# source:trunk/MagicDoku/strategy_mc_ana.tex@780

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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{R. B\"ock, 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
120We propose to define $\Delta \beta$ as a direction difference in the
121local azimuthal angle $\phi$ :
122$\Delta \phi\;=\;\Delta \beta\;/\;sin(\Theta)$. For very small
123$\Theta$ ($\Theta\;<\; 1$ degree) $\Delta \beta$ should be defined
124differently, also to avoid large rotation speeds of the telescope.
125
126Since the radius of the trigger area is 0.8 degrees, we propose
127to choose $\Delta \beta\;=\;0.4$ degrees.
128
129
130\item Pedestals :\\
131Pedestals and their fluctuations are not determined from triggered
132showers but rather from pedestal events. The pedestal events are taken
133'continuously' at a constant rate of 5 Hz. In this way the pedestals
134and their fluctuations are always up to date, and the presence of
135stars and their position in the camera can be monitored continuously.
136
138It is assumed that it is possible to define a gamma/hadron separation
139which is independent
140 \begin{itemize}
141 \item[-] of the level of the light of the night sky (LONS)
142 \item[-] of the presence of stars in the field of view (FOV) of the camera
143 \item[-] of the orientation of the sky image in the camera
144 \item[-] of the source being observed
145 \end{itemize}
146
147It has yet to be proven that this is possible. The corresponding
148procedures have to be developed, which includes a proper treatment of the
149pedestal fluctuations in the image analysis.
150
151The gamma/hadron separation will be given in terms of a set of cuts
152(or certain conditions) on quantities which in general are not
153identical to the measured quantities but which are derived from them. The
154measurable quantities are :
155 \begin{itemize}
156 \item[-] the direction $\Theta$ and $\phi$ the telescope is pointing to
157 \item[-] the image parameters
158 \item[-] the pedestal fluctuations
159 \end{itemize}
160
161Under the above assumption the only dependence to be considered for
162the collection areas (see Section 3) is the dependence on the type of
163the cosmic ray particle (gamma, proton, ...), on its energy and on the
164zenith angle $\Theta$.
165
166It has to be investigated whether also the azimuthal angle $\phi$ has to be
167taken into account, for example because of influences from the earth
168magnetic field.
169
170\item Trigger condition :\\
171
172\item Standard analysis cuts :\\
173
174\end{itemize}
175
176
177\section{Definitions and formulas}
178\subsection{Definitions}
179
180\begin{itemize}
181\item The direction $(\Theta,\phi)$ :\\
182$(\Theta,\phi)$ denotes the direction the telescope is pointing to,
183not the position of the source being observed.
184
185\item Image parameters :\\
186The standard definition of the image parameters is assumed. See for
187example \cite{hillas85,fegan96,reynolds93}. We should also make use of
188additional parameters like asymmetry parameters, number of islands or
189mountains etc.
190\end{itemize}
191
192Quantities which are not directly measurable, but which can be
193estimated from the image parameters are :
194
195\begin{itemize}
196\item Impact parameter :\\
197The impact parameter $p$ is defined as the vertical distance
198of the telescope from the shower axis.
199
200\item The energy of the shower
201\end{itemize}
202
203
204\subsection{Formulas}
205\subsubsection{Differential gamma flux and collection area for a point source}
206
207The differential gamma flux from a point source $s$ is given by
208
209\begin{eqnarray}
210\Phi^{\gamma}_s(E)\;=\;\dfrac{dN^{\gamma}_s}{dE \cdot dF \cdot dt}
211\end{eqnarray}
212
213where $dN^{\gamma}_s$ is the number of gammas from the source $s$ in
214the bin $dE,\;dF,\;dt$ of energy, area and time respectively. We
215denote the probability for 'observing' a gamma shower with energy
216$E$, zenith angle $\Theta$ and position $F$ in a plane perpendicular
217to the source direction by $R^{\gamma}(E,\Theta,F)$. Depending on the
218special study, the term 'observing' may mean triggering,
219reconstructing, etc.
