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\begin{document}
 
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%Novel Technology: 
\title{Proposal: Observations of the Galactic Center \\
Key Programs: Galactic Center / Dark Matter
}
\author{H. Bartko, A. Biland, S. Commichau, P. Flix, S. Stark, W. Wittek}
\date{March 21, 2005\\}
\TDAScode{}%MAGIC 05-xx\\ 04mmdd/HBartko
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\begin{abstract}  
Due to the wealth of sources, the region around
the Galactic Center (GC) is very interesting. Recently, gamma radiation above 
a few hundred GeV has been detected by the Whipple, Cangaroo and HESS 
collaborations. The reconstructed spectra from Cangaroo and HESS show 
significant differences. The acceleration mechanisms have still to be 
identified.

Various possibilities for the acceleration of the very high energy (VHE)
gamma rays are discussed in the literature, like accretion flow onto the 
central black 
hole, supernova shocks in Sgr A East, proton acceleration near the event 
horizon of the black hole, or WIMP dark matter annihilation. Although the 
observed VHE gamma radiation from the GC is most probably not due to 
the annihilation of SUSY-neutralino dark matter (DM) particles, other models 
like Kaluza-Klein dark matter are not ruled out. Moreover, assuming a 
universal DM density profile, the GC is expected to yield the largest DM flux 
amongst the favored candidates, due to its proximity.

At La Palma, the GC culminates at about 58 deg zenith angle (ZA). It can be 
observed with MAGIC at up to 60 deg ZA for about 150 hours per year, between 
April and late August. The expected integral flux above 700 GeV derived from 
the HESS data is $(3.2 \pm 1.0)\cdot 10^{-12}\mathrm{cm}^{-2}\mathrm{s}^{-1}$. 
Comparing this to the expected MAGIC sensitivity from MC simulations, this 
could result in a 5 $\sigma$ detection in about $1.8\pm0.5$ hours. 

The observations have to be conducted as early as possible in order to 
participate in the ongoing discussion about gamma radiation from the GC. 
The main motivations for the observation of the GC are :

\begin{itemize}
\item to measure the gamma flux and its energy dependence (due to the high
zenith angles higher energies are accessible),
\item to inter-calibrate MAGIC and HESS,
\item to help resolving the flux discrepancies between HESS and 
Cangaroo,
\item to gain information about the nature and acceleration mechanism of the 
source, 
\item to set constraints on models for dark-matter-particle annihilation.
\end{itemize}

In order to collect a data sample comparable in size to those of the other 
experiments and to be able to measure the energy spectrum, 40 hours of
observation time are requested. The 40 hours will be split into 20 hours ON
and 20 hours dedicated OFF data or they will be devoted to observations in
the wobble mode. In addition, 60 hours of observation during moonshine are 
applied for.
\end{abstract}

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\section{Introduction}


The Galactic Center (GC) region contains many unusual objects which may be 
responsible for the high energy processes generating gamma rays 
\cite{Aharonian2005,Atoyan2004,Horns2004}. The GC is rich in massive stellar 
clusters with up to 100 OB stars \cite{GC_environment}, immersed in a dense 
gas within a radius of 300 pc and the mass of $2.7 \cdot 10^7 M_{\odot}$, 
young supernova remnants e.g. G0.570-0.018 or Sgr A East, and nonthermal radio arcs. The dynamical center of the Milky Way is associated with the compact radio source Sgr A$^*$, which is believed to be a massive black hole \cite{GC_black_hole,Melia2001}. An overview of the sources in the GC region is given in Figure \ref{fig:GC_sources}. Some data about the GC are summarized in Table \ref{table:GC_properties}.


\begin{table}[h]{\normalsize\center
\begin{tabular}{lc}
 \hline
 (RA, dec), epoch J2000.0 & $(17^h45^m12^s,-29.01$ deg)
\\ heliocentric distance  & $8\pm0.4$ kpc \cite{Eisenhauer2003} 
(1 deg = 140 pc)
\\ mass of the black hole & $2\pm0.5 \cdot 10^6 M_{\odot}$ 
\\ 
\hline
\end{tabular}
\caption{Properties of the Galactic Center.}\label{table:GC_properties}}
\end{table}



