synthetic line profiles · 2007-05-09 · chapter 3 synthetic line profiles 3.1 modelling line...

Chapter 3

Synthetic line profiles

3.1 Modelling line profile variations – the history so far

3.1.1 General remarks

In Sect. 1.3.4 several studies were mentioned, which utilised observational spectroscopic results (assummarised in Sect. 1.3) to infer a stratigraphy of various layers and their movements. By thismethod, one is able to construct empiric velocity structures to interpret the RVs derived fromspectral features originating from different depths. The next step for a deeper understanding ofatmospheric kinematics in LPVs is to attack the problem by numerical modelling.

Historically, approaches with increasing levels of sophistication were pursued:

• The first studies interpreted observed radial velocities of absorption and emission features bya comparison with velocity structures of models.

Wood (1979) found the postshock velocities of his isothermal models (his Fig. 8) to agreereasonably well with velocities of hydrogen emission lines observed in Mira variables (hisFig. 3). Wood also compared RVs of NIR absorption lines (CO, OH) in spectra of R Leo tovelocities at a nominal photosphere of the models (his Fig. 11).

Hill & Willson (1979) created a grid of hydrodynamic atmospheric models and derived ageneralised analytical model for LPV atmospheres which are dominated by pulsation. Theycompared the theoretical velocity structures – including the propagation of shock waves –to observed RVs of R Leo and oCet. Willson et al. (1982) adopted the modelling resultsof Hill & Willson and applied them to observational results (RVs of various features) ofthree other LPVs (RT Cyg, Z Oph, S Car). They discuss the relation between the mode ofpulsation (first overtone vs. fundamental) and the occurence of shocks (one or two at the sameinstance of time). Radius estimates and the strength of hydrogen emission made them favourfundamental mode pulsation. Based on this assumption, they could identify four velocitycomponents in their spectra and explained this fact by the occurence of two shockfronts atthe same time. One large amplitude shock in the lower atmosphere which influences the IRand a second low amplitude shock in the upper layers of the atmosphere which influences thevisual. This means that the propagating shock wave persists more than one pulsation cycleand the two shocks can explain the observed difference in velocity amplitude between thevisual and IR observations. Note that the occurence of two shockfronts can also be reproducedby our dynamical models (see Figs. 1.11 or 2.4).

Bertschinger & Chevalier (1985) presented semi-analytical models for the periodic shockwaves in Mira atmospheres. Observational results for CO ∆v=3 lines (changes in RVs and

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82 Chapter 3. Synthetic line profiles

excitation temperatures) compare rather well to their model results enabling them to inter-pret the dynamic processes in AGB atmospheres.

Tuchman (1991) compared observational results on RVs derived from CO lines to velocityvariations of his non-linear simulations of pulsating Miras (cf. also Ya’Ari & Tuchman 1996)and came up with suggestions on the possible pulsation mode of these stars.

• More sophisticated models for the outer layers of LPVs, including also dust-driven winds,have been developed by different groups since the 1990s (cf. the overview in Sect. 2.1.1 andthe references given there).1 These models comprise more detailed input physics compared toones previously mentioned, leading to more realistic atmospheric structures. This allows thenext important step in the interpretation of observational data on atmospheric kinematics,namely to compare observed line profiles and derived RVs with synthetic spectra computedon the basis of these dynamical models (in the way as described at the beginning of Ch. 2).

Consistent calculations of dynamic model atmospheres and subsequent spectral synthesis helpto proceed further in our understanding of the dynamics in AGB atmospheres. As alreadymentioned in Sect. 1.4, modelling of line profiles is useful for two reasons. On the one hand,such computations may help to interpret the complex multi-component line profiles, causedby the non-monotonic velocity fields throughout LPV atmospheres. On the other hand, repro-ducing spectroscopic observations is a fundamental test for realistic models (line intensities toanalyse the general atmospheric structure; line shapes/shifts to study the velocity structure).

If the behaviour of different (molecular) features in synthetic spectra resembles observations,this would be an indicator of the quality of the models and a confirmation of the correctnessof our ideas about dynamic processes going on in the outer layers of Miras (characteristics ofpulsations; dust-driven winds as the drivers for mass loss, cf. Sect. 4.8.3 of GH04). One majorgoal for the modelling is to reproduce with one model simultaneously the observed behaviourof various types of lines (coming from regions with different velocities within the atmosphere;cf. Sect. 1.3) for several phases during the light cycle, and to compare synthetic RVs withobserved ones. In addition, this may help to constrain stellar parameters (Sect. 2.1.4.4).

As mentioned in Sect. 2.1.1, state-of-the-art dynamic model atmospheres for LPVs (e.g. theones by Hofner et al. 2003a used in this thesis; Sect. 2.1.2) are of reasonable quality andmake line profile modelling, as discussed above, possible. In contrast, only a few of thegroups working on dynamic models (cf. Tab. 1 of Woitke 2003) presented studies dealingwith high-resolution synthetic spectra. The respective synthetic line profiles variations are inqualitative agreement with observations (also pointed out by GH04 in their Sect. 4.8.3.2). Thefew attempts to model individual line profiles and their variations in the past are summarisedin the following Sects. 3.1.2–3.1.4.

