Changeset 777


Ignore:
Timestamp:
May 3, 2001, 8:54:14 AM (20 years ago)
Author:
wittek
Message:
Merge from WOW version with HAKO version on 3.5.2001
File:
1 edited

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  • trunk/MagicDoku/strategy_mc_ana.tex

    r773 r777  
    1414\title{Outline of a standard analysis for MAGIC \\
    1515(including Monte Carlo work)}
    16 \author{W. Wittek, H. Kornmayer\\
    17 \texttt{wittek@mppu.mpg.de, h.kornmayer@web.de }}
     16\author{H. Kornmayer, W. Wittek\\
     17\texttt{h.kornmayer@web.de, wittek@mppmu.mpg.de}}
    1818
    1919\date{ \today}
     
    7979\begin{itemize}
    8080\item Mode of observation :\\
    81 Data are taken in the wobble mode. This means that the telescope is
     81Data are taken in the wobble mode (\cite{konopelko99}).
     82This means that the telescope is
    8283directed not to the position of the selected source but rather to a
    8384position which has a certain offset ($\Delta\beta$) from the source
     
    9596source, although the sky image is rotating in the camera.
    9697
    97 The wobble mode has to be understood as an alternative to taking on
    98 and off data in separate runs. Choosing the wobble mode thus implies
    99 that one is taking on data only, from which also the 'off data' have to be
     98The sky region projected onto the camera would not remain the same
     99during tracking of a source, if $\Delta \beta$ were defined as a fixed
     100angle in the local angles $\Theta$ or $\phi$. This would not
     101necessarily be a disadvantage. In the case $\Delta \beta$ is taken as
     102a fixed angle in $\phi$ a sky region would be selected whose center
     103has the same zenith angle $\Theta$ as the source being observed.
     104
     105The wobble mode has to be understood as an alternative to taking on-
     106and off-data in separate runs. Choosing the wobble mode thus implies
     107that one is taking on-data only, from which also the 'off-data' have to be
    100108obtained by some procedure.
    101109
     
    130138
    131139Under the above assumption the only dependence to be considered for
    132 the collection areas (see Section 3) is the dependence on the energy
    133 of the cosmic ray particle and on the zenith angle $\Theta$.
     140the collection areas (see Section 3) is the dependence on the type of
     141the cosmic ray particle (gamma, proton, ...), on its energy and on the
     142zenith angle $\Theta$.
    134143
    135144It has to be investigated whether also the azimuthal angle $\phi$ has to be
     
    150159\item Image parameters :\\
    151160The standard definition of the image parameters is assumed. See for
    152 example \cite{...}.
     161example \cite{hillas85,fegan96,reynolds93}.
    153162 
    154163\item Impact parameter :\\
     
    168177
    169178\subsection{Formulas}
    170 
    171 \begin{enumerate}
    172 \item Differential gamma flux and collection area for a point source
    173 
    174 The differential gamma flux from as point sourse $s$ is
     179\subsubsection{Differential gamma flux and collection area for a point source}
     180
     181The differential gamma flux from a point sourse $s$ is given by
    175182
    176183\begin{eqnarray}
     
    180187where $dN^{\gamma}_s$ is the number of gammas from the source $s$ in
    181188the bin $dE,\;dF,\;dt$ of energy, area and time respectively. We
    182 denote the probabiolity for reconstructing a gamma shower with energy
    183 $E$, zenith angle $\Theta$ and impact parameter $p$ by
