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It is obvious that the longitude of ascending node ($ \Omega $) of a planet gradually changes with time. For example, the VSOP planetary theory accounts for this. In Astronomical Algorithms (Meeus 1998), one finds for example that, for Mercury, $ \Omega = 48.330893 - 0.1254227 T - 0.00008833 T^2 - 0.000000200 T^3 $ (referred to the standard equinox J2000.0).

While formulas to calculate the precession of the perihelion of a planet are easy to find (e.g., in “Calculation of apsidal precession via perturbation theory” by Barbieri and Talamucci), I am having a hard time finding sets of formulas pertaining to the advance (or regression!) of the ascending node.

Can someone please give me a few references of where I could find them? Thanks in advance!

(Reason: I’d be interested in calculating precise planetary orbits for an imaginary planetary system, or maybe even for known planetary systems such as TRAPPIST-1.)

EDIT: Most of the formulas found online (including those given in answers and comments) are either restricted to the Solar System, or restricted to one body orbiting another one. I’m looking for something that would be in the form of $ \dot \Omega = \sum_{i = 1}^{\infty} k m_i m_0^x $ where $ m_i $ is the mass of the perturbing body, $ m_0 $ is the mass of the perturbed body, and k and x are, well, what I’m looking for! Their determination probably includes G (the constant of gravitation) and R (the distance between the two bodies) and so on…

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  • $\begingroup$ I'm not sure, but there might be something helpful in sources cited in this answer to a different question. $\endgroup$
    – uhoh
    May 1 at 23:10
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    $\begingroup$ Thanks, but that’s a negative. They only mention the precession due to the central body or stuff like that, never that due to other orbiting bodies. $\endgroup$ May 2 at 0:04
  • $\begingroup$ If the rest of the orbital elements are fixed, the precession of perihelion is equal to the precession of the longitude of the ascending node. $\endgroup$
    – Connor Garcia
    May 2 at 4:22
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    $\begingroup$ It’s clearly not the case. The values given by Meeus 1998 are different from those of the precession of the perihelion, and generally of opposite sign as well. $\endgroup$ May 2 at 7:12
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Perhaps you are looking for Lagrange’s planetary equations? Given some perturbation $H_1$ to the Hamiltonian, they give six equations to calculate the change of the orbital elements over time; a thorough discussion is given in Goldstein’s textbook on classical mechanics. The equation for $\Omega$ is $$\frac{d\Omega}{dt}=\frac{1}{\sqrt{GMm^2(1-e^2)\sin i}}\frac{\partial H_1}{\partial i}$$ where $i$, $e$ are orbital elements.

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  • $\begingroup$ Thanks, but how do I find $ H_1 $ ? Also, I’m not very familiar with derivatives, so I have no clue how to calculate that… $\endgroup$ May 2 at 17:43
  • $\begingroup$ Oh! and this kind of formula is for a single body orbiting a central body. There’s no mention in there of the other orbiting bodies… $\endgroup$ May 2 at 17:44
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I think this is what you want: Simon et al. 1994

You can find Meeus formula in it.

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  • $\begingroup$ Thanks, but no; these are for a specific case (the Solar System), whereas I’m looking for a general case—in other words, I want to get to these formulas myself from general-form formulas. $\endgroup$ May 2 at 17:42

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