# Why is rate of apsidal precession not consistent with the first time derivative of the argument of periapsis

The Wikipedia page on 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 (in the earth's orbit page) 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?

• Well, first of all, I see that you did 11612 + 8679 = 20291, but it’s MINUS 8679, no? Commented Mar 16 at 22:56
• Since 2006, the preferred term for planetary precession is precession of the ecliptic. See astronomy.stackexchange.com/q/48903/16685 Commented Mar 17 at 0:38
• @PierrePaquette We're subtracting derivatives, but they have opposite signs. Commented Mar 17 at 0:39
• 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? Commented Mar 19 at 18:17
• @anonymous I think it's just that the person with the right type of expertise hasn't seen it yet. This is not the most active SE in the world, and not everyone who does browse here is able to answer every question Commented Mar 20 at 12:18

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).