Telescope alignment by out-of-focus stellar image analysis

We present an analysis of the familiar method of analyzing out-of-focus ste llar images to align Cassegrain reflector telescopes. We measure the asymme try in the out-of-focus image as the difference in positions of the centers of the bordering circles. By exact raytracing we find that the asymmetry c an be expressed as a linear combination of tip/tilt and displacement of the secondary mirror. Furthermore and surprisingly, the difference in the posi tions of the centers does not depend on the amount of defocus, and so only two parameters of the linear combination are needed to align the telescope. These parameters depend only on the telescope design parameters. Finally, we report experimental results carried out with the 1.5 m telescope of UNAM , which verify the outlined numerical results.