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\title{Outline of a standard analysis for MAGIC \\
(including Monte Carlo work)}
\author{R. B\"ock, H. Kornmayer, W. Wittek\\ 
\texttt{h.kornmayer@web.de, wittek@mppmu.mpg.de}}

\date{ \today}
\TDAScode{MAGIC-TDAS 01-??\\ ??????/W.Wittek}
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\section{Aim of this paper}
The aim of this paper is to describe the procedure to obtain the
absolute energy spectrum of a point source from the data taken with
MAGIC. This includes work on Mont Carlo (MC) data and the analysis of
the real data.

Various steps in the procedure will depend on details of the MC
generation, on the way the real data are taken, etc.. These details
have therefore to be specified, which is done in Section 2.

In Section 3 some basic definitions and formulas are collected in
order to avoid any misunderstanding of the meaning of frequently
used terms.

Section 4 describes the MC work and Section 5 the actual analysis of
the real data.

One aim of this paper is also to define jobs for those who want to
join the activities in the software developments. As will be seen, the
main ingredients both for the MC work and the real data analysis are
available. However, certain parts have yet to be implemented, others
have to be changed, modified, improved or extended. Last not least
extensive tests have to be performed.



\section{Assumptions}
The assumptions for a 'standard analysis' listed below are the result of
discussions in the software group. Some of them are rather arbitrary. 
They should by no means be
understood as final or optimal choices. They should be considered as a
starting point. As our experience with the analysis grows we may
have to revise some of the assumptions.

The aim in all what follows is to define a strategy that is as simple 
and robust as possible. Tests that have yet to be performed will tell
us whether the assumptions are reasonable and realistic.

The assumptions are :

\begin{itemize}
\item Mode of observation :\\
Data are taken in the wobble mode (\cite{konopelko99}). 
This means that the telescope is
directed not to the position of the selected source but rather to a
position which has a certain offset ($\Delta\beta$) from the source
position. Every 20 minutes of observation the sign of $\Delta\beta$ is 
changed. The two wobble positions are called wobble position 1 and 2.

$\Delta \beta$ may be chosen to be a direction difference 
in celestial coordinates
(declination $\delta$, right ascension $\Phi$) or in local coordinates
(zenith angle $\Theta$, azimuthal angle $\phi$).
However the direction $\Delta \beta$ is defined,
the sky region projected onto the camera is different for
wobble positions 1 and 2. 

If $\Delta \beta$ is defined to be a direction difference 
in celestial coordinates,
the sky region projected onto the camera for a fixed wobble position 
remains the same during tracking of a source, although the sky image 
is rotating in the camera.

If $\Delta \beta$ is defined to be a direction difference 
in local coordinates,
the sky region projected onto the camera is changing continuously 
during tracking of a source. The centers of the projected sky regions
lie on a circle, which is centered at the source position.

If $\Delta \beta$ is defined to be a direction difference 
in the local azimuthal 
angle $\phi$, the center of the camera and the source position
would always have the same zenith angle $\Theta$. Since the reconstruction
efficiency of showers mainly depends on $\Theta$, this may be an
advantage of defining $\Delta \beta$ in this way.

The wobble mode has to be understood as an alternative to taking on-
and off-data in separate runs. Choosing the wobble mode thus implies
that one is taking on-data only, from which also the 'off-data' have to be
obtained by some procedure.

We propose to define $\Delta \beta$ as a direction difference in the
local azimuthal angle $\phi$ :
$\Delta \phi\;=\;\Delta \beta\;/\;sin(\Theta)$. For very small
$\Theta$ ($\Theta\;<\; 1$ degree) $\Delta \beta$ should be defined
differently, also to avoid large rotation speeds of the telescope.

Since the radius of the trigger area is 0.8 degrees, we propose 
to choose $\Delta \beta\;=\;0.4$ degrees.
 

\item Pedestals :\\
Pedestals and their fluctuations are not determined from triggered
showers but rather from pedestal events. The pedestal events are taken 
'continuously' at a constant rate of 5 Hz. In this way the pedestals
and their fluctuations are always up to date, and the presence of
stars and their position in the camera can be monitored continuously.

\item Gamma/hadron separation :\\
It is assumed that it is possible to define a gamma/hadron separation
which is independent 
 \begin{itemize}
 \item[-] of the level of the light of the night sky (LONS)
 \item[-] of the presence of stars in the field of view (FOV) of the camera
 \item[-] of the orientation of the sky image in the camera
 \item[-] of the source being observed
 \end{itemize}

It has yet to be proven that this is possible. The corresponding
procedures have to be developed, which includes a proper treatment of the
pedestal fluctuations in the image analysis. 

