Astron. Astrophys. 339, 811-821 (1998)

3. Molecular abundance estimates

The determination of molecular abundances from the observed emission lines is ambiguous, mainly because the spatial distribution of the SiO and HCN emission is not known for these stars. Even so, we can make reasonable assumptions about the physical properties of the circumstellar envelopes (CSEs) to place constraints on the range of molecular abundances consistent with our observed line intensities. The range of derived SiO abundances for our program stars typically spans about one order of magnitude because of the sensitivity to the assumed physical size of the SiO distribution. The chemical equilibrium abundances of species like SiO and HCN, however, are very sensitive to temperature, density, and C/O ratio, spanning many orders of magnitude for reasonable "photospheric" parameter values, as shown in BL. Thus, despite the substantial dependence of the derived abundances on the assumed molecular distribution, it is possible to place some constraints on the physical conditions in the extended stellar atmophere where the SiO forms. For HCN, the formation mechanism is not known with any certainty, and only 2 stars in our sample are detected, so the usefulness of abundance constraints is perhaps more limited.

3.1. Statistical equilibrium model

We used the statistical equilibrium code described in Bieging & Tafalla (1993) to determine the populations of molecules in rotational levels in both the ground and first vibrationally excited states. Briefly, the statistical equilibrium equations are solved iteratively, including both collisional and radiative excitation, for an expanding spherical envelope. Radiative transfer for the exciting photons is treated by an escape probability formalism (Castor 1970). This approach is appropriate for these stars, since the expansion velocity is large compared to the local line width, so the large velocity gradient approximation is valid. The emergent spectrum is calculated by numerical integration of the transfer equation for a series of annuli in the projection of the envelope on the plane of the sky, and then convolved with the telescope beam.

The envelope density is determined by the gas mass loss rate, , and the expansion velocity, , both assumed constant with time. We further assume that the hydrogen is entirely molecular for calculating the collision rates. The outer extent of the envelope is not truncated; however, the drop in density and temperature (and for some models, molecular abundance) with radius effectively cuts off the molecular emission at some radius, beyond which the calculation is terminated.

The stellar radiation field is modelled as a 2000 K blackbody with a total luminosity of . The local (thermal and turbulent) velocity dispersion is assumed to be 1 km s^-1. The kinetic temperature of the envelope is assumed to depend on radius in an analytic form which approximates the results of Kastner (1992), who solved the equation of thermal balance for model CSEs as a function of mass loss rate. We adopt a kinetic temperature law of the form for , and for , where r is the distance from the star, and and depend on the mass loss rate. The exponent, -0.7, seems to be a good approximation to the radial dependence of found by Kastner (1992), for mass loss rates covering the range applicable to our program stars.

Kastner (1992) notes that a significant uncertainty in the adopted temperature law is the nature of the circumstellar dust. The heating rate for gas-grain collisions depends on the grain emissivity and on the dust/gas ratio. Kastner (1992) used values appropriate for carbon stars. If S stars have mainly silicate grains (as indicated by their IRAS LRS spectra-see Sect. 4.2 below), his models based on graphite grains may overestimate the heating rates. If S stars also have a lower dust/gas ratio than carbon stars, on average, then the heating rate would also be too high. Both effects would tend to cause the adopted kinetic temperature law to be too large, by an uncertain factor. (On the other hand, a reduced dust/gas ratio might enhance the radiative heating of the dust, due to a reduction in the envelope opacity to stellar radiation.) The effect on the calculated SiO line intensities is likely to be small for most stars in our sample, however, since radiative excitation through the IR ro-vibrational lines dominates over collisional excitation for all except W Aql, with its relatively high mass loss rate.

3.2. SiO models

The IR ro-vibrational transitions for SiO were calculated using molecular constants from Tipping & Chackerian (1981). Radiative transition rates were calculated from their values for the dipole moments for vibrational and pure rotational transitions. Collisional rates for the ground vibrational state of SiO were derived from the rate coefficients described in Turner et al. (1992). We used an approximation to the temperature dependence of the rate coefficients of the form

where and the three parameters A, B, and C are functions of the difference in initial and final rotational quantum numbers, . A, B, and C were determined by least squares fits to the Turner et al. (1992) rates (which were calculated for a range of temperatures from 20 to 1800 K). Eq. (1) is a modified version of the analytic approximation given by de Jong et al. (1975), and is motivated by the discussion in Albrecht (1983), but with an added term in the exponent to improve the fit over a wider range in . For kinetic temperatures above 100 K, Eq.  (1) gives rates which differ from the quantum mechanical calculations by less than 50%, and are generally better than that at higher . We neglect collisional excitation to the vibrationally excited levels, for which the rates are small compared to the corresponding radiative rates in these models.

Given the uncertainty in the distribution of SiO, we have tried to constrain the range of abundances by considering two types of models. In the simplest case, we take the SiO abundance to be constant throughout the envelope. In reality, SiO molecules are presumably formed in the photosphere but removed from the gas by grain formation and photodissociation. We therefore expect the true SiO distribution to have a maximum abundance close to the star and to fall off with increasing distance. Since our data consist of only beam-averaged spectra, the constant abundance model should constrain the photospheric SiO value as a firm lower limit.

The predicted SiO J=3-2 spectrum (convolved with the SEST beam) for W Aql and a constant abundance model is compared with the observed spectrum in Fig. 3.

