The optimal on-source region size for detections with counting-type telescopes

Source detection in counting type experiments such as Cherenkov telescopes often involves the application of the classical Eq. 17 from the paper of Li & Ma (1983) to discrete on- and off-source regions. The on- source region is typically a circular area with radius {\theta} in which the signal is expected to appear with the shape of the instrument point spread function (PSF). This paper addresses the question of what is the {\theta} that maximises the probability of detection for a given PSF width and background event density. In the high count number limit and assuming a Gaussian PSF profile, the optimum is found to be at $\zeta_\infty^2 \approx 2.51$ times the squared PSF width $\sigma_\mathrm{PSF39}^2$. While this number is shown to be a good choice in many cases, a dynamic formula for cases of lower count numbers, which favour larger on-source regions, is given. The recipe to get to this parametrisation can also be applied to cases with a non-Gaussian PSF. This result can standardise and simplify analysis procedures, reduce trials and eliminate the need for experience-based ad hoc cut definitions or expensive case-by-case Monte Carlo simulations.