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Astron. Astrophys. 318, 535-542 (1997)

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3. Analysis

3.1. Wavelengths

The wavelengths of the envelope emission, as observed in the subtracted spectra, have been measured and agree nicely with the observed CO emission velocity. The systemic velocities relative to the local standard of rest, V (K I), as estimated from the centre of gravity of the K I emission, and the expansion velocities, [FORMULA] (K I), as judged from the width of the emission corrected for the finite resolution of the spectrograph, are presented in Table 1. For comparison the corresponding quantities from the CO observations by Olofsson et al. (1993) are also given. The overall agreement with the measured CO velocities is gratifying.


Table 1. Data for the three stars with detected K I envelopes

Another interesting fact is that for two of the stars, V Aql and X TrA, the wavelength of the deep centre of the absorption line in the on-star spectra agrees with the wavelength of the top of the emission feature in the subtracted spectra minus the half-width of the emission line. This suggests that the absorption is formed in the circumstellar gas expanding in the line of the sight from the star. (Some interference - however, not of great significance for conclusions given in this paper - may also be due to interstellar absorption. E.g., for V Aql, located close to the plane of the Galaxy, interstellar components can be easily traced in the sodium D lines.) For R Scl the centre of the absorption line is shifted a few km/s to the red, suggesting a somewhat smaller expansion velocity in this direction or, possibly, interference from interstellar absorption for this star which is at the largest distance of the programme stars. It is also possible that the CO envelope of this star, which is resolved in the CO(3-2) line observations as a "detached shell source" (Olofsson et al. 1996) is physically distinct from and expands more rapidly than the K I line forming region. Also, note that Olofsson et al. (1996) find a smaller velocity (of [FORMULA]) for the inner regions of the CO envelope.

3.2. Line intensities and mass-loss rates

We have integrated the K I profiles of the three stars for all the subtracted spectra and thus deduced the flux in the emission line as a function of angular distance from the star. The strength of the emission as a function of position angle is different in different directions, an effect which is seen in Fig. 7. However, for the programme stars we have not found any asymmetries as strong as the bipolar jet around V Hya (cf. Plez & Lambert 1994). The dependence of the emission on position angle has not been investigated systematically yet, and one should therefore not take the results displayed in Fig. 7 as more than indicative. A rapid decrease of the emission with the distance from the star seems, however, to be characteristic for all the three envelopes observed. A typical upper limit of the envelope emission from the 14 stars where no detection was made is 10-14 ergs s-1 cm-2 arcseconds-2.

[FIGURE] Fig. 7a and b. The wavelength-integrated circumstellar K I emission [FORMULA] as a function of the angular distance [FORMULA] to the stars. Top panel: the observed values at different slit orientations for the three stars. Bottom panel: [FORMULA] fits are shown together with the mean of the observed values.

We shall now construct a simple spherically symmetric model for the envelope, assumed to be optically thin. Somewhat similar models have been constructed earlier, see, e.g. Bernat (1976), Mauron & Caux (1992) - we give however the expressions needed below for further use. It should be stressed, that a more satisfying treatment of the radiative transfer in spherically symmetric expanding envelopes is needed for more detailed and definitive conclusions. This will be the subject of a forthcoming study. One may easily derive the following expression for the ratio between the wavelength integrated flux [FORMULA] received in a scattered line on Earth from a column with a cross-section area [FORMULA] in the envelope at a minimum distance ("impact parameter") p from the star, relative to the observed stellar flux [FORMULA] (averaged across the line width in the laboratory frame, [FORMULA]):


Here, [FORMULA] is the cross section of the scattering process and [FORMULA] is the number of scattering particles per cm3 at a distance [FORMULA] from the star. For a constant mass loss rate [FORMULA], with a constant velocity [FORMULA] one finds


where [FORMULA] is the number of scatterers per gram matter. Assuming [FORMULA] to be independent of r, performing the integral and measuring [FORMULA] - the wavelength integrated intensity in the scattered line in units of erg s-1 cm-2 arcseconds-2 - with [FORMULA] being the distance in seconds of arc from the star, one finds


