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Astron. Astrophys. 363, 1065-1080 (2000)

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5. Discussion

5.1. Dynamic atmosphere scenario

To understand the behavior of the H2 in Miras it is necessary to start by briefly reviewing the scenario presented in previous papers to explain the features observed in the infrared spectrum (HHR; Fox et al. 1984). Velocity curves of the type measured for H2 and other infrared lines are indicative of an overall stellar pulsation with a shock leading the passage of each pulse through the stellar atmosphere. The pulsation cycle equals the length of the light curve, a period on the order 1 year for a typical Mira. The shock emerges in the infrared photosphere in the premaximum phases. As the shock propagates through the atmosphere the spectrum shows a progressive weakening of lines in the cool, inwardly moving gas. At some phase near maximum light the gas below the shock develops a sufficient column density to be detectable. These lines sample the hot, outwardly moving gas of the next pulse. Hence the observation of doubled spectral lines, with velocities both toward and away from the stellar center-of-mass velocity, exists only after the shock has propagated through a considerable fraction of the stellar atmosphere.

In addition to the above region, which contributes most of the infrared spectrum, Mira atmospheres have a complex, extended structure. This region of the atmosphere can be probed in a variety of strong, low excitation atomic and molecular lines (e.g. HHR). The infalling photospheric layers, seen in the infrared at maximum light, persist well into the next light cycle. At least two distinct circumstellar regions are also present. A pseudo-static, [FORMULA]1000 K molecular region (Ridgway & Friel 1981; Tsuji 19882 and an expanding, Texc [FORMULA] 100 K, classical circumstellar envelope.

5.2. H2 lines

We will focus in the discussion on the M and S type stars. Carbon stars are discussed in Johnson et al. (1983). The above description of the behavior of photospheric lines characterizes the general behavior seen in the H2 lines. The interesting feature of the H2 lines is how their behavior differs from that of other spectral lines. Comparison will be made to CO lines, which are arguably the best infrared spectroscopic probe of the atmosphere. The [FORMULA]v=3 transition has weak spectral lines so that only atmospheric regions near the continuum forming layers are probed. The high excitation lines of the stronger [FORMULA]v=2 transition probe similar atmospheric regions. The low excitation CO [FORMULA]v=2 lines cover extended regions of the atmosphere.

Four aspects of the behavior of the H2 lines need to be explored:

  • In the spectra of Miras, especially those with stronger H2, the H2 lines are notably asymmetric.

  • For Mira variables the H2 lines initially appear (at [FORMULA]0.0) at a velocity near that of CO [FORMULA]v=3 or Ti. They later appear shifted to negative velocities compared to the CO [FORMULA]v=3 and Ti lines. For small amplitude variable stars the H2 line core is velocity shifted relative to atomic lines in that spectral region.

  • For Mira variables H2 lines undergo much more extreme phase dependent intensity changes than lines of other molecules like CO or OH.

  • For Mira variables the H2 lines develop negatively shifted emission near maximum light.

A fraction of the asymmetry and velocity shift can be explained by the low molecular weight and small absorption coefficient of the H2 molecule. H2 weighs 1/14 of CO and 1/24 of Ti, small enough that thermal velocities can be several times larger than the typical cool star microturbulence of a few kilometers per second (Lambert et al. 1986). However, the stronger lines in Miras have FWHM of [FORMULA]20 km s-1, far in excess of the microturbulence, so the thermal broadening has a small rather than large impact on the total line width in these stars.

5.2.1. Line profile

The similar shapes of the low excitation CO and H2 profiles seem to imply that low excitation H2 lines, like low excitation CO lines, are formed over an extended atmospheric region (e.g. HHR). In the case of CO there is considerable evidence from velocities and excitation temperature that strong, low excitation CO lines are formed throughout the atmosphere. The evidence for H2 is more circumstantial. The H2 line profiles are those expected from a very strong line formed in a dynamic atmosphere. Importantly, however, the line nearly disappears at maximum light, suggesting that most of the contribution to the lines occurs in the dynamic part of the atmosphere rather than the stationary molecular layer.

As expected for a line formed high in the atmosphere, the velocity of the S(1) line relative to photospheric lines differs from star to star. In Table 4 and Table 5, the S(1) line velocity is for some stars equal to the photospheric velocity as measured from CN, Ti, or high excitation [FORMULA]v=2 CO. For other stars the S(1) velocity matches that of the CO[FORMULA]v=2 low excitation lines. In other cases it is in between these or even more displaced from the photospheric velocity than are the low excitation CO lines.

