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The angular resolution of long-exposure images from ground-based telescopes equipped with laser guide star adaptive optics systems is fundamentally limited by the the accuracy with which the tip-tilt aberrations introduced by the atmosphere can be corrected. Assuming that a natural star is used as the tilt reference, the residual error due to tilt anisoplanatism can significantly degrade the long-exposure resolution even if the tilt reference star is separated from the object being imaged by a small angle. Given the observed distribution of stars in the sky, the need to find a tilt reference star quite close to the object restricts the fraction of the sky over which long-exposure images with diffraction limited resolution can be obtained. In this paper, the authors present a comprehensive performance analysis of tip-tilt compensation systems that use a natural star as a tilt reference, taking into account properties of the atmosphere and of the Galactic stellar populations, and optimizing over the system operating parameters to determine the fundamental limits to the long-exposure resolution. Their results show that for a ten meter telescope on Mauna Kea, if the image of the tilt reference star is uncorrected, about half the sky can be imaged in the Vmore » band with long-exposure resolution less than 60 milli-arc-seconds (mas), while if the image of the tilt reference star is fully corrected, about half the sky can be imaged in the V band with long-exposure resolution less than 16 mas. Furthermore, V band images long-exposure resolution of less than 16 mas may be obtained with a ten meter telescope on Mauna Kea for unresolved objects brighter than magnitude 22 that are fully corrected by a laser guide star adaptive optics system. This level of resolution represents about 70% of the diffraction limit of a ten meter telescope in the V band and is more than a factor of 45 better than the median seeing in the V band on Mauna Kea.« less