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\documentclass[5p]{elsarticle}
\usepackage{graphicx}
\title{Obtaining the diffusion coefficient for cosmic ray propagation in
the Galactic Centre Ridge through time-dependent simulations of their
$\gamma$-ray emission}
\author[astr]{Stavros Dimitrakoudis\corref{cor1}}
\ead{sdimis@phys.uoa.gr}
\author[astr]{Apostolos Mastichiadis}
\author[nuc]{Athanasios Geranios}
\cortext[cor1]{Corresponding author, sdimis@phys.uoa.gr}
\address[astr]{University of Athens, Physics Department, Section of
Astrophysics, Astronomy and Mechanics, Panepistimioupoli 15771, Greece}
\address[nuc]{University of Athens, Nuclear and Particle Physics
Department, Panepistimioupoli 15771, Greece}
\begin{document}
\begin{abstract}
Recent observations by the H.E.S.S. collaboration of the Galactic Centre
region have revealed what
appears to be $\gamma$-ray emission from the decay of pions produced by
interactions of recently
accelerated cosmic rays with local molecular hydrogen clouds. Synthesizing
a 3-D hydrogen cloud map from
the available data and assuming a diffusion coefficient of the form
$\kappa(E) =
\kappa_0(E/E_0)^\delta$, we performed Monte Carlo simulations of cosmic
ray diffusion for various
propagation times and values of $\kappa_0$ and $\delta$. By fitting the
model $\gamma$-ray spectra to
the observed one we were able to infer the value of the diffusion
coefficient in that environment
($\kappa = 3.0 \pm 0.2 kpc^2Myr^{-1}$ for $E = 10^{12.5}eV$ and for total
propagation time $10^4 yr$) as
well as the source spectrum ($2.1 \leq \gamma \leq 2.3$). Also, we found
that proton losses can be
substantial, which justifies our approach to the problem.
\end{abstract}
\maketitle
\section{Introduction}
The High Energy Stereoscopic System (H.E.S.S.) $\gamma$-ray telescope
began operations in 2004 and
among its first targets was the Galactic Centre region \cite{Hinton2006}.
Images obtained from the H.E.S.S. observations had an angular resolution
$0.1^\circ$, giving us a map of
$\gamma$-ray emission of unprecedented detail. A few point sources were
first detected, which were
compatible with the positions of known objects, like the black hole
Sagittarius A*, the supernova
remnant Sgr A East and the supernova remnant/pulsar wind nebula G0.9+0.1.
By subtracting those known
sources, the H.E.S.S. collaboration was able to produce a mapping of
fainter $\gamma$-ray emission, stretching
over an area of approximately 300pc by 100pc around the Galactic Centre
\cite{Aharonianetal2006}. This
emission seems to follow the contours of molecular gas density, as
measured by its carbon monosulfide
distribution \cite{Tsuboietal1999}, up to a distance of approximately
$1.3^\circ$ in galactic longitude from
the Galactic Centre. Beyond that distance the $\gamma$-ray emission
diminishes to background levels.
This apparent emission correlation points towards possible models, for
which two theories were
initially proposed \cite{Aharonianetal2006}. The first postulated that a
population of electron
accelerators produces the observed emission via inverse Compton
scattering. The objects that would make
up such a population, such as pulsar wind nebulae, would thrive in regions
of high-density molecular
gas; however the large number of sources needed to reproduce the observed
emission renders this
possibility rather unlikely \cite{Aharonianetal2006}.
The other theory claims that the $\gamma$-rays are produced by neutral
pion decay, resulting from
the interaction of locally accelerated cosmic rays with the ambient
molecular gas. The lower energy
threshold of the H.E.S.S. survey was 380 GeV, so cosmic ray protons of
higher energies would be needed
to produce the observed $\gamma$-rays. Furthermore, these cosmic rays
would have to have been
accelerated near the Galactic Centre at some point in the past, yet not
diffused significantly beyond
$1.3^\circ$ from it. Assuming the validity of this theory, one may use
this data to infer the diffusion
properties of cosmic rays in the Galactic Centre region for various
possible sources of cosmic rays.
Considering as a source of the cosmic rays either the supernova remnant
Sgr A East, with an estimated
age of 10 kyr \cite{Uchidaetal1998}, or the black hole Sgr A* with a more
remote age of activity,
Aharonian et al. \cite{Aharonianetal2006} inferred that the diffusion
coefficient should be no more
than $3.5 kpc^2 Myr^{-1}$ in that area. Busching et al.
