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ms.tex ApJ, submitted astro-ph/0602193
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\documentclass[12pt,preprint]{aastex}
\usepackage{graphicx}
\def\gr{$\gamma$-ray }
\def\grs{$\gamma$-rays }
\def\grr{$\gamma$-radiation }
\newcommand{\be}{\begin{eqnarray}}
\newcommand{\ee}{\end{eqnarray}}
\begin{document}
\shorttitle{CR propagation parameters from diffuse VHE \gr emission}
\title{Obtaining cosmic ray propagation parameters from diffuse VHE \gr emi=
ssion from the Galactic center ridge}
\author{I. B\"usching}
\affil{North-West University, Potchefstroom Campus, South Africa}
\author{O.C. de Jager}
\affil{North-West University, Potchefstroom Campus, South Africa}
\author{J. Snyman}
\affil{North-West University, Potchefstroom Campus, South Africa}
\keywords{diffuse emission, cosmic ray propagation, Galactic center}
\begin{abstract}
The recent discovery of diffuse, VHE $\gamma$ radiation from the Galactic
center ridge by the H.E.S.S. telescope allow for the first time the direct=
=20
determination of parameters of galactic cosmic ray propagation models.
In this paper we show that the diffuse \grr near the Galactic center may be
explained by the interaction of VHE cosmic ray (CR) protons with the inters=
tellar
gas located in several giant molecular clouds leading to a measurement of
the cosmic ray diffusion coefficient for the galactic center region
of $\kappa =3D 1.3$ kpc$^2$Myr$^{-1}$ for a mean proton energy of $\sim 3$ =
TeV, if=20
we assume that the CR protons originated from a supernova event (Sgr A East=
), which took off about 10\,kyr ago. This value of $\kappa$ is $\sim 5$ to =
10 times smaller than the locally measured value.
\end{abstract}
\section{Introduction}
The High Energy Stereoscopic System of four telescopes offers currently
the best angular resolution for the study of VHE $\gamma$-rays from cosmic
sources \citep{aharonian06}. With an angular resolution of $\sim 0.08^{\cir=
c}$, the H.E.S.S. Collaboration
was able to resolve $\gamma$-rays associated with molecular clouds in the=20
galactic center region \citep{aharonian06}: Whereas a relatively good
correlation was found between the $\gamma$-ray and CS (with the latter meas=
ured by \citet{tsuboi99}) surface brightness
distributions within 150 pc ($\ell \sim \pm 1^{\circ}$ along galactic longi=
tude) from the galactic center,
this correlation degraded at a distance of 200 pc from the galactic center =
(i.e. at $\ell \sim 1.5^{\circ}$).
A strong indicator that this diffuse component is indirectly associated wit=
h a source at the
galactic center is the similarity of the spectral indices of the point sour=
ce HESS\,J1745-290
at the center and this newly discovered diffuse extended emission. Furtherm=
ore, the relatively good correlation
within $\ell\sim \pm 1^{\circ}$, but with degrading correlation beyond that,
suggests that we are dealing with a source (e.g. the SNR Sgr A East or the =
central Black Hole
Sgr A*) at the GC, which was active for
some time in the past and that we are now seeing the high energetic partic=
les diffusing
from this central source \citep{aharonian06}. =20
The most likely primary species responsible for the gamma-ray emission is t=
hen
protons, since electrons would have to compete against synchrotron losses,
resulting in a spectral steepening towards large distances. Apart from slig=
ht effects
of energy dependent diffusion, protons, in a good approximation, do not loo=
se energy within this environment,=20
resulting in an approximate invariant spectral index with distance.
Whereas these results are important from an Astrophysical viewpoint, it is =
also
of importance from a cosmic ray viewpoint, i.e. the study of cosmic ray pro=
pagation
in our galaxy: \citet{aharonian06} suggested that the diffusion coefficient=
=20
for protons in the 4 to 40 TeV range should be less than $10^{30}$ cm$^2$s$=
^{-1}$
(or 3.5 kpc$^2$Myr$^{-1}$)
as a result of enhanced turbulence and higher magnetic field strengths in t=
he GC region.
The H.E.S.S. data therefore offer a unique possibility to measure the diffu=
sion
coefficient in this part of the galaxy and to compare with other measuremen=
ts
of the galactic diffusion coefficient.=20
This is in particular important as it has been shown \citep{buesching05} th=
at, given=20
SNR are the main sources of CR, the widely used method to obtain propagatio=
n parameters
by fitting secondary to primary data is at least tainted, as the CR primary=
component then shows strong
variations in space and time.
=20
In this paper we will model a transient source at the GC with an
activity time-scale in the past. By solving the transport equation
for proton propagation along the galactic plane, we will obtain
the range of diffusion coefficients, which fit the observed HESS profile be=
st.
