Library

const _EXPONENTIAL_ATMOSPHERE_H

Scale height for the exponential atmospheric model [km].

const _EXPONENTIAL_ATMOSPHERE_H₀

Base altitude for the exponential atmospheric model [km].

const _EXPONENTIAL_ATMOSPHERE_ρ₀

Nominal density for the exponential atmospheric model [kg / m³].

Harris-Priester min-max density (kg/m³) vs. altitude (m) table. These data are valid for a mean solar activity.

const _JACCHIA1977_CONSTANTS

Constants for the Jacchia 1977 atmospheric model.

The species-dependent tuples use the internal order (He, O₂, N₂, Ar, O, H), matching the reference implementation [2].

const _JR1971_CONSTANTS

Constants for the Jacchia-Roberts 1971 atmospheric model.

struct JB2008Output{T<:Number}

Output of the atmospheric model Jacchia-Bowman 2008.

Fields

• total_density::T: Total atmospheric density [kg / m³].
• temperature::T: Temperature at the selected position [K].
• exospheric_temperature::T: Exospheric temperature [K].
• N2_number_density::T: Number density of N₂ [1 / m³].
• O2_number_density::T: Number density of O₂ [1 / m³].
• O_number_density::T: Number density of O [1 / m³].
• Ar_number_density::T: Number density of Ar [1 / m³].
• He_number_density::T: Number density of He [1 / m³].
• H_number_density::T: Number density of H [1 / m³].

struct JR1971Output{T<:Number}

Output of the atmospheric model Jacchia-Roberts 1971.

Fields

• total_density::T: Total atmospheric density [kg / m³].
• temperature::T: Temperature at the selected position [K].
• exospheric_temperature::T: Exospheric temperature [K].
• N2_number_density::T: Number density of N₂ [1 / m³].
• O2_number_density::T: Number density of O₂ [1 / m³].
• O_number_density::T: Number density of O [1 / m³].
• Ar_number_density::T: Number density of Ar [1 / m³].
• He_number_density::T: Number density of He [1 / m³].
• H_number_density::T: Number density of H [1 / m³].

struct Jacchia1977Output{T<:Number}

Output of the atmospheric model Jacchia 1977.

Fields

• total_density::T: Total atmospheric density [kg / m³].
• temperature::T: Temperature at the selected position [K].
• exospheric_temperature::T: Mean exospheric temperature T½ above the selected position [K].
• N2_number_density::T: Number density of N₂ [1 / m³].
• O2_number_density::T: Number density of O₂ [1 / m³].
• O_number_density::T: Number density of O [1 / m³].
• Ar_number_density::T: Number density of Ar [1 / m³].
• He_number_density::T: Number density of He [1 / m³].
• H_number_density::T: Number density of H [1 / m³].

struct Nrlmsise00Flags

Flags to configure NRLMSISE-00.

Fields

• F10_Mean::Bool: F10.7 effect on mean.
• time_independent::Bool: Independent of time.
• sym_annual::Bool: Symmetrical annual.
• sym_semiannual::Bool: Symmetrical semiannual.
• asym_annual::Bool: Asymmetrical annual.
• asym_semiannual::Bool: Asymmetrical semiannual.
• diurnal::Bool: Diurnal.
• semidiurnal::Bool: Semidiurnal.
• daily_ap::Bool: Daily AP.
• all_ut_long_effects::Bool: All UT/long effects.
• longitudinal::Bool: Longitudinal.
• ut_mixed_ut_long::Bool: UT and mixed UT/long.
• mixed_ap_ut_long::Bool: Mixed AP/UT/long.
• terdiurnal::Bool: Terdiurnal.
• departures_from_eq::Bool: Departures from diffusive equilibrium.
• all_tinf_var::Bool: All TINF variations.
• all_tlb_var::Bool: All TLB variations.
• all_tn1_var::Bool: All TN1 variations.
• all_s_var::Bool: All S variations.
• all_tn2_var::Bool: All TN2 variations.
• all_nlb_var::Bool: All NLB variations.
• all_tn3_var::Bool: All TN3 variations.
• turbo_scale_height::Bool: Turbo scale height variations.

struct Nrlmsise00Output{T<:Number}

Output structure for NRLMSISE00 model.

Fields

• total_density::T: Total mass density [kg / m³].
• temperature: Temperature at the selected altitude [K].
• exospheric_temperature: Exospheric temperature [K].
• N_number_density: Nitrogen number density [1 / m³].
• N2_number_density: N₂ number density [1 / m³].
• O_number_density: Oxygen number density [1 / m³].
• aO_number_density: Anomalous Oxygen number density [1 / m³].
• O2_number_density: O₂ number density [1 / m³].
• H_number_density: Hydrogen number density [1 / m³].
• He_number_density: Helium number density [1 / m³].
• Ar_number_density: Argon number density [1 / m³].

Remarks

Anomalous oxygen is defined as hot atomic oxygen or ionized oxygen that can become appreciable at high altitudes (> 500 km) for some ranges of inputs, thereby affecting drag on satellites and debris. We group these species under the term Anomalous Oxygen, since their individual variations are not presently separable with the drag data used to define this model component.

struct Nrlmsise00Structure{T<:Number, T_AP<:Union{Number, AbstractVector}}

Structure with the configuration parameters for NRLMSISE-00 model. T is the floating-number type and T_AP is the type of the AP information, which can be a Number or AbstractVector.

_ccor(h::Number, r::Number, h₁::Number, zh::Number) -> Number

Compute the chemistry / dissociation correction for MSIS models.

