SIS Lens

The singular isothermal sphere (SIS) is one of the most widely used lens models for galaxies, as it naturally reproduces flat rotation curves. Its three-dimensional density profile is given by

\[\begin{equation*} ρ(r) = \frac{σ_v^2}{2π{\rm G}} \frac{1}{r^2}, \end{equation*}\]

where $σ_v$ is the (one-dimensional) velocity dispersion. The corresponding projected surface mass density is

\[\begin{equation*} Σ(R) = \frac{σ_v^2}{2{\rm G}} \frac{1}{R}, \end{equation*}\]

and the lens potential can be written as

\[\begin{equation*} ψ(\pmb{θ}) = 4π \left( \frac{σ_v}{\rm c} \right)^2 |\pmb{θ} - \pmb{θ}_c|, \end{equation*}\]

where $\pmb{θ}_c$ represents the lens center, leading to a deflection angle of constant magnitude,

\[\begin{equation*} \pmb{α}(\pmb{θ}) = 4π \left( \frac{σ_v}{\rm c} \right)^2 \frac{\pmb{θ} - \pmb{θ}_c}{|\pmb{θ} - \pmb{θ}_c|}. \end{equation*}\]

The Einstein angle for the SIS lens is given by

\[\begin{equation*} θ_E = 4π \left( \frac{σ_v}{\rm c} \right)^2 \frac{D_{ds}}{D_s}, \end{equation*}\]

and a source at $|\pmb{β} - \pmb{θ}_c| < θ_E$ produces two images, whereas a source outside the Einstein radius is only singly imaged.

In LensFactory, to define an SIS lens, the user needs to specify three parameters: its position in the image plane ($x_c,~y_c$) and its velocity dispersion ($v_d \equiv σ_v$, in km/s). By default, the lens is placed at the origin, i.e., ($x_c,~y_c$) = (0, 0).

LensFactory.Lenses.init_SISLens — Type

init_SISLens(x_c::Real = 0.0, 
             y_c::Real = 0.0, 
             v_d::Real = NaN)

Initialize a Singular Isothermal Sphere (SIS) lens with the given parameters.

Keyword Arguments

x_c::Real = 0.0: x-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
y_c::Real = 0.0: y-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
v_d::Real = NaN: Velocity dispersion (in $\rm \mathbf{km/s}$).

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LensFactory.Lenses.SISLens.potential! — Function

potential!(ψ::U, θx::S, θy::S, θxc::T, θyc::T, v_d::T) where {U<:Real, S<:Real, T<:Real}

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potential!(ψ::U, θx::S, θy::S, θxc::T, θyc::T, v_d::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate potential at given coordinates for a SIS lens and update the potential (ψ) in place. The lensing potential is given as,

\[ψ(θ_x, θ_y) = 4π \left( \frac{v_d}{{\rm c}} \right)^2 |\pmb{θ} - \pmb{θ}_c|.\]

Arguments

ψ: Potential at given coordinates
θx: x-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θy: y-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θxc: x-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
θyc: y-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
v_d: Velocity dispersion (in $\rm \mathbf{km/s}$).

Returns

nothing: Updates the potential (ψ) in place.

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LensFactory.Lenses.SISLens.deflection! — Function

deflection!(ψx::U, ψy::U, θx::S, θy::S, θxc::T, θyc::T, v_d::T) where {U<:Real, S<:Real, T<:Real}

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deflection!(ψx::U, ψy::U, θx::S, θy::S, θxc::T, θyc::T, v_d::T) where {U<:ROA, S<:ROA, T<:Real}

Calculate deflection at given coordinates for a SIS lens and update the deflection components (ψx, ψy) in place.

Arguments

ψx: x-component of the deflection at given coordinates
ψy: y-component of the deflection at given coordinates
θx: x-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θy: y-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θxc: x-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
θyc: y-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
v_d: Velocity dispersion (in $\rm \mathbf{km/s}$).

Returns

nothing: Updates the deflection (ψx, ψy) in place.

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LensFactory.Lenses.SISLens.jacobian! — Function

jacobian!(ψxx::U, ψyy::U, ψxy::U, θx::S, θy::S, θxc::T, θyc::T, v_d::T) where {U<:Real, S<:Real, T<:Real}

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

Calculate jacobian at given coordinates for a SIS lens and update the jacobian components (ψxx, ψyy, ψxy) in place.

Arguments

ψxx: xx-component of the jacobian at given coordinates
ψyy: yy-component of the jacobian at given coordinates
ψxy: xy-component of the jacobian at given coordinates
θx: x-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θy: y-coordinate(s) (in $\rm \mathbf{arcseconds}$).
θxc: x-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
θyc: y-coordinate of the lens (in $\rm \mathbf{arcseconds}$).
v_d: Velocity dispersion (in $\rm \mathbf{km/s}$).

Returns

nothing: Updates the jacobian (ψxx, ψyy, ψxy) in place.

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LensFactory.Lenses.SISLens.einstein_angle — Function

einstein_angle(;D_ds::Float64=NaN, D_s::Float64=NaN, v_d::Real=NaN)

Calculate the Einstein angle for a SIS lens,

\[\theta_E = 4 \pi \frac{D_{ds}}{D_s} \left( \frac{v_d}{{\rm c}} \right)^2.\]

Keyword Arguments

D_ds: ADD from the observer to the lens (in $\rm \mathbf{meters}$).
D_s: ADD from the observer to the source (in $\rm \mathbf{meters}$).
v_d: Velocity dispersion (in $\rm \mathbf{km/s}$).

Returns

θE: Einstein angle (in $\rm \mathbf{arcseconds}$).

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