#tle #aerospace #omm #sdp4

no-std sgp4

A pure Rust implementation of the SGP4 algorithm for satellite propagation

22 releases (6 stable)

2.2.0 May 25, 2024
2.1.0 Nov 16, 2023
1.2.2 Mar 5, 2023
0.9.1 Jan 30, 2023
0.3.1 Jul 16, 2020

#47 in Algorithms

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The SGP4 algorithm, ported to Rust from the reference Celestrak implementation [1].

The code was entirely refactored to leverage Rust's algebraic data types and highlight the relationship between the implementation and the reference mathematical equations [2].

SGP4 can be called from JavaScript or Python via WebAssembly wrappers. See https://github.com/wasmerio/sgp4 to install and use SGP4 as a WAPM package.

The numerical predictions are almost identical to Celestrak's. The observed differences (less than 2 × 10⁻⁷ km for the position and 10⁻⁹ km.s⁻¹ for the velocity three and a half years after the epoch) are well below the accuracy of the algorithm.

We drew inspiration from the incomplete https://github.com/natronics/rust-sgp4 to write mathematical expressions using UTF-8 characters.

Documentation

The code documentation is hosted at https://docs.rs/sgp4/latest/sgp4/.

Examples can be found in this repository's examples directory:

  • examples/celestrak.rs retrieves the most recent Galileo OMMs from Celestrak and propagates them
  • examples/omm.rs parses and propagates a JSON-encoded OMM
  • examples/space-track.rs retrieves the 20 most recent launches OMMs from Space-Track and propagates them
  • examples/tle.rs parses and propagates a TLE
  • examples/tle_afspc.rs parses and propagates a TLE using the AFSPC compatibility mode
  • examples/advanced.rs leverages the advanced API to (marginally) accelerate the propagation of deep space resonant satellites

To run an example (here examples/celestrak.rs), use:

cargo run --example celestrak

To run the Space-Track example, you must first assign your Space-Track.org credentials to the fields identity and password (see lines 3 and 4 in examples/space-track.rs).

Environments without std or alloc

This crate supports no_std environments. TLE parsing and SGP4 propagation do not require alloc either. We use num-traits with libm for floating point functions when std is not available.

All serde-related features, in particular OMM parsing, require alloc.

Benchmark

The benchmark code is available at https://github.com/neuromorphicsystems/sgp4-benchmark. It compares two SGP4 implementations in different configurations:

  • cpp: the Celestrak implementation [1] in improved mode
  • cpp-afspc: the Celestrak implementation [1] in AFSPC compatibility mode
  • cpp-fastmath: the Celestrak implementation [1] in improved mode with the fast-math compiler flag
  • cpp-afspc-fastmath: the Celestrak implementation [1] in AFSPC compatibility mode with the fast-math compiler flag
  • rust: our Rust implementation in default mode
  • rust-afspc: our Rust implementation in AFSPC compatibility mode

This benchmark must not be confused with the code in this repository's bench directory. The latter considers only a small subset of the Celestrak catalogue (the tests recommended in [1]) and does not measure the original C++ implementation.

The present results were obtained using a machine with the following configuration:

  • CPU - Intel Core i7-8700 @ 3.20GHz
  • RAM - Kingston DDR4 @ 2.667 GHz
  • OS - Ubuntu 16.04
  • Compilers - Rust 1.44.1 and gcc 9.3.0

Accuracy measures the maximum propagation error of each implementation with respect to the reference implementation (cpp-afspc) over the full Celestrak catalogue (1 minute timestep over 24 hours).

implementation maximum position error maximum speed error
cpp-afspc (reference) (reference)
cpp 1.05 km 1.30 × 10⁻³ km.s⁻¹
cpp-fastmath 1.05 km 1.30 × 10⁻³ km.s⁻¹
cpp-afspc-fastmath 4.21 × 10⁻⁸ km 7.51 × 10⁻¹² km.s⁻¹
rust 1.05 km 1.30 × 10⁻³ km.s⁻¹
rust-afspc 4.19 × 10⁻⁸ km 7.46 × 10⁻¹² km.s⁻¹

The Rust and C++ fast-math errors have the same order of magnitude. In both cases, they can be attributed to mathematically identical expressions implemented with different floating-point operations.

Speed measures the time it takes to propagate every satellite in the Celestrak catalogue (1 minute timestep over 24 hours) using a single thread. 100 values are sampled per implementation.

implementation minimum Q1 median Q3 maximum relative difference
cpp-afspc 8.95 s 9.02 s 9.03 s 9.06 s 9.18 s (reference)
cpp 8.95 s 9.01 s 9.04 s 9.06 s 9.25 s + 0 %
cpp-fastmath 7.67 s 7.74 s 7.77 s 7.79 s 7.90 s - 14 %
cpp-afspc-fastmath 7.70 s 7.74 s 7.76 s 7.79 s 7.86 s - 14 %
rust 8.36 s 8.41 s 8.43 s 8.45 s 8.53 s - 7 %
rust-afspc 8.36 s 8.41 s 8.43 s 8.46 s 8.59 s - 7 %

Rust fast-math support is a work in progress (see https://github.com/rust-lang/rust/issues/21690). Similarly to C++, it should have a very small impact on accuracy while providing a substantial speed gain.

Variables and mathematical expressions

Variables

Each variable is used to store the result of one and only one expression. Most variables are immutable, with the exception of the variable (E + ω) used to solve Kepler's equation and the state variables tᵢ, nᵢ and λᵢ used to integrate the resonance effects of Earth gravity.

The following tables list the variables used in the code and their associated mathematical symbol. Where possible, we used symbols from [2]. Partial expressions without a name in [2] follow the convention kₙ, n ∈ ℕ if they are shared between initialization and propagation, and pₙ, n ∈ ℕ if they are local to initialization or propagation.

  1. Initialization variables
  2. Propagation variables
  3. Third-body initialization variables
  4. Third-body propagation variables

Initialization variables

The following variables depend solely on epoch elements.

