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And, if so, will we have to change its name to Distal Centauri ;-) ?

Granted, its orbital period around the other two is about 550,000 Earth years, so we have some time....

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In about 26,000 years, Proxima Centauri will no longer be the closest star. Instead, Alpha Centauri AB will be the closest, but only for about ~7000 years, soon replaced by Ross 248 for 10000 years, and Gliese 445 for 6000 years. Then, Alpha Centauri will continue to be the nearest star for another ~28000 years, then briefly becoming Ross 128. Refer to the diagram below (sourced from Wikipedia).

star distances

Correct me if I'm wrong, and feel free to edit my answer if it needs improvement.

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    $\begingroup$ The plot is out of date (thanks to Gaia), though your conclusions may still be correct. $\endgroup$
    – ProfRob
    Mar 3 at 23:29

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