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 V …