220
221The effective collection area is defined as
222
223\begin{eqnarray}
224F^{\gamma}_{eff}(E,\Theta)\;  &&\int R^{\gamma}(E,\Theta,F)\cdot dF
225\label{eq:form-1}
226\end{eqnarray}
227
228A side remark : The well known behaviour that the effective collection
229area (well above the threshold energy) is larger for larger zenith angles
230$\Theta$, is due to the fact that at higher $\Theta$ the distance of
231the shower maximum (where the majority of Cherenkov photons is
232emitted) from the detector is larger than at smaller $\Theta$. The
233area in which $R^{\gamma}(E,\Theta,F)$ contributes significantly to
234the integral (\ref{eq:form-1}) is therefore larger, resulting in a
235larger $F^{\gamma}_{eff}(E,\Theta)$. For the simulation this means,
236that the maximum impact parameter should be chosen larger for larger $\Theta$.
237
238The number of $\gamma$ showers observed in the bin $\Delta \Theta$ of
239the zenith angle and in the bin $\Delta E$ of the energy is
240then :
241
242\begin{eqnarray}
243\Delta N^{\gamma,obs}_s(E,\Theta&= &\int R^{\gamma}(E,\Theta,F) \cdot
244 \Phi^{\gamma}_s(E) \cdot dE \cdot dF \cdot dt \\
245                                   &= &\Delta T_{on}(\Theta) \cdot
246 \int_{\Delta E}{} \Phi^{\gamma}_s(E)\cdot
247 F^{\gamma}_{eff}(E,\Theta)\cdot dE \\
248\end{eqnarray}
249
250Assuming that $F^{\gamma}_{eff}(E,\Theta)$ depends only weakly on $E$
251in the (sufficiently small) interval $\Delta E$ gives
252
253\begin{eqnarray}
254\Delta N^{\gamma,obs}_s(E,\Theta
255                         &\approx   &\Delta T_{on}(\Theta) \cdot
256  F^{\gamma}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
257 \Phi^{\gamma}_s(E)\cdot dE               \label{eq:form0}\\
258                         &\approx   &\Delta T_{on}(\Theta) \cdot
259  F^{\gamma}_{eff}(E,\Theta) \cdot \Delta E \cdot
260  \overline{\Phi^{\gamma}_s}(E)       \label{eq:form1}
261\end{eqnarray}
262
263Here $\Delta T_{on}(\Theta)$ is the effective on-time for the data
264taken in the zenith angle bin $\Delta \Theta$ and $\overline{\Phi^{\gamma}_s}(E)$
265is the average differential gamma flux in the energy bin $\Delta E$ :
266
267\begin{eqnarray}
268\overline{\Phi^{\gamma}_s}(E)  &
269                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
270                     \Phi^{\gamma}_s(E)\cdot dE
271\end{eqnarray}
272
273By inverting equation (\ref{eq:form1}) and setting
274$\Delta E\;=\;(E^{up}-E^{low})\;\;\;\;\overline{\Phi^{\gamma}_s}(E)$ can
275be written as
276
277\begin{eqnarray}
278  \overline{\Phi^{\gamma}_s}(E)    &=
279  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)}
280{\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta) \cdot
281 (E^{up}-E^{low}) }
282 \label{eq:form2}
283\end{eqnarray}
284
285By means of equation (\ref{eq:form2}) $\overline{\Phi^{\gamma}_s}(E)$
286can be determined
287from the measured $\Delta N^{\gamma,obs}_s(E,\Theta)$ and
288$\Delta T_{on}(\Theta)$, using the collection area
289$F^{\gamma}_{eff}(E,\Theta)$, which is obtained from MC data.