3.1.2 Pulsating model atmospheres (Australia–Heidelberg models)

Four reports on modelling of line profiles of this working group can be found in the literature.Based on simple synthetic atmospheric structures of Bessell et al. (1989a), Bessell et al. (1988)presented NIR CO lines that show asymmetries and line doubling quite similar to the observations.Using the more advanced dynamic models of Bessell & Scholz (1989), Scholz (1992) showed howvelocities in Mira photospheres distort line profiles of different species and affect measurements ofequivalent widths and curves of growth for abundance analyses from these spectra. Starting withthe same models, Bessell et al. (1996) for the first time calculated synthetic line profiles (CO ∆v=2,Fe I) for different phases, which showed qualitative similarities with some observed features (line-shifts, doubling, velocity amplitudes). Scholz& Wood (2000) then presented similar calculations fordifferent lines (CO ∆v=2,3; OH ∆v=2) and derived conversion factors relating RVs derived fromDoppler-shifted lines to actual gas velocities above and below the shock wave running through theatmosphere.

1A common feature of all these dynamic models is that they are computed in spherical geometry; effects ofasymmetry therefore cannot be studied with these 1D-models.

3.1. Modelling line profile variations – the history so far 83

A unique feature of the dynamical atmospheric models by Bessell et al. (1996) and Hofmannet al. (1998) is that the effects of the pulsating stellar interior are introduced through self-excitedpulsation models. The dynamical models should therefore be suitable to represent the inner pul-sating layers of a Mira atmosphere. Bessell et al. (1996) fitted the temporal variations of sub-photospheric layers of the pulsation models with Fourier series and used these fits as inner bound-ary conditions for dynamical calculations of the outermost atmospheric layers with higher spatialresolution. Hofmann et al. (1998) directly used the structures from their new pulsation models butfollowed the same procedure as Bessell et al. for the older models. In both cases, the density andvelocity structures of the dynamical models were used directly for the calculation of observableproperties. Based on the densities, however, new values for the gas temperatures were derived usingradiative transfer in about 70 frequencies before computing the detailed spectra. This was neces-sary because the temperatures resulting from the grey dynamical calculations were unrealistic. Adrawback of this procedure is that it neglects the feedback of non-grey effects on the density anddynamics which may be crucial. Furthermore, different sources of opacity data were used in thedifferent steps which adds to possible inconsistencies. In contrast to that, the dynamical modelsused in this paper are based on a simultaneous solution of hydrodynamics and frequency-dependentradiative transfer leading to consistent dynamical density-temperature structures. It should alsobe pointed out that our models include a time-dependent description of dust condensation andallow for the formation of dust-driven stellar winds whereas the models of Bessell et al. (1996) andHofmann et al. (1998) are pure atmospheric models (no dust2 and therefore no mass loss). Also thelatter are focused on O-rich chemistry; no dynamic models for carbon stars have been computedby Bessell, Hofmann and collaborators so far.

3.1.3 Dust-driven wind models (Berlin models)

After some reports in conference proceedings (Gauger et al. 1996, Gauger et al. 1998, Winters et al.1998, Winters 1999), one paper on line profile modelling based on models of the Berlin group waspublished. Improving the semi–empirical approach of Keady et al. (1988) for spectral synthesis,Winters et al. (2000a) presented synthetic fundamental and first overtone CO line profiles andcompared them with observed ones for the obscured C-rich Mira IRC+10216. Calculating syntheticspectra for different phases of a set of dynamic models with different parameters, they demonstratedthe influence of mass loss rates (and hence dust optical depth) on the shape of the profiles forobservations at different pulsational periods. By suppressing CO absorption for different layers,Winters et al. investigated their contribution to the final profiles.

The models used by Winters et al. (2000a) are based on a detailed description of dust formationand stellar wind dynamics and are well suited for describing stars with heavy mass loss and spectralfeatures coming from the optically thick and dusty outflow. This allowed them to interpret variationsobserved in the first overtone lines of CO as the formation and dynamics of discrete dust shellspredicted by the models. On the other hand, these models contain only a crude description of thestellar atmosphere, using grey radiative transfer and no detailed 3 molecular opacities. This leadsto considerable differences (compared to a full treatment of opacities) in the density-temperaturestructures, in particular in deep photospheric layers and further out in the upper atmospheric layerswhere the dust formation starts. This may be one of the reasons behind the problems of fitting boththe global spectral energy distribution and the line profiles with one consistent model, as mentionedby Winters et al. (2000a). They rescaled the density of the wind to get line profiles comparablewith observations but used the original model to compute the spectral energy distribution. Due tothe uncertainties in mass loss rates derived from observations, no detailed fitting of the observedSED with the purpose of constraining the densities was performed. Compared to the models of

2Bedding et al. (2001) and Ireland & Scholz (2005) investigated possible effects of dust on observable propertiesof Miras. However, they started with the pulsating, dust-free models and introduced aspects of dust (formation,absorption) on top of these. The authors did not include dust in consistent hydrodynamic calculations.

3Most of the Berlin models use the constant value of κν=2·10−4 cm2 g−1 (as given by Bowen 1988) for the gasopacities. Helling et al. (2000) also investigated the influence of different opacity treatments (Rosseland or Planckmeans).


Winters et al., the dynamical models used here combine a more realistic non-grey description ofthe atmosphere with a similar treatment of dust formation and wind dynamics. This allows usto study also stars with moderate mass loss rates that are not completely obscured by dust, andlines which originate in very different geometrical depths in the atmosphere or wind. The modelsused by Winters et al. (2000a) are available only for C-rich stars, no models for M-type stars existdue to the problems for a numerical treatment of the formation and evolution of oxygen-rich dust(as discussed in Sect. 2.1.2). First results in this context will be discussed by Woitke (2006,2007)though.

