4 Discussion

4.1 Correlations among spectral features

Kleinmann & Hall (1986) noted that the CO 2-0 and 3-1 bandhead intensities were highly correlated for the stars in their atlas. We find the same correlation in our data, as can be seen in Table 2. The 3-1 bandhead at our resolution is somewhat blended with the 2-0 R12 and R13 lines, which must affect the measured intensity of the 3-1 bandhead. The 2-0 and 3-1 bandheads have nearly equal absorption depths in all our stars, spanning a range from 27% to 93% of the continuum. The mean ratio of I(3-1 BH)/I(2-0 BH) for all stars in Table 2 is $0.95 \pm 0.08$, where I(F) is the normalized intensity of feature F from Table 2. There is no significant difference between stellar categories. Individual transitions in the v = 2-0 band and shortward of the 3-1 head also show strong correlations - e.g., the R14 and R30 line intensities have a mean ratio I(R30)/I(R14) = $1.04 \pm 0.06$, with no difference between categories. (Note that the R30 line coincides with the R71 line.) The intensity of the 2-0 R0 feature is less well-correlated with that of R14, but the R0 line is blended with the 3-1 R17 line and the ¹³CO 2-0 bandhead, so a larger scatter is not surprising.

Figure 27: v = 2-0 R14 intensity vs. 2-0 bandhead intensity. Symbols denote red supergiants, M giants, S stars, carbon stars, and RV Tauri variables as indicated. Circles indicate Mira variables.

A more interesting relation is found in comparing the intensity of the 2-0 R14 line with the 2-0 bandhead, as shown in Fig. 27. There appears to be a kind of saturation effect in that as the 2-0 bandhead gets deeper ( $I(2-0~BH) < 40\%$), the R14 line approaches a limiting value of $\sim$60% of the continuum. The trend in Fig. 27 seems well-defined for the AGB stars, but the red supergiants generally lie below or close to the AGB stars. The 9 Mira variables in our sample are indicated by circles in Figs. 27-33. In Fig. 27, the Miras show a tight linear correlation, but the non-Mira AGB stars follow the same trend with no more scatter than the Miras. For the plotted quantities, then, there seems to be no difference between variability classes for the AGB stars.

Figure 28: Ratio of R14 absorption depth to (2-0) bandhead absorption depth, vs. (2-0) bandhead intensity. Symbols as in Fig. 27.

An alternative way to compare these features is to plot the ratio of the absorption depths vs. I(2-0 BH), where the absorption depth (in per cent) is defined as 100 - I(F). Figure 28 shows the ratio of the absorption depth of R14 over the 2-0 bandhead. For the stars with the deepest 2-0 bandheads ($I < 40\%$), the depths of the R14 line all cluster near 0.5 of the depth of the 2-0 head. The ratio appears to show an increasing scatter for shallower 2-0 bandheads (>40%). Taking the AGB stars and the red supergiants separately, however, Fig. 28 suggests that there are two different correlations present-one for the giants and another for the red supergiants. The Miras and non-Mira AGB stars in Fig. 28 are not distinguishable in the average trend nor the scatter about it. The two RV Tauri variables do not obviously fit with either trend.

4.2 Correlations with IR colors

The infrared color K-[12] should be a measure of the dust opacity of the circumstellar envelope for cool mass-losing stars. Whitelock et al. (1994) inferred a relation between $\dot M_{\rm dust}$ and K-[12] colors for AGB stars; Josselin et al. (2000) found a similar relation for M supergiants. LeBertre (1997) and LeBertre & Winters (1998) derived relations between the gas mass loss rates and IR colors for carbon stars and oxygen-rich (i.e., M-type) Miras, respectively. Their mass loss rates were determined from radiative transfer models for the 1-100 $\mu$m spectra, and assumptions about the dust properties.

Figure 29: Gas mass loss rate from CO mm-wavelength emission lines (see Table 1), vs. K-[12] color. Symbols as in Fig. 27. Dotted line shows mean relation found by Whitelock et al. (1994) for Miras in South Galactic Cap. Dashed line is for red supergiants assuming a gas-to-dust ratio of 200 and the $\dot M_{\rm dust}$ vs. K-[12] relation from Josselin et al. (2000). Labelled curves show relations derived by LeBertre & Winters (1998) for carbon-rich and oxygen-rich (M-type) Miras.