290
291Equation (\ref{eq:form2}) is for a limited and fixed region of
292the zenith angle. One may calculate $\overline{\Phi^{\gamma}_s}(E)$ from the
293data taken at all $\Theta$, in which case
294
295\begin{eqnarray}
296  \overline{\Phi^{\gamma}_s}(E)    &=
297  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)}
298         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
299          \cdot (E^{up}_i-E^{low}_i) }
300 \label{eq:form3}
301\end{eqnarray}
302
303If a fixed spectral index $\alpha$ is assumed for the differential
304source spectrum
305
306\begin{eqnarray}
307 \Phi^{\gamma}_s(E)  &&\Phi^{\gamma}_0 \cdot
308                       \left(\dfrac{E}{GeV}\right)^{-\alpha}
309\end{eqnarray}
310
311one gets
312
313\begin{eqnarray}
314 \int_{\Delta E}{} \Phi^{\gamma}_s(E) \cdot dE  &
315 &\dfrac{\Phi^{\gamma}_0}{1-\alpha}
316  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
317         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]\cdot GeV
318 \label{eq:form4}
319\end{eqnarray}
320
321Inserting (\ref{eq:form4}) into (\ref{eq:form0}) yields
322
323\begin{eqnarray}
324  \Phi^{\gamma}_0    &=
325  &\dfrac{\Delta N^{\gamma,obs}_s(E,\Theta)}
326         {\Delta T_{on}(\Theta) \cdot F^{\gamma}_{eff}(E,\Theta)
327          \cdot
328  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
329         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]}
330  \cdot \dfrac{1-\alpha}{GeV}
331  \label{eq:form5}
332\end{eqnarray}
333
334which by summing over all $\Theta$ bins gives
335
336\begin{eqnarray}
337  \Phi^{\gamma}_0    &=
338  &\dfrac{\sum_i\Delta N^{\gamma,obs}_s(E,\Theta_i)}
339         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
340          \cdot
341  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
342         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]}
343  \cdot \dfrac{1-\alpha}{GeV}
344  \label{eq:form6}
345\end{eqnarray}
346
347If applied to MC data, for which $\overline{\Phi^{\gamma}_s}(E)$ is known,
348equation (\ref{eq:form1}) can also be used to
349determine the collection area $F^{\gamma}_{eff}(E,\Theta)$ :
350
351\begin{eqnarray}
352F^{\gamma}_{eff}(E,\Theta&
353 &\dfrac{\Delta N^{\gamma,MC,reconstructed}_s(E,\Theta)}
354        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{\gamma}_s}(E) \cdot
355         (E^{up}-E^{low})}
356\end{eqnarray}
357
358This procedure of determining $F^{\gamma}_{eff}(E,\Theta)$ amounts to
359performing the integration in equation (\ref{eq:form-1}) by MC. An
360important precondition is that in the MC simulation all gamma showers for
361which $R^{\gamma}(E,\Theta,F)$ is greater than zero were
362simulated. This means in particular that the MC simulation of gammas
363extends to sufficiently large impact parameters. In reality, in order to save
364computer time showers will be generated up to a maximum
365value of the impact parameter (possibly depending on the zenith
366angle). An appropriate correction for that has to be applied later in
367the analysis.
368
369Knowing $F^{\gamma}_{eff}(E,\Theta)$, the gamma fluxes can be obtained
370from the experimental data using equation (\ref{eq:form2}),
371(\ref{eq:form3}), (\ref{eq:form5}) or (\ref{eq:form6}).
372
373Of course, the MC data sample used for calculating
374$F^{\gamma}_{eff}(E,\Theta)$ and the experimental data sample used for
375determining the gamma flux by means of $F^{\gamma}_{eff}(E,\Theta)$
376have to be defined identically in many respects : in particular
377the set of cuts
378and the offset between source position and telescope orientation have
379to be the same in the MC data and the experimental data sample.
380
381
382
383\subsubsection{Differential flux and collection area for
385
386In the case of hadronic cosmic rays, which arrive from all directions
387$\Omega$, the differential hadron flux is given by
388
389\begin{eqnarray}
390\Phi^{h}(E)\;=\;\dfrac{dN^{h}}{dE \cdot dF \cdot dt \cdot d\Omega}
391\label{eq:form-12}
392\end{eqnarray}
393
394
395In contrast to (\ref{eq:form-1}) the effective collection area for hadrons
396is defined as
397
398\begin{eqnarray}
399F^{h}_{eff}(E,\Theta)\;  &&\int R^{h}(E,\Theta,F,\Omega)\cdot dF
400 \cdot d\Omega
401\label{eq:form-11}
402\end{eqnarray}
403
404Note that for a fixed orientation of the telescope $(\Theta,\phi)$ the
405hadrons are coming from all directions $\Omega$. The reconstruction
406efficiency $R^h(E,\Theta,F,\Omega)$ of hadrons therefore depends also
407on $\Omega$.
