= Spectrum Analysis =

The differential flux \(\phi(E)\) per area, time and energy interval is defined as
\[\phi(E) = \frac{dN}{dA\cdot dt\cdot dE}\]

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:
\[\phi(E) = \frac{1}{A_0\cdot \Delta T}\frac{dN}{d\epsilon(E)\cdot dE}\]

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 

\[\epsilon(E) = \frac{N_\textrm{exc}(E)}{N_0(E)} \]

where \(N_0\) is the number of simulated events in this energy bin and \(N=N_{exc}\) the number of *excess* events that are produced by the analysis chain.

The number of excess events, for data and simulations, is defined as

\[N_\textrm{exc} = N_\textrm{sig} - \hat N_\textrm{bg}\]

where \(N_\textrm{sig}\) is the number of events identified as potential gammas from the source direction ('on-source') and \(N_\textrm{bg}\) the number of gamma-like events measured 'off-source'. Note that for Simulations, \(\hat N_\textrm{bg}\) 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 \(\hat N_\textrm{bg}\) is the total number of background events \(\N_\textrm{bg}\) from all off-regions times the corresponding weight \(\omega\). For five off-regions, this yields

\[\(\hat N_\textrm{bg}\) = \frac{N_\textrm{bg}}{5}\]


== Define Binnings ==

== Get Data File List ==

== Get Observation Time ==

== Get Monte Carlo File List ==

== Get Zenith Angle Histogram ==

== Analyze Data ==

== Analyze Monte Carlo Data ==

== Summarize Corsika Production ==

== Result (Spectrum) ==

== Result (Threshold) ==

== Result (Migration) ==