We have calculated the K-[12] colors for our sample using the IRAS PSC 12 $\mu$m fluxes, S₁₂, and [12] = -2.5 log( S₁₂/28.3 Jy) (see the IRAS Explanatory Supplement - Beichman et al. 1985). In Fig. 29, we show the gas mass loss rates for the 13 AGB stars and supergiants in our sample with published $\dot M_{\rm gas}$ values (see Table 1), plotted against K-[12] color. There is indeed a fairly good correlation as would be expected if the dust mass loss rates determine K-[12] and the gas-to-dust ratios are similar for these stars. The scatter at a given K-[12] color does suggest a factor $\sim$5 spread in $\dot M_{\rm gas}$, which is comparable to the spread in gas-to-dust ratios inferred for samples both of supergiants (Josselin et al. 2000) and of AGB stars (Whitelock et al. 1994). If we consider only the 7 Miras for which CO-derived gas mass loss rates are available (circled symbols in Fig. 29), the data points all lie on a straight line, but with a shallower slope than the relation of Whitelock et al. (1994). The trend for all the Miras in Fig. 29 is actually nearly parallel to the curve of LeBertre & Winters (1998) for carbon stars, even though 4 of the 7 points are M-giants and 2 are S stars. Since there is only 1 carbon star in this small sample, we cannot draw any conclusions about differences between M-giants and carbon stars in the $\dot M$ - (K-[12]) relation. Figure 29 does suggest, however, that the M-giants follow a trend which is significantly shallower than predicted by LeBertre & Winters (1998) for their O-rich dust models.

Figure 30: (2-0) bandhead intensity vs. K-[12] color. Symbols as in Fig. 27.

If the K-[12] colors are indicators of the gas mass loss rate, then the CO band features might be expected to correlate with the infrared colors as well. In Fig. 30, we plot the intensity of the (2-0) bandhead against K-[12] color for all the stars in our sample. Clearly there is no tight correlation between the two quantities. There is, however, a trend in the data that the observed range in (2-0) bandhead intensities depends on the K-[12] color. At the reddest colors, >3 mag, the (2-0) bandhead ranges from 30% to >80% of the continuum. For less reddened stars (K-[12] <3 mag), the observed range is reduced to 27% to 52%. Conversely, the deepest (2-0) bandhead values (<40% of the continuuum) correspond to the widest range in K-[12] colors, from 0.2 to 5 mag, while the shallowest (2-0) bandheads (>60%) have colors in the range 3-6 mag. If the K-[12] color is indeed an indicator of mass loss rate for AGB and RSG stars (but not the RV Tauri stars), as Fig. 29 suggests, then the distribution of points in Fig. 30 implies that stars with the higher mass loss rates exhibit a wider range of CO bandhead absorption than stars with lower mass loss rates. Lower rates correspond to the deepest CO absorption, with (2-0) bandheads 25%-50% of the continuum.

Figure 31: Ratio of R14 absorption depth to (2-0) bandhead absorption depth), vs. K-[12] color. Symbols as in Fig. 27.

A similar trend is found if we compare the ratio of the 2-0 R14 and 2-0 bandhead absorption depths as a function of K-[12] color. Figure 31 shows that for stars with K-[12] < 3 mag, the R14 line depth is between 0.43 and 0.57 of the 2-0 bandhead depth. For stars with K-[12] > 3 mag, the ratio spans 0.45 to 0.92, i.e., about a 3 times larger range. If we interpret the K-[12] colors in terms of mass loss rates, 3 mag corresponds to $\dot M_{\rm gas} \approx 5 \times 10^{-7}$ $M_{\odot}$ y^-1. The data imply that stars with mass loss rates lower than this value show a much narrower range in relative strengths of the CO features than do stars with $\dot M_{\rm gas} > 5 \times 10^{-7}$ $M_{\odot}$ y^-1.

Figure 32: Ratio of R14/(2-0) bandhead intensities vs. K-[12] color. Symbols as in Fig. 27.

An alternative comparison of the CO features with IR color is to plot the ratio of the intensity of the 2-0 R14 line to the intensity of the 2-0 bandhead, vs. K-[12], shown in Fig. 32. An important question in evaluating the spectra is whether the CO absorption features are being filled in by a dust continuum. If this effect were significant, the intensity ratios of features such as [2-0 R14/2-0 BH] should approach unity as the degree of "veiling'' increases. Figure 32 shows a marginal trend in this direction, in that for K-[12] > 3 mag, the mean ratio is lower than for stars with K-[12] < 3 mag. The correlation is weak, however, and at any given K-[12] color, there is a significant spread in the [R14/2-0 BH] intensity ratios. This spread in the ratios suggests that dust continuum is probably not a major factor in determining the relative intensities of the CO absorption features.

