SIE

Set the potential profile to the type SIE.

In set_lens.c:set_dynamics(), the impact parameter is computed as such:

b_0 = \frac{4 \pi \sigma_0^2}{c^2} \frac{D_{LS}}{D_{OS}}

with \frac{\pi}{c^2} = 7.2 10^{-6} arcsec / (km/s)^2. To obtain this value, \pi is converted to 648,000 arcsec.

The ellipticity of the potential ε is proportional to the ellipticity of the mass distribution e_{mass}

\epsilon = e_{mass} / 3

Circular SIS has \epsilon = 0.

A circularised radius is defined as R^2 = (1 - \epsilon)x^2 + (1+\epsilon)y^2.

The projected effective lensing potential in direction \boldsymbol{\theta} of Newtonian potential \Phi is

\psi (\boldsymbol{\theta}) \approx \frac{2}{c^2} \frac{D_{LS}}{D_{L} D_{S}} \int_0^{D_{LS}} \mathrm{d}\chi \Phi(\boldsymbol{\theta} D_A(\chi); \chi)

where D_A is the angular diameter distance. The SIE lensing potential writes

\psi (x, y) = b_0 R

Program e_grad.c computes the first derivatives of this potential

\partial_{x} \psi = \frac{b_0 (1 - \epsilon)}{R} x

\partial_{y} \psi = \frac{b_0 (1 + \epsilon)}{R} y

In file e_grad2.c, the 2nd derivatives of the gradient are computed in 2D in the amplification frame, and rotated afterwards back to the reference frame.

The second derivatives of the lensing potential are

\partial^2_{xx} \psi = \frac{b_0 (1 - \epsilon^2)}{R^3} y^2

\partial^2_{yy} \psi = \frac{b_0 (1 - \epsilon^2)}{R^3} x^2

\partial^2_{xy} \psi = - \frac{b_0 (1-\epsilon^2)}{R^3} xy

From the 2nd derivatives, the convergence is computed in g_mass.c:computeKmass().

\kappa = \frac{1}{2} (\partial^2_{xx} + \partial^2_{yy}) \psi = \frac{b_0(1-\epsilon^2)}{2R^3}(x^2+y^2)

The shear is computed in g_shear.c.

\gamma_1 = \frac{1}{2} (\partial^2_{xx} - \partial^2_{yy}) \psi = \frac{b_0(1-\epsilon^2)}{2R^3} ( y^2 - x^2)

\gamma_2 = - \partial^2_{xy} \psi = \frac{b_0(1-\epsilon^2)}{R^3} xy

\gamma = \sqrt{\gamma_1^2 + \gamma_2^2} = \frac{b_0(1-\epsilon^2)}{2R^3} (x^2 + y^2)