\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=9cm]{GC_sources_1.eps}
\end{center}
\caption[Sources near the Galactic Center.]{Overview about the sources near the Galactic Center \cite{GC_overview}.} \label{fig:GC_sources}
\end{figure}


In fact, EGRET has detected a strong source in direction of the GC, 
3 EG J1745-2852 \cite{GC_egret}, which has a broken power law spectrum 
extending up to at least 10 GeV, with a spectral index of 1.3 below the break at a few 
GeV. Assuming a distance of 8.5 kpc, the gamma ray luminosity of this source 
is very large $~2.2 \cdot 10^{37} \mathrm{erg}/\mathrm{s}$, which is 
equivalent to about 10 Crab pulsars. An independent analysis of the EGRET data 
\cite{Hooper2002} indicates a point source whose position is different from the GC at a confidence level beyond 99.9 \%. %\cite{Hooper2002, A&A 335 (1998) 161} 

Up to now, the GC has been observed at 
energies above 200 GeV by Veritas, Cangaroo and HESS, \cite{GC_whipple,
GC_cangaroo,GC_hess}. Figure \ref{fig:GC_gamma_flux} shows the reconstructed 
spectra by the other IACTs while Figure \ref{fig:GC_source_location} shows the 
different reconstructed positions of the GC source. Recently a second TeV 
gamma source only about 1 degree away from the GC has been 
discovered \cite{SNR_G09+01}. Its integral flux above 200 GeV represents about 
2\% of the gamma flux from the Crab nebula with a spectral index of about 2.4.

\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=6cm]{sgr_figure4.eps}
\end{center}
\caption[Gamma flux from GC.]{The VHE gamma flux as observed by the other IACTs and by the EGRET satellite \cite{GC_hess}.} \label{fig:GC_gamma_flux}
\end{figure}


\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=8cm]{gc_legend.eps}
\end{center}
\caption[Gamma flux from GC.]{The source locations as measured by the other IACTs \cite{Horns2004}.} \label{fig:GC_source_location}
\end{figure}

The discrepancies between the measured flux spectra could indicate inter-calibration problems between the IACTs, a source variability of the order of one year could be due to the different regions in which the signal is integrated.



\section{Investigators and Affiliations}

The investigators of the proposed observations of the GC are stated in Table \ref{table:GC_investigators} together with their assigned analysis tasks. All other interested members of the MAGIC collaboration are invited to join these efforts.


\begin{table}[h]{
\scriptsize{
\centering{
\begin{tabular}{llll}
 \hline
 Investigator & Institution& E-mail & Assigned task\\ \hline
   Hendrik Bartko      & MPI Munich    & hbartko@mppmu.mpg.de & data analysis, spectra, wobble mode
\\ Adrian Biland       & ETH Zurich    & biland@particle.phys.ethz.ch & OFF pointing, Moon observations
\\ Sebastian Commichau & ETH Zurich    & commichau@particle.phys.ethz.ch &
 data analysis, MC generation, spectra
\\ Pepe Flix           & IFAE Barcelona& jflix@ifae.es & data analysis, disp
\\ Sabrina Stark       & ETH Zurich    & lstark@particle.phys.ethz.ch  & data analysis, spectra
\\ Wolfgang Wittek     & MPI Munich    & wittek@mppmu.mpg.de & padding
\\ 
\hline
\end{tabular}
}
\caption{The investigators and the assigned tasks.}\label{table:GC_investigators}}}
\end{table}


\section{Scientific Case} 


In the GC region high energy gamma rays can be produced in different sources:

\begin{itemize}
\item{entire innermost 10 pc region (interaction between cosmic rays and the dense ambient gas)}
\item{non-thermal radio filaments  \cite{Pohl1997}}
\item{young SNR Sgr A East \cite{Fatuzzo2003}}
\item{in the compact radio source Sgr A*}
\item{central part of the dark matter halo.}
\end{itemize}

It is quite possible that some of these potential gamma-ray production sites contribute comparably to the the observed TeV flux.


