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Modified Equinoctial

The Modified Equinoctial Orbital Elements provide a set of parameters that describe an object's orbit without the singularities that arise in Keplerian elements (e.g., near-circular or equatorial orbits). These elements are particularly useful in numerical orbit propagation and optimization because they remain well-defined for nearly all types of orbits (with the exception of retro-grade orbits). They are an alternative to classical orbital elements, offering advantages in continuous, high-precision computations, especially when dealing with low-eccentricity orbits.

image Modified Equinoctial Orbital Element Frame [1]

Components

Similar to the Keplerian state vector, the Modified Equinoctial state vector consists of six elements, divided into size & shape, orientation, and position within the orbit:

  • Orbital Shape and Size

    • semi-latus rectum (p): A measure of the orbit’s size, similar to the semi-major axis, but defined as p=a*(1-e^2). It ensures well-defined behavior even for highly elliptical or circular orbits.
    • first component of eccentricity (f): First of two parameters that replace the scalar eccentricity. f=e*cos(ω + Ω) These components prevent singularities when the eccentricity is zero
    • second component of eccentricity (g): Second of two parameters that replace the scalar eccentricity. g=e*sin(ω + Ω)

  • Orbital Orientation

    • first component of inclination (h): First of two parameters that replace the scalar inclination. h=tan(i/2)*cos(Ω) These components ensures the elements remain well-defined for equatorial orbits
    • second component of inclination (k): Second of two parameters that replace the scalar inclination. h=tan(i/2)*sin(Ω)

  • Position within the Orbit

    • longitude of periapsis (L): This replaces true anomaly and combines information about the satellite’s position within the orbit. L=ω + Ω + f

References

[1]: https://degenerateconic.com/modified-equinoctial-elements.html [2]: https://link.springer.com/article/10.1007/BF01227493 [3]: https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/EquinoctalElements-modified.pdf