J4 Osculating Analytical Orbit Propagator

This algorithm uses the J4 propagator to obtain the secular effects of the Keplerian elements caused by the terms $J_2$, $J_2^2$, and $J_4$of the geopotential field. Afterward, it adds short-term perturbations using only the $J_2$ term. This model is useful when fitting an orbit to a set of mean elements for the J4 orbit propagator.

Algorithm

The algorithm implemented here is based on [1].

Initialization

We can initialize the J4 osculating analytical orbit propagator with the following function:

Propagators.init(Val(:J4osc), orb₀::KeplerianElements; kwargs...) -> OrbitPropagatorJ4Osculating

which creates a J4 osculating propagator structure OrbitPropagatorJ4Osculating with the mean Keplerian elements orb₀.

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

• j4c::J4PropagatorConstants: J4 orbit propagator constants (see J4PropagatorConstants). (Default = j4c_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(:J4osc), orb)OrbitPropagatorJ4Osculating{Float64, Float64}:
   Propagator name : J4 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{:J4osc}, 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 J4 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(:J4), vjd, vr_i, vv_i)ACTION:   Fitting the mean elements for the J4 propagator.
           Iteration        Position RMSE        Velocity RMSE           Total RMSE       RMSE Variation
                                     [km]             [km / s]                  [ ]                     


PROGRESS:          1              2.13353           0.00396406              2133.53                  ---

PROGRESS:          2           7.5338e-05           0.00974534              9.74563             -99.5432 %

PROGRESS:          3          6.15485e-05           0.00974541              9.74561         -0.000202692 %

(KeplerianElements{Float64, Float64}: Epoch = 2.46003e6 (2023-03-24T16:33:40.388), [0.999977188154194 3.33256026080347e-7 … -9.72062490829615e-5 -7.12390352683913e-5; 3.3325912649242495e-7 0.9999779181880107 … 0.003264642607995978 1.7032556508078778e-5; … ; -9.720624907090099e-5 0.003264642607996314 … 2.208211306240958e-5 1.8097580343003953e-8; -7.123903527436024e-5 1.703255648874405e-5 … 1.809758028027451e-8 2.1724922065283353e-5])
julia> orbKeplerianElements{Float64, Float64}:
           Epoch :    2.46003e6 (2023-03-24T16:33:40.388)
 Semi-major axis : 7155.98       km
    Eccentricity :    0.00307271
     Inclination :   98.4357     °
            RAAN :  162.113      °
 Arg. of Perigee :   33.061      °
    True Anomaly :  344.947      °

References

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