Arguments

• h::Number: Altitude.
• r::Number: Target ratio.
• h₁::Number: Transition scale length.
• zh::Number: Altitude of 1/2 r.

_ccor2(h::Number, r::Number, h₁::Number, zh::Number, h₂::Number) -> Number

Compute the O and O₂ chemistry / dissociation correction for MSIS models.

Arguments

• h::Number: Altitude.
• r::Number: Target ratio.
• h₁::Number: Transition scale length.
• zh::Number: Altitude of 1/2 r.
• h₂::Number: Transition scale length 2.

_densm(
    h::Number,
    d₀::Number,
    xm::Number,
    g_lat::Number,
    r_lat::Number,
    tn2::NTuple{4, Number},
    tgn2::NTuple{2, Number},
    tn3::NTuple{5, Number},
    tgn3::NTuple{2, Number}
) -> Number

Compute the temperature and density profiles for the lower atmosphere.

Arguments

• h::Number: Altitude [km].
• d₀::Number: Reference density, returned if h > _ZN2[1].
• xm::Number: Species molecular weight [ ].
• g_lat::Number: Reference gravity at desired latitude [cm / s²].
• r_lat::Number: Reference radius at desired latitude [km].
• tn2::NTuple{4, Number}: Temperature at the nodes for ZN2 scale [K].
• tgn2::NTuple{2, Number}: Temperature gradients at the end nodes for ZN2 scale.
• tn3::NTuple{5, Number}: Temperature at the nodes for ZN3 scale [K].
• tgn3::NTuple{2, Number}: Temperature gradients at the end nodes for ZN3 scale.

Returns

• Number: Density [1 / cm³] if xm is not 0, or the temperature [K] otherwise. The type is the promotion of the types of the numeric inputs.

_densu(
    h::T,
    dlb::T,
    tinf::T,
    tlb::T,
    xm::T,
    α::T,
    zlb::T,
    s2::T,
    g_lat::T,
    r_lat::T,
    tn1::NTuple{5, T},
    tgn1::NTuple{2, T}
) where T<:Number -> T, NTuple{5, T}, NTuple{2, T}

Compute the density [1 / cm³] or temperature [K] profiles according to the new lower thermo polynomial.

Arguments

• h::T: Altitude [km].
• dlb::T: Density at lower boundary [1 / cm³].
• tinf::T: Exospheric temperature [K].
• tlb::T: Temperature at lower boundary [K].
• xm::T: Species molecular weight [ ].
• α::T: Thermal diffusion coefficient.
• zlb::T: Altitude at lower boundary [km].
• s2::T: Slope.
• g_lat::T: Reference gravity at the latitude [cm / s²].
• r_lat::T: Reference radius at the latitude [km].
• tn1::NTuple{5, T}: Temperature at nodes for ZN1 scale [K].
• tgn1::NTuple{2, T}: Temperature gradients at end nodes for ZN1 scale.

Returns

• T: Density [1 / cm³] if xm is not 0, or the temperature [K] otherwise.
• NTuple{5, T}: Updated tn1.
• NTuple{2, T}: Updated tgn1.

_dnet(dd::Number, dm::Number, zhm::Number, xmm::Number, xm::Number) -> Number

Compute the turbopause correction for MSIS models, returning the combined density.

Arguments

• dd::Number: Diffusive density.
• dm::Number: Full mixed density.
• zhm::Number: Transition scale length.
• xmm::Number: Full mixed molecular weight.
• xm::Number: Species molecular weight.

_get_doy(jd::Number) -> Number

Return the day of the year for the Julian day jd.

_glob7s(
    nrlmsise00d::Nrlmsise00Structure{T},
    p::AbstractVector{V}
) where {T<:Number, V<:Number} -> T

Compute the function G(L) with lower atmosphere parameters p and the NRLMSISE-00 structure nrlmsise00d.

_globe7(
    nrlmsise00d::Nrlmsise00Structure{T},
    p::AbstractVector{V}
) where {T<:Number, V<:Number} -> Nrlmsise00Structure{T}, T

Compute the function G(L) with upper thermosphere parameters p and the NRLMSISE-00 structure nrlmsise00.

Returns

• Nrlmsise00Structure{T}: Modified structure nrlmsise00d.
• T: Result of G(L).

_gravity_and_effective_radius(ϕ_gd::T) where T<:Number -> T, T

Compute the gravity [cm / s²] and effective radius [km] at the geodetic latitude ϕ_gd [°].

_gtd7(
    nrlmsise00d::Nrlmsise00Structure{T}
) where T<:Number -> Nrlmsise00Structure{T}, Nrlmsise00Output{T}

Compute the temperatures and densities using the information inside the structure nrlmsise00d without including the anomalous oxygen in the total density.

Returns

• Nrlmsise00Structure{T}: Modified structure nrlmsise00d.
• Nrlmsise00Output{T}: Structure with the output information.

_gtd7d(
    nrlmsise00d::Nrlmsise00Structure{T}
) where T<:Number -> Nrlmsise00Structure{T}, Nrlmsise00Output{T}

Compute the temperatures and densities using the information inside the structure nrlmsise00d including the anomalous oxygen in the total density.

Returns

• Nrlmsise00Structure{T}: Modified structure nrlmsise00d.
• Nrlmsise00Output{T}: Structure with the output information.

