Krang api

• Krang.AbstractCamera
• Krang.AbstractGeometry
• Krang.AbstractLevelSetGeometry
• Krang.AbstractMaterial
• Krang.AbstractMetric
• Krang.AbstractPixel
• Krang.AbstractScreen
• Krang.ConeGeometry
• Krang.ElectronSynchrotronPowerLawIntensity
• Krang.ElectronSynchrotronPowerLawPolarization
• Krang.IntensityCamera
• Krang.IntensityPixel
• Krang.IntensityScreen
• Krang.Kerr
• Krang.Mesh
• Krang.MeshGeometry
• Krang.SlowLightIntensityCamera
• Krang.SlowLightIntensityPixel
• Krang.SlowLightIntensityScreen
• Krang.Gθ
• Krang.I0_inf
• Krang.I1_inf_m_I0_terms
• Krang.I2_inf_m_I0_terms
• Krang.Im_inf_m_I0_terms
• Krang.Ip_inf_m_I0_terms
• Krang.Ir
• Krang.Ir_inf
• Krang.Ir_s
• Krang.It
• Krang.It_inf
• Krang.It_w_I0_terms
• Krang.Iϕ
• Krang.Iϕ_inf
• Krang.Iϕ_w_I0_terms
• Krang._isreal2
• Krang.absGto_Gthat
• Krang.absGθo_Gθhat
• Krang.absGϕo_Gϕhat
• Krang.emission_azimuth
• Krang.emission_coordinates
• Krang.emission_coordinates_fast_light
• Krang.emission_inclination
• Krang.emission_radius
• Krang.get_radial_roots
• Krang.horizon
• Krang.inclination
• Krang.jac_bl_d_zamo_u
• Krang.jac_bl_u_zamo_d
• Krang.jac_fluid_u_zamo_d
• Krang.jac_zamo_d_bl_u
• Krang.jac_zamo_u_bl_d
• Krang.metric
• Krang.metric_dd
• Krang.metric_uu
• Krang.mino_time
• Krang.p_bl_d
• Krang.penrose_walker
• Krang.r_potential
• Krang.radial_inf_integrals_case2
• Krang.radial_inf_integrals_case3
• Krang.radial_inf_integrals_case4
• Krang.radial_inf_integrals_m_I0_terms
• Krang.radial_integrals
• Krang.radial_w_I0_terms_integrals_case2
• Krang.radial_w_I0_terms_integrals_case3
• Krang.radial_w_I0_terms_integrals_case4
• Krang.regularized_Pi
• Krang.roots
• Krang.screen_coordinate
• Krang.screen_polarization
• Krang.synchrotronIntensity
• Krang.synchrotronPolarization
• Krang.total_mino_time
• Krang.α
• Krang.αboundary
• Krang.β
• Krang.βboundary
• Krang.η
• Krang.θ_potential
• Krang.λ

Raytracing Functions

Krang.emission_radius Function

emission_radius(
    pix::Krang.AbstractPixel,
    θs,
    isindir,
    n
) -> Tuple{Any, Any, Any, Any, Bool}

Emission radius for point originating at inclination θs whose nth order image appears at the screen coordinate (α, β). Returns 0 if the emission coordinates do not exist for that screen coordinate.

Arguments

• pix : Pixel information

• θs : Emission inclination

• isindir : Is emission to observer direct or indirect

• n : Image index

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emission_radius(
    pix::Krang.AbstractPixel,
    τ
) -> Tuple{Any, Any, Any, Bool}

Emission radius for point originating at at Mino time τ whose image appears at the screen coordinate (α, β). Returns 0 if the emission coordinates do not exist for that screen coordinate.

Arguments

-pix : Pixel information

• τ : Mino time

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Krang.emission_inclination Function

emission_inclination(pix::Krang.AbstractPixel, rs, νr)

Emission inclination for point originating at inclination rs whose nth order image appears at screen coordinate (α, β).

Arguments

• pix : Pixel information

• rs : Emission radius

• νr : Sign of radial velocity direction at emission. This is always positive for case 3 and case 4 geodesics.

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emission_inclination(
    pix::Krang.AbstractPixel,
    τ
) -> NTuple{6, Any}

Emission inclination for point at Mino time τ whose image appears at screen coordinate (α, β).