% in the non-thermal radio filaments by high-energy leptons which scatter background infrared photons from the nearby ionized clouds \cite{Pohl1997,Aharonian2005}, or by hadrons colliding with dense matter. These high energy hadrons can be accelerated by the massive black hole \cite{GC_black_hole}, associated with the Sgr A$^*$, by supernovae or by energetic pulsars. Alternative mechanisms invoke the hypothetical annihilation of super-symmetric dark matter particles (for a review see \cite{jung96}) or curvature radiation of protons in the vicinity of the central super-massive black hole \cite{GC_black_hole,Melia2001}.


In order to shed new light on the high energy phenomena in the GC region, and to constrain the emission mechanisms and sources, new observations with high sensitivity, good energy and angular resolution are necessary. For the interpretation of the observed gamma flux the following observables are important:

\begin{itemize}
\item{source location, source extension}
\item{time variability of the gamma flux}
\item{energy spectrum.}
\end{itemize} 



\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=8cm]{total_spectrum.eps}
\end{center}
\caption[Total spectrum of the GC.]{Total spectrum of the gamma radiation from the Galactic Center, compiled by \cite{Aharonian2005}.} \label{fig:GC_source_location}
\end{figure}



\subsection{Emission from Sgr A$^*$}

Production of high-energy gamma rays within 10 Schwarzschild radii of a black hole (of any mass) could be copious because of effective acceleration of particles by the rotation-induced electric fields close to the event horizon or by strong shocks in the inner parts of the accretion disk. However, these energetic gamma rays generally cannot escape the source because of severe absorption due to interactions with the dense, low-frequency radiation through photon-photon pair production. But fortunately the supermassive black hole in our Galaxy is an exception because of its unusually low bolometric luminosity. The propagation effects related to the possible cascading in the photon filed may extend the high-energy limit to 10 TeV or even beyond \cite{Aharonian2005}.



\subsubsection{Leptonic Models}

Also advection dominated accretion flow (ADAF) models can describe the production of high energy gamma radiation in the Galactic Center \cite{Atoyan2004}.

A viable site of acceleration of highly energetic electrons could be the compact region within a few gravitational radii of the black hole. In this case the electrons produce not only curvature radiation, which peaks around 1 GeV, but also inverse Compton gamma rays (produced in the Klein-Nishina regime) with the peak emission around 100 TeV. As these high energy gammas cannot escape the source the observed gamma rays would be due to an electromagnetic cascade.

\subsubsection{Hadronic Models}

One scenario is related to protons accelerated to about $10^{18}$ eV \cite{Aharonian2005}. These protons produce gamma rays via photo-meson processes. This scenario also predicts detectable fluxes of  $10^{18}$ eV neutrons and perhaps gamma rays and neutrinos. A hint of an excess of highest energy neutrons from the GC has been reported in \cite{Hayashida1999}.

TeV gamma rays can also be produced by significantly  lower energy protons, accelerated by the electric filed close to the gravitational radius or by strong shocks in the accretion disk. In this case the gamma-ray production is dominated by interactions of $10^{13}$ eV protons with the accretion plasma. This scenario predicts a neutrino flux which should be observable with northern neutrino telescopes like NEMO. It also predicts strong TeV--X-ray--IR correlations.


\subsection{Dark Matter Annihilation}


The presence of a Dark Matter halo of the Galaxy is well established by stellar dynamics \cite{Klypin2002}. At present, the nature of Dark Matter is unknown, but a number of viable candidates have been advocated within different theoretical frameworks, mainly motivated by particle physics (for a review see \cite{jung96}) including the widely studied models of supersymmetric (SUSY) Dark Matter \cite{Ellis1984}. Also models involving extra dimensions are discussed like Kaluza-Klein Dark Matter \cite{Kaluza_Klein,Bergstrom2004}.

