Design constraints on Cherenkov telescopes with Davies-Cotton reflectors

Abstract: This paper discusses the construction of high-performance ground-based gamma-ray Cherenkov telescopes with a Davies-Cotton reflector. For the design of such telescopes, usually physics constrains the field-of-view, while the photo-sensor size is defined by limited options. Including the effect of light-concentrators in front of the photo sensor, it is demonstrated that these constraints are enough to mutually constrain all other design parameters. The dependability of the various design parameters naturally arises once a relationship between the value of the point-spread functions at the edge of the field-of-view and the pixel field-of-view is introduced. To be able to include this constraint into a system of equations, an analytical description for the point-spread function of a tessellated Davies-Cotton reflector is derived from Taylor developments and ray-tracing simulations. Including higher order terms renders the result precise on the percent level. Design curves are provided within the typical phase space of Cherenkov telescopes. The impact of all design parameters on the overall design is discussed. Allowing an immediate comparison of several options with identical physics performance allows the determination of the most cost efficient solution. Emphasize is given on the possible application of solid light concentrators with their typically about two times better concentration allowing the use of small photo sensors such as Geiger-mode avalanche photo diodes. This is discussed in more details in the context of possible design options for the Cherenkov Telescope Array. In particular, a solution for a 60\,mm(^2) photo sensor with hollow cone is compared to a 36\,mm(^2) with solid cone.