Arguments

• pix : Pixel information

• τ : Mino Time

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Krang.emission_coordinates_fast_light Function

emission_coordinates_fast_light(
    pix::Krang.AbstractPixel,
    θs,
    isindir,
    n
) -> Tuple{Any, Any, Any, Any, Bool}

Emission radius and azimuthal angle for point originating at inclination θs whose nth order image appears at the screen coordinate (α, β) for an observer located at inclination θo.

Arguments

• pix : Pixel information

• θs : Emission Inclination

• isindir : Whether emission to observer is direct or indirect

• n : Image index

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Krang.emission_coordinates Function

emission_coordinates(
    pix::Krang.AbstractPixel,
    θs,
    isindir,
    n
) -> Tuple{Any, Any, Any, Any, Any, Bool}

Emission radius and azimuthal angle for point originating at inclination θs whose nth order image appears at the screen coordinate (α, β) for an observer located at inclination θo.

Arguments

• pix : Pixel information

• θs : Emission Inclination

• isindir : Whether emission to observer is direct or indirect

• n : Image index

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emission_coordinates(
    pix::Krang.AbstractPixel,
    τ
) -> Tuple{Any, Any, Any, Any, Any, Any, Bool}

Ray trace a point that appears at the screen coordinate (α, β) for an observer located at inclination θo

Arguments

• pix : Pixel information

• τ : Mino Time

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Krang.mino_time Function

mino_time(pix, rs, isindir) -> Any

Mino time of trajectory between an observer at infinity and point at radius rs

Arguments

• pix : Pixel information

• rs : Emission radius

• isindir : Is the path direct or indirect?

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mino_time(pix::Krang.AbstractPixel, θs, isindir, n) -> Any

Mino time of trajectory between two inclinations for a given screen coordinate

Arguments

• pix : Pixel information

• θs : Emission inclination

• isindir : Is the path direct or indirect?

• n : nth image in orde of amount of minotime traversed

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Metric Functions

Krang.AbstractMetric Type

abstract type AbstractMetric

Abstract Metric Type

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Krang.Kerr Type

struct Kerr{T} <: Krang.AbstractMetric

Kerr Metric in Boyer Lindquist Coordinates

• mass: M = mass

• spin: a = J/M, where J is the angular momentum and M is the mass of the black hole.

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Krang.metric_uu Function

metric_uu(metric::Krang.AbstractMetric, args...) -> Any

Returns the inverse metric in some representation (usually as an nxn matrix).

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metric_uu(metric::Kerr{T}, r, θ) -> Any

Inverse Kerr metric in Boyer Lindquist (BL) coordinates.

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metric_uu(metric::Kerr{T}, coordinates) -> Any

Inverse Kerr metric in Boyer Lindquist (BL) coordinates.

Arguments

• metric : Kerr metric

• coordinates : Coordinates (t, r, θ, ϕ)

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Krang.metric_dd Function

metric_dd(metric::Krang.AbstractMetric, args...) -> Any

Returns the metric in some representation (usually as an nxn matrix).

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metric_dd(metric::Kerr{T}, r, θ) -> Any

Kerr metric in Boyer Lindquist (BL) coordinates.

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metric_dd(metric::Kerr{T}, coordinates) -> Any

Kerr metric in Boyer Lindquist (BL) coordinates.

Arguments

• metric : Kerr metric

• coordinates : Coordinates (t, r, θ, ϕ)

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Krang.horizon Function

horizon(metric::Kerr{T}) -> Any

Outer Horizon for the Kerr metric.

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Krang.λ Function

λ(pix::Krang.AbstractPixel) -> Any

λ(pix::AbstractPixel)

Calculate the λ value for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The λ value of the pixel.

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λ(_::Kerr, α, θo) -> Any

Energy reduced azimuthal angular momentum

Arguments

• metric: Kerr

• α: Horizontal Bardeen screen coordinate

• θo: Observer inclination

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Krang.η Function

η(pix::Krang.AbstractPixel) -> Any

η(pix::AbstractPixel)

Calculate the η value for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The η value of the pixel.