3.1.4 Combined atmosphere and wind models (Vienna models)

First results of the Vienna group on modelling of line profiles were presented by Windsteig (1998),Windsteig et al. (1998a, 1998b, 1999), Lebzelter (1999b), and Aringer (2000, 2005). Based ondynamical models similar to the ones discussed in Sect. 2.1.2 but of previous generations and byapplying the same radiative transfer code used here (Sect. 2.2.4) they demonstrated the influenceof atmospheric velocities on line profiles of (mostly) CO/SiO features. In addition, they presentedcomparisons of observed FTS spectra with synthetic ones.

The high quality of the latest generation of the Vienna dynamic model atmospheres (seeSect. 2.1.2, DMA4) gave reason to a new attempt to investigate synthetic line profiles in greaterdetail. Starting with preliminary results presented in Nowotny et al. (2003b), we used selectedmodels to calculate line profile variations of representative molecular features and compared de-rived RVs for many phases covering the whole lightcycle with observational results. The results ofthis modelling are the subject of this thesis. The findings presented in Sects. 3.2/3.4/3.5.1/3.6 werealready published in Nowotny et al. (2005a, 2005b, 2005c, 2006), the ones of the Sects. 3.3/3.5.2will be the subject of a forthcoming paper (Nowotny et al., in prep.).

The discussion in Sects. 3.1.2+3.1.3 shows that existing dynamical models that were used tostudy line profile variations in AGB stars fall into two groups: pulsating atmospheric models or windmodels. The former deal with lines originating from the various layers of the pulsating atmosphere,while the lines studied with the latter models probe layers from the dust formation zone to theoutflow region (cf. Sect. 2.1.2). The purpose of the new models (presented in DMA3 and used here)is a consistent description of all these phenomena, i.e. simultaneous modelling of lines originatingin various layers (from the deep photosphere out to the dust formation region and beyond to thestellar wind region; cf. Fig 1.11), with one single dynamical model for a given star. To this end,the equations of hydrodynamics, frequency-dependent radiative transfer and time-dependent dustformation and evolution are solved simultaneously to get an adequate description of the highlydynamic AGB atmospheres. Important is the fact, that the formation of polyatomic molecules anddust grains happens within similar temperature regimes (≈1500K).4 It is therefore necessary thatthe occurence and the respective opacities of molecules (not included in the models described inSect. 3.1.3 but significant for the atmospheric structures in deep, dust-free layers) and dust (notincluded in the models described in Sect. 3.1.2 but relevant for the development of a wind and thestructures of the very extended outermost layers) are simultaneously treated in the computationsof the model atmosphere to get realistic atmospheric structures and NIR spectra. This results ina more realistic description of both the dust-free pulsating atmosphere and the dust-driven stellarwind compared to previous models. The Vienna models can be used to simulate pulsating, dust-free (no mass loss) atmospheres for both types of chemistries (O-, C-rich). On top of that, modelsincluding dust and therefore mass loss are available. While a consistent modelling of the wind ispossible in the C-rich case, the O-rich dust is included so far only in a simple parameterised form(cf. Sect. 2.1.2.1). The implementation of O-rich dust is planned for the near future (Hofner 2007).

4This is especially important for C-rich stars (Loidl et al. 1999, Fig.3). In contrast, the formation of dust in theO-rich case takes place at rather large distances from the star with low temperatures (≈800 K). One may assumethat the formation of polyatomic molecules (H2O) occurs already at larger temperatures (and therefore at largeratmospheric depths) and is rather completed in the dust-forming layers. Thus, modelling with limited consistency(combining molecular spectra based on a dust-free photospheric model with dust spectra resulting from a dustyenvelope on top) may rather be justified in this case.

3.2. Reproducing the global (velocity) structures of Miras 85

3.2 Reproducing the global (velocity) structures of Miras

The main goal for this thesis was to see if the Vienna models are able to reproduce the globalatmospheric structure of Miras, especially the velocities at different depths as it was derived fromspectroscopic observations (cf. Sect. 1.3). As the circumstellar dust and its influence on the atmo-spheric structure (as well as on the spectrum) is at the moment only consistently included in thecarbon-rich models (cf. Sect. 2.1.2), we started with C-type LPVs. In fact, there is only one objectwith a reasonable time series of high-resolution spectroscopy, namely the C-rich Mira S Cep. Thisstar was therefore chosen as the reference object for our line profile modelling, the correspondingobservational results were summarised in Sect. 1.3.5.3. A dynamic model atmosphere was calcu-lated which parameters (cf. Tab. 2.1) were chosen to be close to the stellar parameters derived fromobservations of S Cep (Tab. 1.3). The model was described in detail in Sect. 2.1.2.2 and is denotedby model S there. Atmospheric structures from three different, separated cycles of the dynamicalmodel – several snapshots from the temporal evolution – were used to synthesise high-resolutionspectra. This was done for all molecular lines itemised in Sect. 2.2.1. As stated in Sect. 1.4, threedifferent types of results can be analysed. Both, the temporal variations of synthetic line profiles(computed with spectral resolutions of 300.000 but also binned down to 70.000, which is compa-rable to observed high-resolution spectra) and the derived RVs were compared to observationalresults of S Cep as well as other Miras (observed FTS spectra, velocity data in the literature).In addition, the regions of line formation were compared (excitation temperatures derived fromobservation vs. gas temperatures in the model at τν≈1). In the following, we present the results(starting from the inner pulsating region and proceeding outwards to the wind region), which werealready published in Nowotny et al. (2005a, 2005b, 2005c).

