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Elliptical Power Law (EPL)

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The Elliptical Power Law (EPL) profile is the standard mass model for galaxy-scale gravitational lenses. It describes a smooth, elliptically-symmetric power-law convergence profile with a very small softening core. It is the preferred choice for most single-galaxy lens modelling.

note

The EPL profile requires a very small softening radius to avoid a central divergence. Set Rsoft 1e-5 in your configfile when using this profile.

Parameters​

#NameDescription
1x-coordx-centroid of the lens (arcsec)
2y-coordy-centroid of the lens (arcsec)
3b/aAxis ratio (minor/major), range (0, 1]
4thetaPosition angle (radians, counter-clockwise from +x)
5theta_eEinstein radius (arcsec)
6gamSlope parameter; γ′=2γ−1\gamma' = 2\gamma - 1 where density ρ∝r−γ′\rho \propto r^{-\gamma'}

Usage​

    epl
    4.006098  #x-coord   noprior:  step:0.001 
    4.006098  #y-coord   noprior:  step:0.001
    0.800000  #b/a       flat:0.3,1  step:0.01
    3.141590  #theta     noprior:  step:0.05
    1.500000  #theta_e   flat:0.75,2.5  step:0.01
    2.400000  #gam       exact:  step:0.01

Surface Mass Density​

The surface mass density κ(r)\kappa(r) is given by:

κ(r)=3−γ2(bqR2+rsoft2)γ−1\kappa(r) = \frac{3 - \gamma}{2} \left( \frac{b}{q \sqrt{R^2 + r_{\text{soft}}^2}} \right)^{\gamma - 1}

where:

  • R2=x2+y2q2R^2 = x^2 + \frac{y^2}{q^2}
  • rsoft=10−5r_{\text{soft}} = 10^{-5} (must set Rsoft 1e-5 in config)
  • γ=γ′−12\gamma = \frac{\gamma^\prime - 1}{2}
  • b=(21+q)1γ−1⋅q⋅θEb = \left( \frac{2}{1 + q} \right)^{\frac{1}{\gamma - 1}} \cdot q \cdot \theta_E

Important Notes​

  • This profile assumes a very small softening core (via rsoft=10−5r_{\text{soft}} = 10^{-5}), suitable for lensing-only analysis.
    It diverges at the centre, so not ideal for dynamical modelling.
  • For extended core usage (e.g. in dark matter halos for group/cluster lenses), use the SPEMD profile instead.
  • θE\theta_E should represent the observed ring radius from imaging — not a free scaling factor.

Slope Conversion​

To convert between EPL and SPEMD slope conventions: γEPL′=2⋅γSPEMD+1\gamma^\prime_{\text{EPL}} = 2 \cdot \gamma_{\text{SPEMD}} + 1

Citation​

@ARTICLE{2015A&A...580A..79T,
       author = {{Tessore}, Nicolas and {Metcalf}, R. Benton},
        title = "{The elliptical power law profile lens}",
      journal = {\aap},
     keywords = {gravitational lensing: strong, methods: analytical, Astrophysics - Cosmology and Nongalactic Astrophysics},
         year = 2015,
        month = aug,
       volume = {580},
          eid = {A79},
        pages = {A79},
          doi = {10.1051/0004-6361/201526773},
archivePrefix = {arXiv},
       eprint = {1507.01819},
 primaryClass = {astro-ph.CO},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2015A&A...580A..79T},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}