To do this we will also model the gas distribution as traced by the CS emis=
sion.
\citet{tsuboi99}
was the first to obtain a full coverage of the Galactic Center
Bow (GCB) and the molecular cloud structures in CS in the region of interes=
t. Since the
gamma-ray surface brightness is reflected by the line-of-sight integral of =
the
product of the cosmic ray and gas densities, we will introduce 3D structures
in the GC region (even though not strictly modelling the GCB itself), such =
that
the line-of-sight integral through these model structures reproduce the true
line-of-sight integrals through the gas density within $\sim 5\%$ accuracy,=
as described
in more detail below.
=20
Using the diffusive model of CR propagation, together
with observations made by the High Energy Stereoscopic System
(H.E.S.S.) \citep{aharonian06} it is possible to get an
estimation of the diffusion coefficient controlling cosmic ray (CR)
transport in the galactic center region.
\subsection{\grs from pion decay}
The omnidirectional (i.e. integrated over solid angle) differential
\gr source function $q_{\pi^0}(E_{\gamma},\vec{r})$ at the position=20
$\vec{r}=3D(l,b,r)$=20
for the decay $\pi^0\rightarrow 2\gamma$ is given by \citep{buesching01}
\be
q_{\pi^0}(E_{\gamma},\vec{r})&=3D& 2\int_{\eta}^{\infty}=20
\frac{Q_{\pi^0}(\gamma_{\pi},\vec{r})}{\sqrt{\gamma^2_{\pi}-1}}d\gamma_{\pi}
\label{grs:srcfkt}
\ee
where $\gamma_{\pi}$ is the pion Lorentz factor.
The lower boundary of the integration is given by
\be
\eta&=3D&\frac{E_{\gamma}}{m_{\pi}c^2}+\frac{m_{\pi}c^2}{4\,E_{\gamma}}.
\ee
The pion source function then is=20
\be
Q_{\pi^0}(\gamma_{\pi},\vec{r})&=3D&\rho_{\rm gas}(\vec{r})\,c=20
\int_{\gamma_{\rm thr}}^{\infty} \beta\sigma^{\pi^0}_{pp}(\gamma_p,\gamma_{=
\pi^0})N_p(\gamma_p,\vec{r})d\gamma_p
\label{grs:losi}
\ee
where $\sigma^{\pi^0}_{pp}$ is the total cross section for pion production =
in $pp$ collisions and $N_p$ the CR proton spectrum.
The differential photon flux from the decay of CR induced $\pi^0$s from the=
=20
direction $(l,b)$ is given by integrating Eq.\ref{grs:srcfkt} along the lin=
e of sight=20
\be
\frac{dN(E_{\gamma},l,b)}{dt\,dE_{\gamma}\,d\Omega} &=3D&
\frac{1}{4\pi}\int q_{\pi^0}(E_{\gamma},\vec{r}) dr
\ee
The above calculation is for pion production from $pp$ interactions only.
The effect of the known chemical composition of the ISM can be taken into a=
ccount by=20
increasing the total pion production cross section by a factor of 1.30=20
\citep{ms94}.=20
\section{Reproducing the diffusive \gr emission from the Galactic center}
=46or our studies we reproduce the diffusive \gr emission calculating the l=
ine of sight integral
Eq.~\ref{grs:losi} for various $(l,b)$ combinations. As we are only interes=
ted in relative=20
intensities, we can use instead of Eq.~\ref{grs:losi} the relative emissi=
vity
\be
\epsilon(l,b)&\propto&\int_{r=3D0}^{\infty}\rho(l,b,r)N_{CR}(l,b,r)\,dr
\label{calc:emissivity}
\ee
where $\rho(l,b,r)$ is the target material density as a function of
galactic coordinates and $N_{CR}(l,b,r)$ is the calculated CR
density at the point defined by triplet $(l,b,r)$.
\subsection{Gas distribution near the Galactic center}
The inner $\pm 150$\,pc region of the Galaxy contain interstellar $H_2$ gas
of about 2 to 5$\times10^7$ solar masses=20
\citep{tsuboi99}=20
in a rather complex setup of molecular clouds.
=46or our analysis we assumed that the target material density function can=
be
adequately described by the superposition of five spherical Gaussian
functions upon an asymmetric Gaussian base. These functions represent the=20
molecular clouds associated with the radio arc of Sgr A, Sgr B, as well as =
the
longitude varying line-of-sight projection effect of the "Galactic Center B=
ow" as described
by \citet{tsuboi99}. Even though we are able to reproduce the observed line=
=2Dof-sight
gas densities within about 5\% from the observed values, uncertainties in t=
he exact depth=20
distribution along the r-coordinate in Eq. \ref{calc:emissivity}, is expect=
ed
to result in a $\sim 50$\% systematic uncertainty in the final estimate of =
the diffusion coefficient. =20
=20
\subsection{Cosmic ray distribution near the Galactic center}
The CR distribution as a
function of spatial coordinates is calculated using=20
the diffusive model of CR propagation.