The gamma/hadron separation will be given in terms of a set of cuts
(or certain conditions) on quantities which in general are not
identical to the measured quantities but which are derived from them. The
measurable quantities are :
 \begin{itemize}
 \item[-] the direction $\Theta$ and $\phi$ the telescope is pointing to
 \item[-] the image parameters
 \item[-] the pedestal fluctuations 
 \end{itemize}

Under the above assumption the only dependence to be considered for
the collection areas (see Section 3) is the dependence on the type of
the cosmic ray particle (gamma, proton, ...), on its energy and on the
zenith angle $\Theta$.

It has to be investigated whether also the azimuthal angle $\phi$ has to be
taken into account, for example because of influences from the earth
magnetic field.

\item Trigger condition :\\

\item Standard analysis cuts :\\ 
 
\end{itemize}


\section{Definitions and formulas}
\subsection{Definitions}

\begin{itemize} 
\item The direction $(\Theta,\phi)$ :\\
$(\Theta,\phi)$ denotes the direction the telescope is pointing to,
not the position of the source being observed.

\item Image parameters :\\
The standard definition of the image parameters is assumed. See for
example \cite{hillas85,fegan96,reynolds93}. We should also make use of
additional parameters like asymmetry parameters, number of islands or
mountains etc.
\end{itemize}
 
Quantities which are not directly measurable, but which can be
estimated from the image parameters are :

\begin{itemize} 
\item Impact parameter :\\
The impact parameter $p$ is defined as the vertical distance 
of the telescope from the shower axis. 

\item The energy of the shower
\end{itemize}


\subsection{Formulas}
\subsubsection{Differential gamma flux and collection area for a point source}

The differential gamma flux from a point source $s$ is given by

\begin{eqnarray}
\Phi^{\gamma}_s(E)\;=\;\dfrac{dN^{\gamma}_s}{dE \cdot dF \cdot dt} 
\end{eqnarray} 

where $dN^{\gamma}_s$ is the number of gammas from the source $s$ in
the bin $dE,\;dF,\;dt$ of energy, area and time respectively. We
denote the probability for 'observing' a gamma shower with energy
$E$, zenith angle $\Theta$ and position $F$ in a plane perpendicular
to the source direction by $R^{\gamma}(E,\Theta,F)$. Depending on the
special study, the term 'observing' may mean triggering,
reconstructing, etc. 

The effective collection area is defined as

\begin{eqnarray}
F^{\gamma}_{eff}(E,\Theta)\;  &=  &\int R^{\gamma}(E,\Theta,F)\cdot dF  
\label{eq:form-1}
\end{eqnarray} 

A side remark : The well known behaviour that the effective collection 
area (well above the threshold energy) is larger for larger zenith angles
$\Theta$, is due to the fact that at higher $\Theta$ the distance of
the shower maximum (where the majority of Cherenkov photons is
emitted) from the detector is larger than at smaller $\Theta$. The
area in which $R^{\gamma}(E,\Theta,F)$ contributes significantly to
the integral (\ref{eq:form-1}) is therefore larger, resulting in a
larger $F^{\gamma}_{eff}(E,\Theta)$. For the simulation this means,
that the maximum impact parameter should be chosen larger for larger $\Theta$.

The number of $\gamma$ showers observed in the bin $\Delta \Theta$ of
the zenith angle and in the bin $\Delta E$ of the energy is
then :

\begin{eqnarray}
\Delta N^{\gamma,obs}_s(E,\Theta)  &= &\int R^{\gamma}(E,\Theta,F) \cdot
 \Phi^{\gamma}_s(E) \cdot dE \cdot dF \cdot dt \\
                                   &= &\Delta T_{on}(\Theta) \cdot
 \int_{\Delta E}{} \Phi^{\gamma}_s(E)\cdot
 F^{\gamma}_{eff}(E,\Theta)\cdot dE \\
\end{eqnarray} 

Assuming that $F^{\gamma}_{eff}(E,\Theta)$ depends only weakly on $E$
in the (sufficiently small) interval $\Delta E$ gives

\begin{eqnarray}
\Delta N^{\gamma,obs}_s(E,\Theta)  
                         &\approx   &\Delta T_{on}(\Theta) \cdot 
  F^{\gamma}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
 \Phi^{\gamma}_s(E)\cdot dE               \label{eq:form0}\\
                         &\approx   &\Delta T_{on}(\Theta) \cdot 
  F^{\gamma}_{eff}(E,\Theta) \cdot \Delta E \cdot 
  \overline{\Phi^{\gamma}_s}(E)       \label{eq:form1}
\end{eqnarray} 

