Functions for blackbody thermal emission (and its derivatives) calculation

PlanckFunctions

A comprehensive suite of tools for evaluating properties of the Planck thermal emission spectrum, including spectral intensity (radiance), integrated band power, and their ratios.

The module provides exact analytical derivatives with respect to both wavelength (λ) and temperature (T), optimized for high-performance physics and pyrometry calculations.

Core Design Principles

• Derivative Naming Convention: For any base function foo, its first and second temperature derivatives are available via ∇ₜfoo and ∇²ₜfoo.
• Combined Evaluation: Functions prefixed with Dₜ (e.g., Dₜfoo) evaluate the base value, the first derivative, and the second derivative simultaneously, returning them as a single Tuple to prevent redundant calculations.
• Functional Operators: The symbols ∇ₜ ,Dₜ and ∇²ₜ can be used directly as operators on functions, mapping a base function to its corresponding derivative function (e.g., PlanckFunctions.∇ₜ(PlanckFunctions.ibb) -> ∇ₜibb). The symbol ∫ₗ applied on function maps it to its wavelength integrator

Physical Units

• Wavelength (λ): Microns (μm)
• Temperature (T): Kelvins (K)

API Reference

Spectral Radiance & Core Functions

• ibb: Spectral intensity (spectral radiance) in [W/m²⋅sr⋅μm].
• ∇ₜibb: First derivative of spectral intensity with respect to temperature [W/m²⋅sr⋅μm·K].
• ∇²ₜibb: Second derivative of spectral intensity with respect to temperature [W/m²⋅sr⋅μm·K²].

All this functions has also and integrated versions , which can be obtained both by direct name or by applying the ∫ₗ ∫ₗ operator

• Dₜibb: Base functions, its first and second derivative.
• ibb!, ∇ₜibb!, ∇²ₜibb!, Dₜibb! - in-place versions
• ∇ₗibb: First derivative of spectral intensity with respect to wavelength [W/m²⋅sr⋅μm²].
• ∇²ₗibb: Second derivative of spectral intensity with respect to wavelength [W/m²⋅sr⋅μm³].

Integrated Band Power

• band_power, ∇ₜband_power, ∇²ₜband_power: Total integrated intensity within a specified wavelength region along with its temperature derivatives [W/m²⋅sr].

Analytical Ratios (Continuous Pyrometry)

• spectral_ratio, ∇ₜspectral_ratio, ∇²ₜspectral_ratio: Ratio of spectral intensities at two distinct discrete wavelengths (useful for ratio pyrometers).
• spectral_band_ratio, ∇ₜspectral_band_ratio, ∇²ₜspectral_band_ratio: Ratio of integrated spectral intensities for two distinct wavelength bands.

Averaging & Continuum Mechanics

• planck_averaged: Planck-averaged spectral quantity (e.g., total emittance).
• planck_averaged_attenuation: Planck-averaging applied specifically to spectral attenuation coefficients.
• rosseland_averaged_attenuation: Rosseland-averaging applied to spectral attenuation coefficients.
• weighted_average: General-purpose function to compute a custom weighted average over a spectrum.

Discrete/Tabular Integration (Experimental Data Pyrometry)

• planck_weighted, ∇ₜplanck_weighted, ∇²ₜplanck_weighted, Dₜplanck_weighted: Integrates a discrete, wavelength-dependent experimental quantity (such as emissivity or detector response) weighted by the Planck function and its derivatives.
• planck_weighted_ratio, ∇ₜplanck_weighted_ratio, ∇²ₜplanck_weighted_ratio, Dₜplanck_weighted_ratio: Computes the ratio of two independent Planck-weighted discrete quantities and its analytical temperature derivatives.

Operators

• ∫ₗ : Integration
• ∇ₜ , ∇²ₜ : Differentiation

C₁ constant for Planck function multiplier in [W⋅μm⁴/(m²⋅sr)]
source https://physics.nist.gov/

C₂ constant for Planck spectral intensity exponent in [μm⋅K]
source https://physics.nist.gov/

C₃ constant of Wien's displacement law [μm⋅K]
source https://physics.nist.gov/

Constant of the first derivaitve Wien's displacement

Constant of the second derivaitve Wien's displacement

C₄ constant in equation for maximum blackbody intensity [W/(m²⋅μm⋅sr*K⁵)]
source https://physics.nist.gov/

Speed of light in vacuum
`c` [https://physics.nist.gov/] , m s^-1

Planck const
`h` [https://physics.nist.gov/] , J Hz^-1

Boltzmann const 
`kb` [https://physics.nist.gov/] , J K^-1

Stefan-Boltzmann constant [W/(m²*K⁴)]

source https://physics.nist.gov/

frule((Δself, ΔT), ::typeof(band_power), T; λₗ=0.0, λᵣ=Inf, tol=1e-8)

Forward-mode AD rule for band_power.

frule((Δself, ΔT), ::typeof(band_power), T; λₗ=0.0, λᵣ=Inf, tol=1e-8)

Forward-mode AD rule for band_power.

frule((Δself, Δλ, ΔT), ::typeof(ibb), λ::Number, T::Number)

Custom forward-mode automatic differentiation rule for the Planck function ibb(λ, T)

rrule(::typeof(ibb), λ::Number, T::Number)

Custom reverse-mode automatic differentiation rule (pullback) for ibb(λ, T).