We calculate the CR density assuming the CR=20
coming from a single SNR event 10\,kyr ago, accelerating particles for a ce=
rtain time.=20
As was pointed out by \citet{maeda02}, the SNR Sgr A East has an estimated=
=20
age of 10\,kyr, but also younger ages have been given for that SNR \citep{r=
ockefeller05}.
=46or our study, we assume that Sgr A East has an age of 10\,kyr and was =20
accelerating CR on various timescales less than 10\,kyr.=20
We are interested in the CR distribution near the source and thus neglect=
=20
the effects of spatial inhomogeneities in the interstellar gas density (whi=
ch are small for CR protons with the=20
energies we are dealing with) by assuming a mean gas density,
and also boundary effects by imposing boundary conditions for the CR propa=
gation problem at infinity.=20
In this case one can find a solution for the propagation equation in the=20
literature \citep{syrovatskii59}.
Using the integral in Eq.~\ref{calc:emissivity} the emission seen by H.E.S.=
S.
\citep{aharonian06} can be recreated for different diffusion
coefficients as shown in Fig.~\ref{fig:res} (top). As noted by
\citet{aharonian06}, the excess counts observed by H.E.S.S.
follow the known target material density fairly well, except for the
region $l\ge1^\circ$. This provides a gauge for approximating the
diffusion coefficient governing the CR propagation near the galactic
center.
To obtain better counting statistics, we compare the model and observed
H.E.S.S. excess counts, integrated over Galactic latitude.=20
The excess counts observed by H.E.S.S. then are
proportional to the integral
\be
\epsilon(l)&\propto&\int_{r=3D0}^{\infty}\int_{b=3D-\pi2}^{\pi/2}\rho(l,b,r=
)N_{CR}(l,b,r)r\,db\,dr,
\label{calc:int}
\ee
where the CR density $N_{CR}$ has been=20
calculated for different diffusion coefficients, as shown in Fig.~\ref{fig:=
res}
(bottom).=20
The obtained fit is normalised to the total numbers=20
of excess counts before calculating the corresponding $\chi^2$ value.=20
The results of the analysis are shown
in Fig.~\ref{fig:chisq} below. The optimal value for the diffusion
coefficient $k$ was found to be
\begin{equation}
\kappa=3D1.3\,{\rm kpc}^2{\rm Myr}^{-1},
\label{calc:kfinal}
\end{equation}
which is about 40\% smaller than the diffusion coefficient estimated by
\citet{aharonian06}. We also note that the minimum reduced $\chi^2$ is 1.5
(for $\sim 25$ d.o.f.), which indicates that the data are marginally
described by the model. Note that we do observe
a well-defined minimum in Fig.~\ref{fig:chisq}, leading to the abovemention=
ed
measurement of $\kappa$. It would be interesting to see if more sophisticat=
ed
3D modelling of the total gas distribution result in even smaller $\chi^2$ =
values. Because of this uncertainty, we estimate a systematic uncertainty o=
f $\sim 50\%$ on the measured value of $\kappa$.
\begin{figure*}
\resizebox{\hsize}{!}{\includegraphics{f1.ps}}=20
\caption{
Calculated emission sky maps (top) together with the calculated excess
counts (bottom) shown as the solid lines. The histogram indicate
the excess counts from the H.E.S.S. observation \citep{aharonian06}.=20
The figures (a), (b) and (c) where generated for diffusion
coefficients with values $0.3\ kpc^2Myr^{-1}$, $1.3\ kpc^2Myr^{-1}$ (best f=
it)
and $15.1\ kpc^2Myr^{-1}$ respectively.}
\label{fig:res}
\end{figure*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{f2.ps}}=20
\caption{
Reduced $\chi^2$ values plotted for different diffusion coefficients. We fi=
nd a
well defined minimum for a diffusion coefficient of $k=3D1.3\ kpc^2Myr^{-1}=
$.=20
}
\label{fig:chisq}
\end{figure}
\subsection{Mean CR energy}
In the last section, we have shown that the diffuse \gr emission from the=20
Galcatic center ridge can be explained by CR hadrons which propagation can =
be described by a diffusion coefficient of=20
$1.3$\,kpc$^2$Myr$^{-1}$.
