External Effects

A lens galaxy is rarely isolated: nearby galaxies and the large-scale environment perturb the lensing signal. To the lowest order, these perturbations can be described by a constant external convergence ($κ$) and a constant external shear ($γ$) with position angle ($ϕ$). The corresponding lens potential is given by

\[\begin{equation*} ψ(\pmb{θ}) = \frac{κ + γ_1}{2} θ_x^2 + \frac{κ - γ_1}{2} θ_y^2 + γ_2 \, θ_x θ_y, \end{equation*}\]

where the two shear components are defined as

\[\begin{equation*} γ_1 = γ \cos(2ϕ), \qquad γ_2 = γ \sin(2ϕ). \end{equation*}\]

The external convergence produces an isotropic (de-)magnification, whereas the external shear stretches the images along the direction $ϕ$. Since the external convergence is degenerate with the lens mass (mass-sheet degeneracy), it is often fixed to zero in lens modeling.

In LensFactory, to define external effects, the user needs to specify three parameters: the external convergence ($κ$), the shear amplitude ($γ$), and the shear angle ($ϕ$, in degrees). The external effects are always centered at the origin of the image plane.

LensFactory.Lenses.init_ExternalEffects — Type

init_ExternalEffects(kappa::Real = NaN, 
                     gamma::Real = NaN, 
                     angle::Real = NaN)

Initialize constant external effects with the given parameters.

Keyword Arguments

kappa::Real = NaN: Convergence (dimensionless).
gamma::Real = NaN: Shear amplitude (dimensionless).
angle::Real = NaN: Shear angle (in $\rm \mathbf{degrees}$).

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LensFactory.Lenses.ExternalEffects.potential! — Function

potential!(ψ::U, θx::S, θy::S, kappa::T, gamma::T, angle::T) where {U<:Real, S<:Real, T<:Real}

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potential!(ψ::U, θx::S, θy::S, kappa::T, gamma::T, angle::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate potential at given coordinates for constant external convergence and shear 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 $\rm \mathbf{arcseconds}$).
κ : External convergence.
γ : External shear value.
ϕ : External Shear angle (in $\rm \mathbf{degrees}$).

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LensFactory.Lenses.ExternalEffects.deflection! — Function

deflection!(ψx::U, ψy::U, θx::S, θy::S, kappa::T, gamma::T, angle::T) where {U<:Real, S<:Real, T<:Real}

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deflection!(ψx::U, ψy::U, θx::S, θy::S, kappa::T, gamma::T, angle::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate deflection at given coordinates for constant external convergence and shear 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 $\rm \mathbf{arcseconds}$).
κ : External convergence.
γ : External shear value.
ϕ : External Shear angle (in $\rm \mathbf{degrees}$).

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LensFactory.Lenses.ExternalEffects.jacobian! — Function

jacobian!(ψxx::U, ψyy::U, ψxy::U, θx::S, θy::S, kappa::T, gamma::T, angle::T) where {U<:Real, S<:Real, T<:Real}

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

Calculate Jacobian at given coordinates for constant external convergence and shear 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 $\rm \mathbf{arcseconds}$).
κ : External convergence.
γ : External shear value.
ϕ : External Shear angle (in $\rm \mathbf{degrees}$).

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