    184 $R^{\gamma}(E,\;\Theta,\;p)$. The effective collection area is defined as
    185 
    186 \begin{eqnarray}{lll}
    187 F^{\gamma}_{eff}\;  &=  &\int R^{\gamma}(E,\Theta,p)\;dF  \\
    188                     &=  &2\pi\;\int R^{\gamma}(E,\Theta,p)\;p\;dp
     189denote the probability for reconstructing a gamma shower with energy
     190$E$, zenith angle $\Theta$ and position $F$ in a plane perpendicular
     191to the source direction by
     192$R^{\gamma}(E,\Theta,F)$. The effective collection area is defined as
     193
     194\begin{eqnarray}
     195F^{\gamma}_{eff}(E,\Theta)\;  &=  &\int R^{\gamma}(E,\Theta,F)\cdot dF 
     196\label{eq:form-1}
    189197\end{eqnarray}
    190198
    191199
    192200The number of $\gamma$ showers observed in the bin $\Delta \Theta$ of
    193 the zenith angle $\Theta$ and in the bin $|Delta E$ of the energy is
     201the zenith angle and in the bin $\Delta E$ of the energy is
    194202then :
    195203
    196 \begin{eqnarray}{lll}
    197 \Delta N^{\gamma,obs}_s  &=         &\Delta T_{on}(\Theta) \cdot
    198  \int_{\Delta E}{} \Phi^{\gamma}_s(E)\;F^{\gamma}_{eff}(E,\Theta)\;dE \\
    199                          &\approx=  &\Delta T_{on}(\Theta) \cdot
    200   F^{\gamma}_{eff}(E),\Theta) \cdot \int_{\Delta E}{}
    201  \Phi^{\gamma}_s(E)\;dE \\
    202                          &=         &\Delta T_{on}(\Theta) \cdot
    203   F^{\gamma}_{eff}(E),\Theta) \cdot \Delta E \cdot
    204   \overline{\Phi^{\gamma}_s}(E) \\
    205 \end{eqnarray}
    206 
    207 \end{enumerate}
     204\begin{eqnarray}
     205\Delta N^{\gamma,obs}_s(E,\Theta)  &= &\int R^{\gamma}(E,\Theta,F) \cdot
     206 \Phi^{\gamma}_s(E) \cdot dE \cdot dF \cdot dt \\
     207                                   &= &\Delta T_{on}(\Theta) \cdot
     208 \int_{\Delta E}{} \Phi^{\gamma}_s(E)\cdot
     209 F^{\gamma}_{eff}(E,\Theta)\cdot dE \\
     210                         &\approx   &\Delta T_{on}(\Theta) \cdot
     211  F^{\gamma}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
     212 \Phi^{\gamma}_s(E)\cdot dE               \label{eq:form0}\\
     213                         &\approx   &\Delta T_{on}(\Theta) \cdot
     214  F^{\gamma}_{eff}(E,\Theta) \cdot \Delta E \cdot
     215  \overline{\Phi^{\gamma}_s}(E)       \label{eq:form1}
     216\end{eqnarray}
     217
     218Here $\Delta T_{on}(\Theta)$ is the effective on-time for the data
     219taken in the zenith angle bin $\Delta \Theta$ and $\overline{\Phi^{\gamma}_s}(E)$
     220is the average differential gamma flux in the energy bin $\Delta E$ :
     221
     222\begin{eqnarray}
     223\overline{\Phi^{\gamma}_s}(E)  &= 
     224                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
     225                     \Phi^{\gamma}_s(E)\cdot dE
     226\end{eqnarray}
     227
     228By inverting equation (\ref{eq:form1}) and setting
     229$\Delta E\;=\;(E^{up}-E^{low})\;\;\;\;\overline{\Phi^{\gamma}_s}(E)$ can
     230be written as
     231
     232\begin{eqnarray}
     233  \overline{\Phi^{\gamma}_s}(E)    &=
     234  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)} 
     235{\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta) \cdot
     236 (E^{up}-E^{low}) }
     237 \label{eq:form2}
     238\end{eqnarray}
     239
     240By means of equation (\ref{eq:form2}) $\overline{\Phi^{\gamma}_s}(E)$
     241can be determined
     242from the measured $\Delta N^{\gamma,obs}_s(E,\Theta)$ and
     243$\Delta T_{on}(\Theta)$, using the collection area
     244$F^{\gamma}_{eff}(E,\Theta)$, which is obtained from MC data.
