<p>Turns out I completely missed the fact that the linked paper, when giving values for <span class="math-container">$\Omega$</span> etc, is not taking the reference plane as the sun’s equatorial plane or the solar system’s invariable plane. But instead it’s taking the ecliptic as the reference plane (for all the planets it lists values for). In the case of earth, this means an inclination of zero. And even though it would not make sense to define <span class="math-container">$\Omega$</span> for a perfectly zero inclination orbit, there is a (small) variation in inclination that the paper also provides formulae for, and this variation (however small) is inherently tied to some well defined ascending node which would make <span class="math-container">$\Omega$</span> well defined. And given a well defined <span class="math-container">$\Omega$</span> it makes perfect sense that at an inclination of zero, apsidal precession is the time derivative of <span class="math-container">$\varpi$</span> and not <span class="math-container">$\omega$</span>.</p>
<p>While this answers the immediate discrepancy I had in calculated and looked up values for period of apsidal precession, this leaves me even more confused about apsidal precession in general. Why was it called to be equal to time derivative of <span class="math-container">$\omega$</span> when even for non zero inclinations, there is a contribution from time derivative of <span class="math-container">$\Omega$</span> in changing the ‘actual’ longitude at which periapsis happens. If it’s not the time derivative of <span class="math-container">$\omega$</span>, what is it defined as exactly? Can’t be time derivative of <span class="math-container">$\varpi$</span> as it’s not exact for non zero inclinations. Is it the time derivative of some ‘longitude’? With respect to the reference plane or perhaps the orbital plane. I’m also unable to visualise, for comparable rates of change of <span class="math-container">$\Omega$</span> and <span class="math-container">$\omega$</span>, what it means for a sidereal year to be completed (so that it may be contrasted with an anomalistic year in the sense of how much is this difference and how do you relate it to time derivative of <span class="math-container">$\Omega$</span> etc).</p>
<p>In light of this I am leaving this answer without the green check. Hopefully someone can help</p>

CC BY-SA 4.0
2024-03-24T23:45:06.990Z
2024-03-24T23:45:06.990Z
anonymous
<p>The Wikipedia page on <a href="https://en.wikipedia.org/wiki/Apsidal_precession" rel="nofollow noreferrer">apsidal precession</a> states that the rate of apsidal precession is the first time derivative of the argument of periapsis (<span class="math-container">$\omega$</span>), which is related to the longitude of ascending node (<span class="math-container">$\Omega$</span>) and longitude of periapsis (<span class="math-container">$\varpi$</span>) by <span class="math-container">$\omega = \varpi - \Omega$</span>.</p>
<p>The <a href="https://ui.adsabs.harvard.edu/abs/1994A&amp;A...282..663S" rel="nofollow noreferrer">Wikipedia reference</a> (in the <a href="https://en.wikipedia.org/wiki/Earth%27s_orbit#cite_note-VSOP87-15" rel="nofollow noreferrer">earth's orbit page</a>) for these values gives the following expressions (on page 13 of 21) for earth's orbit where <span class="math-container">$t$</span> is &quot;TDB measured in thousands of Julian years from J2000&quot; (I truncated these till the first order terms only when writing here):
<span class="math-container">$$\varpi = 102.93734808^\circ + 11612.35290'' t + \dots$$</span>
<span class="math-container">$$\Omega = 174.87317577^\circ - 8679.27034'' t + \dots$$</span></p>
<p>meaning that the current rate of apsidal precession (current would mean <span class="math-container">$t=0$</span> or at J2000) is <span class="math-container">$\frac{d\omega}{dt} = 20291.62324$</span> (in arc-seconds per 1000 years) which translates to a full <span class="math-container">$360^\circ$</span> in <span class="math-container">$63868$</span> years which is way off from the supposed <span class="math-container">$112000$</span> years I read everywhere else. In fact, <span class="math-container">$\frac{d\varpi}{dt}$</span> gets much closer and corresponds to a full <span class="math-container">$360^\circ$</span> in <span class="math-container">$111605$</span> years</p>
<p>So where exactly am I going wrong? It did make intuitive sense for apsidal precession to be the rate of change of <span class="math-container">$\omega$</span> instead of <span class="math-container">$\varpi$</span>. Also, likely unrelated question, is the rate of change of <span class="math-container">$\Omega$</span> what is referred to as planetary precession?</p>

CC BY-SA 4.0
2024-03-16T22:56:33.127Z
Well, first of all, I see that you did 11612 + 8679 = 20291, but it’s MINUS 8679, no?
Pierre Paquette
CC BY-SA 4.0
2024-03-17T00:38:38.647Z
Since 2006, the preferred term for planetary precession is precession of the ecliptic. See https://astronomy.stackexchange.com/q/48903/16685
PM 2Ring
CC BY-SA 4.0
2024-03-17T00:39:37.867Z
@PierrePaquette We're subtracting derivatives, but they have opposite signs.
PM 2Ring
CC BY-SA 4.0
2024-03-17T08:52:06.683Z
@PM2Ring after reading the question you had linked, specifically the second comment of the accepted answer, my understanding is that the rate of the precession of the ecliptic would  indeed be equal to $\frac{d\Omega}{dt}$ had the reference plane been taken as the solar system's invariable plane (in contrast to the linked article which gives values taking sun's equatorial plane as the reference plane). Did I understand it correctly?
anonymous
CC BY-SA 4.0
2024-03-19T18:17:32.063Z
By the way, is there something unclear in the way I have phrased my question? If so I'd like to know so I can edit it. When posting this question I expected to be quickly told what obvious oversight I am making because from where I stand this seems such a simple situation and that I must just be interpreting things wrongly. If that's not the case, is that my logic is wrong? Is the reference paper's values wrong? Am I trying to equate unrelated quantities? Am I interpreting the definitions wrong? Or is it just that my question is not upto SE code or that I haven't shown enough effort of my own?
anonymous
<orbital-elements><precession><apsidal-precession>
Why is rate of apsidal precession not consistent with the first time derivative of the argument of periapsis
anonymous
# Why is rate of apsidal precession not consistent with the first time derivative of the argument of periapsis