variable symbol description
Elements::datetime.year() yᵤ Gregorian calendar year
Elements::datetime.month() mᵤ Gregorian calendar month in the range [1, 12]
Elements::datetime.day() dᵤ Gregorian calendar day in the range [1, 31]
Elements::datetime.hour() hᵤ Hours since midnight in the range [0, 23]
Elements::datetime.minute() minᵤ Minutes since the hour in the range [0, 59]
Elements::datetime.second() sᵤ Seconds since the minute in the range [0, 59]
Elements::datetime.nanosecond() nsᵤ Nanoseconds since the second in the range [0, 10[
epoch y₂₀₀₀ Julian years since UTC 1 January 2000 12h00 (J2000)
d1900 d₁₉₀₀ Julian days since UTC 1 January 1900 12h00 (J1900)
d1970 d₁₉₇₀ Julian days since UTC 1 January 1970 12h00 (J1970)
c2000 c₂₀₀₀ Julian centuries since UTC 1 January 2000 12h00 (J2000)
geopotential.ae aₑ equatorial radius of the earth in km
geopotential.ke kₑ square root of the earth's gravitational parameter in earth radii³ min⁻²
geopotential.j2 J₂ un-normalised second zonal harmonic
geopotential.j3 J₃ un-normalised third zonal harmonic
geopotential.j4 J₄ un-normalised fourth zonal harmonic
kozai_mean_motion n₀ mean number of orbits per day (Kozai convention) at epoch in rad.min⁻¹
a1 a₁ semi-major axis at epoch (Kozai convention)
p0 p₀ partial expression of 𝛿₀ and 𝛿₁
d1 𝛿₁ used in the Kozai to Brouwer conversion
d0 𝛿₀ used in the Kozai to Brouwer conversion
B* B* radiation pressure coefficient in earth radii⁻¹
orbit_0.inclination I₀ angle between the equator and the orbit plane at epoch in rad
orbit_0.right_ascension Ω₀ angle between vernal equinox and the point where the orbit crosses the equatorial plane at epoch in rad
orbit_0.eccentricity e₀ shape of the orbit at epoch
orbit_0.argument_of_perigee ω₀ angle between the ascending node and the orbit's point of closest approach to the earth at epoch in rad
orbit_0.mean_anomaly M₀ angle of the satellite location measured from perigee at epoch in rad
orbit_0.mean_motion n₀" mean number of orbits per day (Brouwer convention) at epoch in rad.min⁻¹
p1 p₁ cosine of the inclination at epoch used in multiple expressions during initialization (θ in [2], renamed to avoid confusion with the sidereal time)
p2 p₂ partial expression of multiple initialization expressions
a0 a₀" semi-major axis at epoch (Brouwer convention)
p3 p₃ perigee in earth radii
p4 p₄ height of perigee in km
p5 p₅ partial expression of s
s s altitude parameter of the atmospheric drag expression
p6 p₆ partial expression of the atmospheric drag
xi ξ partial expression of multiple initialization expressions
p7 p₇ partial expression of multiple initialization expressions
eta η partial expression of multiple initialization expressions and of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation
p8 p₈ partial expression of multiple initialization expressions
p9 p₉ partial expression of multiple initialization expressions
c1 C₁ partial expression of multiple initialization and propagation expressions
p10 p₁₀ partial expression of multiple initialization expressions
b0 β₀ partial expression of multiple initialization expressions
p11 p₁₁ partial expression of multiple initialization expressions
p12 p₁₂ partial expression of multiple initialization expressions
p13 p₁₃ partial expression of multiple initialization expressions
p14 p₁₄ partial expression of multiple initialization expressions
p15 p₁₅ partial expression of multiple initialization expressions
k14 k₁₄ first order coefficient of the argument of perigee before adding solar and lunar perturbations
c4 C₄ partial expression of multiple initialization and propagation expressions (differs from the C₄ expression in [2] by a factor B*)
right_ascension_dot Ω̇ first order coefficient of the right ascension
argument_of_perigee_dot ω̇ first order coefficient of the argument of perigee
mean_anomaly_dot first order coefficient of the mean anomaly
k0 k₀ second order coefficient of the right ascension before adding perturbations
k1 k₁ partial expression of the second order coefficient of the mean anomaly
k2 k₂ partial expression of aᵧₙ in near earth propagation
k3 k₃ partial expression of rₖ, ṙₖ and rḟₖ in near earth propagation
k4 k₄ partial expression of uₖ in near earth propagation
k5 k₅ partial expression of the initial Kepler variable p₃₈ in near earth propagation
k6 k₆ partial expression of multiple initialization expressions and of rₖ and rḟₖ in near earth propagation
d2 D₂ partial expression of multiple near earth initialization expressions and of the semi-major axis in near earth propagation
p16 p₁₆ partial expression of multiple near earth initialization expressions
d3 D₃ partial expression of multiple near earth initialization expressions and of the semi-major axis in near earth propagation
d4 D₄ partial expression of multiple near earth initialization expressions and of the semi-major axis in near earth propagation
c5 C₅ partial expression of multiple initialization and propagation expressions (differs from the C₅ expression in [2] by a factor B*)
k7 k₇ sine of the mean anomaly at epoch
k8 k₈ partial expression of the mean anomaly third order coefficient in high altitude near earth propagation
k9 k₉ partial expression of the mean anomaly fourth order coefficient in high altitude near earth propagation
k10 k₁₀ partial expression of the mean anomaly fifth order coefficient in high altitude near earth propagation
k11 k₁₁ partial expression of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation
k12 k₁₂ partial expression of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation
k13 k₁₃ partial expression of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation
lunar_right_ascension_epsilon Ωₗₑ lunar right ascension of the ascending node
lunar_right_ascension_sine sin Ωₗ sine of the lunar right ascension of the ascending node referred to the equator
lunar_right_ascension_cosine cos Ωₗ cosine of the lunar right ascension of the ascending node referred to the equator
lunar_argument_of_perigee ωₗ lunar argument of perigee
sidereal_time_0 θ₀ Greenwich sidereal time at epoch
lambda_0 λ₀ Earth gravity resonance variable at epoch
lambda_dot_0 λ̇₀ time derivative of the Earth gravity resonance variable at epoch
p17 p₁₇ partial expression of 𝛿ᵣ₁, 𝛿ᵣ₂ and 𝛿ᵣ₃
dr1 𝛿ᵣ₁ first Earth gravity resonance coefficient for geosynchronous orbits (𝛿₁ in [2], renamed to avoid confusion with 𝛿₁ used in the Kozai to Brouwer conversion)
dr2 𝛿ᵣ₂ second Earth gravity resonance coefficient for geosynchronous orbits (𝛿₂ in [2], renamed to match 𝛿ᵣ₁)
dr3 𝛿ᵣ₃ third Earth gravity resonance coefficient for geosynchronous orbits (𝛿₃ in [2], renamed to match 𝛿ᵣ₁)
p18 p₁₈ partial expression of D₂₂₀₋₁ and D₂₂₁₁
p19 p₁₉ partial expression of D₃₂₁₀ and D₃₂₂₂
p20 p₂₀ partial expression of D₄₄₁₀ and D₄₄₂₂
p21 p₂₁ partial expression of D₅₂₂₀, D₅₂₃₂, D₅₄₂₁ and D₅₄₃₃
f220 F₂₂₀ partial expression of D₂₂₀₋₁ and D₄₄₁₀
g211 G₂₁₁ partial expression of D₂₂₁₁
g310 G₃₁₀ partial expression of D₃₂₁₀
g322 G₃₂₂ partial expression of D₃₂₂₂
g410 G₄₁₀ partial expession of D₄₄₁₀
g422 G₄₂₂ partial expession of D₄₄₂₂
g520 G₅₂₀ partial expression of D₅₂₂₀
g532 G₅₃₂ partial expression of D₅₂₃₂
g521 G₅₂₁ partial expression of D₅₄₂₁
g533 G₅₃₃ partial expression of D₅₄₃₃
d220₋1 D₂₂₀₋₁ gravity resonance coefficient for Molniya orbits (the Dₗₘₚₖ expression in [2] is missing a factor l - 2p + k from the original equation in [4] with k = -1 instead of 1)
d2211 D₂₂₁₁ gravity resonance coefficient for Molniya orbits (the Dₗₘₚₖ expression in [2] is missing a factor l - 2p + k from the original equation in [4])
d3210 D₃₂₁₀ see D₂₂₁₁
d3222 D₃₂₂₂ see D₂₂₁₁
d4410 D₄₄₁₀ see D₂₂₁₁
d4422 D₄₄₂₂ see D₂₂₁₁
d5220 D₅₂₂₀ see D₂₂₁₁
d5232 D₅₂₃₂ see D₂₂₁₁
d5421 D₅₄₂₁ see D₂₂₁₁
d5433 D₅₄₃₃ see D₂₂₁₁

Propagation variables

The following expressions depend on the propagation time t.