This diffusion coefficient is valid for CR generating the bulk of
the \gr emission in the H.E.S.S. energy range.=20
To compare this value=20
with that obtained by fitting local CR data, as derived e.g. by \citet{mosk=
alenko02}, \citet{jones01} or \citet{maurin02}, we have to estimate the e=
nergy of the CR probed here.
=46or this investigation, we approximate the sensitivity of the H.E.S.S. t=
elescope array by a box function from 0.4\,TeV to 20\,TeV. For the CR proto=
n spectrum=20
we assumed a power law with the observed photon index $\Gamma\,=3D\,2.29\=
pm 0.27$ \citep{aharonian06}.
We used the Pythia \citep{pythia1,pythia2} event generator package to calcu=
late the number=20
of \grs with $E_{\gamma}>0.4$\,TeV, for different CR proton energies.
The result of this calculation is shown in Fig.~\ref{fig:gl400}.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{f3.ps}}=20
\caption{
Number of photons with $E_{\gamma}>400\,GeV$ for different CR proton energi=
es
assuming a CR proton spectral index of $s\,=3D\,2.29$.}
\label{fig:gl400}
\end{figure}
The maximum number of photons is produced by CR protons with energies of ab=
out 2.2\,TeV, given a proton spectral index of 2.29. Note that the location=
of the maximum =20
depends on the proton spectral index. Given the uncertainties stated by \c=
itet{aharonian06},=20
we find that CR protons with energies in the range 1.7\,TeV to 3\,TeV contr=
ibute=20
mostly to the \grs seen by H.E.S.S.
Assuming a diffusion coefficient of the form=20
\be
\kappa&=3D&\kappa_0\left(\frac{\xi}{\xi_0}\right)^{0.6},
\ee
where $\xi_0=3D1\,$GV and $\xi$ the particle rigidity
we find $\kappa_0=3D0.013$\,kpc$^2$Myr$^{-2}$,
which is significantly smaller than the values found =20
by fitting local CR data. However, $\kappa$ for the latter range from 0.053=
5\,kpc$^2$Myr$^{-2}$ to 0.201\,kpc$^2$Myr$^{-2}$, as compiled by \citet{mau=
rin02} (with references therein).
This finding can be well explained by enhanced turbulence and a higher fiel=
d strength of the interstellar magnetic field in the =20
Galactic center region. We note however that the rigidity dependence of 0.6=
can only apply to a limited
energy (rigidity) range, since $\kappa$ must always be larger than the Bohm=
limit.
\section{Summary and Discussion}
We have shown that the progress in the imaging Cherenkov technique now make=
s it=20
possible for the first time to measure the CR diffusion coefficient =20
in other parts of the Galaxy. A diffusion coefficient of $\kappa\sim 1.3$ k=
pc$^2$Myr$^{-1}$
appears to be well measured from the data, although we have to add a $\sim =
50\%$ systematic
undertainty arising from uncertainties in the actual 3D gas density distrib=
ution, as well
as uncertainties on the epoch when the central source activity started, whi=
ch was assumed to be 10 kyr
in this paper. The most likely central source is Sgr A East, for which the =
epoch of onset of
activity is $\sim 10$ kyr, but if the central source was the central massiv=
e black hole Sgr A*,
the timescale of activity would be much less certain, resulting in even lar=
ger uncertainties
on $\kappa$. Note however that, given an initial epoch of onset of activity=
(i.e. 10 kyr),
estimates of $\kappa$ appear to be fortunately robust
against uncertainties in the detailed time profile of particle acceleration=
within 5 kyr after the SN explosion. For example,=20
reducing the ``on''-time of the source by a factor of five, decreases the d=
iffusion coefficient by 40\%. Such detailed studies will however be treated=
in a subsequent paper.=20
Thus, if Sgr A East was the source of CR, then we have a relatively
accurate masurement of $\kappa$.=20
In general, following the notion of \citet{aharonian06} to constrain the co=
smic ray diffusion
coefficient $\kappa$ from the spatial distribution of diffuse $\gamma$-rays=
resulting
from impulsive injection of a CR source at some time in the past, we have
shown that one can obtain unique measurements of the diffusion coefficient,
provided that the gas density distribution, as well as the epoch of onset of
central source activity is known.=20
We also showed that the CR diffusion coefficient in the Galactic center reg=
ion is=20
significantly smaller than that obtained by fitting local CR data. Our unde=
rstanding
of turbulence theory is however still too limited to understand how diffusi=
on coefficients
scale with the turbulence $\delta B$, the correlation length associated wit=
h this
turbulence, the total magnetic field strength $B$ and rigidity
dependence for perpendicular and parallel diffusion. Hopefully our new
measurement of $\kappa$ will help to constrain results from turbulence theo=
ry.
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\end{document}
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