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Astron. Astrophys. 324, 51-64 (1997)

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4. Column densities and abundance ratios

4.1. Basic assumptions

In order to derive accurate column densities from the absorption lines, we have to make three assumptions. These are

  1. assume a value for the covering factor f of the absorbing molecular gas with respect to the extent of the radio core in Cen A.
  2. assume a value for the excitation temperature [FORMULA] of the absorbing gas.
  3. assume that each molecular species is in local thermodynamical equilibrium, i.e. that all the rotational levels are characterized by the same excitation temperature (called then the rotational temperature [FORMULA]).

The covering factor f is unknown for Cen A, but due to the small size of the radio core ([FORMULA] 2.0 mas [FORMULA] 6000 AU), it is likely to be close to unity for the molecular gas. Also, the extinction towards the core of Cen A at optical wavelengths has been estimated to be [FORMULA] mag (Eckart et al. 1990), corresponding to a substantial amount of obscuring material. The depth of the deepest [FORMULA] (1-0) absorption line (no. 1 in Table 2) is 0.95 when the continuum level is unity. If the line is saturated, this means that 95% of the continuum source is covered by molecular gas. If, on the other hand, the absorption line is not saturated, the covering factor is larger. The abundance ratio of [FORMULA] / [FORMULA] for the deepest absorption line is [FORMULA] 70. This ratio would be lower than the real [FORMULA] / [FORMULA] ratio if (1) the [FORMULA] line was saturated and (2) if isotopic fractionation augments the [FORMULA] abundance. The high [FORMULA] / [FORMULA] ratio therefore shows that the [FORMULA] line is unlikely to be strongly saturated. The nondetection of [FORMULA] is consistent with this scenario. The observed optical depth is nevertheless 3.0 for the deepest absorption component. The other absorption components are not as deep, but the upper limits to the [FORMULA] / [FORMULA] ratio at the corresponding velocities (Table 7) implies that these lines are not saturated and, hence, that their covering factors are always larger than their depth in the normalized spectra. In the following we will assume that [FORMULA].


Table 6. Column densities and abundance ratios for the main absorption components [FORMULA].


Table 7. Column densities of [FORMULA] in the High Velocity absorption complex

The excitation temperature defines the relative population of two levels and can be derived if the velocity integrated optical depths of two rotational transitions of the same molecule can be determined. This is not the case for Cen A. Furthermore, in order to derive the total column density we must link the fractional level population to all the levels. This is done by invoking a weak LTE-approximation and assuming that [FORMULA], where [FORMULA] is a temperature which governs the fractional population of all rotational levels in a given molecule. The LTE approximation is weak in the sense that it does not imply that [FORMULA] equals the kinetic temperature and it allows for different molecules to have different [FORMULA]. With this approximation we can use the partition function [FORMULA] to express the total column density [FORMULA] as


where [FORMULA] is the observed optical depth integrated over the line for a given transition, [FORMULA] the statistical weight for rotational level J, [FORMULA] the Einstein radiative transition coefficient for levels J and [FORMULA], and


The excitation temperature is most likely low for the gas seen in absorption. The reason is that a high [FORMULA] quickly depopulates the lower rotational levels and decreases their opacity. For [FORMULA] K, [FORMULA]. Hence, along a line of sight with a mixture of molecular gas components of similar column density but with different excitation temperatures, absorption lines of ground transitions will preferentially sample the excitationally coldest gas. Excitationally cold gas does not necessarily imply that the kinetic temperature is low. [FORMULA], HCN and HNC are thermalized at H2 densities of 105 cm-3. A molecular gas with densities lower than this would therefore give these molecules a low excitation temperature. In a survey of [FORMULA] (1-0) absorption and emission in our Galaxy, Lucas & Liszt (1996) find only one case out of eighteen where emission is associated with the absorption. This shows that the excitation temperature in the gas sampled through absorption is indeed very low. Also denser molecular gas seen in absorption has a low excitation temperature. Greaves & Williams (1992) measured [FORMULA] using CS(2-1) and CS(3-2) for several clouds towards Sgr B2 and found [FORMULA] to be always [FORMULA] 4 K.

The detection of a relatively strong absorption of CS(2-1) does not necessarily imply either a high [FORMULA] or a high density. The J [FORMULA] 1 level of CS has an energy corresponding to 2.35 K and should be significantly populated by the cosmic microwave background radiation. We searched for the CS(3-2) line and came up with a very weak detection of the main line at 552 km s-1 (line no. 1 in Table 2). The line is too weak to allow a determination of the excitation temperature, but the mere difficulty in detecting this line is a strong indication that the excitation temperature is low. The J [FORMULA] 2 level of CS has an energy corresponding to 7.05 K. The CO molecule is detected in both the J [FORMULA] 1-0, J [FORMULA] 2-1 and J [FORMULA] 3-2 transitions. The energy J [FORMULA] 1 and J [FORMULA] 2 levels correspond to temperatures of 5.53 and 16.6 K, respectively. The HV complex is tentatively detected in CO J [FORMULA] 3-2 (Israel et al. 1991) 2. CO has an electric dipole moment more than [FORMULA] times lower than either [FORMULA], HCN, HNC and CS, and is thermalized at H2 densities which are [FORMULA] times lower. Hence, the excitation temperature of CO is likely to be higher. This is consistent with the weak LTE assumption made above, where [FORMULA] can vary between different molecular species.

Since we have observed molecules with a large electric dipole moment, we will use [FORMULA] K for all the lines when deriving column densities. If the excitation temperature is close to the cosmic microwave background temperature, we overestimate the column densities of [FORMULA], HCN and HNC by a factor 2.0. For CS the factor is 1.5. If, on the other hand, the excitation temperature is 10 K, we would underestimate the [FORMULA], HCN and HNC column densities by a factor 3.1 and the CS by a factor 2.5.

A low [FORMULA] means that only the lowest rotational levels are significantly populated and that the assumption that one single temperature governs the overall population distribution, i.e. [FORMULA] is likely to be valid. The abundance ratios are very insensitive to the assumed excitation temperature, only depending on the weak LTE assumption. Hence, the column densities should be accurate to within a factor of 2, while the abundance ratios are very robust estimates.

4.2. Qualitative results

We calculated column densities for the four absorption lines in the LV complex by using the gaussian components given in Table 2. For the HV complex we derived the velocity integrated optical depth directly from the spectra 3. The results are presented in Table 6. For HCN we used the decomposition of the hyperfine components as given in Table 3. For all lines we used an excitation temperature of 5 K. Upper limits to the velocity integrated optical depth of [FORMULA] was derived as


where [FORMULA] is the rms of the opacity (0.012), [FORMULA] the velocity resolution (0.15 km s-1) and [FORMULA] the assumed line width. For the latter we used 50% of the corresponding HCO [FORMULA] line (compare with the line widths of [FORMULA] and [FORMULA] for line no. 1). A 3 [FORMULA] upper limit to the [FORMULA] column density, not given in Table 6, is [FORMULA]. This is derived in the same way as above, but with a [FORMULA] of 5 km s-1 in order to compare it with line no. 1.

In Table 7 we present column densities and abundance ratios for 4 clearly defined [FORMULA] (1-0) components in the High Velocity complex. The components can only be identified in the [FORMULA] spectrum and are designated 5a-d. They are defined in a smoothed spectrum and correspond to gaussian components 10, 11-13, 14-16 and 17 as given in Table 5. Integrated optical depths were derived by integrating over the velocity intervals given in Table 7.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998