Is Proxima bound to α Centauri?

Adapted from IOAA 2011 (Katowice), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A planetarium program gives these data for two stars, each of about one solar mass.

The proper motions in right ascension are already multiplied by the cosine of the declination, so they are true angles on the sky. Using these data, decide whether the two stars are bound to each other by gravity. Write YES if they are and NO if they are not.

AnswerNO. The relative speed, at least 1.58 km/s, is more than three times the escape speed of 0.49 km/s at their separation of about 15 000 au. The official solution leaves out the factor $\cos \delta$ in the right-ascension difference, getting 3.09° for the separation, and quotes a speed of about 15 km/s, ten times too large, though its answer is the same. The stars are Proxima and α Centauri. More precise distances and radial velocities measured since suggest that Proxima is in fact bound to the pair, on an orbit lasting more than half a million years, so the answer belongs to the data given.