High-contrast imaging observations are fundamentally limited by the spatially and temporally correlated noise source called speckles. Suppression of speckle noise is the key goal of wavefront control and adaptive optics (AO), coronagraphy, and a host of post-processing techniques. Speckles average at a rate set by the statistical speckle lifetime, and speckle-limited integration time in long exposures is directly proportional to this lifetime. As progress continues in post-coronagraph wavefront control, residual atmospheric speckles will become the limiting noise source in high-contrast imaging, so a complete understanding of their statistical behavior is crucial to optimizing high-contrast imaging instruments. Here we present a novel power spectral density (PSD) method for calculating the lifetime, and develop a semi-analytic method for predicting intensity PSDs behind a coronagraph. Considering a frozen-flow turbulence model, we analyze the residual atmosphere speckle lifetimes in a MagAO-X-like AO system as well as 25--39 m giant segmented mirror telescope (GSMT) scale systems. We find that standard AO control shortens atmospheric speckle lifetime from ~130 ms to ~50 ms, and predictive control will further shorten the lifetime to ~20 ms on 6.5 m MagAO-X. We find that speckle lifetimes vary with diameter, wind speed, seeing, and location within the AO control region. On bright stars lifetimes remain within a rough range of ~20 ms to ~100 ms. Due to control system dynamics there are no simple scaling laws which apply across a wide range of system characteristics. Finally, we use these results to argue that telemetry-based post-processing should enable ground-based telescopes to achieve the photon-noise limit in high-contrast imaging.
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