3.2.1 Probing the pulsating layers

3.2.1.1 CO ∆v=3 lines

The second overtone CO lines were used most frequently to investigate deep photospheric layers inAGB stars and they show a very typical behaviour as already described in detail in Sect. 1.3.2.1.Although some observations of these lines were reported by Barnbaum & Hinkle (1995), identifica-tions are difficult in C star spectra due to contamination by other molecules (mainly CN, C2). Noextensive time series studies have been published yet, the FTS spectra for S Cep of HB96 were allof poor quality in the H-band. Nevertheless, we synthesised such line profiles, since the behaviourof CO ∆v=3 lines for C stars is supposed to be similar to M- and S-type Miras. We then comparedour results to observations of the latter. Especially, line profiles and radial velocities for differentphases of the S-type Mira χ Cyg (HHR82, HSH84), as shown in Fig. 1.13, are used for the com-parison here. In addition, the left panel of Fig. 3.20 shows a representative RV curve for a largesample of Miras (Lebzelter & Hinkle 2002b).

The CO 5–2 P30 line at 6033.8967cm−1 (1.6573µm) was chosen for modelling. CN and C2, whichare also prominent in this spectral region, were not taken into account. Using the OS data, wouldmean to overestimate the opacity by far (as described in Sect. 2.2.3). Including opacities from linelists (only CN implemented so far) of these lines could (due to uncertainties in positions) influencethe profiles by blending.5 This specific synthetic CO line may then not be directly comparable tothe corresponding observed (and by CN/C2 influenced) one. But since all CO ∆v=3 lines in thisregion have similar excitation potentials and gf values, the synthetic line should be comparable toan observed average line profile (where the various influences smooth out). The synthetic spectrahere are computed in a slightly different way than the preliminary results presented and analysedin Nowotny et al. (2005a). Dust opacities are from Rouleau & Martin (1991) instead of Maron(1990). The former contains more complete data over a larger spectral range and was also used

5Only from synthetic spectra of this spectral region, the CO 5–2 P30 line appears to be in a ”window” of theCN-line-forest. Some C2 contamination still left, this may allow a comparison with observed high-resolution spectrain the future.


for the modelling of the atmospheric structures (important for consistency reasons Aringer 2005).In addition, the pseudo-continuous opacity of C2H2 is also included here, leading to smaller linedepth due to a depressed ”continuum”.

The left panel of Fig. 3.1 shows time series of the synthetic second overtone CO line profiles fortwo different spectral resolutions. To get an idea of the typical line width, a spectrum computed forthe phase of light maximum and without taking velocities into account in the radiative transfer isplotted at the top. Due to the limited number of spectral points, the profile looks asymmetric andnot centered on the rest wavelength (RV =0) at lower resolution. In principle, the typical observedbehaviour of CO ∆v=3 lines in Miras (Fig. 1.13) can be recognised from the synthetic spectra.Some deviations from observations of Mira variables can be found though. Most noticeable is thefact that the transition from blue- to red-shift during phases φbol≈0.2–0.5 is only visible in thelower-resolution spectra. The line profiles look more complicated at higher spectral resolutions.There, the movement of the original component stops at RV ≈ 0 km s−1 where it disappears, whileanother red-shifted component develops from φbol≈0.2 on. On the other hand, line splitting is morepronounced at higher resolution (φbol=0.75 ). From the synthetic spectra it can be noticed thatthe (pseudo-) continuous opacity is strongest6 for φbol≈0.4, leading to a depressed ”continuum”and apparently weaker lines. As it would be expected for lines being formed in the very inner partsof the atmosphere with periodic movements (Fig. 2.1), line shapes duplicate for all three chosenperiods. Therefore, only profiles from the first one are shown.

The synthetic profiles shall be compared to the averaged line profiles of χ Cyg shown inFigs. 1.13+3.11. To match profiles of the same shape, a difference in phase of ≈0.25 between φv

of the χ Cyg observations and φbol of the model is needed. Line strengths are comparable and thesynthetic profiles reproduce the typical pattern reasonably well, at least for the lower spectral reso-lution. However, the profiles do show some substructures at the spectral resolution of 300 000. Theshifts in wavelength are too small compared with observations and the doubling is less pronounced(compare phases φv=0.01/φbol=0.75 ), but still clearly visible. This indicates that the difference invelocity behind and in front of the shock front is too low in the atmospheric model (see Sect. 3.5.2).A blue-shifted emission feature appearing before the line doubling phases (0.86/0.47 ) can also beseen in the modelled profiles.