ChainRulesCore.rrule(
                    ::typeof(band_power), 
                    T::TT; λₗ::Number , λᵣ::Number
                ) where { TT<:Number}

Custom reverse-mode automatic differentiation rule (pullback) for band_power(T; λₗ = λₗ , λᵣ = λᵣ).

Dₗibb(λ,T)

Returns a three-element tuple of (1.bb intensity,2.its first and 3.second derivative with respect to the wavelentgh)

Arguments:

λ - wavelength, μm T - temperature, K

Dₜ(f) 

Differentiation operator returns the tuple of :

(value of base function, its first derivative , its second derivative)

ibb , band_power , power , spectral_ratio , spectral_band_ratio , planck_weighted,planck_weighted_ratio

Must be implemented on user defined [`AbstractSpectralQuatity`](@ref) interface to perform

fast continuous function weighting derivatives evaluation

Examples

 F = Dₜ(ibb)          # -> Dₜibb
 F(1273.15) == (ibb(1273.15) , ∇ₜibb(1273.15) , ∇²ₜibb(1273.15))
 Dₜ(planck_weighted)   # ->  Dₜplanck_weighted

Dₜband_power(T , skip_second_derivative::Val{true} ; λₗ=0.0 , λᵣ=Inf , tol=1e-8)

returns the value and its derivative

Dₜband_power(T ; λₗ=0.0 , λᵣ=Inf , tol=1e-8)

returns all derivatives

    Dₜibb!(input_tuple, λ::AbstractVector,T)

In-place filling the tuple of (bb intensity, its first ,and second ) derivatives with respect to temperature

Arguments:

input_tuple, {Nx0 vector or nothing , Nx0 vector or nothing, Nx0 vector or nothing} λ - wavelength, μm, [Nx0] T - temperature, K Out: input_tuple filled with ibb , dIbb/dT , d²Ibb/dT²

This version is slightly faster than calling each derivative separately

Dₜibb( λ::Number, T::Number , skip_second_derivative::Val{true})

Skips the second derivative evaluation Dₜibb( λ, T , Val(true))

Dₜibb( λ::Number, T::Number)

returns a tuple of (value, first derivative, second derivative)

Dₜibb(λ::AbstractVector,T::AbstractVector)

Calculates tuple of (Ibb,dIbb/dT,d²Ibb/dT²) calculated according to:

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T³))*[(C₂/(λ*T))*(2*eᵃ¹/(eᵃ¹-1)-1)-2], d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(a₁/(λ⁵*T²))*[a₁*(2*eᵃ¹/(eᵃ¹-1) -1)-2], a₁=C₂/(λ*T), a₂ = 1/(eᵃ¹-1) , a₃ = eᵃ¹/(eᵃ¹-1) , d²Ibb/dT² = C₁*a₂*a₃*(a₁/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2] as far as Ibb = (λ⁻⁵)* C₁*a₂ and dIbb/dT = C₁*a₃*a₂*a₁*(1/(λ⁵*T)) = a₃*a₁*Ibb/T hense d²Ibb/dT² = C₁*a₂*a₃*a₁*(1/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2] = [a₃*a₁*Ibb/T^2]*[a₁*(2*a₃ - 1))-2] = [(dIbb/dT)/T]*[a₁*(2*a₃ - 1))-2]

Arguments:

λ - wavelength region, μm T - temperature, Kelvins

Returns:

(Ibb , dIbb/dT , d²Ibb/dT²)

Dₜplanck_weighted(e::AbstractVector{Q}, λ::AbstractVector{L}, T::D) where {Q <: Number, L <: Number, D <: Number}

Evaluates the integral of e(λ) over three weighting functions of Planck's function , its first- and second- derivaitve: ibb, ∇ₜibb и ∇²ₜibb

Returns three-element tuple : ( ∫α(λ)ibb(λ,T)dλ , ∫α(λ)∇ₜibb(λ,T)dλ , ∫α(λ)∇²ₜibb(λ,T)dλ)

    Dₜplanck_weighted_ratio(
                            α::AbstractSpectralQuantity, 
                            band1::NTuple{2}, 
                            band2::NTuple{2},
                            T::D) where { D <: Number}

Returns the tuple of the value, first and second derivatives of two weighted values ratio :

Returns

`( ∫α(λ)ibb(λ,T)dλ1 / ∫α(λ)ibb(λ,T)dλ2,

d/dT [ (∫α(λ)ibb(λ,T)dλ1 / ∫α(λ)ibb(λ,T)dλ2) ] ,

d²/dT² [ (∫α(λ)ibb(λ,T)dλ1 / ∫α(λ)ibb(λ,T)dλ2) ] )`

Arguments

• α: An AbstractSpectralQuantity (e.g., analytical or autodiff emissivity).
• band1: Tuple (λ_min, λ_max) for the first spectral band (numerator), μm.
• band2: Tuple (λ_min, λ_max) for the second spectral band (denominator), μm.
• T: Temperature, K.

