Lens Models Hans Georg Schaathun
NTNU---Norwegian University of Science and Technology
Ben David Normann
NTNU---Norwegian University of Science and Technology
At present, CosmoSim provides three analytic lens models:
Point Mass, SIS, and SIE.
The critical point is to provide the lens potential ψ R \psi^R ψ R Definition: Raytrace Equation
Point Mass ¶ Stars and other celestial objects may sometimes be modelled as
so-called point-masses.
The gravitational potential is in this case
ψ P M R ( θ 1 , θ 2 ) = θ E 2 ln θ = θ E 2 ⋅ ln θ 1 2 + θ 2 2 , \psi_{\mathrm{PM}}^R(\theta_1,\theta_2) = \theta_E^2\ln\theta
= \theta_E^2\cdot\ln\sqrt{\theta_1^2+\theta_2^2}, ψ PM R ( θ 1 , θ 2 ) = θ E 2 ln θ = θ E 2 ⋅ ln θ 1 2 + θ 2 2 , where θ E \theta_E θ E θ E = R E / D L \theta_E=R_E/D_L θ E = R E / D L θ = θ 1 2 + θ 2 2 . \theta=\sqrt{\theta_1^2+\theta_2^2}. θ = θ 1 2 + θ 2 2 .
Note that if one starts from ψ = ln x \psi=\ln{x} ψ = ln x
ψ R = θ E 2 ln θ θ E = θ E 2 ln θ − θ E 2 ln θ E \begin{aligned}
\psi^R = \theta_E^2\ln\frac{\theta}{\theta_E}= \theta_E^2\ln\theta-\theta_E^2\ln\theta_E
\end{aligned} ψ R = θ E 2 ln θ E θ = θ E 2 ln θ − θ E 2 ln θ E But the latter term is a constant, and the potential is always defined only up to a constant term, which shows that these two expressions are the same, as far as physics is concerned.
SIS (Singular isothermal sphere) ¶ It is often necessary to model the lens as an extended object.
The simplest such model,
useful for instance in the modelling of a galaxy surrounded by dark amtter,
is the so-called Singular isothermal sphere , or SIS-model for short.
In the roulette formalism, the lens potential is given as,
ψ S I S R ( θ 1 , θ 2 ) = θ E ⋅ θ = θ E ⋅ θ 1 2 + θ 2 2 , \psi^R_\mathrm{SIS}(\theta_1,\theta_2) = \theta_E\cdot\theta
= \theta_E\cdot\sqrt{\theta_1^2+\theta_2^2}, ψ SIS R ( θ 1 , θ 2 ) = θ E ⋅ θ = θ E ⋅ θ 1 2 + θ 2 2 , where notation is as for the point-mass case.
The partial derivatives are readily calculated as
∂ ψ R ∂ θ i = θ E ⋅ θ i θ 1 2 + θ 1 2 = θ E θ i θ for i = 1 , 2. \frac{\partial\psi^R}{\partial \theta_i} = \theta_E\cdot\frac{\theta_i}{\sqrt{\theta_1^2+\theta_1^2}}=\theta_E\frac{\theta_i}{\theta}\quad\text{for}\quad i=1,2. ∂ θ i ∂ ψ R = θ E ⋅ θ 1 2 + θ 1 2 θ i = θ E θ θ i for i = 1 , 2. Thus the reduced deflection angle (see (19)
α = ∇ θ ψ R = θ E θ ( θ 1 , θ 2 ) . \boldsymbol{\alpha}=\nabla_\theta\psi^R
=\frac{\theta_E}{\theta}\left(\theta_1,\theta_2\right). α = ∇ θ ψ R = θ θ E ( θ 1 , θ 2 ) . Note that the norm of this vector is constant; α = ∣ α ∣ = θ E . \alpha=|\boldsymbol{\alpha}|=\theta_E. α = ∣ α ∣ = θ E .