variable symbol description
t t minutes elapsed since epoch (can be negative)
p22 p₂₂ right ascension of the ascending node with neither Earth gravity resonance nor Sun and Moon contributions
p23 p₂₃ argument of perigee with neither high altitude drag effects, Earth gravity resonance nor Sun and Moon contributions
orbit.inclination I inclination at epoch plus t without the short-period effects of Earth gravity
orbit.right_ascension Ω right ascension of the ascending node at epoch plus t without the short-period effects of Earth gravity
orbit.eccentricity e eccentricity at epoch plus t without the short-period effects of Earth gravity
orbit.argument_of_perigee ω argument of perigee at epoch plus t without the short-period effects of Earth gravity
orbit.mean_anomaly M mean anomaly at epoch plus t without the short-period effects of Earth gravity
orbit.mean_motion n mean motion at epoch plus t without the short-period effects of Earth gravity
a a semi-major axis
p32 p₃₂ partial expression of aᵧₙ
p33 p₃₃ partial expression of rₖ, ṙₖ and rḟₖ
p34 p₃₄ partial expression of uₖ
p35 p₃₅ partial expression of the initial Kepler variable p₃₈
p36 p₃₆ partial expression of rₖ and rḟₖ
p37 p₃₇ partial expression of aᵧₙ and the initial Kepler variable p₃₈
axn aₓₙ normalized linear eccentricity projected on the line of nodes
ayn aᵧₙ normalized linear eccentricity projected on the normal to the line of nodes
p38 p₃₈ initial Kepler variable (U in [2], renamed to avoid confusion with the true anomaly plus argument of perigee u)
ew (E + ω) Kepler variable used in an iterative process to estimate the eccentric anomaly E
delta Δ(E + ω) correction to the Kepler variable at iteration i
p39 p₃₉ eccentricity at epoch plus t
pl pₗ semi-latus rectum
p40 p₄₀ normalized linear eccentricity projected on the semi-minor axis
r r radius (distance to the focus) without the short-period effects of Earth gravity
r_dot radius time derivative without the short-period effects of Earth gravity
b β semi-minor axis over semi-major axis
p41 p₄₁ partial expression of p₄₂ and p₄₃
p42 p₄₂ sine of u
p43 p₄₃ cosine of u
u u true anomaly plus argument of perigee without the short-period effects of Earth gravity
p44 p₄₄ sin(2 u), partial expression of uₖ, Ωₖ and ṙₖ
p45 p₄₅ cos(2 u), partial expression of rₖ, Iₖ and rḟₖ
p46 p₄₆ partial expression of rₖ, uₖ, Iₖ and Ωₖ
rk rₖ radius (distance to the focus)
uk uₖ true anomaly plus argument of perigee
inclination_k Iₖ inclination at epoch plus t
right_ascension_k Ωₖ right ascension at epoch plus t
rk_dot ṙₖ radius time derivative
rfk_dot rḟₖ radius times the true anomaly derivative
u0 u₀ x component of the position unit vector
u1 u₁ y component of the position unit vector
u2 u₂ z component of the position unit vector
prediction.position[0] r₀ x component of the position vector in km (True Equator, Mean Equinox (TEME) of epoch reference frame)
prediction.position[1] r₁ y component of the position vector in km (True Equator, Mean Equinox (TEME) of epoch reference frame)
prediction.position[2] r₂ z component of the position vector in km (True Equator, Mean Equinox (TEME) of epoch reference frame)
prediction.velocity[0] ṙ₀ x component of the velocity vector in km.s⁻¹ (True Equator, Mean Equinox (TEME) of epoch reference frame)
prediction.velocity[1] ṙ₁ y component of the velocity vector in km.s⁻¹ (True Equator, Mean Equinox (TEME) of epoch reference frame)
prediction.velocity[2] ṙ₂ z component of the velocity vector in km.s⁻¹ (True Equator, Mean Equinox (TEME) of epoch reference frame)
p24 p₂₄ mean anomaly without drag contributions in near earth propagation
p25 p₂₅ partial expression of ω and M in near earth propagation
p26 p₂₆ mean anomaly with elliptic correction and without drag contributions in near earth propagation
p27 p₂₇ non-clamped eccentricity in near earth propagation
p28 p₂₈ semi-major axis with resonance correction in deep space propagation
p29 p₂₉ mean anomaly with resonance correction in deep space propagation
p31 p₃₁ non-clamped eccentricity in deep space propagation
sidereal_time θ sidereal time at epoch plus t
delta_t Δt time step used in the integration of resonance effects of Earth gravity in min (either 720 or -720)
lambda_dot λ̇ᵢ resonance effects of Earth gravity variable's time derivative at epoch plus i Δt
ni_dot ṅᵢ mean motion time derivative at epoch plus i Δt
ni_ddot n̈ᵢ mean motion second time derivative at epoch plus i Δt
ResonanceState::t tᵢ resonance effects of Earth gravity integrator time (i Δt)
ResonanceState::mean_motion nᵢ mean motion time derivative at epoch plus Δt i
ResonanceState::lambda λᵢ resonance effects of Earth gravity variable at epoch plus i Δt
p30 p₃₀ non-normalised Ω in Lyddane deep space propagation

Third-body initialization variables

The contribution of the Sun and the Moon to the orbital elements are calculated with a unique set of expressions. src/third_body.rs provides a generic implementation of these expressions. Variables specific to the third body (either the Sun or the Moon) are annotated with x. In every other file, these variables are annotated with s if they correspond to solar perturbations, and l if they correspond to lunar perturbations.

The aₓₙ, Xₓₙ, Zₓₙ (n ∈ ℕ), Fₓ₂ and Fₓ₃ variables correspond to the aₙ, Xₙ, Zₙ, F₂ and F₃ variables in [2]. The added x highlights the dependence on the perturbing third body.

The following variables depend solely on epoch elements.

variable symbol description
third_body_inclination_sine sin Iₓ sine of the inclination of the Sun (sin Iₛ) or the Moon (sin Iₗ)
third_body_inclination_cosine cos Iₓ cosine of the inclination of the Sun (cos Iₛ) or the Moon (cos Iₗ)
delta_right_ascension_sine sin(Ω₀ - Ωₓ) sine of the difference between the right ascension of the ascending node of the satellite at epoch and the Sun's (sin(Ω₀ - Ωₛ)) or the Moon's (sin(Ω₀ - Ωₗ))
delta_right_ascension_cosine cos(Ω₀ - Ωₓ) cosine of the difference between the right ascension of the ascending node of the satellite at epoch and the Sun's (cos(Ω₀ - Ωₛ)) or the Moon's (cos(Ω₀ - Ωₗ))
third_body_argument_of_perigee_sine sin ωₓ sine of the argument of perigee of the Sun (sin ωₛ) or the Moon (sin ωₗ)
third_body_argument_of_perigee_cosine cos ωₓ cosine of the argument of perigee of the Sun (sin ωₛ) or the Moon (cos ωₗ)
third_body_mean_anomaly_0 Mₓ₀ mean anomaly at epoch of the Sun (Mₛ₀) or the Moon (Mₗ₀)
ax1 aₓ₁ partial expression of multiple Xₓₙ and Zₓₙ expressions
ax3 aₓ₃ partial expression of multiple Xₓₙ and Zₓₙ expressions
ax7 aₓ₇ partial expression of multiple aₓ₂ and aₓ₅
ax8 aₓ₈ partial expression of multiple aₓ₂ and aₓ₅
ax9 aₓ₉ partial expression of multiple aₓ₄ and aₓ₆
ax10 aₓ₁₀ partial expression of multiple aₓ₄ and aₓ₆
ax2 aₓ₂ partial expression of multiple Xₓₙ and Zₓₙ expressions
ax4 aₓ₄ partial expression of multiple Xₓₙ and Zₓₙ expressions
ax5 aₓ₅ partial expression of multiple Xₓₙ and Zₓₙ expressions
ax6 aₓ₆ partial expression of multiple Xₓₙ and Zₓₙ expressions
xx1 Xₓ₁ partial expression of multiple Zₓₙ expressions, kₓ₀, kₓ₁ and ėₓ
xx2 Xₓ₂ partial expression of multiple Zₓₙ expressions, kₓ₀, kₓ₁ and ėₓ
xx3 Xₓ₃ partial expression of multiple Zₓₙ expressions, kₓ₀, kₓ₁ and ėₓ
xx4 Xₓ₄ partial expression of multiple Zₓₙ expressions, kₓ₀, kₓ₁ and ėₓ
xx5 Xₓ₅ partial expression of multiple Zₓₙ expressions
xx6 Xₓ₆ partial expression of multiple Zₓₙ expressions
xx7 Xₓ₇ partial expression of multiple Zₓₙ expressions
xx8 Xₓ₈ partial expression of multiple Zₓₙ expressions
zx31 Zₓ₃₁ partial expression of Zₓ₃, kₓ₈ and ω̇ₓ
zx32 Zₓ₃₂ partial expression of Zₓ₂, kₓ₇ and ω̇ₓ
zx33 Zₓ₃₃ partial expression of Zₓ₃, kₓ₈ and ω̇ₓ
zx11 Zₓ₁₁ partial expression of kₓ₃ and İₓ
zx13 Zₓ₁₃ partial expression of kₓ₃ and İₓ
zx21 Zₓ₂₁ partial expression of kₓ₁₁ and Ω̇ₓ
zx23 Zₓ₂₃ partial expression of kₓ₁₁ and Ω̇ₓ
zx1 Zₓ₁ partial expression of kₓ₅ and Ṁₓ
zx3 Zₓ₃ partial expression of kₓ₅ and Ṁₓ
px0 pₓ₀ partial expression of multiple kₓₙ expressions and Ṁₓ
px1 pₓ₁ partial expression of multiple kₓₙ expressions and İₓ
px2 pₓ₂ partial expression of multiple kₓₙ expressions and ω̇ₓ
px3 pₓ₃ partial expression of multiple kₓₙ expressions and ėₓ
kx0 kₓ₀ Fₓ₂ coefficient of δeₓ
kx1 kₓ₁ Fₓ₃ coefficient of δeₓ
kx2 kₓ₂ Fₓ₂ coefficient of δIₓ
kx3 kₓ₃ Fₓ₃ coefficient of δIₓ
kx4 kₓ₄ Fₓ₂ coefficient of δMₓ
kx5 kₓ₅ Fₓ₃ coefficient of δMₓ
kx6 kₓ₆ sin fₓ coefficient of δMₓ
kx7 kₓ₇ Fₓ₂ coefficient of pₓ₄
kx8 kₓ₈ Fₓ₃ coefficient of pₓ₄
kx9 kₓ₉ sin fₓ coefficient of pₓ₄
kx10 kₓ₁₀ Fₓ₂ coefficient of pₓ₅
kx11 kₓ₁₁ Fₓ₃ coefficient of pₓ₅
third_body_dots.inclination İₓ secular contribution of the Sun (İₛ) or the Moon (İₗ) to the inclination
third_body_right_ascension_dot Ω̇ₓ secular contribution of the Sun (Ω̇ₛ) or the Moon (Ω̇ₗ) to the right ascension of the ascending node
third_body_dots.eccentricity ėₓ secular contribution of the Sun (ėₛ) or the Moon (ėₗ) to the eccentricity
third_body_dots.agument_of_perigee ω̇ₓ secular contribution of the Sun (ω̇ₛ) or the Moon (ω̇ₗ) to the argument of perigee
third_body_dots.mean_anomaly Ṁₓ secular contribution of the Sun (Ṁₛ) or the Moon (Ṁₗ) to the mean anomaly