_gts7(
    nrlmsise00d::Nrlmsise00Structure{T}
) where T<:Number -> Nrlmsise00Structure{T}, Nrlmsise00Output{T}

Compute the temperatures and densities using the information inside the structure nrlmsise00d for altitudes higher than 72.5 km (thermospheric portion of NRLMSISE-00). Notice that the anomalous oxygen is not included in the total density.

Returns

• Nrlmsise00Structure{T}: Modified structure nrlmsise00d.
• Nrlmsise00Output{T}: Structure with the output information.

_g₀(a::Number, abs_p25::Number, p26::Number) -> Number

Compute g₀ function (see Eq. A24d) using the coefficients abs_p25 = abs(p[25]) and p26 = p[26].

_harris_priester_mod_density_poly(F10ₐ::Number, i::Integer, c::Integer) -> Number

Evaluate the cubic polynomial in the flux F10ₐ [sfu] whose coefficients are stored in the columns c to c + 3 of the i-th row of _HARRIS_PRIESTER_MOD_COEFS, returning the related density [g / km³]. Notice that we build the coefficient tuple explicitly since splatting a runtime view into @evalpoly leads to dynamic dispatch and allocations.

_jacchia1977_diurnal(
    T½::Number,
    Ωp::Number,
    Ωs::Number,
    δs::Number,
    ϕ::Number,
    z::Number,
    M̄::Number
) -> NTuple{6, T}, NTuple{6, T}, T

Compute the base-10 logarithm of the number densities considering the diurnal variation (routine DIVARI of [2] and eqs. 24 to 27 of [1]).

Arguments

• T½::Number: Mean exospheric temperature [K].
• Ωp::Number: Right ascension of the selected location [rad].
• Ωs::Number: Right ascension of the Sun [rad].
• δs::Number: Declination of the Sun [rad].
• ϕ::Number: Latitude [rad].
• z::Number: Altitude [km].
• M̄::Number: Local mean molecular mass [g / mol].

Returns

• NTuple{6, T}: Base-10 logarithm of the number densities [1 / m³] evaluated at the per-species pseudo exospheric temperatures, in the internal order (He, O₂, N₂, Ar, O, H), where T is the promotion of the input types.
• NTuple{6, T}: Base-10 logarithm of the number densities [1 / m³] of the static model evaluated at the hydrogen pseudo exospheric temperature.
• T: Pseudo exospheric temperature of the hydrogen [K], used as the quiet temperature by the geomagnetic variation.

_jacchia1977_dynamic(
    z::Number,
    ϕ::Number,
    Ωp::Number,
    Ωs::Number,
    δs::Number,
    λ::Number,
    Φ::Number,
    F10::Number,
    F10ₐ::Number,
    Kp::Number
) -> Jacchia1977Output

Compute the Jacchia 1977 dynamic model (routine ISDAMO of [2]).

Arguments

• z::Number: Altitude [km].
• ϕ::Number: Latitude [rad].
• Ωp::Number: Right ascension of the selected location [rad].
• Ωs::Number: Right ascension of the Sun [rad].
• δs::Number: Declination of the Sun [rad].
• λ::Number: Longitude of the selected location [rad].
• Φ::Number: Fraction of the tropic year starting on January 1st [-].
• F10::Number: 10.7-cm solar flux [sfu].
• F10ₐ::Number: 10.7-cm averaged solar flux [sfu].
• Kp::Number: Kp geomagnetic index [-].

Returns

• Jacchia1977Output: Structure containing the results obtained from the model. Its element type is the promotion of the types of the numeric inputs.

_jacchia1977_geomagnetic(
    T_quiet::Number,
    Kp::Number,
    ϕ::Number,
    λ::Number,
    z::Number
) -> NTuple{6, T}

Compute the base-10 logarithm of the number densities of the static model evaluated at the exospheric temperature increased by the geomagnetic activity Kp [-], together with the homopause displacement and equatorial wave corrections (routine GEOACI of [2] and eqs. 28 to 35 of [1]), given the quiet exospheric temperature T_quiet [K], the latitude ϕ [rad], the longitude λ [rad], and the altitude z [km].

Returns

• NTuple{6, T}: Base-10 logarithm of the number densities [1 / m³] in the internal order (He, O₂, N₂, Ar, O, H), where T is the promotion of the input types. The caller must subtract the static model evaluated at T_quiet to obtain the geomagnetic variation.

_jacchia1977_profile_params(T∞::Number) -> NTuple{7, Number}

Compute the parameters of the temperature profile (eqs. 1 to 4 of [1]) related to the exospheric temperature T∞ [K].

See also: _jacchia1977_temperature

_jacchia1977_seasonal_latitudinal(
    Φ::Number,
    δs::Number,
    ϕ::Number,
    z::Number
) -> NTuple{6, T}

Compute the seasonal-latitudinal variation of the base-10 logarithm of the number densities (routine SEALAT of [2] and eqs. 36 to 39 of [1]) given the fraction of the tropic year Φ [-], the Sun declination δs [rad], the latitude ϕ [rad], and the altitude z [km]. The result is in the internal order (He, O₂, N₂, Ar, O, H), and T is the promotion of the input types.

_jacchia1977_semiannual(Φ::Number, z::Number) -> Number

Compute the semiannual variation of the base-10 logarithm of the number densities (routine SEMIAN of [2] and eqs. 40 to 44 of [1]) given the fraction of the tropic year Φ [-] and the altitude z [km]. The variation is the same for all species.

_jacchia1977_static(T∞::T1, z::T2) where {T1<:Number, T2<:Number} -> NTuple{6, T}, T, T

Compute the Jacchia 1977 static model (routine IMOWEI of [2]) for the exospheric temperature T∞ [K] and altitude z [km].

