Hartmann testing of x-ray telescopes is a simple test method to retrieve and analyze alignment errors and low-order circumferential errors of x-ray telescopes and their components. A narrow slit is scanned along the circumference of the telescope in front of the mirror and the centroids of the images are calculated. From the centroid data, alignment errors, radius variation errors, and cone-angle variation errors can be calculated. Mean cone-angle, mean radial height (average radius), and the focal length of the telescope can also be estimated if the centroid data is measured at multiple focal plane locations. This test is the only viable way of verifying the alignment of tightly nested x-ray telescopes.
In this paper we present the basic equations that are used in the analysis process. These equations can be applied to full circumference or segmented x-ray telescopes. We use the Optical Surface Analysis Code (OSAC) to model a segmented x-ray telescope and show that the derived equations and accompanying analysis retrieves the alignment errors and low order circumferential errors accurately.