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Spectrum Analysis
The differential flux per area, time and energy interval is defined as
Often is also referred to as as observation time and effective collection area is a constant. The effective area is then defined as . Note that at large distances the efficiency vanishes, so that the effective area is an (energy dependent) constant while and the efficiency are mutually dependent.
For an observation with an effective observation time , this yields in a given Energy interval :
For simplicity, in the following, will be replaced by just but always refers to an average quantity in an given energy interval. Note that strictly speaking the average quantity is not independent of the event distribution in that interval so that the calculation are only exact if the correct distribution is assumed!
The total area and the corresponding efficiency are of course only available for simulated data. For simulated data, is the production area and the corresponding energy dependent efficiency of the analysis chain. For a given energy bin, the efficiency is then defined as
where is the number of simulated events in this energy bin and the number of *excess* events that are produced by the analysis chain.
The number of excess events, for data and simulations, is defined as
where is the number of events identified as potential gammas from the source direction ('on-source') and the number of gamma-like events measured 'off-source'. Note that for Simulations, is not necessarily zero for wobble-mode observations as an event can survive the analysis for on- and off-events, if this is not protected by the analysis chain.
The average number of background events is the total number of background events from all off-regions times the corresponding weight . For five off-regions, this yields