Dₜplanck_weighted_ratio(
                    α1::AbstractVector{Q}, 
                    λ1::AbstractVector{L},
                    α2::AbstractVector{Q}, 
                    λ2::AbstractVector{L},
                    T::D ) where { D <: Number , Q <: Number , L <: Number}

Returns the tuple of the value, first and second derivatives of two weighted values ratio :

Arguments

α1 - first wavelength dependent quantity, λ1 - first wavelengths range, μm α2- second wavelength dependent quantity, λ2- second wavelengths range, μm T - temperature , K

Returns

`( ∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2,

d/dT [ (∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2) ] ,

d²/dT² [ (∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2) ] )`

Dₜplanck_weighted_values(e::AbstractVector{Q}, λ::AbstractVector{L}, T::D) where {Q <: Number, L <: Number, D <: Number}

Evaluates the integral of e(λ) over three weighting functions of Planck's function , its first- and second- derivaitve: ibb, ∇ₜibb и ∇²ₜibb

Returns six-elements tuple (integrals and normalizers) : ( ∫α(λ)ibb(λ,T)dλ , ∫α(λ)∇ₜibb(λ,T)dλ , ∫α(λ)∇²ₜibb(λ,T)dλ , ∫ibb(λ,T)dλ , ∫∇ₜibb(λ,T)dλ , ∫∇²ₜibb(λ,T)dλ)

Dₜpower(T)

Tuple of total intensity , fisrt and second derivative with respect to temeperature

Dₜspectral_band_ratio(λ1::NTuple{2, TL}, λ2::NTuple{2,TL}, T::Number , skip_second_derivative::Val{true};  tol = 1e-6) where TL <: Number

Ignores the second derivative evaluation

Dₜspectral_band_ratio(λ1::NTuple{2, TL}, λ2::NTuple{2,TL}, T::Number;  tol = 1e-6) where TL <: Number

All spectral band ratio derivatives are in one tuple

Arguments

λ1- tuple of left and right wavelength of the first band, in μm. λ2- tuple of left and right wavelength of the second band, in μm. T- Absolute temperature, in K.

Dₜspectral_ratio(λ1::Number, λ2::Number, T::Number)

All spectral ratio derivatives in one tuple

Arguments

λ1- First wavelength (usually the shorter one), in μm. λ2- Second wavelength (usually the longer one), in μm. T- Absolute temperature, in K.

    attenuated_band_power(T, τ::AbstractVector, λ::AbstractVector;  tol=1e-6)

Evaluates the integrated band power of BB with temperature T reaching the detector, accounting for the spectral transmittance profile τ(λ).

Arguments:

T - temeperature , K τ - transmittance λ - wavelength , μm

a₁₂₃!(amat::AbstractMatrix,λ::AbstractVector,T::Number)

In-place filling of the intermediate matrix a₁=C₂/(λ*T) - amat first column a₂ = 1/(eᵃ¹-1) - amat second column a₃ = eᵃ¹/(eᵃ¹-1) - amat third column

Arguments:

amat - matrix of intermediate coefficients size [Nx3], λ - wavelength , μm, [Nx0], T - temperature, K,

a₁₂₃(λ::Float64,T::Float64)

Arguments:

amat - matrix of intermediate coefficients size [Nx3] λ - wavelength , μm, [Nx0] T - temperature, K

Returns tuple:

(a1 = C₂/(λ*T), a2 = 1/expm1(a1), a3 = 1 + a2)

    band_power(T;λₗ=0.0 , λᵣ=Inf , tol=1e-8)

Total bb with temperature T integral intensity (∫ibb(λ , T)dλ) within (in-band radiance), [W/(m²⋅sr)] the spectral range λₗ...λᵣ (by default the range is 0...inf) tol - tolerance of intehration

Arguments:

T - temperature,Kelvins (optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm tol - intergation tolerance

bb_max(T::Number)

Blackbody instensity at maximum value

bb_max(bb_function , T)

Returns the value of function at a maximum bb_function can be ibb , ∇ₜibb , ∇²ₜibb

bright_temperature(i, λ; ϵ=1.0)

Evaluates bright temperature for single wavelength pyrometer i - measured intensity , λ - wavelength, μm ϵ - emissivity

    ibb!(i::AbstractVector , λ::AbstractVector , amat::AbstractMatrix)

In-place blackbody intensity with intermediate coefficients provided externally, [W/m2-sr-mkm] Ibb = C₁*(λ⁻⁵)*a₂ , where a₁=C₂/(λ*T) - amat first column a₂ = 1/(eᵃ¹-1) - amat second column

Arguments:

i - BB intensity, [Nx0] λ - wavelength in μm, [Nx0] amat - matrix of intermediate coefficients, [Nx3]

    ibb!(i::AbstractVector , λ::AbstractVector , T::Number)