Here $\Delta T_{on}(\Theta)$ is the effective on-time for the data
taken in the zenith angle bin $\Delta \Theta$ and $\overline{\Phi^{\gamma}_s}(E)$
is the average differential gamma flux in the energy bin $\Delta E$ :

\begin{eqnarray}
\overline{\Phi^{\gamma}_s}(E)  &=  
                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
                     \Phi^{\gamma}_s(E)\cdot dE
\end{eqnarray} 

By inverting equation (\ref{eq:form1}) and setting 
$\Delta E\;=\;(E^{up}-E^{low})\;\;\;\;\overline{\Phi^{\gamma}_s}(E)$ can
be written as

\begin{eqnarray}
  \overline{\Phi^{\gamma}_s}(E)    &=
  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)}  
{\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta) \cdot 
 (E^{up}-E^{low}) }
 \label{eq:form2}
\end{eqnarray} 

By means of equation (\ref{eq:form2}) $\overline{\Phi^{\gamma}_s}(E)$
can be determined
from the measured $\Delta N^{\gamma,obs}_s(E,\Theta)$ and 
$\Delta T_{on}(\Theta)$, using the collection area
$F^{\gamma}_{eff}(E,\Theta)$, which is obtained from MC data.

Equation (\ref{eq:form2}) is for a limited and fixed region of
the zenith angle. One may calculate $\overline{\Phi^{\gamma}_s}(E)$ from the
data taken at all $\Theta$, in which case

\begin{eqnarray}
  \overline{\Phi^{\gamma}_s}(E)    &=
  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)}  
         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
          \cdot (E^{up}_i-E^{low}_i) }
 \label{eq:form3}
\end{eqnarray} 

If a fixed spectral index $\alpha$ is assumed for the differential 
source spectrum

\begin{eqnarray}
 \Phi^{\gamma}_s(E)  &=  &\Phi^{\gamma}_0 \cdot 
                       \left(\dfrac{E}{GeV}\right)^{-\alpha}
\end{eqnarray} 

one gets

\begin{eqnarray}
 \int_{\Delta E}{} \Phi^{\gamma}_s(E) \cdot dE  &=  
 &\dfrac{\Phi^{\gamma}_0}{1-\alpha} 
  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]\cdot GeV 
 \label{eq:form4}
\end{eqnarray} 

Inserting (\ref{eq:form4}) into (\ref{eq:form0}) yields

\begin{eqnarray}
  \Phi^{\gamma}_0    &=
  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)}  
         {\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta)
          \cdot 
  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]} 
  \cdot \dfrac{1-\alpha}{GeV}
  \label{eq:form5}
\end{eqnarray} 

which by summing over all $\Theta$ bins gives

\begin{eqnarray}
  \Phi^{\gamma}_0    &=
  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)}  
         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
          \cdot 
  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]} 
  \cdot \dfrac{1-\alpha}{GeV}
  \label{eq:form6}
\end{eqnarray} 

If applied to MC data, for which $\overline{\Phi^{\gamma}_s}(E)$ is known,
equation (\ref{eq:form1}) can also be used to
determine the collection area $F^{\gamma}_{eff}(E,\Theta)$ : 

\begin{eqnarray}
F^{\gamma}_{eff}(E,\Theta)  &=  
 &\dfrac{\Delta N^{\gamma,MC,reconstructed}_s(E,\Theta)}
        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{\gamma}_s}(E) \cdot
         (E^{up}-E^{low})}
\end{eqnarray} 

This procedure of determining $F^{\gamma}_{eff}(E,\Theta)$ amounts to
performing the integration in equation (\ref{eq:form-1}) by MC. An
important precondition is that in the MC simulation all gamma showers for
which $R^{\gamma}(E,\Theta,F)$ is greater than zero were
simulated. This means in particular that the MC simulation of gammas 
extends to sufficiently large impact parameters. In reality, in order to save
computer time showers will be generated up to a maximum
value of the impact parameter (possibly depending on the zenith
angle). An appropriate correction for that has to be applied later in
the analysis.

Knowing $F^{\gamma}_{eff}(E,\Theta)$, the gamma fluxes can be obtained
from the experimental data using equation (\ref{eq:form2}),
(\ref{eq:form3}), (\ref{eq:form5}) or (\ref{eq:form6}).