The supersymmetric particle dark matter candidates might self-annihilate into boson or fermion pairs yielding very high energy gammas in subsequent decays and from hadronisation. The gamma flux above an energy threshold $E_{\mathrm{thresh}}$ per solid angle $\Omega$ is given by:

\begin{equation*}
\frac{\text{d} N_{\gamma}(E_{\gamma}>E_{\mathrm{thresh}})}{\text{d}t\  \text{d}A\  \text{d}\Omega }= N_{\gamma}(E_{\gamma}>E_{\mathrm{thresh}}) \cdot \frac{1}{2} \cdot \frac{\langle \sigma v \rangle}{4 \pi m_{\chi}^2} \cdot  \int_{\text{los}}\rho_{\chi}^2(\vec{r}(s,\Omega)) \text{d}s \ ,
\end{equation*}


where $\langle \sigma v \rangle$ is the thermally averaged annihilation cross section, $m_{\chi}$ the mass and $\rho_{\chi}$ the spatial density distribution of the hypothetical dark matter particles. $N_{\gamma}(E_{\gamma}>E_{\mathrm{thresh}})$ is the gamma yield above the threshold energy per annihilation. The predicted flux depends on the SUSY parameters and on the spatial distribution of the dark matter. The energy spectrum of the produced gamma radiation has a very characteristic feature : a sharp cut-off at the mass of the dark matter particle. Also the flux should be absolutely stable in time.

Numerical simulations of cold dark matter \cite{NFW1997,Stoehr2002,Hayashi2004,Moore1998} predict universal DM halo profiles with a density enhancement in the center of the dark halos. In the very center the dark matter density can be even more enhanced through an adiabatic compression due to the baryons \cite{Prada2004} present. All dark matter distributions that predict observable fluxes are cusped, yielding an approximately point-like source.

Using fits of these dark matter profiles to the rotation data of the Milky Way predictions for the density profile $\rho_{\chi}$ of the dark matter can be made \cite{Fornego2004,Evans2004}. On the other hand, for a given choice of SUSY parameters $m_{\chi},\;\langle \sigma v \rangle$ and $N_{\gamma}$ are determined. 
%Assuming parameters for the SUSY models determine the neutralino mass, the thermally averaged annihilation cross section and the gamma yield. Combining both models about the dark matter distribution and SUSY 
Combining the SUSY predictions with the predictions for the DM density profile
predictions for the gamma flux from SUSY particle dark matter annihilation are derived.


Figure \ref{fig:exclusion_lmits} shows exclusion limits for MAGIC (solid straight lines) for the four most promising sources,
in the plane $N_{\gamma}(E_{\gamma}>E_{\mathrm{thresh}})\langle \sigma v \rangle$ vs. $m_{\chi}$. The energy threshold $E_th$ has been assumed to be 100 GeV. Due to its proximity the GC yields the largest expected flux from particle dark matter annihilation and thus the lowest exclusion limit. Nevertheless, this minimum measurable flux is more than one order of magnitude above the highest fluxes predicted by SUSY models (full circles). Also the flux measured by the HESS experiment is far above the theoretical expectation.


\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=6cm]{plot_DM_exclusion.eps}%{Dark_exclusion_limits.eps}
\end{center}
\caption[DM exclusion limits.]{Exclusion limits (solid straight lines) for the four most promising sources of dark matter annihilation radiation. The GC is expected to give the largest flux from all sources. For energies above 700 GeV, the flux from the GC as observed by the HESS experiment (dotted line) is within the reach of MAGIC. The full circles represent flux predictions from some typical SUSY models.  -- Figure to be updated --} \label{fig:exclusion_lmits}
\end{figure}



Detailed discussions of the observed gamma fluxes from the GC can be found in \cite{Hooper2004,Horns2004}. The observed spectrum extends to more than 18 TeV, well beyond the favored mass region of the lightest SUSY particle, and the observed flux is larger than the flux expected in most theoretical models. This leads to the conclusion that most likely the dominating part of the observed gamma flux from the GC is not due to SUSY particle Dark Matter annihilation. Other dark matter scenarios like Kaluza-Klein Dark Matter can not be excluded.