In-place blackbody intensity, [W/m²⋅sr⋅μm] Ibb = (λ⁻⁵)* C₁/(eᵃ¹-1) , where a₁=C₂/(λT)

Arguments:

i - bb intensity vector, [Nx0] λ - wavelength in μm, [Nx0] T - temperature in Kelvins

ibb(λ::AbstractVector,amat::AbstractMatrix)

Blackbody spectral intensity (spectral radiance) with intermediate matrix provided externally, [W/m²⋅sr⋅μm] Ibb = C₁*(λ⁻⁵)*a₂ , a₁=C₂/(λ*T) - amat first column, a₂ = 1/(eᵃ¹-1) - amat second column

Arguments:

amat - matrix of intermediate coefficients, [Nx3] λ - wavelength in μm, [Nx0]

    ibb(λ::AbstractVector,T::AbstractVector)

Blackbody spectral intensity (spectral radiance), [W/m²⋅sr⋅μm] Ibb = (λ⁻⁵)* C₁/(eᵃ¹-1) , where a₁=C₂/(λ*T)

Arguments:

λ - wavelength in μm, [Nx0] T - temperature in Kelvins [Mx0]

ibb(λ,T)

Blackbody spectral intensity (spectral radiance), [W/m²⋅sr⋅μm] Ibb = (λ⁻⁵)* C₁/(eᵃ¹-1) , a₁=C₂/(λ*T)

Arguments:

λ - wavelength in μm T - temperature in Kelvins

planck_averaged(x::AbstractVector, λ::AbstractVector,T::Number)

Evaluates the Planck-averaged value of x(λ) for temperature T:

xᵣ = ∫x(λ)ibb(λ,T)dλ/∫ibb(λ,T)dλ

E.g. can be used to evaluate the integral from spectral emissivity

planck_averaged_attenuation(α::AbstractVector, λ::AbstractVector,T::Number)

Planck-averaged spectral attenuation coefficient (the summation of spectral scattering and absorption coefficients) α(λ) for temperature T:

αᵣ = (∫(1/α(λ))⋅ibb(λ,T)dλ/∫ibb(λ,T)dλ)⁻¹

planck_weighted(α::AbstractVector{Q}, 
                    λ::AbstractVector{L},
                    T::D) where { D <: Number , Q <: Number , L <: Number}

Evaluates ∫α(λ)⋅ibb(λ,T)dλ using numeric integration

planck_weighted_ratio(
                    α::AbstractSpectralQuantity, 
                    band1::NTuple{2}, 
                    band2::NTuple{2},
                    T::D) where { D <: Number}

Evaluates the ratio of planck_weighted quantity α integrated over two bands band1 and band2 at temperature T:

R(T) = ∫α(λ)ibb(λ,T)dλ1 / ∫α(λ)ibb(λ,T)dλ2

Arguments

• α: An AbstractSpectralQuantity (e.g., analytical or autodiff emissivity).
• band1: Tuple (λ_min, λ_max) for the first spectral band (numerator), μm.
• band2: Tuple (λ_min, λ_max) for the second spectral band (denominator), μm.
• T: Temperature, K.

planck_weighted_ratio(
                    α1::AbstractVector{Q}, 
                    λ1::AbstractVector{L},
                    α2::AbstractVector{Q}, 
                    λ2::AbstractVector{L},
                    T::D) where { D <: Number , Q <: Number , L <: Number}

Returns the ratio of two weighted values (∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2)

power(T)

Integral (over the wavelength) intensity of BB (radiance) at temperature T

Units: W/(m²⋅sr)

Arguments:

T - temperature, K

rosseland_averaged_attenuation(α::AbstractVector, λ::AbstractVector,T::Number)

Evaluates the Rosseland-averaged spectral attenuation coefficient (the summation of spectral scattering and absorption coefficients) α(λ) for temperature T:

αᵣ = (∫(1/α(λ))⋅∇ₜibb(λ,T)dλ/∫∇ₜibb(λ,T)dλ)⁻¹

second_order_polynomial_fit(x1,x2,x3,g1,g2,g3)

Hardcoded second order polynomial lsqr fitting

spectral_band_ratio(λ1::NTuple{2, TL}, λ2::NTuple{2,TL}, T::Number;   tol = 1e-6) where TL <: Number

The same as spectral_ratio, but now the band can be wide (not a single wavelength). This function may be useful for two-color pyrometers in case when their working regions width cannot be ignored

Arguments

λ1- tuple of left and right wavelength of the first band, in μm. λ2- tuple of left and right wavelength of the second band, in μm. T- Absolute temperature, in K.

spectral_ratio(λ1::Number, λ2::Number, T::Number)

Calculate the theoretical intensity ratio R = ( Ibb1/Ibb2) between two wavelengths λ1 and λ2 at temperature T

Arguments

λ1- First wavelength (usually the shorter one), in μm. λ2- Second wavelength (usually the longer one), in μm. T- Absolute temperature, in K.

    temperature(i)

