J2 Osculating Analytical Orbit Propagator

This algorithm uses the J2 propagator to obtain the secular effects of the Keplerian elements caused only by the J2 term of the geopotential field. Afterward, it adds short-term perturbations. This model is useful when fitting an orbit to a set of mean elements for the J2 orbit propagator. Hence, we can use it to verify, for example, how close a satellite is from a Sun-Synchronous orbit.

Algorithm

The algorithm implemented here is based on [1].

Initialization

We can initialize the J2 osculating analytical orbit propagator with the following function:

Propagators.init(Val(:J2osc), orb₀::KeplerianElements; kwargs...) -> OrbitPropagatorJ2Osculating

which creates a J2 osculating propagator structure OrbitPropagatorJ2Osculating with the mean Keplerian elements orb₀.

The following keyword selects the gravitational constants for the propagation algorithm:

• j2c::J2PropagatorConstants: J2 orbit propagator constants (see J2PropagatorConstants). (Default: J2C_EGM2008)

This package contains some pre-built propagation constants for this propagator:

julia> orb = KeplerianElements(
           date_to_jd(2023, 1, 1, 0, 0, 0),
           7190.982e3,
           0.001111,
           98.405 |> deg2rad,
           100    |> deg2rad,
           90     |> deg2rad,
           19     |> deg2rad
       )KeplerianElements{TrueAnomaly, Float64, Float64}:
  Epoch             : 2.45995e6 (2023-01-01T00:00:00)
  Semi-Major Axis   : 7190.982 km
  Eccentricity      : 0.001111
  Inclination       : 98.405°
  RA of Asc. Node   : 100.0°
  Arg. of Periapsis : 90.0°
  True Anomaly      : 19.0°
julia> orbp = Propagators.init(Val(:J2osc), orb)OrbitPropagatorJ2Osculating{Float64, Float64} (J2 Osculating Orbit Propagator):
  ├─ Mean Elements
  │    Epoch             : 2.45995e6 (2023-01-01T00:00:00)
  │    Semi-Major Axis   : 7190.982 km
  │    Eccentricity      : 0.001111
  │    Inclination       : 98.405°
  │    RA of Asc. Node   : 100.0°
  │    Arg. of Periapsis : 90.0°
  │    Mean Anomaly      : 18.95858415°
  ├─ Secular Rates
  │    Mean Motion            : 14.22855327 rev/day
  │    RAAN Rate              : 0.9565368878 °/day
  │    Arg. of Periapsis Rate : -2.922474801 °/day
  ├─ Constants
  │    R₀ : 6378.137 km
  │    μm : 0.001239447462 rad/s
  │    J₂ : 0.001082626174
  └─ Propagation
       Last Instant : 0.0 s

Fitting Mean Elements

We can use the function:

Propagators.fit_mean_elements(::Val{:J2osc}, vjd::AbstractVector{Tjd}, vr_i::AbstractVector{Tv}, vv_i::AbstractVector{Tv}; kwargs...) -> KeplerianElements{MeanAnomaly, Float64, T}, SMatrix{6, 6, T}, NamedTuple

to fit a set of mean Keplerian elements for the J2 osculating orbit propagator using the osculating elements represented by a set of position vectors vr_i [m] and a set of velocity vectors vv_i [m / s] represented in an inertial reference frame at instants in the array vjd [Julian Day].

It returns the fitted Keplerian elements, the final covariance matrix of the least-square algorithm, and a NamedTuple with its statistics: converged, iterations, position_rmse [m], velocity_rmse [m / s], and total_rmse.

The following keywords are available to configure the fitting process:

• j2c::J2PropagatorConstants{T}: J2 orbit propagator constants (see J2PropagatorConstants), whose number type T is used in the fitting. (Default: J2C_EGM2008)
• atol::Number: Tolerance for the residual absolute value. If the residual is lower than atol at any iteration, the computation loop stops. (Default: 2e-4)
• rtol::Number: Tolerance for the relative difference between the residuals. If the relative difference between the residuals in two consecutive iterations is lower than rtol, the computation loop stops. (Default: 2e-4)
• initial_guess::Union{Nothing, KeplerianElements}: Initial guess for the mean elements fitting process. If it is nothing, the algorithm will obtain an initial estimate from the osculating elements in vr_i and vv_i. (Default: nothing)
• jacobian_method::AbstractJacobianMethod: Method used to compute the Jacobian matrix. It can be FiniteDiffJacobian() for finite differences or ForwardDiffJacobian() for ForwardDiff.jl automatic differentiation. (Default: FiniteDiffJacobian())
• jacobian_perturbation::Number: Initial state perturbation to compute the finite-difference when calculating the Jacobian matrix. Only used with FiniteDiffJacobian(). (Default: 1e-3)
• jacobian_perturbation_tol::Number: Tolerance to accept the perturbation when calculating the Jacobian matrix. If the computed perturbation is lower than jacobian_perturbation_tol, we increase it until its absolute value is higher than jacobian_perturbation_tol. Only used with FiniteDiffJacobian(). (Default: 1e-7)
• max_iterations::Int: Maximum number of iterations allowed for the least-square fitting. (Default: 50)
• mean_elements_epoch::Union{Number, DateTime}: Epoch of the fitted mean elements, represented by a Julian Day [UTC] or a DateTime [UTC]. (Default: vjd[end])
• verbose::Bool: If true, the algorithm prints debugging information to stdout. (Default: true)
• weight_vector::AbstractVector: Vector with the measurements weights for the least-square algorithm. We assemble the weight matrix W as a diagonal matrix with the elements in weight_vector at its diagonal. (Default: @SVector(ones(Bool, 6)))

julia> vr_i = [
           [-6792.402703741442, 2192.6458461287293, 0.18851758695295118] .* 1000,
           [-6357.88873265975, 2391.9476768911686, 2181.838771262736] .* 1000
       ];
julia> vv_i = [
           [0.3445760107690598, 1.0395135806993514, 7.393686131436984] .* 1000,
           [2.5285015912807003, 0.27812476784300005, 7.030323100703928] .* 1000
       ];
julia> vjd = [
           2.46002818657856e6,
           2.460028190050782e6
       ];
julia> orb, P, stats = Propagators.fit_mean_elements(Val(:J2osc), vjd, vr_i, vv_i)ACTION:   Fitting the mean elements for the J2 osculating propagator.
           Iteration        Position RMSE        Velocity RMSE           Total RMSE       RMSE Variation
                                     [km]             [km / s]                  [ ]


PROGRESS:          1              6.24887           0.00466183              6248.87                  ---

PROGRESS:          2          0.000816963           0.00260117              2.72644             -99.9564 %

PROGRESS:          3          1.69098e-05           0.00260088              2.60093             -4.60352 %

PROGRESS:          4          1.69072e-05           0.00260088              2.60093          1.10532e-09 %

(KeplerianElements{MeanAnomaly, Float64, Float64}: Epoch = 2.46003e6 (2023-03-24T16:33:40.388), [0.9999427807011695 -0.004901152258826149 … -0.00012021282782821985 -0.00011550583200755688; -0.00490115226000162 1.000732580079742 … 0.003266156907073989 3.8567168871770624e-5; … ; -0.00012021282783057524 0.003266156907074014 … 2.2048267812795964e-5 5.382766445309007e-8; -0.00011550583200733415 3.8567168867268407e-5 … 5.3827664438749715e-8 2.1657420758757818e-5], (converged = true, iterations = 4, position_rmse = 0.016907178972967667, velocity_rmse = 2.6008770843978817, total_rmse = 2.6009320369526865))
julia> orbKeplerianElements{MeanAnomaly, Float64, Float64}:
  Epoch             : 2.46003e6 (2023-03-24T16:33:40.388)
  Semi-Major Axis   : 7135.801151 km
  Eccentricity      : 0.001353825508
  Inclination       : 98.43042292°
  RA of Asc. Node   : 162.1131669°
  Arg. of Periapsis : 64.92555131°
  Mean Anomaly      : 313.1984092°
julia> stats(converged = true, iterations = 4, position_rmse = 0.016907178972967667, velocity_rmse = 2.6008770843978817, total_rmse = 2.6009320369526865)

References

• [1] Vallado, D. A (2013). Fundamentals of Astrodynamics and Applications. 4th ed. Microcosm Press, Hawthorn, CA, USA.