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Softened Power-law Elliptical Mass Distribution (SPEMD)

Usage​

spemd
  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  (= Einstein radius if "scale:" keyword given)
  0.200  #r_core  (= Core radius; appears as `s = r_core*2/(1+q)` in equations)
  0.500  #gam  (= Slope parameter; related to γ' where 3D density ρ ∝ r^{-γ'}, with γ' = 2*gam + 1)

The surface mass density distribution follows:

κ(x1,x2)=E(u2+s2)−γ\kappa(x_1, x_2) = E (u^2 + s^2)^{-\gamma}

where:

u2=x12+x22q2u^2 = x_1^2 + \frac{x_2^2}{q^2} and s2=s⋅swheres=rcore2⋅2q2s^2=s⋅s \quad\text{where}\quad s=r_{core}^2\cdot\frac{2}{q^2}

  • For an isothermal profile: γ=0.5\gamma = 0.5
  • Without the scale: keyword:
    • E=θEE=\theta_E (theta_e parameter in GLEE config)
  • With the scale: keyword:
    • E=21+q(1−γ)re2((re2+s2)1−γ−s2(1−γ))E = \frac{2}{1+q} (1 - \gamma) \frac{r_e^2}{( (r_e^2+s^2)^{1-\gamma} - s^{2(1-\gamma)} )} If γ=1\gamma = 1, the above expression is undefined (division by zero), using L'Hôpital’s rule we get:
      E=21+q⋅(1−γ)re2log⁡e(re2+s2)−loge(s2)E=\frac{2}{1+q}\cdot\frac{(1-\gamma)r_e^2}{\log_e(r_e^2+s^2)-log_e(s^2)} E=s2E = s^2 when γ=1\gamma=1 and re=0r_e=0

Notes​

  • Converting PIEMD to SPEMD (without scale: keyword):

    1. Keep the same values for xx, yy, ba\frac{b}{a}, θ\theta, and rcorer_{\text{core}}.

    2. Adjust θE\theta_E using:

      θE,SPEMD=θE,PIEMD1+ba\theta_{E,\text{SPEMD}}=\frac{\theta_{E,\text{PIEMD}}}{1+\frac{b}{a}}

    3. Add the γ\gamma parameter, setting γ=0.5\gamma = 0.5.

Spherical Equivalent Einstein Radius:​

The Einstein radius θE,sph.eq.\theta_{E,\text {sph.eq.}} in Eq.(12) of the Suyu et al. (2013) parameterisation

κpl(θ1,θ2)=3−γ′2(θE,sph.eq.qθ12+θ22/q)γ′−1\kappa_{\text{pl}}(\theta_1, \theta_2) = \frac{3 - \gamma'}{2} \left( \frac{\theta_{E,\text {sph.eq.}}}{\sqrt{q\theta_1^2 + \theta_2^2/q}} \right)^{\gamma' - 1}

is the most robust and least dependent/correlated on other power-law mass parameter.

To convert this to our parameterisation κ(x1,x2)\kappa(x_1,x_2)

  • With scale: and zero core (s=0s = 0): The equivalent spherical Einstein radius is given by: θE,sph, eq=(21+q)12γq ⋅re\theta_{E, \text{sph, eq}} = \left( \frac{2}{1+q} \right)^{\frac{1}{2\gamma}} \sqrt{q} \, \cdot r_e where rer_e is the fifth parameter (theta_e) of the profile in the GLEE configuration.

    Notably, when q=1q = 1, this reduces to θE,sph, eq=re\theta_{E, \text{sph, eq}} = r_e , which aligns with the design of the "scale:" keyword, ensuring that it provides the Einstein radius in the circular case q=1q = 1. For cases where q≠1q \neq 1, θE,sph, eq\theta_{E, \text{sph, eq}} is defined to be as independent of qq as possible.

  • Without scale: and zero core (s=0s = 0): The equivalent spherical Einstein radius is: θE,sph, eq=(2re2−2γ)12γq\theta_{E, \text{sph, eq}} = \left( \frac{2r_e}{2 - 2\gamma} \right)^{\frac{1}{2\gamma}} \sqrt{q} where rer_e is again the fifth parameter (theta_e) of the profile in the GLEE configuration.

Citation for Profile​

@article{Barkana1998,
doi = {10.1086/305950},
url = {https://dx.doi.org/10.1086/305950},
year = {1998},
month = {8},
publisher = {},
volume = {502},
number = {2},
pages = {531},
author = {Rennan Barkana},
title = {Fast Calculation of a Family of Elliptical Gravitational Lens Models},
journal = {The Astrophysical Journal}
}