3. Radiative transfer
In order to model the circumstellar line emission, and to determine accurate 12CO/13CO abundance ratios, we have used a non-LTE radiative transfer code based on the Monte Carlo method [see Schöier (2000) for details]. Assuming a spherically symmetric CSE expanding at a constant velocity, the code calculates the molecular excitation, i.e., the level populations, required to solve the radiative transfer exactly.
The excitation of the CO molecules were calculated taking into account 30 rotational levels in each of the ground and first vibrational states. The transition probabilities and energy levels are taken from Chandra et al. (1996), and the rotational collisional rate coefficients (CO-H2) are based on the results in Flower & Launay (1985) (they are extrapolated for J11 and for temperatures higher than 250 K). Collisional transitions between vibrational levels are not important due to low densities and short radiative lifetimes.
The basic physical parameters of the CSE, e.g., the mass loss rate, expansion velocity, and temperature structure, are determined from the analysis of the observed 12CO radio line emission. CO is well suited for this purpose since it is difficult to photodissociate and easy to excite through collisions, and thus is a very good tracer of the molecular gas and its temperature. The kinetic temperature of the gas is derived in a self-consistent manner, i.e., the calculations include the most important heating and cooling mechanisms for the gas, e.g., heating due to dust-gas collisions and molecular line cooling from CO. Once the characteristics of the CSE have been determined, the 13CO excitation analysis is performed. The 13CO abundance is varied until a satisfactory fit to the observations is obtained. In this way the circumstellar 12CO/13CO-ratio is estimated.
The spatial extent of the molecular envelope is an important input parameter, and the derived mass loss rate and, to a lesser extent, the 12CO/13CO-ratio will depend on this. The size of the circumstellar CO envelope, assumed to be the same for 12CO and 13CO, was estimated based on the modelling presented in Mamon et al. (1988). It includes photodissociation, self-shielding and H2-shielding, and chemical exchange reactions. Here we assume the radial fractional abundance distribution to follow
where is the initial (photospheric) abundance with respect to H2, is the photodissociation radius (where the abundance has dropped to ), and is a parameter describing the rate at which the abundance declines. Both and depend on the mass loss rate, the expansion velocity, and . When modelling the 12CO emission we assume 110-3. This is an average of the :s estimated by Olofsson et al. (1993b) for a sample of optically bright carbon stars.
In our models we include both a central source of radiation and the cosmic background radiation at 2.7 K. The central radiation emanates from the star itself, which may be approximated by a blackbody. For heavily dust-enshrouded objects, like CW Leo (a.k.a. IRC+10216), most of the stellar light is re-emitted by dust at longer wavelenghts. This emission source is also approximated by a blackbody. For low mass loss rate objects, the stellar blackbody temperature, , was estimated from a fit to the SED of the object. For stars of intermediate to high mass loss rates, two blackbodies were used, one representing the stellar contribution and one representing the dust. A fit to the SED gives the two blackbody temperatures, and , and the relative luminosities of the two blackbodies, /. The method is described in Kerschbaum (1999). The temperatures and luminosities used in the modelling are presented in Table 3. The inner boundary of the CSE was set to be outside the radius of the central blackbody(s), but never lower than 11014 cm.
Table 3. Input parameters and derived 12CO/13CO-ratios from the radiative transfer calculations. A colon (:) marks an uncertain 12CO/13CO-ratio estimate that is based on observations of only one 13CO transition.
The distances, presented in Table 3, were estimated using one of the following methods: the observed Hipparcos parallax, a period-luminosity relation (Groenewegen & Whitelock 1996), or an assumed bolometric luminosity. In the two former cases the luminosities were estimated using apparent bolometric magnitudes and the distances.
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000