\cite{Buschingetal2006} went one step further
with an analytical reproduction of the H.E.S.S. observations by
calculating the emission for different
diffusion coefficients. Using relativistic protons of mean energy $\sim 3
TeV$ to represent the cosmic
rays and a small number of Gaussian functions to represent the molecular
clouds they arrived at a
diffusion coefficient of $\kappa = 1.3 kpc^2 Myr^{-1}$. B\"usching and de
Jager
\cite{BuschingJager2008}
subsequently expanded their results for different ages and on-times for
the source. More recently,
Wommer at al. \cite{Wommeretal2008} employed a different approach, in
which the motion of individual
test particles is computed by solving the Lorentz force equation for short
time periods, and the
resulting distributions provide the coefficients for the diffusion
equation. From their initial
conditions for the turbulent magnetic field, it is derived that no single
source can account for the
observed emission, but that a more continuous source of cosmic ray protons
produces a better
correlation.
In this paper we present a series of time-dependent simulations of proton
propagation in the Galactic
Centre in a detailed 3D distribution of molecular hydrogen concentrations
for different diffusion
parameters. By utilizing a wide array of data on molecular clouds in the
Galactic Centre from various
observations we construct a fairly accurate density grid of the area of
diffusion, as shown in \S 2.1.
We can then inject cosmic rays, assumed to be protons, from an origin of
our choice, which propagate
according to the diffusion model and its free parameters described in \S
2.2. In \S 2.3 we deal with the
interactions of these protons with hydrogen molecules and with the
production of photons from the
resulting neutral pion decays. In \S 2.4 we briefly discuss the likely
sources of cosmic ray origin, and
the implications on their age. In \S 2.5 we describe the more technical
aspects of the simulation
program, such as the initial choice of free parameters and the methods
used to raise the efficiency of
the numerical code. Finally, in \S 3 we present the results of the
simulations and in \S 4 we summarize
and discuss the effectiveness of our approach and the likely significance
of our results. The present
paper expands upon the initial results of Dimitrakoudis et al.
\cite{Dimitrakoudis2008a}.
\section{Simulations}
\subsection{Synthesizing a hydrogen cloud map }
The area of the Galactic Centre in which we simulated the diffusion of
cosmic rays is rich in $H_2$
gas, contained in a complex setup of high-density clouds, ridges and
streams that comprise about 10\% of
our galaxy˘s interstellar molecular gas, i.e. about 2 to 5 $\times 10^7$
solar masses
\cite{Tsuboietal1999}. Due to the high densities involved, tracer
molecules were used to determine the
mass of each gas cloud. To create a realistic 3-D map of that environment
we first obtained the data for
the 159 distinct molecular cloud clumps, as compiled by Miyazaki \& Tsuboi
\cite{MiyazakiTsuboi2000},
i.e. we assigned to each one a galactic longitude, latitude, radius and
density. Their radial distances
are unknown, so we assumed a random function to simulate them, within
reasonable limits. To these data
we added the locations and densities of the radio-sources Sgr A
\cite{Shuklaetal2004}, Sgr B1, Sgr B2
and Sgr C \cite{LawYusefZadeth2004}, which are rich in atomic hydrogen. We
then broke down the entire CS
map of the Galactic Centre by Tsuboi et al. \cite{Tsuboietal1999} into
blocks of uniform density, which
we treated as clouds of equal radius and of random radial distances.
Finally we added the larger CO
clouds by Oka et al. \cite{Okaetal1998} in order to expand the spatial
distribution of our map. Every
time we added a new set of clouds we checked the possibility that clouds
from previous sets were sharing
their projected locations with the new ones, and we subtracted their
masses accordingly. The result was
584 clouds of hydrogen gas with a high uncertainty as to their radial
distances. These clouds were then
turned into a 3-D grid of $120 \times 60 \times 60$ boxes of uniform
density, which form the volume of
our diffusion model, $5.4 \cdot 10^7 pc^3$. The projection of that grid
constitutes an area far greater
than the extent of the observed ă-ray emission by
\cite{Aharonianetal2006}, comprising a total mass of
$1.2 \cdot 10^8 M_\odot$, so particle interactions with clouds well
outside the observed area of
emission are accounted for.
