source: trunk/MagicDoku/strategy_mc_ana.tex @ 779

Last change on this file since 779 was 779, checked in by wittek, 20 years ago
*** empty log message ***
File size: 31.3 KB
Line 
1\documentclass[12pt]{article}
2
3\usepackage{magic-tdas} 
4
5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
6%% BEGIN DOCUMENT
7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
8\begin{document}
9
10%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
11%% Please, for the formatting just include here the standard
12%% elements: title, author, date, plus TDAScode
13%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14\title{Outline of a standard analysis for MAGIC \\
15(including Monte Carlo work)}
16\author{H. Kornmayer, W. Wittek\\ 
17\texttt{h.kornmayer@web.de, wittek@mppmu.mpg.de}}
18
19\date{ \today}
20\TDAScode{MAGIC-TDAS 01-??\\ ??????/W.Wittek}
21%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
22
23%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
24\maketitle
25
26%% abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
27\begin{abstract}
28
29\end{abstract}
30
31%% contents %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
32\thetableofcontents
33
34\newpage
35
36%% body %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37
38%------------------------------------------------------------
39\section{Aim of this paper}
40The aim of this paper is to describe the procedure to obtain the
41absolute energy spectrum of a point source from the data taken with
42MAGIC. This includes work on Mont Carlo (MC) data and the analysis of
43the real data.
44
45Various steps in the procedure will depend on details of the MC
46generation, on the way the real data are taken, etc.. These details
47have therefore to be specified, which is done in Section 2.
48
49In Section 3 some basic definitions and formulas are collected in
50order to avoid any misunderstanding of the meaning of frequently
51used terms.
52
53Section 4 describes the MC work and Section 5 the actual analysis of
54the real data.
55
56One aim of this paper is also to define jobs for those who want to
57join the activities in the software developments. As will be seen, the
58main ingredients both for the MC work and the real data analysis are
59available. However, certain parts have yet to be implemented, others
60have to be changed, modified, improved or extended. Last not least
61extensive tests have to be performed.
62
63
64
65\section{Assumptions}
66The assumptions for a 'standard analysis' listed below are the result of
67discussions in the software group. Some of them are rather arbitrary.
68They should by no means be
69understood as final or optimal choices. They should be considered as a
70starting point. As our experience with the analysis grows we may
71have to revise some of the assumptions.
72
73The aim in all what follows is to define a strategy that is as simple
74and robust as possible. Tests that have yet to be performed will tell
75us whether the assumptions are reasonable and realistic.
76
77The assumptions are :
78
79\begin{itemize}
80\item Mode of observation :\\
81Data are taken in the wobble mode (\cite{konopelko99}).
82This means that the telescope is
83directed not to the position of the selected source but rather to a
84position which has a certain offset ($\Delta\beta$) from the source
85position. Every 20 minutes of observation the sign of $\Delta\beta$ is
86changed. The two wobble positions are called wobble position 1 and 2.
87
88$\Delta \beta$ may be chosen to be a direction difference
89in celestial coordinates
90(declination $\delta$, right ascension $\Phi$) or in local coordinates
91(zenith angle $\Theta$, azimuthal angle $\phi$).
92However the direction $\Delta \beta$ is defined,
93the sky region projected onto the camera is different for
94wobble positions 1 and 2.
95
96If $\Delta \beta$ is defined to be a direction difference
97in celestial coordinates,
98the sky region projected onto the camera for a fixed wobble position
99remains the same during tracking of a source, although the sky image
100is rotating in the camera.
101
102If $\Delta \beta$ is defined to be a direction difference
103in local coordinates,
104the sky region projected onto the camera is changing continuously
105during tracking of a source. The centers of the projected sky regions
106lie on a circle, which is centered at the source position.
107
108If $\Delta \beta$ is defined to be a direction difference
109in the local azimuthal
110angle $\phi$, the center of the camera and the source position
111would always have the same zenith angle $\Theta$. Since the reconstruction
112efficiency of showers mainly depends on $\Theta$, this may be an
113advantage of defining $\Delta \beta$ in this way.
114
115The wobble mode has to be understood as an alternative to taking on-
116and off-data in separate runs. Choosing the wobble mode thus implies
117that one is taking on-data only, from which also the 'off-data' have to be
118obtained by some procedure.