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η(metric::Kerr, α, β, θo) -> Any

Energy reduced Carter integral

Arguments

• metric: Kerr

• α: Horizontal Bardeen screen coordinate

• β: Bardeen vertical coordinate

• θo: Observer inclination

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Radial Integrals

Krang.r_potential Function

r_potential(metric::Kerr{T}, η, λ, r) -> Any

Radial potential of spacetime

Arguments

• metric: Kerr{T} metric

• η : Reduced Carter constant

• λ : Reduced azimuthal angular momentum

• r : Boyer Lindquist radius

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Krang.get_radial_roots Function

get_radial_roots(metric::Kerr{T}, η, λ) -> NTuple{4, Any}

Returns roots of $r^4+(a^2-\eta -{\lambda }^2)r^2+2(\eta +(a-\lambda {)}^2)r-a^2\eta$

Arguments

• metric: Kerr{T} metric

• η : Reduced Carter constant

• λ : Reduced azimuthal angular momentum

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Krang.Ir Function

Ir(pix::Krang.AbstractPixel, νr::Bool, rs) -> Any

Returns the antiderivative $I_r=\int \frac{dr}{\sqrt{\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}$. See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• pix : Pixel information

• νr : Sign of radial velocity direction at emission. This is always positive for case 3 and case 4 geodesics.

• rs : Emission radius

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Angular Integrals

Krang.θ_potential Function

θ_potential(metric::Kerr{T}, η, λ, θ) -> Any

Theta potential of a Kerr black hole

Arguments

• metric: Kerr{T} metric

• η : Reduced Carter constant

• λ : Reduced azimuthal angular momentum

• θ : Boyer Lindquist inclination

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Krang.Gθ Function

Gθ(
    pix::Krang.AbstractPixel,
    θs,
    isindir,
    n
) -> Tuple{Any, Any, Any, Any, Any, Bool}

Returns the antiderivative $G_{\theta }=\int \frac{d\theta }{\sqrt{\mathrm{\Theta }(\theta )}}$. See θ_potential(x) for an implementation of $\mathrm{\Theta }(heta)$.

Arguments

• pix : Pixel information

• θs : Emission inclination

• isindir : Is the path direct or indirect?

• n : nth image ordered by minotime

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Screen Coordinates

Krang.α Function

α(_::Kerr, λ, θo) -> Any

Horizontal Bardeen Screen Coordinate

Arguments

• metric: Kerr

• α: Horizontal Bardeen screen coordinate

• θo: Observer inclination

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Krang.β Function

β(metric::Kerr, λ, η, θo) -> Any

Vertical Bardeen Screen Coordinate

Arguments

• metric: Kerr

• λ: Energy reduced Azimuthal angular momentul

• η: Energy reduced Carter integral

• θo: Observer inclination

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Krang.αboundary Function

αboundary(metric::Kerr, θs) -> Any

Defines a horizontal boundary on the assymptotic observer's screen that emission that from θs must fall within.

Arguments

• metric: Kerr metric

• θs : Emission Inclination

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Krang.βboundary Function

βboundary(metric::Kerr{T}, α, θo, θs) -> Any

Defines a vertical boundary on the assymptotic observer's screen that emission that from θs must fall within.

Arguments

• metric: Kerr{T} metric

• α : Horizontal Bardeen screen coordinate

• θo : Observer inclination

• θs : Emission Inclination

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Local Frame transformations

Krang.p_bl_d Function

p_bl_d(
    metric::Kerr{T},
    r,
    θ,
    η,
    λ,
    νr::Bool,
    νθ::Bool
) -> Any

Returns the momentum form in the Boyer-Lindquist basis.

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Krang.jac_bl_u_zamo_d Function

jac_bl_u_zamo_d(metric::Kerr{T}, r, θ) -> Any

Jacobian which converts ZAMO vector to a Boyer-Lindquist basis

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Krang.jac_zamo_u_bl_d Function

jac_zamo_u_bl_d(metric::Kerr{T}, r, θ) -> Any

Jacobian which converts Boyer-Lindquist vector to a ZAMO basis

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Krang.jac_bl_d_zamo_u Function

jac_bl_d_zamo_u(metric::Kerr{T}, r, θ) -> Any

Jacobian which converts ZAMO covector to a Boyer-Lindquist basis

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Krang.jac_zamo_d_bl_u Function

jac_zamo_d_bl_u(metric::Kerr{T}, r, θ) -> Any

Returns the Jacobian which converts a Boyer-Lindquist covector to ZAMO basis.

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Krang.jac_fluid_u_zamo_d Function

jac_fluid_u_zamo_d(_::Kerr{T}, β, θ, φ) -> Any

Jacobian which expreases ZAMO vector in the fluid frame

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Polarization

Krang.screen_polarization Function

screen_polarization(
    metric::Kerr{T},
    κ::Complex,
    θ,
    α,
    β
) -> Tuple{Any, Any}

Returns the screen polarization associated with a killing spinor κ as seen seen by an asymptotic observer.

