Focusing mirrors for X-ray telescopes in grazing incidence, introduced in the
70s, are characterized in terms of their performance by their imaging quality
and effective area, which in turn determines their sensitivity. Even though the
on-axis effective area is assumed in general to characterize the collecting
power of an X-ray optic, the telescope capability of imaging extended X-ray
sources is also determined by the variation in its effective area with the
off-axis angle. [...] The complex task of designing optics for future X-ray
telescopes entails detailed computations of both imaging quality and effective
area on- and off-axis. Because of their apparent complexity, both aspects have
been, so far, treated by using ray-tracing routines aimed at simulating the
interaction of X-ray photons with the reflecting surfaces of a given focusing
system. Although this approach has been widely exploited and proven to be
effective, it would also be attractive to regard the same problem from an
analytical viewpoint, to assess an optical design of an X-ray optical module
with a simpler calculation than a ray-tracing routine. [...] We have developed
useful analytical formulae for the off-axis effective area of a
double-reflection mirror in the double cone approximation, requiring only an
integration and the standard routines to calculate the X-ray coating
reflectivity for a given incidence angle. [...] Algebraic expressions are
provided for the mirror geometric area, as a function of the off-axis angle.
Finally, the results of the analytical computations presented here are
validated by comparison with the corresponding predictions of a ray-tracing
code.Comment: 12 pages, 11 figures, accepted for publication in "Astronomy &
Astrophysics", section "Instruments, observational techniques, and data
processing". Updated version after grammatical revision and typos correctio