\subsection{Diffusion model }
We used the diffusive model of CR propagation, assuming a diffusion
coefficient of the form:
$$\kappa = \kappa_0(E/E_0)^\delta$$
where E is the energy of the CR protons, $E_0=1GeV$, while $\kappa_0$ and
$\delta$ are free parameters,
whose values we will try to infer through our simulations. The index
$\delta$ is a measure of the
turbulence of the magnetic field and is assumed to be $0.3 \leq \delta
\leq 0.6$ \cite{Strongetal2007},
while $\kappa_0$ is a constant whose value we will attempt to estimate
once $\delta$ has been obtained.
The diffusion coefficient in our simulations is represented by the mean
free path $\ell$, which is given
by the usual expression
$$\ell = 3\kappa/c$$
where c is the velocity of light.
Thus, the test protons move in straight lines of length equal to $\ell$,
after which their directions
change randomly. At the end of each such free walk, the box number of the
hydrogen density grid is
calculated and a check is made for collisions with hydrogen protons. If
such a collision occurs, the
diffusion continues with a lower proton energy, as shown in the next section.
\subsection{Production of $\gamma$-rays }
We have assumed that all $\gamma$-rays forming the observed emission are
produced from neutral pion
decay. Moreover, we have assumed that pions are produced in proton-proton
collisions through two main
channels
\
a) $p + p \rightarrow p + p + \pi^+ + \pi^- + \pi^0$
\
b) $p + p \rightarrow n + p + \pi^+$
\
In case (a) the initial proton loses part of its energy and a multiplicity
of pions is created that take
equal amount of the energy lost from the proton. The positive and negative
pions break down into muons,
and ultimately into electrons, positrons and neutrinos, while the neutral
pions decay into $\gamma$-
rays. Assuming an inelasticity $k_{pp}=0.45$ \cite{MastichiadisKirk1995},
15\% of the initial proton
energy will go to the produced $\gamma$-rays, while the original proton
will continue its diffusion with
55\% of its initial energy. This approach, while simplified, gives results
which are in good agreement
with the more detailed spectra produced by B\"usching et al.
\cite{Buschingetal2006}.
Channel (b) produces no photons and thus does not contribute directly to
our observed emission. However,
the initial proton is turned into a neutron, having lost some of its
energy in the collision, and will
thus continue its propagation in a straight line, unaffected by the
magnetic fields that regulated its
trajectory as a proton. It will revert, though, back into a proton after
half life $\tau = 886.7 \pm 0.8
sec$ \cite{Yaoetal2006}, and then it will continue its diffusion as
before. Accounting for time
dilation, the distances traveled by neutrons are always much smaller than
the mean free path for each
proton energy (0.028pc for $E=10^{12.5}eV$; 0.92pc for $E=10^{14}eV$; the
radii of molecular gas clouds
range between 2pc and 40pc), so we can safely assume that this case will
not affect the diffusion
process.
Using the above we calculate the optical depth for each random walk. This
is calculated as the product
of the interaction cross section with the mean free path $\ell$ and the
density n of hydrogen molecules.
That density is retrieved from the grid box where each step ends, a fairly
effective approximation as
most random walks are contained within single grid boxes. We have assumed
that the cross section is
$\sigma_{pp} = 4 \cdot 10^{-26} cm^2$ \cite{Begelmanetal1990}, while an
increase by a factor of 1.3 is
sufficient to account for the known chemical composition of the
Interstellar Medium
\cite{MannheimSchlickeiser1994}. The reaction probability would then be $P
= 1 - e^{-\tau}$, where
$\tau$ is the optical depth. In practice though, the optical depth is a
very good approximation of $P$ for
low probability values ($P < 0.15$). Thus, to speed up our simulations, we
can simply define the
reaction probability as the optical depth, having first checked that its
value is always low enough to
warrant the approximation. At every step, that probability is checked
against a random number. If the
random number is smaller than the reaction probability, then there is a
collision and $\gamma$-rays are
produced, and the proton continues its propagation with diminished energy.
In the case where its final
energy is so low that we are no longer interested in its resulting
photons, the proton is removed from
the simulation.
Apart from proton-proton collisions, CR protons could lose energy due to
ionization losses, coulomb
collisions, Compton scattering, synchrotron emission, Bethe-Heitler pair
production and photo-pion
production. Wommer et al. have demonstrated that energy loss rates due to
the four latter processes are
insignificant compared to energy loss rates from proton-proton collisions
\cite{Wommeretal2008}. As for
ionization and Coulomb losses, Mannheim and Schlickeiser
\cite{MannheimSchlickeiser1994} clearly show
that their effect is negligible compared to that of proton-proton
collisions in the local interstellar
medium. In the higher density environment of the galactic centre, and
especially within molecular
hydrogen clouds, their comparative effect would be minimal.