Of course, the MC data sample used for calculating 
$F^{\gamma}_{eff}(E,\Theta)$ and the experimental data sample used for
determining the gamma flux by means of $F^{\gamma}_{eff}(E,\Theta)$
have to be defined identically in many respects : in particular
the set of cuts
and the offset between source position and telescope orientation have
to be the same in the MC data and the experimental data sample.


 
\subsubsection{Differential flux and collection area for
hadronic cosmic rays}

In the case of hadronic cosmic rays, which arrive from all directions 
$\Omega$, the differential hadron flux is given by

\begin{eqnarray}
\Phi^{h}(E)\;=\;\dfrac{dN^{h}}{dE \cdot dF \cdot dt \cdot d\Omega} 
\label{eq:form-12}
\end{eqnarray} 


In contrast to (\ref{eq:form-1}) the effective collection area for hadrons
is defined as

\begin{eqnarray}
F^{h}_{eff}(E,\Theta)\;  &=  &\int R^{h}(E,\Theta,F,\Omega)\cdot dF
 \cdot d\Omega  
\label{eq:form-11}
\end{eqnarray} 

Note that for a fixed orientation of the telescope $(\Theta,\phi)$ the
hadrons are coming from all directions $\Omega$. The reconstruction
efficiency $R^h(E,\Theta,F,\Omega)$ of hadrons therefore depends also
on $\Omega$.

With the definitions (\ref{eq:form-12}) and (\ref{eq:form-11})
very similar formulas are obtained for hadrons as
were derived for photons in the previous section. For clarity they
are written down explicitely :

\begin{eqnarray}
\Delta N^{h,obs}(E,\Theta)  &= &\int R^{h}(E,\Theta,F) \cdot
 \Phi^{h}(E) \cdot dE \cdot dF \cdot dt \\
                                   &= &\Delta T_{on}(\Theta) \cdot
 \int_{\Delta E}{} \Phi^{h}(E)\cdot
 F^{h}_{eff}(E,\Theta)\cdot dE \\
\end{eqnarray} 

\begin{eqnarray}
\Delta N^{h,obs}(E,\Theta) 
                         &\approx   &\Delta T_{on}(\Theta) \cdot 
  F^{h}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
 \Phi^{h}(E)\cdot dE               \label{eq:form10}\\
                         &\approx   &\Delta T_{on}(\Theta) \cdot 
  F^{h}_{eff}(E,\Theta) \cdot \Delta E \cdot 
  \overline{\Phi^{h}}(E)       \label{eq:form11}
\end{eqnarray} 


\begin{eqnarray}
\overline{\Phi^{h}}(E)  &=  
                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
                     \Phi^{h}(E)\cdot dE
\end{eqnarray} 


\begin{eqnarray}
  \overline{\Phi^{h}}(E)    &=
  &\dfrac{\Delta N^{h,obs}(E,\Theta)}  
{\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta) \cdot 
 (E^{up}-E^{low}) }
 \label{eq:form12}
\end{eqnarray} 



\begin{eqnarray}
  \overline{\Phi^{h}}(E)    &=
  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)}  
         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
          \cdot (E^{up}_i-E^{low}_i) }
 \label{eq:form13}
\end{eqnarray} 


\begin{eqnarray}
 \Phi^{h}(E)  &=  &\Phi^{h}_0 \cdot 
                       \left(\dfrac{E}{GeV}\right)^{-\beta}
\end{eqnarray} 


\begin{eqnarray}
 \int_{\Delta E}{} \Phi^{h}(E) \cdot dE  &=  
 &\dfrac{\Phi^{h}_0}{1-\beta} 
  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]\cdot GeV 
 \label{eq:form14}
\end{eqnarray} 


\begin{eqnarray}
  \Phi^{h}_0    &=
  &\dfrac{\Delta N^{h,obs}(E,\Theta)}  
         {\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta)
          \cdot 
  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]} 
  \cdot \dfrac{1-\beta}{GeV}
  \label{eq:form15}
\end{eqnarray} 


\begin{eqnarray}
  \Phi^{h}_0    &=
  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)}  
         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
          \cdot 
  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]} 
  \cdot \dfrac{1-\beta}{GeV}
  \label{eq:form16}
\end{eqnarray} 


Note that $\Phi^{h}(E)$, $\Phi^h_0$ and $F^{h}_{eff}(E,\Theta)$ differ
from      $\Phi^{\gamma}(E)$, $\Phi^{\gamma}_0$ and 
$F^{\gamma}_{eff}(E,\Theta)$ by the dimension of the
solid angle, due to the additional factor $d\Omega$ in
(\ref{eq:form-12}) and (\ref{eq:form-11}).