Here, d is the distance to the star from the Earth. For atomic scattering, one finds


and [FORMULA], leading to


Here, [FORMULA] is the abundance of neutral potassium relative to hydrogen, assumed to stay constant through the envelope. e and [FORMULA] are the electron charge and mass, respectively, [FORMULA] the hydrogen atomic mass, [FORMULA] the wavelength, f the oscillator strength of the line, µ the mass density of matter relative to the density of hydrogen. As a test of this relation we have plotted the measured flux ratios in Fig. 7 for the three envelopes as a function of the angular distance to the central star. In this figure the means of the fluxes in two different slit directions are plotted. Fig. 7 suggests that there are differences in intensity between different slit directions on the sky, but in view of the uncertainties in the measurements and calibrations we still consider these indications tentative. Obviously the [FORMULA] dependence gives a reasonably good description of the observations, although there are indications that the intensity increase falls below this relation in the inner shell region. Eq. (3) offers a possibility to make independent estimates of the mass loss rates. Adopting a value of [FORMULA] of [FORMULA], which is the solar abundance of potassium (Anders & Grevesse, 1989 - which is not inconsistent with the meagre knowledge available on metal abundances for these stars, cf. Lambert et al. 1986 - we tentatively assume all potassium to be in neutral atomic form) and with relevant numbers one obtains


where [FORMULA] is given in [FORMULA], d in parsec, [FORMULA] in seconds of arc, [FORMULA] in ergs s-1 cm-2 arcseconds-2 and [FORMULA] in ergs s-1 cm-2 Å-1.

In using Eq. (6) for estimating mass-loss rates we have determined [FORMULA] relatively far out in the K I envelope, at [FORMULA] values given in Table 1. We have taken velocities from the CO observations (Olofsson et al. 1993, OEGC); however they are compatible with the optical observations, see above, and those could have been adopted as alternatives. The stellar distances were also taken from Olofsson et al. and the resulting mass loss rates are compared with their mass loss rates, estimated from CO emission, in Table 1.

A reasonably good agreement is obtained between the resulting mass loss rates and those based on CO observations, suggesting that the basic assumptions behind our model are essentially correct. We note in particular that severe ionization of potassium, a lower mass loss rate after the ejection of the (outer) CO envelope or an optically thick envelope in the K I lines should show up as systematically lower mass-loss estimates from the optical observations.

3.3. Are the envelopes optically thin?

As a more direct test of the assumption of the optical thinness of the envelope we estimate the optical depth of a column of the envelope at a minimum distance p from the centre. We find


where the integral is to be extended across a part of the envelope where the differential Doppler shift is small enough as compared with the line width, i.e.


Eqs. (7) and (8) then give for the "impact parameter" p at which [FORMULA],


with [FORMULA] in [FORMULA] and [FORMULA] in [FORMULA]. One should note that the line profile width, [FORMULA], does not enter here, since the range of the integral diminishes in proportion to the increase in the peak value of the line profile with decreasing [FORMULA].

The impact parameters [FORMULA] and the corresponding angular distances [FORMULA], where the optical depths along a column through the envelopes become 1, are listed in Table 1. Obviously, the assumption that the envelope is optically thin in the regions where the fluxes are measured for the mass estimate is verified, i.e., [FORMULA] is indeed smaller than [FORMULA] for two stars, R Scl and V Aql. For X TrA the case is marginal in the sense that [FORMULA]. It should be noted that this does not mean that the envelopes are optically thin radially into an angular distance of [FORMULA] ; instead the radial depths are one order of magnitude greater than 1, as a result of the lack of a velocity gradient along the radii. However, this does not invalidate our mass loss estimate according to Eq. (6) as long as [FORMULA], which we estimate from the measured on-star spectrum, can be taken to be representative for the stellar spectrum that reaches the envelope gas at a point at angular distance around [FORMULA].