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Krang.penrose_walker Function

penrose_walker(
    metric::Kerr{T},
    r,
    θ,
    p_u::AbstractVector,
    f_u::AbstractVector
) -> Tuple{Any, Any}

Returns the Penrose walker constant for a photon with momentum p_u emitted from a fluid particle with momentum f_u.

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Krang.synchrotronIntensity Function

synchrotronIntensity(
    metric::Kerr{T},
    α,
    β,
    ri,
    θs,
    θo,
    magnetic_field::StaticArraysCore.SArray{Tuple{3}, T, 1, 3},
    βfluid::StaticArraysCore.SArray{Tuple{3}, T, 1, 3},
    νr::Bool,
    θsign::Bool
) -> Tuple{Any, Any, Any}

Calculates the intensity of a photon emitted from a fluid particle with momentum f_u and observed by an asymptotic observer.

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Krang.synchrotronPolarization Function

synchrotronPolarization(
    metric::Kerr{T},
    α,
    β,
    ri,
    θs,
    θo,
    magnetic_field::StaticArraysCore.SArray{Tuple{3}, T, 1, 3},
    βfluid::StaticArraysCore.SArray{Tuple{3}, T, 1, 3},
    νr::Bool,
    θsign::Bool
) -> NTuple{4, Any}

Calculates the polarization of a photon emitted from a fluid particle with momentum f_u and observed by an asymptotic observer.

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Meshes, Geometries and Materials

Krang.Mesh Type

struct Mesh{G<:Krang.AbstractGeometry, M<:Krang.AbstractMaterial}

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Krang.MeshGeometry Type

Load a mesh from a file.

# Arguments
- `filename::String`: The path to the file containing the mesh.

# Returns
- A `Mesh` object representing the mesh.

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Krang.AbstractLevelSetGeometry Type

Level Sets should be a functor

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Krang.ElectronSynchrotronPowerLawIntensity Type

struct ElectronSynchrotronPowerLawIntensity{N, T} <: Krang.AbstractMaterial

A struct representing the linear polarization intensity of synchrotron radiation from electrons following a power-law energy distribution. This model is based on the material described in https://doi.org/10.3847/1538-4357/abf117.

Fields

- `magnetic_field::SVector{3, T}`: The magnetic field vector components (x, y, z).
- `fluid_velocity::SVector{3, T}`: The fluid velocity vector components (speed, inclination angle, azimuthal angle).
- `spectral_index::T`: The spectral index of the electron energy distribution.
- `R::T`: The characteristic radius of the emissivity profile.
- `p1::T`: The first power-law index.
- `p2::T`: The second power-law index.
- `subimgs::NTuple{N, Int}`: The sub-images to ray trace.

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Krang.ElectronSynchrotronPowerLawPolarization Type

struct ElectronSynchrotronPowerLawPolarization{N, T} <: Krang.AbstractMaterial

A struct representing the linear polarization of synchrotron radiation from electrons following a power-law energy distribution. This model is based on the material described in https://doi.org/10.3847/1538-4357/abf117.

Fields

- `magnetic_field::SVector{3, T}`: The magnetic field vector components (x, y, z).
- `fluid_velocity::SVector{3, T}`: The fluid velocity vector components (speed, inclination angle, azimuthal angle).
- `spectral_index::T`: The spectral index of the electron energy distribution.
- `R::T`: The characteristic radius of the emissivity profile.
- `p1::T`: The first power-law index.
- `p2::T`: The second power-law index.
- `subimgs::NTuple{N, Int}`: The sub-images to ray trace.

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Pixel Related Functions

Krang.absGθo_Gθhat Function

absGθo_Gθhat(
    pix::Krang.AbstractPixel
) -> Tuple{T, T} where T

absGθo_Gθhat(pix::AbstractPixel)

Calculate the absolute value of Gθo divided by Gθhat for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The absolute value of Gθo divided by Gθhat of the pixel.

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Krang.I1_inf_m_I0_terms Function

I1_inf_m_I0_terms(pix::Krang.AbstractPixel) -> Any

I1_inf_m_I0_terms(pix::AbstractPixel)

Calculate the I1 infinity minus I0 terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The I1 infinity minus I0 terms of the pixel.

