I have some orbit, with a given semi-major axis, inclination, eccentricity, longitude of ascending node, argument of periapsis, and true anomaly. How can I, from this, calculate the position and velocity as Cartesian vectors?

For the sake of this, assume that the parent body is stationary at the origin.

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    $\begingroup$ Does this answer your question? Converting Orbital Elements to Cartesian State Vectors $\endgroup$ Apr 18, 2020 at 22:49
  • $\begingroup$ Or this: space.stackexchange.com/q/12107/6944 $\endgroup$ Apr 18, 2020 at 22:54
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    $\begingroup$ @uhoh sounds reasonable, retracted vote $\endgroup$ Apr 19, 2020 at 3:36
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    $\begingroup$ About the above links: The first one contains links to pdfs which explain the algorithm, but they're difficult to parse (and I think SE wants the answer to not depend on external links). The top answer depends on dead links. The second one talks about how to compute osculating vs. mean elements (more general than my question, at the cost of being much more complicated by forgoing orbital dynamics). The only answer depends entirely on links to things behind a paywall (not a good answer for this site). $\endgroup$ Apr 19, 2020 at 4:05
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    $\begingroup$ The third one is about going the other direction, is just someone who wants their code debugged, and has no answers posted. The fourth link has a lot of equations, and the answer only explains some of them. The fifth link talks about non-Keplerian mechanics, which is not what I was talking about and, therefore, the answers don't answer my question. If a wiki of some kind to link all questions on this topic is being made, they may be useful things to link to, but none of them answer the question. $\endgroup$ Apr 19, 2020 at 4:05

1 Answer 1


I've started a Community Wiki answer so that we can consolidate all the best answers as links here, accompanied by short explanations.

I've chosen this question to do so because the question is simple and short and so it requires the most general answer.

From @OrganicMarble's comments:

I believe there are other answers that contain actual step-by-step instructions how to go from a full set of Keplerian elements to a state vector, but I haven't found them yet.

One way to find more answers will be to search for "eccentricity vector" since I think that that's central to the conversion in either direction.

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    $\begingroup$ You may remove the line which says "rotted PDFs", I got that fixed. $\endgroup$ Apr 20, 2020 at 13:11
  • $\begingroup$ @WilliamR.Ebenezer That's great! This is a community wiki answer, anybody can and is encouraged to edit this to improve it. Go for it! $\endgroup$
    – uhoh
    Apr 20, 2020 at 16:57
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    $\begingroup$ Ah, alright. Editing done! $\endgroup$ Apr 20, 2020 at 17:57

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