The function numerically integrates the barometric equation between 90 km and 100 km and the diffusion equations above 100 km using the Boole rule, as in the reference implementation [2]. The atomic hydrogen is anchored at 500 km and integrated with its flux term for other altitudes.

Returns

• NTuple{6, T}: Base-10 logarithm of the number densities [1 / m³] in the internal order (He, O₂, N₂, Ar, O, H), where T is the promotion of T1 and T2.
• T: Mean molecular mass at the selected altitude [g / mol].
• T: Total density at the selected altitude [kg / m³].

References

• [1] Jacchia, L. G (1977). Thermospheric temperature, density and composition: New models. SAO Special Report #375.
• [2] de Matos, B. S., Carrara, V (1985-1987). Fortran implementation of the Jacchia 1977 model. INPE, São José dos Campos, BR.

_jacchia1977_step(z_ini::Number, z_end::Number, base_step::Number) -> Number

Compute the integration step [km] so that the interval between z_ini and z_end [km] is divided into an integer number of Boole rule applications with a step close to base_step [km], as in the reference implementation [2].

_jacchia1977_temperature(z::Number, c::NTuple{7, T}) where {T<:Number} -> Number

Compute the temperature [K] at the altitude z [km] using the profile parameters c obtained from _jacchia1977_profile_params (eqs. 3 and 4 of [1]).

_jb2008_gravity(z::Number) -> Number

Compute the gravity [m / s²] at altitude z [km] according to the model Jacchia 1971 [3].

_jb2008_high_altitude(h::Number, F10ₐ::Number) -> Number

Compute the high altitude exospheric density correction factor [-] at the altitude h [km] given the averaged 10.7-cm solar flux F10ₐ [sfu], which must be obtained using an 81-day window centered on the input time, using the model in Section 6.2 of [2].

_jb2008_mean_molecular_mass(z::Number; kwargs...) -> Number

Compute the mean molecular mass [g / mol] at the altitude z [km] using the empirical profile in eq. 1 [3], which is valid only between 90 km and 105 km.

Keywords

• verbose::Val: Set to Val(true) to emit a warning when the altitude is outside the validity range, or to Val(false) to suppress it. (Default: Val(true))

_jb2008_semiannual(
    doy::Number,
    h::Number,
    F10ₐ::Number,
    S10ₐ::Number,
    M10ₐ::Number
) -> Number, Number, Number

Compute the semiannual density variation considering the JB2008 model [1].

Arguments

• doy::Number: Day of the year plus the fraction of the day [-].
• h::Number: Height [km].
• F10ₐ::Number: Averaged 10.7-cm flux, which must use an 81-day window centered on the input time [sfu].
• S10ₐ::Number: EUV 81-day averaged centered index.
• M10ₐ::Number: MG2 81-day averaged centered index.

Returns

• Number: Semiannual F(z) height function, clamped to be equal or higher than 10⁻⁶.
• Number: Semiannual G(t) yearly periodic function.
• Number: Semiannual variation of the density Δsalog₁₀ρ.

_jb2008_temperature(z::Number, Tx::Number, T∞::Number) -> Number

Compute the temperature [K] at height z [km] given the temperature Tx [K] at the inflection point, and the exospheric temperature T∞ [K] according to the theory of the model Jacchia 1971 [3]. The inflection point is considered to be z = 125 km.

The function throws an ArgumentError if z is lower than 90 km or if T∞ is negative.

_jb2008_ΔTc(F10::Number, lst::Number, ϕ_gd::Number, h::Number) -> Number

Compute the correction ΔTc [K] in the exospheric temperature for the Jacchia-Bowman model.

This correction is mentioned in [2]. However, the equations do not seem to match those in the source-code. The ones implemented here are exactly the same as in the source-code.

Arguments

• F10::Number: 10.7-cm solar flux [sfu].
• lst::Number: Local solar time (0 - 24) [hr].
• ϕ_gd::Number: Geodetic latitude [rad].
• h::Number: Altitude [km].

_jb2008_δf1(z::Number, Tx::Number, T∞::Number; kwargs...) -> Number

Compute the integrand M̄´ g / T used by _jb2008_∫ between 90 km and 105 km at the altitude z [km] given the temperature Tx [K] at the inflection point and the exospheric temperature T∞ [K].

Keywords

• verbose::Val: Set to Val(true) to emit warnings from the internal functions, or to Val(false) to suppress them. (Default: Val(true))

_jb2008_δf2(z::Number, Tx::Number, T∞::Number; kwargs...) -> Number

Compute the integrand g / T used by _jb2008_∫ above 105 km at the altitude z [km] given the temperature Tx [K] at the inflection point and the exospheric temperature T∞ [K].

Keywords

• verbose::Val: Set to Val(true) to emit warnings from the internal functions, or to Val(false) to suppress them. (Default: Val(true))

_jb2008_∫(
    z₀::Number,
    z₁::Number,
    R::Number,
    Tx::Number,
    T∞::Number,
    δf::Function;
    kwargs...
) -> Number, Number

Compute the integral of the function δf between the altitudes z₀ and z₁ [km] using the Newton-Cotes 4th degree method, where R defines the step size [-], Tx is the temperature at the inflection point [K], and T∞ is the exospheric temperature [K]. The integrand function δf must be _jb2008_δf1 or _jb2008_δf2.

