3.3.2 Power-law spectrum example

Consider a distant AGN with a power law spectrum of energy slope 0.5 which is at a luminosity distance of 1000 Mpc and is viewed through an absorption column of 5 x 10²⁰ cm^-2 (log(N _H) = 20.7). The AGN luminosity in the ROSAT band is L_x = 10⁴⁴ ergs s^-1. We want to determine how long an observation is required for a 10$\sigma$ detection. The unabsorbed flux is F_{S, 0} = 8.37 x 10^-13 ergs cm^-2 s^-1. The ECF for this spectral model is taken from Table 8 or Figure 30, it is 1.75 x 10¹⁰, which gives a count rate 0.0146 ct s^-1. Using the sensitivity formula from Section 3.2.1 with R=0.0073 ct s^-1 (50% of the total so as to refer to the 5  '' x 5'' detect cell), we calculate that the observation time needed to detect this source at the 10$\sigma$ level is about 13,700 seconds.