408
409With the definitions (\ref{eq:form-12}) and (\ref{eq:form-11})
410very similar formulas are obtained for hadrons as
411were derived for photons in the previous section. For clarity they
412are written down explicitely :
413
414\begin{eqnarray}
415\Delta N^{h,obs}(E,\Theta&= &\int R^{h}(E,\Theta,F) \cdot
416 \Phi^{h}(E) \cdot dE \cdot dF \cdot dt \\
417                                   &= &\Delta T_{on}(\Theta) \cdot
418 \int_{\Delta E}{} \Phi^{h}(E)\cdot
419 F^{h}_{eff}(E,\Theta)\cdot dE \\
420\end{eqnarray}
421
422\begin{eqnarray}
423\Delta N^{h,obs}(E,\Theta)
424                         &\approx   &\Delta T_{on}(\Theta) \cdot
425  F^{h}_{eff}(E,\Theta) \cdot \int_{\Delta E}{}
426 \Phi^{h}(E)\cdot dE               \label{eq:form10}\\
427                         &\approx   &\Delta T_{on}(\Theta) \cdot
428  F^{h}_{eff}(E,\Theta) \cdot \Delta E \cdot
429  \overline{\Phi^{h}}(E)       \label{eq:form11}
430\end{eqnarray}
431
432
433\begin{eqnarray}
434\overline{\Phi^{h}}(E)  &
435                     &\dfrac{1}{\Delta E}\;\int_{\Delta E}{}
436                     \Phi^{h}(E)\cdot dE
437\end{eqnarray}
438
439
440\begin{eqnarray}
441  \overline{\Phi^{h}}(E)    &=
442  &\dfrac{\Delta N^{h,obs}(E,\Theta)}
443{\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta) \cdot
444 (E^{up}-E^{low}) }
445 \label{eq:form12}
446\end{eqnarray}
447
448
449
450\begin{eqnarray}
451  \overline{\Phi^{h}}(E)    &=
452  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)}
453         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
454          \cdot (E^{up}_i-E^{low}_i) }
455 \label{eq:form13}
456\end{eqnarray}
457
458
459\begin{eqnarray}
460 \Phi^{h}(E)  &&\Phi^{h}_0 \cdot
461                       \left(\dfrac{E}{GeV}\right)^{-\beta}
462\end{eqnarray}
463
464
465\begin{eqnarray}
466 \int_{\Delta E}{} \Phi^{h}(E) \cdot dE  &
467 &\dfrac{\Phi^{h}_0}{1-\beta}
468  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
469         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]\cdot GeV
470 \label{eq:form14}
471\end{eqnarray}
472
473
474\begin{eqnarray}
475  \Phi^{h}_0    &=
476  &\dfrac{\Delta N^{h,obs}(E,\Theta)}
477         {\Delta T_{on}(\Theta) \cdot F^{h}_{eff}(E,\Theta)
478          \cdot
479  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
480         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]}
481  \cdot \dfrac{1-\beta}{GeV}
482  \label{eq:form15}
483\end{eqnarray}
484
485
486\begin{eqnarray}
487  \Phi^{h}_0    &=
488  &\dfrac{\sum_i\Delta N^{h,obs}(E,\Theta_i)}
489         {\sum_i\Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
490          \cdot
491  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
492         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]}
493  \cdot \dfrac{1-\beta}{GeV}
494  \label{eq:form16}
495\end{eqnarray}
496
497
498Note that $\Phi^{h}(E)$, $\Phi^h_0$ and $F^{h}_{eff}(E,\Theta)$ differ
499from      $\Phi^{\gamma}(E)$, $\Phi^{\gamma}_0$ and
500$F^{\gamma}_{eff}(E,\Theta)$ by the dimension of the
501solid angle, due to the additional factor $d\Omega$ in
502(\ref{eq:form-12}) and (\ref{eq:form-11}).
503
504Like in the case of gammas from point sources, the effective area
505$F^h_{eff}(E,\Theta)$ for
506hadrons can be calculated by applying equation (\ref{eq:form11}) to MC
507data, for which $\overline{\Phi^h}(E)$ is known :
508
509\begin{eqnarray}
510F^{h}_{eff}(E,\Theta&
511 &\dfrac{\Delta N^{h,MC,reconstructed}(E,\Theta)}
512        {\Delta T_{on}(\Theta) \cdot \overline{\Phi^{h}}(E) \cdot
513         (E^{up}-E^{low})}
514\end{eqnarray}
515
516Similar to the case of gammas from point sources,
517this procedure of determining $F^h_{eff}(E,\Theta)$ amounts to
518performing the integrations in equation (\ref{eq:form-11}) by MC. The
519precondition in the case of hadrons is that in the
520MC simulation all hadron showers for
521which $R^{h}(E,\Theta,F,\Omega)$ is greater than zero were
522simulated. So the simulation should not only include large enough
523impact parameters but also a sufficiently large range of $\Omega$ at
524fixed orientation $(\Theta,\phi)$ of the telescope.