5.2.2. Resonant scattering and postshock recombination

Schmid-Burgk & Scholz (1975) and Schmid-Burgk et al. (1981) demonstrated that M giant and supergiant atmospheres are extended. In the case of pulsating late-type stars, Jones et al. (1981), Ukita (1982), Bowen (1988), and Bessell et al. (1989) have shown that the atmosphere becomes extended by an additional factor of at least 2. HHR, Hinkle & Barnes (1979a,b) and Bessell et al. (1989) have shown that a large atmospheric extent compared to the stellar radius appears necessary to explain the behavior of the infrared CO lines, infrared atomic lines, and H2O bands in Miras.

In an extended atmosphere, a non-negligible contribution could be made to the line profile from gas beyond the photospheric limb. The spectrum from this gas contributes emission to the line profile as continuum photons undergo resonant scattering and are redirected into the line of sight. The gas over the limb has a very small velocity component from stellar pulsation along the line of sight and emission appears near the center-of-mass velocity. The observed emission, strongest near maximum light, is asymmetric presumably because it covers and partly fills the absorption line. The emission appears near, or slightly negative of the center-of-mass velocity (-19 km s-1 for R And [Lo & Bechis 1977]; -8 km s-1 for [FORMULA] Cyg [Lo & Bechis 1977; Knapp et al. 1982]).

Fox et al. (1984) and Gillet (1988) have shown that the atomic hydrogen line emission, conspicuous in the near maximum light spectra, is the result of recombination behind the shock. Hinkle & Barnes (1979b) suggested that infrared atomic emission lines also could be explained by recombination behind the shock. However, in the case of H2, the very small oscillator strength and low density of the Mira atmosphere requires recombination over a path length that is too long for this scenario possibly to be correct. Furthermore rapid recombination of the H2 would not be expected in the hot, post shock gas, where excitation temperatures in excess of 3500 K have been measured for CO (HSH). The relative contributions to the atomic emission lines from post-shock recombination and resonant scattering in the extended cool infalling gas need to be reexamined.

The resonant scattering that we propose to explain the H2 emission is slightly different in physical origin. We propose that the H2 profiles are formed in a spherically extended "photosphere" where resonant scattering makes a contribution to the photospheric line profile. This contribution will have velocity near the center-of-mass velocity since the resonant scattering takes place in gas seen near the stellar limb.

The contribution to the H2 line by the classic expanding circumstellar shell generally should be negligible as a result of the low H2 oscillator strength and the relatively modest column density expected. For example, in the case of [FORMULA] [FORMULA] yr-1, a mass loss rate which probably exceeds that of any star discussed in this paper by at least an order of magnitude, Keady & Ridgway (1991) predict an S(1) line 10% deep for an IRC+10216 model circumstellar shell. The circumstellar H2 line depth will scale approximately as the mass loss rate since the lines are unsaturated.

5.3. Role in atmospheric structure

An estimate of the column density of H2 would be revealing of the role of H2 in the Mira atmospheres. However, there are obvious difficulties in converting the observed line profiles or equivalent widths into a column density. In the 2 µm region, lines with central depths of more than about 40% are saturated (Hinkle et al. 1976) while lines can be no stronger than about 10% to be on the linear portion of the curve of growth (Tsuji 1983). Table 4 reveals that in most cases the H2 lines are strongly saturated. Also as described above, the atmospheres of stars with H2 are dynamic and highly extended, requiring spherical models. The complex line profile and atmospheric geometry demand detailed modeling and this will be carried out in Paper II. Here we simply note that typical CO column densities of [FORMULA]1024 CO/cm-1 have been reported by HSH, so assuming solar C/H H2 column densities of [FORMULA]1027 would not be unexpected.

H2 Rayleigh scattering opacity increases as [FORMULA]. Using the H2 Rayleigh scattering cross section of Dalgarno & Williams (1962), optical depth greater than unity can be expected from the substantial H2 column density present. However, in the infrared this opacity will be negligible. It follows that due to the continuum optical depth the infrared line forming region is not easily observable in the visual. The low dissociation potential H2 molecule must be concentrated in the cooler regions of the atmosphere, so the H2 opacity conceals even more of the atmosphere than might otherwise be assumed. We note that a large H2 Rayleigh scattering contribution to the blue continuum is not limited to phases when the H2 column density is at a maximum. A column density of 2[FORMULA]1026 in the infrared corresponds to optical depth unity at 4000 Å. From Fig. 2, we can see that the H2 Rayleigh scattering opacity from infalling gas near maximum light could be significant in the blue and ultraviolet even when H2 lines are difficult to detect in the infrared. This is in agreement with the observation that the infalling outer layers of Miras are observed for a much longer time in the blue-ultraviolet than in the infrared (Hinkle & Barnes 1979b; Barbier et al. 1988).