The bb temperature for intensity i corresponding to wavelength λ

tₘ(λ)

The temperature of BB having maximum at wavelength λ in Kelvins

units(f::Function)

returns units string of output quantity return

weighted(α::AbstractVector{Q}, 
                    λ::AbstractVector{L},
                    T::D, 
                    g::Dfunctions) where { D <: Number , Q <: Number , L <: Number}

∫α⋅g(λ,T)dλ where g ∈ {ibb , ∇ₜibb , ∇²ₜibb}

weighted_average(α::AbstractVector, 
                    λ::AbstractVector,
                    T::Number, 
                    g::Dfunctions,
                    f::F = identity) where F

Generic function to evaluate the averaged value of some f(x) function of variable x dependent on wavelength λ for temperature T. Uses linear approximation for the discrete variable and square polynomial for the g function

xᵣ = ∫f(x)g(λ,T)dλ/∫g(λ,T)dλ the default value of f is identity, e.g. if f = inv: xᵣ = ∫g(λ,T)/x(λ)dλ/∫g(λ,T)dλ

weighted_value(     α::AbstractVector, 
                    λ::AbstractVector,
                    T , 
                    g::Union{typeof(ibb),typeof(∇ₜibb),typeof(∇²ₜibb)},
                    f::F = identity) where F <: Function

Returns the tuple of (∫f(α)g(λ,T)dλ , ∫g(λ,T)dλ)

weighted_values_ratio(
                    α1::AbstractVector{Q}, 
                    λ1::AbstractVector{L},
                    α2::AbstractVector{Q}, 
                    λ2::AbstractVector{L},
                    T::D, 
                    g::Dfunctions,
                    f::F = identity) where {F   , D <: Number , Q <: Number , L <: Number}

Returns the ratio of two weighted values (∫f(α1)g(λ,T)dλ1 / ∫f(α2)g(λ,T)dλ2)

λₘ(T::Number)

The wavelength (in μm) of bb intensity maximum vs temperature T argmax(Planck(T)) - Wien's displacement law

Arguments:

T - temperature in Kelvins

λₘ(::Union{typeof(ibb) , typeof(∇ₜibb) , typeof(∇²ₜibb)} , T::Number)

The wavelength (in μm) of bb intensity , its first or second derivaitve maximum vs temperature T

Arguments:

T - temperature in Kelvins

∇²ₗibb(λ,T)

BB intensity second derivative with respect to the wavelength

Arguments:

λ - wavelength, μm T - temperature, K

∇²ₜ(f) 

Differentiation operator returns the second derivatives of function

ibb , band_power , power , spectral_ratio , spectral_band_ratio , planck_weighted,planck_weighted_ratio

Must be implemented on user defined [`AbstractSpectralQuatity`](@ref) interface to perform

fast continuous function weighting derivatives evaluation

Examples

 ∇²ₜ(ibb)          # -> ∇²ₜibb
 ∇²ₜ(band_power)   # ->  ∇²ₜband_power

∇²ₜband_power(T , band_power_value; λₗ=0.0 , λᵣ=Inf , tol=1e-8)

Band power second derivative with respect to temeprature with band_power value evaluated externally

Arguments:

T - temperature,Kelvins

(optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm tol - intergation tolerance

∇²ₜband_power(T ; λₗ=0.0 , λᵣ=Inf , tol=1e-8)

Band power second derivative with respect to temeprature

Arguments:

T - temperature,Kelvins

(optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm tol - intergation tolerance

∇²ₜibb!(h::AbstractMatrix{Float64} , λ::AbstractVector{Float64}, T::AbstractVector{Float64})

In-place bb intensity second order derivative with respect to temperature

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T³))*[(C₂/(λ*T))*(2*eᵃ¹/(eᵃ¹-1)-1)-2],

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(a₁/(λ⁵*T²))*[a₁*(2*eᵃ¹/(eᵃ¹-1) -1)-2], a₁=C₂/(λ*T), a₂ = 1/(eᵃ¹-1) , a₃ = eᵃ¹/(eᵃ¹-1) ,

d²Ibb/dT² = C₁*a₂*a₃*(a₁/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2]

Arguments :

h - to be filled, [Nx0] λ - wavelength in μm, [Nx0] T- tmperature in Kelvins

∇²ₜibb!(h::AbstractVector{Float64}, λ::AbstractVector{Float64} , T::Float64 ,amat::AbstractMatrix{Float64})::Nothing

In-place bb intensity second order derivative with respect to temperature with intermediate matrix provided externally

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T³))*[(C₂/(λ*T))*(2*eᵃ¹/(eᵃ¹-1)-1)-2],

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(a₁/(λ⁵*T²))*[a₁*(2*eᵃ¹/(eᵃ¹-1) -1)-2], a₁=C₂/(λ*T), a₂ = 1/(eᵃ¹-1) , a₃ = eᵃ¹/(eᵃ¹-1) ,

d²Ibb/dT² = C₁*a₂*a₃*(a₁/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2]

