Index: /trunk/MagicDoku/strategy_mc_ana.tex
===================================================================
 /trunk/MagicDoku/strategy_mc_ana.tex (revision 779)
+++ /trunk/MagicDoku/strategy_mc_ana.tex (revision 780)
@@ 14,5 +14,5 @@
\title{Outline of a standard analysis for MAGIC \\
(including Monte Carlo work)}
\author{H. Kornmayer, W. Wittek\\
+\author{R. B\"ock, H. Kornmayer, W. Wittek\\
\texttt{h.kornmayer@web.de, wittek@mppmu.mpg.de}}
@@ 118,6 +118,13 @@
obtained by some procedure.
Open questions :  how should $\Delta \beta$ be defined
  how big should $\Delta \beta$ be chosen
+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 :\\
@@ 143,5 +150,7 @@
The gamma/hadron separation will be given in terms of a set of cuts
on quantities which are derived from the measurable quantities, which are :
+(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
@@ 170,20 +179,24 @@
\begin{itemize}
\item Image parameters :\\
The standard definition of the image parameters is assumed. See for
example \cite{hillas85,fegan96,reynolds93}.

\item Impact parameter :\\
The impact parameter $p$ is defined as the vertical distance
of the telescope from the shower axis. It is not directly
measurable. It may be estimated from the image parameters.

\item Energy :\\
The energy of the shower is not directly measurable either, but may be
estimated from the image parameters too.

\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}
@@ 192,5 +205,5 @@
\subsubsection{Differential gamma flux and collection area for a point source}
The differential gamma flux from a point sourse $s$ is given by
+The differential gamma flux from a point source $s$ is given by
\begin{eqnarray}
@@ 200,8 +213,11 @@
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 reconstructing a gamma shower with energy
+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
$R^{\gamma}(E,\Theta,F)$. The effective collection area is defined as
+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}
@@ 210,4 +226,13 @@
\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:form1}) 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
@@ 221,4 +246,11 @@
\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}{}
@@ 329,5 +361,9 @@
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.
+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
@@ 382,4 +418,8 @@
\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}{}
@@ 841,4 +881,53 @@
\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. $\gamma$/hseparation. 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 collaborationwide 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 offaxis,
+ 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 firstlevel 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 $\gamma$/hseparation, 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}