Equinoctial Elements

Nonsingular orbital elements: a replacement for the Classical Orbital Elements that stays well defined for circular and equatorial orbits.

Need for Nonsingular Elements

• Classical elements break down in two common cases:
  • Circular orbits : there’s no perigee, so the argument of perigee and true anomaly are undefined
  • Equatorial orbits : the line of nodes does not exist, so RAAN and the argument of perigee are undefined
• Circular and near-circular orbits:
  • Very small eccentricity: imagine making a small change to the orbit
  • Sketch: the tip of sits near the origin of the - plane; a small change can move it to the other side of the origin, so the direction of perigee () jumps by a large angle
• One partial fix is the argument of latitude

Equinoctial Orbital Elements (EqOE)

• References:
  • Section 2.1 of Danielson, D. A., C. P. Sagovac, B. Neta, and L. W. Early. Semianalytic Satellite Theory. NPS-95-MA-002. Naval Postgraduate School, 1995. DTIC ADA294557.
  • Broucke, R. A., and P. J. Cefola. “On the Equinoctial Orbit Elements.” Celestial Mechanics 5, no. 3 (1972): 303-10. DOI 10.1007/BF01228432.
  • Walker, M.J.H., Ireland, B. & Owens, J. “A set modified equinoctial orbit elements.” Celestial Mechanics 36, 409-419 (1985). DOI 10.1007/BF01227493
  • Horwood, Joshua T., Nathan D. Aragon, and Aubrey B. Poore. “Gaussian Sum Filters for Space Surveillance: Theory and Simulations.” Journal of Guidance, Control, and Dynamics 34, no. 6 (2011): 1839-51. DOI 10.2514/1.53793.
  • Baù, Giulio, Javier Hernando-Ayuso, and Claudio Bombardelli. “A Generalization of the Equinoctial Orbital Elements.” Celestial Mechanics and Dynamical Astronomy 133, no. 11 (2021): 50. DOI 10.1007/s10569-021-10049-1.
• Definitions:
  • is the mean longitude
  • : the eccentricity vector’s components; : the inclination/node “vector”

Corrected from the notes

• The 9/10 notes write , (with the argument of latitude just defined)
• The typed table on 9/24 (below) and the standard definition use the longitude of periapsis , which is used here

• Properties:
  • No large changes in these elements with small changes in the orbit
  • Mean longitude is like mean anomaly in that it increases linearly in time
  • Computational efficiency
• Remaining singularity: (); see the comparison table in Variants of Orbital Elements

Classical Equinoctial Elements

• Singular conditions of the classical set: circular and equatorial orbits
• Classical equinoctial elements, Broucke & Cefola in 1972:
  • Semimajor axis
  • component of the eccentricity vector:
  • component of the eccentricity vector:
    • handle
  • component of the inclination/node:
  • component of the inclination/node:
    • handle
  • Mean longitude:
• Equinoctial transformations, if the classical elements are defined:

Classical equinoctial	Equinoctial classical

• The last entry is cut off at the edge of the slide after “atan2(h,”; completed as

Equinoctial Kepler Equation

• = equinoctial eccentric longitude, the analogue of the eccentric anomaly (think )

Equinoctial Kepler Equation

• Analogue of Kepler’s equation ; solve for given

Sign correction

• In the lecture of 9/24 this was first written . The lecture of 10/6 announced “Corrections to lecture 8: signs on Kepler’s equation” (see announcements on ELMS); the mean-longitude step of state to equinoctial later in the same lecture already uses the corrected signs shown above.

Equinoctial Elements to Orbital State

Beta

• Used to compute the in-plane coordinates below

Sign correction

• Written in the lecture of 9/24 as ; the lecture of 10/6 announced a “definition of beta sign error” (see announcements on ELMS). With the sign, as .

• Radial distance:
• Cartesian coordinates of position and velocity in the (unrotated) orbital plane:
  • Position components:
  • Velocity components ():
• Rotate to the inertial frame with the equinoctial basis vectors:

Equinoctial State

• Summary (equinoctial orbital elements to orbital state):
  1. Solve the equinoctial Kepler equation for
  2. Compute the radial distance
  3. , , , as above
  4. ,

Orbital State to Equinoctial Elements

1. Compute the angular momentum vector, eccentricity vector, semimajor axis (as in Orbital Elements from State Vector)
2. Equinoctial bases:
   • ; , in the orbital plane
3. Calculate , , , :
4. Calculate the mean longitude:
   • from and with the two-argument arctangent; no iteration needed in this direction

Variants of Equinoctial Elements

Formulation	Primary size element	Position angle	Primary strength / use case
EqOE (Broucke & Cefola 1972)	Semimajor axis	Mean longitude ()	General theoretical development, analytical perturbation theories
MEE (Walker et al. 1985)	Semilatus rectum	True longitude ()	Low-thrust trajectory optimization, steering law designs
AEqOE (Horwood et al. 2011)	Mean motion	Mean longitude ()	High-accuracy orbit determination and non-linear uncertainty propagation
GEqOE (Baù et al. 2021)	(generalized )	Generalized mean longitude	High-fidelity propagation under dominant oblateness

• Modified equinoctial elements:
  • Semilatus rectum instead of sma:
  • True longitude (if these elements exist):
• Alternate equinoctial elements: use the mean motion instead of sma
• Generalized equinoctial elements: , the “generalized mean motion”
  • Based on the total energy (kinetic + potential energy with perturbation forces)
• Modified Rodriguez parameters version of the equinoctial elements ( in place of ):

Corrections Announced 10/6

• The lecture of 10/6 opened with corrections to the lecture of 9/24 (see announcements on ELMS):
  • Signs on Kepler’s equation: Equinoctial Kepler Equation
  • Definition of sign error: Equinoctial Elements to Orbital State