Index: /old/MagicDoku/strategy_mc_ana.tex =================================================================== --- /old/MagicDoku/strategy_mc_ana.tex (revision 15423) +++ /old/MagicDoku/strategy_mc_ana.tex (revision 15423) @@ -0,0 +1,976 @@ +\documentclass[12pt]{article} + +\usepackage{magic-tdas} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% BEGIN DOCUMENT +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Please, for the formatting just include here the standard +%% elements: title, author, date, plus TDAScode +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\maketitle + +%% abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{abstract} + +\end{abstract} + +%% contents %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\thetableofcontents + +\newpage + +%% body %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%------------------------------------------------------------ +\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} + + + +% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& +% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& +% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& + + +\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} +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Upper-case A B C D E F G H I J K L M N O P Q R S T U V W X Y Z +%%% Lower-case a b c d e f g h i j k l m n o p q r s t u v w x y z +%%% Digits 0 1 2 3 4 5 6 7 8 9 +%%% Exclamation ! Double quote " Hash (number) # +%%% Dollar $ Percent % Ampersand & +%%% Acute accent ' Left paren ( Right paren ) +%%% Asterisk * Plus + Comma , +%%% Minus - Point . 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