Keywords

• verbose::Val: Set to Val(true) to emit warnings from the internal functions, or to Val(false) to suppress them. (Default: Val(true))

Returns

• Number: Value of the integral.
• Number: Last value of z [km] used in the numerical algorithm.

_jr1971_S(
    z::Number,
    B₀::Number,
    B₁::Number,
    B₂::Number,
    B₃::Number,
    B₄::Number,
    B₅::Number
) -> Number

Compute the S(z) polynomial [1, p. 371] at the altitude z [km] given its coefficients B₀ to B₅.

_jr1971_U(ν::Number, Ra::Number, x::Number, y::Number, r₁::Number, r₂::Number) -> Number

Compute the partial-fraction helper function U(ν) [1, p. 372] given the mean Earth radius Ra [km], the real roots r₁ and r₂ [km], and the real and imaginary parts x and y [km] of the complex root of the quartic polynomial.

_jr1971_V(ν::Number, x::Number, y::Number, r₁::Number, r₂::Number) -> Number

Compute the partial-fraction helper function V(ν) [1, p. 372] given the real roots r₁ and r₂ [km], and the real and imaginary parts x and y [km] of the complex root of the quartic polynomial.

_jr1971_W(ν::Number, Ra::Number, x::Number, y::Number, r₁::Number, r₂::Number) -> Number

Compute the partial-fraction helper function W(ν) [1, p. 372] given the mean Earth radius Ra [km], the real roots r₁ and r₂ [km], and the real and imaginary parts x and y [km] of the complex root of the quartic polynomial. Notice that the equation for W in [1] is incorrect and the corrected form from [3, 4] is used.

_jr1971_mean_molecular_mass(z::Number; kwargs...) -> Number

Compute the mean molecular mass [g / mol] at the altitude z [km] using the empirical profile in eq. 1 [3, 4], which is valid only between 90 km and 100 km.

Keywords

• verbose::Val: Set to Val(true) to emit a warning when the altitude is outside the validity range, or to Val(false) to suppress it. (Default: Val(true))

_jr1971_roots(c₀::Number, c₁::Number, c₂::Number, c₃::Number) -> T, T, T, T

Compute the roots of the monic quartic polynomial:

P(z) = z⁴ + c₃ ⋅ z³ + c₂ ⋅ z² + c₁ ⋅ z + c₀,

which is necessary to compute the density below 125 km. The model theory states that this polynomial always has two distinct real roots and one complex conjugate pair [1].

The algorithm uses the Ferrari method: the depressed quartic is split into two quadratic factors whose coefficients are obtained from the largest root of the resolvent cubic, computed by the Cardano method. The real roots are polished with Newton iterations and the complex pair is recovered from the Vieta relations, keeping the accuracy close to the machine precision. Since only closed-form expressions are used, this function does not allocate and is compatible with automatic differentiation.

Returns

• T: Highest real root r₁, where T is the promotion of the input types to float.
• T: Lowest real root r₂.
• T: Real part x of the complex conjugate pair.
• T: Positive imaginary part y of the complex conjugate pair.

_jr1971_temperature(z::Number, Tx::Number, T∞::Number) -> Number

Compute the temperature [K] at height z [km] according to the theory of the model Jacchia-Roberts 1971 [1, 3, 4] given the temperature Tx [K] at the inflection point and the exospheric temperature T∞ [K]. The inflection point is considered to be z = 125 km.

The function throws an ArgumentError if z is lower than 90 km or if T∞ is negative.

_scale_height(h::Number, xm::Number, temp::Number, g_lat::Number, r_lat::Number) -> Number

Compute the scale height.

Arguments

• h::Number: Altitude [km].
• xm::Number: Species molecular weight [ ].
• temp::Number: Temperature [K].
• g_lat::Number: Reference gravity at desired latitude [cm / s²].
• r_lat::Number: Reference radius at desired latitude [km].

_scale_height_junk(
    xbar::AbstractVector{<:Number},
    h::Number,
    H_ρ_min::AbstractVector{<:Number},
    H_ρ_max::AbstractVector{<:Number}
) -> Number, Number, Number

Compute the scale heights at the altitude h [km] using the Junkins / Jancaitis weighting method for third-order continuity [1, Section 3.2], given the abscissas xbar [km] of the blending interval and the minimum and maximum density scale heights H_ρ_min and H_ρ_max [km] at its borders.

Returns

• Number: Weighting function value [-].
• Number: Blended minimum density scale height [km].
• Number: Blended maximum density scale height [km].

_sg₀(ex::Number, ap::AbstractVector, abs_p25::Number, p26::Number) -> Number

Compute the sg₀ function (see Eq. A24a) using the ap vector and the coefficients abs_p25 and p26.

_spline(
    x::NTuple{N, T},
    y::NTuple{N, T},
    ∂²y::NTuple{N, T},
    xᵢ::T
) where {N, T<:Number} -> float(T)

Compute the interpolation of the cubic spline y(x) with second derivatives ∂²y at xᵢ.

Arguments

• x::NTuple{N, T}: X components of the tabulated function in ascending order.
• y::NTuple{N, T}: Y components of the tabulated function evaluated at x.
• ∂²y::NTuple{N, T}: Second derivatives of y(x) ∂²y/∂x² evaluated at x.
• xᵢ::T: Point to compute the interpolation.

_spline_∂²(
    x::NTuple{N, T},
    y::NTuple{N, T},
    ∂²y₁::T,
    ∂²yₙ::T
) where {N, T<:Number} -> NTuple{N, T}

Compute the 2nd derivatives of the cubic spline interpolation y(x) given the 2nd derivatives at x[1] (∂²y₁) and at x[N] (∂²yₙ). This functions return a tuple with the evaluated 2nd derivatives at each point in x.