%\newpage

\section{Preparatory Work}

The Sgr A$^*$ data that has been taken in September 8, 9 and 10 2004, is
still being analyzed. Preliminary results were presented at the MAGIC
collaboration meeting in Berlin, 21-25th February 2005.\\
Up to now only 2.9 hours of ON data are available, at zenith angles between 60.3 and 67.8 degrees. Some details of the data set are shown in Table \ref{table:GC_dataset}.\\

\begin{table}[!ht]{
\centering{
\begin{tabular}{l|l|l|l}
 \hline
   Date       & Time          & Az $[^\circ]$ & ZA $[^\circ]$\\ \hline
   09/08/2004 & 21:00 - 22:00 & 198.3 - 214.7 & 60.3 - 67.8
\\ 09/09/2004 & 21:17 - 22:12 & 203.4 - 214.7 & 62.2 - 67.7
\\ 09/10/2004 & 21:06 - 22:03 & 202.2 - 213.7 & 61.6 - 67.1
\\
\hline
\end{tabular}
\caption{The Sgr A$^*$ data set from September 2004.}\label{table:GC_dataset}}}
\end{table}

In our preliminary analysis we used the Random Forest method for the gamma
hadron separation. For this purpose high
ZA (65$^\circ$ ZA and 205$^\circ$ Az) Monte Carlo gammas were generated, 
99500 events in all, with energies between 200
and 30,000 GeV. The differential spectral index of the generated spectrum is $-2.6$, conforming with the energy spectrum of the Crab nebula.

The MC sample was divided into a training
and a test sample. Since no dedicated OFF data were available, we used a
subsample of Sgr A$^*$ ON data to represent the hadronic background in the Random Forest training. As training
parameters we used SIZE, DIST, WIDTH, LENGTH, CONC, and M3Long...


%\begin{figure}[!h]
%\centering
%\subfigure[The Hadronness distribution.]{
%\includegraphics[scale= .3]{hadronness}}
%\subfigure[SIZE $> 300$ Phe]{
%\includegraphics[scale= .3]{size300}}
%\subfigure[SIZE $> 500$ Phe]{
%\includegraphics[scale= .3]{size500}}
%\subfigure[SIZE $> 1000$ Phe]{
%\includegraphics[scale= .3]{size1000}}
%\caption{Hadronness distribution and ALPHA plots for three different lower SIZE cuts. The
%  Hadronness cut is made at 0.4.}\label{fig:prelresults}
%\end{figure}


\section{Feasibility} 

The HESS collaboration has observed the GC for 16.5 hours, at zenith angles around 20 degrees, with energy thresholds between 165 and 255 GeV. The total number of excess events amounts to $\sim$300, and the differential gamma flux is measured as \cite{GC_hess}:

\begin{equation}
\frac{\mathrm{d}N_{\gamma}}{\mathrm{d}A\mathrm{d}t\mathrm{d}E} = (2.50 \pm 0.21 \pm 0.6) \cdot 10^{-12} \frac{1}{\mathrm{cm}^2\mathrm{sTeV}} \left(\frac{E}{\mathrm{TeV}}\right)^{-2.21\pm 0.09 \pm 0.15}
\end{equation}

A fit to the published Cangaroo flux data points \cite{GC_cangaroo} yields:

\begin{equation}
\frac{\mathrm{d}N_{\gamma}}{\mathrm{d}A\mathrm{d}t\mathrm{d}E} = (3.4 \pm 3.8) \cdot 10^{-12} \frac{1}{\mathrm{cm}^2\mathrm{sTeV}} \left(\frac{E}{\mathrm{TeV}}\right)^{-4.4\pm 1.1}
\end{equation}