119
120Open questions : - how should $\Delta \beta$ be defined
121                 - how big should $\Delta \beta$ be chosen
122
123\item Pedestals :\\
124Pedestals and their fluctuations are not determined from triggered
125showers but rather from pedestal events. The pedestal events are taken
126'continuously' at a constant rate of 5 Hz. In this way the pedestals
127and their fluctuations are always up to date, and the presence of
128stars and their position in the camera can be monitored continuously.
129
130\item Gamma/hadron separation :\\
131It is assumed that it is possible to define a gamma/hadron separation
132which is independent
133 \begin{itemize}
134 \item[-] of the level of the light of the night sky (LONS)
135 \item[-] of the presence of stars in the field of view (FOV) of the camera
136 \item[-] of the orientation of the sky image in the camera
137 \item[-] of the source being observed
138 \end{itemize}
139
140It has yet to be proven that this is possible. The corresponding
141procedures have to be developed, which includes a proper treatment of the
142pedestal fluctuations in the image analysis.
143
144The gamma/hadron separation will be given in terms of a set of cuts
145on quantities which are derived from the measurable quantities, which are :
146 \begin{itemize}
147 \item[-] the direction $\Theta$ and $\phi$ the telescope is pointing to
148 \item[-] the image parameters
149 \item[-] the pedestal fluctuations
150 \end{itemize}
151
152Under the above assumption the only dependence to be considered for
153the collection areas (see Section 3) is the dependence on the type of
154the cosmic ray particle (gamma, proton, ...), on its energy and on the
155zenith angle $\Theta$.
156
157It has to be investigated whether also the azimuthal angle $\phi$ has to be
158taken into account, for example because of influences from the earth
159magnetic field.
160
161\item Trigger condition :\\
162
163\item Standard analysis cuts :\\ 
164 
165\end{itemize}
166
167
168\section{Definitions and formulas}
169\subsection{Definitions}
170
171\begin{itemize} 
172\item Image parameters :\\
173The standard definition of the image parameters is assumed. See for
174example \cite{hillas85,fegan96,reynolds93}.
175 
176\item Impact parameter :\\
177The impact parameter $p$ is defined as the vertical distance
178of the telescope from the shower axis. It is not directly
179measurable. It may be estimated from the image parameters.
180
181\item Energy :\\
182The energy of the shower is not directly measurable either, but may be
183estimated from the image parameters too.
184
185\item The direction $(\Theta,\phi)$ :\\
186$(\Theta,\phi)$ denotes the direction the telescope is pointing to,
187not the position of the source being observed.
188\end{itemize}
189
190
191\subsection{Formulas}
192\subsubsection{Differential gamma flux and collection area for a point source}
193
194The differential gamma flux from a point sourse $s$ is given by
195
196\begin{eqnarray}
197\Phi^{\gamma}_s(E)\;=\;\dfrac{dN^{\gamma}_s}{dE \cdot dF \cdot dt} 
198\end{eqnarray} 
199
200where $dN^{\gamma}_s$ is the number of gammas from the source $s$ in
201the bin $dE,\;dF,\;dt$ of energy, area and time respectively. We
202denote the probability for reconstructing a gamma shower with energy
203$E$, zenith angle $\Theta$ and position $F$ in a plane perpendicular
204to the source direction by
205$R^{\gamma}(E,\Theta,F)$. The effective collection area is defined as
206
207\begin{eqnarray}
208F^{\gamma}_{eff}(E,\Theta)\;  &&\int R^{\gamma}(E,\Theta,F)\cdot dF 
209\label{eq:form-1}
210\end{eqnarray} 
211
212
213The number of $\gamma$ showers observed in the bin $\Delta \Theta$ of
214the zenith angle and in the bin $\Delta E$ of the energy is
215then :
216
217\begin{eqnarray}
218\Delta N^{\gamma,obs}_s(E,\Theta&= &\int R^{\gamma}(E,\Theta,F) \cdot
219 \Phi^{\gamma}_s(E) \cdot dE \cdot dF \cdot dt \\
220                                   &= &\Delta T_{on}(\Theta) \cdot
221 \int_{\Delta E}{} \Phi^{\gamma}_s(E)\cdot
222 F^{\gamma}_{eff}(E,\Theta)\cdot dE \\
223                         &\approx   &\Delta T_{on}(\Theta) \cdot 
224  F^{\gamma}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
225 \Phi^{\gamma}_s(E)\cdot dE               \label{eq:form0}\\