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Krang.radial_inf_integrals_m_I0_terms Function

radial_inf_integrals_m_I0_terms(
    pix::Krang.AbstractPixel
) -> NTuple{4, Any}

radial_inf_integrals_m_I0_terms(pix::AbstractPixel)

Calculate the radial infinity minus I0 terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The radial infinity minus I0 terms of the pixel.

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Krang.inclination Function

inclination(pix::Krang.AbstractPixel) -> Any

inclination(pix::AbstractPixel)

Get the inclination angle (θo) of a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The inclination angle (θo) of the pixel.

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Krang.metric Function

metric(pix::Krang.AbstractPixel) -> Any

metric(pix::AbstractPixel)

Get the space time metric.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The space time metric

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Krang.absGto_Gthat Function

absGto_Gthat(
    pix::Krang.AbstractPixel
) -> Tuple{T, T} where T

absGto_Gthat(pix::AbstractPixel)

Calculate the absolute value of Gto divided by Gthat for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The absolute value of Gto divided by Gthat of the pixel.

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Krang.screen_coordinate Function

screen_coordinate(
    pix::Krang.AbstractPixel
) -> Tuple{T, T} where T

screen_coordinate(pix::AbstractPixel)

Get the screen coordinates of a given pixel.

Arguments

• pix::AbstractPixel: The pixel for which to get the screen coordinates.

Returns

• The screen coordinates of the pixel.

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Krang.Ip_inf_m_I0_terms Function

Ip_inf_m_I0_terms(pix::Krang.AbstractPixel) -> Any

Ip_inf_m_I0_terms(pix::AbstractPixel)

Calculate the Ip infinity minus I0 terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The Ip infinity minus I0 terms of the pixel.

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Krang.I2_inf_m_I0_terms Function

I2_inf_m_I0_terms(pix::Krang.AbstractPixel) -> Any

I2_inf_m_I0_terms(pix::AbstractPixel)

Calculate the I2 infinity minus I0 terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The I2 infinity minus I0 terms of the pixel.

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Krang.roots Function

roots(
    pix::Krang.AbstractPixel
) -> NTuple{4, Complex{T}} where T

roots(pix::AbstractPixel)

Calculate the radial roots for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The radial roots of the pixel.

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Krang.Im_inf_m_I0_terms Function

Im_inf_m_I0_terms(pix::Krang.AbstractPixel) -> Any

Im_inf_m_I0_terms(pix::AbstractPixel)

Calculate the Im infinity minus I0 terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The Im infinity minus I0 terms of the pixel.

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Krang.absGϕo_Gϕhat Function

absGϕo_Gϕhat(
    pix::Krang.AbstractPixel
) -> Tuple{T, T} where T

absGϕo_Gϕhat(pix::AbstractPixel)

Calculate the absolute value of Gϕo divided by Gϕhat for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The absolute value of Gϕo divided by Gϕhat of the pixel.

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Krang.I0_inf Function

I0_inf(pix::Krang.AbstractPixel) -> Any

I0_inf(pix::AbstractPixel)

Calculate the I0 infinity value for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The I0 infinity value of the pixel.

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Krang.total_mino_time Function

total_mino_time(pix::Krang.AbstractPixel) -> Any

total_mino_time(pix::AbstractPixel)

Return the total possible Mino time for a ray associated with a pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The total possible Mino time for a ray associated with the pixel.

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total_mino_time(
    metric::Kerr{T},
    roots::NTuple{4, T} where T
) -> Any

Returns the maximum mino time that can be accrued along a ray.

Arguments

• metric : Kerr metric

• roots : Roots of the radial potential

• I0_inf : Mino time at infinity

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Misc

Krang._isreal2 Function

_isreal2(num::Complex{T}) -> Any

Checks if a complex number is real to some tolerance

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Krang.regularized_Pi Function

regularized_Pi(n, ϕ, k) -> Any

Regularized elliptic integral of the third kind

Arguments

• n: Parameter

• ϕ: Arguments

• k: Parameter

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Krang.radial_inf_integrals_case4 Function

radial_inf_integrals_case4(
    metric::Kerr{T},
    roots::NTuple{4, T} where T
) -> NTuple{4, Any}

Returns the radial integrals for the case where there are no real roots in the radial potential

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Krang.It_w_I0_terms Function

It_w_I0_terms(
    metric::Kerr{T},
    rs,
    τ,
    roots::NTuple{4, T} where T,
    λ,
    νr
) -> Any

Returns the antiderivative $I_t=\int \frac{r^2\mathrm{\Delta }+2Mr(r^2+a^2-a\lambda )}{\sqrt{\mathrm{\Delta }\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}dr$ with I0 terms.

