|Version 46 (modified by 17 months ago) ( diff ),|
The differential fluxper area, time and energy interval is defined as
Oftenis 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 to a given energy interval.
The total areaand 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
whereis the number of simulated events in this energy bin and the number of *excess* events that are produced by the analysis chain.
Note that the exact calculation of the efficiencydepends on prior knowledge of the correct source spectrum . Therefore, it is strictly speaking only correct if the simulated spectrum and the real spectrum are identical. As the real spectrum is unknown, special care has to be taken of the systematic introduced by the assumption of .
The number of excess events, for data and simulations, is defined as
whereis 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 eventsis the total number of background events from all off-regions times the corresponding weight . For five off-regions, this yields
Theta and Spectral Weights
While the source spectrumis of course independent of the zenith angle of the observation, the efficiency is not, so that
In addition, the observed differential flux depends on the zenith angle, so that
That means that for a correct calculation of the efficiency, the number of simulated events per zenith angle interval has to match the observation time distribution. This can be achieved by applying zenith angle dependent weightsto each simulated event. Similarly, the discussed requirement of the match between result spectrum and simulated spectrum can be achieved applying energy dependent weights .
As the simulated energy spectrum is independent of zenith angle, it can be expressed as
with the differential energy spectrumand the zenith angle distribution , for the total production range to and to
andthe total number of generated Monte Carlo events. For the generated number of events in the energy interval and zenith angle interval this is
with the weights chosen such that the sum over all intervals