For a 60 deg ZA we conservatively estimate the analysis energy threshold to be about 700 GeV. The integrated flux of the HESS spectrum is:

\begin{equation}
\frac{\mathrm{d}N_{\gamma}(E>700 \mathrm{GeV})}{\mathrm{d}A\mathrm{d}t}=(3.2 \pm 1.0)\cdot 10^{-12}\frac{1}{\mathrm{cm}^2\mathrm{s}}
\end{equation}


while the integrated flux above 700 GeV obtained from the Cangaroo spectrum is given by:

\begin{equation}
\frac{\mathrm{d}N_{\gamma}(E>700 \mathrm{GeV})}{\mathrm{d}A\mathrm{d}t}=(3 \pm 5)\cdot 10^{-12}\frac{1}{\mathrm{cm}^2\mathrm{s}} \ .
\end{equation}


Thus the integral fluxes above 700 GeV based on the HESS and Cangaroo data agree within errors.

Using MC simulations \cite{MC-Camera} for small zenith angles we conservatively estimate MAGICs sensitivity \cite{MC-Sensitivity} to the integral flux to be:

\begin{equation}
\frac{\mathrm{d}N_{\gamma}(E>700 \mathrm{GeV})}{\mathrm{d}A\mathrm{d}t}\vline_{\mathrm{min}} \approx 6\cdot 10^{-13}\frac{1}{\mathrm{cm}^2\mathrm{s}} \ .
\end{equation}

Assuming this sensitivity and using the integrated flux of $3.2\cdot 10^{-12} \mathrm{cm}^{-2}\mathrm{s}^{-1}$ MAGIC will obtain an excess at the 5
$\sigma$ significance level in $1.8 \pm 0.5$ h observation time. 

The observed Cangaroo and HESS spectra differ
substantially in the spectral index. While the Cangaroo spectrum only extends
to about 2 TeV, the recently published HESS spectrum goes up to about 9 TeV. Figure \ref{fig:MAGIC_flux_limits} shows the HESS and Cangaroo observed fluxes together with the minimum flux detectable by MAGIC in 20 hours observation time.

MAGIC will be able to solve the obvious discrepancy between the observed fluxes. Due to the observation under high zenith angle of about 60 deg MAGIC will be able to extend the source spectrum to higher energies.


\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=8cm]{MAGIC_flux_limits.eps}
\end{center}
\caption[Flux limits.]{Observed gamma spectra of the HESS and Cangaroo experiments compared to the minimum flux detectable by the MAGIC telescope in 20 hours observation time.} \label{fig:MAGIC_flux_limits}
\end{figure}



%?? How long do we have to observe to get a good spectrum above 7 TeV??



\section{Observational Constraints}


The GC culminates at about 58 deg ZA in La Palma. Below 60 deg ZA, it is visible between April and late August for about 150 hours. The GC has a quite large LONS background. This together with the large ZA requires to take either dedicated OFF data or to take data in the wobble mode. Since the LONS level is in any case very large moon observations are considered in addition to the normal observations.


\section{Requested Observation Time}

Based on the above estimates a 5 $\sigma$ excess is expected to be observed in about 2 hours. To acquire a data set which is comparable in size to those of the other experiments at least 40 hours of observation time are requested. These 40 hours may be either split into 20 hours ON and 20 hours OFF data taking or be devoted exclusively to data taking in the wobble mode. It is still being investigated which of the observation modes is to be preferred.

To get the lowest possible threshold all data shall be taken at the
smallest possible zenith angles between culmination at about 58 deg and 60
deg. This limits the data taking interval to about 1 hour per night between
April and August. In order to have the most appropriate OFF data we propose to
take OFF data each night directly before or after the ON observations under
the same condition, i.e. ZA and azimuth. At such high zenith angles the effect
of the earth's magnetic field can be non-negligible. This depends of course on
ZA and azimuth under which the data is taken.

To extend the available observation time we propose to take moon ON and OFF data in addition. Also in this case, the proposed maximum ZA of 60 deg should not be exceeded during moon observations.

In order to take part in exploring the exciting physics of the GC
we propose to start taking data as soon as possible, beginning in April. In this way first results may be available at the time of the summer conferences 2005.


