In years, what is the time it takes our sun to orbit around our solar system's barycenter?
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$\begingroup$ There isn't a stable location of the barycenter, so.... $\endgroup$ – Carl Witthoft Oct 16 '18 at 17:29
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As you can see from the diagrams below of the motion of the Solar System's barycenter relative to the Sun (courtesy of Wikipedia), the Sun's motion with respect to the system's barycenter is not a simple closed curve, and it doesn't have a simple period.
(Click the above image for a SVG version).
It is primarily dominated by Jupiter's period (11.862 years) because Jupiter has most of the non-solar mass of the Solar System. (On a related note, Jupiter has most of the angular momentum of the solar system). Jupiter's mass is approximately 0.0009543 solar masses, so the Sun is almost 1048 times more massive than Jupiter.
According to Wikipedia
To calculate the actual motion of the Sun, you only need to consider the motions of the four giant planets (Jupiter, Saturn, Uranus, Neptune). The contributions of all other planets, dwarf planets, etc. are negligible.
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2$\begingroup$ @sunSis It varies quite a bit, but it's roughly 12 years. $\endgroup$ – PM 2Ring Oct 16 '18 at 1:16
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7$\begingroup$ @SunSis - The correct answer is in this answer: "the Sun's motion with respect to the system's barycenter is not a simple closed curve, and it doesn't have a simple period." By insisting on a simple period, you are demanding an answer that doesn't exist. $\endgroup$ – David Hammen Oct 16 '18 at 10:30
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