     245
     246Equation (\ref{eq:form2}) is for a limited and fixed region of
     247the zenith angle. One may calculate $\overline{\Phi^{\gamma}_s}(E)$ from the
     248data taken at all $\Theta$, in which case
     249
     250\begin{eqnarray}
     251  \overline{\Phi^{\gamma}_s}(E)    &=
     252  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)} 
     253         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
     254          \cdot (E^{up}_i-E^{low}_i) }
     255 \label{eq:form3}
     256\end{eqnarray}
     257
     258If a fixed spectral index $\alpha$ is assumed for the differential
     259source spectrum
     260
     261\begin{eqnarray}
     262 \Phi^{\gamma}_s(E)  &=  &\Phi^{\gamma}_0 \cdot
     263                       \left(\dfrac{E}{GeV}\right)^{-\alpha}
     264\end{eqnarray}
     265
     266one gets
     267
     268\begin{eqnarray}
     269 \int_{\Delta E}{} \Phi^{\gamma}_s(E) \cdot dE  &= 
     270 &\dfrac{\Phi^{\gamma}_0}{1-\alpha}
     271  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
     272         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]\cdot GeV
     273 \label{eq:form4}
     274\end{eqnarray}
     275
     276Inserting (\ref{eq:form4}) into (\ref{eq:form0}) yields
     277
     278\begin{eqnarray}
     279  \Phi^{\gamma}_0    &=
     280  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)} 
     281         {\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta)
     282          \cdot
     283  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
     284         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]}
     285  \cdot \dfrac{1-\alpha}{GeV}
     286  \label{eq:form5}
     287\end{eqnarray}
     288
     289which by summing over all $\Theta$ bins gives
     290
     291\begin{eqnarray}
     292  \Phi^{\gamma}_0    &=
     293  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)} 
     294         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
     295          \cdot
     296  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
     297         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]}
     298  \cdot \dfrac{1-\alpha}{GeV}
     299  \label{eq:form6}
     300\end{eqnarray}
     301
     302If applied to MC data, for which $\overline{\Phi^{\gamma}_s}(E)$ is known,
     303equation (\ref{eq:form1}) can also be used to
     304determine the collection area $F^{\gamma}_{eff}(E,\Theta)$ :
     305
     306\begin{eqnarray}
     307F^{\gamma}_{eff}(E,\Theta)  &= 
     308 &\dfrac{\Delta N^{\gamma,MC,reconstructed}_s(E,\Theta)}
     309        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{\gamma}_s}(E) \cdot
     310         (E^{up}-E^{low})}
     311\end{eqnarray}
     312
     313This procedure of determining $F^{\gamma}_{eff}(E,\Theta)$ amounts to
     314performing the integration in equation (\ref{eq:form-1}) by MC. An
     315important precondition is that in the MC simulation all gamma showers for
     316which $R^{\gamma}(E,\Theta,F)$ is greater than zero were
     317simulated. This means in particular that the MC simulation of gammas
     318extends to sufficiently large impact parameters.
     319
     320Knowing $F^{\gamma}_{eff}(E,\Theta)$, the gamma fluxes can be obtained
     321from the experimental data using equation (\ref{eq:form2}),
     322(\ref{eq:form3}), (\ref{eq:form5}) or (\ref{eq:form6}).
     323
     324Of course, the MC data sample used for calculating
     325$F^{\gamma}_{eff}(E,\Theta)$ and the experimental data sample used for
     326determining the gamma flux by means of $F^{\gamma}_{eff}(E,\Theta)$
     327have to be defined identically in many respects : in particular
     328the set of cuts
     329and the offset between source position and telescope orientation have
     330to be the same in the MC data and the experimental data sample.
