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The wobble of a star around its barycenter is interesting (the wobble caused by the pull of the rest of the solar-system on the sun).

This question explains why we don't see this wobble from within our own solar-system (because we are mostly orbiting the sun, and not our solar-system's barycenter): What point does Earth actually orbit?

But is it possible to detect from here on Earth, the wobble of neighbouring solar-systems relative to the wobble of our solar-system?

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Yes it is. The Gaia astrometric satellite is and will be doing exactly that. The subtle movement of the photocentre of a target star due to any orbiting planets can be separated from the proper motion and the annual parallax due to our motion around the Sun.

The technique is most sensitive to massive planets that are widely separated from parent stars that are close to the Sun and have low mass themselves, since these provide the biggest angular shift in the star's position with respect to the barycentre. However, the separations cannot be too wide or the orbital period of the star/planet system will be too long for the motion to be detectable on the timescales of the mission. In practice therefore, Gaia will be most sensitive to planets with orbital periods of 1-10 years (or separations of 1-5 au). Simulations suggest that tens of thousands of exoplanets may be revealed in this way (Perryman et al. 2014).

ESA have even provided a little animation to show how it works.

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  • $\begingroup$ Go Gaia!! In my opinion, the most underrated space science mission of our era. $\endgroup$ – Jack R. Woods Jun 9 at 20:12

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