PNFW (Prolate NFW)

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Overview

The PNFW (Prolate Navarro-Frenk-White) profile is a special case of the TNFW (Triaxial NFW) profile, specifically for prolate (cigar-shaped) halos where $a = b \leq c$.

Due to the axial symmetry of prolate halos, the viewing angle $\phi$ (which matters for fully triaxial halos) becomes irrelevant. All projected prolate halos have PA = 90° when measured from the observer's x-axis. Instead of $\phi$, PNFW directly specifies the observed position angle PA.

Mathematical Definition

See TNFW for full mathematical details. The key difference:

Triaxial case (TNFW): $a \leq b \leq c$ (three independent axes)
Prolate case (PNFW): $a = b \leq c$ (axisymmetric around long axis)

Simplifications for Prolate Halos

1. Axial symmetry: Rotation about the long axis (c-axis) doesn't change the halo
2. Viewing angle $\phi$ → PA: Since $\phi$ is degenerate, replace it with the directly observable position angle
3. Fixed PA behavior: All projections have PA = 90° in the natural frame

Projected Convergence

The convergence formula is identical to TNFW, but with simplified geometry due to $a = b$:

$$\kappa(x', y') = \frac{b_{\text{PNFW}}}{2} \cdot f_{\text{GNFW}}(\zeta)$$

where $\zeta$ depends on $(a/c, \theta, \text{PA})$ instead of the full set $(a/c, b/c, \theta, \phi)$.

Softening Radius

Like TNFW, a softening radius avoids singularities:

$$R^2 \rightarrow R^2 + r_{\text{soft}}^2$$

Default: $r_{\text{soft}} = 10^{-4}$ (adjustable via Rsoft keyword).

GLEE Configuration

Basic Usage

pnfw
  4.000000  #x-coord   exact:
  4.000000  #y-coord   exact:
  0.600000  #a/c       exact:
  0.750000  #theta     exact:
  1.570796  #PA        exact:
  10.000000  #theta_e   exact:
  25.000000  #r_scale   exact:
  0.740028  #q         exact:

Parameter Summary

#	Parameter	Description	Input/Output	Notes
1	x	x-coordinate of centroid (arcsec)	Input
2	y	y-coordinate of centroid (arcsec)	Input
3	a/c	Short-to-long axis ratio (= b/c for prolate)	Input	$0 < a/c \leq 1$
4	theta	Viewing angle (radians)	Input	Angle from line-of-sight
5	PA	Position angle (radians, CCW from +x axis)	Input	Replaces $\phi$ from TNFW
6	theta_e	Einstein radius $r_E$ (arcsec)	Input	For equivalent sphere
7	r_scale	Scale radius $R_0$ (arcsec)	Input
8	q	Output axis ratio of projected κ	Output	Can impose prior

:::tip Output Parameter Parameter 8 ($q$) is an output computed from the 3D→2D projection. It is included in the parameter list to allow imposing priors based on observed 2D light distributions (e.g., from photometry). :::

Key Differences from TNFW

Feature	TNFW	PNFW
Shape	Triaxial ($a \leq b \leq c$)	Prolate ($a = b \leq c$)
Axis ratios	2 independent (a/c, b/c)	1 independent (a/c)
Viewing angles	2 angles ($\theta, \phi$)	1 angle ($\theta$) + PA
Position angle	Computed output	Direct input parameter
Parameters	10 total	8 total (simpler)

Use Cases

Observationally Motivated Models:

• Prolate halos are common in simulations for isolated elliptical galaxies
• Measure PA directly from 2D imaging (e.g., HST)
• Impose prior on output $q$ based on observed light axis ratio

Simplified Modeling:

• When full triaxiality is not required (e.g., visual inspection suggests prolate geometry)
• Reduces parameter space by 2 compared to TNFW
• Faster MCMC convergence due to fewer parameters

Physical Interpretation

Input Parameters (3D Halo)

• a/c: How "cigar-like" the halo is (smaller = more elongated)
• theta: Inclination angle (0 = edge-on, 90° = face-on)
• PA: Orientation of long axis on the sky

Output Parameter (2D Projection)

• q: Observed axis ratio depends on both a/c and theta
  • Face-on ($\theta \approx 90°$): $q \approx 1$ (circular)
  • Edge-on ($\theta \approx 0°$): $q \approx a/c$ (most elongated)

Related Profiles

• TNFW - General triaxial NFW (includes PNFW as special case)
• NFW - Standard spherical NFW
• GNFW - Generalized NFW with variable inner slope
• eNFW - Elliptical NFW with ellipticity in κ

References

• Oguri, M., Lee, J., & Suto, Y. (2003). Detailed Cluster Lensing Profiles at Large Radii and the Impact on Cluster Weak Lensing Studies. ApJ, 599, 7. ADS Link