Arguments :

h - to be filled, [Nx0] λ- wavelength in μm, [Nx0] T - temperature in Kelvins amat - matrix of intermediate coefficients, [Nx3]

∇²ₜibb!(h::AbstractVector , λ::AbstractVector , T::Number)

In-place bb intensity second order derivative with respect to temperature

Arguments :

h - to be filled, [Nx0] λ - wavelength in μm, [Nx0] T - tmperature in Kelvins

∇²ₜibb!(h::AbstractVector{Float64},T::Float64,amat::AbstractMatrix{Float64},∇i::AbstractVector{Float64})::Nothing

In-place bb intensity second order derivative with respect to temperature with provided both the intermediate matrix amat and the the Planck function first derivative

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T³))*[(C₂/(λ*T))*(2*eᵃ¹/(eᵃ¹-1)-1)-2]

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(a₁/(λ⁵*T²))*[a₁*(2*eᵃ¹/(eᵃ¹-1) -1)-2] a₁=C₂/(λ*T) a₂ = 1/(eᵃ¹-1) , a₃ = eᵃ¹/(eᵃ¹-1) , d²Ibb/dT² = C₁*a₂*a₃*(a₁/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2] as far as Ibb = (λ⁻⁵)* C₁*a₂ and dIbb/dT = C₁*a₃*a₂*a₁*(1/(λ⁵*T)) = a₃*a₁*Ibb/T hense

d²Ibb/dT² = C₁*a₂*a₃*a₁*(1/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2] = [a₃*a₁*Ibb/T^2]*[a₁*(2*a₃ - 1))-2] = [(dIbb/dT)/T]*[a₁*(2*a₃ - 1))-2]

Arguments :

h - to be filled, [Nx0] λ - wavelength in μm, [Nx0] amat - matrix of intermediate coefficients, [Nx3] ∇i - vector of bb intensity first derivatives, [Nx0]

∇²ₜibb(λ,T)

BB intensity second derivative with respect to temperature

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T³))*[(C₂/(λ*T))*(2*eᵃ¹/(eᵃ¹-1)-1)-2] ,

d²Ibb/dT² = C₁*(eᵃ¹/(eᵃ¹-1)²)*(a₁/(λ⁵*T²))*[a₁*(2*eᵃ¹/(eᵃ¹-1) -1)-2], a₁=C₂/(λ*T), a₂ = 1/(eᵃ¹-1), a₃ = eᵃ¹/(eᵃ¹-1) ,

d²Ibb/dT² = C₁*a₂*a₃*(a₁/(λ⁵*T²))*[a₁*(2*a₃ - 1))-2]

Arguments :

λ - wavelength in μm T - tmperature in Kelvins

∇²ₜplanck_weighted(α::AbstractVector{Q}, 
                    λ::AbstractVector{L},
                    T::D) where { D <: Number , Q <: Number , L <: Number}

Evaluates ∫α(λ)⋅∇²ₜibb(λ,T)dλ using numeric integration

    ∇²ₜplanck_weighted_ratio(
                        α::AbstractSpectralQuantity, 
                        band1::NTuple{2}, 
                        band2::NTuple{2},
                        T::D) where { D <: Number}

Returns the ratio of two weighted values d²/dT² [ (∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2) ]

Arguments

• α: An AbstractSpectralQuantity (e.g., analytical or autodiff emissivity).
• band1: Tuple (λ_min, λ_max) for the first spectral band (numerator), μm.
• band2: Tuple (λ_min, λ_max) for the second spectral band (denominator), μm.
• T: Temperature, K.

∇²ₜplanck_weighted_ratio(
                    α1::AbstractVector{Q}, 
                    λ1::AbstractVector{L},
                    α2::AbstractVector{Q}, 
                    λ2::AbstractVector{L},
                    T::D) where { D <: Number , Q <: Number , L <: Number}

Returns the ratio of two weighted values d²/dT² [ (∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2) ]

∇²ₜpower(T)

Total intensity second derivative of BB (radiance) at temperature T

Units: W/(m²⋅sr⋅K)

Arguments:

T - temperature, K

∇²ₜspectral_band_ratio(λ1::NTuple{2, TL}, λ2::NTuple{2,TL}, T::Number;  tol = 1e-6) where TL <: Number

Spectral band ratio second derivative

Arguments

λ1- tuple of left and right wavelength of the first band, in μm. λ2- tuple of left and right wavelength of the second band, in μm. T- Absolute temperature, in K.

∇²ₜspectral_ratio(λ1::Number, λ2::Number, T::Number)

Spectral ratio second derivative

Arguments

λ1- First wavelength (usually the shorter one), in μm. λ2- Second wavelength (usually the longer one), in μm. T- Absolute temperature, in K.

