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Astron. Astrophys. 345, 925-935 (1999)

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4. Results and discussion

In Figs. 1 and 2 we plot the H2O model spectra superimposed on the observed spectra of observations I and II for comparison. The fit for the long-wavelength portion (i.e. detectors LW1-5) of the LWS spectrum (lower panel of Figs. 1 and 2) seems to be better than for the short-wavelength detectors (upper panels). This is, to a large extent, due to the high continuum/line flux ratio toward short wavelengths, causing the difficulty in determining the baseline among the continuum fluctuations. Fig. 4 represents the spectra with continuum: the rapid rise of the continuum toward 29 µm, the low line/continuum ratios, and blending are the main causes of uncertainty. At some wavelength positions, the forward and backward scans of SWS (Fig. 3, see Sect. 2) do not clearly show the same lines when the continuum fluctuations can be comparable to the line heights. Uncertainties in line fluxes, determined after continuum removal, are estimated to be better than [FORMULA] 40% for LW. They increase toward SW and can reach a factor 2-3 for SWS. The measured line fluxes are listed in Tables 1-3 (see Sect. 2).

Our best fit to the observed FIR continuum and H2O line fluxes (Observations I-III) was obtained with a mass loss rate of [FORMULA] [FORMULA] 3.4[FORMULA][FORMULA] [FORMULA]  [FORMULA] , and a H2O total initial relative abundance of [FORMULA]1.1[FORMULA]10-5 (7.9[FORMULA]10-6 for ortho - and 2.7[FORMULA]10-6 for para - H2O). The scaling factor for FIR excitation, [FORMULA] (Sect. 3), was found to be [FORMULA] 0.61 and 1 for Observations I and II-III, respectively. SW and SWS lines are excited closer to the star ([FORMULA] 1.4[FORMULA][FORMULA]  cm for line [FORMULA]m) than LW's ([FORMULA] 6.3[FORMULA][FORMULA]  cm for line [FORMULA]m). Increasing the mass loss rate by 20% would lead to an increase of the line fluxes of [FORMULA] 25% and 40% for the 174.62 µm and 45.11 µm lines, respectively. In an outward integration over a large range of radius, from [FORMULA] 1.0[FORMULA][FORMULA] cm to 2.5[FORMULA][FORMULA] cm, we determined the contributions to the line and continuum fluxes at each step. This allowed us to estimate an overall best fit for [FORMULA] =3.8[FORMULA][FORMULA] cm. A smaller value of [FORMULA] would overestimate the SWS+SW line fluxes, as reported in Tables 1 and 2. Here H2O line fluxes, [FORMULA] and [FORMULA], were calculated for 2 values of [FORMULA] : [FORMULA] 3.8[FORMULA][FORMULA] cm and 3.0[FORMULA][FORMULA] cm, respectively. In the following we adopt [FORMULA] = 3.8[FORMULA][FORMULA] cm. We note that most of the H2O line excitation arises at radii [FORMULA] 5[FORMULA][FORMULA] cm. Within observational uncertainty, an outer radius [FORMULA] chosen greater than 1[FORMULA][FORMULA] cm makes little difference to the line fluxes. The outer shell from r= 1[FORMULA][FORMULA] cm to 2.5[FORMULA][FORMULA] cm contributes [FORMULA] 16% to the FIR continuum at 29 µm, i.e. , within the observation uncertainty. The calculated intensities for H2O vibrationally-excited lines of [FORMULA] were found to be weak compared to those in the ground vibrational state, in agreement with the D-NQR model. In order to see the influence of collisional excitation from the ground to the vibrational levels, we multiplied the assumed vibrational collisional rates by a factor of 10. However the calculated H2O line fluxes in the fundamental state are not appreciably increased. On the other hand, when we decreased the FIR radiation field by a factor of 0.5, we have found that the H2O line fluxes decrease by e.g., 35% and 46% for the [FORMULA] 174.62 µm and 99.49 µm lines. If we take out the vibrational levels, the line fluxes would decrease by [FORMULA] 61% and 57% for [FORMULA] 174.62 µm and 45.11 µm, respectively. From this new level configuration, if we vary the collisional rates by a factor of 2 these line fluxes would increase by [FORMULA] 6%. These tests indicate that FIR radiative effects are more important than collisions in the excitation of H2O . This finding opposes that of D-NQR and Barlow et al. (1996) which found that the FIR H2O lines are mainly excited by collisions. However it is not easy to compare these models with the present because they are fundamentally different and they did not constrain FIR continuum fluxes.

Fig. 6 gives the averaged optical depth of dust grains at 80 µm in the outflow. It decreases from [FORMULA] 0.1 to [FORMULA] 0.035 as r increases from [FORMULA] to [FORMULA] 1[FORMULA] cm.

[FIGURE] Fig. 5. The FIR fluxes (in W m-2µm-1) at 29, 40, 50, 62, 72, 86, 108, 130, 155 and 190 µm as functions of radius (from top to bottom, respectively)

[FIGURE] Fig. 6. The averaged optical depth of dust grains at 80 µm from r to [FORMULA]

In Fig. 7, we plot the expansion velocity, [FORMULA], of the circumstellar gas, the drift velocity, [FORMULA], of dust grains relative to the gas, and the logarithmic velocity gradient, [FORMULA], as functions of radius r. Here we have set [FORMULA] for [FORMULA] lower than this value as discussed by D-NQR for the large velocity gradient approximation, but numerical tests show that the modelling results are insensitive to the adopted limiting value of [FORMULA].

