PIEMDMFL (Mass-Follows-Light PIEMD)

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Overview

The PIEMDMFL (Pseudo-Isothermal Elliptical Mass Distribution - Mass Follows Light) profile is almost identical to PIEMD, but specifically designed as a mass-follows-light (MFL) model for baryonic components.

PIEMDMFL has one additional parameter multfac to multiply the theta_e parameter. This is particularly useful when:

• A PIEMD profile models the lens light (e.g., as part of a chameleon profile)
• The chameleon profile (difference of 2 PIEMDs mimicking a Sérsic profile) is used as the baryonic κ profile
• The first six parameters of PIEMDMFL are identical to the PIEMD light profile
• The 7th parameter multfac converts the light amplitude (theta_e) to κ amplitude

Mathematical Definition

The convergence is given by:

$$\kappa(x,y) = \frac{E_0}{2\sqrt{w^2 + r_{\text{em}}^2}}$$

where:

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

• $E_0$ = lens strength
• $w$ = core radius
• $e$ = ellipticity, $e = \frac{1-q}{1+q}$ (inverting: $q = \frac{1-e}{1+e}$)
• $q$ = axis ratio (minor/major)

In the limiting case where $w = e = 0$, $E_0$ is the Einstein radius of the SIS.

Parameter Calculation

Without scale:

$$E_0 = \theta_E \times m_f$$

where $\theta_E$ is the 5th parameter and $m_f$ is the 7th parameter (multfac).

With scale:

$$E_0 = \theta_E \times m_f$$ $$E_0 = \frac{E_0^2}{\sqrt{E_0^2 + w^2} - w}$$

:::warning Softening Radius If $w < r_{\text{soft}}$, then $w$ will be set to $r_{\text{soft}}$ (default: $r_{\text{soft}} = 10^{-4}$) :::

Logarithmic Parameters

The parameters $E_0$ and $w$ (5th and 6th parameters) support the log: option:

• $E_0$ is computed first as $E_0 = \theta_E \times m_f$
• Then treated as a logarithmic value (base 10)

Combined log: and scale:

When both options are used:

1. First: $E_0 = \theta_E \times m_f$ is exponentiated ($10^{E_0}$) to account for log:
2. Then: The scaling $E_0 = \frac{E_0^2}{\sqrt{E_0^2 + w^2} - w}$ is applied for scale:

GLEE Configuration

Basic Usage

piemdmfl  
  0.000  #x-coord  (x centroid)
  0.000  #y-coord  (y centroid)  
  0.800  #b/a      (q, axis ratio related to e in kappa definition)
  0.500  #theta    (position angle, ccw from +x in radians; 0 = elongated along x)
  2.600  #theta_e  (Einstein radius or amplitude)
  0.200  #r_core   (w, core radius)  
  1.500  #multfac  (mf, multiplication factor for theta_e)

Parameter Summary

#	Parameter	Description	Options
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 $w$	log:
7	multfac	Multiplication factor for theta_e

Alternative Formulation

To express the PIEMD κ distribution in terms of $(x^2 + y^2/q^2)$ instead of $r_{\text{em}}$, see the PIEMD profile description.

Use Cases

Chameleon Profile Example:

1. Model lens light with 2 PIEMD profiles (difference mimics Sérsic)
2. Use PIEMDMFL for baryonic mass:
   • Copy parameters 1–6 from the PIEMD light profile
   • Set multfac (parameter 7) as the mass-to-light ratio
   • This enforces mass-follows-light with a constant M/L ratio

Related Profiles

• PIEMD - Standard pseudo-isothermal elliptical mass distribution
• SPEMD - Softened power-law elliptical mass distribution
• dPIE - Dual pseudo-isothermal elliptical (truncated version)