3.4. Are the envelopes ionized?

We shall now address the question what ionization conditions are to be expected in the envelopes. The model calculations of Glassgold & Huggins (1986) for the circumstellar envelope of [FORMULA] Ori suggest fractions of neutral K I relative to K I +K II of the order of 1% in the envelope. This reflects the high photoionization rate due to chromospheric UV radiation from the star. In particular radiation in the wavelength interval from 170 nm to 250 nm is of significance. At longer wavelengths - approaching the ionization threshold at 285.5 nm - the photoionization cross section gets very small due to overlapping wave functions. The knowledge about the ultraviolet fluxes from the chromospheres of N-type stars is very restricted; the best studied case being TX Psc (Eriksson et al. 1984, Johnson et al. 1996) but also for this star nothing is known about the flux shortwards of 230 nm. Also, the circumstellar absorption of that flux may be expected to be severe; we note that the IUE spectrum of the hot central star in the Red Rectangle object suggests that a carbon-rich envelope may produce a very strong absorption shortwards of 145 nm (Sitko et al. 1981, Mauron & Caux 1992). Extrapolating the TX Psc flux to shorter wavlengths Mauron & Caux (1992) estimate that, at a distance of [FORMULA] for that star (corresponding to [FORMULA] cm) the stellar photoionization rate is of the same order of magnitude as the interstellar one which they estimate to be [FORMULA] [FORMULA] s-1. This interstellar flux is great enough to significantly reduce the K I density in our observed envelopes within their timescales. Since the stellar photoionization rate scales as [FORMULA], where r is the distance from the star, one would obtain as a rough estimate for the photoionization rate [FORMULA] due to the UV flux from a TX Psc-like chromosphere:


In order to explore the role of the recombinations, we have complicated our model by adopting the basics of the model developed for the envelope of TX Psc of Mauron & Caux (1992), with their extrapolated chromospheric TX Psc flux, the electron density [FORMULA] relative to hydrogen set to [FORMULA], and a characteristic envelope temperature of 30K, with a smooth temperature variation with radius (the precise form of which, however, is not very significant). The envelope emission is proportional to a quantity [FORMULA] (see Mauron & Caux 1992, Eqs. (2) and (A1)) which is somewhat dependent on the temperature distribution but in particular on the ratio of stellar to interstellar K I photoionization rates. From Eq. (2) of Mauron & Caux (1992) we derive, with the same symbols and units as in Eq. (6) above,


(The difference between this relation and Eq. (6) as regards the different dependence on [FORMULA] originates in the density dependence of the recombination rate.) Following Mauron and Caux in their choice of [FORMULA] =0.4 for TX Psc, we derive the mass loss rates listed in the last row of Table 1. It is obvious that these mass loss rates are significantly higher than those derived from Eq. (6), basically reflecting the effects of ionization by a factor of about 10. This, however, does not explain the approximate [FORMULA] dependence inside [FORMULA] in Table 1 since the increased ionization would be compensated for by higher estimates of [FORMULA] and envelope density, so that the transparency of the envelope would only increase marginally.

The mass loss rates derived from Eq. (11) are significantly larger than those derived from the CO observations. The latter may, however, be systematically underestimated as a result of, e.g., the neglect of the possible clumpiness of the shells (Olofsson et al. 1993, 1996; note that in the former paper the mass loss rate of R Scl is estimated to be [FORMULA] [FORMULA] using a simple formula, while in the latter paper, where the clumpiness is taken into account and the radiative transfer is treated in some detail, the result is [FORMULA] [FORMULA]. For this star, however, it may well be that the scattered light comes from the inner envelope, while the CO emission originates in an outer shell, with higher expansion velocity, as the radial velocity difference may suggest). Conversely, a clumpy structure will increase the recombination rates and thus diminish [FORMULA] (Eq. (11)) in proportion. Anyhow, it is not possible to discard the application of the Mauron & Caux (1992) model for these envelopes on the basis of the discrepancy in Table 1 between its last two rows. One should also note that the photoionisation cross-section of the K I ground state is less well known than one would desire. We have considered alternatives to the experimental data of Hudson & Carter (1965, 1967) that were used by Mauron & Caux (1992). The Hudson & Carter data is superseded by the experiments of Sandner et al. (1981) around the cross-section minimum at 270 nm but our calculations show that this has minimal impact on the rates. The computed values of Rahman-Attia et al. (1986) are significantly lower (by almost a factor of 2) than the Hudson and Carter cross sections in the most important wavelength region between 250 and 180 nm, but Rahman-Attia et al. consider the experimental data to be more reliable here.

