External Shear

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External shear (shear) models the tidal gravitational influence of the large-scale mass distribution surrounding the primary lens (e.g., a galaxy group or cluster). It is almost always included alongside a mass profile such as EPL or PIEMD to account for line-of-sight structure.

Parameters

#	Name	Description
1	magnitude	Shear strength $\gamma_\text{ext}$ (dimensionless; typically $\lesssim 0.3$)
2	theta	Shear position angle $\phi_\text{ext}$ in radians, counter-clockwise from +x

Usage

shear
0.030851  #magnitude  γ_ext, shear strength; flat:0,0.3  step:0.01)
0.800128  #theta  φ_ext, shear position angle in radians; noprior: step:0.03)

The lensing potential for external shear is given by:

$\psi = \frac{1}{2} \gamma_{\text{ext}} r^2 \cos\left(2(\phi - \phi_{\text{ext}})\right)$

which can be rewritten as:

$\psi = \frac{1}{2} \gamma_{\text{ext}} \left[ \cos(2\phi_{\text{ext}})(x^2 - y^2) + 2\sin(2\phi_{\text{ext}}) xy \right]$

where:

• $r^2 = x^2 + y^2$, assuming the shear centre is set to $(0,0)$ without loss of generality.

• $\gamma_{\text{ext}}$ is the shear strength.

• $\phi_{\text{ext}}$ is the shear position angle:

  • $\phi_{\text{ext}} = 0^\circ$ → images are stretched horizontally.
  • $\phi_{\text{ext}} = 90^\circ$ → images are stretched vertically.

• $\phi$ is the polar angle of the position $(x, y)$, given by:

  $\phi = \arccos \left(\frac{x}{r}\right)$

Notes:

• In GLEE configuration:
  • magnitude corresponds to $\gamma_{\text{ext}}$.
  • theta corresponds to $\phi_{\text{ext}}$ in radians.