3. A model for H2O emission
Our previous modelling of H2O emission from W Hya (Barlow et al. 1996) (i) neglected excited vibrational states, (ii) did not use the observed FIR continuum as a constraint (iii) adopted the kinetic temperature distribution of Goldreich & Scoville (1976, hereafter GS) (iv) assumed a two-shell model of H2O abundances.
The present treatment abandons these assumptions. Here we add the first vibrational state to allow for 4-10 µm photon excitation. The H2O abundance starts from a guessed value at the inner radius with an initial ortho/para H2O ratio of 3. We solve radiative transfer and thermal balance to derive a radial kinetic temperature , H2O line fluxes, and the FIR continuum consistently in order to compare with the observations.
For R Cas, the Hipparcos parallax (van Leeuwen et al. 1997) corresponds to a distance of 107 12 pc and = -3.74. Adopting the angular diameter E-model of Haniff et al. (1995) gives: the stellar radius 2 cm, stellar effective temperature 2415 K, stellar luminosity L 2420 , and stellar mass M 1.2 . From CO observations, the outflow terminal velocity is = 12 (Bujarrabal et al. 1989, Loup et al. 1993). We assumed an atomic-to-molecular hydrogen ratio of f = 1. Collisional rates of ortho - and of para -H2O excited by He impact (Green et al. 1993) were corrected by a factor 1.38 to account for "2 encounters. We treat 45 rotational levels of the ground vibrational state, , whose highest energy is 2000 K. To account for 4-10 µ photon excitation, as suggested by Barlow et al. (1996), we considered also the 45 lowest rotational levels of the first excited vibrational state, . We assumed that and have the same rotational collision rates, - ), while between and the vib-rotational collision rates are lower by a factor 0.01 (Deguchi & Nguyen-Q-Rieu 1990; hereafter D-NQR). This factor was subsequently varied by a factor of 10 to determine its effects. As in D-NQR, we found that uncertainties in the vibrational collisional rates have no effect on the calculated H2O line fluxes.
The FIR continuum radiation comes from the central star, the 2.7 K background, and dust. It was scaled by a factor, , to account for the stellar phase dependence. The dust temperature varies as (Elitzur et al. 1976). The FIR emissivity of dust grains was taken to vary as (80/(µm))q (Bujarrabal et al. 1980), where we have found q 0.80 is appropriate to reproduce the observed wavelength dependence of the FIR continuum spectra (Fig. 4). We adopted a grain radius of =0.1 µm and a dust mass density =3 g cm-3. A dust-to-gas mass ratio of 0.005 seems to be appropriate (Groenewegen et al. 1998). If is the grain number density, and is the grain geometrical cross section, uncertainties in the grain characteristics can be absorbed in the momentum transfer efficiency factor Q, a free parameter in the dust absorption coefficient, which is used to fit the observed H2O line fluxes and continuum:
Here is the averaged optical depth of dust grains at 80 µm from - to z along the line of sight at impact parameter p; p and z are the components of the radius vector in the Castor (1970) coordinate system (p=0 and z=r for the on-source direction).
In order to reduce the calculation time, we have fitted the model to a sample of 10 points of the observed FIR continuum spectrum (see Table 4). The entire continuum distribution can be obtained by interpolation, using a spline function (see Fig. 4).
Table 4. A 10-point sample of the observed FIR continuum fluxes used for the fit and their corresponding calculated values (in 10-15 W mm-1)
Since the ortho states do not mix with the para states (Herzberg 1962, Townes & Schawlow 1975), we have distinguished two systems whose transition probabilities, collisional rates, and statistical weights 3(2J+1) for an ortho and 2J+1 for a para level J are different. We assumed an initial ortho-to-para H2O abundance ratio of 3, which is expected to be equal to the high-temperature limit for H2O molecules newly formed on a grain surface (Tielens & Allamandola 1987). The gas velocity, , was obtained by solving the equation of motion for the ejected gas (GS). The original expression for the drift velocity, , of the moving grains relative to the gas particles is not valid close to the stellar atmosphere (GS). We used a modified expression for (Truong-Bach et al. 1991) based on the equation of motion of the grains (Kwok 1975). The molecular densities of H2O , OH, and O were derived from the exothermic chemical equilibrium reactions
Photodissociation by the interstellar UV field operates in the outer envelope according to
Here the unshielded photodissociation rates were taken to be (Roberge et al. 1991)
H2O self-shielding is negligible. Dust shielding against the UV interstellar field is represented by a dust opacity at 1000 Å (Morris & Jura 1983). Because the local line width due to turbulence and thermal broadening ( 0.3-0.7 ) is much lower than the terminal velocity (12 ), one can expect that photons which escape from a local volume are not reabsorbed, and use the escape probability formalism (Castor 1970) to perform the statistical equilibrium calculations. The deduced level populations were used to determine molecular line opacities, excitation temperatures, and fluxes. We have included heating and cooling and photodissociation by FUV interstellar field (e.g. Truong-Bach et al. 1990, 1991) together with chemical reactions (GS) in our calculations.
Heating of gas is due to collisions with dust grains and also by exothermic (chemical and photodissociating) reactions. Cooling is due to: (i) the gas adiabatic expansion (GS); (ii) H2O spontaneous emission between rotational levels (of and ) whose fractional level populations are determined by solving the statistical equilibrium equations (see Truong-Bach et al. 1990); (iii) "2 decay from the first excited to the ground vibrational state (GS). We also estimated cooling by CO emission in a three-level scheme: two rotational levels in the ground vibrational state and one in the excited vibrational state (e.g. GS; Justtanont et al. 1994). With a CO/"2 abundance of 4 at the inner radius (Mamon et al. 1988), we found that CO cooling operates farther out than the H2O excitation region. H2O molecules formed in the vibrationally-excited state (and the reverse, dissociation, reaction) were found to have a negligible effect on the energy balance of the envelope, in agreement with the results of Justtanont et al. (1994).
The net energy variation is related to the temperature gradient throughout the shell. Thus, when the best fit to the observations was achieved, the molecular abundances and densities, the heating and cooling rates, and the kinetic temperature, , were simultaneously derived.
© European Southern Observatory (ESO) 1999
Online publication: April 28, 1999