525
526Knowing $F^{h}_{eff}(E,\Theta)$, the hadron fluxes can be obtained
527from the experimental data using equation (\ref{eq:form12}),
528(\ref{eq:form13}), (\ref{eq:form15}) or (\ref{eq:form16}).
529
530
531\subsubsection{Measurement of the absolute differential flux of gammas
532from a point source by normalizing to the flux of hadronic cosmic rays}
533
534In section 3.2.1 a procedure was described for measuring the absolute
535differential flux of gammas from a point source. The result for
536$\overline{\Phi^{\gamma}_s}(E)$ depends on a reliable determination of
537the collection area $F^{\gamma}_{eff}(E,\Theta)$ by MC and the
538measurement of the on-time $\Delta T_{on}(\Theta)$.
539
540The dependence on the MC simulation may be reduced by normalizing to
541the known differential flux of hadronic cosmic rays. Combining
542equations (\ref{eq:form2}) and (\ref{eq:form12}), and assuming that
543$\Delta T_{on}(\Theta)$ is the same for the gamma and the hadron
544sample, yields
545
546\begin{eqnarray}
547\dfrac{\overline{\Phi^{\gamma}_s}(E)}
548      {\overline{\Phi^{h}}(E)}          &&
549\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
550      {\Delta N^{h,obs}(E,\Theta)}      \cdot
551\dfrac{F^{h}_{eff}(E,\Theta)}
552      {F^{\gamma}_{eff}(E,\Theta)}
553\label{eq:form20}
554\end{eqnarray}
555
556If $\overline{\Phi^{h}}(E)$ is assumed to be known from other
557experiments, equation (\ref{eq:form20}) allows to determine
558$\overline{\Phi^{\gamma}_s}(E)$ from
559the experimental number of gamma and hadron showers using the
560collection areas for gammas and hadrons from the MC. Since only the
561ratio of the collection areas enters the dependence on the
562MC simulation is reduced.
563
564If data from all zenith angles are to be used the corresponding
565expression for $\overline{\Phi^{\gamma}_s}(E)$ is (see equations
566(\ref{eq:form3}) and (\ref{eq:form13}))
567
568\begin{eqnarray}
569\dfrac{\overline{\Phi^{\gamma}_s}(E)}
570      {\overline{\Phi^{h}}(E)}          &&
571\dfrac{\sum_i \Delta N^{\gamma,obs}(E,\Theta_i)}
572      {\sum_i \Delta N^{h,obs}(E,\Theta_i)}      \cdot
573\dfrac{\sum_i \Delta T_{on}(\Theta_i) \cdot F^{h}_{eff}(E,\Theta_i)
574                                      \cdot (E^{up}_i-E^{low}_i)}
575      {\sum_i \Delta T_{on}(\Theta_i) \cdot F^{\gamma}_{eff}(E,\Theta_i)
576                                      \cdot (E^{up}_i-E^{low}_i)}
577\label{eq:form21}
578\end{eqnarray}
579
580Clearly, the set of cuts defining the gamma sample is different from
581the set of cuts defining the hadron sample. However,
582$\Delta N^{\gamma,obs}$ and $\Delta N^{h,obs}$ can still be measured
583simultaneously, in which case $\Delta T_{on}(\Theta_i)$ is the same for
585simultaneously has the advantage that variations of the detector
586properties or of the atmospheric conditions during the observation
587partly cancel in (\ref{eq:form20}) and (\ref{eq:form21}).
