Issue 
A&A
Volume 568, August 2014



Article Number  A122  
Number of page(s)  16  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201322639  
Published online  04 September 2014 
Online material
Appendix A: Virialized gas state
The gravitational potential of the densely packed stars and the nearby supermassive nuclear engine in the central region of a massive galaxy may cause significant velocity gradients along lines of sight, which can be well in excess of what would be found in a normal cloud near virial equilibrium. Therefore the velocity gradient expected in the virialized gas motion can be taken as a lower limit. The ratio between the measured velocity gradient and that calculated from virial equilibrium is defined by (A.1)The virialized velocity gradient is given by (A.2)where μ is the mean particle mass, G is the gravitational constant, ⟨ n ⟩ is the mean number density of the cloud, and α is a constant between 0.5 to 3 depending on the assumed density profile (Bryant & Scoville 1996). For a cloud with assumed density of 10^{5} cm^{3}, and with the largest value of α = 3, the estimated (dv/dr) is about 10 kms^{1} pc^{1}. For more diffuse gas with a density of 10^{3} cm^{3}and α = 0.5, (dv/dr) is around 0.5kms^{1} pc^{1}. Molecular gas close to the central massive black hole will be strongly influenced by gravity (e.g., Bonnell & Rice 2008), thus could be subvirialized (K_{vir}< 1). However such an effect is likely obvious only within a few tenths pc in the center. On the other hand, the tidal shear produced by the black hole would also increase the instability of molecular gas, where K_{vir}> 1.
Appendix B: Likelihood analysis of the singlecomponent fitting
In the following, we analyze possible solution ranges for the central 18′′ of the Circinus galaxy and corresponding physical conditions satisfying maximum likelihood achievable in the set of all combinations parameters (see also Sect. 4.2.1). We caution, however, that these findings – in particular the numbers shown below – are rather uncertain, and will be only indicative.
Instead of the Bayesian probability, which is the integral of all probabilities in the parameter space (e.g., Weiß et al. 2005; HaileyDunsheath et al. 2012; Kamenetzky et al. 2011; Rangwala et al. 2011), we analyze the trend of the solutions with the highest likelihood. The maximum likelihood function of a given parameter (or given parameters) is based on the best fitting results in the whole parameter space.
Figure B.1 (upper panel) shows the maximum likelihood as a function of velocity gradient in a range of 1kms^{1} pc^{1}≤dν/dr≤ 10^{3} kms^{1} pc^{1}. We plot the corresponding values of n_{H2} and T_{kin} of the best fit for each given velocity gradient (lower panel). Over the modeled dν/dr range, T_{kin} and n_{H2} vary by about an order of magnitude. The likelihood drops below half of the peak value when dν/dr is beyond 10^{1.8} kms^{1} pc^{1}, where the solutions have relative high temperature and low density. This suggests that reasonable models are not likely to have a dν/dr higher than 60 kms^{1} pc^{1} because then even the best fitting result shows poor fits to the measurements.
Fig. B.1
Upper panel: maximum likelihood as a function of the velocity gradient dν/dr for singlecomponent LVG fitting. Lower panel: bestfit values of density (blue circles) and temperature (red squares) for a given velocity gradient as functions of velocity gradient, with error bars showing a 1σ range of the likelihood distribution. The gray regions have K_{vir}~ 1, 10, and 100. 

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Fig. B.2
Upper panel: maximum likelihood as a function density for singlecomponent LVG fitting. Lower panel: bestfit velocity gradient for a given density as a function of density. The gray regions have K_{vir}~ 1, 10, and 100. 

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Fig. B.3
Upper panel: maximum likelihood as a function of temperature for singlecomponent LVG fitting. Lower panel: bestfit density for a given temperature as a function of temperature. The dashed lines show thermal pressure, log (n_{H2}×T_{kin}), in units of K cm^{3}. 

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Fig. B.4
The contours show the distributions of the maximum likelihood for a given density and temperature in the singlecomponent LVG fitting. Contours are drawn from 0.1 to 0.9 by 0.2. Background gray scale levels show the velocity gradient associated with the best LVG fitting results, for each given temperature and density. The dashed lines indicate the thermal pressure log (n_{H2}×T_{kin}) in units of K cm^{3}. 