     331
     332
     333 
     334\subsubsection{Differential flux and collection area for
     335hadronic cosmic rays}
     336
     337In the case of hadronic cosmic rays, which arrive from all directions
     338$\Omega$, the differential hadron flux is given by
     339
     340\begin{eqnarray}
     341\Phi^{h}(E)\;=\;\dfrac{dN^{h}}{dE \cdot dF \cdot dt \cdot d\Omega}
     342\label{eq:form-12}
     343\end{eqnarray}
     344
     345
     346In contrast to (\ref{eq:form-1}) the effective collection area for hadrons
     347is defined as
     348
     349\begin{eqnarray}
     350F^{h}_{eff}(E,\Theta)\;  &=  &\int R^{h}(E,\Theta,F,\Omega)\cdot dF
     351 \cdot d\Omega 
     352\label{eq:form-11}
     353\end{eqnarray}
     354
     355Note that for a fixed orientation of the telescope $(\Theta,\phi)$ the
     356hadrons are coming from all directions $\Omega$. The reconstruction
     357efficiency $R^h(E,\Theta,F,\Omega)$ of hadrons therefore depends also
     358on $\Omega$.
     359
     360With the definitions (\ref{eq:form-12}) and (\ref{eq:form-11})
     361very similar formulas are obtained for hadrons as
     362were derived for photons in the previous section. For clarity they
     363are written down explicitely :
     364
     365\begin{eqnarray}
     366\Delta N^{h,obs}(E,\Theta)  &= &\int R^{h}(E,\Theta,F) \cdot
     367 \Phi^{h}(E) \cdot dE \cdot dF \cdot dt \\
     368                                   &= &\Delta T_{on}(\Theta) \cdot
     369 \int_{\Delta E}{} \Phi^{h}(E)\cdot
     370 F^{h}_{eff}(E,\Theta)\cdot dE \\
     371                         &\approx   &\Delta T_{on}(\Theta) \cdot
     372  F^{h}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
     373 \Phi^{h}(E)\cdot dE               \label{eq:form10}\\
     374                         &\approx   &\Delta T_{on}(\Theta) \cdot
     375  F^{h}_{eff}(E,\Theta) \cdot \Delta E \cdot
     376  \overline{\Phi^{h}}(E)       \label{eq:form11}
     377\end{eqnarray}
     378
     379
     380\begin{eqnarray}
     381\overline{\Phi^{h}}(E)  &= 
     382                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
     383                     \Phi^{h}(E)\cdot dE
     384\end{eqnarray}
     385
     386
     387\begin{eqnarray}
     388  \overline{\Phi^{h}}(E)    &=
     389  &\dfrac{\Delta N^{h,obs}(E,\Theta)} 
     390{\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta) \cdot
     391 (E^{up}-E^{low}) }
     392 \label{eq:form12}
     393\end{eqnarray}
     394
     395
     396
     397\begin{eqnarray}
     398  \overline{\Phi^{h}}(E)    &=
     399  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)} 
     400         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
     401          \cdot (E^{up}_i-E^{low}_i) }
     402 \label{eq:form13}
     403\end{eqnarray}
     404
     405
     406\begin{eqnarray}
     407 \Phi^{h}(E)  &=  &\Phi^{h}_0 \cdot
     408                       \left(\dfrac{E}{GeV}\right)^{-\beta}
     409\end{eqnarray}
     410
     411
     412\begin{eqnarray}
     413 \int_{\Delta E}{} \Phi^{h}(E) \cdot dE  &= 
     414 &\dfrac{\Phi^{h}_0}{1-\beta}
     415  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
     416         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]\cdot GeV
     417 \label{eq:form14}
     418\end{eqnarray}
     419
     420
     421\begin{eqnarray}
     422  \Phi^{h}_0    &=
     423  &\dfrac{\Delta N^{h,obs}(E,\Theta)} 
     424         {\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta)
     425          \cdot
     426  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
     427         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]}
     428  \cdot \dfrac{1-\beta}{GeV}
     429  \label{eq:form15}
     430\end{eqnarray}
     431
     432
     433\begin{eqnarray}
     434  \Phi^{h}_0    &=
     435  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)} 
     436         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
     437          \cdot
     438  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
     439         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]}
     440  \cdot \dfrac{1-\beta}{GeV}
     441  \label{eq:form16}
     442\end{eqnarray}
     443
     444
     445Note that $\Phi^{h}(E)$, $\Phi^h_0$ and $F^{h}_{eff}(E,\Theta)$ differ
     446from      $\Phi^{\gamma}(E)$, $\Phi^{\gamma}_0$ and
     447$F^{\gamma}_{eff}(E,\Theta)$ by the dimension of the
     448solid angle, due to the additional factor $d\Omega$ in
     449(\ref{eq:form-12}) and (\ref{eq:form-11}).
