4.3 Data analysis and results
4.3.5 The optical depth of the MCs
For optically thin clouds, the Fe Kα brightness is proportional to the column density of
104 line reverberation in Galactic Centre molecular clouds
τF e; theoretically, this parameter can be written as
τFe = NFe σFe= 3×10−5 ZFe NH σFe, (4.4)
where NF e is the column density of Fe atoms inside the MC andσF eis the cross section
for photoabsorption of Fe at the K-edge (σF e=4.03×10−20 per Fe atom). The NF e value
can be further written in terms of the H column density (NH) using the standard solar
Fe:H ratio and assuming that the Fe abundance relative to solar within the cloud is ZF e.
In the XRN scenario, both τF e and the distance of the cloud from the illuminating
source are the crucial parameters when estimating the X-ray luminosity required to produce a given Fe-Kαflux. Previous studies have often relied upon estimates of NH combined with
assumptions of ZF ein such calculations, where the former is inferred from intensity maps of
CO, CS or other molecular tracers of high density material in the inner Galaxy. However, the estimation of the cloud NH is often a very difficult measurement. For example, the
G0.11-0.11 cloud (our region F) has been the subject of numerous studies in the last decade. Adopting two different ways of measuring the NH of this MC, respectively using
the intensity of the CS and the H13CO+ emission lines, Amo-Baladr´on et al. (2009) and
Handa et al. (2006) measured two extreme values for the NH of this MC: 2×1022 and 1024
cm−2, respectively.
Here, we suggest that a much more direct approach is to use the measured optical depth of the absorption at the Fe-K edge directly. In the spectral analysis of the stacked spectra in Section 4.3.3, we were able to determine the Fe-K edge optical depth (the EDGE component in XSPEC) with reasonable precision, as is evident from the results reported in the second last row of Table 4.5. Taking the G0.11-0.11 cloud as an example, if we assume the average Fe abundance determined earlier (ZF e=1.56), we estimate the NH to
be 1.8×1023 cm−2, a value between the two extreme numbers cited in the literature.
Technically the X-ray spectrum does not give a direct measure of the column density of neutral Fe atoms through the cloud, but instead records the Fe-K edge imprinted on the reflected continuum by absorption. However, our simulations (see later) show that for optically thin clouds and simple cloud geometries, these two quantities are of very similar magnitude. In the limit when the cloud becomes optically thick, the depth of the edge will reach a maximum value (i.e. the value predicted by reflection models such as pexrav).
In order to test whether the value of NH that we measured are a good set, we run some
simulations in order to describe the relation between the spectral shape of the scattered high energy continuum (power law in the stacked spectra) and the column density of the cloud. In a Thomson scattering process, the incident spectrum and the reflected one have the same slope; but two caveats are to be considered. Theoretically high energy photons escape from the absorption zone (the cloud) more easily than low energy ones. Moreover, a 30 keV photon will suffer less scattering than a 10 keV photon in crossing a certain amount of cold matter (e.g. George & Fabian, 1991). For these reasons, the spectrum reflected by a MC has a different shape from the incident one; it will be harder than the incident spectrum and this hardness is a function of NH (or τF e) within the MC.
4.3 Data analysis and results 105
Figure 4.8: The spectral slope of the reflected spectrum as a function of the measured column density (derived from the Fe-K edge and the average value for ZF e). The green
points show the results of our simulations (see text), with the best fit exponential function overplotted in green. The blue points relate to the nine clouds studied in this work, where the blue line represents the best fit exponential function to the data. The red points are based on the data from Ponti et al. (2010). The dotted black line shows the best fit constant function to the blue data (see text).
emergent from the cloud assuming an incident Γ=2.0, ZF e=1.5 and θ=90 deg. The results
of this simulation are plotted as green points in Fig.4.8; as earlier explained, the spectrum of the reflected continuum flattens towards higher column densities, because of the absorption occurring in the cloud both up to the scattering point and as the scattered photons emerge from the MC. In order to model the flattening of the reflected continuum, we fitted the green points with an exponential function of the form Γ=A·exp(−B·NH)+C; we are aware of
the fact that the exponential fit does not have any physical basis,apart from the functional dependance of the absorption (i.e. exp(−τ)), we however chose it because it better fits the data points, whereas a linear decrease or a power law does not fit the data as well. The best fit function is described by A=41.4 and B=1.8×10−25 cm2 and C=-39.0, and it
is shown as a green dashed-dotted line in Fig.4.8.
