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eNFW (Elliptical NFW)

:::info Migrated from Old Wiki This content was partially migrated from the old wiki using GitHub Copilot and should be double-checked for accuracy. :::

Usage​

enfw  
  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        #rho_s  
  20.0       #r_s  ($x_s$ above, scale radius in arcsec)  

...

Rsoft  1e-5   # See the additional configuration section at the end of the page!!!

Definition of r0′r_0' and RR​

We define:

r0′=rs+rsoftqandR=x2+y2q2r_0' = \frac{r_s + r_{\text{soft}}}{\sqrt{q}} \quad\text{and}\quad R = \sqrt{x^2 + \frac{y^2}{q^2}}

such that

:::warning Work in Progress The surface mass density equations for the eNFW below are currently being verified. Please refer to the original paper (Oguri 2021) or consult a GLEE developer for the authoritative formulation. :::

κ={0.5⋅13,if R2+rsoft2=r0′0.5((R2+rsoft2r0′)2−1)(1−11−(R2+rsoft2r0′)2arctanh⁡(1−(R2+rsoft2r0′)2)),if R2+rsoft2>r0′0.5((R2+rsoft2r0′)2−1)(1−1(R2+rsoft2r0′)2−1arctan⁡((R2+rsoft2r0′)2−1)),if R2+rsoft2>r0′\kappa = \left\{\begin{array}{ll}0.5\cdot\frac{1}{3}, & \quad \text{if } \sqrt{R^2 + r_{\text{soft}}^2} = r_0' \\[12pt] \frac{0.5}{\left(\left(\frac{\sqrt{R^2 + r_{\text{soft}}^2}}{r_0'}\right)^2 - 1\right)} \left(1 - \frac{1}{\sqrt{1 - \left(\frac{\sqrt{R^2 + r_{\text{soft}}^2}}{r_0'}\right)^2}} \operatorname{arctanh}\left(\sqrt{1 - \left(\frac{\sqrt{R^2 + r_{\text{soft}}^2}}{r_0'}\right)^2}\right)\right), & \quad \text{if } \sqrt{R^2 + r_{\text{soft}}^2} > r_0' \\[10pt] \frac{0.5}{\left(\left(\frac{\sqrt{R^2 + r_{\text{soft}}^2}}{r_0'}\right)^2 - 1\right)} \left(1 - \frac{1}{\sqrt{\left(\frac{\sqrt{R^2 + r_{\text{soft}}^2}}{r_0'}\right)^2 - 1}} \arctan\left(\sqrt{\left(\frac{\sqrt{R^2 + r_{\text{soft}}^2}}{r_0'}\right)^2 - 1}\right)\right),& \quad \text{if } \sqrt{R^2 + r_{\text{soft}}^2} > r_0' \end{array}\right.

Normalisation Factor​

To make κ\kappa consistent with the NFW profile in GLEE (when q=1q=1), we apply a normalisation factor:

κ=κ⋅norm_factor\kappa = \kappa \cdot \text{norm\_factor}

where:

nor_factor=χc2gnfw(χc)\text{nor\_factor} = \frac{\chi_c^2}{g_{\text{nfw}}(\chi_c)}

with:

χc=ρsrs\chi_c = \frac{\rho_s}{r_s}

and gnfw(χc)g_{\text{nfw}}(\chi_c) is given by:

gnfw(χc)={log⁡(0.5χc)+arccosh(1/χc)1−χc2,χc<10.306852819440,χc=1log⁡(0.5χc)+arccos(1/χc)χc2−1,χc>1g_{\text{nfw}}(\chi_c) = \left\{ \begin{array}{ll} \log(0.5\chi_c) + \frac{\text{arccosh}(1/\chi_c)}{\sqrt{1 - \chi_c^2}}, & \chi_c < 1 \\[10pt] 0.306852819440, & \chi_c = 1 \\[10pt] \log(0.5\chi_c) + \frac{\text{arccos}(1/\chi_c)}{\sqrt{\chi_c^2 - 1}}, & \chi_c > 1\end{array}\right.

Deflection Angle Calculation​

The deflection angle is calculated by decomposing the κ\kappa profile into a set of Core Steep Ellipsoids (CSEs), from which an analytical solution for the deflection angle is derived. Details of this method are presented in Oguri 2019.

Scaling Options in GLEE​

If using GLEE installed after September 2024:

  • scale: after ρs\rho_s means κˉ(ρs)=1\bar{\kappa}(\rho_s) = 1
  • Without scale:, ρs\rho_s is simply a normalisation factor

If using older versions of GLEE:

  • Both "scale:" and non-"scale:" cases imply κˉ(ρs)=1\bar{\kappa}(\rho_s) = 1

Additional Configuration​

To use the eNFW profile in GLEE, set:

Rsoft=10−5R_{\text{soft}} = 10^{-5}

in the configuration file.

Profile Citation​

@article{Oguri2021,
  title = {Fast Calculation of Gravitational Lensing Properties of Elliptical Navarro–Frenk–White and Hernquist Density Profiles},
  volume = {133},
  ISSN = {1538-3873},
  url = {http://dx.doi.org/10.1088/1538-3873/ac12db},
  DOI = {10.1088/1538-3873/ac12db},
  number = {1025},
  journal = {Publications of the Astronomical Society of the Pacific},
  publisher = {IOP Publishing},
  author = {Oguri,  Masamune},
  year = {2021},
  month = jul,
  pages = {074504}
}