     450
     451Like in the case of gammas from point sources, the effective area
     452$F^h_{eff}(E,\Theta)$ for
     453hadrons can be calculated by applying equation (\ref{eq:form11}) to MC
     454data, for which $\overline{\Phi^h}(E)$ is known :
     455
     456\begin{eqnarray}
     457F^{h}_{eff}(E,\Theta)  &= 
     458 &\dfrac{\Delta N^{h,MC,reconstructed}(E,\Theta)}
     459        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{h}}(E) \cdot
     460         (E^{up}-E^{low})}
     461\end{eqnarray}
     462
     463Similar to the case of gammas from point sources,
     464this procedure of determining $F^h_{eff}(E,\Theta)$ amounts to
     465performing the integrations in equation (\ref{eq:form-11}) by MC. The
     466precondition in the case of hadrons is that in the
     467MC simulation all hadron showers for
     468which $R^{h}(E,\Theta,F,\Omega)$ is greater than zero were
     469simulated. So the simulation should not only include large enough
     470impact parameters but also a sufficiently large range of $\Omega$ at
     471fixed orientation $(\Theta,\phi)$ of the telescope.
     472
     473Knowing $F^{h}_{eff}(E,\Theta)$, the hadron fluxes can be obtained
     474from the experimental data using equation (\ref{eq:form12}),
     475(\ref{eq:form13}), (\ref{eq:form15}) or (\ref{eq:form16}).
     476
     477
     478\subsubsection{Measurement of the absolute differential flux of gammas
     479from a point source by normalizing to the flux of hadronic cosmic rays}
     480
     481In section 3.2.1 a procedure was described for measuring the absolute
     482differential flux of gammas from a point source. The result for
     483$\overline{\Phi^{\gamma}_s}(E)$ depends on a reliable determination of
     484the collection area $F^{\gamma}_{eff}(E,\Theta)$ by MC and the
     485measurement of the on-time $\Delta T_{on}(\Theta)$.
     486
     487The dependence on the MC simulation may be reduced by normalizing to
     488the known differential flux of hadronic cosmic rays. Combining
     489equations (\ref{eq:form2}) and (\ref{eq:form12}), and assuming that
     490$\Delta T_{on}(\Theta)$ is the same for the gamma and the hadron
     491sample, yields
     492
     493\begin{eqnarray}
     494\dfrac{\overline{\Phi^{\gamma}_s}(E)}
     495      {\overline{\Phi^{h}}(E)}          &=  &
     496\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
     497      {\Delta N^{h,obs}(E,\Theta)}      \cdot
     498\dfrac{F^{h}_{eff}(E,\Theta)}
     499      {F^{\gamma}_{eff}(E,\Theta)}
     500\label{eq:form20}
     501\end{eqnarray}
     502
     503If $\overline{\Phi^{h}}(E)$ is assumed to be known from other
     504experiments, equation (\ref{eq:form20}) allows to determine
     505$\overline{\Phi^{\gamma}_s}(E)$ from
     506the experimental number of gamma and hadron showers using the
     507collection areas for gammas and hadrons from the MC. Since only the
     508ratio of the collection areas enters the dependence on the
     509MC simulation is reduced.