Fig. 3. Sample of SiO (v=0, J=3-2) model spectrum (dashed line) compared with observed spectrum (solid line) for W Aql

A second, more realistic model for the distribution is an exponential fall-off in SiO abundance with increasing distance from the star. We have calculated exponential models of the form where is the abundance of SiO with respect to H₂ (assumed to be the only significant form of hydrogen), is the central or photospheric abundance, and is the e-folding distance. This distribution is a reasonable approximation to the form of the distribution derived for 3 stars, including the S star Cyg, by Sahai & Bieging (1993) based on interferometer images of the SiO J=2-1 emission. We further restrict the exponential model by fixing the e-folding distance, , at cm, a value which produces satisfactory agreement with the observed SiO J=2-1 brightness distribution for Cyg (Lucas et al. 1992; Sahai & Bieging 1993).

The parameters of the excitation models, and the abundances which reproduce the observed beam-smoothed brightness temperatures in the model spectra, are summarized in Table 3. Columns 2 and 3 give the parameters of the adopted kinetic temperature law for each star, which depend on the mass loss rate. This rate is given in column 4, and is taken from the CO results of BL, Sahai & Liechti (1995), or Sahai (1992) (for Gru). The value of the envelope expansion velocity, given in column 5, is from Table 1 for the SEST sample, or from BL for the 4 additional stars in Table 3(b). Column 6 gives the value, , for a constant SiO abundance model which should be a lower limit to the photospheric value. Column 7 gives the corresponding central abundance, , for an exponential model with an e-folding distance of cm.

Table 3. Excitation model parameters and derived abundances for SiO and HCN

Besides the stars observed at SEST in the SiO v=0, J=3-2 line, we have also calculated SiO abundance models for the 4 northern S stars detected in the v=0, J=2-1 line with the NRAO 12-m telescope by BL. Values for these stars are listed in the second part of Table 3. The adopted values of mass loss rate and envelope expansion velocity for R And, S Cas, and W And are taken from BL. For Cyg we adopt the mass loss rate and expansion velocity from Jorissen & Knapp (1998), which are derived from observations of several CO transitions and assume the Hipparcos distance of 106 pc.

The range of values for the constant abundance models is relatively small for 10 of the 11 stars, with values between and . The star Gru is an exception, with a value of = . The exponential models yield central (i.e., photospheric) abundances, , between and , except for Gru, which has , more than an order of magnitude lower than for the other 10 stars. We note that in the case of Cyg, our exponential model yields a central SiO abundance = , which is similar to that derived by Bujarrabal et al. (1989) for a comparable model calculation, though the functional forms of the SiO abundance differ. In their model, the limiting value of the SiO abundance at large radii is , close to our best-fit constant abundance model at = .

The exponential models for SiO produce (v=0, J=3-2) line opacities which are only moderately optically thick. For example, the best-fit exponential model for RT Sco reaches a peak tangential optical depth of 0.9 at a radius of cm. RT Sco has a moderate mass loss rate, typical of most stars in our sample. In the case of very high mass loss rates, such as W Aql, the SiO model opacities may be large enough that the derived SiO abundance should be considered lower a limit, as indicated in Table 3. For most of the stars in our sample, however, the SiO lines are not very optically thick and our derived abundances should be reliable (within the context of the assumed model abundance distribution).

3.3. HCN models

Models for HCN emission followed the analysis applied to the carbon star IRC+10^o216 by Dayal & Bieging (1995). The same statistical equilibrium code was used as for the SiO analysis described in the previous section. Collision cross sections for HCN were derived from Green & Thaddeus (1974), and molecular constants were from Maki (1974) and Evans et al. (1991). The stellar properties are the same as those used in the SiO models, including mass loss rates, wind velocities, and kinetic temperature law.

The abundance distribution of HCN is unknown since the formation mechanism is uncertain for S stars. In this analysis, we assume that HCN is formed close to the star (rather than in an extended photochemical region) and is carried out in the stellar wind. At sufficiently large radii, the HCN is photodissociated by ambient UV photons penetrating the dusty circumstellar envelope. With this assumption, the HCN abundance distribution should be similar to that of carbon stars such as IRC+10216. Following Olofsson et al. (1998) we parametrize the photodissociation radius for HCN as a function of the gas mass loss rate, (in solar masses per year), and wind velocity, (in km/s), as

where s^-1 is the unshielded photodissociation rate of HCN (van Dishoeck 1988). The second term in this formula assumes that the dust in the circumstellar envelope provides shielding typical for carbon stars. Given this assumption, the HCN photodissociation radii for the detected stars are given in column 8 of Table 3, calculated from Eq. (2) and the values in columns 4 and 5.

The model abundance distribution is taken to be a gaussian centered at the star with an e-folding radius as given by Eq. (2), i.e., . The model central or "photospheric" abundance of HCN is then given by . Statistical equilibrium models were calculated, varying to obtain a predicted line intensity (convolved to the SEST telescope beam) which is equal to the observed value (see Table 2). We have also calculated HCN abundance models for the northern S stars detected in the HCN J=1-0 line by BL. The model spectra for these stars were convolved to the 72" HPBW of the NRAO 12-m telescope. Results for stars in both samples are given in the last two columns of Table 3.

For RT Sco, the signal to noise ratio of the HCN spectrum is not high, so the model fit is uncertain by of order a factor of 2. For W Aql, the HCN line is strong and has good signal to noise ratio, but is clearly parabolic in shape, indicating optically thick emission. The fitted HCN abundance must therefore be considered a lower limit. The same comment applies to S Cas and Cyg, detected by BL.

It should be emphasized that these models assume a near-photospheric origin for HCN, with destruction by photodissociation at a characteristic radius given by Eq. (2). If HCN in these stars were produced by photochemical processes in the outer envelope (e.g., involving dissociation of N₂ and CH₄-cf. Willacy & Millar 1997; Charnley et al. 1995), then the foregoing analysis would be inappropriate. Without a resolved image of the HCN emission, however, there are too many free parameters to make alternative (i.e., photochemically-produced) HCN abundance models of much use at present.

© European Southern Observatory (ESO) 1998

Online publication: October 22, 1998
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