Third-body propagation variables

The following variables depend on the propagation time t.

variable symbol description
third_body_mean_anomaly Mₓ mean anomaly of the Sun (Mₛ) or the Moon (Mₗ)
fx fₓ third body true anomaly
fx2 Fₓ₂ partial expression of the third body long-period periodic contribution
fx3 Fₓ₃ partial expression of the third body long-period periodic contribution
third_body_delta_eccentricity δeₓ long-period periodic contribution of the Sun (δeₛ) or the Moon (δeₗ) to the eccentricity
third_body_delta_inclination δIₓ long-period periodic contribution of the Sun (δIₛ) or the Moon (δIₗ) to the inclination
third_body_delta_mean_mootion δMₓ long-period periodic contribution of the Sun (δMₛ) or the Moon (δMₗ) to the mean motion
px4 pₓ₄ partial expression of the long-period periodic contribution of the Sun (pₛ₄) or the Moon (pₗ₄) to the right ascension of the ascending node and the argument of perigee
px5 pₓ₅ partial expression of the long-period periodic contribution of the Sun (pₛ₅) or the Moon (pₗ₅) to the right ascension of the ascending node

Mathematical expressions

UT1 to Julian conversion

The epoch (Julian years since UTC 1 January 2000 12h00) can be calculated with either the AFSPC formula:

y₂₀₀₀ = (367 yᵤ -7 (yᵤ +(mᵤ + 9) / 12) / 4+ 275 ⌊mᵤ / 9+ dᵤ
        + 1721013.5
        + (((nsᵤ / 10+ sᵤ) / 60 + minᵤ) / 60 + hᵤ) / 24
        - 2451545)
        / 365.25

or the more accurate version of the same formula:

y₂₀₀₀ = (367 yᵤₜ₁ -7 (yᵤₜ₁ +(mᵤₜ₁ + 9) / 12) / 4+ 275 ⌊mᵤₜ₁ / 9+ dᵤₜ₁ - 730531) / 365.25
        + (3600 hᵤₜ₁ + 60 minᵤₜ₁ + sᵤₜ₁ - 43200) / (24 × 60 × 60 × 365.25)
        + nsᵤₜ₁ / (24 × 60 × 60 × 365.25 × 10)

Common initialization

a₁ = (kₑ / n₀)²ᐟ³

      3      3 cos²I₀ - 1
 p₀ = - J₂ ---------------
      4      (1 − e₀²)³ᐟ²

𝛿₁ = p₀ / a₁²

𝛿₀ = p₀ / (a₁ (1 - ¹/₃ 𝛿₁ - 𝛿₁² - ¹³⁴/₈₁ 𝛿₁³))²

n₀" = n₀ / (1 + 𝛿₀)

p₁ = cos I₀

p₂ = 1 − e₀²

k₆ = 3 p₁² - 1

a₀" = (kₑ / n₀")²ᐟ³

p₃ = a₀" (1 - e₀)

p₄ = aₑ (p₃ - 1)

p₅ =20      if p₄ < 98
     │ p₄ - 78 if 98 ≤ p₄ < 15678      otherwise

s = p₅ / aₑ + 1

p₆ = ((120 - p₅) / aₑ)⁴

ξ = 1 / (a₀" - s)

p₇ = p₆ ξ⁴

η = a₀" e₀ ξ

p₈ = |1 - η²|

p₉ = p₇ / p₈⁷ᐟ²

C₁ = B* p₉ n₀" (a₀" (1 + ³/₂ η² + e₀ η (4 + η²))
     + ³/₈ J₂ ξ k₆ (8 + 3 η² (8 + η²)) / p₈)

p₁₀ = (a₀" p₂)⁻²

β₀ = p₂¹ᐟ²

p₁₁ = ³/₂ J₂ p₁₀ n₀"

p₁₂ = ¹/₂ p₁₁ J₂ p₁₀

p₁₃ = - ¹⁵/₃₂ J₄ p₁₀² n₀"

p₁₄ = - p₁₁ p₁ + (¹/₂ p₁₂ (4 - 19 p₁²) + 2 p₁₃ (3 - 7 p₁²)) p₁

k₁₄ = - ¹/₂ p₁₁ (1 - 5 p₁²) + ¹/₁₆ p₁₂ (7 - 114 p₁² + 395 p₁⁴)

p₁₅ = n₀" + ¹/₂ p₁₁ β₀ k₆ + ¹/₁₆ p₁₂ β₀ (13 - 78 p₁² + 137 p₁⁴)

C₄ = 2 B* n₀" p₉ a₀" p₂ (
     η (2 + ¹/₂ η²)
     + e₀ (¹/+ 2 η²)
     - J₂ ξ / (a p₈) (-3 k₆ (1 - 2 e₀ η + η² (³/- ¹/₂ e₀ η))
     + ³/(1 - p₁²) (2 η² - e₀ η (1 + η²)) cos 2 ω₀)

k₀ =/₂ p₂ p₁₁ p₁ C₁

k₁ = ³/₂ C₁

Ω̇ = │ p₁₄            if n₀" > 2π / 225
    │ p₁₄ + (Ω̇ₛ + Ω̇ₗ) otherwise

ω̇ = │ k₁₄            if n₀" >/ 225
    │ k₁₄ + (ω̇ₛ + ω̇ₗ) otherwise

Ṁ = │ p₁₅            if n₀" > 2π / 225
    │ p₁₅ + (Ṁₛ + Ṁₗ) otherwise

Near earth initialization

Defined only if n₀" > 2π / 225 (near earth).

