The SIE Lens

SIE (Singular isothermal ellipsoid)¶

The singular isothermal ellipsoid is a simple example of a non-spherical source.

The dimensionless projected surface-mass density $\kappa$ is given as

\kappa(\xi_1,\xi_2)=\frac{\sqrt{f}\xi_0}{2\sqrt{\xi_1^2+f^2\xi_2^2}},

(1)

where the axis ratio $f$ obeys $0\lt f\le1$ and $\xi_0$ is a constant parameter related to the total mass (analogous to the Einstein radius $R_E$ ).

Remark The relationship between $\kappa$ and the lens potential $\psi^{\mathrm{R}}$ is

\kappa(\xi_1,\xi_2) = \frac12D_L^2\left( \psi_{\xi_1\xi_1}^{\mathrm{R}}(\xi_1,\xi_2) + \psi_{\xi_2\xi_2}^{\mathrm{R}}(\xi_1,\xi_2) \right)

(2)

where $\psi^{\mathrm{R}}$ is defined in Lens Potential.

Remark Note that the normalization is chosen such that the mass inside an elliptical iso-density contour for fixed $\Sigma$ is independent of the axis ratio $f$ .

Solving the Poisson equation, this gives, according to Kormann (1994) the following. \begin{aligned} \begin{split} \psi_{\xi_0,f,\theta,D_\mathrm{L}}^\textrm{SIE(R)}(R,\phi) = \frac{\xi_0}{D_\textrm{L}^2}\sqrt{\frac{f}{1-f^2}}R\cdot &\Bigg([\sin(\phi-\theta)]\cdot\sin^{-1}\left(\sqrt{1-f^2}\cdot \sin{(\phi-\theta)}\right) \\& +[\cos(\phi-\theta)]\cdot\sinh^{-1}\left(\frac{\sqrt{1-f^2}}{f}\cos(\phi-\theta)\right)\Bigg). \end{split} \end{aligned} where $(R,\phi)$ are the polar coordinates in the lens plane, whereas $\theta$ is the orientation of the ellipse. That is, the major axis of the ellipse is $\theta$ from the $x$ -axis, counter-clockwise. Thus $\phi$ should take a constant value across the image. Note the co-ordinate relation

\boldsymbol{\xi}= R\cdot (\sin\phi,\cos\phi).

(3)

In the following, we use the following shorthands: \begin{aligned} f’ & =\sqrt{1-f^2}, \\\\ C_0 & = \frac{\xi_0}{D_L^2} \end{aligned}

Normalisation of the Lens Potential¶

Recall from Lens Potential that the lens potential $\psi^{\mathrm{R}}$ used in the Roulette formalism differs from the more standard normalisation by a factor of $\xi_0^2/D_L^2$ or $C_0\xi_0$ . If we also mormalise $R$ , and write $R=\xi_0r$ , we can rewrite $\psi$ as \begin{split} \psi_{\xi_0,f,\theta,D_\mathrm{L}}^\textrm{SIE(R)}(r,\phi) = \frac{\xi_0^2}{D_\textrm{L}^2}\sqrt{\frac{f}{1-f^2}}\cdot r\cdot &\Bigg([\sin(\phi-\theta)]\cdot\sin^{-1}\left(\sqrt{1-f^2}\cdot \sin{(\phi-\theta)}\right) \\& +[\cos(\phi-\theta)]\cdot\sinh^{-1}\left(\frac{\sqrt{1-f^2}}{f}\cos(\phi-\theta)\right)\Bigg). \end{split} Now the standard normalisation reads \begin{split} \psi_{\xi_0,f,\theta,D_\mathrm{L}}^\textrm{SIE}(r,\phi) = \sqrt{\frac{f}{1-f^2}}\cdot r\cdot &\Bigg([\sin(\phi-\theta)]\cdot\sin^{-1}\left(\sqrt{1-f^2}\cdot \sin{(\phi-\theta)}\right) \\& +[\cos(\phi-\theta)]\cdot\sinh^{-1}\left(\frac{\sqrt{1-f^2}}{f}\cos(\phi-\theta)\right)\Bigg). \end{split}

Derivation with $\theta=0$ ¶

In this section we assume $\theta=0$ , aligning the elliptical lens with the primary axis. Then $\psi^{\mathrm{R}}$ becomes

\psi^{\mathrm{R}}(r,\phi) = C_0\cdot \frac{\sqrt{f}}{f'}R\cdot\left[ \sin\phi\sin^{-1}(f'\cdot\sin\phi) + \cos\phi\sinh^{-1}\left(\frac{f'}{f}\cdot\cos\phi\right)\right]

(4)

In Cartesian coordinates, we get

\psi^{\mathrm{R}}(x,y) = C_0\cdot \frac{\sqrt{f}}{f'}\cdot\left[ y\sin^{-1}\left(f'\cdot\frac{y}{R}\right) + x\sinh^{-1}\left(\frac{f'}{f}\cdot\frac{x}{R}\right) \right]

(5)

where

R=\sqrt{x^2+y^2}

(6)

The deflection is given by the gradient $\nabla\psi^{\mathrm{R}}$ . We need the following basic rules.