5.4. Comparison with atmospheric models

The behavior of the H2 absorption observed in Fig. 2 and Fig. 3 clearly shows strong variations in intensity. These are probably due to the dissociation and recombination of molecular hydrogen during the Mira variation cycle. Other effects that could modify the line intensity can be excluded for the following reasons: We found no indication of unusual effects on the line profiles, especially the line cores, that might indicate significant filling of the absorption by emission. An excitation effect would modify the profile between the different H2 lines, but we observe the same phase dependent variation in all lines of H2. Finally, a variation of the continuum can be excluded as well by other spectral lines, e.g. Ti lines. This significantly constrains the physical conditions in the atmosphere, and the applicability of numerical models.

From Fig. 3, it is clear that the strong and variable photospheric H2 component arises in the post-shock gas. The delayed onset of H2 relative to Ti and CO is most likely due to the small H2 oscillator strength - the post shock gas column must increase to a much greater column density for detectable H2 absorption than for Ti and CO. However, there is no evidence for a significant temporal delay in formation of H2 relative to, for instance, CO. If the H2 formation were significantly delayed, the absorption core might be expected to appear red shifted relative to CO and Ti, since the photospheric gas decelerates throughout the cycle. The observed appearance of the H2 component with a relative blue shift is puzzling, until the extended geometry is considered. The observed line profile will include the sum of contributions from the absorption profile (observed against a limb-darkened source) and a most likely limb brightened distribution of emission. The emission will tend to shift the resultant absorption profile center to the blue. The apparent "phase shift" between H2 and Ti/CO may then be due to differences in detail in the distribution and proportion of absorption and emission components. Certainly an interpretation will require careful models of the line formation, or eventually spatial resolution of the Mira surface.

Can the rapid formation of H2 in the post-shock gas be understood? It has been suggested (Bowen 1988), based on the highly density dependent three-body association rates for the reaction H + H + H [FORMULA] H2 + H and densities predicted by pulsation models, that molecular H2 will not form in Mira atmospheres. This conclusion was based on models which predicted typical photospheric densities in the range 105 to 1010 H2 cm-3. The H2 formation rate at T=2500 K would then be in the range 10-17 to 10-12 sec-1 cm-3, and in the most favorable case the H2 formation time would be thousands of years. However, we can now approach the question empirically. The formation of H2 is observed to occur rapidly, in no more than 0.1 of the period, or about 40 days. At 2500 K, via three-body reaction, the required density of H is greater than or equal to 1012 cm-3. Is this reasonable? A typical H density of 1012 cm-3 and a scale height of 150 [FORMULA] would give an H2 column of about 5[FORMULA]1024 cm-2. This is near the smallest H2 column which we might expect to detect by absorption in the quadrupole lines. Hence any star in which we detect H2 will probably have a column density consistent with a high photospheric density, sufficient for rapid H2 formation. A corollary conclusion is that the Bowen models do not predict a photospheric density sufficiently high to describe the Mira stars observed in our program.

Latter & Black (1991) outline paths to form H2 in addition to three body reactions. Of particular relevance to low density regions of Mira atmospheres are processes requiring either H+ or H-. Since the shock both dissociates H2 and ionizes hydrogen (Fox & Wood 1985), ample H+ will be present. Alternately, in the 3000-4000 K post shock gas the supply of free electrons guarantees H- is present. In any case, the rapid reformation of H2 in the cooling post shock gas does not appear to be a theoretical embarrassment.

5.5. Outlook

Dynamical model atmospheres as they have been published by Höfner & Dorfi (1997) and Höfner et al. (1998) predict quite strong temporal variations of the H2 density. This is demonstrated in Fig. 9 where we plot the gas temperature and the partial pressure of H2 as a function of the radius. Three phases of an oxygen-rich non-grey dynamical atmosphere (Höfner 1999) calculated with a piston velocity of 2 km/s and a period of 525 d and the corresponding hydrostatic initial model ([FORMULA], [FORMULA]) are shown. No dust has been included for the oxygen-rich stars. Due to the different atmospheric extensions and the shocks moving through the atmosphere the H2 partial pressure changes significantly in a given radial layer. At certain phases it may be much higher or smaller than in the hydrostatic case. This will also give rise to variations of the H2 features as they have been observed in the Mira stars. Synthetic H2 spectra calculated from dynamical model atmospheres will be discussed in a following publication.

[FIGURE] Fig. 9. Gas temperature and partial pressure of H2 as a function of the radius for three phases (phases 0.34, 0.56 and 0.93 of the bolometric light curve) of a dynamical atmosphere calculated with a piston velocity of 2 km/s and a period of 525 d and for the hydrostatic initial model ([FORMULA], [FORMULA]).

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Online publication: December 5, 2000