226                         &\approx   &\Delta T_{on}(\Theta) \cdot 
227  F^{\gamma}_{eff}(E,\Theta) \cdot \Delta E \cdot 
228  \overline{\Phi^{\gamma}_s}(E)       \label{eq:form1}
229\end{eqnarray} 
230
231Here $\Delta T_{on}(\Theta)$ is the effective on-time for the data
232taken in the zenith angle bin $\Delta \Theta$ and $\overline{\Phi^{\gamma}_s}(E)$
233is the average differential gamma flux in the energy bin $\Delta E$ :
234
235\begin{eqnarray}
236\overline{\Phi^{\gamma}_s}(E)  &
237                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
238                     \Phi^{\gamma}_s(E)\cdot dE
239\end{eqnarray} 
240
241By inverting equation (\ref{eq:form1}) and setting
242$\Delta E\;=\;(E^{up}-E^{low})\;\;\;\;\overline{\Phi^{\gamma}_s}(E)$ can
243be written as
244
245\begin{eqnarray}
246  \overline{\Phi^{\gamma}_s}(E)    &=
247  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)} 
248{\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta) \cdot 
249 (E^{up}-E^{low}) }
250 \label{eq:form2}
251\end{eqnarray} 
252
253By means of equation (\ref{eq:form2}) $\overline{\Phi^{\gamma}_s}(E)$
254can be determined
255from the measured $\Delta N^{\gamma,obs}_s(E,\Theta)$ and
256$\Delta T_{on}(\Theta)$, using the collection area
257$F^{\gamma}_{eff}(E,\Theta)$, which is obtained from MC data.
258
259Equation (\ref{eq:form2}) is for a limited and fixed region of
260the zenith angle. One may calculate $\overline{\Phi^{\gamma}_s}(E)$ from the
261data taken at all $\Theta$, in which case
262
263\begin{eqnarray}
264  \overline{\Phi^{\gamma}_s}(E)    &=
265  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)} 
266         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
267          \cdot (E^{up}_i-E^{low}_i) }
268 \label{eq:form3}
269\end{eqnarray} 
270
271If a fixed spectral index $\alpha$ is assumed for the differential
272source spectrum
273
274\begin{eqnarray}
275 \Phi^{\gamma}_s(E)  &&\Phi^{\gamma}_0 \cdot 
276                       \left(\dfrac{E}{GeV}\right)^{-\alpha}
277\end{eqnarray} 
278
279one gets
280
281\begin{eqnarray}
282 \int_{\Delta E}{} \Phi^{\gamma}_s(E) \cdot dE  &
283 &\dfrac{\Phi^{\gamma}_0}{1-\alpha} 
284  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
285         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]\cdot GeV
286 \label{eq:form4}
287\end{eqnarray} 
288
289Inserting (\ref{eq:form4}) into (\ref{eq:form0}) yields
290
291\begin{eqnarray}
292  \Phi^{\gamma}_0    &=
293  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)} 
294         {\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta)
295          \cdot 
296  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
297         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]} 
298  \cdot \dfrac{1-\alpha}{GeV}
299  \label{eq:form5}
300\end{eqnarray} 
301
302which by summing over all $\Theta$ bins gives
303
304\begin{eqnarray}
305  \Phi^{\gamma}_0    &=
306  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)} 
307         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
308          \cdot 
309  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
310         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]} 
311  \cdot \dfrac{1-\alpha}{GeV}
312  \label{eq:form6}
313\end{eqnarray} 
314
315If applied to MC data, for which $\overline{\Phi^{\gamma}_s}(E)$ is known,
316equation (\ref{eq:form1}) can also be used to
317determine the collection area $F^{\gamma}_{eff}(E,\Theta)$ :
318
319\begin{eqnarray}
320F^{\gamma}_{eff}(E,\Theta&
321 &\dfrac{\Delta N^{\gamma,MC,reconstructed}_s(E,\Theta)}
322        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{\gamma}_s}(E) \cdot
323         (E^{up}-E^{low})}
324\end{eqnarray} 
325
326This procedure of determining $F^{\gamma}_{eff}(E,\Theta)$ amounts to
327performing the integration in equation (\ref{eq:form-1}) by MC. An
328important precondition is that in the MC simulation all gamma showers for
329which $R^{\gamma}(E,\Theta,F)$ is greater than zero were
330simulated. This means in particular that the MC simulation of gammas
331extends to sufficiently large impact parameters.
332
333Knowing $F^{\gamma}_{eff}(E,\Theta)$, the gamma fluxes can be obtained
334from the experimental data using equation (\ref{eq:form2}),
335(\ref{eq:form3}), (\ref{eq:form5}) or (\ref{eq:form6}).
