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I'm simulating the motion of a body in a Geocentric orbit by integrating the equation of motion.

I'm using the following equations for the acceleration of an object in a gravitational field defined by the central mass given by $\mu$ and an axisymmetric oblateness described by $J_2$:

enter image description here

Motion will be a Keplerian orbit, perturbed by the effects of $J_2$.

For increased accuracy, I should also consider other effects of Earth geometry, i.e J22, J3, J4. How could I modify the equation?

I would appreciate if you give links to papers.

uhoh
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Tarlan Mammadzada
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  • +1 While I've touched on the spherical harmonics of the Geopotential in this answer this new question needs a more thorough and mathematical answer than I can easily provide. Hopefully a "gravity person" will be able to address this better. – uhoh Feb 12 '18 at 08:23
  • @uhoh Do you know someone to help with this? – Tarlan Mammadzada Feb 14 '18 at 12:35
  • @uhoh I mean, how to use J22, J3, J4 in equation? Edited the question – Tarlan Mammadzada Feb 14 '18 at 18:13
  • The other terms are much smaller than J2, don't worry about them now. Why don't you re-read the first section Newtonian Gravity in my answer and then ask specific questions about whatever is there that's not clear enough. The Wikipedia article I link to there is pretty good. – uhoh Feb 14 '18 at 18:19
  • @uhoh I read all your answers. The things I'm worried about are smaller terms. If you think, that the equation in my question is good enough, please post it in answer – Tarlan Mammadzada Feb 14 '18 at 18:28
  • I will comment now, try to write a good answer later. In Stack Exchange most people don't write answers until they are able to write good answers (although some people don't worry about it). I can't say something is good enough because I don't know what your idea of good enough is of course! Usually if high accuracy is needed, people will use a gravity model with a much higher order and many more coefficients. I only added terms beyond J2 to let you know they exist, not to recommend that you use them. it's late; t I will just say ignore them. – uhoh Feb 14 '18 at 18:40
  • @uhoh I need very high accuracy, because I will propagate a real satellite. That's why I'm worried about smallest terms. Later, I would ask also about the Solar Radiation – Tarlan Mammadzada Feb 14 '18 at 18:58
  • @uhoh what a bizarre suggestion to someone specifically asking about adding the effects of higher order gravity terms. The only thing more bizarre than that is that the answer you linked to includes GR, which is near the bottom if the list of "things to worry about when simulating a low Earth orbit". Don't feel compelled to comment on every question you have an opinion on - if you can't answer, it's okay to say nothing! – Chris Feb 15 '18 at 01:49
  • @Chris I will agree there is a bit of surreality here, but look more carefully. Check the timeline and edit history. When I wrote that comment, the question asked "Should I consider for better accuracy also other effects of Earth geometry, i.e J22, J3, J4? If so, how could I modify the equation?" and the comment above asks "If you think, that the equation in my question is good enough, please post it in answer." Of course no one can answer 'what is good enough' or how much accuracy someone else should use... – uhoh Feb 15 '18 at 02:16
  • @Chris and in the previous comment I say very specifically and using bold to only "re-read the first section Newtonian Gravity". If you look at that section there are three numbered references and the answer to how to use these coefficients is really there. I've asked several times in several places for the OP to read my previous answers more carefully first. Why don't you review several of the OPs questions and all comments and compare them to how we generally encourage people to write questions. Also take note of the kinds of answers I've written for OP vs what others have done. – uhoh Feb 15 '18 at 02:20
  • @Chris if after a careful look at the big picture, you feel you can help better, consider doing so! – uhoh Feb 15 '18 at 02:27
  • @TarlanMammadzada after seeing your newest comment "I need very high accuracy, because I will propagate a real satellite." I have to say that this is asking way too much for a Stack Exchange question. You're asking for a whole textbook. People just use an SGP4 propagator, generally people don't write them. I thought you were just trying to learn orbital mechanics. If you really need accurate satellite propagation, use a standard program. Search this SE site or the web for "SGP4". – uhoh Feb 15 '18 at 02:31
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    @uhoh Yes, I'm learning, but checking results on real tests. Thanks, I'll look on SGP4. – Tarlan Mammadzada Feb 15 '18 at 05:03
  • @uhoh Does the SGP4 give highly accurate result? And, do you think that trying to write the program for high accuracy is bad idea? – Tarlan Mammadzada Feb 15 '18 at 07:39
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    I just ran across this answer. You may find it very interesting and possibly helpful. The links/references are great! – uhoh Mar 08 '18 at 09:04

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I would recommend picking up a basic Astrodynamics book such as Vallado's "Fundamentals of Astrodynamics and Applications". Depending on what your orbit altitude is (and what level of fidelity you are trying to get to), you will need to model many more gravity terms plus third body perturbations (sun and moon), drag, and solar radiation pressure.

Tom Johnson
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