The discovery of the exoplanet Proxima b highlights the potential for the
coming generation of giant segmented mirror telescopes (GSMTs) to characterize
terrestrial --- potentially habitable --- planets orbiting nearby stars with
direct imaging. This will require continued development and implementation of
optimized adaptive optics systems feeding coronagraphs on the GSMTs. Such
development should proceed with an understanding of the fundamental limits
imposed by atmospheric turbulence. Here we seek to address this question with a
semi-analytic framework for calculating the post-coronagraph contrast in a
closed-loop AO system. We do this starting with the temporal power spectra of
the Fourier basis calculated assuming frozen flow turbulence, and then apply
closed-loop transfer functions. We include the benefits of a simple predictive
controller, which we show could provide over a factor of 1400 gain in raw PSF
contrast at 1 λ/D on bright stars, and more than a factor of 30 gain on
an I = 7.5 mag star such as Proxima. More sophisticated predictive control can
be expected to improve this even further. Assuming a photon noise limited
observing technique such as High Dispersion Coronagraphy, these gains in raw
contrast will decrease integration times by the same large factors. Predictive
control of atmospheric turbulence should therefore be seen as one of the key
technologies which will enable ground-based telescopes to characterize
terrrestrial planets.Comment: Accepted to JATI