336
337Of course, the MC data sample used for calculating
338$F^{\gamma}_{eff}(E,\Theta)$ and the experimental data sample used for
339determining the gamma flux by means of $F^{\gamma}_{eff}(E,\Theta)$
340have to be defined identically in many respects : in particular
341the set of cuts
342and the offset between source position and telescope orientation have
343to be the same in the MC data and the experimental data sample.
344
345
346 
347\subsubsection{Differential flux and collection area for
348hadronic cosmic rays}
349
350In the case of hadronic cosmic rays, which arrive from all directions
351$\Omega$, the differential hadron flux is given by
352
353\begin{eqnarray}
354\Phi^{h}(E)\;=\;\dfrac{dN^{h}}{dE \cdot dF \cdot dt \cdot d\Omega} 
355\label{eq:form-12}
356\end{eqnarray} 
357
358
359In contrast to (\ref{eq:form-1}) the effective collection area for hadrons
360is defined as
361
362\begin{eqnarray}
363F^{h}_{eff}(E,\Theta)\;  &&\int R^{h}(E,\Theta,F,\Omega)\cdot dF
364 \cdot d\Omega 
365\label{eq:form-11}
366\end{eqnarray} 
367
368Note that for a fixed orientation of the telescope $(\Theta,\phi)$ the
369hadrons are coming from all directions $\Omega$. The reconstruction
370efficiency $R^h(E,\Theta,F,\Omega)$ of hadrons therefore depends also
371on $\Omega$.
372
373With the definitions (\ref{eq:form-12}) and (\ref{eq:form-11})
374very similar formulas are obtained for hadrons as
375were derived for photons in the previous section. For clarity they
376are written down explicitely :
377
378\begin{eqnarray}
379\Delta N^{h,obs}(E,\Theta&= &\int R^{h}(E,\Theta,F) \cdot
380 \Phi^{h}(E) \cdot dE \cdot dF \cdot dt \\
381                                   &= &\Delta T_{on}(\Theta) \cdot
382 \int_{\Delta E}{} \Phi^{h}(E)\cdot
383 F^{h}_{eff}(E,\Theta)\cdot dE \\
384                         &\approx   &\Delta T_{on}(\Theta) \cdot 
385  F^{h}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
386 \Phi^{h}(E)\cdot dE               \label{eq:form10}\\
387                         &\approx   &\Delta T_{on}(\Theta) \cdot 
388  F^{h}_{eff}(E,\Theta) \cdot \Delta E \cdot 
389  \overline{\Phi^{h}}(E)       \label{eq:form11}
390\end{eqnarray} 
391
392
393\begin{eqnarray}
394\overline{\Phi^{h}}(E)  &
395                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
396                     \Phi^{h}(E)\cdot dE
397\end{eqnarray} 
398
399
400\begin{eqnarray}
401  \overline{\Phi^{h}}(E)    &=
402  &\dfrac{\Delta N^{h,obs}(E,\Theta)} 
403{\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta) \cdot 
404 (E^{up}-E^{low}) }
405 \label{eq:form12}
406\end{eqnarray} 
407
408
409
410\begin{eqnarray}
411  \overline{\Phi^{h}}(E)    &=
412  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)} 
413         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
414          \cdot (E^{up}_i-E^{low}_i) }
415 \label{eq:form13}
416\end{eqnarray} 
417
418
419\begin{eqnarray}
420 \Phi^{h}(E)  &&\Phi^{h}_0 \cdot 
421                       \left(\dfrac{E}{GeV}\right)^{-\beta}
422\end{eqnarray} 
423
424
425\begin{eqnarray}
426 \int_{\Delta E}{} \Phi^{h}(E) \cdot dE  &
427 &\dfrac{\Phi^{h}_0}{1-\beta} 
428  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
429         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]\cdot GeV
430 \label{eq:form14}
431\end{eqnarray} 
432
433
434\begin{eqnarray}
435  \Phi^{h}_0    &=
436  &\dfrac{\Delta N^{h,obs}(E,\Theta)} 
437         {\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta)
438          \cdot 
439  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
440         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]} 
441  \cdot \dfrac{1-\beta}{GeV}
442  \label{eq:form15}
443\end{eqnarray} 
444
445
446\begin{eqnarray}
447  \Phi^{h}_0    &=
448  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)} 
449         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
450          \cdot 
451  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
452         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]} 
453  \cdot \dfrac{1-\beta}{GeV}
454  \label{eq:form16}
455\end{eqnarray} 
456
457
458Note that $\Phi^{h}(E)$, $\Phi^h_0$ and $F^{h}_{eff}(E,\Theta)$ differ
459from      $\Phi^{\gamma}(E)$, $\Phi^{\gamma}_0$ and
460$F^{\gamma}_{eff}(E,\Theta)$ by the dimension of the
461solid angle, due to the additional factor $d\Omega$ in
462(\ref{eq:form-12}) and (\ref{eq:form-11}).
