NFW (Navarro-Frenk-White)

NFW (Navarro-Frenk-White) Profile

Usage

nfw  
  4.0000     #x-coord  (= x centroid)  
  3.0000     #y-coord  (= y centroid)  
  0.7        #b/a  (= q, axis ratio)  
  3.6        #theta  (= position angle, ccw from +x in radians. 0 means distribution is elongated along x.)  
  0.2        #theta_e  (see $\theta_e$ definition below)  
  20.0       #r_scale  ($x_s$ above, scale radius in arcsec)  

The deflection angle components ($\alpha_1, \alpha_2$) and lens potential ($\psi$) for the NFW profile are given by:

$\alpha_1 = \theta_e \frac{g(\chi)}{\chi^2} \cdot q \cdot x_1 \quad\quad \alpha_2 = \theta_e \frac{g(\chi)}{\chi^2} \cdot \frac{x_2}{q}$

$\psi = \frac{1}{2} \theta_e \cdot \theta_s^2 \cdot h(\chi)$

where: $\chi = \frac{\sqrt{q x_1^2 + \frac{x_2^2}{q}}}{x_s}$

Here, $(x_1, x_2)$ are the angular coordinates on the image plane (not to be confused with $\chi_1, \chi_2$, which are scaled by $x_s$).

The function $g(\chi)$ is defined as:

$$g(\chi) = \left\{\begin{array}{ll}0.306852819440, & \text{if } \chi = 1 \\ \log(0.5\chi) + \frac{\operatorname{acosh}(1/\chi)}{\sqrt{1-\chi^2}}, & \text{if } \chi < 1 \\ \log(0.5\chi) + \frac{\operatorname{acos}(1/\chi)}{\sqrt{\chi^2 - 1}}, & \text{if } \chi > 1 \end{array} \right.$$

Normalisation and Scaling

• Without the scale: option:
  The parameter $\theta_e$ serves as the strength/normalization of the profile.
  Comparing to Equation (6) of Golse & Kneib (2002):
  $\theta_e = 4 \kappa_s$
  where:
  $\kappa_s = \frac{\rho_c r_s D_d}{\Sigma_{\text{crit}}}$
  (for $r_s$ in angular coordinates).
• With the scale: option:
  $\theta_e \to \theta_E \quad \text{with} \quad \bar{\kappa}(\theta_E) = 1$
  This means $\theta_E$ is no longer equal to $4 \kappa_s$.

Notes:

• Ellipticity Implementation:
  • The ellipticity is applied in the lens potential rather than in the surface mass density $\kappa$.
  • This means that:
    $\psi(r) \to \psi (q x_1^2 + \frac{x_2^2}{q})$

Citation for Profile:

@ARTICLE{Navarro1997,
       author = {{Navarro}, Julio F. and {Frenk}, Carlos S. and {White}, Simon D.~M.},
        title = "{The Structure of Cold Dark Matter Halos}",
      journal = {The Astrophysical Journal},
         year = 1996,
        month = may,
       volume = {462},
        pages = {563},
          doi = {10.1086/177173}
}