The right panel of Fig 3.1 shows RVs derived from the synthetic CO lines for various phasesduring one pulsation period. The development of a few spectral components can be followed atthe higher resolution of 300 000. A weak component with RV ≈ 0 kms−1 seen at phases of 0.6–0.7may also (weakly) be recognised in the observed χ Cyg profiles (phase 0.90 in Fig. 1.13) as wellas in the composite RV-curve of Fig. 3.20. It is conspicuous that line splitting is not only foundaround luminosity maximum, but also (more moderately though) during phases ≈0.2–0.5, whichis not being observed. This splitting turns into a continuous transition from blue- to red-shiftsfor a lower resolution of 70 000, as the two components of the line profile melt into one broadfeature (phase 0.30 ). Generally, the observed discontinuous RV-curve (Fig. 1.13) is reproduced bythe low-resolution profiles. Only the RVs from the lower resolution were thus used in the followingcomparison (Figs. 3.9 and 3.10). The behaviour pattern is highly periodic (Fig. 3.10), as found inobservations and as also is expected from the model. Taking into account the above-mentionedshift of ≈ 0.25 between φv and φbol, the ”S-shape” is reproduced as well as the asymmetry7 w.r.t.RV =0km s−1. Line doubling appears during phases ≈0.7–0.85, a similar time interval as observed(∆φ≈ 0.2). Zero-crossing happens at φbol=0.3, while it is observed at φv=0.4. Although the am-plitude8 ∆RV ≈14 km s−1 is too low (typically 25 km s−1 in Mira observations), the characteristicbehaviour could be reproduced successfully by consistent calculations.

6Around minimum phase, the rate of dust production is highest and also the C2H2 features are strongest for adusty model like the one used here.

7RV-curves extend to more positive values and appear ”shifted” to RV =0km s−1. This is seen in all observationsof second overtone CO lines (HSH84, Fig. 3.20). As one would expect to see equal infall and outflow for radialpulsations, this asymmetry is interpreted as due to line components that do not come from the same depth duringthe lightcycle. Note that also in the structures of the model atmosphere (Fig. 2.4), infall velocities are larger thanoutflow velocities in the pulsating layers.

8maximum difference in RV of the two components during phases of line doubling


A more detailed analysis shows that the synthetic CO ∆v=3 lines emerge from atmosphericdepths of R=0.8–1.3R∗ with gas temperatures of ≈2200–3500K. This seems consistent with theobservational results of HHR82 or HSH84 (compare also Fig. 1.13), listing excitation temperaturesof ≈2000–4500K.

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Figure 3.1: Left: Time series of synthetic second overtone CO line profiles (5–2 P30) during onelightcycle with different spectral resolutions (based on model S ). To be compared with observedprofiles of CO ∆v=3 lines as, e.g., given in Fig. 1.13. Right: Radial velocities for φbol=0.0–1.0 asderived from the synthetic profiles, plotted repeatedly for spectral resolutions of 300 000 (+) and70 000 (squares) respectively.

3.2.1.2 CN lines

As discussed in Sect. 1.3.5.2, CN lines are also suited for investigating kinematics in pulsatinglayers. The most thorough study of S Cep by HB96 (Sect. 1.3.5.3) used lines around 2.14µm, whichare electronic transitions of the red system of CN with ∆v=–2. Figure 1.21 shows a sequence oftheir FTS spectra for selected phases with the relatively unblended 1–3 P238.5 line (most otherregions of the K-band are much more crowded). It reveals that this line shows a similar behaviouras CO ∆v=3 lines resulting in a discontinuous RV curve as shown in Fig. 1.22. For most of the CNlines there, this cannot be recognised that clearly due to heavy blending.

For the spectral synthesis it turned out that the pseudo-continuous opacity of C2H2 in thisregion is very strong compared to the virtual line. Therefore a similar (gf , Eexc) one, the 1–3Q24.5 line at 4871.340cm−1 (2.0528µm) was chosen for modelling. The influence of C2H2 is ratherweak there, but observations are difficult due to telluric contamination. No molecules other than


C2H2 are relevant for the overall continuous opacity in this region. As in the case of CO, thebehaviour of this CN line should be similar to those typically used in observational studies.

Figure 3.2: Left: Observed CN ∆v=–2 lines profiles (1–3 P238.5) in the FTS spectra of S Cep byHB96 for several phases (cf. Fig. 1.21). Observed heliocentric velocities were converted to systemicvelocities RV ∗ of the star by assuming a CMRV heliocent of –26.7 km·s−1 (HB96). Middle, right:Temporal evolution of synthetic profiles (based on model S ) of similar CN lines (1–3 Q24.5) duringa pulsation period at higher and lower resolution.

The two right panels of Fig. 3.2 show the results of the spectral synthesis for several phasesduring the lightcycle. As the line profiles duplicate also from one period to the next, spectra areagain shown only for one lightcycle (φbol≈0.0–1.0 ). Observed profile variations can qualitativelybe reproduced by the calculations. The pattern with blue-/red-shifts and line doubling aroundlight maximum is discernible. However, the line shapes appear somewhat complex. The splittingis even less pronounced than for the CO ∆v=3 lines; the two components are visible but theymerge. The strength of the red component does not decrease continuously. Several components arevisible at higher resolution (e.g. φbol=0.80 ), measuring RVs can be difficult (e.g. 0.1–0.25 ). Again,the picture becomes clearer by rebinning the synthetic spectra down to resolutions comparable toobserved FTS spectra (70 000). The line profiles are smoothed and especially the transition fromblue- to red-shift during phases of ≈0.2–0.8 appears more distinct. While for example at phase0.80 the two visible red-shifted components merge into one broad feature, it becomes difficult tomeasure the weak blue-shifted component. The (pseudo-) continuous opacity (C2H2, dust) is againstrongest for φbol≈0.4.