588
589If fixed spectral indices $\alpha$ and $\beta$ are assumed for the
590differential
591gamma and the hadron fluxes respectively one obtains for the ratio
592$\Phi^{\gamma}_0\;/\;\Phi^h_0$
593(see (\ref{eq:form5}) and (\ref{eq:form15}))
594
595\begin{eqnarray}
596\dfrac{\Phi^{\gamma}_0}
597      {\Phi^{h}_0}          &&
598\dfrac{\Delta N^{\gamma,obs}(E,\Theta)}
599      {\Delta N^{h,obs}(E,\Theta)}      \cdot
600\dfrac{F^{h}_{eff}(E,\Theta) \cdot
601  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\alpha} -
602         \left(\dfrac{E^{low}}{GeV}\right)^{1-\alpha} \right]}
603      {F^{\gamma}_{eff}(E,\Theta)
604  \left[ \left(\dfrac{E^{up}} {GeV}\right)^{1-\beta} -
605         \left(\dfrac{E^{low}}{GeV}\right)^{1-\beta} \right]} \cdot
606\dfrac{1-\beta}{1-\alpha}
607\label{eq:form22}
608\end{eqnarray}
609
610or, when using the data from all zenith angles,
611(see (\ref{eq:form6}) and (\ref{eq:form16}))
612
613\begin{eqnarray}
614\dfrac{\Phi^{\gamma}_0}
615      {\Phi^{h}_0}          &&
616\dfrac{\sum_i\Delta N^{\gamma,obs}(E,\Theta_i)}
617      {\sum_i\Delta N^{h,obs}(E,\Theta_i)}      \cdot
618\dfrac{\sum_i F^{h}_{eff}(E,\Theta_i) \cdot
619  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\alpha} -
620         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\alpha} \right]}
621      {\sum_i F^{\gamma}_{eff}(E,\Theta_i)
622  \left[ \left(\dfrac{E^{up}_i} {GeV}\right)^{1-\beta} -
623         \left(\dfrac{E^{low}_i}{GeV}\right)^{1-\beta} \right]} \cdot
624\dfrac{1-\beta}{1-\alpha}
625\label{eq:form23}
626\end{eqnarray}
627
628
629
630% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
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632% &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
633
634
635\section{MC work}
636
637\subsection{Overview of the MC and analysis chain}
638
639After a few iterations to improve the programs in speed,
640reliability, ... there is a sample of available programs
641to simulate the behaviour of the MAGIC telescope.
642Due to the big amount of diskspace needed for this simulation
643it was decided, that not only one program will generate
644the MAGIC telescope, but a subsequent chain of different
645programs. In figure \ref{MC_progs} you can see a overview of
646the existing programs and their connections.
647\begin{figure}[h]
648\setlength{\unitlength}{1.cm}
649  \begin{picture}(18.,12.)
650        \put (0., 0.){\framebox(18.,12.){}}
651
652        \put (1, 11.5){{\sl Air shower programs}}
653        \put (1., 10.){\framebox(3.,1.){MMCS}}
654        \put (2., 10.){\vector(0,-1){.9} }
655        \put (1., 8.){\framebox(3.,1.){reflector}}
656        \put (2., 8.){\vector(0,-1){.9}}
657
658        \put (6, 10.){{\sl star background programs}}
659        \put (6.,8.){\framebox(3.,1.){starresponse}}
660        \put (6., 8.){\line(0, -1){1.5}}
662        \put (10., 8.){\line(0, -1){1.5}}
663        \put (10., 6.5){\vector(-1,0){6.} }
664
665        \put (1., 6.){\framebox(3.,1.){camera}}
666        \put (2., 6.){\vector(3,-1){5.} }
667
668
669
670        \put (14, 11.5){{\sl real data programs}}
671        \put (14, 8.){\framebox(3,1){MAGIC DAQ}}
672        \put (15, 8.){\vector(0,-1){.9} }
673        \put (14, 6.){\framebox(3.,1.){MERPP}}
674        \put (15., 6.){\vector(-3,-1){5.} }
675
676        \put (8.75, 3.7){\oval(4.,1.)}
677        \put (7., 3.5){MAGIC root file}
678        \put (8., 3.2){\vector(0, -1){1.0}}
679
680        \put (7, 1.){\framebox(3.,1.){MARS}}
681
682        \thicklines
683        \put (5., 11.){\line(0, -1){6.5}}
684        \put (13., 12.){\line(0, -1){7.5}}
685
686  \end{picture}
687\caption{Overview of the existing programs in the MC of
688MAGIC.}
689\label{MC_progs}
690\end{figure}
691A detailed description of the properties of the different programs can be found
692in section \ref{sec_exist_progs}.
693From that diagram you can see the following features of the simulation and
694analysis chain of MAGIC.
695\begin{enumerate}
696  \item The simulation of Air showers and the simulation of the night sky
697        background (NSB) is seperated.
698
699  \item The NSB is seperated in two parts, the contribution from the starfield
700        and from a diffuse part.
701
702  \item To speed up the production the starresponse program creates a databases
703        for stars of different magnitude.
704
705  \item The join of air showers and NSB is done in the camera program.
706
707  \item The analysis of MC \underline{and} real data will be done with only one program.
708        This program is called MARS (Magic Analysis and Reconstruction Software).
709        The output of the camera program from Monte Carlo data and the output of
710        the MERPP (MERging and PreProcessing) program for the real data are the same.
711        So there is no need to use different programs for the analysis. The file
712        generated by this program used the root package from CERN for data storage.
