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{Float64, Float64}:
           Epoch :    2.45995e6 (2023-01-01T00:00:00)
 Semi-major axis : 7190.98     km
    Eccentricity :    0.001111
     Inclination :   98.405    °
            RAAN :  100.0      °
 Arg. of Perigee :   90.0      °
    True Anomaly :   19.0      °
julia> orbp = Propagators.init(Val(:J2osc), orb)OrbitPropagatorJ2Osculating{Float64, Float64}:
   Propagator name : J2 Osculating Orbit Propagator
  Propagator epoch : 2023-01-01T00:00:00
  Last propagation : 2023-01-01T00:00:00

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{Float64, Float64}, SMatrix{6, 6, Float64}

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 and the final covariance matrix of the least-square algorithm.

The following keywords are available to configure the fitting process:

• atol::Number: Tolerance for the residue absolute value. If the residue is lower than atol at any iteration, the computation loop stops. (Default = 2e-4)
• rtol::Number: Tolerance for the relative difference between the residues. If the relative difference between the residues 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_perturbation::Number: Initial state perturbation to compute the finite-difference when calculating the Jacobian matrix. (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 it absolute value is higher than jacobian_perturbation_tol. (Default = 1e-7)
• max_iterations::Int: Maximum number of iterations allowed for the least-square fitting. (Default = 50)
• mean_elements_epoch::Number: Epoch for the fitted mean elements. (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 = Propagators.fit_mean_elements(Val(:J2), vjd, vr_i, vv_i)ACTION:   Fitting the mean elements for the J2 propagator.
           Iteration        Position RMSE        Velocity RMSE           Total RMSE       RMSE Variation
                                     [km]             [km / s]                  [ ]                     


PROGRESS:          1              2.13455           0.00396508              2134.56                  ---

PROGRESS:          2           7.5425e-05           0.00975016              9.75045             -99.5432 %

PROGRESS:          3          6.15798e-05           0.00975023              9.75043           -0.0002039 %

(KeplerianElements{Float64, Float64}: Epoch = 2.46003e6 (2023-03-24T16:33:40.388), [0.9999771881586103 3.3330538199638824e-7 … -9.720588376900117e-5 -7.124383306089359e-5; 3.3330159687496843e-7 0.9999779181614062 … 0.0032646452140992063 1.701808112036195e-5; … ; -9.720588377946891e-5 0.0032646452140986325 … 2.2082152622482682e-5 1.7944993096216286e-8; -7.124383307731214e-5 1.701808111721979e-5 … 1.7944993088607555e-8 2.172473646254285e-5])
julia> orbKeplerianElements{Float64, Float64}:
           Epoch :    2.46003e6 (2023-03-24T16:33:40.388)
 Semi-major axis : 7155.99       km
    Eccentricity :    0.00307382
     Inclination :   98.4357     °
            RAAN :  162.113      °
 Arg. of Perigee :   33.0518     °
    True Anomaly :  344.956      °

References

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