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The discovery of the exoplanet Proxima b highlights the potential for thecoming generation of giant segmented mirror telescopes (GSMTs) to characterizeterrestrial --- potentially habitable --- planets orbiting nearby stars withdirect imaging. This will require continued development and implementation ofoptimized adaptive optics systems feeding coronagraphs on the GSMTs. Suchdevelopment should proceed with an understanding of the fundamental limitsimposed by atmospheric turbulence. Here we seek to address this question with asemi-analytic framework for calculating the post-coronagraph contrast in aclosed-loop AO system. We do this starting with the temporal power spectra ofthe Fourier basis calculated assuming frozen flow turbulence, and then applyclosed-loop transfer functions. We include the benefits of a simple predictivecontroller, which we show could provide over a factor of 1400 gain in raw PSFcontrast at 1 $\lambda/D$ on bright stars, and more than a factor of 30 gain onan I = 7.5 mag star such as Proxima. More sophisticated predictive control canbe expected to improve this even further. Assuming a photon noise limitedobserving technique such as High Dispersion Coronagraphy, these gains in rawcontrast will decrease integration times by the same large factors. Predictivecontrol of atmospheric turbulence should therefore be seen as one of the keytechnologies which will enable ground-based telescopes to characterizeterrrestrial planets.

Ground-based adaptive optics coronagraphic performance under closed-loop predictive control