Arguments

• x::NTuple{N, T}: X components of the tabulated function in ascending order.
• y::NTuple{N, T}: Y components of the tabulated function evaluated at x.
• ∂²y₁::T: Second derivative of y(x) ∂²y/∂x² evaluated at x[1].
• ∂²yₙ::T: Second derivative of y(x) ∂²y/∂x² evaluated at x[N].

_spline_∫(
    x::NTuple{N, T},
    y::NTuple{N, T},
    ∂²y::NTuple{N, T},
    xf::Number
) where {N, T<:Number} -> float(T)

Compute the integral of the cubic spline function y(x) from x[1] to xf, where the function second derivatives evaluated at x are ∂²y.

Arguments

• x::NTuple{N, T}: X components of the tabulated function in ascending order.
• y::NTuple{N, T}: Y components of the tabulated function evaluated at x.
• ∂²y::NTuple{N, T}: Second derivatives of y(x) ∂²y/∂x² evaluated at x.
• xf::Number: Abscissa endpoint for integration.

_ζ(r_lat::Number, zz::Number, zl::Number) -> Number

Compute the zeta function.

exponential(h::Number) -> Number

Compute the atmospheric density [kg / m³] at the altitude h [m] above the ellipsoid using the exponential atmospheric model:

                ┌            ┐
                │    h - h₀  │
ρ(h) = ρ₀ . exp │ - ──────── │ ,
                │      H     │
                └            ┘

in which ρ₀, h₀, and H are parameters obtained from tables that depend only on h.

harrispriester(
    instant::DateTime,
    ϕ_gd::Number,
    λ::Number,
    h::Number;
    kwargs...
) -> Number
harrispriester(jd::Number, ϕ_gd::Number, λ::Number, h::Number; kwargs...) -> Number

Compute the atmospheric density [kg / m³] using the Harris-Priester model.

The model is valid only inside the altitude range of the density profile alt_ρ (100 km to 1000 km for the default profile). The function throws an ArgumentError if the altitude h is lower than the minimum altitude in the profile, and returns zero if it is higher than the maximum altitude.

Arguments

• instant::DateTime: Instant to compute the model represented using DateTime.
• jd::Number: Julian day to compute the model.
• ϕ_gd::Number: Geodetic latitude [rad].
• λ::Number: Geodetic longitude [rad].
• h::Number: Geodetic altitude [m].

Keywords

• n::Int: Cosine exponent in the diurnal bulge modeling (2 <= n <= 6). If n is 2, it models a smooth transition, whereas if n is 6, it models a sharp transition. (Default: 4)
• alt_ρ::AbstractMatrix: Matrix containing the minimum and maximum density profiles, where the columns are the altitude [m], the minimum density [kg / m³], and the maximum density [kg / m³], respectively. (Default: _HARRIS_PRIESTER_ALT_RHO, related to the mean solar activity)

Returns

• Number: Atmospheric density [kg / m³]. The type is the promotion of the types of the numeric inputs.

harrispriester_modified(
    instant::DateTime,
    ϕ_gd::Number,
    λ::Number,
    h::Number[, F10ₐ::Number];
    kwargs...
) -> Number
harrispriester_modified(
    jd::Number,
    ϕ_gd::Number,
    λ::Number,
    h::Number[, F10ₐ::Number];
    kwargs...
) -> Number

Compute the atmospheric density [kg / m³] using the modified Harris-Priester model.

This model is a Julia translation of the Fortran code harris_priester_mod_dist.f90 developed by Noble Hatten and Ryan P. Russell [1]. It ensures continuous first derivatives, eliminates singularities, and uses a cubic dependency on the 81-day centered average of the F10.7 solar flux index (F10ₐ) to model density variations.

If F10ₐ is not provided, it will be automatically fetched using the SpaceIndices package. In this case, the initialization of the space indices package with SpaceIndices.init() is required.

Arguments

• instant::DateTime: Instant to compute the model represented using DateTime.
• jd::Number: Julian day to compute the model.
• ϕ_gd::Number: Geodetic latitude [rad].
• λ::Number: Geodetic longitude [rad].
• h::Number: Geodetic altitude [m].
• F10ₐ::Number: 81-day centered average of the F10.7 solar flux index [sfu].

Keywords

• n::Number: Cosine exponent in the diurnal bulge modeling. The original Fortran implementation computes n from the orbital inclination as n = 2.001 + 4 sin²(inclination). Hence, use n = 2.001 for equatorial orbits, n = 6.001 for polar orbits, and interpolate for intermediate inclinations. If the orbital inclination is unknown, the default provides a reasonable approximation. This functionality was purposefully removed here to avoid needing an additional dependency. (Default: 4)

Returns

• Number: Atmospheric density [kg / m³]. The type is the promotion of the types of the numeric inputs.

References

• [1] Hatten, N., & Russell, R. P. (2017). A smooth and robust Harris-Priester atmospheric density model for low Earth orbit applications. Advances in Space Research, 59(2), 571-586.

jacchia1977(
    instant::DateTime,
    ϕ_gd::Number,
    λ::Number,
    h::Number[, F10::Number, F10ₐ::Number, Kp::Number];
    kwargs...
) -> Jacchia1977Output
jacchia1977(
    jd::Number,
    ϕ_gd::Number,
    λ::Number,
    h::Number[, F10::Number, F10ₐ::Number, Kp::Number];
    kwargs...
) -> Jacchia1977Output

Compute the atmospheric density using the Jacchia 1977 model.

