External Effects: Third Order

Going one order beyond the constant convergence and shear described in External Effects, the environment also introduces third-order (flexion-like) perturbations in the lens potential. Assuming the perturber to be an SIS, the third-order perturbation takes a "restricted" one-parameter form,

\[\begin{equation*} ψ(\pmb{θ}) = δ \, θ^3 \cos(φ - ϕ) \sin^2(φ - ϕ), \end{equation*}\]

where ($θ,~φ$) are the polar coordinates in the image plane, $δ$ is the amplitude of the perturbation, and $ϕ$ is its direction (i.e., the direction towards the SIS perturber, which coincides with the external shear angle it produces).

In LensFactory, to define third-order external effects, the user needs to specify two parameters: the amplitude of the perturbation ($δ$) and its direction ($ϕ$, in degrees). The perturbation is always centered at the origin of the image plane.

LensFactory.Lenses.init_ExternalEffects3 — Type

init_ExternalEffects3(delta::Real = NaN, 
                      angle::Real = NaN)

Initialize "restricted" third order perturbations assuming SIS as our perturber.

Keyword Arguments

delta::Real = NaN: Amplitude of third order perturbations (dimensionless).
angle::Real = NaN: Direction of the perturbation (in $\rm \mathbf{degrees}$).

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LensFactory.Lenses.ExternalEffects3.potential! — Function

potential!(ψ::U, θx::S, θy::S, δ::T, ϕ::T) where {U<:Real, S<:Real, T<:Real}

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potential!(ψ::U, θx::S, θy::S, δ::T, ϕ::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate potential at given coordinates for "restricted" third order perturbations corresponding to SIS lens model and update the potential in place.

Arguments

ψ : Potential at given coordinates
θx: x-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θy: y-coordinate(s) (in $\rm \mathbf{arcseconds}$).
δ : Amplitude of third order perturbations.
ϕ : External Shear angle (in $\rm \mathbf{degrees}$).

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LensFactory.Lenses.ExternalEffects3.deflection! — Function

deflection!(ψx::U, ψy::U, θx::S, θy::S, δ::T, ϕ::T) where {U<:Real, S<:Real, T<:Real}

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deflection!(ψx::U, ψy::U, θx::S, θy::S, δ::T, ϕ::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate deflection at given coordinates for "restricted" third order perturbations corresponding to SIS lens model and update the deflection in place.

Arguments

ψx: x-component of deflection at given coordinates
ψy: y-component of deflection at given coordinates
θx: x-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θy: y-coordinate(s) (in $\rm \mathbf{arcseconds}$).
δ : Amplitude of third order perturbations.
ϕ : External Shear angle (in $\rm \mathbf{degrees}$).

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LensFactory.Lenses.ExternalEffects3.jacobian! — Function

jacobian!(ψxx::U, ψyy::U, ψxy::U, θx::S, θy::S, δ::T, ϕ::T) where {U<:Real, S<:Real, T<:Real}

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jacobian!(ψxx::U, ψyy::U, ψxy::U, θx::S, θy::S, δ::T, ϕ::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate jacobian at given coordinates for "restricted" third order perturbations corresponding to SIS lens model and update the jacobian in place.

Arguments

ψxx: xx-component of jacobian at given coordinates
ψyy: yy-component of jacobian at given coordinates
ψxy: xy-component of jacobian at given coordinates
θx: x-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θy: y-coordinate(s) (in $\rm \mathbf{arcseconds}$).
δ : Amplitude of third order perturbations.
ϕ : External Shear angle (in $\rm \mathbf{degrees}$).

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