Figure 3.2 also allows a comparison between observed and synthetic spectral lines. Althoughnot exactly the same line – observed CN 1–3 P238.5 vs. synthetic CN 1–3 Q24.5 – the behaviouris expected to be very similar (Sect. 1.3.5.2). The plotted profiles were chosen to show maximumcongruence, which occurs not necessarily at the same phase/phase-shift. The change with phase


(red/blue-shifts, line doubling) can in principle be recognised from the synthetic profiles. While theones at higher-resolution show more fine structures than known from observations, complexity isreduced by rebinning the spectra to lower resolutions and they become more similar to the observedones. Line strengths are comparable, but again the shifts in wavelength are too small. The splittingof the two components around maximum phases (φv=0.00/φbol=0.91 ) is much too weak for thesynthetic profiles. They merge and form one broad feature, while observations show two distinctabsorption lines. This is even more pronounced here than for CO ∆v=3 lines in Sect. 3.2.1.1.

Figure 3.3 shows RVs derived from synthetic CN lines during the first pulsation period examined.A plot of values measured from single profiles at highest spectral resolution looks even more complexthan for CO lines due to the multi-component line profiles. Some components obviously present asa distortion of the line profile do not give a clearly measurable velocity. Opposite to the expectation(motivated by observed and R=70 000 synthetic spectra) of one component (appearing at blue-shifts, moving toward red-shifts and disappearing again), the development of several components,which appear to be independent of each other, can be followed. But profiles rebinned to a lowerresolution of 70 000 lead to RV-curves (•) that are in notable qualitative agreement with S Cepobservations (Fig. 1.22). Even the discontinuity in the blue-shifted component at φbol=0.15 issmoothed out.

Figure 3.3: Radial velocities forφbol=0.0–1.0 as derived from syntheticprofiles of CN ∆v=–2 lines as shownin Fig. 3.2 Measured components ofindividual lines in spectra with reso-lutions of 300 000 (+) and 70 000 (•)are plotted as well as results of thecross-correlation (♦).

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Another way to extract velocity curves is not to calculate one single line profile but rather awhole region occupied by CN lines and derive RVs by cross-correlation. This method is not able toresolve very weak components recognisable in single profiles as blended and unblended lines mix inthe wavelength range used. HB96 used this technique because of the crowded CN spectra. We havetried to simulate this approach with synthetic spectra. Figure 3.4 shows the spectral region used,occupied by many CN ∆v=–2 lines of the red system. It was chosen to have not too strong blending,but rather several strong individual lines. The positions of CN lines in the line list were not correctedfor wavelength shifts between theoretical and true values; a direct comparison with any observedspectra may therefore not be straightforward. Relative velocities should still be correct. Also forthis region spectra with resolutions of 300 000 were calculated for all phases, the line shapes lookexactly like the ones in Fig. 3.2. RVs were then measured by cross-correlating with the templatespectrum shown in Fig. 3.4 (φbol=0.0, no velocities taken into account in the radiative transfer).Results are also plotted in Fig. 3.3. The common characteristic of line doubling can be recognisedand values similar to the ones from the low-resolution profiles are derived for phases φbol≈0.25–0.8.However, cross-correlation is not able to resolve the full amplitude in splitting for phases 0.8–1.25 ;the blue-shifted component (RV ≈–2.5 km s−1 instead of –7) is especially suppressed.


Figure 3.4: Synthetic spectrum domi-nated by CN 0–2 lines of the red system(here for phase 0.0 of model S with-out velocities taken into account, reso-lution R=300000), which was used astemplate for deriving CN ∆v=–2 radialvelocities by cross-correlation (Fig. 3.3).

As they are more similar to observed ones, only RVs from low-resolution single profiles will beconsidered in the following and are plotted in Figs. 3.9 and 3.10, the latter demonstrating the highlyperiodic behaviour. The observed RV-curve of S Cep in Fig. 1.22 can be reproduced qualitativelyby the synthetic ones. The scenario of line doubling due to shock fronts can be followed by CN lineprofiles, too. No global shift (as for CO ∆v=3 lines) between observed φv and φbol of the models isneeded to match the RV-curve of S Cep and the model. Line doubling appears almost for the sameinterval (φbol≈ 0.85–1.1 ). Zero-crossing occurs at the same phase (0.5). The synthetic RV-curve issymmetric w.r.t. RV =0km s−1. The velocity amplitude of ∆RV ≈13.5 kms−1 is too low comparedwith the observed one of 22 km s−1 for S Cep. Similar large amplitudes from CN lines in the opticalhave also been reported for a few other Miras by Barnbaum (1992b).

With line profiles being even more complex, CN lines qualitatively show the same behaviour asCO ∆v=3 lines. But – still sampling deep layers driven by pulsation – they originate in slightlysmaller optical depths. This can be deduced from the radial gradient of τ and from line doublingat later phases.

In addition, a CN ∆v=–1 line of the 0–1 red system observable in the H-band was calculated.This 0–1 Q176.5 line at 6107.012cm−1 (1.6374µm) shows almost the same behaviour (profiles,RVs) as described above, being only different in a very small shift in φbol. The same was found fora Ti line in the K-band where observed line doubling around light maximum (HB96) could clearlybe reproduced by the modelling.

