Changeset 777 for trunk/MagicDoku

Timestamp:

05/03/01 08:54:14 (24 years ago)

Author:

wittek

Message:

Merge from WOW version with HAKO version on 3.5.2001

File:

• trunk/MagicDoku/strategy_mc_ana.tex (modified) (9 diffs)

trunk/MagicDoku/strategy_mc_ana.tex

@@ -14 +14 @@
 \title{Outline of a standard analysis for MAGIC \\
 (including Monte Carlo work)}
-\author{W. Wittek, H. Kornmayer\\ 
-\texttt{wittek@mppu.mpg.de, h.kornmayer@web.de }}
+\author{H. Kornmayer, W. Wittek\\ 
+\texttt{h.kornmayer@web.de, wittek@mppmu.mpg.de}}
 
 \date{ \today}
@@ -79 +79 @@
 \begin{itemize}
 \item Mode of observation :\\
-Data are taken in the wobble mode. This means that the telescope is
+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
@@ -95 +96 @@
 source, although the sky image is rotating in the camera.
 
-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
+The sky region projected onto the camera would not remain the same
+during tracking of a source, if $\Delta \beta$ were defined as a fixed
+angle in the local angles $\Theta$ or $\phi$. This would not
+necessarily be a disadvantage. In the case $\Delta \beta$ is taken as
+a fixed angle in $\phi$ a sky region would be selected whose center
+has the same zenith angle $\Theta$ as the source being observed.
+
+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.
 
@@ -130 +138 @@
 
 Under the above assumption the only dependence to be considered for
-the collection areas (see Section 3) is the dependence on the energy
-of the cosmic ray particle and on the zenith angle $\Theta$.
+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
@@ -150 +159 @@
 \item Image parameters :\\
 The standard definition of the image parameters is assumed. See for
-example \cite{...}.
+example \cite{hillas85,fegan96,reynolds93}.
  
 \item Impact parameter :\\
@@ -168 +177 @@
 
 \subsection{Formulas}
-
-\begin{enumerate}
-\item Differential gamma flux and collection area for a point source
-
-The differential gamma flux from as point sourse $s$ is
+\subsubsection{Differential gamma flux and collection area for a point source}
+
+The differential gamma flux from a point sourse $s$ is given by
 
 \begin{eqnarray}
@@ -180 +187 @@
 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 probabiolity for reconstructing a gamma shower with energy
-$E$, zenith angle $\Theta$ and impact parameter $p$ by 
-$R^{\gamma}(E,\;\Theta,\;p)$. The effective collection area is defined as
-
-\begin{eqnarray}{lll}
-F^{\gamma}_{eff}\;  &=  &\int R^{\gamma}(E,\Theta,p)\;dF  \\
-                    &=  &2\pi\;\int R^{\gamma}(E,\Theta,p)\;p\;dp
+denote the probability for reconstructing a gamma shower with energy
+$E$, zenith angle $\Theta$ and position $F$ in a plane perpendicular
+to the source direction by 
+$R^{\gamma}(E,\Theta,F)$. 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} 
 
 
 The number of $\gamma$ showers observed in the bin $\Delta \Theta$ of
-the zenith angle $\Theta$ and in the bin $|Delta E$ of the energy is
+the zenith angle and in the bin $\Delta E$ of the energy is
 then :
 
-\begin{eqnarray}{lll}
-\Delta N^{\gamma,obs}_s  &=         &\Delta T_{on}(\Theta) \cdot
- \int_{\Delta E}{} \Phi^{\gamma}_s(E)\;F^{\gamma}_{eff}(E,\Theta)\;dE \\
-                         &\approx=  &\Delta T_{on}(\Theta) \cdot 
-  F^{\gamma}_{eff}(E),\Theta) \cdot \int_{\Delta E}{}
- \Phi^{\gamma}_s(E)\;dE \\
-                         &=         &\Delta T_{on}(\Theta) \cdot 
-  F^{\gamma}_{eff}(E),\Theta) \cdot \Delta E \cdot 
-  \overline{\Phi^{\gamma}_s}(E) \\
-\end{eqnarray} 
-
-\end{enumerate}
+\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 \\
+                         &\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.
+
+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 \\
+                         &\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}
@@ -461 +830 @@
 \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 1 (1985) 155
+\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}
 %
@@ -486 +864 @@
 %% End:
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