Saturn's tropical year? What is the value of Saturn's "tropical year" (as opposed to its "sidereal year").

I am seeing it listed as 10 days shorter (10,747 days) than the sidereal year (10,759.22 days).

How could this be if the precession of the axis (equinox) is so slow (~1.86M years)? Is this an "transient/osculating" value? Frame of reference (e.g. J2000)? Am I completely missing something?

I think I can calculate the tropical year by (close to, this works for Earth):

$P_{tropical} = P_{sidereal} \cdot (1-\frac{P_{sidereal} }{P_{precession}})$


Using NASA planetary facts data on Saturn
(these figures are re-used on many websites):

$10759.22\cdot(1-\frac{10759.22}{1.86\times10^6 \space\times\space 365.256})=10759.05$
Using JPL Horizons data:

$10755.698\cdot(1-\frac{10755.698}{1.86\times10^6 \space\times\space 365.256})=10755.528$


about 0.170 days less.

Also, interestingly, see the instantaneous sidereal year of Saturn from the graph of JPL Horizon's data, using the sun's centre (blue) and solar system barycentre (red). I think the instantaneous tropical year (at epoch) would always be 0.170 days less.

I think this is where the confusion is: texts have used particular values, and further the same text has picked a sidereal from one epoch (or average), and a tropical from another...



Saturn's instantaneous length of sidereal year in days, at epoch.
A quick skim of a few papers on Earth's tropical year, e.g. M.J.White, suggests to me that existing formulas are best-fit to data, not analytic.  How Saturn's ellipticity of orbit &amp; axial precession(s) play into this is beyond my learning, but I'm getting the feeling that "it just is," and that any number of cofactors such as planetary layers, densities, and so on might affect what's being observed.