Another conclusion from Table 1 is that the chromospheric UV fluxes from the programme stars should not be greater, and are probably smaller, than those adopted for TX Psc on the basis of extrapolations by Mauron & Caux (1992); see their Fig. 4. These, in turn, are one order of magnitude or more smaller than the fluxes observed for late type giants and supergiants. The discrepancy between real and assumed UV fluxes may well amount to a factor of 10 - the mass-loss rate scales as the square-root of the UV flux in Mauron & Caux Eq. (2), and a factor of [FORMULA] in mass loss is needed to make the two last rows in Table 1 compatible.

One may wonder whether the fact that no K I emission could be traced from the majority of our 17 programme stars indicates that they have stronger chromospheric UV fluxes. That conclusion may, however, not be drawn; typical upper limits for their wavelength integrated K I envelope fluxes are only marginally below those observed for the three stars with clear detections. Moreover, smaller, or greater (see below), mass loss rates might also weaken the brightness of the envelopes.

3.5. Are the envelopes clumpy?

It is noteworthy that the envelope K I emission of R Scl varies roughly as [FORMULA] further inside the estimated [FORMULA] point, i.e. inside the point where the envelope is expected to be optically thick, which is contrary to the expected flattening of the profile. One interesting way of explaining the [FORMULA] dependence inside [FORMULA] in Table 1 would be to invoke a clumpy circumstellar gas - after all, clumps were traced in the CO observations of the more extended and detached shells (Olofsson et al. 1992, 1996, Bergman et al. 1993) and it would be of great interest if one could argue that the present observations indicate the existence of clumps in these smaller envelopes, much closer to the stars. If these clumps are of suitable size, with a radius [FORMULA], one might think that each of them could be optically thick at this relatively close distance to the central star, and act as an individual scatterer - we disregard from destruction of photons by collisional deexcitation - while the clumps could be rare enough to leave free paths for light in between them all the way out into empty space. This would preserve the [FORMULA] dependence.

Another virtue of an inhomogeneous model is that because the recombination rate scales with the density it is considerably enhanced locally as compared with the homogeneous envelope model. Thus, the high mass loss estimates that result from efficient photoionization are moderated although the mass loss rate scales as the density in the clumps (cf. Eq. (13)).

It is easy to see that the condition for each clump to be optically thick is easily satisfied. The optical depth of a clump with density [FORMULA] is on the order of


We can now use Eq. (3) to derive a mass-loss rate for the clumpy medium, assuming all clumps to have the same radius and density. The cross section of each clump is [FORMULA] and the number of clumps per mass unit is [FORMULA], leading to


From Eqs (5), (12) and (13) one finds the following condition for the individual clumps to be optically thick


which is as expected.

The system should be optically thin, in the sense that the clouds do not overlap considerably along the line of sight, in order to preserve the [FORMULA] dependence of the scattered emission. One clump has a mass along a column of 1 cm2 cross section of typically [FORMULA] grams. This mass should be larger than the mean mass in a column across the range L (cf. Eq. (8)) which is on the order of [FORMULA]. Denoting the smallest angular distance from the star for which overlap does not occur by [FORMULA] and expressing p in [FORMULA] and d one then obtains


Eqs. (13) and (15) now lead to an expression which is independent of most parameters,


With the fluxes of Table 1 this leads to values of [FORMULA] ranging from [FORMULA] (V Aql) to [FORMULA] (R Scl). I.e., independently of most stellar parameters one can obtain the observed fluxes from a circumstellar medium arranged in clumps with so few clumps that they do not overlap on the sky for the outer parts of the observed envelopes. However, the [FORMULA] values obtained makes it less probable that the problem with the [FORMULA] dependence in the inner spherically symmetric envelope may be solved by adopting such a more inhomogeneous model.

Above, we have assumed all clumps to have the same radius which is certainly unrealistic. With an increasing clump size with radius (and time) which is physically most reasonable, one gets a slower decrease of the shell emission with radius. E.g., an expansion of the clump radius in proportion to the distance from the star would, for optically thick clumps, lead to a [FORMULA] dependence, which is far from the observed slope. This may suggest that inhomogeneities are less pronounced in these inner regions than in the outer detached CO shells.

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Online publication: July 8, 1998