Lenses

get_meshgrid(θx::RV, θy::RV, dθ::RV)

Generate a meshgrid of coordinates on which various quantities can be evaluated. At present, this function only generates square pixels. In future if the need arises, it can be extended to generate rectangular pixels as well.

get_critical_density(Dd::RV, Dds::RV, Ds::RV; unit::Symbol=:kg_m2)

Calculate the critical surface density,

\[Σ_{\rm cr} = \frac{c^2}{4 π {\rm G}} \frac{D_s}{D_d D_{ds}},\]

given the angular diameter distances. The result can be returned in different units,

• :kg_m2 $\Rightarrow{\rm kg/m^2}$,
• :msun_pc2 $\Rightarrow{\rm M_⊙/pc^2}$,
• :msun_arcsec2 $\Rightarrow{\rm M_⊙/arcsec^2}$.

get_potential(lens::AbstractLens, θx::ROA, θy::ROA) --> ROA

get_deflection(lens::AbstractLens, θx::ROA, θy::ROA) --> Tuple{ROA, ROA}

Calculates the deflection angles (i.e., the gradient of the potential) for a given lens model. Returns a tuple of deflection components, i.e., $(ψ_x, ψ_y)$.

get_jacobian(lens::AbstractLens, θx::ROA, θy::ROA) --> Tuple{ROA, ROA, ROA}

Calculates the jacobian (i.e., deformation tensor) of the lens mapping for a given lens model. The jacobian is a $2\times2$ matrix composed of the second derivatives of the potential, which is given as,

\[\mathcal{A} = \begin{pmatrix} ψ_{xx} & ψ_{xy} \\ ψ_{xy} & ψ_{yy} \end{pmatrix}.\]

Since the jacobian is symmetric (for single lens plane), only three components are returned, i.e., $(ψ_{xx}, ψ_{yy}, ψ_{xy})$.

get_time_delay(lens::AbstractLens, θ_x::ROA, θ_y::ROA) --> ROA

Calculates the time delay for a given lens model. The corresponding expression is given as,

\[t_d(\pmb{θ}; \pmb{β}) = \frac{1+z_l}{\rm c} \frac{D_d D_s}{D_{ds}} \left[ \frac{(\pmb{θ} - \pmb{β})^2}{2} - \frac{D_{ds}}{D_s} \psi(\pmb{θ}) \right]\]

get_magnification(lens::AbstractLens, θx::ROA, θy::ROA) --> ROA

Calculates the magnification for a given lens model. The corresponding expression is given as,

\[\mu = \frac{1}{\det \mathcal{A}} = \frac{1}{(1 - κ)^2 - γ^2}\]

get_image(lens::AbstractLens, θx::ROA, θy::ROA, adis::Float64, β::NTuple{2, RV})::Vector{NTuple{2, RV}}
get_image(lens::AbstractLens, θx::ROA, θy::ROA, adis::Float64, β::Matrix{<:RV})::Matrix{<:RV}

Calculates the image position for a given lens model by solving the lens equation,

\[\pmb{β} = \pmb{θ} - \frac{D_{ds}}{D_s} ∇\psi(\pmb{θ})\]