       1 J₃
k₂ = - - -- sin I₀
       2 J₂

k₃ = 1 - p₁²

k₄ = 7 p₁² - 11 J₃        3 + 5 p₁
k₅ =- - -- sin I₀ --------    if |1 + p₁| > 1.5 × 10⁻¹²
     │   4 J₂         1 + p₁
     │   1 J₃         3 + 5 p₁
     │ - - -- sin I₀ ----------- otherwise
     │   4 J₂        1.5 × 10⁻¹²

High altitude near earth initialization

Defined only if n₀" > 2π / 225 (near earth) and p₃ ≥ 220 / (aₑ + 1) (high altitude).

D₂ = 4 a₀" ξ C₁²

p₁₆ = D₂ ξ C₁ / 3

D₃ = (17 a + s) p₁₆

D₄ = ¹/₂ p₁₆ a₀" ξ (221 a₀" + 31 s) C₁

C₅ = 2 B* p₉ a₀" p₂ (1 + 2.75 (η² + η e₀) + e₀ η³)

k₁₁ = (1 + η cos M₀)³

k₇ = sin M₀

k₈ = D₂ + 2 C₁²

k₉ = ¹/(3 D₃ + C₁ (12 D₂ + 10 C₁²))

k₁₀ = ¹/(3 D₄ + 12 C₁ D₃ + 6 D₂² + 15 C₁² (2 D₂ + C₁²))

Elliptic high altitude near earth initialization

Defined only if n₀" > 2π / 225 (near earth), p₃ ≥ 220 / (aₑ + 1) (high altitude) and e₀ > 10⁻⁴ (elliptic).

                    J₃ p₇ ξ  n₀" sin I₀
k₁₂ = - 2 B* cos ω₀ -- ----------------
                    J₂        e₀

        2 p₇ B*
k₁₃ = - - -----
        3 e₀ η

Deep space initialization

Defined only if n₀" ≤ 2π / 225 (deep space).

e₁₉₀₀ = 365.25 (t₀ + 100)

sin Iₛ = 0.39785416

cos Iₛ = 0.91744867

sin(Ω₀ - Ωₛ) = sin Ω₀

cos(Ω₀ - Ωₛ) = cos Ω₀

sin ωₛ = -0.98088458

cos ωₛ = 0.1945905

Mₛ₀ = (6.2565837 + 0.017201977 e₁₉₀₀) rem 2π

Ωₗₑ = 4.523602 - 9.2422029 × 10⁻⁴ e₁₉₀₀ rem 2π

cos Iₗ = 0.91375164 - 0.03568096 Ωₗₑ

sin Iₗ = (1 - cos²Iₗ)¹ᐟ²

sin Ωₗ = 0.089683511 sin Ωₗₑ / sin Iₗ

cos Ωₗ = (1 - sin²Ωₗ)¹ᐟ²

ωₗ = 5.8351514 + 0.001944368 e₁₉₀₀
                    0.39785416 sin Ωₗₑ / sin Iₗ
     + tan⁻¹ ------------------------------------------ - Ωₗₑ
             cos Ωₗ cos Ωₗₑ + 0.91744867 sin Ωₗ sin Ωₗₑ

sin(Ω₀ - Ωₗ) = sin Ω₀ cos Ωₗ - cos Ω₀ sin Ωₗ

cos(Ω₀ - Ωₗ) = cos Ωₗ cos Ω₀ + sin Ωₗ sin Ω₀

Mₗ₀ = (-1.1151842 + 0.228027132 e₁₉₀₀) rem 2π

Third body perturbations

Defined only if n₀" ≤ 2π / 225 (deep space).

The following variables are evaluated for two third bodies, the Sun (solar perturbations s) and the Moon (lunar perturbations l). Variables specific to the third body are annotated with x. In other sections, x is either s or l.

aₓ₁ = cos ωₓ cos(Ω₀ - Ωₓ) + sin ωₓ cos Iₓ sin(Ω₀ - Ωₓ)

aₓ₃ = - sin ωₓ cos(Ω₀ - Ωₓ) + cos ωₓ cos Iₓ sin(Ω₀ - Ωₓ)

aₓ₇ = - cos ωₓ sin(Ω₀ - Ωₓ) + sin ωₓ cos Iₓ cos(Ω₀ - Ωₓ)

aₓ₈ = sin ωₓ sin Iₓ

aₓ₉ = sin ωₓ sin(Ω₀ - Ωₓ) + cos ωₓ cos Iₓ cos(Ω₀ - Ωₓ)

aₓ₁₀ = cos ωₓ sin Iₓ

aₓ₂ = aₓ₇ cos i₀ + aₓ₈ sin I₀

aₓ₄ = aₓ₉ cos i₀ + aₓ₁₀ sin I₀

aₓ₅ = - aₓ₇ sin I₀ + aₓ₈ cos I₀

aₓ₆ = - aₓ₉ sin I₀ + aₓ₁₀ cos I₀

Xₓ₁ = aₓ₁ cos ω₀ + aₓ₂ sin ω₀

Xₓ₂ = aₓ₃ cos ω₀ + aₓ₄ sin ω₀

Xₓ₃ = - aₓ₁ sin ω₀ + aₓ₂ cos ω₀

Xₓ₄ = - aₓ₃ sin ω₀ + aₓ₄ cos ω₀

Xₓ₅ = aₓ₅ sin ω₀

Xₓ₆ = aₓ₆ sin ω₀

Xₓ₇ = aₓ₅ cos ω₀

Xₓ₈ = aₓ₆ cos ω₀

Zₓ₃₁ = 12 Xₓ₁² - 3 Xₓ₃²

Zₓ₃₂ = 24 Xₓ₁ Xₓ₂ - 6 Xₓ₃ Xₓ₄

Zₓ₃₃ = 12 Xₓ₂² - 3 Xₓ₄²

Zₓ₁₁ = - 6 aₓ₁ aₓ₅ + e₀² (- 24 Xₓ₁ Xₓ₇ - 6 Xₓ₃ Xₓ₅)

Zₓ₁₃ = - 6 aₓ₃ aₓ₆ + e₀² (-24 Xₓ₂ Xₓ₈ - 6 Xₓ₄ Xₓ₆)

Zₓ₂₁ = 6 aₓ₂ aₓ₅ + e₀² (24.0 Xₓ₁ Xₓ₅ - 6 Xₓ₃ Xₓ₇)

Zₓ₂₃ = 6 aₓ₄ aₓ₆ + e₀² (24 Xₓ₂ Xₓ₆ - 6 Xₓ₄ Xₓ₈)

Zₓ₁ = 2 (3 (aₓ₁² + aₓ₂²) + Zₓ₃₁ e₀²) + p₁ Zₓ₃₁

Zₓ₃ = 2 (3 (aₓ₃² + aₓ₄²) + Zₓ₃₃ e₀²) + p₁ Zₓ₃₃

pₓ₀ = Cₓ / n₀"