\section{Outlook and Conclusions}

The GC is an interesting target in all wavelengths. A great wealth of scientific publications is available, over 600 since 1999. First detections of the GC by the other IACTs Whipple, Cangaroo and HESS are made. Nevertheless the reconstructed fluxes differ significantly. This can be explained by calibration problems, time variations of the source or different integrated sources due to different point spread functions. The nature of the source of the VHE gamma rays is not yet been agreed on. 

Conventional acceleration mechanisms for the VHE gamma radiation utilize the accretion onto the black hole and supernova remnants. The GC is expected to be the brightest source of VHE gammas from particle dark matter annihilation. Although the observed gamma radiation is most probably not due to dark matter annihilation, it is interesting to investigate and characterize the observed gamma radiation as it is not excluded that a part of the flux is due to dark matter annihilation.

The MAGIC data could help to determine the nature of the source and to solve the flux discrepancies between the measurements by other experiments. Due to the large Zenith angle MAGIC will have a large energy threshold but also a large collection area and good statistics at the highest energies. The observation results can also be used to inter-calibrate the different IACTs.






%------------------------------------------------------------------------------

\appendix

\section{Acknowledgements}

The authors thank A. Moralejo for helpful discussions about the Monte Carlo simulations. 


\bibliography{bibbib}
\bibliographystyle{GC}










\end{document}



\appendix


\subsection{Dark Matter Halo Modeling}


The tidal radius $r_t$ is that distance from the center of Draco, beyond which tidal effects due to the gravitational field of the Milky Way are expected to become important.

\subsection{Star Distribution}

Stars are tracer particles in the combined potential from the stars and the DM halo. As the Draco dSph has a negligible ISM component the luminosity is due to stars. The star distributions are modeled in the literature. %The models fit the data well.


\subsection{DM Profiles}

We use DM halo profiles which are suggested or compatible with numerical simulations of cold dark matter halo simulations, see \cite{NFW1997,Stoehr2002,Hayashi2004}. The Moore et al. profile \cite{Moore1998} has not been considered because it is not compatible with the measured velocity profiles of low surface brightness galaxies \cite{Stoehr2004}.


Cusped spherical power law \cite{NFW1997,Evans2004} for the DM density:

\begin{equation} \label{eq:NFW_profile}
\rho_{\mathrm{cusp}}(r)=\frac{A}{r^{\gamma}(r+r_s)^{3-\gamma}}
\end{equation}


Cusped spherical power law with exponential cut-off \cite{Kazantzidis2004a,Kazantzidis2004b}:

\begin{equation} \label{eq:Kazantzidis_profile}
\rho_{\mathrm{cusp}}(r)=\frac{C}{r}\exp\left(-\frac{r}{r_b}\right)
\end{equation}



Intermediate profile \cite{Stoehr2002} of the circular velocity $V_c$ as a function of the distance $r$ from the center of Draco:

\begin{equation} \label{eq:Stoehr_profile}
\log\left(V_c/V_{max}\right) = - a\left[ \log(r/r_{max})\right]^2
\end{equation}


Intermediate profile \cite{Hayashi2004} of the dark matter density $\rho(r)$ as a function of the distance from the center of Draco:

\begin{equation} \label{eq:Hayashi_profile}
\ln(\rho_{\alpha}/\rho_{-2}) = (-2 / \alpha) \left[(r/r_{-2})^{\alpha} -1 \right] 
\end{equation} 


Cored spherical power law from \cite{Wilkinson2002} 

\begin{equation} \label{eq:Wilkinson_Profile}
\psi(r) = \frac{\psi_0}{[1+r^2]^{\alpha/2}} = \frac{G_N M(r)}{r} \quad \alpha \neq 0 ,
\end{equation}

where $G_N$ is Newtons gravitation constant.


Cored spherical power law from \cite{Evans1994} for the DM density:
 
\begin{equation} \label{eq:Evans_Profile}
\rho_{\mathrm{pow}}(r)=\frac{v_a^2 r_c^{\alpha}}{4 \pi G} \frac{3 r_c^2 + r^2(1-\alpha)}{(r_c^2 + r^2)^{2+\alpha/2}}
\end{equation}


























































































