As Jupiter and Saturn have very fast rotation periods, about 10 and 11 hours, respectively, I am assuming that their poles will precess much faster then Earth. Since Earth's poles precess every 26000 years, I think Jupiter and Saturn would have a precession time of about ~10000 years or so. Is this estimate correct or am I way off?
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You’ll find the answers you want in the Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements 2015, published by Archinal et al in 2018. I found a copy at https://astropedia.astrogeology.usgs.gov/download/Docs/WGCCRE/WGCCRE2015reprint.pdf. A new version might be available this year, as the committee meets every three years… and take about three years to publish its report!
What you’re looking for are on pages 22 and 23 and are the mathematical formulas for the direction of the North poles of Jupiter and Saturn. These start by $ \alpha_0 $ and $ \delta_0 $ and are rather long for Jupiter, quite short for Saturn.
As for how long it would take their pole to fully precess by 360°, it’s exactly 1 million years for Saturn… as for Jupiter, because the formula is more complex, it’s more difficult to say, but I’d guess around 5.5 million years from the first term should be a good ballpark.
Clear skies!
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$\begingroup$ But why? Saturn's axial tilt is higher than Earth's and has a faster rotation speed. $\endgroup$ Commented Feb 9, 2021 at 15:41
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$\begingroup$ Think what causes the precession. On Earth, the most important factor is the Moon. The Moon is way bigger in regard to Earth than all the Jupiter and Saturn satellites are in regard to their host planets. $\endgroup$– fraxinusCommented Feb 9, 2021 at 22:29
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$\begingroup$ Also, the orbital planes of satellites of other planets are all (basically) aligned with their planet’s equator, whereas the Moon’s orbital plane is tilted 5° to the ecliptic (18–28° to Earth’s equator; it varies, due to orbital precession). $\endgroup$ Commented Feb 9, 2021 at 22:37
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1$\begingroup$ @fasterthanlight I'm curious if "but why" constitutes a separate question or if it's close enough to under the umbrella of this question. Probably a new question like "Why do planets precess". My understanding is that it has to do with lumpiness. Earth (Mars even more so) is lumpy, not uniform and Earth, in addition, is shifting it's mass due to plate tectonics and ice ages. Mars has that impressively large Olympus Mons. Jupiter and Saturn are basically soup that get much more viscous as you go deeper inside. There's much less for other gravitational bodies to tug on. $\endgroup$– userLTKCommented Feb 9, 2021 at 23:23