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.

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.

A circularized radius is defined as $R^2 = (1 - \epsilon)x^2 + (1+\epsilon)y^2$, and the derivatives are

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

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

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

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

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

The shear is computed in g_shear.c.

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

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

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