(Asked by: username_0 on 2024-03-16)

The Wikipedia page on [apsidal precession](https://en.wikipedia.org/wiki/Apsidal_precession) states that the rate of apsidal precession is the first time derivative of the argument of periapsis ($\omega$), which is related to the longitude of ascending node ($\Omega$) and longitude of periapsis ($\varpi$) by $\omega = \varpi - \Omega$.

The [Wikipedia reference](https://ui.adsabs.harvard.edu/abs/1994A&A...282..663S) (in the [earth's orbit page](https://en.wikipedia.org/wiki/Earth%27s_orbit#cite_note-VSOP87-15)) for these values gives the following expressions (on page 13 of 21) for earth's orbit where $t$ is "TDB measured in thousands of Julian years from J2000" (I truncated these till the first order terms only when writing here):
$$\varpi = 102.93734808^\circ + 11612.35290'' t + \dots$$
$$\Omega = 174.87317577^\circ - 8679.27034'' t + \dots$$

meaning that the current rate of apsidal precession (current would mean $t=0$ or at J2000) is $\frac{d\omega}{dt} = 20291.62324$ (in arc-seconds per 1000 years) which translates to a full $360^\circ$ in $63868$ years which is way off from the supposed $112000$ years I read everywhere else. In fact, $\frac{d\varpi}{dt}$ gets much closer and corresponds to a full $360^\circ$ in $111605$ years

So where exactly am I going wrong? It did make intuitive sense for apsidal precession to be the rate of change of $\omega$ instead of $\varpi$. Also, likely unrelated question, is the rate of change of $\Omega$ what is referred to as planetary precession?

**Tags:** orbital-elements, precession, apsidal-precession

> **Comments:**
> **username_1**: Well, first of all, I see that you did 11612 + 8679 = 20291, but it’s MINUS 8679, no?  
> **username_2**: Since 2006, the preferred term for planetary precession is precession of the ecliptic. See https://astronomy.stackexchange.com/q/48903/16685  
> **username_2**: @username_1 We're subtracting derivatives, but they have opposite signs.  
> **username_0**: @username_2 after reading the question you had linked, specifically the second comment of the accepted answer, my understanding is that the rate of the precession of the ecliptic would  indeed be equal to $\frac{d\Omega}{dt}$ had the reference plane been taken as the solar system's invariable plane (in contrast to the linked article which gives values taking sun's equatorial plane as the reference plane). Did I understand it correctly?  
> **username_0**: By the way, is there something unclear in the way I have phrased my question? If so I'd like to know so I can edit it. When posting this question I expected to be quickly told what obvious oversight I am making because from where I stand this seems such a simple situation and that I must just be interpreting things wrongly. If that's not the case, is that my logic is wrong? Is the reference paper's values wrong? Am I trying to equate unrelated quantities? Am I interpreting the definitions wrong? Or is it just that my question is not upto SE code or that I haven't shown enough effort of my own?  

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## 1 answers

### Answer by username_0 on 2024-03-24. Score: 3

Turns out I completely missed the fact that the linked paper, when giving values for $\Omega$ etc, is not taking the reference plane as the sun’s equatorial plane or the solar system’s invariable plane. But instead it’s taking the ecliptic as the reference plane (for all the planets it lists values for). In the case of earth, this means an inclination of zero. And even though it would not make sense to define $\Omega$ for a perfectly zero inclination orbit, there is a (small) variation in inclination that the paper also provides formulae for, and this variation (however small) is inherently tied to some well defined ascending node which would make $\Omega$ well defined. And given a well defined $\Omega$ it makes perfect sense that at an inclination of zero, apsidal precession is the time derivative of $\varpi$ and not $\omega$.

While this answers the immediate discrepancy I had in calculated and looked up values for period of apsidal precession, this leaves me even more confused about apsidal precession in general. Why was it called to be equal to time derivative of $\omega$ when even for non zero inclinations, there is a contribution from time derivative of $\Omega$ in changing the ‘actual’ longitude at which periapsis happens. If it’s not the time derivative of $\omega$, what is it defined as exactly? Can’t be time derivative of $\varpi$ as it’s not exact for non zero inclinations. Is it the time derivative of some ‘longitude’? With respect to the reference plane or perhaps the orbital plane. I’m also unable to visualise, for comparable rates of change of $\Omega$ and $\omega$, what it means for a sidereal year to be completed (so that it may be contrasted with an anomalistic year in the sense of how much is this difference and how do you relate it to time derivative of $\Omega$ etc).

In light of this I am leaving this answer without the green check. Hopefully someone can help


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