# From osculating to mean equinoctial elements and back

I'm struggling with orbital elements...

I'm trying, at the same time as finding a clear definition to them, to write a piece of code to switch from osculator elements to mean equinoctial elements.

I've found this topic but it did not really help me to distinguish the two.

Could somebody give me a hint?

And is it possible the other way around?

Thank you!

Benoist

• Are you saying that you have read this answer but you don't understand it, and are looking for a simpler description of each and understanding of the difference? Because that would be an excellent stackexchange question if you wrote it like that. You can ask as many questions as you like btw, just keep them clear and describe what you do understand and then what you'd like some help with. Welcome to stackexchange, why not take the tour! – uhoh Jul 27 '17 at 15:49
• I think the questioner here is looking for formulas to convert from osculating elements to mean elements and back. I'm looking for the same thing, but to be more precise I'm looking for the relatively simple formulas that are first-order in J^2. There was a PDF I found at one point which had fairly simple formulas to convert between osculating and mean elements with that precision. Most of the literature is concerned with a level of accuracy and complexity well beyond what I want... – lamont Feb 10 '19 at 22:10
• space.stackexchange.com/a/22155/14420 has a reference to the BLST with only $J_2$ which is what i was looking for. – lamont Feb 18 '19 at 8:52

## 2 Answers

According to Analytical Mechanics of Space Systems, Osculating Elements are the one containing different perturbations in the Orbit. When you omit the short time perturbation (J2) you gain the mean orbital element. The mean orbital elements change not much during the short time so they are a good choice for long term orbital maneuvers (when you have small thruster and are interested to see the changes in orbit in a long term basis).

The whole concept of osculating to mean came with two paper from 1959:

• The motion of a close earth satellite Kozai
• Solution of the problem of artificial satellite theory without drag Brouwer

They did not cover all the orbits, so Lyddane done some correction to them:

• Numerical comparison between Brouwer's theory and solution by Cowell's method for the orbit of an artificial satellite Lyddane

These are relevant for Keplerian Elements.

Books and Papers which can help you for Equinoctial Elements:

• Analytical Mechanics of Space Systems Schaub and Jenkins
• An Introduction to Astrodynamics Battin
• On the Equinoctial Orbit Elements Burke and Cefola
• The Long-Term prediction of artificial Satellite Orbit Cefola
• A computerized algebraic utility for the construction of nonsingular satellite theories Zeis

If you want to test your implementation or the mathematics use Orekit, it has most of these orbit types implemented.

I hadn't heard about equinoctial elements until just now, but this is too long for a comment.

This doesn't quite do everything you want, and there are several caveats (in some cases, converting a state vector to osculating elements yields a nonsense results, particularly if one body doesn't actually orbit the other -- eg, if you pretend Charon revolves around Earth, for example), but, if you look at the source code, it might help.