Optics for X-ray telescopes: analytical treatment of the off-axis effective area of mirrors in optical modules

Optical modules for X-ray telescopes comprise several double-reflection mirrors operating in grazing incidence. The concentration power of an optical module, which determines primarily the telescope's sensitivity, is in general expressed by its on-axis effective area as a function of the X-ray energy. Nevertheless, the effective area of X-ray mirrors in general decreases as the source moves off-axis, with a consequent loss of sensitivity. To make matters worse, the dense nesting of mirror shells in an optical module results in a mutual obstruction of their aperture when an astronomical source is off-axis, with a further effective area reduction. [...] While the effective area of an X-ray mirror is easy to predict on-axis, the same task becomes more difficult for a source off-axis. It is therefore important to develop an appropriate formalism to reliably compute the off-axis effective area of a Wolter-I mirror, including the effect of obstructions. Most of collecting area simulation for X-ray optical modules has been so far performed along with numerical codes, involving ray-tracing routines, very effective but in general complex, difficult to handle, time consuming and affected by statistical errors. In contrast, in a previous paper we approached this problem from an analytical viewpoint, to the end of simplifying and speeding up the prediction of the off-axis effective area of unobstructed X-ray mirrors with any reflective coating, including multilayers.In this work we extend the analytical results obtained: we show that the analytical formula for the off-axis effective area can be inverted, and we expose in detail a novel analytical treatment of mutual shell obstruction in densely nested mirror assemblies, which reduces the off-axis effective area computation to a simple integration. The results are in excellent agreement with the findings of a detailed ray-tracing routine.