        1 pₓ₀
pₓ₁ = - - ---
        2 β₀

pₓ₂ = pₓ₀ β₀

pₓ₃ = - 15 e₀ pₓ₂

Ω̇ₓ = │ 0                               if I₀ < 5.2359877 × 10⁻²
     │                                 or I₀ > π - 5.2359877 × 10⁻²
     │ - nₓ pₓ₁ (Zₓ₂₁ + Zₓ₂₃) / sin I₀ otherwise

kₓ₀ = 2 pₓ₃ (Xₓ₂ Xₓ₃ + Xₓ₁ Xₓ₄)

kₓ₁ = 2 pₓ₃ (Xₓ₂ Xₓ₄ - Xₓ₁ Xₓ₃)

kₓ₂ = 2 pₓ₁ (- 6 (aₓ₁ aₓ₆ + aₓ₃ aₓ₅) + e₀² (- 24 (Xₓ₂ Xₓ₇ + Xₓ₁ Xₓ₈) - 6 (Xₓ₃ Xₓ₆ + Xₓ₄ Xₓ₅)))

kₓ₃ = 2 pₓ₁ (Zₓ₁₃ - Zₓ₁₁)

kₓ₄ = - 2 pₓ₀ (2 (6 (aₓ₁ aₓ₃ + aₓ₂ aₓ₄) + Zₓ₃₂ e₀²) + p₁ Zₓ₃₂)

kₓ₅ = - 2 pₓ₀ (Zₓ₃ - Zₓ₁)

kₓ₆ = - 2 pₓ₀ (- 21 - 9 e₀²) eₓ

kₓ₇ = 2 pₓ₂ Zₓ₃₂

kₓ₈ = 2 pₓ₂ (Zₓ₃₃ - Zₓ₃₁)

kₓ₉ = - 18 pₓ₂ eₓ

kₓ₁₀ = - 2 pₓ₁ (6 (aₓ₄ aₓ₅ + aₓ₂ aₓ₆) + e₀² (24 (Xₓ₂ Xₓ₅ + Xₓ₁ Xₓ₆) - 6 (Xₓ₄ Xₓ₇ + Xₓ₃ Xₓ₈)))

kₓ₁₁ = - 2 pₓ₁ (Zₓ₂₃ - Zₓ₂₁)

İₓ = pₓ₁ nₓ (Zₓ₁₁ + Zₓ₁₃)

ėₓ = pₓ₃ nₓ (Xₓ₁ Xₓ₃ + Xₓ₂ Xₓ₄)

ω̇ₓ = pₓ₂ nₓ (Zₓ₃₁ + Zₓ₃₃ - 6) - cos I₀ Ω̇ₓ

Ṁₓ = - nₓ pₓ₀ (Zₓ₁ + Zₓ₃ - 14 - 6 e₀²)

Resonant deep space initialization

Defined only if n₀" ≤ 2π / 225 (deep space) and either:

  • 0.0034906585 < n₀" < 0.0052359877 (geosynchronous)
  • 8.26 × 10⁻³ ≤ n₀" ≤ 9.24 × 10⁻³ and e₀ ≥ 0.5 (Molniya)

The sidereal time θ₀ at epoch can be calculated with either the IAU formula:

c₂₀₀₀ = y₂₀₀₀ / 100

θ₀ = ¹/₂₄₀ (π / 180) (- 6.2 × 10⁻⁶ c₂₀₀₀³ + 0.093104 c₂₀₀₀²
     + (876600 × 3600 + 8640184.812866) c₂₀₀₀ + 67310.54841) mod

or the AFSPC formula:

d₁₉₇₀ = 365.25 (y₂₀₀₀ + 30) + 1

θ₀ = 1.7321343856509374 + 1.72027916940703639 × 10⁻² ⌊d₁₉₇₀ + 10⁻⁸⌋
     + (1.72027916940703639 × 10⁻² +) (d₁₉₇₀ - ⌊d₁₉₇₀ + 10⁻⁸⌋)
     + 5.07551419432269442 × 10⁻¹⁵ d₁₉₇₀² mod
λ₀ = │ M₀ + Ω₀ + ω₀ − θ₀ rem 2π if geosynchronous
     │ M₀ + 2 Ω₀ − 2 θ₀ rem 2π  otherwise

λ̇₀ = │ p₁₅ + (k₁₄ + p₁₄) − θ̇ + (Ṁₛ + Ṁₗ) + (ω̇ₛ + ω̇ₗ) + (Ω̇ₛ + Ω̇ₗ) - n₀" if geosynchronous
     │ p₁₅ + (Ṁₛ + Ṁₗ) + 2 (p₁₄ + (Ω̇ₛ + Ω̇ₗ) - θ̇) - n₀"                otherwise

Geosynchronous deep space initialization

Defined only if n₀" ≤ 2π / 225 (deep space) and 0.0034906585 < n₀" < 0.0052359877 (geosynchronous orbit).

p₁₇ = 3 (n / a₀")²

𝛿ᵣ₁ = p₁₇ (¹⁵/₁₆ sin²I₀ (1 + 3 p₁) - ³/₄ (1 + p₁))
          (1 + 2 e₀²) 2.1460748 × 10⁻⁶ / a₀"²

𝛿ᵣ₂ = 2 p₁₇ (³/(1 + p₁)²)
     (1 + e₀² (-/+ ¹³/₁₆ e₀²)) 1.7891679 × 10⁻⁶

𝛿ᵣ₃ = 3 p₁₇ (¹⁵/(1 + p₁)³) (1 + e₀² (- 6 + 6.60937 e₀²))
      2.2123015 × 10⁻⁷ / a₀"²

Molniya deep space initialization

Defined only if n₀" ≤ 2π / 225 (deep space) and 8.26 × 10⁻³ ≤ n₀" ≤ 9.24 × 10⁻³ and e₀ ≥ 0.5 (Molniya).

p₁₈ = 3 n₀"² / a₀"²

p₁₉ = p₁₈ / a₀"

p₂₀ = p₁₉ / a₀"

p₂₁ = p₂₀ / a₀"

F₂₂₀ = ³/₄ (1 + 2 p₁ + p₁²)

G₂₁₁ = │ 3.616 - 13.247 e₀ + 16.29 e₀²                     if e₀ ≤ 0.65
       │ - 72.099 + 331.819 e₀ - 508.738 e₀² + 266.724 e₀³ otherwise

G₃₁₀ = │ - 19.302 + 117.39 e₀ - 228.419 e₀² + 156.591 e₀³      if e₀ ≤ 0.65
       │ - 346.844 + 1582.851 e₀ - 2415.925 e₀² + 1246.113 e₀³ otherwise

G₃₂₂ = │ - 18.9068 + 109.7927 e₀ - 214.6334 e₀² + 146.5816 e₀³ if e₀ ≤ 0.65
       │ - 342.585 + 1554.908 e₀ - 2366.899 e₀² + 1215.972 e₀³ otherwise

G₄₁₀ = │ - 41.122 + 242.694 e₀ - 471.094 e₀² + 313.953 e₀³      if e₀ ≤ 0.65
       │ - 1052.797 + 4758.686 e₀ - 7193.992 e₀² + 3651.957 e₀³ otherwise

G₄₂₂ = │ - 146.407 + 841.88 e₀ - 1629.014 e₀² + 1083.435 e₀³   if e₀ ≤ 0.65
       │ - 3581.69 + 16178.11 e₀ - 24462.77 e₀² + 12422.52 e₀³ otherwise

G₅₂₀ = │ - 532.114 + 3017.977 e₀ - 5740.032 e₀² + 3708.276 e₀³ if e₀ ≤ 0.65
       │ 1464.74 - 4664.75 e₀ + 3763.64 e₀²                    if 0.65 < e₀ < 0.715
       │ - 5149.66 + 29936.92 e₀ - 54087.36 e₀² + 31324.56 e₀³ otherwise

G₅₃₂ = │ - 853.666 + 4690.25 e₀ - 8624.77 e₀² + 5341.4 e₀³         if e₀ < 0.7
       │ - 40023.88 + 170470.89 e₀ - 242699.48 e₀² + 115605.82 e₀³ otherwise

G₅₂₁ = │ - 822.71072 + 4568.6173 e₀ - 8491.4146 e₀² + 5337.524 e₀³  if e₀ < 0.7
       │ - 51752.104 + 218913.95 e₀ - 309468.16 e₀² + 146349.42 e₀³ otherwise

G₅₃₃ = │ - 919.2277 + 4988.61 e₀ - 9064.77 e₀² + 5542.21 e₀³      if e₀ < 0.7
       │ - 37995.78 + 161616.52 e₀ - 229838.2 e₀² + 109377.94 e₀³ otherwise

D₂₂₀₋₁ = p₁₈ 1.7891679 × 10⁻⁶ F₂₂₀ (- 0.306 - 0.44 (e₀ - 0.64))

