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I'm trying to write code to go back and forth between equatorial KOE (Keplerian Orbital Elements) and ECEI (Cartesian).

ECEI→KOE I found here: How to programmatically calculate orbital elements using position/velocity vectors?

KOE→ECEI I found on the Wikipedia page for the Orbital Elements which has the equations for x,y of the ellipse which can then be passed through a DCM transform from the orbital plane to the ECEI system.

So this is all well-and-good for a spherical earth. However, the purpose of this is to replace my RK4 integration which takes WGS84 into account with KOE parameters for performance.

I'd greatly appreciate any pointers leading to a method to account for a non-spherical earth.

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    $\begingroup$ Keplerian elliptical orbits are predicated on the assumption of an ideal point-source mass, which creates a spherically-symmetric gravitational field. The WGS84 uses the asymmetric field defined by the EGM96 gravitational model which results in (slightly) non-Keplerian orbits, with orbital elements that evolve over time. $\endgroup$ – Caleb Hines Sep 27 '15 at 19:07
  • $\begingroup$ They evolve with time? Does this imply that gravity is not rotationally-symmetric about the z-axis? $\endgroup$ – iAdjunct Sep 28 '15 at 1:32
  • $\begingroup$ The potential field of non-homogenous masses such as Earth are typically described as a linear combination of a series of spherical harmonic functions. See: en.wikipedia.org/wiki/Geopotential_model $\endgroup$ – Caleb Hines Sep 28 '15 at 4:44
  • $\begingroup$ I'm thinking about "... the purpose of this is to replace my RK4 integration which takes WGS84 into account with KOE parameters for performance." and wondering how this could improve performance. When you say "...KOE→ECEI I found on the Wikipedia page for the Orbital Elements which has the equations for x,y of the ellipse." are these $x(t)$ and $y(t)$ using some approximation or series solution? I don't know which Wikipedia article you are referring to. Something like that would be faster than a cartesian RK for a given accuracy mostly because it assumes a pure $1/r$ potential. $\endgroup$ – uhoh Dec 8 '16 at 10:15
  • $\begingroup$ Are you asking for something that has the speed benefits of a 2D planar pure $1/r$ potential solution (or approximation) but incorporates the lumpy 3D $(r, \theta, \phi)$ gravity model? There may be some intriguing analytical approximations for a few higher order terms (e.g. $J_2, J_4$) but for the full "lumpy earth", maybe not, since computers were coming along nicely by the time satellites could spatially resolve the Earths gravity field with higher accuracy. It's possible, some people just like math! Am I understanding your question correctly? $\endgroup$ – uhoh Dec 8 '16 at 10:21
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If you are talking about osculating Keplerian orbital elements, then they are independent of the gravity field spherical harmonics, and of non-keplerian perturbations in general, because they are defined as the instantaneous ellipse matching the (r,v) state vector at this instant. The conversions ECEI→KOE (named RV2COE in Fundamentals of astrodynamics and applications by David Vallado) and KOE→ECEI (named COE2RV) can therefore be done back and forth without any integration, except a small loop to solve Kepler's equation if you need to compute the mean anomaly. The implementation of these functions by Vallado (for instance in MATLAB) can be found here.

If you want to compute mean elements, then you indeed need a numerical integrator to integrate the perturbation forces, as you are probably already doing.

Recommended read about the distinction osculating/mean elements: Nuances of the terms (mean / osculating / Keplerian / orbital) elements

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