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Generalized Equinoctial Orbital Elements

The Generalized Equinoctial Orbital Elements (GEqOE) extend the classical equinoctial elements to account for perturbations derived from a disturbing potential. Unlike the standard modified equinoctial elements, which are only valid under Keplerian dynamics, GEqOE absorbs the effect of conservative perturbations (e.g., gravitational harmonics) directly into the element definitions. This makes them particularly well-suited for high-fidelity orbit propagation and uncertainty quantification when the dominant perturbations are potential-based.

The formulation replaces the classical two-body integrals (energy, eccentricity vector, angular momentum) with their generalized counterparts that include the disturbing potential W, yielding elements that vary more slowly under perturbed motion.

Components

The GEqOE state vector [ν, p₁, p₂, L, q₁, q₂] consists of six elements:

  • Orbital Shape and Size

    • generalized mean motion (ν): Defined as ν = (1/μ)(-2E)^(3/2), where E is the total energy including the disturbing potential. This generalizes the Keplerian mean motion.
    • first generalized eccentricity component (p₁): p₁ = g sin(Ψ), where g is the magnitude of the generalized Laplace vector and Ψ is the generalized longitude of pericenter.
    • second generalized eccentricity component (p₂): p₂ = g cos(Ψ), analogous to the equinoctial eccentricity components but defined using the generalized eccentricity.

  • Position within the Orbit

    • generalized mean longitude (L): Replaces the true longitude and encodes the satellite's position along its orbit in a generalized sense.

  • Orbital Orientation

    • first inclination component (q₁): q₁ = tan(i/2) sin(Ω), identical to the classical equinoctial definition.
    • second inclination component (q₂): q₂ = tan(i/2) cos(Ω), identical to the classical equinoctial definition.

Computed Properties

The following derived quantities are available on a GEqOE instance:

  • g: Generalized eccentricity magnitude, √(p₁² + p₂²)
  • Ψ: Generalized longitude of pericenter, atan(p₁, p₂)

Additional helper functions:

  • L₀(geq, t): Generalized mean longitude at epoch, L₀ = L - νt
  • computed_a(geq, μ): Generalized semi-major axis, a = (μ/ν²)^(1/3)
  • computed_ρ(geq, μ): Generalized semi-latus rectum, ρ = a(1 - g²)

Usage

GEqOE uses a RegularizedCoordinateConfig to pass the perturbing potential W. For pure Keplerian orbits, set W = 0. For perturbed orbits, precompute the disturbing potential (Vtotal - Vkeplerian) externally and pass it via the config.

using AstroCoords

μ = 3.986004415e5
cart_state = Cartesian(-1076.225, -6765.896, -332.309, 9.357, -3.312, -1.188)

# Keplerian dynamics (W = 0, equivalent to modified equinoctial)
config = RegularizedCoordinateConfig(; W=0.0)

geqoe_state = GEqOE(cart_state, μ, config)

# Round-trip back to Cartesian
cart_roundtrip = Cartesian(geqoe_state, μ, config)

For non-Keplerian dynamics, precompute the disturbing potential externally:

# W is the disturbing potential at the current state, computed externally
# e.g., W = V_total_gravity(state) - V_keplerian(state)
W = compute_disturbing_potential(state, μ, gravity_model)  # user-defined
config = RegularizedCoordinateConfig(; W=W)

geqoe_state = GEqOE(cart_state, μ, config)

Singularities

  • Non-singular for circular orbits (g = 0)
  • Non-singular for equatorial orbits (i = 0)
  • Singular for retrograde equatorial orbits (i = π) and rectilinear motion
  • Requires negative total energy (E < 0), i.e., bound orbits only

References

[1]: Baù, G., Hernando-Ayuso, J., & Bombardelli, C. (2021). "A generalization of the equinoctial orbital elements." Celestial Mechanics and Dynamical Astronomy, 133(9), 1-32.