Figure 33 makes a direct comparison between the 2-0 bandhead and the CO-derived gas mass loss rates for the 13 stars with $\dot M_{\rm gas}$ in Table 1. The trends are very similar to those in Fig. 30, i.e., that stars with the deepest CO bandheads ($I < 40\%$) span a wide range in mass loss rates, while the shallower bandheads ($I > 60\%$) are limited to the highest mass loss rates. This similarity in the trends between Figs. 30 and 33 is to be expected, given the rather good correlation between K-[12] and $\dot M_{\rm gas}$ seen in Fig. 29.

The Mira variables in Figs. 30-33 (circled symbols) show the trend noted above. For the reddest stars (K-[12] > 3 mag), the ranges in CO line intensities and ratios are very large, while the less reddened Miras have more restricted values of CO line properties. Since the K-[12] color correlates well with $\dot M$ for the Miras (see Fig. 29), this result implies that Miras with $\dot M >$ (5-10) $\times 10^{-7}$ $M_{\odot}$ y^-1 exhibit a larger range in atmospheric structure than do the Miras with lower $\dot M$. This dichotomy may possibly be related to effects found by Winters et al. (2000) in their dust-driven hydrodynamic models. At high mass loss rates (> $3\times 10^{-7}~M_{\odot}$ y^-1), they find a large range in variation of the envelope structure with time - e.g., cycle-to-cycle and multiperiodic behavior. In contrast, for lower mass loss rates, the envelopes are nearly stationary with time, so that more uniform atmospheric absorption line properties would be expected. In this theoretical context, then, the large range of CO properties at high $\dot M$ or large K-[12] color (Figs. 30-33) is in fact a symptom of the large variation in atmospheric dynamical structure with time. This connection reinforces the case for the CO bands as diagnostics for hydrodynamic models.

Figure 33: (2-0) bandhead intensity vs. gas mass loss rate. Symbols as in Fig. 27.

4.3 Spectral variability

There is previous evidence for variability in CO and other molecular bands in the near IR spectra of cool giants which, it has been thought, were modulated with the phase of the light cycle, though the connection with phase is more an assumption than a demonstrated correlation. Frogel (1971) conducted an extensive study of 18 Mira variables, mostly of late-M spectral type. His data typically extend over one period of stellar variation. Changes in CO absorption strength were detected for most of the stars. The bands were systematically weaker at stellar temperature minima than at maxima, but there was a wide range of behavior relative to precise phase of the light cycle. Time variations in the CO 2-0 bandhead of the S star $\chi$ Cyg were reported by Wallace & Hinkle (1997), who found that the absorption depth varied over the light cycle from 0.49 to 0.66 of the continuum when observed at R = 2700. Aringer et al. (1999) found that for a sample of 8 Miras, the equivalent width of the SiO first overtone bandheads varied systematically with phase.

We plotted the CO 2-0 bandhead intensity versus phase for the 9 Miras in our sample with allegedly accurate periods, where the phase was determined from periods and reference epochs in the GCVS. There is no evidence of a correlation in our data. Three factors could mask any modulation of the CO features with the light cycle however. First, there may be a large intrinsic spread in the CO 2-0 bandhead depth from star to star, both in the mean and the extreme values over a cycle. Second, the phases we derived (see Table 1) may have some accumulated error over the time interval from the reference epoch, or the period may have changed or be modulated by a second period. Third, long-term cycle-to-cycle variations in the atmospheric structure may also modulate the CO absorption lines. Recent theoretical calculations find that the envelope structure for some models may change over a timescale longer than the stellar pulsation. Hofmann et al. (1998) see such cycle-to-cycle behavior in their pulsational models for M-type Miras. Winters et al. (2000) find that hydrodynamic models for the carbon star IRC+10216 show individual CO lines varying in shape and intensity over several light cycles, as a consequence of episodes of dust shell formation and ejection. These models suggest that there may be no simple correlation in spectral variations with the light cycle, at least for the Miras. Rather, longer term variations are to be expected, as the observations of IRC+10216 presented by Winters et al. seem to indicate.

Comparison of the spectra of the 4 supergiants in our sample which were also observed by Wallace & Hinkle (1997) shows no evidence for variability. The 2-0 and 3-1 bandheads are resolved in both data sets, and in all 4 stars, the depth and width of the bandheads are the same within the noise for each pair of spectra. The individual R-branch lines differ only to the degree expected due to the different reolutions of the two sets of spectra. The Wallace & Hinkle (1997) spectra were obtained in 1981 June ($\mu$ Cep and SU Per) and 1984 April (KY Cyg and PZ Cas), so on a timescale of more than a decade, we see no evidence for spectral variability in the CO bands for these red supergiants.