\frac{d}{dx}\sin^{-1} x = \frac{1}{\sqrt{1-x^2}}

(7)

\frac{d}{dx}\sinh^{-1} x = \frac{1}{\sqrt{1+x^2}}

(8)

Differentiation with respect to $x$ ¶

Firstly, we differentiate the $\sin^{-1}$ -term \begin{aligned} \begin{split} t_1 = \frac{d}{dx} y\sin^{-1}\left(f’\cdot\frac{y}{R}\right) &= \frac{y}{\sqrt{1 - (f’\frac{y}{R})^2}}(f’y) \frac{d}{dx}\frac{1}{R} \\& = \frac{y}{\sqrt{1 - (f’\frac{y}{R})^2}}(f’y) \frac{-1}{R^2} \frac{dR}{dx} \\& = \frac{y}{\sqrt{1 - (f’\frac{y}{R})^2}}(f’y) \frac{-1}{R^2} \frac{x}{R} \\& = -\frac{f’xy^2}{\sqrt{1 - (f’\frac{y}{R})^2}} \frac{1}{R^3} \\& = -\frac{f’xy^2}{R^3}\cdot \frac{1}{\sqrt{1 - (f’\frac{y}{R})^2}} \end{split} \end{aligned} Secondly, we differentiate the $\sinh^{-1}$ -term \begin{aligned} \begin{split} \frac{d}{dx} x\sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x}{R}\right) &= \sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x}{R}\right) + \frac{x}{\sqrt{1 + (\frac{f’}{f}\frac{x}{R})^2}}\frac{f’}{f} \cdot \frac{d}{dx}\frac{x}{R} \\& = \sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x}{R}\right) + t_2 \end{split} \end{aligned} where \begin{aligned} \begin{split} t_2 &= \frac{x}{\sqrt{1 + (\frac{f’}{f}\frac{x}{R})^2}}\frac{f’}{f} \cdot \frac{d}{dx}\frac{x}{R} \\& = \frac{x}{\sqrt{1 + (\frac{f’}{f}\frac{x}{R})^2}}\frac{f’}{f} \cdot \left[ \frac{1}{R} - x\frac{1}{R^2}\frac{x}{R} \right] \\& = \frac{x}{\sqrt{1 + (\frac{f’}{f}\frac{x}{R})^2}}\frac{f’}{f} \cdot \frac{x^2+y^2 - x^2}{R^3} \\& = \frac{f’}{f}\cdot\frac{xy^2}{\sqrt{1 + (\frac{f’}{f}\frac{x}{R})^2}} \cdot \frac{1}{R^3} \\& = \frac{f’xy^2}{R^3} \cdot \frac{1}{\sqrt{f^2 + (f’\frac{x}{R})^2}} \end{split} \end{aligned} The partial derivative of the lens potential is then

\frac{d}{dx}\psi^{\mathrm{R}}(x,y) = C_0\frac{\sqrt{f}}{f'}\cdot\left[ \sinh^{-1}\left(\frac{f'}{f}\cdot\frac{x}{R}\right) + t_1 + t_2 \right]

(9)

We can prove that $t_1+t_2=0$ .

t_1 + t_2 = \frac{f'xy^2}{R^3} \cdot \left[ \frac{1}{\sqrt{f^2 + (f'\frac{x}{R})^2}} - \frac{1}{\sqrt{1 - (f'\frac{y}{R})^2}} \right]

(10)

It suffices to prove that $T=0$ where

T = \frac{1}{\sqrt{f^2 + (f'\frac{x}{R})^2}} - \frac{1}{\sqrt{1 - (f'\frac{y}{R})^2}}

(11)

We have

T = \frac{R}{\sqrt{f^2R^2 + (1-f^2)x^2}} - \frac{R}{\sqrt{R^2 - (1-f^2)y^2}}

(12)

Using the fact that $R^2=x^2+y^2$ we get

T = = \frac{R}{\sqrt{f^2y^2+x^2}} - \frac{R}{\sqrt{x^2 + f^2y^2}} = 0,

(13)

as required. We conclude that

\frac{d}{dx}\psi^{\mathrm{R}}(x,y) = C_0\frac{\sqrt{f}}{f'} \sinh^{-1}\left(\frac{f'}{f}\cdot\frac{x}{R}\right)

(14)

Differentiation with respect to $y$ ¶

The differentiation with respect to $y$ is similar and gives

\frac{d}{dy}\psi^{\mathrm{R}}(x,y) = C_0\frac{\sqrt{f}}{f'} \sin^{-1}\left(f'\cdot\frac{y}{R}\right)

(15)

Deflection for arbitrary orientation¶

For Raytrace simulation, we only require the first order derivatives. We find it easiest to calculate the deflection vector in a local co-ordinate system where $\theta=0$ , and then rotate the deflection vector to get the deflectio in the global co-ordinate system. This is derived as [Approach 1](Differentiating SIE Approach 1).

In Roulette simulation, we require higher order derivatives. The recursive formulæ for roulette amplitudes proved intractible, but the non-recursive formulæ are straight forward if higher-order partial derivatives of $\psi$ can be calculated.

The rotation trick which worked for first-order derivativs does not easily generalise to higher orders. We ended up using a co-ordinate substitution and chain rule differentiation. This is elaborated as [Approach 2](Differentiation of SIE).

A possible . Approach 3 would differentiate the general formula for $\psi^{\mathrm{R}}$ , but believing this to be computationally intractible, we have not developed this any further.

Other properties¶

[Critical Curves for SIE](Critical Curves for SIE)
[Implementation of SIE](Implementation of SIE)