463
464Like in the case of gammas from point sources, the effective area
465$F^h_{eff}(E,\Theta)$ for
466hadrons can be calculated by applying equation (\ref{eq:form11}) to MC
467data, for which $\overline{\Phi^h}(E)$ is known :
468
469\begin{eqnarray}
470F^{h}_{eff}(E,\Theta&
471 &\dfrac{\Delta N^{h,MC,reconstructed}(E,\Theta)}
472        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{h}}(E) \cdot
473         (E^{up}-E^{low})}
474\end{eqnarray} 
475
476Similar to the case of gammas from point sources,
477this procedure of determining $F^h_{eff}(E,\Theta)$ amounts to
478performing the integrations in equation (\ref{eq:form-11}) by MC. The
479precondition in the case of hadrons is that in the
480MC simulation all hadron showers for
481which $R^{h}(E,\Theta,F,\Omega)$ is greater than zero were
482simulated. So the simulation should not only include large enough
483impact parameters but also a sufficiently large range of $\Omega$ at
484fixed orientation $(\Theta,\phi)$ of the telescope.
485
486Knowing $F^{h}_{eff}(E,\Theta)$, the hadron fluxes can be obtained
487from the experimental data using equation (\ref{eq:form12}),
488(\ref{eq:form13}), (\ref{eq:form15}) or (\ref{eq:form16}).
489
490
491\subsubsection{Measurement of the absolute differential flux of gammas
492from a point source by normalizing to the flux of hadronic cosmic rays}
493
494In section 3.2.1 a procedure was described for measuring the absolute
495differential flux of gammas from a point source. The result for
496$\overline{\Phi^{\gamma}_s}(E)$ depends on a reliable determination of
497the collection area $F^{\gamma}_{eff}(E,\Theta)$ by MC and the
498measurement of the on-time $\Delta T_{on}(\Theta)$.
499
500The dependence on the MC simulation may be reduced by normalizing to
501the known differential flux of hadronic cosmic rays. Combining
502equations (\ref{eq:form2}) and (\ref{eq:form12}), and assuming that
503$\Delta T_{on}(\Theta)$ is the same for the gamma and the hadron
504sample, yields
505
506\begin{eqnarray}
507\dfrac{\overline{\Phi^{\gamma}_s}(E)}
508      {\overline{\Phi^{h}}(E)}          &&
509\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
510      {\Delta N^{h,obs}(E,\Theta)}      \cdot
511\dfrac{F^{h}_{eff}(E,\Theta)}
512      {F^{\gamma}_{eff}(E,\Theta)}
513\label{eq:form20}
514\end{eqnarray} 
515
516If $\overline{\Phi^{h}}(E)$ is assumed to be known from other
517experiments, equation (\ref{eq:form20}) allows to determine
518$\overline{\Phi^{\gamma}_s}(E)$ from
519the experimental number of gamma and hadron showers using the
520collection areas for gammas and hadrons from the MC. Since only the
521ratio of the collection areas enters the dependence on the
522MC simulation is reduced.
523
524If data from all zenith angles are to be used the corresponding
525expression for $\overline{\Phi^{\gamma}_s}(E)$ is (see equations
526(\ref{eq:form3}) and (\ref{eq:form13}))
527
528\begin{eqnarray}
529\dfrac{\overline{\Phi^{\gamma}_s}(E)}
530      {\overline{\Phi^{h}}(E)}          &&
531\dfrac{\sum_i \Delta N^{\gamma,obs}(E,\Theta_i)}
532      {\sum_i \Delta N^{h,obs}(E,\Theta_i)}      \cdot
533\dfrac{\sum_i \Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
534                                      \cdot (E^{up}_i-E^{low}_i)}
535      {\sum_i \Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
536                                      \cdot (E^{up}_i-E^{low}_i)}
537\label{eq:form21}
538\end{eqnarray} 
539
540Clearly, the set of cuts defining the gamma sample is different from
541the set of cuts defining the hadron sample. However,
542$\Delta N^{\gamma,obs}$ and $\Delta N^{h,obs}$ can still be measured
543simultaneously, in which case $\Delta T_{on}(\Theta_i)$ is the same for
544the gamma and the hadron sample. Measuring gammas and hadrons
545simultaneously has the advantage that variations of the detector
546properties or of the atmospheric conditions during the observation
547partly cancel in (\ref{eq:form20}) and (\ref{eq:form21}).