The two RV Tauri variables, which are believed to be in a post-AGB evolutionary stage, are more interesting candidates for spectral variability. Oudmaijer et al. (1995) note that 3 of 5 known post-AGB stars with 2.3 $\mu$m CO in emission show spectral variability. One of these is AC Her (= HD 170756). Oudmaijer et al. argue that the CO overtone bands are in emission at the optical (V-band) minimum, and go into absorption during the decline after secondary maximum. Their data clearly show dramatic changes in the CO first overtone spectra, but the sampling of the light cycle is too sparse and irregular to draw firm conclusions. In our spectrum of AC Her (Fig. 25), which was taken at about phase 0.3 near or just after the secondary minimum in the light cycle, the CO 2-1 bandhead depth is about 7% of the continuum (i.e., I(2-0 BH) = 0.93). This value is only half the absorption depth measured by Oudmaijer et al. (1995) for a spectrum taken at phase 0.90. Figure 25 also shows an indication of weak emission in the vicinity of the 2-0 R0 and P1 lines, at a few per cent of the continuum, but the bandheads are definitely in absorption. Our spectrum evidently shows the CO bands in a state intermediate between the deep CO absorption and emission reported by Oudmaijer et al. They argued that the variability was a result of episodic mass ejection during the pulsational cycle, with emission occurring just after ejection of gas while it is still close to the star in a dense warm state. Post-AGB mass loss is probably closely related to the formation and shaping of planetary nebulae (e.g., Kwok 1993), but the physical mechanisms are poorly understood. Well-sampled monitoring of the 2.3 $\mu$m CO bands over the light cycle of AC Her and related stars could be a very useful diagnostic for the mass loss. The relatively short periods ($\sim$ $70^{\rm d}$) of these stars would help make such a monitoring program practical.

4.4 CO isotopomers, atomic lines, and other molecular lines

The 2-0 bandhead of ¹³CO at 2.3448 $\mu$m is a relatively prominent feature in most of the spectra in Figs. 2-25. It is important to note, however, that at a resolution of R = 3500 the bandhead is affected by other lines close in wavelength. The 2-0 R0 and 3-1 R17 lines of the main CO isotopomer, which are almost coincident, lie within one resolution element of the ¹³CO 2-0 bandhead. The individual R-branch lines of the main isotopomer are typically quite strong, and these two clearly contribute to the absorption at the wavelength of the ¹³CO bandhead. Interpreting the ¹³CO 2-0 bandhead strength in terms of a ¹²CO/¹³CO abundance ratio must include a calculation of the main isotopomer R-branch lines near the bandhead.

In this connection, there is also a very close coincidence between the ¹³CO 2-0 bandhead and a line of Ti I at 2.34479 $\mu$m. This line is not separated from the ¹³CO bandhead even at very high resolution in the spectrum of Arcturus by Hinkle et al. (1995). Another Ti I line at 2.2970 $\mu$m is prominent on the side of the CO 2-0 bandhead in most of our spectra, typically absorbing $\sim$5% of the continuum at R = 3500. The strength of this line suggests that the Ti I line coincident with the ¹³CO 2-0 bandhead also contributes significantly to the observed absorption feature. As with the R-branch lines discussed above, the effect of Ti I absorption must be included in comparing the ¹³CO 2-0 bandhead with ¹²CO.

The spectra of our sample of carbon stars (Figs. 18-22) show remarkable variations in the depth and shapes of the CO 2-0 bandheads and main R-branch lines (compare for example HV Cas - Fig. 18 - with V460 Cyg - Fig. 22). The main features in all cases do appear to be the CO overtone band lines, but it is clear that the bands are modulated by other molecules with features in the observed range of wavelengths. It is well known that carbon stars have a rich organic chemistry in their atmospheres and circumstellar envelopes. A recent study of ISO-SWS spectra by Jørgensen et al. (2000) identified C₂, CN, CH, CS, HCN, C₃, and C₂H₂ in the spectra of V460 Cyg over the 2.4 to 45 $\mu$m range. The theoretical models of Helling et al. (2000) for carbon-rich AGB stars indicate that the main contributors to molecular opacity in the 2.3 $\mu$m region, besides CO, include C₂, CN, C₃, and HCN. The contribution of each to the absorption lines in our observed wavelength band depends on the temperature and density structure, and on the C/O ratio. Differences in these properties among our sample of carbon stars could easily explain the variations in the observed spectra. We have not attempted to identify other molecular lines in these spectra, and in fact most of the individual absorption features can be identified with CO lines. The differences in the detailed shapes of the R-branch lines in the 2-0 band as compared with the very regular structure in the M-giants or supergiants, however, indicates the presence of lines from other carbon-bearing molecules. Accurate models for the carbon star spectra in this wavelength region must include these species in addition to CO.