Pseudo-Isothermal Elliptical Mass Distribution (PIEMD)

Usage

piemd
  0.000  #x-coord  (= x centroid)
  0.000  #y-coord  (= y centroid)
  0.800  #b/a    (= q axis ratio)
  0.500  #theta  (= position angle, ccw from +x in radians.  0 means distribution is elongated along x.)
  2.600  #theta_e [scale:] (= Lens strength; = Einstein radius if "scale:" keyword given)
  0.200  #r_core  (= w core radius)

The surface mass density distribution is given by:

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

where:

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

• $E_0$ is the lens strength.
• $w$ is the core radius.
• $e$ is the ellipticity, defined as: $e = \frac{1 - q}{1 + q}$, where $q$ is the axis-ratio. Inverting this gives $q = \frac{1 - e}{1 + e}$

Notes

• When $w = 0$ and $e = 0$ : $E_0$ corresponds to the Einstein radius of the Singular Isothermal Sphere (SIS).
• Without scale: keyword $E_0 = \theta_E$ where $\theta_E$ is the 5th parameter in the config file
• With scale: keyword $E_0 = \frac{\theta_E^2}{\sqrt{\theta_E^2 + w^2} - w}$ where $\theta_E$ is the 5th parameter in the config file
• If $w < r_{\text{soft}}$, then $w$ is set to $r_{\text{soft}}$.
• The two parameters $E_0$ and $w$ (5th and 6th parameters) allow the log: option.

Alternative Formulation of $\kappa(x, y)$

Rewriting $\kappa$ in terms of $x^2 + \frac{y^2}{q^2}$ instead of $r_{em}$:

$$\begin{aligned} \kappa(x,y) &= \frac{E_0}{2\sqrt{w^2 + r_{\text{em}}^2}} \\ &= \frac{E_0 (1+e)}{(1+e) \cdot 2\sqrt{w^2 + r_{\text{em}}^2}} \\ &= \frac{E_0 (1+e)}{2\sqrt{w^2 (1+e)^2 + x^2 + \frac{y^2 (1+e)^2}{1-e^2}}} \\ &= \frac{E_0 (1+e)}{2\sqrt{w^2 (1+e)^2 + x^2 + \frac{y^2}{q^2}}} \\ &= \frac{E_0 \frac{2}{1+q}}{2\sqrt{\frac{4w^2}{(1+q)^2} + x^2 + \frac{y^2}{q^2}}} \\ &= \frac{E_0}{(1+q)} \cdot \frac{1}{\sqrt{\frac{4w^2}{(1+q)^2} + x^2 + \frac{y^2}{q^2}}} \end{aligned}$$

Citation for Profile:

@ARTICLE{Kassiola1993,
       author = {{Kassiola}, Aggeliki and {Kovner}, Israel},
        title = "{Elliptic Mass Distributions versus Elliptic Potentials in Gravitational Lenses}",
      journal = {The Astrophysical Journal},
         year = 1993,
        month = nov,
       volume = {417},
        pages = {450},
          doi = {10.1086/173325}
}