713\end{enumerate}
714In this section we will only describe the usage of the Monte Carlo programs. The
715descriptions of the MERPP and MARS can be found somewhere else\footnote{Look on the
717
718\subsection{Existing programs}
719\label{sec_exist_progs}
720\subsubsection{MMCS - Magic Monte Carlo Simulation}
721
722This program is based on a CORSIKA simulation. It is used to generate
723air showers for the MAGIC telecope. At the start one run of the
724program, one has to define the details of the simulation.
725One can specify the following parameters of an shower
727%
728\begin{enumerate}
729  \item the type of the particles in one run ($PartID$)
730  \item the energy range of the particles ($E_1, E_2$)
731  \item the slope of the Energy spectra
732  \item the range of the shower core on the ground $r_{core}$.
733  \item the direction of the shower by setting the range of
734        zenith angle ($\Theta_1, \Theta_2$) and
735        azimuth angle  ($\phi_1, \phi_2$)
736\end{enumerate}
737%
738\begin{figure}[h]
739\setlength{\unitlength}{1.5cm}
740\begin{center}
741  \begin{picture}(9.,6.)
742        \put (0., 0.){\framebox(9.,6.){}}
743
744        \thicklines
745        % telescope
746        \put (5., .5){\oval(.75, .75)[t]}
747        \put (3., 1.){{\sl Telesope position}}
748        \put (4.5, 1.){\vector(1, -1){0.5}}
749        % observation level
750        \put (.5, .5){\line(1, 0){8}}
751        \put (.5, .6){{\sl Observation level}}
752
753        % air shower
754        \put (4. , 5.5 ){\line(2, -3){3.3}}
755        \put (4.5, 5.5 ){{\sl Particle Type ($PartId$)}}
756        \put (4.5, 5.25){{\sl Energy ($E_1 < E < E_2$)}}
757        \put (4.5, 5.  ) {$\Theta_1 < \Theta < \Theta_2$}
758        \put (4.5, 4.75) {$\phi_1 < \phi < \phi_2$}
759        \put (7.5, .75){{\sl shower core}}
760
761        \thinlines
762        \put (5., .25){\line(1,0){2.3}}
763        \put (6.1, .25){{\sl $r_{Core}$}}
764
765        \put (5., .5){\line(4,3){1.571}}
766        \put (6., 1.35){{\sl $p$}}
767
768  \end{picture}
769\end{center}
770  \caption {The parameter of an shower that are possible to define
771at the begin of an MMCS run.}
772\label{pic_shower}
773\end{figure}
774Other parameters, that will be important in the analysis later,
775can be calculated. I.e. the impact parameter $p$ is defined by
776the direction
777of the shower ($\Theta, \phi$) and the core position
778($x_{core}, y_{core}$).
779
780The program MMCS will track the whole shower development
781through the atmosphere. All the cerenkov particles that hit a
782sphere around the telesope (in the figure \ref{pic_shower}
783drawn as the circle around the telecope position) are stored
784on disk. It is important to recognize, that up to now no
785information of the pointing of the telescope was taking into
786account.
787This cerenkov photons are the input for the next program,
788called reflector.
789
790
791\subsubsection{reflector}
792
793The aim of the reflector program is the
794tracking of the cerenkov photons to the camera
795of the MAGIC telescope. So this
796is the point where we introduce a specific pointing of
797the telescope ($\Theta_{MAGIC}, \phi_{MAGIC}$).
798For all cerenkov photons the program
799tests if the mirrors are hitten, calculates the
800probability for the reflection and tracks them to the
801mirror plane. All the photons that are hitting the
802camera are written to disk (*.rfl)
803with their important parameters
804($x_{camera}, y_{camera}, \lambda, t_{arrival}$).
805These parameters are the input from the shower simulation
806for the next program in the
807MC simulation chain, the camera program.
808
809\subsubsection{camera}
810
811The camera program simulates the behaviour of the
812PMTs and the electronic of the trigger and FAC system.
813For each photon out of the reflector file (*.rfl) the
814camera program calculates the probability to generate
815an photo electron out of the photo cathode. If a photo
816electrons was ejected, this will create a signal in the
817trigger and FADC system of the hitted pixel.
818You have to specify the
819parameter of the signal shaping
820(shape, Amplitude, FWHM of signal)
821at the beginning of the
822camera, seperatly for the trigger and the FADC system.
