<p>I'm trying to propagate the orbit of the ISS, but it became very clear to me very quickly that the J2 perturbations were nothing to scoff at.</p>
<p>A quick google search gave me the equation for the change in longitude of the ascending node over time, given <a href="https://ai-solutions.com/_freeflyeruniversityguide/j2_perturbation.htm" rel="nofollow noreferrer">here</a>, but in the site and also through comparing my model to Celestrak it's very clear that the change in the argument of periapsis over time is also significant.</p>
<p>For some reason, only the equation for the change in RAAN over time is given, but not the one for the argument of periapsis. Is there an equation for this?</p>

CC BY-SA 4.0
2021-08-14T15:02:11.670Z
Although orbital mechanics questions are on-topic here, questions about artificial satellites are better suited to our sister site https://space.stackexchange.com/questions/tagged/orbit
PM 2Ring
CC BY-SA 4.0
2021-08-14T15:10:52.813Z
Bear in mind that the ISS orbit is re-boosted frequently. https://space.stackexchange.com/q/9087/38535
PM 2Ring
CC BY-SA 4.0
2021-08-14T15:26:23.840Z
This question does seem ontopic since the OP is asking for the equation that governs the $J_2$ general relativistic correction to the trajectory of a satellite.
Daddy Kropotkin
CC BY-SA 4.0
2021-08-14T22:24:54.927Z
For example, In Space Exploration SE see [this answer to *Juno's original orbit around Jupiter - is this apsidal precession? If so, need expression*](https://space.stackexchange.com/a/19188/12102) and also [Does apsidal precession rate have any dependence on the argument of periapsis? Perhaps higher order?](https://space.stackexchange.com/q/49629/12102) and its answers. It's a great exercise to use perturbations to to evolve a Keplerian orbit! Remember that these equations are only approximate up to some order, *they are not exact,* not even for pure J2, and J3, J4 have some effects as well.
uhoh
CC BY-SA 4.0
2021-08-14T22:27:07.990Z
Also note that J2 has an effect on the period since $GM$ underestimates the average force of gravity: [Equation for orbital period around oblate bodies, based on J2?](https://space.stackexchange.com/q/25868/12102) If you'd like to get even more accurate, you can look for higher order perturbation corrections and add atmospheric drag which is what SGP4 does when given TLEs, or for your next project start to do some numerical solution; set up the initial velocity and position and integrate the orbit directly.
uhoh
CC BY-SA 4.0
2021-08-15T15:42:23.347Z
Thank you for all these replies! @uhoh those answers that you listed are quite comprehensive, after having a slight glance yes, that's totally what I was looking for, thank you!
Martin Leung
CC BY-SA 4.0
2021-08-15T23:16:12.693Z
@MartinLeung *Welcome to Stack Exchange!* Please feel free to write an answer to your own question if you like. You can simply summarize what you found there as it applies to your question. If you click "edit" on another post you can copy/paste text and equations directly into your answer here. Just include the link where they're from and use a `>` at the beginning of each paragraph or equation to indicate a "block quote". This helps future readers who find your question to also gain from your experience, and is a quick way to pick up a few more reputation points (need >50 to leave comments).
uhoh
<orbit><orbital-mechanics><orbital-elements><mathematics>
What's the formula for change in argument of periapsis over time due to J2 perturbations for a geocentric satellite?
Martin Leung
# What's the formula for change in argument of periapsis over time due to J2 perturbations for a geocentric satellite?

(Asked by: username_0 on 2021-08-14)

I'm trying to propagate the orbit of the ISS, but it became very clear to me very quickly that the J2 perturbations were nothing to scoff at.

A quick google search gave me the equation for the change in longitude of the ascending node over time, given [here](https://ai-solutions.com/_freeflyeruniversityguide/j2_perturbation.htm), but in the site and also through comparing my model to Celestrak it's very clear that the change in the argument of periapsis over time is also significant.

For some reason, only the equation for the change in RAAN over time is given, but not the one for the argument of periapsis. Is there an equation for this?

**Tags:** orbit, orbital-mechanics, orbital-elements, mathematics
> **Comments:**
> **username_1**: Although orbital mechanics questions are on-topic here, questions about artificial satellites are better suited to our sister site https://space.stackexchange.com/questions/tagged/orbit  
> **username_1**: Bear in mind that the ISS orbit is re-boosted frequently. https://space.stackexchange.com/q/9087/38535  
> **username_2**: This question does seem ontopic since the OP is asking for the equation that governs the $J_2$ general relativistic correction to the trajectory of a satellite.  
> **username_3**: For example, In Space Exploration SE see [this answer to *Juno's original orbit around Jupiter - is this apsidal precession? If so, need expression*](https://space.stackexchange.com/a/19188/12102) and also [Does apsidal precession rate have any dependence on the argument of periapsis? Perhaps higher order?](https://space.stackexchange.com/q/49629/12102) and its answers. It's a great exercise to use perturbations to to evolve a Keplerian orbit! Remember that these equations are only approximate up to some order, *they are not exact,* not even for pure J2, and J3, J4 have some effects as well.  
> **username_3**: Also note that J2 has an effect on the period since $GM$ underestimates the average force of gravity: [Equation for orbital period around oblate bodies, based on J2?](https://space.stackexchange.com/q/25868/12102) If you'd like to get even more accurate, you can look for higher order perturbation corrections and add atmospheric drag which is what SGP4 does when given TLEs, or for your next project start to do some numerical solution; set up the initial velocity and position and integrate the orbit directly.  
> **username_0**: Thank you for all these replies! @uhoh those answers that you listed are quite comprehensive, after having a slight glance yes, that's totally what I was looking for, thank you!  
> **username_3**: @MartinLeung *Welcome to Stack Exchange!* Please feel free to write an answer to your own question if you like. You can simply summarize what you found there as it applies to your question. If you click "edit" on another post you can copy/paste text and equations directly into your answer here. Just include the link where they're from and use a `>` at the beginning of each paragraph or equation to indicate a "block quote". This helps future readers who find your question to also gain from your experience, and is a quick way to pick up a few more reputation points (need >50 to leave comments).  

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