548
549If fixed spectral indices $\alpha$ and $\beta$ are assumed for the
550differential
551gamma and the hadron fluxes respectively one obtains for the ratio
552$\Phi^{\gamma}_0\;/\;\Phi^h_0$
553(see (\ref{eq:form5}) and (\ref{eq:form15}))
554
555\begin{eqnarray}
556\dfrac{\Phi^{\gamma}_0}
557      {\Phi^{h}_0}          &&
558\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
559      {\Delta N^{h,obs}(E,\Theta)}      \cdot
560\dfrac{F^{h}_{eff}(E,\Theta) \cdot
561  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
562         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]} 
563      {F^{\gamma}_{eff}(E,\Theta)
564  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
565         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]} \cdot
566\dfrac{1-\beta}{1-\alpha}
567\label{eq:form22}
568\end{eqnarray} 
569
570or, when using the data from all zenith angles,
571(see (\ref{eq:form6}) and (\ref{eq:form16}))
572
573\begin{eqnarray}
574\dfrac{\Phi^{\gamma}_0}
575      {\Phi^{h}_0}          &&
576\dfrac{\sum_i\Delta N^{\gamma,obs}(E,\Theta_i)}
577      {\sum_i\Delta N^{h,obs}(E,\Theta_i)}      \cdot
578\dfrac{\sum_i F^{h}_{eff}(E,\Theta_i) \cdot
579  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
580         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]} 
581      {\sum_i F^{\gamma}_{eff}(E,\Theta_i)
582  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
583         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]} \cdot
584\dfrac{1-\beta}{1-\alpha}
585\label{eq:form23}
586\end{eqnarray} 
587
588
589
590% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
591% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
592% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
593
594
595\section{MC work}
596
597\subsection{Overview of the MC and analysis chain}
598
599After a few iterations to improve the programs in speed,
600reliability, ... there is a sample of available programs
601to simulate the behaviour of the MAGIC telescope.
602Due to the big amount of diskspace needed for this simulation
603it was decided, that not only one program will generate
604the MAGIC telescope, but a subsequent chain of different
605programs. In figure \ref{MC_progs} you can see a overview of
606the existing programs and their connections.
607\begin{figure}[h]
608\setlength{\unitlength}{1.cm}
609  \begin{picture}(18.,12.)
610        \put (0., 0.){\framebox(18.,12.){}}
611
612        \put (1, 11.5){{\sl Air shower programs}}
613        \put (1., 10.){\framebox(3.,1.){MMCS}}
614        \put (2., 10.){\vector(0,-1){.9} }
615        \put (1., 8.){\framebox(3.,1.){reflector}}
616        \put (2., 8.){\vector(0,-1){.9}}
617
618        \put (6, 10.){{\sl star background programs}}
619        \put (6.,8.){\framebox(3.,1.){starresponse}}
620        \put (6., 8.){\line(0, -1){1.5}}
621        \put (10.,8.){\framebox(3,1){starfieldadder}}
622        \put (10., 8.){\line(0, -1){1.5}}
623        \put (10., 6.5){\vector(-1,0){6.} }
624
625        \put (1., 6.){\framebox(3.,1.){camera}}
626        \put (2., 6.){\vector(3,-1){5.} }
627
628       
629       
630        \put (14, 11.5){{\sl real data programs}}
631        \put (14, 8.){\framebox(3,1){MAGIC DAQ}}       
632        \put (15, 8.){\vector(0,-1){.9} }
633        \put (14, 6.){\framebox(3.,1.){MERPP}} 
634        \put (15., 6.){\vector(-3,-1){5.} }
635       
636        \put (8.75, 3.7){\oval(4.,1.)} 
637        \put (7., 3.5){MAGIC root file} 
638        \put (8., 3.2){\vector(0, -1){1.0}}
639
640        \put (7, 1.){\framebox(3.,1.){MARS}}
641
642        \thicklines
643        \put (5., 11.){\line(0, -1){6.5}}       
644        \put (13., 12.){\line(0, -1){7.5}}     
645
646  \end{picture} 
647\caption{Overview of the existing programs in the MC of
648MAGIC.}
649\label{MC_progs} 
650\end{figure} 
651A detailed description of the properties of the different programs can be found
652in section \ref{sec_exist_progs}.
