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The closest star system after our sun is Alpha Centauri, a three star system with Proxima Centauri, a red dwarf, slightly closer to us than the other 2, around 4.3 light years. I tried Wikipedia and other articles that describe Alpha Centauri and there is plenty of information, but no where could a find an answer to said question.

My presumption is that if the distance is measurable, so should the variance in this distance.

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Yes, it's moving closer.

Per Wikipedia.

Source.

In about 30,000 years Alpha Centauri will be 3 (and change) light years away, then it will start moving further away.

If anyone wants to provide a more detailed explanation on stellar movement and charts, please feel free.

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Yes the Alpha Centauri system is moving closer and it's not the only one. Barnard's Star, the fastest moving star in our skies, will get close sooner. The major concern is whether or not they will get close enough to disrupt our Oort cloud.

Near stars, past & future

This diagram shows the relative distance of various nearby stars over time. So Alpha Centauri was close to 6LY away 20,000 years ago and will be at its closest position at around 3LY about 30,000 years from now.

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    $\begingroup$ I have edited the picture into your text, but have a problem interpreting it. What are we seeing here exactly? More importantly: where does this picture come from. Please edit your answer and give proper attribution. $\endgroup$
    – user1569
    Commented Nov 12, 2018 at 9:29

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