\subsection{Possible sources}
Assuming a single source for the origin of the diffuse cosmic rays in the
Galactic Centre, then the most
likely candidates would be either the supernova remnant Sgr A East or the
black hole Sgr A*. The former
has an estimated age of $10^4 yr$ \cite{Uchidaetal1998} while the latter
could have had a burst of
activity even further in the past. We have conducted simulations for both
scenarios, selecting an age of
$10^4$ yr for the former case and an age of $10^5$ yr for the latter. In
both cases, the galactic
coordinates for the source are set to $l = 0^\circ$, $b = 0^\circ$, which
correspond well to the
location of Sgr A* and to the approximate centre of the extended source
Sgr A East.
\subsection{Simulation parameters}
Except for the time available for diffusion (which corresponds to the
elapsed time since activity at the
source), all other parameters are the same for both potential sources. We
have assumed that the source
is active for 100 years and that it produces a constant flux of cosmic
rays during that period. During
this time cosmic rays are assumed to constantly escape from the source and
to start diffusing. The
proton injection spectrum is divided into six logarithmic bins, ranging
from $E = 10^{12.5} eV$ to $E =
10^{15} eV$. Lower energies would produce $\gamma$-rays below the
H.E.S.S. threshold, while higher
energies would have a negligible impact on the resulting emission, due to
their low numerical density
and their large mean free paths. Each bin has the same number of test
particles, which is fixed for each
run, depending on the interaction probabilities for a given diffusion
coefficient. Those numbers are
then weighted after each simulation according to a power law distribution
that fits best the power law
of the observed $\gamma$-rays from H.E.S.S., see \cite{Aharonianetal2006}
($\Gamma = 2.29 \pm 0.07_{stat}
\pm 0.20_{sys}$). Since the weighting procedure is applied to the photon
spectrum at the end of the
simulation, the resulting initial proton spectrum is generated in
deference to its modification during
the diffusion process.
To improve the efficiency of the simulations, we treat the process of
continuous production of cosmic
rays in the following way. Only one burst of cosmic rays of all energies
is created in the simulation,
starting at the beginning of the source˘s activity. The photons produced
at the end of the simulation
are, naturally, the ones observed from protons that have propagated for
the duration of our simulation.
In addition to them we also take into account all the photons produced in
a time period before the end
of the simulation that is equal to the production time at the source
($10^2 yr$). These correspond to
the photons that would have been produced at the end of our simulation by
newer populations of protons
produced within that time range at their source. The actual production
time is of little importance, as
long as it much shorter than the propagation time (as is also seen in
\cite{BuschingJager2008}). Had we
considered all the protons to have been produced instantaneously, the
results would have been the same,
provided the number of protons was increased to provide the same
statistical robustness. On the other
end, a production time as large as $10^3 yr$ wouldn˘t significantly alter
our results.
Other parameters besides the two ages are the index $\delta$ and the
proportionality factor $\kappa_0$
of the diffusion expression. For the index $\delta$ we have chosen the
values 0.3, 0.4, 0.5 and 0.6,
while $\kappa_0$ takes on a series of test values by increments of
log(0.1) around the roughly expected
value required for a diffusion coefficient that would allow a mean
propagation of $^2 = \kappa \cdot
\Delta t$, in accordance with each total time and the mean propagation
distance inferred from the
observations.
Once each simulation is completed and the results normalised, the
resulting $\gamma$-rays are compared
against the results from H.E.S.S. using the reduced $\chi^2$ criterion.
\section{Results}
The resulting reduced $\chi^2$ values for cosmic rays originating from the
supernova remnant Sgr A East,
assumed to be produced $10^4 yr$ ago, are shown in Fig.1 for various
values of the diffusion
coefficient. The minimum value of $\chi^2$ calculated is 1.8, and
corresponds to $\delta = 0.3$ and
$\kappa_0 = 0.25 kpc^2Myr^{-1}$. However we can find minima which are less
than 2 for all values of
$\delta$. If we use those values of $\delta$ and $\kappa_0$ to calculate
the diffusion coefficient for
protons of energy $10^{12.5}eV$ (the lowest energy in our sample and also
the most important due to
their relative abundance over those at higher energies), we arrive at the
results illustrated in Fig.2.
We can see that, for each value of $\delta$, the diffusion coefficient
displays the same minimum at
$\kappa = 3.0 \pm 0.2 kpc^2Myr^{-1}$. This value is higher than that
calculated by B\"usching et al.
