3. Molecular excitation temperatures
Based on the results of the time-dependent model (velocity field, gas density and temperature), several molecular excitation temperatures have been calculated. The non-LTE calculations are performed in a similar way as described by Woitke et al. (1996), but have been extended to polyatomic molecules and improved in order to include the ro-vibrational pumping effect.
Detailed level energies and transition probabilities have been deduced from the HITRAN-database (Rothman et al. 1987), yielding ro-vibrational levels and lines for H2O, CO2 and SO2 each. The radiative data for CO have been kindly provided by Jorgensen (1997, priv. comm.). The collisional rates for vibrational and rotational (de-)excitation are calculated by analytical formulae (Millikan & White (1964) and Hollenbach & McKee (1979), respectively) with more recent data for CO and H2O. The code calculates two excitation temperatures ( and ) in order to achieve constancy in time of the total respective energies contained in the molecule in form of rotational and vibrational excitation, as demanded by the non-LTE statistical equations. The continuous background radiation field is assumed to be given by a radially diluted Planckian of stellar temperature, i.e. the outer atmosphere is assumed to be optically thin in the continuum . Optical depth effects in the lines are accounted for by applying Sobolev theory according to the local velocity gradient present in the model. More details will be described in a forthcoming paper.
Fig. 3 shows the results obtained for CO, H2O, CO2 and SO2. The figure indicates that LTE () only holds very close to the star. As the gas density decreases with increasing distance from the star, first the vibrational temperatures decouple from as soon as the gas density approaches a critical value of . The rotational temperatures stay longer coupled to , but finally deviate also substantially from below a critical density of , dependent on the molecule. This behavior is a consequence of the shorter radiative lifetimes of the vibrational levels and of the smaller collisional rates for vibrational de-excitation as compared to the respective rotational quantities.
In order to allow for a quantitative comparison to the results of radiative transfer calculations which adopt a static, uniform layer (see references in Table 1), we have calculated straight means of the calculated excitation temperatures 1 as defined by . The integrations are carried out from the first shock front () to the outer boundary of the model (). The results are summarized in Table 2. While the column densities of CO and H2O agree with the observations within a factor of 3, the amounts of CO2 and SO2 molecules are too low, by factors of and , respectively. The calculated excitation temperatures of H2O are consistent with the observations. For CO, the calculated mean rotational temperature is a bit too low, but this value - only in case of CO - depends strongly on and would increase to 1930 K if was assumed. The mean rotational temperatures of CO2 and SO2 are too high in the model, roughly by a factor of 2.
Table 2. Calculated molecular properties (see text).
Table 2 also shows the maximum of the calculated Sobolev optical depths of the individual ro-vibrational lines (see Woitke et al. 1996) of some vibrational bands at 1.7 , taking into account the actual velocity gradient. The individual lines are mostly found to be optically thin, but some strong lines of CO and H2O can exceed . If the molecular concentrations of CO2 and SO2 were multiplied by 50 and (see above), the same would be found for these molecules.
© European Southern Observatory (ESO) 1999
Online publication: July 16, 1999