     510
     511If data from all zenith angles are to be used the corresponding
     512expression for $\overline{\Phi^{\gamma}_s}(E)$ is (see equations
     513(\ref{eq:form3}) and (\ref{eq:form13}))
     514
     515\begin{eqnarray}
     516\dfrac{\overline{\Phi^{\gamma}_s}(E)}
     517      {\overline{\Phi^{h}}(E)}          &=  &
     518\dfrac{\sum_i \Delta N^{\gamma,obs}(E,\Theta_i)}
     519      {\sum_i \Delta N^{h,obs}(E,\Theta_i)}      \cdot
     520\dfrac{\sum_i \Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
     521                                      \cdot (E^{up}_i-E^{low}_i)}
     522      {\sum_i \Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
     523                                      \cdot (E^{up}_i-E^{low}_i)}
     524\label{eq:form21}
     525\end{eqnarray}
     526
     527Clearly, the set of cuts defining the gamma sample is different from
     528the set of cuts defining the hadron sample. However,
     529$\Delta N^{\gamma,obs}$ and $\Delta N^{h,obs}$ can still be measured
     530simultaneously, in which case $\Delta T_{on}(\Theta_i)$ is the same for
     531the gamma and the hadron sample. Measuring gammas and hadrons
     532simultaneously has the advantage that variations of the detector
     533properties or of the atmospheric conditions during the observation
     534partly cancel in (\ref{eq:form20}) and (\ref{eq:form21}).
     535
     536If fixed spectral indices $\alpha$ and $\beta$ are assumed for the
     537differential
     538gamma and the hadron fluxes respectively one obtains for the ratio
     539$\Phi^{\gamma}_0\;/\;\Phi^h_0$
     540(see (\ref{eq:form5}) and (\ref{eq:form15}))
     541
     542\begin{eqnarray}
     543\dfrac{\Phi^{\gamma}_0}
     544      {\Phi^{h}_0}          &=  &
     545\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
     546      {\Delta N^{h,obs}(E,\Theta)}      \cdot
     547\dfrac{F^{h}_{eff}(E,\Theta) \cdot
     548  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
     549         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]}
     550      {F^{\gamma}_{eff}(E,\Theta)
     551  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
     552         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]} \cdot
     553\dfrac{1-\beta}{1-\alpha}
     554\label{eq:form22}
     555\end{eqnarray}
     556
     557or, when using the data from all zenith angles,
     558(see (\ref{eq:form6}) and (\ref{eq:form16}))
     559
     560\begin{eqnarray}
     561\dfrac{\Phi^{\gamma}_0}
     562      {\Phi^{h}_0}          &=  &
     563\dfrac{\sum_i\Delta N^{\gamma,obs}(E,\Theta_i)}
     564      {\sum_i\Delta N^{h,obs}(E,\Theta_i)}      \cdot
     565\dfrac{\sum_i F^{h}_{eff}(E,\Theta_i) \cdot
     566  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
     567         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]}
     568      {\sum_i F^{\gamma}_{eff}(E,\Theta_i)
     569  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
     570         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]} \cdot
     571\dfrac{1-\beta}{1-\alpha}
     572\label{eq:form23}
     573\end{eqnarray}
     574
     575
    208576
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     580
    212581
    213582\section{MC work}
     
    461830\section{Analysis of the real data}
    462831
     832\begin{thebibliography}{xxxxxxxxxxxxxxx}
     833\bibitem{fegan96}D.J.Fegan, Space Sci.Rev. 75 (1996)137
     834\bibitem{hillas85}A.M.Hillas, Proc. 19th ICRC, La Jolla 1 (1985) 155
     835\bibitem{konopelko99}A.Konopelko et al., Astropart. Phys. 10 (1999)
     836275
     837\bibitem{reynolds93}P.T.Reynolds et al., ApJ 404 (1993) 206
     838\end{thebibliography}
     839
     840
    463841\end{document}
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