We use more than a decade of radial velocity measurements for $alpha$ Cen A, B, and Proxima Centauri from HARPS, CHIRON, and UVES to identify the $M sin i$ and orbital periods of planets that could have been detected if they existed. At each point in a mass-period grid, we sample a simulated, Keplerian signal with the precision and cadence of existing data and assess the probability that the signal could have been produced by noise alone. Existing data places detection thresholds in the classically defined habitable zones at about $M sin i$ of 53 M$_{oplus}$ for $alpha$ Cen A, 8.4 M$_{oplus}$ for $alpha$ Cen B, and 0.47 M$_{oplus}$ for Proxima Centauri. Additionally, we examine the impact of systematic errors, or red noise in the data. A comparison of white- and red-noise simulations highlights quasi-periodic variability in the radial velocities that may be caused by systematic errors, photospheric velocity signals, or planetary signals. For example, the red-noise simulations show a peak above white-noise simulations at the period of Proxima Centauri b. We also carry out a spectroscopic analysis of the chemical composition of the $alpha$ Centauri stars. The stars have super-solar metallicity with ratios of C/O and Mg/Si that are similar to the Sun, suggesting that any small planets in the $alpha$ Cen system may be compositionally similar to our terrestrial planets. Although the small projected separation of $alpha$ Cen A and B currently hampers extreme-precision radial velocity measurements, the angular separation is now increasing. By 2019, $alpha$ Cen A and B will be ideal targets for renewed Doppler planet surveys.
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