Unlike the Jacchia-Roberts 1971 model, the Jacchia 1977 model does not have a closed-form solution. Hence, the barometric and diffusion equations are numerically integrated here, making this model considerably slower than jr1971.

If we omit all space indices, the system tries to obtain them automatically for the selected day jd or instant using the prescriptions in the report [1]: the daily flux is evaluated with a lag that depends on the solar hour angle, the averaged flux is a Gaussian-weighted mean with a standard width of 71 days centered on the input time, and the Kp is delayed by an interval that depends on the geomagnetic latitude. However, the indices must be already initialized using the function SpaceIndices.init(). Notice that the Gaussian-weighted mean requires the indices to be available up to 213 days after the input time.

The function throws an ArgumentError if the altitude h is outside the interval [90, 2000] km.

Arguments

• jd::Number: Julian day to compute the model.
• instant::DateTime: Instant to compute the model represented using DateTime.
• ϕ_gd::Number: Geodetic latitude [rad].
• λ::Number: Longitude [rad].
• h::Number: Altitude [m].
• F10::Number: 10.7-cm solar flux [sfu], evaluated with the lag prescribed in [1] (from 0.9 day to 1.6 days, depending on the local solar time).
• F10ₐ::Number: 10.7-cm averaged solar flux, Gaussian-weighted mean centered on the input time with a standard width of 71 days [sfu].
• Kp::Number: Kp geomagnetic index, delayed by 0.1 day to 0.3 day depending on the geomagnetic latitude, as prescribed in [1].

Keywords

• verbose::Val: Set to Val(true) to emit debug messages related to the automatic space index fetching, or to Val(false) to suppress them. Notice that this keyword must be a Val object, not a Bool, and it is only available in the methods that fetch the space indices automatically. (Default: Val(true))

Returns

• Jacchia1977Output: Structure containing the results obtained from the model. Its element type is the promotion of the types of the numeric inputs.

References

• [1] Jacchia, L. G (1977). Thermospheric temperature, density and composition: New models. SAO Special Report #375.
• [2] de Matos, B. S., Carrara, V (1985-1987). Fortran implementation of the Jacchia 1977 model. INPE, São José dos Campos, BR.

jb2008(instant::DateTime, ϕ_gd::Number, λ::Number, h::Number; kwargs...) -> JB2008Output
jb2008(
    instant::DateTime,
    ϕ_gd::Number,
    λ::Number,
    h::Number,
    F10::Number,
    F10ₐ::Number,
    S10::Number,
    S10ₐ::Number,
    M10::Number,
    M10ₐ::Number,
    Y10::Number,
    Y10ₐ::Number,
    DstΔTc::Number;
    kwargs...
) -> JB2008Output
jb2008(jd::Number, ϕ_gd::Number, λ::Number, h::Number; kwargs...) -> JB2008Output
jb2008(
    jd::Number,
    ϕ_gd::Number,
    λ::Number,
    h::Number,
    F10::Number,
    F10ₐ::Number,
    S10::Number,
    S10ₐ::Number,
    M10::Number,
    M10ₐ::Number,
    Y10::Number,
    Y10ₐ::Number,
    DstΔTc::Number;
    kwargs...
) -> JB2008Output

Compute the atmospheric density using the Jacchia-Bowman 2008 (JB2008) model.

This model is a product of the Space Environment Technologies, please, refer to the following website for more information:

https://spacewx.com/JB2008/

If we omit all space indices, the system tries to obtain them automatically for the selected day jd or instant. However, the indices must be already initialized using the function SpaceIndices.init().

Arguments

• jd::Number: Julian day to compute the model.
• instant::DateTime: Instant to compute the model represented using DateTime.
• ϕ_gd: Geodetic latitude [rad].
• λ: Longitude [rad].
• h: Altitude [m].
• F10: 10.7-cm solar flux [sfu] obtained 1 day before jd.
• F10ₐ: 10.7-cm averaged solar flux using a 81-day window centered on input time obtained 1 day before jd.
• S10: EUV index (26-34 nm) scaled to F10.7 obtained 1 day before jd.
• S10ₐ: EUV 81-day averaged centered index obtained 1 day before jd.
• M10: MG2 index scaled to F10.7 obtained 2 days before jd.
• M10ₐ: MG2 81-day averaged centered index obtained 2 days before jd.
• Y10: Solar X-ray & Ly-α index scaled to F10.7 obtained 5 days before jd.
• Y10ₐ: Solar X-ray & Ly-α 81-day averaged centered index obtained 5 days before jd.
• DstΔTc: Temperature variation related to the Dst.

Keywords

• verbose::Val: Set to Val(true) to emit debug messages related to the automatic space index fetching, or to Val(false) to suppress them. Notice that this keyword must be a Val object, not a Bool. (Default: Val(true))

Returns

• JB2008Output: Structure containing the results obtained from the model. Its element type is the promotion of the types of the numeric inputs.

jr1971(
    instant::DateTime,
    ϕ_gd::Number,
    λ::Number,
    h::Number;
    kwargs...
) -> JR1971Output
jr1971(
    instant::DateTime,
    ϕ_gd::Number,
    λ::Number,
    h::Number,
    F10::Number,
    F10ₐ::Number,
    Kp::Number;
    kwargs...
) -> JR1971Output
jr1971(jd::Number, ϕ_gd::Number, λ::Number, h::Number; kwargs...) -> JR1971Output
jr1971(
    jd::Number,
    ϕ_gd::Number,
    λ::Number,
    h::Number,
    F10::Number,
    F10ₐ::Number,
    Kp::Number;
    kwargs...
) -> JR1971Output

Compute the atmospheric density using the Jacchia-Roberts 1971 model.