See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• metric: Kerr{T} metric

• roots : Radial roots

• λ : Reduced azimuthal angular momentum

• νr : Radial emission direction (Only necessary for case 1&2 geodesics)

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Krang.Ir_inf Function

Ir_inf(pix::Krang.AbstractPixel) -> Any

Ir_inf(pix::AbstractPixel)

Calculate the Ir infinity value for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The Ir infinity value of the pixel.

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Ir_inf(metric::Kerr{T}, roots) -> Any

Returns the antiderivative $I_r=\int \frac{dr}{\sqrt{\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}$ evaluated at infinity. See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• metric: Kerr{T} metric

• roots : Roots of the radial potential

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Krang.emission_azimuth Function

emission_azimuth(
    pix::Krang.AbstractPixel,
    θs,
    rs,
    τ,
    νr,
    isindir,
    n
) -> Any

Emission azimuth for point at Mino time τ whose image appears at screen coordinate (α, β).

Arguments

• pix : Pixel information

• θs : Emission Inclination

• rs :Emission radius

• τ : Mino Time

• νr : Sign of radial velocity direction at emission. This is always positive for case 3 and case 4 geodesics.

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Krang.ConeGeometry Type

struct ConeGeometry{T, A} <: Krang.AbstractGeometry

Cone Geometry with half opening angle opening_angle.

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Krang.Iϕ Function

Iϕ(pix::Krang.AbstractPixel, rs, τ, νr) -> Any

Returns the antiderivative $I_{\varphi }=\int \frac{a(2Mr-a\lambda )}{\sqrt{\mathrm{\Delta }\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}dr$. See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• pix: SlowLightIntensityPixel

• rs : Emission radius

• τ : Mino time

• νr : Sign of radial velocity direction at emission. This is always positive for case 3 and case 4 geodesics.

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Krang.SlowLightIntensityCamera Type

struct SlowLightIntensityCamera{T, A} <: Krang.AbstractCamera

Camera that caches slow light raytracing information for an observer sitting at radial infinity. The frame of this observer is alligned with the Boyer-Lindquist frame.

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Krang.Iϕ_w_I0_terms Function

Iϕ_w_I0_terms(metric::Kerr{T}, rs, τ, roots, νr, λ) -> Any

Returns the antiderivative $I_{\varphi }=\int \frac{a(2Mr-a\lambda )}{\sqrt{\mathrm{\Delta }\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}dr$ with full I0 terms.

See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• metric: Kerr{T} metric

• τ: Minotime

• roots : Radial roots

• νr : Radial emission direction (Only necessary for case 1&2 geodesics)

• λ : Reduced azimuthal angular momentum

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Krang.SlowLightIntensityPixel Type

struct SlowLightIntensityPixel{T} <: Krang.AbstractPixel{T}

Intensity Pixel Type.

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Krang.Iϕ_inf Function

Iϕ_inf(pix::Krang.AbstractPixel) -> Any

Iϕ_inf(pix::AbstractPixel)

Calculate the Iϕ infinity terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The Iϕ infinity terms of the pixel.

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Iϕ_inf(metric::Kerr{T}, roots, λ) -> Any

Returns the antiderivative $I_{\varphi }=\int \frac{a(2Mr-a\lambda )}{\sqrt{\mathrm{\Delta }\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}dr$ evaluated at infinity without I0 terms.

See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• metric: Kerr{T} metric

• roots : Radial roots

• λ : Reduced azimuthal angular momentum

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Krang.radial_inf_integrals_case3 Function

radial_inf_integrals_case3(
    metric::Kerr{T},
    roots::NTuple{4, T} where T
) -> NTuple{4, Any}

Returns the radial integrals for the case where there are two real roots in the radial potential

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Krang.AbstractPixel Type

abstract type AbstractPixel{T}

Abstract Pixel Type

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Krang.radial_w_I0_terms_integrals_case2 Function

radial_w_I0_terms_integrals_case2(
    metric::Kerr{T},
    rs,
    roots::NTuple{4, T} where T,
    τ,
    νr
) -> Tuple

Returns the radial integrals for the case where there are four real roots in the radial potential, with roots outside the horizon.

