Index: trunk/MagicDoku/strategy_mc_ana.tex =================================================================== --- trunk/MagicDoku/strategy_mc_ana.tex (revision 776) +++ trunk/MagicDoku/strategy_mc_ana.tex (revision 777) @@ -14,6 +14,6 @@ \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} @@ -79,5 +79,6 @@ \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 @@ -95,7 +96,14 @@ 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. @@ -130,6 +138,7 @@ 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 @@ -150,5 +159,5 @@ \item Image parameters :\\ The standard definition of the image parameters is assumed. See for -example \cite{...}. +example \cite{hillas85,fegan96,reynolds93}. \item Impact parameter :\\ @@ -168,9 +177,7 @@ \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} @@ -180,34 +187,396 @@ 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} @@ -461,4 +830,13 @@ \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} % @@ -486,2 +864,11 @@ %% End: %% EOF + + + + + + + + +