dPIE (Dual Pseudo-Isothermal Elliptical)

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Overview

The dPIE (Dual Pseudo-Isothermal Elliptical) profile, also known as a truncated PIEMD, is a mass distribution with both a core radius and a truncation/scale radius. This profile is particularly useful for modeling satellite galaxies or dark matter subhalos that have been tidally truncated.

The dPIE profile approaches zero at large radii (unlike PIEMD which extends to infinity), making it physically motivated for substructures in a host halo.

Mathematical Definition

Convergence

$$\kappa(x,y) = \frac{E_{\text{limit}}}{2} \cdot \frac{s^2}{s^2 - w^2} \left( \frac{1}{\sqrt{w^2 + r_{\text{em}}^2}} - \frac{1}{\sqrt{s^2 + r_{\text{em}}^2}} \right)$$

where:

$$r_{\text{em}} = \sqrt{\frac{x^2}{(1+e)^2} + \frac{y^2}{(1-e)^2}}$$

• $E_{\text{limit}}$ = strength (Einstein radius of SIS in limiting case $s \to \infty, w \to 0$)
• $w$ = core radius
• $s$ = truncation/scale radius (must satisfy $s > w$)
• $e$ = ellipticity, $e = \frac{1-q}{1+q}$
• $q$ = axis ratio (minor/major), $q \leq 1$

Corresponding 3D Density

The convergence corresponds to the following 3D density distribution (without $\Sigma_{\text{crit}}$ factor):

$$\rho(x,y,z) = \frac{\rho_0}{(1 + r_{3D}^2/w^2)(1 + r_{3D}^2/s^2)}, \quad s > w$$

where:

• $r_{3D} = \sqrt{r_{\text{em}}^2 + z^2}$
• $\rho_0$ and $E_{\text{limit}}$ are related by: $E_{\text{limit}} = 2\pi\rho_0 w^2$

Parameter Calculation

Without scale:

$$E_{\text{limit}} = \theta_E \quad \text{(5th parameter)}$$

With scale:

$$E_0 = \theta_E$$ $$E_{\text{limit}} \cdot \frac{s^2}{s^2-w^2} = \frac{E_0^2}{(\sqrt{w^2+E_0^2}-w) - (\sqrt{s^2+E_0^2}-s)}$$

This ensures that the effective Einstein radius (area-averaged) equals $E_0$.

Logarithmic Parameters

The parameters $\theta_E$ (5th), $w$ (6th), and $s$ (7th) all support the log: option, allowing sampling in logarithmic space for parameters spanning multiple orders of magnitude.

GLEE Configuration

Basic Usage

dpie
  30.000000  #x-coord   exact:
  30.000000  #y-coord   exact:
   0.750000  #b/a       exact:
   0.800000  #theta     exact:
  10.000000  #theta_e   exact:
   5.000000  #r_core    exact:  (w parameter in kappa equation)
 100.000000  #r_trunc   exact:  (s parameter in kappa equation)

Parameter Summary

#	Parameter	Description	Options	Constraints
1	x	x-coordinate of centroid (arcsec)
2	y	y-coordinate of centroid (arcsec)
3	q	Axis ratio (b/a, minor/major)		$0 < q \leq 1$
4	theta	Position angle (radians, CCW from +x axis)
5	theta_e	Einstein radius / strength	log:, scale:
6	r_core	Core radius $w$ (arcsec)	log:	$w > 0$
7	r_trunc	Truncation radius $s$ (arcsec)	log:	$s > w$

:::warning Radius Constraint The truncation radius $s$ must be greater than the core radius $w$. GLEE will produce errors or unphysical results if $s \leq w$. :::

Physical Interpretation

Core Radius ($w$)

• Central region with approximately constant density
• Suppresses central cusp (like PIEMD)
• Typical range: 0.1–5 arcsec for galaxy-scale lenses

Truncation Radius ($s$)

• Outer boundary where mass distribution drops to zero
• Physically motivated by tidal truncation radius
• For satellites: set by tidal radius $r_t \propto (M_{\text{sat}}/M_{\text{host}})^{1/3} r_{\text{orbit}}$

Limiting Cases

Limit	Result
$s \to \infty$	Approaches PIEMD (no truncation)
$w \to 0$	Singular isothermal truncated profile
$s \to \infty, w \to 0$	Approaches SIS
$w \approx s$	Very compact, strongly truncated profile

Use Cases

Satellite Galaxies:

• Model companion galaxies in group/cluster environments
• Truncation radius constrained by tidal stripping
• Example: Perturbers in quad lens systems

Dark Matter Subhalos:

• Substructure in strong lens galaxies
• $s/w$ ratio from N-body simulations: typically $s/w \sim 10$–$20$

Compact Components:

• Bulge + disk decomposition in spiral lenses
• Bulge component with finite extent

Comparison with PIEMD

Feature	PIEMD	dPIE
Extent	Infinite	Finite (truncated)
Parameters	6	7 (adds truncation radius)
Mass	Diverges at large $r$	Finite total mass
Use case	Main lens galaxies	Satellites, subhalos
Complexity	Simpler	More physical for substructure

Related Profiles

• PIEMD - Pseudo-isothermal elliptical (no truncation)
• PIEMDMFL - Mass-follows-light version of PIEMD
• SPEMD - Softened power-law elliptical

References

• Elíasdóttir, Á. et al. (2007). Where is the dark matter? Probing structure with strong lensing. arXiv:0710.5636
• Suyu, S. H. & Halkola, A. (2010). The halos of satellite galaxies: the companion of the massive elliptical lens SL2S-J08544-0121. A&A, 524, A94