Like in the case of gammas from point sources, the effective area 
$F^h_{eff}(E,\Theta)$ for
hadrons can be calculated by applying equation (\ref{eq:form11}) to MC
data, for which $\overline{\Phi^h}(E)$ is known :

\begin{eqnarray}
F^{h}_{eff}(E,\Theta)  &=  
 &\dfrac{\Delta N^{h,MC,reconstructed}(E,\Theta)}
        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{h}}(E) \cdot
         (E^{up}-E^{low})}
\end{eqnarray} 

Similar to the case of gammas from point sources,
this procedure of determining $F^h_{eff}(E,\Theta)$ amounts to
performing the integrations in equation (\ref{eq:form-11}) by MC. The
precondition in the case of hadrons is that in the 
MC simulation all hadron showers for
which $R^{h}(E,\Theta,F,\Omega)$ is greater than zero were
simulated. So the simulation should not only include large enough
impact parameters but also a sufficiently large range of $\Omega$ at
fixed orientation $(\Theta,\phi)$ of the telescope.

Knowing $F^{h}_{eff}(E,\Theta)$, the hadron fluxes can be obtained
from the experimental data using equation (\ref{eq:form12}),
(\ref{eq:form13}), (\ref{eq:form15}) or (\ref{eq:form16}).


\subsubsection{Measurement of the absolute differential flux of gammas 
from a point source by normalizing to the flux of hadronic cosmic rays}

In section 3.2.1 a procedure was described for measuring the absolute
differential flux of gammas from a point source. The result for 
$\overline{\Phi^{\gamma}_s}(E)$ depends on a reliable determination of
the collection area $F^{\gamma}_{eff}(E,\Theta)$ by MC and the
measurement of the on-time $\Delta T_{on}(\Theta)$.

The dependence on the MC simulation may be reduced by normalizing to
the known differential flux of hadronic cosmic rays. Combining
equations (\ref{eq:form2}) and (\ref{eq:form12}), and assuming that
$\Delta T_{on}(\Theta)$ is the same for the gamma and the hadron
sample, yields

\begin{eqnarray}
\dfrac{\overline{\Phi^{\gamma}_s}(E)}
      {\overline{\Phi^{h}}(E)}          &=  &
\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
      {\Delta N^{h,obs}(E,\Theta)}      \cdot
\dfrac{F^{h}_{eff}(E,\Theta)}
      {F^{\gamma}_{eff}(E,\Theta)}
\label{eq:form20}
\end{eqnarray} 

If $\overline{\Phi^{h}}(E)$ is assumed to be known from other
experiments, equation (\ref{eq:form20}) allows to determine 
$\overline{\Phi^{\gamma}_s}(E)$ from
the experimental number of gamma and hadron showers using the
collection areas for gammas and hadrons from the MC. Since only the
ratio of the collection areas enters the dependence on the
MC simulation is reduced.

If data from all zenith angles are to be used the corresponding
expression for $\overline{\Phi^{\gamma}_s}(E)$ is (see equations
(\ref{eq:form3}) and (\ref{eq:form13}))

\begin{eqnarray}
\dfrac{\overline{\Phi^{\gamma}_s}(E)}
      {\overline{\Phi^{h}}(E)}          &=  &
\dfrac{\sum_i \Delta N^{\gamma,obs}(E,\Theta_i)}
      {\sum_i \Delta N^{h,obs}(E,\Theta_i)}      \cdot
\dfrac{\sum_i \Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i) 
                                      \cdot (E^{up}_i-E^{low}_i)}
      {\sum_i \Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i) 
                                      \cdot (E^{up}_i-E^{low}_i)}
\label{eq:form21}
\end{eqnarray} 

Clearly, the set of cuts defining the gamma sample is different from
the set of cuts defining the hadron sample. However, 
$\Delta N^{\gamma,obs}$ and $\Delta N^{h,obs}$ can still be measured 
simultaneously, in which case $\Delta T_{on}(\Theta_i)$ is the same for
the gamma and the hadron sample. Measuring gammas and hadrons
simultaneously has the advantage that variations of the detector 
properties or of the atmospheric conditions during the observation 
partly cancel in (\ref{eq:form20}) and (\ref{eq:form21}).