SIE ¶ The extension of the SIS to an ellipsoid is well described by
Kormann et al. (1994) Singular Isothermal Ellipsoid , or SIE-lens for short.
It is a three-parameter family of lens models:
The Einstein radius (θ E \theta_E θ E f f f λ L \lambda_L λ L
ψ SIE R ( θ E , f , λ L ; θ , ϕ ) = θ E 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^R_\textrm{SIE}(\theta_E,f,\lambda_L;\theta,\phi) & =
\theta_E\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 SIE R ( θ E , f , λ L ; θ , ϕ ) = θ E 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 )
θ = θ E x = θ ( sin ϕ , cos ϕ ) . \boldsymbol{\theta}=\theta_E\mathbf{x}=\theta(\sin\phi,\cos\phi). θ = θ E x = θ ( sin ϕ , cos ϕ ) . The deflection may now be calculated using that
∂ ψ ∂ θ 1 = θ E ⋅ f 1 − f 2 ⋅ ( cos λ L ⋅ sinh − 1 ( 1 − f 2 f θ 1 ′ θ ) − sin λ L ⋅ sin − 1 ( 1 − f 2 θ 2 ′ θ ) ) ∂ ψ ∂ θ 2 = θ E ⋅ f 1 − f 2 ⋅ ( sin λ L ⋅ sinh − 1 ( 1 − f 2 f θ 1 ′ θ ) + cos λ L ⋅ sin − 1 ( 1 − f 2 θ 2 ′ θ ) ) \begin{aligned}
\frac{\partial\psi}{\partial \theta_1} &=
\theta_E\cdot\frac{\sqrt{f}}{\sqrt{1-f^2}}
\\&
\cdot\left(
\cos\lambda_L\cdot\sinh^{-1}(\frac{\sqrt{1-f^2}}{f}\frac{\theta_1^\prime}{\theta})
- \sin\lambda_L\cdot\sin^{-1}(\sqrt{1-f^2}\frac{\theta_2^\prime}{\theta})
\right)
\\
\frac{\partial\psi}{\partial \theta_2} &=
\theta_E\cdot\frac{\sqrt{f}}{\sqrt{1-f^2}}
\\&
\cdot\left(
\sin\lambda_L\cdot\sinh^{-1}(\frac{\sqrt{1-f^2}}{f}\frac{\theta_1^\prime}{\theta})
+ \cos\lambda_L\cdot\sin^{-1}(\sqrt{1-f^2}\frac{\theta_2^\prime}{\theta})
\right)
\end{aligned} ∂ θ 1 ∂ ψ ∂ θ 2 ∂ ψ = θ E ⋅ 1 − f 2 f ⋅ ( cos λ L ⋅ sinh − 1 ( f 1 − f 2 θ θ 1 ′ ) − sin λ L ⋅ sin − 1 ( 1 − f 2 θ θ 2 ′ ) ) = θ E ⋅ 1 − f 2 f ⋅ ( sin λ L ⋅ sinh − 1 ( f 1 − f 2 θ θ 1 ′ ) + cos λ L ⋅ sin − 1 ( 1 − f 2 θ θ 2 ′ ) ) where
θ 1 ′ = cos λ L ⋅ θ 1 + sin λ L ⋅ θ 2 θ 2 ′ = − sin λ L ⋅ θ 1 + cos λ L ⋅ θ 2 \begin{aligned}
\theta_1^\prime &= \cos\lambda_L\cdot \theta_1 + \sin\lambda_L\cdot \theta_2
\\
\theta_2^\prime &= -\sin\lambda_L\cdot \theta_1 + \cos\lambda_L\cdot \theta_2
\end{aligned} θ 1 ′ θ 2 ′ = cos λ L ⋅ θ 1 + sin λ L ⋅ θ 2 = − sin λ L ⋅ θ 1 + cos λ L ⋅ θ 2 and θ = θ 1 2 + θ 2 2 \theta = \sqrt{\theta_1^2+\theta_2^2} θ = θ 1 2 + θ 2 2
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