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

Question

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?

Answer 1

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