Supplementary Notes
In our paper we present an interpretation of the diffuse very high energy (VHE) -ray
emission of the Galactic Centre (GC) region in terms of the interactions of cosmic rays in
giant molecular clouds in the region. Here we provide additional discussion of the
uncertainties related to the measurement of the -ray flux and spectral index and to the
calculation of the energy density and spectrum of cosmic rays in the Galactic Centre
region.
1. The total gas mass of the Galactic Centre region
The argument that the cosmic ray (CR) flux in the GC region is enhanced over the
local density rests on an estimate of the total mass of target material (for CR interactions)
in the GC region. The bulk of the gas in the GC region is likely to be molecular
hydrogen (H2), which is very difficult to detect directly. Therefore indirect methods of
estimating the mass using tracer molecules (or dust particles) must be applied. Tracer
molecules are typically rare relative to molecular hydrogen but much easier to detect and
with an approximately known ratio to H2. The three mass estimates quoted in our paper
are derived via different measurement channels: sub-millimetre emission from dust [16]
and line emission from transitions of the CS [14] and C18O [17] molecules. These
independent estimates result in CR enhancement factors of 3-9 (CS), 5-13 (C18O) and 46 (dust). Therefore, despite the considerable uncertainties in the mass estimate we find
that the hypothesis of a CR density equal to that in the solar neighbourhood (i.e. an
enhancement factor of 1) is excluded at a high level of confidence.
2. Uncertainties in the inferred cosmic ray spectrum
a) The relationship of the -decay -ray spectrum to the primary hadron spectrum
Hadronic interactions of cosmic rays in the interstellar medium lead to the production of
neutral pions. The energy spectrum of the -rays produced in the decay of these pions is
directly linked to the energy spectrum of the primary cosmic rays. Several authors have
calculated the spectral index of the resulting -ray spectrum assuming a power-law for the
primary cosmic rays. In the approximation used here [3], the -ray spectral index is
identical to that of the primary spectrum. There have been suggestions that the -ray
spectrum may be somewhat harder than the primary spectrum, due to scale-breaking
effects and the slight energy dependence of the interaction cross-section [S1]. The
hardening seen in Figure 6 of [S1] amounts to a difference of 0.04 between the proton
and -ray spectral indices. Our own Pythia [S2] simulations confirm that the expected ray spectrum is flatter than the input proton spectrum by about 0.05 in the energy range
from 100 GeV to 10 TeV. This difference is a factor of 10 smaller than the observed
difference between local and GC CRs observed by H.E.S.S.. Figure 11 of [S3] compares
the parameterisation we use ([3], labelled Aharonian Approx.) with that of [S1]. The two
curves are in extremely good agreement in the relevant energy range.
b) Measurement errors on the -ray spectral index
The uncertainty on the -ray spectrum measured by H.E.S.S. includes a statistical and a
systematic part. The systematic errors we quote reflect all the non-statistical errors on the
spectrum that we are aware of. As the measurement of -rays by a system of Cherenkov
telescopes is rather indirect, it is worthwhile to consider the steps involved in the
calculation of the -ray energy spectrum and what systematic errors may arise.
1. Production of an electromagnetic cascade by an incident -ray in the atmosphere
Electromagnetic interactions in the GeV-TeV energy range are extremely well
understood. To confirm that our numerical simulations describe properly the underlying
physics, three independent shower simulation codes were compared within the HESS
collaboration and no significant differences were found.
2. Production and propagation of Cherenkov light
The production of Cherenkov light is also well understood. The uncertainty here stems
from the absorption of light in the atmosphere. Comparing our best estimate atmospheric
model with a model only marginally inconsistent with our measurements leads to a shift
in flux of ~10% but a change in spectral index of only 0.05. These considerations are
discussed in detail in [S4] and [S5].
3. Measurement of a signal in a Cherenkov camera
The absolute calibration of the H.E.S.S. cameras has been performed using muon ring
images. As the yield of Cherenkov light of a single muon passing through the telescope
dish is well known, this signal can be used to calibrate the conversion of Cherenkov
photons to electrical signals. Variations of ~10% in the optical efficiency were measured
for the data taken on the GC, producing an uncertainty in the absolute energy scale but
having little effect on the shape of the reconstructed spectrum. Two independent
calibration chains are applied in the analysis of all H.E.S.S. sources and give essentially
identical results.
4. Spectral analysis and background subtraction
One way a change in spectral slope could arise is due to background subtraction
problems. Comparison of different background subtraction models shows differences of
<1%, leading to a systematic effect on the spectral index of less than 0.05. Completely
independent analyses of the GC (using different shower reconstruction [S6] and spectral
reconstruction [S7] techniques) show index variations of < 0.1.
Our overall quoted systematic error of 0.2 on the spectral index is a very conservative
estimate based on all 4 sources of uncertainty.
Considering the field of VHE -ray astronomy as a whole, we note the very good
agreement on the Crab Nebula spectrum between the CAT, HEGRA, Whipple, MAGIC
and H.E.S.S. instruments as a strong indication that spectra can be well measured at VHE
energies. We also note that measurements of the cosmic-ray proton spectrum by the
HEGRA instrument at TeV energies [S8] showed good agreement with direct
measurements made using satellites and balloons. This overlap with direct measurements
provides an additional cross-check of the ground-based air-Cherenkov technique.
These notes are the result of discussions with two anonymous referees, whose assistance
we acknowledge here.
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