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Figure B.2 shows the maximum likelihood as a function of n_{H2} in a H_{2} density range from 10^{2} cm^{2}to 10^{6} cm^{2}. We plot the corresponding dν/dr of the best fits as a function of density. The solutions are found over a broad range of velocity gradients, which increase almost linearly as density increases. Solutions with high densities also have high velocity gradients. But the density is not likely to be higher than 10^{4.5} cm^{3}where the likelihood is dropping below half of the peak value and K_{vir} exceeds ten. This implies that models with high density solutions are highly supervirialized and are not bound by selfgravity.
Figure B.3 shows the maximum likelihood as a function of T_{kin} from 10 K to 10^{3} K. The thermal pressure P = n_{H2} × T_{kin} of the best fitting results is presented for given temperatures. The thermal pressure decreases by an order of magnitude when T_{kin} increases from a few tens of K to about 200 K. This indicates that the solutions of high temperature will have low thermal pressure because of the corresponding low density of these solutions.
In Fig. B.4, we show as contours the densitytemperature likelihood distribution of the LVG modeling. The gray scale background displays the velocity gradient associated with the best fitting results, at given temperatures and densities. The contours present a bananashaped likelihood distribution, which is mainly caused by the degeneracy between temperature and density. In the contour map, thermal pressure almost stays constant along the ridge of the distribution. Both density and temperature vary by two orders of magnitude within the 50% contour. The likelihood distribution covers a range of thermal pressure from 10^{4.8} to 10^{6.5} Kcm^{3} and peaks at ~10^{5.2} Kcm^{3}. From the map of the associated velocity gradients in the background, dν/dr increases with n_{H2}, and decreases with T_{kin}. Most good solutions have small dν/dr between 1 kms^{1} pc^{1} and 10 kms^{1} pc^{1}.
Appendix C: Likelihood analysis of the twocomponent fitting
Fig. C.1
Background gray scale images show the maximum likelihood distributions of temperature and density, derived from the twocomponent LVG modelings. The gray levels are from 0.1 to 0.9 with a spacing of 0.2. The low and highexcitation components are plotted in blue and red contours. The dashed lines indicate thermal pressure, log (n_{H2}×T_{kin}), in units of K cm^{3}. 

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Fig. C.2
Upper panel: velocity gradients of the best fitting results as functions of density, derived from the twocomponent LVG modeling. The gray regions have K_{vir}~ 1 and 10. Lower panel: maximum likelihood as functions of the densities of both excitation components. The low and highexcitation components are plotted in blue and red shadows, respectively. 

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Fig. C.3
Upper panel: thermal pressure (log (n_{H2}×T_{kin})) of the best fitting results as functions of density, derived from the twocomponent LVG modeling. Lower panel: maximum likelihood as functions of temperatures for both excitation components. The low and highexcitation components are plotted in blue and red shadows, respectively. 

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In Fig. C.1, we show the maximum likelihood distribution of both components in our modeling, with contours of the enclosed probability. Both distributions have banana shapes that are mainly caused by the degeneracy between T_{kin} and n_{H2}. We find that both distributions are characterized by component specific thermal pressures. The thermal pressure of the HE component is about one order of magnitude higher than that of the LE component.
The HE component shows a steep slope in the high density and low temperature regime, and a flat slope at the high temperature side with a very broad range of T_{kin} solutions. This indicates that the density is not tightly constrained for the HE component. The HE component has a bestfit dν/dr of ~ 50kms^{1} pc^{1}, which is about 10 times higher than that of the LE component, where ~6kms^{1} pc^{1} is the best fitting result. General fitting results of both excitation components are listed in Table 9.
FigureC.2 shows the density likelihood as functions of both excitation components (lower panel), and the corresponding velocity gradient of the best fittings for given densities (upper panel). The higher the density, the larger the velocity gradient for both components. Solutions with higher densities also have higher K_{vir}. Molecular gas in such conditions has very violent motions and high temperature. Unless the HE component adopt a low density solution of ~10^{3.5} cm^{3}, K_{vir} is always higher than unity. The density range of the HE component is much wider than that of the LE component, which is due to the high degeneracy between n_{H2}, T_{kin}, and dν/dr, and less constrained for the highJ transitions. In Fig. C.3, the temperatures of both components are not well constrained although the likelihood curve of the LE component looks narrower and peaks at lower temperature (~40 K) than that of the higher temperature (~50 K). The thermal pressure drops when the temperature increases, and stays nearly constant when the temperatures of both components are higher than 100 K.
© ESO, 2014
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