If fixed spectral indices $\alpha$ and $\beta$ are assumed for the 
differential 
gamma and the hadron fluxes respectively one obtains for the ratio
$\Phi^{\gamma}_0\;/\;\Phi^h_0$
(see (\ref{eq:form5}) and (\ref{eq:form15}))

\begin{eqnarray}
\dfrac{\Phi^{\gamma}_0}
      {\Phi^{h}_0}          &=  &
\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
      {\Delta N^{h,obs}(E,\Theta)}      \cdot
\dfrac{F^{h}_{eff}(E,\Theta) \cdot
  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]} 
      {F^{\gamma}_{eff}(E,\Theta)
  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]} \cdot
\dfrac{1-\beta}{1-\alpha}
\label{eq:form22}
\end{eqnarray} 

or, when using the data from all zenith angles,
(see (\ref{eq:form6}) and (\ref{eq:form16}))

\begin{eqnarray}
\dfrac{\Phi^{\gamma}_0}
      {\Phi^{h}_0}          &=  &
\dfrac{\sum_i\Delta N^{\gamma,obs}(E,\Theta_i)}
      {\sum_i\Delta N^{h,obs}(E,\Theta_i)}      \cdot
\dfrac{\sum_i F^{h}_{eff}(E,\Theta_i) \cdot
  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]} 
      {\sum_i F^{\gamma}_{eff}(E,\Theta_i)
  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]} \cdot
\dfrac{1-\beta}{1-\alpha}
\label{eq:form23}
\end{eqnarray} 



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\section{MC work}

\subsection{Overview of the MC and analysis chain}

After a few iterations to improve the programs in speed, 
reliability, ... there is a sample of available programs 
to simulate the behaviour of the MAGIC telescope. 
Due to the big amount of diskspace needed for this simulation
it was decided, that not only one program will generate 
the MAGIC telescope, but a subsequent chain of different 
programs. In figure \ref{MC_progs} you can see a overview of 
the existing programs and their connections. 
\begin{figure}[h]
\setlength{\unitlength}{1.cm}
  \begin{picture}(18.,12.)
	\put (0., 0.){\framebox(18.,12.){}}

	\put (1, 11.5){{\sl Air shower programs}}
	\put (1., 10.){\framebox(3.,1.){MMCS}}
	\put (2., 10.){\vector(0,-1){.9} }
	\put (1., 8.){\framebox(3.,1.){reflector}}
	\put (2., 8.){\vector(0,-1){.9}}

	\put (6, 10.){{\sl star background programs}}
	\put (6.,8.){\framebox(3.,1.){starresponse}}
	\put (6., 8.){\line(0, -1){1.5}}
	\put (10.,8.){\framebox(3,1){starfieldadder}}
	\put (10., 8.){\line(0, -1){1.5}}
	\put (10., 6.5){\vector(-1,0){6.} }

	\put (1., 6.){\framebox(3.,1.){camera}}
	\put (2., 6.){\vector(3,-1){5.} }

	
	
	\put (14, 11.5){{\sl real data programs}}
	\put (14, 8.){\framebox(3,1){MAGIC DAQ}}	
	\put (15, 8.){\vector(0,-1){.9} }
	\put (14, 6.){\framebox(3.,1.){MERPP}}	
	\put (15., 6.){\vector(-3,-1){5.} }
	
	\put (8.75, 3.7){\oval(4.,1.)}	
	\put (7., 3.5){MAGIC root file}	
	\put (8., 3.2){\vector(0, -1){1.0}}

	\put (7, 1.){\framebox(3.,1.){MARS}}

	\thicklines
	\put (5., 11.){\line(0, -1){6.5}}	
	\put (13., 12.){\line(0, -1){7.5}}	

  \end{picture}  
\caption{Overview of the existing programs in the MC of
MAGIC.}
\label{MC_progs} 
\end{figure} 
A detailed description of the properties of the different programs can be found
in section \ref{sec_exist_progs}. 
From that diagram you can see the following features of the simulation and
analysis chain of MAGIC.  
\begin{enumerate} 
  \item The simulation of Air showers and the simulation of the night sky 
	background (NSB) is seperated.  

  \item The NSB is seperated in two parts, the contribution from the starfield
	and from a diffuse part. 

  \item To speed up the production the starresponse program creates a databases 
	for stars of different magnitude. 

  \item The join of air showers and NSB is done in the camera program. 