D₂₂₁₁ = p₁₈ 1.7891679 × 10⁻⁶ (³/₂ sin²I₀) G₂₁₁

D₃₂₁₀ = p₁₉ 3.7393792 × 10⁻⁷ (¹⁵/₈ sin I₀ (1 - 2 p₁ - 3 p₁²)) G₃₁₀

D₃₂₂₂ = p₁₉ 3.7393792 × 10⁻⁷ (- ¹⁵/₈ sin I₀ (1 + 2 p₁ - 3 p₁²)) G₃₂₂

D₄₄₁₀ = 2 p₂₀ 7.3636953 × 10⁻⁹ (35 sin²I₀ F₂₂₀) G₄₁₀

D₄₄₂₂ = 2 p₂₀ 7.3636953 × 10⁻⁹ (³¹⁵/₈ sin⁴I₀) G₄₂₂

D₅₂₂₀ = p₂₁ 1.1428639 × 10⁻⁷ (³¹⁵/₃₂ sin I₀
        (sin²I₀ (1 - 2 p₁ - 5 p₁²)
        + 0.33333333 (- 2 + 4 p₁ + 6 p₁²))) G₅₂₀

D₅₂₃₂ = p₂₁ 1.1428639 × 10⁻⁷ (sin I₀
        (4.92187512 sin²I₀ (- 2 - 4 p₁ + 10 p₁²)
        + 6.56250012 (1 + p₁ - 3 p₁²))) G₅₃₂

D₅₄₂₁ = 2 p₂₁ 2.1765803 × 10⁻⁹ (⁹⁴⁵/₃₂ sin I₀
        (2 - 8 p₁ + p₁² (- 12 + 8 p₁ + 10 p₁²))) G₅₂₁

D₅₄₃₃ = 2 p₂₁ 2.1765803 × 10⁻⁹ (⁹⁴⁵/₃₂ sin I₀
        (- 2 - 8 p₁ + p₁² (12 + 8 p₁ - 10 p₁²))) G₅₃₃

Common propagation

The following values depend on the propagation time t (minutes since epoch).

Named conditions have the following meaning:

  • near earth: n₀" ≤ 2π / 225
  • low altitude near earth: near earth and p₃ < 220 / (aₑ + 1)
  • high altitude near earth: near earth and p₃ ≥ 220 / (aₑ + 1)
  • elliptic high altitude near earth: high altitude near earth and e₀ > 10⁻⁴
  • non-elliptic near earth: low altitude near earth or high altitude near earth and e₀ ≤ 10⁻⁴
  • deep space: n₀" > 2π / 225
  • non-Lyddane deep space: deep space and I ≥ 0.2
  • Lyddane deep space: deep space and I < 0.2
  • AFSPC Lyddane deep space: Lyddane deep space and use the same expression as the original AFSPC implementation, with an ω discontinuity at p₂₂ = 0
p₂₂ = Ω₀ + Ω̇ t + k₀ t²

p₂₃ = ω₀ + ω̇ t

I = │ I₀                    if near earth
    │ I₀ + İ t + (δIₛ + δIₗ) otherwise

Ω = │ p₂₂                      if near earth
    │ p₂₂ + (pₛ₅ + pₗ₅) / sin I if non-Lyddane deep space
    │ p₃₀ +if Lyddane deep space and p₃₀ + π < p₂₂ rem 2π
    │ p₃₀ -if Lyddane deep space and p₃₀ - π > p₂₂ rem 2π
    │ p₃₀                      otherwise

e =10⁻⁶              if near earth and p₂₇ < 10⁻⁶
    │ p₂₇               if near earth and p₂₇ ≥ 10⁻⁶
    │ 10⁻⁶ + (δeₛ + δeₗ) if deep space and p₃₁ < 10⁻⁶
    │ p₃₁ + (δeₛ + δeₗ)  otherwise

ω = │ p₂₃ - p₂₅                                   if elliptic high altitude near earth
    │ p₂₃                                         if non-elliptic near earth
    │ p₂₃ + (pₛ₄ + pₗ₄) - cos I (pₛ₅ + pₗ₅) / sin I if non-Lyddane deep space
    │ p₂₃ + (pₛ₄ + pₗ₄) + cos I ((p₂₂ rem 2π) - Ω)- (δIₛ + δIₗ) (p₂₂ mod) sin I             if AFSPC Lyddane deep space
    │ p₂₃ + (pₛ₄ + pₗ₄) + cos I ((p₂₂ rem 2π) - Ω)- (δIₛ + δIₗ) (p₂₂ rem 2π) sin I             otherwise

M = │ p₂₆ + n₀" (k₁ t² + k₈ t³ + t⁴ (k₉ + t k₁₀) if high altitude near earth
    │ p₂₄ + n₀" k₁ t²                            if low altitude near earth
    │ p₂₉ + (δMₛ + δMₗ) + n₀" k₁ t²               otherwise


a = │ a₀" (1 - C₁ t - D₂ t² - D₃ t³ - D₄ t⁴)² if high altitude near earth
    │ a₀" (1 - C₁ t)²                         if low altitude near earth
    │ p₂₈ (1 - C₁ t)²                         otherwise

n = kₑ / a³ᐟ²

p₃₂ = │ k₂           if near earth
      │   1 J₃
      │ - - -- sin I othewise
      │   2 J₂

p₃₃ = │ k₃        if near earth
      │ 1 - cos²I othewise

p₃₄ = │ k₄          if near earth
      │ 7 cos²I - 1 otherwise

p₃₅ = │ k₅                       if near earth
      │   1 J₃       3 + 5 cos I
      │ - - -- sin I ----------- if deep space and |1 + cos I| > 1.5 × 10⁻¹²
      │   4 J₂        1 + cos I
      │   1 J₃       3 + 5 cos I
      │ - - -- sin I ----------- otherwise
      │   4 J₂       1.5 × 10⁻¹²

p₃₆ = │ k₆          if near earth
      │ 3 cos²I - 1 otherwise

p₃₇ = 1 / (a (1 - e²))

aₓₙ = e cos ω

aᵧₙ = e sin ω + p₃₇ p₃₂

p₃₈ = M + ω + p₃₇ p₃₅ aₓₙ rem 2π

(E + ω)₀ = p₃₈

            p₃₈ - aᵧₙ cos (E + ω)ᵢ + aₓₙ sin (E + ω)ᵢ - (E + ω)ᵢ
Δ(E + ω)ᵢ = ---------------------------------------------------
                  1 - cos (E + ω)ᵢ aₓₙ - sin (E + ω)ᵢ aᵧₙ

(E + ω)ᵢ₊₁ = (E + ω)ᵢ + Δ(E + ω)ᵢ|[-0.95, 0.95]

E + ω = │ (E + ω)₁₀ if ∀ j ∈ [0, 9], Δ(E + ω)ⱼ ≥ 10⁻¹²
        │ (E + ω)ⱼ  otherwise, with j the smallest integer | Δ(E + ω)ⱼ < 10⁻¹²

p₃₉ = aₓₙ² + aᵧₙ²

pₗ = a (1 - p₃₉)

p₄₀ = aₓₙ sin(E + ω) - aᵧₙ cos(E + ω)

r = a (1 - (aₓₙ cos(E + ω) + aᵧₙ sin(E + ω)))

