<p>Perhaps you are looking for <a href="https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Book%3A_Celestial_Mechanics_(Tatum)/14%3A_General_Perturbation_Theory/14.04%3A_Lagrange%27s_Planetary_Equations" rel="nofollow noreferrer">Lagrange’s planetary equations?</a> Given some perturbation <span class="math-container">$H_1$</span> 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 <span class="math-container">$\Omega$</span> is <span class="math-container">$$\frac{d\Omega}{dt}=\frac{1}{\sqrt{GMm^2(1-e^2)\sin i}}\frac{\partial H_1}{\partial i}$$</span> where <span class="math-container">$i$</span>, <span class="math-container">$e$</span> are orbital elements.</p>

CC BY-SA 4.0
2021-05-02T17:43:03.173Z
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…
Pierre Paquette
CC BY-SA 4.0
2021-05-02T17:44:05.943Z
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…
Pierre Paquette
CC BY-SA 4.0
2021-05-02T08:01:08.513Z
2021-05-02T08:01:08.513Z
John Dumancic
<p>I think this is what you want: <a href="https://ui.adsabs.harvard.edu/abs/1994A%26A...282..663S/abstract" rel="nofollow noreferrer">Simon et al. 1994</a></p>
<p>You can find Meeus formula in it.</p>

CC BY-SA 4.0
2021-05-02T17:42:29.763Z
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.
Pierre Paquette
CC BY-SA 4.0
2021-05-02T16:53:47.053Z
2021-05-02T18:56:08.213Z
2021-05-02T18:56:08.213Z
Huy Pham
<p>It is obvious that the longitude of ascending node (<span class="math-container">$ \Omega $</span>) of a planet gradually changes with time. For example, the VSOP planetary theory accounts for this. In <em>Astronomical Algorithms</em> (Meeus 1998), one finds for example that, for Mercury, <span class="math-container">$ \Omega = 48.330893 - 0.1254227 T - 0.00008833 T^2 - 0.000000200 T^3 $</span> (referred to the standard equinox J2000.0).</p>
<p>While formulas to calculate the precession of the <em>perihelion</em> of a planet are easy to find (e.g., in “Calculation of apsidal precession via perturbation theory” by <a href="https://ui.adsabs.harvard.edu/abs/2018arXiv180207115B/abstract" rel="nofollow noreferrer">Barbieri and Talamucci</a>), I am having a hard time finding sets of formulas pertaining to the advance (or regression!) of the ascending node.</p>
<p>Can someone please give me a few references of where I could find them? Thanks in advance!</p>
<p>(Reason: I’d be interested in calculating precise planetary orbits for an imaginary planetary system, or maybe even for known planetary systems such as TRAPPIST-1.)</p>
<p>EDIT: Most of the formulas found online (including those given in answers and comments) are either restricted to the Solar System, or restricted to one body orbiting another one. I’m looking for something that would be in the form of <span class="math-container">$ \dot \Omega = \sum_{i = 1}^{\infty} k m_i m_0^x $</span> where <span class="math-container">$ m_i $</span> is the mass of the perturbing body, <span class="math-container">$ m_0 $</span> 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…</p>

CC BY-SA 4.0
2021-05-01T23:10:06.860Z
I'm not sure, but there might be something helpful in sources cited in [this answer](https://space.stackexchange.com/a/49929/12102) to a different question.
uhoh
CC BY-SA 4.0
2021-05-02T00:04:05.627Z
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.
Pierre Paquette
CC BY-SA 4.0
2021-05-02T04:22:50.153Z
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.
Connor Garcia
CC BY-SA 4.0
2021-05-02T07:12:40.797Z
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.
Pierre Paquette
<orbital-mechanics><orbital-elements>
Nodal Precession of Planets
Pierre Paquette
# Nodal Precession of Planets

(Asked by: username_0 on 2021-05-01)

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](https://ui.adsabs.harvard.edu/abs/2018arXiv180207115B/abstract)), 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…

**Tags:** orbital-mechanics, orbital-elements

> **Comments:**
> **username_1**: I'm not sure, but there might be something helpful in sources cited in [this answer](https://space.stackexchange.com/a/49929/12102) to a different question.  
> **username_0**: 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.  
> **username_2**: 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.  
> **username_0**: 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.  

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## 2 answers

### Answer by username_3 on 2021-05-02. Score: 3

Perhaps you are looking for [Lagrange’s planetary equations?](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Book%3A_Celestial_Mechanics_(Tatum)/14%3A_General_Perturbation_Theory/14.04%3A_Lagrange%27s_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.

> **Comments:**
> **username_0**: 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…  
> **username_0**: 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…  

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### Answer by username_4 on 2021-05-02. Score: 1

I think this is what you want: [Simon et al. 1994](https://ui.adsabs.harvard.edu/abs/1994A%26A...282..663S/abstract)

You can find Meeus formula in it.

> **Comments:**
> **username_0**: 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.  

---

