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Optical modules for X-ray telescopes comprise several double-reflectionmirrors operating in grazing incidence. The concentration power of an opticalmodule, which determines primarily the telescope's sensitivity, is in generalexpressed 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 thesource moves off-axis, with a consequent loss of sensitivity. To make mattersworse, the dense nesting of mirror shells in an optical module results in amutual obstruction of their aperture when an astronomical source is off-axis,with a further effective area reduction. [...] While the effective area of anX-ray mirror is easy to predict on-axis, the same task becomes more difficultfor a source off-axis. It is therefore important to develop an appropriateformalism to reliably compute the off-axis effective area of a Wolter-I mirror,including the effect of obstructions. Most of collecting area simulation forX-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. Incontrast, in a previous paper we approached this problem from an analyticalviewpoint, to the end of simplifying and speeding up the prediction of theoff-axis effective area of unobstructed X-ray mirrors with any reflectivecoating, including multilayers.In this work we extend the analytical resultsobtained: we show that the analytical formula for the off-axis effective areacan be inverted, and we expose in detail a novel analytical treatment of mutualshell obstruction in densely nested mirror assemblies, which reduces theoff-axis effective area computation to a simple integration. The results are inexcellent agreement with the findings of a detailed ray-tracing routine.

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