\cite{Buschingetal2006} but close to that suggested by Aharonian et al.
\cite{Aharonianetal2006} (less
than $3.5 kpc^2 Myr^{-1}$).
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig1.eps}
\caption{Reduced $\chi^2$ values for different values of $\kappa_0$ and
$\delta$ for total propagation
time $10^4 yr$.}\label{fig1}
\end{figure}
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig2.eps}
\caption{The curves represent the reduced $\chi^2$ values for different
diffusion coefficients,
calculated for protons of energy $10^{12.5}eV$, for each value of
$\delta$. We see in all cases a
minimum for $\kappa \approx 3 kpc^2 Myr^{-1}$. The grey line shows the
equivalent results from B\"usching
et al. [3] for comparison.}\label{fig2}
\end{figure}
For higher values of $\kappa$, we also see a disparity between the way the
reduced $\chi^2$ values
increase in our results and those of B\"usching et al. The reason for this
is that for these values the
mean free path becomes very large and comparable to the dimensions of the
area of simulation, therefore
the simulation becomes inefficient, resulting in poor statistics even with
increased numbers of test
particles.
Fig.3 shows the reduced $\chi^2$ values that correspond to a total
propagation time of $10^5 yr$ – a
possible past activity of Sgr A*. If we once again use the values of
$\delta$ and $\kappa_0$ which
correspond to a minimum to calculate the diffusion coefficient for protons
of energy $10^{12.5}eV$ we
arrive at the results illustrated in Fig. 4. The resulting best choices
for the diffusion coefficients
appear in Table 1.
\begin{table}[where]
\centering % used for centering table
\begin{tabular}{c c} % centered columns (4 columns)
$\delta$ & $\kappa [kpc^2Myr^{-1}]$ \\% inserts table
%heading
\hline % inserts single horizontal line
0.3 & 0.45 \\ % inserting body of the table
0.4 & 0.4 \\
0.5 & 0.56 \\
0.6 & 0.32 \\ % [1ex] adds vertical space
\hline %inserts single line
\end{tabular}
\caption{Diffusion coefficients for different values of $\delta$ for total
propagation time $10^5 yr$.}
\end{table}
This variation for different values of $\delta$ is more pronounced than in
the case of propagation for
$10^4 yr$, but if we were to derive a mean value of $\kappa$ like before,
we would find $\kappa = 0.43
\pm 0.05 kpc^2Myr^{-1}$. The higher propagation time requires the cosmic
rays to diffuse more slowly
than in the case of $10^4 yrs$, which is why the resulting diffusion
coefficient is considerably
smaller.
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig3.eps}
\caption{Reduced $\chi^2$ values for different values of $\kappa_0$ and
$\delta$ for total propagation
time $10^5 yr$.}\label{fig3}
\end{figure}
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig4.eps}
\caption{Plot of the reduced $\chi^2$ as a function of the diffusion
coefficient $\kappa$ for different
values of $\delta$. The total propagation time was assumed to be $10^5
yr$. Higher values of $\delta$
tend to favor lower values of $\kappa$. }\label{fig4}
\end{figure}
Fig.5 and 6 plot the proton spectral indices $\gamma$, as these were
inferred from the simulations, for
the two total propagation times versus $\kappa_0$ for different values of
$\delta$.
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig5.eps}
\caption{Plot of spectral index $\gamma$ inferred from simulations as a
function of the diffusion
normalization $\kappa_0$ for different values of $\delta$. The total
propagation time was taken to be
$10^4 yr$.}\label{fig5}
\end{figure}
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig6.eps}
\caption{Plot of spectral index $\gamma$ inferred from simulations as a
function of the diffusion
normalization $\kappa_0$ for different values of $\delta$. The total
propagation time was taken to be
$10^5 yr$. }\label{fig6}
\end{figure}
In Fig.5 we can see a fluctuation of $\gamma$ values, most likely due to
irregularities in the
interactions of the different energy populations with the complex setup of
hydrogen clouds in the small
propagation time. A progression to lower $\gamma$ values as we raise the
value of $\kappa_0$ is,
however, evident. The values of $\gamma$ for the minimum reduced $\chi^2$
values are in the range $2.1
\leq \gamma \leq 2.3$. For the larger propagation time, we can see in
Fig.6 a similar progression to
lower $\gamma$ values as a function of $\kappa_0$. In the range of our
best results, the proton indices
are in the range $2.0 \leq \gamma \leq 2.1$, so the proton spectrum is
clearly slightly steeper than the
resulting photon spectrum. These results are in general agreement with
leaky box model predictions
\cite{Hillas2005}.