  \item The analysis of MC \underline{and} real data will be done with only one program. 
	This program is called MARS (Magic Analysis and Reconstruction Software). 
	The output of the camera program from Monte Carlo data and the output of 
	the MERPP (MERging and PreProcessing) program for the real data are the same. 
	So there is no need to use different programs for the analysis. The file 
	generated by this program used the root package from CERN for data storage. 
\end{enumerate} 
In this section we will only describe the usage of the Monte Carlo programs. The 
descriptions of the MERPP and MARS can be found somewhere else\footnote{Look on the 
MAGIC home page for more information.}. 

\subsection{Existing programs} 
\label{sec_exist_progs}
\subsubsection{MMCS - Magic Monte Carlo Simulation} 
 
This program is based on a CORSIKA simulation. It is used to generate
air showers for the MAGIC telecope. At the start one run of the 
program, one has to define the details of the simulation. 
One can specify the following parameters of an shower 
(see also figure \ref{pic_shower}): 
%
\begin{enumerate}
  \item the type of the particles in one run ($PartID$)
  \item the energy range of the particles ($E_1, E_2$) 
  \item the slope of the Energy spectra 
  \item the range of the shower core on the ground $r_{core}$. 
  \item the direction of the shower by setting the range of 
	zenith angle ($\Theta_1, \Theta_2$) and 
	azimuth angle  ($\phi_1, \phi_2$)
\end{enumerate}
%
\begin{figure}[h]
\setlength{\unitlength}{1.5cm}
\begin{center} 
  \begin{picture}(9.,6.)
	\put (0., 0.){\framebox(9.,6.){}}

	\thicklines
	% telescope
	\put (5., .5){\oval(.75, .75)[t]} 
	\put (3., 1.){{\sl Telesope position}}	
	\put (4.5, 1.){\vector(1, -1){0.5}}
	% observation level 
	\put (.5, .5){\line(1, 0){8}}
	\put (.5, .6){{\sl Observation level}}	

	% air shower
	\put (4. , 5.5 ){\line(2, -3){3.3}}
	\put (4.5, 5.5 ){{\sl Particle Type ($PartId$)}}
	\put (4.5, 5.25){{\sl Energy ($E_1 < E < E_2$)}}
	\put (4.5, 5.  ) {$\Theta_1 < \Theta < \Theta_2$}
	\put (4.5, 4.75) {$\phi_1 < \phi < \phi_2$}
	\put (7.5, .75){{\sl shower core}}
	
	\thinlines
	\put (5., .25){\line(1,0){2.3}} 
	\put (6.1, .25){{\sl $r_{Core}$}}
	
	\put (5., .5){\line(4,3){1.571}} 	
	\put (6., 1.35){{\sl $p$}}

  \end{picture}  
\end{center} 
  \caption {The parameter of an shower that are possible to define
at the begin of an MMCS run.}
\label{pic_shower} 
\end{figure} 
Other parameters, that will be important in the analysis later,
can be calculated. I.e. the impact parameter $p$ is defined by 
the direction 
of the shower ($\Theta, \phi$) and the core position 
($x_{core}, y_{core}$). 

The program MMCS will track the whole shower development 
through the atmosphere. All the cerenkov particles that hit a 
sphere around the telesope (in the figure \ref{pic_shower} 
drawn as the circle around the telecope position) are stored 
on disk. It is important to recognize, that up to now no 
information of the pointing of the telescope was taking into
account.  
This cerenkov photons are the input for the next program, 
called reflector. 


\subsubsection{reflector} 

The aim of the reflector program is the 
tracking of the cerenkov photons to the camera 
of the MAGIC telescope. So this
is the point where we introduce a specific pointing of 
the telescope ($\Theta_{MAGIC}, \phi_{MAGIC}$).
For all cerenkov photons the program 
tests if the mirrors are hitten, calculates the 
probability for the reflection and tracks them to the 
mirror plane. All the photons that are hitting the 
camera are written to disk (*.rfl)  
with their important parameters 
($x_{camera}, y_{camera}, \lambda, t_{arrival}$). 
These parameters are the input from the shower simulation
for the next program in the 
MC simulation chain, the camera program. 

