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Changes between Version 18 and Version 19 of DatabaseBasedAnalysis/Spectrum

-              v18
+              v19
 Often \(\phi(E)\) is also referred to as \(\frac{dN}{dE}\) as observation time and effective collection area is a constant. The effective area is then defined as \(A_\textrm{eff}(E)=\epsilon(E)\cdot A_0\). Note that at large distances the efficiency vanishes, so that the effective area is an (energy dependent) constant while \(A_0\) and the efficiency \(\epsilon(E)\) are mutually dependent.
+For an observation with an effective observation time \(\Delta T\), this yields:
+\[\phi(E) = \frac{1}{A_0\cdot \Delta T}\frac{dN}{d\epsilon(E)\cdot dE}\]
+For an observation with an effective observation time \(\Delta T\), this yields in a given Energy interval \(\Delta E\):
+\[\phi(E) = \frac{1}{A_0\cdot \Delta T}\frac{dN}{\epsilon(\Delta E)\cdot \Delta E}\]
+For simplicity, in the following, \(\Delta E\) will be replaced by just \(E\) 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 \(A_0\) and the corresponding efficiency \(\epsilon(E)\) are of course only available for simulated data. For simulated data, \(A_0\) is the production area and \(\epsilon(E)\) the corresponding energy dependent efficiency of the analysis chain. For a given energy bin, the efficiency is then defined as