A final, very central concept in lensing, is the convergence κ \kappa κ ψ \psi ψ
κ ( θ ) = 1 2 ∇ θ ψ R \boxed{\kappa(\boldsymbol{\theta})= \frac12\nabla_\theta\psi^R} κ ( θ ) = 2 1 ∇ θ ψ R Writing it out more explicitely, we find
κ ( θ ) = 1 2 ( ∂ 2 ψ R ∂ θ 1 2 + ∂ 2 ψ R ∂ θ 2 2 ) \kappa(\boldsymbol{\theta})= \frac12\left(
\frac{\partial^2\psi^{\mathrm{R}}}{\partial\theta_1^2} +
\frac{\partial^2\psi^{\mathrm{R}}}{\partial\theta_2^2}
\right) κ ( θ ) = 2 1 ( ∂ θ 1 2 ∂ 2 ψ R + ∂ θ 2 2 ∂ 2 ψ R ) SIE (Singular isothermal ellipsoid) ¶ The singular isothermal ellipsoid is a simple example of a non-spherical source.
The dimensionless projected surface-mass density κ \kappa κ
κ ( ξ 1 , ξ 2 ) = f ξ 0 2 ξ 1 2 + f 2 ξ 2 2 , \kappa(\xi_1,\xi_2)=\frac{\sqrt{f}\xi_0}{2\sqrt{\xi_1^2+f^2\xi_2^2}}, κ ( ξ 1 , ξ 2 ) = 2 ξ 1 2 + f 2 ξ 2 2 f ξ 0 , where the axis ratio f f f 0 < f ≤ 1 0\lt f\le1 0 < f ≤ 1 ξ 0 \xi_0 ξ 0 R E R_E R E
Solving the Poisson equation, this gives, according to
Kormann et al. (1994)
ψ ξ 0 , f , θ , D L SIE(R) ( θ , ϕ ) = ξ 0 D L 2 f 1 − f 2 θ ⋅ ( [ sin ( ϕ − λ L ) ] ⋅ sin − 1 ( 1 − f 2 ⋅ sin ( ϕ − λ L ) ) + [ cos ( ϕ − λ L ) ] ⋅ sinh − 1 ( 1 − f 2 f cos ( ϕ − λ L ) ) ) . \begin{aligned}
\begin{split}
\psi_{\xi_0,f,\theta,D_\mathrm{L}}^\textrm{SIE(R)}(\theta,\phi) =
\frac{\xi_0}{D_\textrm{L}^2}\sqrt{\frac{f}{1-f^2}}\theta\cdot
&\Bigg([\sin(\phi-\lambda_L)]\cdot\sin^{-1}\left(\sqrt{1-f^2}\cdot \sin{(\phi-\lambda_L)}\right)
\\\\&
+[\cos(\phi-\lambda_L)]\cdot\sinh^{-1}\left(\frac{\sqrt{1-f^2}}{f}\cos(\phi-\lambda_L)\right)\Bigg).
\end{split}
\end{aligned} ψ ξ 0 , f , θ , D L SIE(R) ( θ , ϕ ) = D L 2 ξ 0 1 − f 2 f θ ⋅ ( [ sin ( ϕ − λ L )] ⋅ sin − 1 ( 1 − f 2 ⋅ sin ( ϕ − λ L ) ) + [ cos ( ϕ − λ L )] ⋅ sinh − 1 ( f 1 − f 2 cos ( ϕ − λ L ) ) ) . where ( θ , ϕ ) (\theta,\phi) ( θ , ϕ ) λ L \lambda_L λ L
Kormann, R., Schneider, P., & Bartelmann, M. (1994). Isothermal elliptical gravitational lens models. Astronomy and Astrophysics (ISSN 0004-6361), Vol. 284, No. 1, p. 285-299 , 284 , 285–299. https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994A&A...284..285K&classic=YES