} + I_{22}} \; \;. math Various instrumental effects, including but not limited to a position-dependent PSF, must then be removed to obtain a corrected polarization for each faint galaxy.

The weak-lensing signal measured from the net galaxy polarization under the assumption that the galaxies are randomly oriented can be quantified by computing the azimuthally averaged tangential shear math \gamma_T. math Shear is related to the surface mass density through math \langle \gamma_T \rangle (r) = \frac { \bar{\Sigma}( < r) - \bar{\Sigma}(r)} {\Sigma_{\rm crit}} math where math \bar{\Sigma}( < r) math is the mean surface density within radius math r, \, \, \bar{\Sigma}(r) math is the mean surface density at radius math r, math and math \Sigma_{\rm crit} = \frac {c^2} {4 \pi G} \frac {D_s} {{D_l} {D_{ls}}} math is the critical surface density computed from the observer-lens angular diameter distance math D_l , math the observer-source angular diameter distance math D_s , math and the lens-source angular diameter distance math D_{ls}. math It can also be quantified in terms of the convergence math \kappa (r) , math equal to the surface density divided by the critical surface density. Note that the critical surface density, and by extension the convergence, both depend on the distance to the source and therefore on the redshift distribution of the lensed objects used to compute the shear. And in practice, the directly observed quantity is not math \gamma_T math but rather the reduced shear math g_T (r) = \frac {\gamma_T (r)} {1 - \kappa (r)} \, , math which must be corrected for mass-sheet degeneracy.

Azimuthally averaged quantities such as math \langle \gamma_T \rangle (r) math depend on the algorithm used to identify the cluster's center. An offset between the adopted center and the halo's ``true" center will lead to an underestimate of the cluster's mass. Substructure in the cluster core can therefore cause bias in mass estimates (see [|Hoekstra et al. 2002]). Using wide-field lensing data to establish the halo's center can minimize these biases. When such data are not available, X-ray imaging or the location of the brightest cluster galaxy (BCG) can be used to estimate the halo's center, but the BCG in particular may be substantially offset from the halo's true center.

Once the azimuthally averaged shear has been measured, it can be fit with a specific mass model as detailed below. All such weak-lensing mass measurements are subject to uncertainty arising from cosmic variance in the amount of matter along the line of sight to the cluster but not belonging to the cluster itself (see [|Hoekstra 2001], [|Hoekstra 2003]).


 * __Links to Software:__** Add links to any publicly available software. This will be a great way to get others to use it.


 * __Strengths:__**
 * Make these brief bullet-like points.


 * __Weaknesses:__**
 * Make these brief bullet-like points.


 * __Discussion:__** Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)

4.2.1. Isothermal-Sphere Lensing Masses
math M_{\Delta, {\rm lens,iso}} math (The density contrast factor $\Delta$ gives the mean density contrast adopted for the mass definition, along with an "m" or a "c" to specify the reference density, as described in Section 1.2 on spherical-overdensity masses.)
 * __Notation:__** [[image:cue_mathinline.gif]]

math M( < r) \propto r. math The mass and radius corresponding to a given overdensity math \Delta math can then be computed from the best fitting isothermal mass model.
 * __Introduction:__** The simplest spherical mass model one can fit to weak-lensing data is a singular isothermal sphere, in which[[image:cue_mathinline.gif]]

math \gamma_T (r) = \frac {r_E} {2r} , math where the Einstein radius (in radians) can be expressed in terms of the line-of-sight velocity dispersion math \sigma math as math r_E = 4 \pi \left( \frac {\sigma} {c} \right)^2 \frac {D_{ls}} {D_s} , math assuming that halo orbits are isotropic. Alternatively, the velocity-dispersion parameter math \sigma math can be related to shear through math \sigma = c \, \sqrt{ \frac {\theta \gamma_T(\theta)} {2 \pi} \frac {D_s} {D_{ls}} } , math where math \theta math is the angular distance from the cluster center at which the average shear has been measured. A spherical-overdensity cluster mass defined with respect to the critical density is then math M_{\Delta{\rm c,lens,iso}} = \frac {4 \sigma} {G H(z) \Delta^{1/2}} , math and using the mean density as the reference density gives math M_{\Delta{\rm m,lens,iso}} = \frac {4 \sigma} {G H(z) \Omega_{\rm M} \Delta^{1/2}}. math [Algorithmic description currently lacks an explanation of how to average over the distribution of source redshifts.]
 * __Algorithmic Description:__** Azimuthally-averaged tangential shear must first be measured as described in section 4.2.0. This procedure requires an algorithm for determining the cluster center, which must be specified explicitly and can introduce uncertainty and bias into the mass measurement. Once the shear has been measured, it can be related to the mass model through


 * __Links to Software:__** Add links to any publicly available software. This will be a great way to get others to use it.


 * __Strengths:__**
 * Make these brief bullet-like points.


 * __Weaknesses:__**
 * Make these brief bullet-like points.


 * __Discussion:__** Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)

4.2.2. NFW-Profile Lensing Masses
math M_{\Delta, {\rm lens,NFW}} math (The density contrast factor math \Delta math gives the mean density contrast adopted for the mass definition, along with an "m" or a "c" to specify the reference density, as described in Section 1.2 on spherical-overdensity masses.)
 * __Notation:__** [[image:cue_mathinline.gif]]

One of the most widely used mass models for clusters is the spherically-symmetric NFW profile ([|Navarro et al. 1997]). In this model the matter density is math \rho_{\rm M} (r) \propto r^{-1} (r+r_s)^{-2} \; , math where math r_s math is a scale radius at which the power-law slope of the density profile is math \rho_{\rm M} (r) \propto r^{-2}. math Integrating over this density profile gives the mass profile math M( < r) \propto \ln \left( 1 + \frac {r} {r_s} \right) - \frac {r} {r + r_s}. math After fitting this mass profile to the lensing data, one can calculate the cluster's mass within a chosen overdensity threshold math \Delta. math
 * __Introduction:__**

math r_s , math or equivalently the concentration math c_\Delta \equiv r_\Delta / r_s math (e.g. [|Okabe et al. 2010]). Otherwise, one must assume a relationship between halo mass and concentration (see [|Hoekstra 2007] for an example). The derived lensing masses will then depend on the mass-concentration relation adopted in the fitting procedure. [Algorithmic description currently lacks an explanation of how to average over the distribution of source redshifts.]
 * __Algorithmic Description:__** Azimuthally-averaged tangential shear must first be measured as described in section 4.2.0. This procedure requires an algorithm for determining the cluster center, which must be specified explicitly and can introduce uncertainty and bias into the mass measurement. Once the shear has been measured, an NFW mass model can be fit to the shear signal. If the lensing data are of very high quality, it may be possible for the fitting procedure to constrain the scale radius [[image:cue_mathinline.gif]]


 * __Links to Software:__** Add links to any publicly available software. This will be a great way to get others to use it.


 * __Strengths:__**
 * Make these brief bullet-like points.


 * __Weaknesses:__**
 * Make these brief bullet-like points.


 * __Discussion:__** Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)