Strehl ratio and modulation transfer function for segmented mirror telescopes as functions of segment phase error

We derive the Strehl ratio for a segmented mirror telescope as a function of the rms segment phase error and the observing wavelength, with and without the effects of the atmosphere. A simple analytical expression is given for the atmosphere-free case. Although our specific results are in the context of the Keck telescope, they are presented in a way that should be readily adaptable to other segmented geometries. We also derive the corresponding modulation transfer functions. These results are useful in determining how accurately a segmented mirror telescope needs to be phased for a variety of observing applications.