3.2.2 Probing the dust-forming region – CO ∆v=2

Figure 3.5 shows parts of the K-band high-resolution spectra of S Cep from HB96 containing theprominent first overtone CO lines. Also shown are the synthetic spectra in the same range as theycan be calculated on the basis of our dynamic model atmosphere. For the line profile modellingand this comparison, a few suitable lines were chosen, especially the CO 2–0 R19 low-excitationline at 4322.0657cm−1 (2.3137µm) and the CO 2–0 R82 high-excitation line at 4321.2240cm−1

(2.3142µm). Concerning the treatment of other molecules, the statements of Sect. 3.2.1.1 are alsovalid here.

As described in Sect. 1.3.2.2 (+1.3.5), the behaviour of CO ∆v=2 lines is two-fold. The weakerhigh-excitation lines (e.g. R82 in Fig. 3.5) act in the same way as second overtone CO lines andalmost duplicate their S-shaped RV-curve. While this is clearly visible for M- (Hinkle 1978) orS-type (HHR82) stars, contamination by many other lines (mainly CN) in this spectral regionhampers the line identification for C-rich stars (HB96). This fact can be recognised in the shownFTS spectra of S Cep given in Fig. 3.5. However, this figure also illustrates that our model canreproduce the scenario (compare Fig. 1.14) for high-excitation lines reasonably well.


Figure 3.5: Upper panel: Observed CO∆v=2 lines in the FTS spectra of S Cep(from HB96; R≈70.000) for selectedphases. While the strong low-excitationlines (e.g. 2–0 R19 and R18) can clearlybe identified, the high-excitation lines(e.g. 2–0 R82) are heavily blended byother features.Lower panel: The corresponding syn-thetic line profiles (R=300000) basedon model S for comparison. Thelow-excitation lines sample the dust-forming region, they consist of severalcomponents and appear always blue-shifted. The high-excitation line probesthe pulsating layers (as CO ∆v=3 lines)and show the characteristic behaviourof line doubling (cf. Fig. 1.14).

In the following, only the stronger low-excitation lines (e.g. R19 in Fig. 3.5) will be considered.Easier to identify in the spectra, they show a somewhat different behaviour. Variability is lesspronounced than for second overtone CO lines (which appear shifted/split with approximatelythe same line width). They show complex, asymmetric shapes and seem to consist of severalcomponents, which are not further separable though. This is reproduced by our model; Fig. 3.6presents a series of line profiles. Again the pseudo-continuous opacity (C2H2, dust) is strongest forφbol≈0.4. A comparison with the profile calculated without taking velocities into account suggeststhat various layers with different velocities contribute to the final, broadened shapes. The variationis not repeating for the same phase of different lightcycles. This is also illustrated in Fig. 3.7, wherevelocity structures of approximately the same phase of different lightcycles are plotted. While theyduplicate in the inner regions, considerable deviations can be found from the dust-forming layersoutwards. As discussed in Sect. 2.1.2.2, this is due to the interplay of the two independent timescales of pulsation and dust formation. Velocity fields within the line forming region varying fromone cycle to another result in differing line profiles for similar phases, which is also shown in theinsert of Fig. 3.7.

While the profiles differ somewhat for different lines (compare e.g. R19 and R18) in spectraobserved at a given time (due to contamination and observational uncertainties), they are exactlyreproduced for synthetic ones. Line depths (70–80%) are comparable for synthetic and observedspectra as well as the general appearance (see also Fig. 1.16). These lines have complex shapes withvarious components present, although no clear line splitting is seen most of the time. Line profileschange with time, but these changes are not coupled to the lightcycle. This complex behaviour hasalso been found in observations of R Leo by Hinkle (1978) and of χ Cyg by HHR82 (their Fig. 8).Profile variations in the CO first overtone low-excitation lines that are not related to the light-cyclewere also found earlier, e.g. by Winters et al. (2000a), who used a very similar approach.


Figure 3.6: Comparison of synthetic line profiles of the CO first overtone low-excitation line 2–0 R19for similar phases of different lightcycles (model S ). Substructures that are visible at higher spectralresolution (upper panel) are smoothed out at lower resolution (lower panel).

Figure 3.7: Three velocity structures ofmodel S for the same phase of differ-ent lightcycles as given. While the ve-locity pattern repeats in the same wayin the deep pulsating layers, consider-able differences can be found from thelayers where the dust-driven wind istriggered on outwards (cf. Sect. 2.1.2.2).This is reflected in the CO ∆v=2 low-excitation lines which are formed in thisregions and show different line profilesat similar phases.


Radial velocities were derived from the deepest points of the synthetic line profiles at a resolutionof 300 000 (Fig. 3.6) and are shown in Figs. 3.9 and 3.10. Weak line doubling was found only insome cases, disappearing in rebinning to resolutions of 70 000. At all instances of time one blue-shifted main component around ≈5 km s−1 is visible. This would be expected from the region ofline formation estimated from radial gradients of the optical depth (Fig. 2.9 –B) and the velocitiesthere (Fig. 2.4). In these depths of ≈2–3R∗, gas temperatures of ≈800–1500K are found. Thisseems consistent with the excitation temperatures of 800±100K derived for the same lines inχCyg spectra by HHR82 or values of 1100–1200K dervied from R Leo spectra by Hinkle (1978).Low-excitation first overtone CO lines can therefore be used to probe dynamics in layers wheredust is being formed and the outflow starts (Fig. 2.4c,d). Similar conclusions were also drawn byWinters et al. (2000a).