∇²ₜ∫ibbₗ(T; λₗ=0.0, λᵣ=Inf)

Relative (with respect to the integral power in the whole spectrum) integral intensity derivative (analytic) of bb intensity fraction in the spectral range λₗ...λᵣ (by default the range is 0...inf)

Arguments:

T - temperature,Kelvins (optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm

∇ₗibb(λ,T)

BB intensity first derivative with respect to the wavelength

Arguments:

λ - wavelength, μm T - temperature, K

∇ₜ(f) 

Differentiation operator returns the derivatives function for the input function

ibb , band_power , power , spectral_ratio , spectral_band_ratio , planck_weighted,planck_weighted_ratio and their derivatives e.g ∇ₜband_power etc.

Must be implemented on user defined [`AbstractSpectralQuatity`](@ref) interface to perform

fast continuous function weighting derivatives evaluation

Examples

∇ₜ(ibb)        # -> ∇ₜibb
F = (∇ₜ∘∇ₜ)(spectral_band_ratio)   #  ->  ∇²ₜspectral_band_ratio
F(1253.5) # returns the second derivative with respect to temperature of the ratio of two spectral band at temperature 1253.5 K

∇ₜband_power(T , band_power_value; λₗ=0.0 , λᵣ=Inf)

Evaluates bandpower derivative with respect to temperature with `bandpowervalue` provided externally (this may be usefull if one already has calculated the bandpower itself)

Arguments:

T - temperature,Kelvins band_power_value - band_power calculated elswhere

(optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm tol - intergation tolerance

∇ₜband_power(T;λₗ=0.0,λᵣ=Inf,tol=1e-6)

Total bb with temperature T integral intensity derivative within (in-band radiance), [W/(m²⋅sr⋅K)] the spectral range λₗ...λᵣ (by default the range is 0...inf) tol - tolerance of integration

Arguments:

T - temperature,Kelvins (optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm tol - intergation tolerance

∇ₜibb!(g::AbstractMatrix , λ::AbstractVector , T::AbstractVector)

In-place BB intensity first derivative with respect to temperature a₁=C₂/(λ*T) a₂ = 1/(eᵃ¹-1) a₃ = eᵃ¹/(eᵃ¹-1) dIbb/dT = C₁*a₃*a₂*a₁*(1/(λ⁵*T))

Arguments:

g - vector to be filled, [Nx0] λ - wavelength in μm, [Nx0] T - temperature in Kelvins

∇ₜibb!(g::AbstractVector , λ::AbstractVector , T , amat::AbstractMatrix)

In-place bb intensity first derivative with respect to temperature with externally provided amat - matrix with columns a₁,a₂,a₃

dIbb/dT = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T²)), a₁=C₂/(λ*T) , a₂ = 1/(eᵃ¹-1) , a₃ = eᵃ¹/(eᵃ¹-1) , dIbb/dT = C₁*a₃*a₂*a₁*(1/(λ⁵*T)),

Arguments:

g - to be filled, [Nx0] λ - wavelength in μm, [Nx0] T - temperature in Kelvins amat - matrix of intermediate coefficients, [Nx3]

∇ₜibb!(g::AbstractVector , T , amat::AbstractMatrix , bb_intensity::AbstractVector)

In-place bb intensity first derivative with respect to temperature with externally provided both amat - matrix with columns a₁,a₂,a₃ and bb_intensity

dIbb/dT = C₁(eᵃ¹/(eᵃ¹-1)²)(C₂/(λ⁶T²)) a₁=C₂/(λT) a₂ = 1/(eᵃ¹-1) # 1/expm1(a1) a₃ = eᵃ¹/(eᵃ¹-1) # exp(a)/expm1(a) dIbb/dT = C₁a₃a₂a₁(1/(λ⁵T)) as far as Ibb = C₁a₂/λ⁵ dIbb/dT = a₃a₁C₁(a₂/λ⁵)(1/T)=a₃a₁Ibb/T

Arguments:

g - to be filled, [Nx0] λ - wavelength in μm, [Nx0] T - temperature in Kelvins amat - matrix of intermediate coefficients, [Nx3]

∇ₜibb(λ::AbstractVector,T,amat::AbstractMatrix)

BB intensity first derivative with respect to temperature with externally provided matrix of intermediate coefficients dIbb/dT = C₁*(eᵃ¹/(eᵃ¹-1)²)*(C₂/(λ⁶*T²)) a₁=C₂/(λ*T) a₂ = 1/(eᵃ¹-1) a₃ = eᵃ¹/(eᵃ¹-1) dIbb/dT = C₁*a₃*a₂*a₁*(1/(λ⁵*T))

Arguments:

λ - wavelength in μm, [Nx0] T - temperature in Kelvins amat - matrix of intermediate coefficients, [Nx3]

        ∇ₜibb(λ,T)

BB intensity first derivative with respect to temperature dIbb/dT = C₁(eᵃ¹/(eᵃ¹-1)²)(C₂/(λ⁶T²)) `a₁=C₂/(λT)a₂ = 1/(eᵃ¹-1)# 1/expm1(a1)a₃ = eᵃ¹/(eᵃ¹-1)` # exp(a)/expm1(a) dIbb/dT = C₁a₃a₂a₁(1/(λ⁵*T))

Arguments:

λ - wavelength, μm T - temperature, K

planck_weighted(α::AbstractVector{Q}, 
                    λ::AbstractVector{L},
                    T::D) where { D <: Number , Q <: Number , L <: Number}

Evaluates ∫α(λ)⋅∇ₜibb(λ,T)dλ using numeric integration

    ∇ₜplanck_weighted_ratio(
                    α::AbstractSpectralQuantity, 
                    band1::NTuple{2}, 
                    band2::NTuple{2},
                    T::D) where { D <: Number}

Returns the ratio of two weighted values d/dT [ ∫α(λ)ibb(λ,T)dλ1 / ∫α(λ)ibb(λ,T)dλ2]

Arguments

• α: An AbstractSpectralQuantity (e.g., analytical or autodiff emissivity).
• band1: Tuple (λ_min, λ_max) for the first spectral band (numerator), μm.
• band2: Tuple (λ_min, λ_max) for the second spectral band (denominator), μm.
• T: Temperature, K.

∇ₜplanck_weighted_ratio(
                    α1::AbstractVector{Q}, 
                    λ1::AbstractVector{L},
                    α2::AbstractVector{Q}, 
                    λ2::AbstractVector{L},
                    T::D) where { D <: Number , Q <: Number , L <: Number}

Returns the ratio of two weighted values d/dT [ (∫α1(λ)ibb(λ,T)dλ1 / ∫α2(λ)ibb(λ,T)dλ2) ]

∇ₜpower(T)

Total intensity first derivative of BB (radiance) at temperature T

Units: W/(m²⋅sr⋅K)

Arguments:

T - temperature, K

∇ₜspectral_band_ratio(λ1::NTuple{2, TL}, λ2::NTuple{2,TL}, T::Number;   tol = 1e-6) where TL <: Number

First derivative of two wide spectral band ratio

Arguments

λ1- tuple of left and right wavelength of the first band, in μm. λ2- tuple of left and right wavelength of the second band, in μm. T- Absolute temperature, in K

∇ₜspectral_ratio(λ1::Number, λ2::Number, T::Number)

Calculate the theoretical intensity ratio derivative dR/T =d/dT ( Ibb1/Ibb2 ) between two wavelengths λ1 and λ2 at temperature T

Arguments

λ1- First wavelength (usually the shorter one), in μm. λ2- Second wavelength (usually the longer one), in μm. T- Absolute temperature, in K.

    ∇ₜ∫ibbₗ(T; λₗ=0.0, λᵣ=Inf)

Relative (with respect to the integral power in the whole spectrum) integral intensity derivative (analytic) of bb intensity fraction in the spectral range λₗ...λᵣ (by default the range is 0...inf)

Arguments:

T - temperature,Kelvins (optional) λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm

∫ibbₗ(T;λₗ=0.0,λᵣ=Inf,tol=1e-6)

Relative (with respect to the integral power in the whole spectrum) integral intensity of bb in the spectral range λₗ...λᵣ (by default the range is 0...inf)

Arguments:

`T - temperature,Kelvins

(optional)

λₗ - left wavelength boundary, μm λᵣ - right wavelength boundary, μm tol - intergation tolerance

∫ₗ(g)
∫ₗ(g, λ₁, λ₂)
∫ₗ(g, λ::AbstractVector, α::AbstractVector)

The spectral integration operator for Planck radiation quantities.

Depending on the provided argument signature, this operator returns different highly optimized, type-stable callable objects (functors) designed to minimize runtime allocations.

Arguments

• g::Dfunctions: The base Planck function layer (ibb, ∇ₜibb, ∇²ₜibb, or Dₜibb).
• λ₁, λ₂::Number: Spectral band boundaries in microns (μm).
• λ::AbstractVector, α::AbstractVector: Discrete experimental data arrays (wavelength grid and spectral coefficients).

Full Spectrum Integration (Unary)

When applied directly to a single argument, it returns a functor representing the analytical total integrated power over the entire spectrum ([0, ∞)) governed by the Stefan-Boltzmann law.

f = ∫ₗ ibb
f(1273.5) # Returns σ * T^4

Spectral Band Integration

When given a function and two numeric boundaries, it returns a functor that evaluates the integrated band power within the specific interval [λ₁, λ₂].

f = ∫ₗ(ibb, 2.3, 4.5)
f(1273.5) # Returns integrated power between 2.3 and 4.5 μm

Tabular Data Integration

When given a function and two matching discrete arrays, it returns a functor designed for experimental data pyrometry (e.g., integrating a wavelength-dependent emissivity vector α against the Planck distribution).

f = ∫ₗ(ibb, λ_grid, e_data)
f(1273.5) # Returns the discrete Planck-weighted integral 

∫ₗ(f::F , λ₁::Number, λ₂::Number) where F <: Union{typeof(ibb), typeof(∇²ₜibb), typeof(∇ₜibb)}

Returns a callable object which returns the f function integral as a function of temperature

f = ∫ₗ(ibb , 2.3 , 4.5)
f(1273.5) # returns the value of Planck function integrated over 2.3 - 4.5 spectral range at temperature 1273,5 K