Furthermore, it is possible to make an estimate of the total energy of the
protons accelerated at their
source, by comparing the number of observed photons to the number of
diffusing protons. Doing so for the
lowest observable proton energy and extrapolating for the energy range
$10^9-10^{15} eV$ yields a total
energy of $(8 \pm 1) \times 10^{49} erg$, for our different time and
optimal diffusion parameters. That
represents approximately 10\% of the energy output of a typical supernova
explosion.
The Fermi Gamma-ray Space Telescope should be able to observe that area of
space in a lower energy range
(from 20 MeV up to 300 GeV). We have repeated our simulations for the
optimal diffusion coefficients we
have found, for a total propagation time $10^4 yr$, with the proton
injection spectrum extended to $10^9
eV$. In these cases, the emission is dominated by the lower energy
protons, and thus extends from about
$-0.3^\circ$ to $0.2^\circ$ in galactic longitude.
\section{Summary/Discussion}
We have presented the results of a series of time-dependent simulations of
the diffusion of cosmic rays
in the Galactic Centre region. In the first scenario it was assumed that a
burst of cosmic rays occurred
$10^4 yrs$ ago, while in the second $10^5 yrs$ ago. Likely candidates
could have been the SNR Sgr A East
or the black hole candidate Sgr A*. Their $\gamma$-ray emission was
compared with observations from the
H.E.S.S. collaboration \cite{Aharonianetal2006}, in order to determine the
diffusion coefficient in that
region. For that purpose, a detailed 3-D map of hydrogen concentrations
was synthesized from various observations \cite{LawYusefZadeth2004,
MiyazakiTsuboi2000, Okaetal1998,
Shuklaetal2004, Tsuboietal1999}, and two scenarios for the origin of the
cosmic rays were taken into
account. For the SNR Sgr A East, $10^4 yrs$ is the estimated upper limit
on its age, so the resulting
diffusion coefficient should be taken as a lower limit. There is much more
uncertainty concerning the
possible age of activity of Sgr A*, but it is clear that should that have
been in the order of $10^5
yrs$ ago or earlier, the random component of the interstellar magnetic
field would have to be
considerably more pronounced than in the first case.
The need for such elaborate simulations as were described in the preceding
chapters arises from the
considerable losses of protons during their propagation, losses which are
inextricably connected to the
local densities of hydrogen gas. In Fig.7 one can compare the resulting
reduced $\chi^2$ values from a
series of test runs where proton energy losses are not taken into account,
as opposed to the same values
from our actual simulations. An underestimation of the diffusion
coefficient is evident in the
simulations without losses. Furthermore, this underestimation explains the
discrepancy between our
results and those of B\"usching et al. \cite{Buschingetal2006}, as the
diffusion coefficient calculated
without losses is about two times smaller than when losses are taken into
account in the propagation.
\begin{figure}
\includegraphics [width=0.48\textwidth]{fig7.eps}
\caption{Reduced $\chi^2$ values for different values of $\kappa_0$ and
$\delta = 0.6$ for total propagation time $10^4 yr$ with and without
proton energy losses taken into account during simulations.}\label{fig7}
\end{figure}
The above results were derived under the assumption that the acceleration
of the cosmic rays responsible
for the observed $\gamma$-rays occurred $10^4$ or $10^5$ yrs ago at a
single source, with no subsequent
periods of activity at that source. Recent papers
\cite{ErlykinWolfendale2007} have noted that this may
be a very simplified approach, as there have been many supernovae in the
Galactic Centre region in the
past millennia. Also, the uncertainty in the radial distances of hydrogen
concentrations may have had a
significant impact on the final results. Finally, these simulations
assumed that the diffusion
coefficient remains constant throughout the whole region of propagation,
and local orderings of magnetic
fields (including their correlation with molecular density within the gas
clouds) were not taken into
account.
Those limitations notwithstanding, the results of these simulations are
useful in providing an estimate
of the diffusion coefficient in the Galactic Centre, taking the different
magnetic turbulence theories
(that become manifest in the different values of $\delta$) into account.
\section{Acknowledgments}
This project is co-funded by the European Social Fund and National
Resources – (EPEAEK II) PYTHAGORAS II.
We would like to thank the anonymous referee for the useful comments which
helped us improve this paper.
We would also like to thank John Kirk for helpful discussion.
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