GRALE: grale::NFWLens Class Reference

Lensing by a spherically symmetric Navarro-Frenk-White profile (NFW). More...

#include <nfwlens.h>

Inheritance diagram for grale::NFWLens:

[legend]

Collaboration diagram for grale::NFWLens:

[legend]

List of all members.

Additional Inherited Members
Protected Member Functions inherited from grale::SymmetricLens
	SymmetricLens (GravitationalLens::LensType t)
	Constructor meant to be used by subclasses.
Protected Member Functions inherited from grale::GravitationalLens
	GravitationalLens (LensType t)
	Meant to be used by a specific lens implementation.

Detailed Description

Lensing by a spherically symmetric Navarro-Frenk-White profile (NFW).

$F(x) = \left\{ \begin{array}{ll} \frac{1}{\sqrt{1-x^2}}\textrm{atanh}{\sqrt{1-x^2}} & x < 1 \\ 1 & x = 1 \\ \frac{1}{\sqrt{x^2-1}}\textrm{atan}{\sqrt{x^2-1}} & x > 1 \\ \end{array} \right.$

$\Rightarrow \frac{d F}{d x} = \frac{1-x^2 F(x)}{x(x^2-1)}$

$G(x) = \frac{1-F(x)}{x^2-1}$

$H(x) = \textrm{ln}\left(\frac{x}{2}\right) + F(x) \Rightarrow \frac{d H}{d x} = G(x) x$

$A(x) = \frac{d H}{d x}$

$B(x) = \left\{ \begin{array}{ll} \textrm{ln}^2\left(\frac{x}{2}\right) - \textrm{atanh}^2\sqrt{1-x^2} & x < 1 \\ \textrm{ln}^2 2 & x = 1 \\ \textrm{ln}^2\left(\frac{x}{2}\right) + \textrm{atan}^2\sqrt{x^2-1} & x > 1 \\ \end{array} \right.$

$\Rightarrow \frac{1}{2} \frac{d B}{d x} = A(x)$

$\Sigma(\theta) = 2 r_s \rho_s G\left(\frac{\theta}{\theta_s}\right)$

$\textrm{M}(\theta) = 4\pi r_s^3\rho_s H\left(\frac{\theta}{\theta_s}\right)$

$\Rightarrow \vec{\hat{\alpha}}\left(\vec{\theta}\right) = \frac{16 \pi G r_s^2 \rho_s}{c^2} A\left(\frac{\theta}{\theta_s}\right) \frac{\vec{\theta}}{\theta}$

$\psi(\theta) = \frac{D_{ds}}{D_s} \frac{8 \pi G r_s^2 \rho_s \theta_s}{c^2} B\left(\frac{\theta}{\theta_s}\right)$

References

The documentation for this class was generated from the following file:

src/lens/nfwlens.h

Additional Inherited Members

Detailed Description