Are Proxima and Alpha Centauri Gravitationally Bound?
  Using the most recent kinematic and radial velocity data in the literature, we calculate the binding energy of Proxima Centauri relative to the center of mass of the Alpha Centauri system. When we adopt the centroids of the observed data, we find that the three stars constitute a bound system, albeit with a semi-major axis that is on order the same size as Alpha Centauri AB's Hill radius in the galactic potential. We carry out a Monte Carlo simulation under the assumption that the errors in the observed quantities are uncorrelated. In this simulation, 44% of the trial systems are bound, and systems on the 1-3 sigma tail of the radial velocity distribution can have Proxima currently located near the apastron position of its orbit. Our analysis shows that a further, very significant improvement in the characterization of the system can be gained by obtaining a more accurate measurement of the radial velocity of Proxima Centauri. 
Jeremy G. Wertheimer and Gregory Laughlin