Alpha Centauri A is the closest solar-type star to the Sun and offers an excellent opportunity to detect the thermal emission of a mature planet heated by its host star. The MIRI coronagraph on JWST can search the 1-3 AU (1-2) region around alpha Cen A which is predicted to be stable within the alpha Cen AB system. We demonstrate that with reasonable performance of the telescope and instrument, a 20 hr program combining on-target and reference star observations at 15.5 um could detect thermal emission from planets as small as ~5 RE. Multiple visits every 3-6 months would increase the geometrical completeness, provide astrometric confirmation of detected sources, and push the radius limit down to ~3 RE. An exozodiacal cloud only a few times brighter than our own should also be detectable, although a sufficiently bright cloud might obscure any planet present in the system. While current precision radial velocity (PRV) observations set a limit of 50-100 ME at 1-3 AU for planets orbiting alpha Cen A, there is a broad range of exoplanet radii up to 10 RE consistent with these mass limits. A carefully planned observing sequence along with state-of-the-art post-processing analysis could reject the light from alpha Cen A at the level of ~10^-5 at 1-2 and minimize the influence of alpha Cen B located 7-8 away in the 2022-2023 timeframe. These space-based observations would complement on-going imaging experiments at shorter wavelengths as well as PRV and astrometric experiments to detect planets dynamically. Planetary demographics suggest that the likelihood of directly imaging a planet whose mass and orbit are consistent with present PRV limits is small, ~5%, and possibly lower if the presence of a binary companion further reduces occurrence rates. However, at a distance of just 1.34 pc, alpha Cen A is our closest sibling star and certainly merits close scrutiny.
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