# 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:

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

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.

• This was originally posted in Space SE but the OP has decided to ask here instead. It's not a deliberate cross post. Now, I hope someone here can actually answer this, otherwise we'll feel silly recommending the OP move it here! ;-) – uhoh Jun 11 '18 at 4:06
• The dates in this Planetary Society blog post suggest a tropical year length of 10754.4 days, based on whatever Saturn model is built into JPL HORIZONS. – Mike G Jun 11 '18 at 18:38
• How are you determining the "instantaneous" sidereal year length from HORIZONS? – Mike G Jun 14 '18 at 11:10
• @MikeG "osculating orbital elements". ssd.jpl.nasa.gov/horizons.cgi?s_type=1 JPL Horizons: Ephemeris Type = Orbital ELEMENTS, Body=Saturn, Centre=Sun ... Look for "PR = " numeric – PaulS Jun 14 '18 at 12:38