653From that diagram you can see the following features of the simulation and
654analysis chain of MAGIC. 
655\begin{enumerate} 
656  \item The simulation of Air showers and the simulation of the night sky
657        background (NSB) is seperated. 
658
659  \item The NSB is seperated in two parts, the contribution from the starfield
660        and from a diffuse part.
661
662  \item To speed up the production the starresponse program creates a databases
663        for stars of different magnitude.
664
665  \item The join of air showers and NSB is done in the camera program.
666
667  \item The analysis of MC \underline{and} real data will be done with only one program.
668        This program is called MARS (Magic Analysis and Reconstruction Software).
669        The output of the camera program from Monte Carlo data and the output of
670        the MERPP (MERging and PreProcessing) program for the real data are the same.
671        So there is no need to use different programs for the analysis. The file
672        generated by this program used the root package from CERN for data storage.
673\end{enumerate} 
674In this section we will only describe the usage of the Monte Carlo programs. The
675descriptions of the MERPP and MARS can be found somewhere else\footnote{Look on the
676MAGIC home page for more information.}.
677
678\subsection{Existing programs} 
679\label{sec_exist_progs}
680\subsubsection{MMCS - Magic Monte Carlo Simulation} 
681 
682This program is based on a CORSIKA simulation. It is used to generate
683air showers for the MAGIC telecope. At the start one run of the
684program, one has to define the details of the simulation.
685One can specify the following parameters of an shower
686(see also figure \ref{pic_shower}):
687%
688\begin{enumerate}
689  \item the type of the particles in one run ($PartID$)
690  \item the energy range of the particles ($E_1, E_2$)
691  \item the slope of the Energy spectra
692  \item the range of the shower core on the ground $r_{core}$.
693  \item the direction of the shower by setting the range of
694        zenith angle ($\Theta_1, \Theta_2$) and
695        azimuth angle  ($\phi_1, \phi_2$)
696\end{enumerate}
697%
698\begin{figure}[h]
699\setlength{\unitlength}{1.5cm}
700\begin{center} 
701  \begin{picture}(9.,6.)
702        \put (0., 0.){\framebox(9.,6.){}}
703
704        \thicklines
705        % telescope
706        \put (5., .5){\oval(.75, .75)[t]} 
707        \put (3., 1.){{\sl Telesope position}} 
708        \put (4.5, 1.){\vector(1, -1){0.5}}
709        % observation level
710        \put (.5, .5){\line(1, 0){8}}
711        \put (.5, .6){{\sl Observation level}} 
712
713        % air shower
714        \put (4. , 5.5 ){\line(2, -3){3.3}}
715        \put (4.5, 5.5 ){{\sl Particle Type ($PartId$)}}
716        \put (4.5, 5.25){{\sl Energy ($E_1 < E < E_2$)}}
717        \put (4.5, 5.  ) {$\Theta_1 < \Theta < \Theta_2$}
718        \put (4.5, 4.75) {$\phi_1 < \phi < \phi_2$}
719        \put (7.5, .75){{\sl shower core}}
720       
721        \thinlines
722        \put (5., .25){\line(1,0){2.3}} 
723        \put (6.1, .25){{\sl $r_{Core}$}}
724       
725        \put (5., .5){\line(4,3){1.571}}       
726        \put (6., 1.35){{\sl $p$}}
727
728  \end{picture} 
729\end{center} 
730  \caption {The parameter of an shower that are possible to define
731at the begin of an MMCS run.}
732\label{pic_shower} 
733\end{figure} 
734Other parameters, that will be important in the analysis later,
735can be calculated. I.e. the impact parameter $p$ is defined by
736the direction
737of the shower ($\Theta, \phi$) and the core position
738($x_{core}, y_{core}$).
739
740The program MMCS will track the whole shower development
741through the atmosphere. All the cerenkov particles that hit a
742sphere around the telesope (in the figure \ref{pic_shower} 
743drawn as the circle around the telecope position) are stored
744on disk. It is important to recognize, that up to now no
745information of the pointing of the telescope was taking into
746account. 
747This cerenkov photons are the input for the next program,
748called reflector.
749
750
751\subsubsection{reflector} 
752
753The aim of the reflector program is the
754tracking of the cerenkov photons to the camera
755of the MAGIC telescope. So this
756is the point where we introduce a specific pointing of
757the telescope ($\Theta_{MAGIC}, \phi_{MAGIC}$).
