# Changeset 777

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May 3, 2001, 8:54:14 AM (20 years ago)
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Merge from WOW version with HAKO version on 3.5.2001

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 r773 \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} \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 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. 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 \item Image parameters :\\ The standard definition of the image parameters is assumed. See for example \cite{...}. example \cite{hillas85,fegan96,reynolds93}. \item Impact parameter :\\ \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} 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} % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& \section{MC work} \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} % %% End: %% EOF