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If Sun-Jupiter baricentre is outside the Sun, does that mean the Sun is orbiting around that point too right?
Then the Sun shouldn't be nearer to the Earth at some point during the Sun-Jupiter orbit (and also farther at the opposite point)?
Sun-Jupiter orbit completes every 11.86 Earth years, so why we don't (or yes we do?) have cycles with hot/cold weather every 6 years?
If you want to model the entire solar system, it's best to model the Sun, the planets, the asteroids, etc. with respect to the barycenter. That choice reduces the equations of motion to their simplest form. Note well: That "simplest form" is not all that simple if one chooses to incorporate relativistic effects into the equations of motion.
On the other hand, if you want to model the behavior of a single planet, and especially the inner terrestrial planets, it's best to model the planet as orbiting the Sun, with the other planets slightly perturbing the orbit of the planet in question. Stealing from an answer to a related question I wrote on physics.SE, the image below depicts the orbit of Venus about the Sun (red curve) and the orbit of Venus about the solar system barycenter (black curve). The horizontal axis is time in days from noon on 1 January 2000. The vertical axis is distance from the Sun (red curve) and distance from the barycenter (black curve).
It's pretty clear from the above graph that it is better, at least for the short term, to model terrestrial planets as orbiting the Sun. Over longer spans of time, the other planets in the solar system perturb the orbit of a terrestrial planet. The greatest of these perturbations is to make the orbit precess. The relativistic precession of Mercury is less than a tenth of the Mercury's non-relativistic precession due to perturbations from other planets.
We would see some variation. The sun orbits the barycenter of our solar system (roughly the barycenter of Sun-Juptiter) at a radius of 742,000 km. This is a variation of roughly 0.5% of the orbital distance of the Earth from the sun.
The Earth is also in an elliptical orbit. The perihelion and aphelion are 5,000,000km apart, about 3.3% of the orbital distance from the Earth to the sun. And its worth remembering that this is not what causes most of our cycles of weather. We are furthest from the sun in June, which is summer in the Northern hemisphere. Weather cycles are more driven by the tilt of the planet.
It would be difficult to find solid environmental evidence of this cycle. The sun happens to go through a 11 year cycle where its output varies by 0.1%. We do indeed see a warming and cooling effect at the extremes of the cycle. The variation from maximum to minimum is about 0.18K, and this effect is applied for a peak that is measured in years rather than one that is measured in months. Research on the solar effects is ongoing and complicated. For instance, there are hypotheses that the solar cycle may change cloud cover, which would make detecting the nearby 11.86 year cycle more tricky.