ṙ = a¹ᐟ² p₄₀ / r

β = (1 - p₃₉)¹ᐟ²

p₄₁ = p₄₀ / (1 + β)

p₄₂ = a / r (sin(E + ω) - aᵧₙ - aₓₙ p₄₁)

p₄₃ = a / r (cos(E + ω) - aₓₙ + aᵧₙ p₄₁)

          p₄₂
u = tan⁻¹ ---
          p₄₃

p₄₄ = 2 p₄₃ p₄₂

p₄₅ = 1 - 2 p₄₂²

p₄₆ = (¹/₂ J₂ / pₗ) / pₗ

rₖ = r (1 - ³/₂ p₄₆ β p₃₆) + ¹/₂ (¹/₂ J₂ / pₗ) p₃₃ p₄₅

uₖ = u - ¹/₄ p₄₆ p₃₄ p₄₄

Ωₖ = Ω + ³/₂ p₄₆ cos I p₄₄

Iₖ = I + ³/₂ p₄₆ cos I sin I p₄₅

ṙₖ = ṙ + n (¹/₂ J₂ / pₗ) p₃₃ / kₑ

rḟₖ = pₗ¹ᐟ² / r + n (¹/₂ J₂ / pₗ) (p₃₃ p₄₅ + ³/₂ p₃₆) / kₑ

u₀ = - sin Ωₖ cos Iₖ sin uₖ + cos Ωₖ cos uₖ

u₁ = cos Ωₖ cos Iₖ sin uₖ + sin Ωₖ cos uₖ

u₂ = sin Iₖ sin uₖ

r₀ = rₖ u₀ aₑ

r₁ = rₖ u₁ aₑ

r₂ = rₖ u₂ aₑ

ṙ₀ = (ṙₖ u₀ + rḟₖ (- sin Ωₖ cos Iₖ cos uₖ - cos Ωₖ sin uₖ)) aₑ kₑ / 60

ṙ₁ = (ṙₖ u₁ + rḟₖ (cos Ωₖ cos Iₖ cos uₖ - sin Ωₖ sin uₖ)) aₑ kₑ / 60

ṙ₂ = (ṙₖ u₂ + rḟₖ (sin Iₖ cos uₖ)) aₑ kₑ / 60

Near earth propagation

Defined only if n₀" > 2π / 225 (near earth).

p₂₄ = M₀ + Ṁ t

p₂₇ = | e₀ - (Ct + C₅ (sin p₂₆ - k)) if high altitude
      | e₀ - C₄ t                       otherwise

High altitude near earth propagation

Defined only if n₀" > 2π / 225 (near earth) and p₃ ≥ 220 / (aₑ + 1) (high altitude).

elliptic means e₀ > 10⁻⁴.

p₂₅ = k₁₃ ((1 + η cos p₂₄)³ - k₁₁) + k₁₂ t

p₂₆ = │ p₂₄ + p₂₅ if elliptic
      │ p₂₄       otherwise

Deep space propagation

Defined only if n₀" ≤ 2π / 225 (deep space).

p₂₈ =(kₑ / (nⱼ + ṅⱼ (t - tⱼ) + ¹/₂ n̈ⱼ (t - tⱼ)²))²ᐟ³ if geosynchronous or Molniya
      │ a₀"                                            otherwise

p₂₉ = │ λⱼ + λ̇ⱼ (t - tⱼ) + ¹/₂ ṅᵢ (t - tⱼ)² - p₂₂ - p₂₃ + θ if geosynchronous
      │ λⱼ + λ̇ⱼ (t - tⱼ) + ¹/₂ ṅᵢ (t - tⱼ)² - 2 p₂₂ + 2 θ   if Molniya
      │ M₀ + Ṁ t                                            otherwise

j is │ the largest positive integer | tⱼ ≤ t  if t > 0
     │ the smallest negative integer | tⱼ ≥ t if t < 0
     │ 0                                      otherwise

p₃₁ = e₀ + ė t - C₄ t

Third body propagation

Defined only if n₀" ≤ 2π / 225 (deep space).

The following variables are evaluated for two third bodies, the Sun (solar perturbations s) and the Moon (lunar perturbations l). Variables specific to the third body are annotated with x. In other sections, x is either s or l.

Mₓ = Mₓ₀ + nₓ t

fₓ = Mₓ + 2 eₓ sin Mₓ

Fₓ₂ = ¹/₂ sin²fₓ - ¹/₄

Fₓ₃ = - ¹/₂ sin fₓ cos fₓ

δeₓ = kₓ₀ Fₓ₂ + kₓ₁ Fₓ₃

δIₓ = kₓ₂ Fₓ₂ + kₓ₃ Fₓ₃

δMₓ = kₓ₄ Fₓ₂ + kₓ₅ Fₓ₃ + kₓ₆ sin fₓ

pₓ₄ = kₓ₇ Fₓ₂ + kₓ₈ Fₓ₃ + kₓ₉ sin fₓ

pₓ₅ = kₓ₁₀ Fₓ₂ + kₓ₁₁ Fₓ₃

Resonant deep space propagation

Defined only if n₀" ≤ 2π / 225 (deep space) and either:

  • 0.0034906585 < n₀" < 0.0052359877 (geosynchronous)
  • 8.26 × 10⁻³ ≤ n₀" ≤ 9.24 × 10⁻³ and e₀ ≥ 0.5 (Molniya)
θ = θ₀ + 4.37526908801129966 × 10⁻³ t rem 2π

Δt =|Δt|  if t > 0-|Δt| if t < 00     otherwise

λ̇ᵢ = nᵢ + λ̇₀

ṅᵢ = │ 𝛿ᵣ₁ sin(λᵢ - λ₃₁) + 𝛿ᵣ₂ sin(2 (λᵢ - λ₂₂)) + 𝛿ᵣ₃ sin(3 (λᵢ - λ₃₃)) if geosynchronous
     │ Σ₍ₗₘₚₖ₎ Dₗₘₚₖ sin((l - 2 p) ωᵢ + m / 2 λᵢ - Gₗₘ)                    otherwise

n̈ᵢ =(𝛿ᵣ₁ cos(λᵢ - λ₃₁) + 𝛿ᵣ₂ cos(2 (λᵢ - λ₂₂)) + 𝛿ᵣ₃ cos(3 (λᵢ - λ₃₃))) λ̇ᵢ if geosynchronous
     │ (Σ₍ₗₘₚₖ₎ m / 2 Dₗₘₚₖ cos((l - 2 p) ωᵢ + m / 2 λᵢ - Gₗₘ)) λ̇ᵢ               otherwise

(l, m, p, k){(2, 2, 0, -1), (2, 2, 1, 1), (3, 2, 1, 0),
    (3, 2, 2, 2), (4, 4, 1, 0), (4, 4, 2, 2), (5, 2, 2, 0),
    (5, 2, 3, 2), (5, 4, 2, 1), (5, 4, 3, 3)}

tᵢ₊₁ = tᵢ + Δt

nᵢ₊₁ = nᵢ + ṅᵢ Δt + n̈ᵢ (Δt² / 2)

λᵢ₊₁ = λᵢ + λ̇ᵢ Δt + ṅᵢ (Δt² / 2)

Lyddane deep space propagation

Defined only if n₀" ≤ 2π / 225 (deep space) and I < 0.2 (Lyddane).

            sin I sin p₂₂ + (pₛ₅ + pₗ₅) cos p₂₂ + (δIₛ + δIₗ) cos I sin p₂₂
p₃₀ = tan⁻¹ -------------------------------------------------------------
            sin I cos p₂₂ - (pₛ₅ + pₗ₅) sin p₂₂ + (δIₛ + δIₗ) cos I cos p₂₂

References

David A. Vallado, Paul Crawford, R. S. Hujsak and T. S. Kelso, "Revisiting Spacetrack Report #3", presented at the AIAA/AAS Astrodynamics Specialist Conference, Keystone, CO, 2006 August 21–24, https://doi.org/10.2514/6.2006-6753

F. R. Hoots, P. W. Schumacher Jr. and R. A. Glover, "History of Analytical Orbit Modeling in the U. S. Space Surveillance System", Journal of Guidance, Control, and Dynamics, 27(2), 174–185, 2004, https://doi.org/10.2514/1.9161

F. R. Hoots and R. L. Roehrich, "Spacetrack Report No. 3: Models for propagation of NORAD element sets", U.S. Air Force Aerospace Defense Command, Colorado Springs, CO, 1980, https://www.celestrak.com/NORAD/documentation/

R. S. Hujsak, "A Restricted Four Body Solution for Resonating Satellites Without Drag", Project SPACETRACK, Rept. 1, U.S. Air Force Aerospace Defense Command, Colorado Springs, CO, Nov. 1979, https://doi.org/10.21236/ada081263

Dependencies

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~29K SLoC