\subsubsection{camera} 

The camera program simulates the behaviour of the 
PMTs and the electronic of the trigger and FAC system. 
For each photon out of the reflector file (*.rfl) the 
camera program calculates the probability to generate
an photo electron out of the photo cathode. If a photo
electrons was ejected, this will create a signal in the 
trigger and FADC system of the hitted pixel. 
You have to specify the
parameter of the signal shaping 
(shape, Amplitude, FWHM of signal)
at the beginning of the 
camera, seperatly for the trigger and the FADC system. 
All signal from all photoelectrons are superimposed for
each pixel. As an example you can see the output of 
the trigger and FADC system in figure \ref{fig_trigger_fadc}. 
\begin{figure}[h]

 \caption{The response of one shower from the trigger (left) and 
fadc system (right).}
\label{fig_trigger_fadc}
\end{figure}

All these analog signals going into the trigger system are used 
to check if for a given event a trigger signal was generated or 
not. But before the start of the camera program on also has to
set a few parameters of the trigger system like: 
\begin{itemize}
  \item diskriminator threshold
  \item mulitplicity
  \item topology 
\end{itemize} 
With this set of parameter the camera program will analyse
if one event has triggered. For the triggered event all the FADC
content will be writen on the file (*.root). In addition all the 
information about the event ($PartID, E, \Theta$,...) and 
information of trigger (FirstLevel, SecondLevel, ..) are also 
be written to the file.

One of the nice features of the camera program is the possiblity
so simulate the NSB, the diffuse and the star light part of it. 
But before doing this, on has to start other programs 
(called starresponse and starfieldadder) that are describe 
below.

\subsubsection{starresponse}

This program will simulate the analog response for stars of
a given brightness $B$. 


\subsubsection{starfieldadder}







\subsection{What to do} 

\begin{itemize} 
  \item pedestal fluctuations
  \item trigger
  \item rates (1st level, 2nd level, .... ) 
  \item discriminator thresholds
  \item Xmax
  \item collection area
  \item $\gamma$/h-Seperation
  \item magnetic field studies ($\phi$-dependence) 
  \item rotating star field
\end{itemize} 



\subsection{A suggestion for an initial workplan} 
We propose in the following a list of tasks whose common goal
it is to provide and use data files with a definition of data suitable for
initial studies, e.g. trigger rates, and for subsequent further 
analysis in MARS, e.g. $\gamma$/h-separation. We consider this list to be
minimal and a first step only.
Given the amount of work that will have to be invested, the detailed 
assumptions below should be backed up by collaboration-wide agreement; also, some
input from groups is essential, so PLEASE REACT. 
 
Event generation should be done with the following conditions:
\begin{itemize}
  \item Signal definition: we will use the Crab, over a range of zenith angles
  (define!!). A minimum of 20,000 (can we get that?) triggers will be 
  generated, starting from existing MMCS files;
  \item Observation mode: observations are assumed off-axis,
  with an offset of $\pm 0.4 \deg $ in $\Delta \beta$ along the direction of the
  local azimuthal angle $\phi$,
  switching sign every 500 events (see 'Assumptions' above);
  \item Adding star field: adapt starfieldadder and starresponse to the
  Crab. Ignore star field rotation problems for the moment, until a separate study
  is available (??);
  \item Pedestal fluctuations: all pixel values are smeared by a Gaussian
  centered at zero with a sigma of 1.5 photoelectrons;
  \item Trigger:  Padova to define (!!) the grouping of pixels, the
  trigger thresholds, and a method to avoid triggering on stars. We assume 
  only a first-level trigger.
\end{itemize}
With this event sample available, we suggest to embark on several studies,
which will help us in understanding better the MAGIC performance, and will
also pave our way into future analysis.
\begin{itemize}
  \item determine trigger rates (1st level only), as function of energy and
  zenith angle (also of impact parameter?);
  \item determine gamma acceptance, 
  as function of energy and zenith angle (also of impact parameter?);  
  \item determine effective collection area (gammas and hadrons), 
  as function of energy and zenith angle (also of impact parameter?);  
  \item show the position of the shower maximum (Xmax);
  \item start comparing methods for $\gamma$/h-separation, i.e. the generation
  of ON and OFF samples from the observations;
  \item start magnetic field studies ($\phi$-dependence); 
  \item eventually, study the effect of the rotating star field.
\end{itemize} 



\section{Analysis of the real data}

\begin{thebibliography}{xxxxxxxxxxxxxxx}
\bibitem{fegan96}D.J.Fegan, Space Sci.Rev. 75 (1996)137
\bibitem{hillas85}A.M.Hillas, Proc. 19th ICRC, La Jolla 3 (1985) 445
\bibitem{konopelko99}A.Konopelko et al., Astropart. Phys. 10 (1999)
275
\bibitem{reynolds93}P.T.Reynolds et al., ApJ 404 (1993) 206
\end{thebibliography}


\end{document}
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