[FIGURE] Fig. 7. (Top ): the logarithmic velocity derivative [FORMULA] (in cgs units). (Bottom ): the gas expansion velocity [FORMULA] and the drift velocity of dust grains [FORMULA] as functions of r

The H2O line fluxes were calculated neglecting any possible non-local self-absorption effect (Nguyen-Q-Rieu et al. 1984, Morris et al. 1985). This is a good approximation since the change to the (optically thin/thick) line profile due to that effect is small, as it would enhance the redshifted wing while depressing the blueshifted one (Morris et al. 1985, Truong-Bach et al. 1991). Another characteristic of the self-absorbed line is the appearance of a weak shelf on the blue wing when self-absorption does not reduce the intensity entirely to zero. The width of this shelf increases with the line opacity (Morris et al. 1985). The low spectral resolutions of our observations do not allow us to discern the line profiles. It would be of interest to make higher-resolution observations of the IR line profiles. The optical depths at line centre (radial velocity [FORMULA], Castor 1970) and along the line of sight at projected distance in the plane of the sky p [FORMULA] 1[FORMULA][FORMULA] cm are [FORMULA] 3.56, 1.40, and 0.15 for the 174.62 µm, 99.49 µm (505-411), and 45.11 µm lines, respectively.

In Fig. 8, we plot the heating and cooling rates. The momentum transfer efficiency used is Q [FORMULA] 0.073. We recall that Q is linked to the grain characteristic factor [FORMULA] in the dust absorption coefficient. A lower (or higher) value of Q means a higher (or lower) value of [FORMULA] for the fit. Grain-gas collisions dominate the heating effects. Cooling by gas adiabatic expansion is important throughout the envelope. Cooling by "2 [FORMULA]1-0 photons is important close to the central star. CO cooling is weak and operates outside of the H2O excitation region.

[FIGURE] Fig. 8. The heating and cooling rates scaled by ([FORMULA] cm)4 (in erg cm-3 s-1). a , grain-gas heating. Cooling due to: b , adiabatic expansion of gas; c , "2 vibrational excitation; d , H2O rotational excitation. Solid , ortho; dashed , para H2O

Fig. 9 shows the radial abundances of H2O in chemical equilibrium with O and OH. The ortho -to-para abundance ratio which starts from the value [FORMULA] 3 in the centre remains unchanged through the entire shell. Fig. 10 represents the densities of "2, H2O , OH, and O.

[FIGURE] Fig. 9. The radial abundances of H2O , O, and OH as functions of r for Observations II. Solid , ortho; dotted , para H2O

[FIGURE] Fig. 10. The radial density distributions of H2O , O, and OH for Observations II. Solid , ortho; dotted , para H2O

Fig. 11 shows the kinetic temperature distribution, [FORMULA] , in the envelope. It decreases outward from [FORMULA] 1300 K. At radius [FORMULA] [FORMULA] cm, the kinetic temperature is still [FORMULA] 950 K. At r [FORMULA] 5[FORMULA][FORMULA] cm, it becomes [FORMULA] 380 K. From here the excitation becomes too low to contribute significantly to the observed line fluxes.

[FIGURE] Fig. 11. The radial kinetic temperature distribution [FORMULA]

In Fig. 12 we plot the radial abundances of H2O (ortho ), O, and OH for two cases: with and without photodissociation. The weak dust shielding would allow interstellar FUV to penetrate deeply into the inner shell, since photoproduction of O and OH begins at [FORMULA]7[FORMULA][FORMULA] cm. However their abundance variations are weak compared to the H2O abundance itself. The figure shows that the H2O abundance is perceptibly decreased by photodissociation from [FORMULA] 2.2[FORMULA][FORMULA] cm. Here H2O excitation becomes low and the outer layers contribute typically [FORMULA] 14-20% to LW ([FORMULA] 174.62 µm and 108.07 µm), but insignificantly to SWS ([FORMULA] 45.11 µm and 29.87µm) line fluxes.

[FIGURE] Fig. 12. The radial abundances of H2O (ortho ), O, and OH. Solid , with UV photodissociation; dotted , without photodissociation

Uncertainty of the derived mass loss and H2O abundance due to uncertain thermodynamics can be estimated. We recall that the kinetic temperature depends on the drift velocity which varies as [FORMULA]. If we varied this quantity, for example by decreasing Q by [FORMULA] = 50% Q, the calculated line fluxes would noticeably decrease by, e.g., 80%, 92%, and 98% for the LW 174.62 µm, SW 82.03 µm, and SWS 34.549 µm lines, respectively. For a moderate decrease of Q, [FORMULA] = 30%Q, the latter line intensities would decrease only by 30%, 31%, and 50%, respectively, i.e., by about their observed uncertainties. We tried to compensate this line flux diminution by increasing the mass-loss rate by [FORMULA] = 25% [FORMULA], and the intensity deviations were reduced to 0.6%, 8%, and 19%, respectively. This means that the new fit obtained is still good for LW but worse for shorter wavelengths. Therefore the upper limit of uncertainty for the mass loss can be estimated as 25%. Accordingly, the uncertainties for the H2O abundance and the kinetic temperature at the inner radius were derived to be 47% and 33%, respectively. Naturally the final uncertainty should include that from the Hipparcos distance determination (van Leuuwen et al. 1997).

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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