The blue points in Fig.4.8 show the Γ-NH combinations measured in the MCs studied
in this work. For comparison, we also plotted in red the points which refer to the same clouds as studied in Ponti et al. (2010), where different values of the NH were assumed
and the relative slopes Γ of the reflected spectrum measured (Table 4 in this reference). As a general comment, we can see that in our measurements show a general decrease
106 line reverberation in Galactic Centre molecular clouds
Table 4.6: Values for the Fe-K edge optical depth at 7.1 keV (as measured from the time averaged X-ray spectra, see Table 4.5), listed together with the NH values measured form
the optical depth of the Fe K edge absorption (assuming Z=1.56, see text) and the ones assumed in (Ponti et al., 2010, NH,old in units of 1022 cm−2).
MC τF e NH NH,old A 0.6 32.1 4.0 B1 0.2 10.7 ≤2.0 B2 ≤0.07 3.7 9.0 C 0.23 12.3 - D ≤0.4 21.4 9.0 E 0.22 11.8 9.0 F 0.34 18.2 2.0 DS1 0.4 21.4 - DS2 0.53 28.3 -
of the power law slope (an hardening of the reflected spectrum) towards higher NH, as
expected. To quantify the flattening of the reflected continuum as seen in our data, we fitted the measured Γ-NH points (blue data in Fig.4.8) with an exponential form as the
one used before. The resulting best fit function is shown in Fig.4.8 as a blue dashed-dotted line, whose parameters have been measured to be A=90.3, B=1.6×10−25 cm2 and C=-
89.0 (χ2
red=0.96). Moreover, when fitted with a constant function (i.e. Γ=constant, black
dotted line in Fig.4.8), the reduced χ2 for the blue data results 1.87, too large to consider the constant behaviour as a good model for our data. We can therefore safely conclude that also in our data we see a significant flattening of the reflected spectra towards higher absorptions.
On the other hand, the Γ-NH points measured in Ponti et al. (2010, red points in
Fig.4.8) are randomly distributed in the plot area and no decreasing trend is evident. We also notice that the error bars of the blue points are much smaller than the red respective measurements; this is due to the use of the Cash statistics (instead of the standard χ2
statistics), which better measures the spectral parameters and therefore allow us to place a better constrain on the spectral shape of the non thermal reflected continuum.
We remind the reader that the simulations were performed for a fixed geometry of the MCs, i.e. all the clouds projected into the plane of Sgr A* (θ=90 deg); this is very unlikely to be the case for the actual distribution of the clouds studied in this work, which are more likely distributed along the line of sight as previously discussed (see Sections 4.3.4 and 4.3.6). Because of this dislocation, and since the effective slope of the reflected continuum depends on the geometry of the illumination, the decreasing trend for the actual data could be slightly smeared with respect to simulated ones (i.e. the green points in Fig.4.8), resulting in a more spread distribution of the data points into the Γ-NH plot, like we
4.3 Data analysis and results 107
effectively measured. As a confirmation of this, we notice that the three points that more significantly differ from the simulated exponential decay are the ones for the B2, DS2 and A clouds, which we will find to likely have a significant distribution behind the plane of Sgr A* (see Section 4.3.6). More generally, the spread measured in the Γ-NH distribution
with respect to a simulated decay will also depend on all the clouds parameters, i.e. NH
and ZF e.
To conclude, we are confident that the NH values inferred from the measurement of the
optical depth of the absorption at the Fe-K edge through equation 4.4 (see third column in Table 4.6) are more suitable for the calculations of all the other MCs properties we will develop in this paper.