823All signal from all photoelectrons are superimposed for
824each pixel. As an example you can see the output of
826\begin{figure}[h]
827
828 \caption{The response of one shower from the trigger (left) and
831\end{figure}
832
833All these analog signals going into the trigger system are used
834to check if for a given event a trigger signal was generated or
835not. But before the start of the camera program on also has to
836set a few parameters of the trigger system like:
837\begin{itemize}
838  \item diskriminator threshold
839  \item mulitplicity
840  \item topology
841\end{itemize}
842With this set of parameter the camera program will analyse
843if one event has triggered. For the triggered event all the FADC
844content will be writen on the file (*.root). In addition all the
845information about the event ($PartID, E, \Theta$,...) and
846information of trigger (FirstLevel, SecondLevel, ..) are also
847be written to the file.
848
849One of the nice features of the camera program is the possiblity
850so simulate the NSB, the diffuse and the star light part of it.
851But before doing this, on has to start other programs
852(called starresponse and starfieldadder) that are describe
853below.
854
855\subsubsection{starresponse}
856
857This program will simulate the analog response for stars of
858a given brightness $B$.
859
860
862
863
864
865
866
867
868
869\subsection{What to do}
870
871\begin{itemize}
872  \item pedestal fluctuations
873  \item trigger
874  \item rates (1st level, 2nd level, .... )
875  \item discriminator thresholds
876  \item Xmax
877  \item collection area
878  \item $\gamma$/h-Seperation
879  \item magnetic field studies ($\phi$-dependence)
880  \item rotating star field
881\end{itemize}
882
883
884
885\subsection{A suggestion for an initial workplan}
886We propose in the following a list of tasks whose common goal
887it is to provide and use data files with a definition of data suitable for
888initial studies, e.g. trigger rates, and for subsequent further
889analysis in MARS, e.g. $\gamma$/h-separation. We consider this list to be
890minimal and a first step only.
891Given the amount of work that will have to be invested, the detailed
892assumptions below should be backed up by collaboration-wide agreement; also, some
893input from groups is essential, so PLEASE REACT.
894
895Event generation should be done with the following conditions:
896\begin{itemize}
897  \item Signal definition: we will use the Crab, over a range of zenith angles
898  (define!!). A minimum of 20,000 (can we get that?) triggers will be
899  generated, starting from existing MMCS files;
900  \item Observation mode: observations are assumed off-axis,
901  with an offset of $\pm 0.4 \deg$ in $\Delta \beta$ along the direction of the
902  local azimuthal angle $\phi$,
903  switching sign every 500 events (see 'Assumptions' above);
905  Crab. Ignore star field rotation problems for the moment, until a separate study
906  is available (??);
907  \item Pedestal fluctuations: all pixel values are smeared by a Gaussian
908  centered at zero with a sigma of 1.5 photoelectrons;
909  \item Trigger:  Padova to define (!!) the grouping of pixels, the
910  trigger thresholds, and a method to avoid triggering on stars. We assume
911  only a first-level trigger.
912\end{itemize}
913With this event sample available, we suggest to embark on several studies,
914which will help us in understanding better the MAGIC performance, and will
915also pave our way into future analysis.
916\begin{itemize}
917  \item determine trigger rates (1st level only), as function of energy and
918  zenith angle (also of impact parameter?);
919  \item determine gamma acceptance,
920  as function of energy and zenith angle (also of impact parameter?);
921  \item determine effective collection area (gammas and hadrons),
922  as function of energy and zenith angle (also of impact parameter?);
923  \item show the position of the shower maximum (Xmax);
924  \item start comparing methods for $\gamma$/h-separation, i.e. the generation
925  of ON and OFF samples from the observations;
926  \item start magnetic field studies ($\phi$-dependence);
927  \item eventually, study the effect of the rotating star field.
928\end{itemize}
929
930
931
932\section{Analysis of the real data}
933
934\begin{thebibliography}{xxxxxxxxxxxxxxx}
935\bibitem{fegan96}D.J.Fegan, Space Sci.Rev. 75 (1996)137
936\bibitem{hillas85}A.M.Hillas, Proc. 19th ICRC, La Jolla 3 (1985) 445
937\bibitem{konopelko99}A.Konopelko et al., Astropart. Phys. 10 (1999)
938275
939\bibitem{reynolds93}P.T.Reynolds et al., ApJ 404 (1993) 206
940\end{thebibliography}
941
942
943\end{document}
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