758For all cerenkov photons the program
759tests if the mirrors are hitten, calculates the
760probability for the reflection and tracks them to the
761mirror plane. All the photons that are hitting the
762camera are written to disk (*.rfl) 
763with their important parameters
764($x_{camera}, y_{camera}, \lambda, t_{arrival}$).
765These parameters are the input from the shower simulation
766for the next program in the
767MC simulation chain, the camera program.
768
769\subsubsection{camera} 
770
771The camera program simulates the behaviour of the
772PMTs and the electronic of the trigger and FAC system.
773For each photon out of the reflector file (*.rfl) the
774camera program calculates the probability to generate
775an photo electron out of the photo cathode. If a photo
776electrons was ejected, this will create a signal in the
777trigger and FADC system of the hitted pixel.
778You have to specify the
779parameter of the signal shaping
780(shape, Amplitude, FWHM of signal)
781at the beginning of the
782camera, seperatly for the trigger and the FADC system.
783All signal from all photoelectrons are superimposed for
784each pixel. As an example you can see the output of
785the trigger and FADC system in figure \ref{fig_trigger_fadc}.
786\begin{figure}[h]
787
788 \caption{The response of one shower from the trigger (left) and
789fadc system (right).}
790\label{fig_trigger_fadc}
791\end{figure}
792
793All these analog signals going into the trigger system are used
794to check if for a given event a trigger signal was generated or
795not. But before the start of the camera program on also has to
796set a few parameters of the trigger system like:
797\begin{itemize}
798  \item diskriminator threshold
799  \item mulitplicity
800  \item topology
801\end{itemize} 
802With this set of parameter the camera program will analyse
803if one event has triggered. For the triggered event all the FADC
804content will be writen on the file (*.root). In addition all the
805information about the event ($PartID, E, \Theta$,...) and
806information of trigger (FirstLevel, SecondLevel, ..) are also
807be written to the file.
808
809One of the nice features of the camera program is the possiblity
810so simulate the NSB, the diffuse and the star light part of it.
811But before doing this, on has to start other programs
812(called starresponse and starfieldadder) that are describe
813below.
814
815\subsubsection{starresponse}
816
817This program will simulate the analog response for stars of
818a given brightness $B$.
819
820
821\subsubsection{starfieldadder}
822
823
824
825
826
827
828
829\subsection{What to do} 
830
831\begin{itemize} 
832  \item pedestal fluctuations
833  \item trigger
834  \item rates (1st level, 2nd level, .... )
835  \item discriminator thresholds
836  \item Xmax
837  \item collection area
838  \item $\gamma$/h-Seperation
839  \item magnetic field studies ($\phi$-dependence)
840  \item rotating star field
841\end{itemize} 
842
843\section{Analysis of the real data}
844
845\begin{thebibliography}{xxxxxxxxxxxxxxx}
846\bibitem{fegan96}D.J.Fegan, Space Sci.Rev. 75 (1996)137
847\bibitem{hillas85}A.M.Hillas, Proc. 19th ICRC, La Jolla 3 (1985) 445
848\bibitem{konopelko99}A.Konopelko et al., Astropart. Phys. 10 (1999)
849275
850\bibitem{reynolds93}P.T.Reynolds et al., ApJ 404 (1993) 206
851\end{thebibliography}
852
853
854\end{document}
855%
856%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
857%%% 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
858%%% 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
859%%% Digits        0 1 2 3 4 5 6 7 8 9
860%%% Exclamation   !           Double quote "          Hash (number) #
861%%% Dollar        $           Percent      %          Ampersand     &
862%%% Acute accent  '           Left paren   (          Right paren   )
863%%% Asterisk      *           Plus         +          Comma         ,
864%%% Minus         -           Point        .          Solidus       /
865%%% Colon         :           Semicolon    ;          Less than     <
866%%% Equals        =           Greater than >          Question mark ?
867%%% At            @           Left bracket [          Backslash     \
868%%% Right bracket ]           Circumflex   ^          Underscore    _
869%%% Grave accent  `           Left brace   {          Vertical bar  |
870%%% Right brace   }           Tilde        ~
871%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
872%% Local Variables:
873%% mode:latex
874%% mode:font-lock
875%% mode:auto-fill
876%% time-stamp-line-limit:100
877%% End:
878%% EOF
879
880
881
882
883
884
885
886
887
Note: See TracBrowser for help on using the repository browser.