Implementation of Critical Curves

The critical curve is the set of points $\theta$ where the determinant of the magnification matrix $\mathcal{A}(\boldsymbol{\theta})$ vanishes. We have

\mathcal{A}(\boldsymbol{\theta}) = \begin{bmatrix} 1 - \psi_{\theta_1\theta_1} & -\psi_{\theta_1\theta_2} \\ -\psi_{\theta_1\theta_2} & 1 - \psi_{\theta_2\theta_2} \end{bmatrix}

(1)

and

\det\mathcal{A}(\boldsymbol{\theta}) = 1 - \psi_{\theta_1\theta_1} - \psi_{\theta_2\theta_2} + \psi_{\theta_1\theta_1}\psi_{\theta_2\theta_2} + 2\psi_{\theta_1\theta_2}

(2)

It is straight forward to compute this given the second derivatives.

The API¶

We see the critical curve as a set of points $(r,\phi)$ in polar coordinates. The Lens::criticalXi( double phi ) method returns $r$ for a given $\phi$ (phi).

As of v3.2.3 the implementation is incomplete. We should have a generic, but probably slow implementation in Lens, and faster implementations in subclasses where this is possible.

Critical Curves for Analytics Lenses¶

For circular lenses (SIS and Point Mass) the critical curve is a circle with radius $\theta_E$ . For these two lenses criticalXi() returns the Einstein radius.

For SIE, centred at the origin, the critical curve is given in polar coordinates $(r,\phi)$ where

r = \frac{\sqrt{f}\theta_E}{\sqrt{ \cos^2\phi + f^2\sin^2\phi}}

(3)

where $f$ is the elliptic ratio. This is impemented in SIE::criticalXi().