init_NSISMDLens(x_c::Real=0.0, y_c::Real=0.0, v_d::Real=NaN, x_s::Real=NaN)

Initialize a Non-Singular Isothermal Sphere mass distribution (NSISMD) lens with the given parameters.

potential!(ψ::T, θx::T, θy::T, θxc::RV, θyc::RV, vd::RV, θs::RV) where T <: ROA

\[ψ(\pmb{θ}) = 4 π \left(\frac{v_d}{c} \right)^2 \left[ \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} - θ_s \ln \left( \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} + θ_s \right) \right]\]

deflection!(ψx::T, ψy::T, θx::T, θy::T, θxc::RV, θyc::RV, vd::RV, θs::RV) where T <: ROA

\[\pmb{\hat{α}} (\pmb{θ}) = 4 π \left(\frac{v_d}{c} \right)^2 \frac{\pmb{θ} - \pmb{θ}_c}{θ_s + \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2}}\]

jacobian!(ψxx::T, ψyy::T, ψxy::T, θx::T, θy::T, θxc::RV, θyc::RV, vd::RV, θs::RV) where T <: ROA

\[\begin{align*} ψ_{xx} (\pmb{θ}) &= 4 π \left(\frac{v_d}{c} \right)^2 \left[ \frac{1}{θ_s + \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2}} - \frac{(\pmb{θ}_x - \pmb{θ}_{xc})^2}{\sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} \left( θ_s + \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} \right)^2 } \right] \\[5pt] ψ_{yy} (\pmb{θ}) &= 4 π \left(\frac{v_d}{c} \right)^2 \left[ \frac{1}{θ_s + \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2}} - \frac{(\pmb{θ}_y - \pmb{θ}_{yc})^2}{\sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} \left( θ_s + \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} \right)^2 } \right] \\[5pt] ψ_{xy} (\pmb{θ}) &= 4 π \left(\frac{v_d}{c} \right)^2 \left[ \frac{-(\pmb{θ}_x - \pmb{θ}_{xc}) \, (\pmb{θ}_y - \pmb{θ}_{yc})}{\sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} \left( θ_s + \sqrt{θ_s^2 + |\pmb{θ} - \pmb{θ}_c|^2} \right)^2 } \right] \\[5pt] \end{align*}\]

einstein_angle(Dds::RV, Ds::RV, vd::RV, θs::RV)::RV

\[θ_E = \sqrt{ 4 π \frac{D_{ds}}{D_s} \left(\frac{v_d}{c} \right)^2 } \sqrt{4 π \frac{D_{ds}}{D_s} \left(\frac{v_d}{c} \right)^2 - 2\,θ_s}\]