If we omit all space indices, the system tries to obtain them automatically for the selected day jd or instant. However, the indices must be already initialized using the function SpaceIndices.init().

The function throws an ArgumentError if the altitude h is lower than 90 km.

Arguments

• jd::Number: Julian day to compute the model.
• instant::DateTime: Instant to compute the model represented using DateTime.
• ϕ_gd::Number: Geodetic latitude [rad].
• λ::Number: Longitude [rad].
• h::Number: Altitude [m].
• F10::Number: 10.7-cm solar flux [sfu].
• F10ₐ::Number: 10.7-cm averaged solar flux, 81-day centered on input time [sfu].
• Kp::Number: Kp geomagnetic index with a delay of 3 hours.

Keywords

• verbose::Val: Set to Val(true) to emit debug messages related to the automatic space index fetching, or to Val(false) to suppress them. Notice that this keyword must be a Val object, not a Bool. (Default: Val(true))
• roots_container::Union{Nothing, AbstractVector}: This keyword is not used anymore and is kept only for backward compatibility. The quartic polynomial roots are now computed using a closed-form algorithm that does not allocate. (Default: nothing)

Returns

• JR1971Output: Structure containing the results obtained from the model. Its element type is the promotion of the types of the numeric inputs.

nrlmsise00(
    instant::DateTime,
    h::Number,
    ϕ_gd::Number,
    λ::Number;
    kwargs...
) -> Nrlmsise00Output
nrlmsise00(
    instant::DateTime,
    h::Number,
    ϕ_gd::Number,
    λ::Number,
    F10ₐ::Number,
    F10::Number,
    ap::Union{Number, AbstractVector};
    kwargs...
) -> Nrlmsise00Output
nrlmsise00(
    jd::Number,
    h::Number,
    ϕ_gd::Number,
    λ::Number;
    kwargs...
) -> Nrlmsise00Output
nrlmsise00(
    jd::Number,
    h::Number,
    ϕ_gd::Number,
    λ::Number,
    F10ₐ::Number,
    F10::Number,
    ap::Union{Number, AbstractVector};
    kwargs...
) -> Nrlmsise00Output

Compute the atmospheric density using the NRLMSISE-00 model.

If we omit all space indices, the system tries to obtain them automatically for the selected day jd or instant. However, the indices must be already initialized using the function SpaceIndices.init().

Arguments

• instant::DateTime: Instant to compute the model represented using DateTime.
• jd::Number: Julian day to compute the model.
• h::Number: Altitude [m].
• ϕ_gd::Number: Geodetic latitude [rad].
• λ::Number: Longitude [rad].
• F10ₐ::Number: 10.7-cm averaged solar flux, 81-day centered on input time [sfu].
• F10::Number: 10.7-cm solar flux [sfu].
• ap::Union{Number, AbstractVector}: Magnetic index, see the section AP for more information.

Keywords

• flags::Nrlmsise00Flags: A list of flags to configure the model. For more information, see Nrlmsise00Flags. (Default: Nrlmsise00Flags())
• include_anomalous_oxygen::Bool: If true, the anomalous oxygen density will be included in the total density computation. (Default: true)
• P::Union{Nothing, Matrix}: If the user passes a matrix with dimensions equal to or greater than 8 × 4, it will be used when computing the Legendre associated functions, reducing allocations and improving the performance. If it is nothing, the matrix is allocated inside the function. (Default: nothing)
• verbose::Val: Set to Val(true) to emit debug messages related to the automatic space index fetching, or to Val(false) to suppress them. Notice that this keyword must be a Val object, not a Bool, and it is only available in the methods that fetch the space indices automatically. (Default: Val(true))

Returns

• Nrlmsise00Output: Structure containing the results obtained from the model. Its element type is the promotion of the types of the numeric inputs.

AP

The input variable ap contains the magnetic index. It can be a Number or an AbstractVector.

If ap is a number, it must contain the daily magnetic index.

If ap is an AbstractVector, it must be a vector with 7 elements as described below:

Extended help

1. The densities of O, H, and N are set to 0 below 72.5 km.
2. The exospheric temperature is set to global average for altitudes below 120 km. The 120 km gradient is left at global average value for altitudes below 72.5 km.
3. Anomalous oxygen is defined as hot atomic oxygen or ionized oxygen that can become appreciable at high altitudes (> 500 km) for some ranges of inputs, thereby affecting drag on satellites and debris. We group these species under the term Anomalous Oxygen, since their individual variations are not presently separable with the drag data used to define this model component.

Notes on Input Variables

F10 and F10ₐ values used to generate the model correspond to the 10.7 cm radio flux at the actual distance of the Earth from the Sun rather than the radio flux at 1 AU. The following site provides both classes of values:

ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SOLAR_RADIO/FLUX/

F10, F10ₐ, and ap effects are neither large nor well established below 80 km and these parameters should be set to 150, 150, and 4 respectively.

If include_anomalous_oxygen is false, the total_density field in the output is the sum of the mass densities of the species He, O, N₂, O₂, Ar, H, and N, but does not include anomalous oxygen.

If include_anomalous_oxygen is true, the total_density field in the output is the effective total mass density for drag and is the sum of the mass densities of all species in this model including the anomalous oxygen.