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Krang.SlowLightIntensityScreen Type

struct SlowLightIntensityScreen{T, A<:(AbstractMatrix)} <: Krang.AbstractScreen

Screen made of SlowLightIntensityPixels.

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Krang.radial_w_I0_terms_integrals_case4 Function

radial_w_I0_terms_integrals_case4(
    metric::Kerr{T},
    rs,
    roots::NTuple{4, T} where T,
    τ
) -> NTuple{4, Any}

Returns the radial integrals for the case where there are no real roots in the radial potential

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Krang.radial_integrals Function

radial_integrals(
    pix::Krang.AbstractPixel,
    rs,
    τ,
    νr
) -> NTuple{5, Any}

Return the radial integrals

• pix: SlowLightIntensityPixel

• rs : Emission radius

• τ : Mino time

• νr : Sign of radial velocity direction at emission. This is always positive for case 3 and case 4 geodesics.

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Krang.AbstractCamera Type

abstract type AbstractCamera

Abstract Observer Type

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Krang.IntensityScreen Type

struct IntensityScreen{T, A<:(AbstractMatrix)} <: Krang.AbstractScreen

Screen made of IntensityPixels.

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Krang.It Function

It(pix::Krang.AbstractPixel, rs, τ, νr) -> Any

Returns the antiderivative $I_t=\int \frac{a(2Mr-a\lambda )}{\sqrt{\mathrm{\Delta }\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}dr$. See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• pix: SlowLightIntensityPixel

• rs : Emission radius

• τ : Mino time

• νr : Sign of radial velocity direction at emission. This is always positive for case 3 and case 4 geodesics.

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Krang.IntensityPixel Type

struct IntensityPixel{T} <: Krang.AbstractPixel{T}

Intensity Pixel Type. Each Pixel is associated with a single ray, and caches some information about the ray.

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Krang.AbstractMaterial Type

abstract type AbstractMaterial

Abstract Material

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Krang.AbstractScreen Type

abstract type AbstractScreen

Abstract Screen Type

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Krang.radial_w_I0_terms_integrals_case3 Function

radial_w_I0_terms_integrals_case3(
    metric::Kerr{T},
    rs,
    roots::NTuple{4, T} where T,
    τ
) -> NTuple{4, Any}

Returns the radial integrals for the case where there are two real roots in the radial potential

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Krang.IntensityCamera Type

struct IntensityCamera{T, A} <: Krang.AbstractCamera

Camera that caches fast light raytracing information for an observer sitting at radial infinity. The frame of this observer is alligned with the Boyer-Lindquist frame.

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Krang.Ir_s Function

Ir_s(metric::Kerr{T}, rs, roots, νr) -> Any

Returns the antiderivative $I_r=\int \frac{dr}{\sqrt{\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}$. See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• metric: Kerr{T} metric

• rs : Emission radius

• roots : Roots of the radial potential

• νr : Radial emission direction (Only necessary for case 1&2 geodesics)

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Krang.AbstractGeometry Type

abstract type AbstractGeometry

Abstract Geometry

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Krang.radial_inf_integrals_case2 Function

radial_inf_integrals_case2(
    metric::Kerr{T},
    roots::NTuple{4, T} where T
) -> NTuple{4, Any}

Returns the radial integrals for the case where there are four real roots in the radial potential, with roots outside the horizon.

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Krang.It_inf Function

It_inf(pix::Krang.AbstractPixel) -> Any

It_inf(pix::AbstractPixel)

Calculate the It infinity terms for a given pixel.

Arguments

• pix::AbstractPixel: The pixel of a screen.

Returns

• The It infinity terms of the pixel.

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It_inf(
    metric::Kerr{T},
    roots::NTuple{4, T} where T,
    λ
) -> Any

Returns the antiderivative $I_t=\int \frac{r^2\mathrm{\Delta }+2Mr(r^2+a^2-a\lambda )}{\sqrt{\mathrm{\Delta }\mathcal{R}\mathcal{(}\mathcal{r}\mathcal{)}}}dr$ evaluated at infinity without I0 terms.

See r_potential(x) for an implementation of $\mathcal{R}(r)$.

Arguments

• metric: Kerr{T} metric

• roots : Radial roots

• λ : Reduced azimuthal angular momentum

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