Sampling the region of the onset of the wind, the main components of our synthetic low-excitation CO ∆v=2 lines appear slightly blue-shifted by ≈5 km s−1 most of the time. RVs forthe same lines in observed spectra (cf. Sect. 1.3.2.2) were found to be either roughly at the CMRV(χ Cyg) or show some variation (R Leo, S Cep or Lebzelter et al. 1999), but with smaller amplitudesthan lines sampling pulsating layers (CO ∆v=3, CN, CO ∆v=2 high-excitation). These threebehaviours can be understood from the velocity structures of Fig. 2.4, if the lines probe layers of3, 2 or 1.5R∗, respectively. One might suspect that a change in optical depth would influence thebehaviour of the RVs accordingly. For all stars studied, RVs from CO ∆v=2 lines were not coupledto the lightcycle. This may reflect the irregular motions within the dust-forming region, where CO∆v=2 lines originate.

3.2.3 Probing the outflow – CO ∆v=1

As described in Sect. 1.3.2.3, lines of fundamental CO bands are usable to study the outflow ofmass-losing Miras. The CO 1–0 R1 line at 2150.856cm−1 (4.6493µm) was chosen for modelling ofthe line profiles. Only the pseudo-continuous opacity of C2H2 and dust is taken into account, sinceno other molecules contribute significantly in this region.

Figure 3.8 shows a time series of synthetic spectra compared to one calculated without velocitiestaken into account. The line profile looks qualitatively the same for all phases of the model andshows a typical PCygni-type shape with a deep, blue-shifted absorption component (from theoutflowing material in the line of sight) and a superimposed red-shifted emission component (fromthe extended regions around the star). The strength of the emission is variable and provides clueson the optical thickness of the surrounding shell; in Fig. 3.8 the extreme cases are shown. Similarsynthetic profiles were presented by Winters et al. (2000a).

Unfortunately, all of the few existing LM-band spectra for SCep are of poor quality. A com-parison can therefore only be done with other results which were already described in Sect. 1.3.2.3.HHR82 reported on fundamental CO lines in a few spectra of χCyg (cf. their Fig. 11). Being af-fected by several photospheric as well as telluric CO lines, they still show a constant blue-shift overtime. One spectrum of the 4µm region containing the CO ∆v=1 lines was obtained for IRC+10216(a C-rich object with heavy mass loss) by Keady et al. (1988). These spectra were also used byWinters et al. (2000a) who compared them to synthetic spectra based on their dynamic models.Observed line profiles are shown in the lower right panel of Fig. 1.16. Except for the saturated ab-sorption, these profiles compare well to the ones we calculated here and which are shown in Fig. 3.8.They show PCygni-type shapes, which is typical9 for lines sampling the outermost regions andindicating stellar winds.

RVs were calculated from the minima of the blue-shifted absorption in profiles with highest res-olution (300 000); the results are shown in Figs. 3.9+3.10. The sometimes very broad or asymmetricabsorption features can complicate the RV measurements. Note, that the RVs of the deepest point

9This is however not necessarily the case for all lines being formed in the outflow. CO ∆v=2 lines for example,although sampling the region where the wind starts, are not expected to show PCygni-emission as the de-excitationv=2→ 0 has a lower transition probability than cascading via the v=1-state.


in absorption can be larger than the velocities in the outer layers of the atmospheric model due tothe asymmetry of the complex profile caused by the superposed emission. Apart from some smalllong-term variation in the terminal velocity, CO ∆v=1 line velocities show a steady outflow overtime.

From radial gradients of the optical depth τ it is found that these lines originate as expected inthe wind region at ≈15R∗. Low gas temperature of ≈350–500K in the line forming regions thereappear comparable with excitation temperatures of 300±200K given by HHR82.

Figure 3.8: Synthetic fundamental CO lines(1–0 R1) based on model S for selected phasesand two different spectral resolutions. At anyinstance of time, these lines show characteris-tic P Cygni-type profiles indicative of a stellarwind.

3.2.4 The overall picture concerning RVs

Figure 3.9 shows a compilation of all RVs derived from synthetic line profiles which were computedon the basis of model S and were presented in the previous sections. Calculations were done for threeseparated periods (23 instances of time each). Figure 3.10 then shows a plot for direct comparisonwith the observations of S Cep in Fig. 1.22. For CO ∆v=3 and CN ∆v=–2 lines RVs (R=70.000)from all periods were combined into one composite light-cycle and then plotted repeatedly forbetter illustration (RV-curves from different periods are almost identical to each other, becauseof the very regular movement of the inner regions, see Fig. 2.1). For CO ∆v=2 and ∆v=1 linesonly measurements from one period (φbol=7–8 and 14–15 respectively) are plotted for a clearerpicture. It can be recognised that (with restrictions) our model calculations can reproduce thevelocity pattern found for S Cep (and other Miras).

The first two types of lines sample deep pulsating layers and show the typical discontinuousRV-curves, reflecting shock fronts running through the line-forming regions. Synthetic line profilescould generally reproduce this characteristic behaviour, at least for the lower spectral resolutionof 70 000 (only these results are adopted here). Some aspects (S-shape, asymmetry w.r.t. RV =0,line doubling interval) are realistically replicated with synthetic CO ∆v=3 lines, but the apparentphase shift is conspicuous. While the mean phase of the line doubling interval is φv≈1.0 in mostobservations (HSH84), it is clearly shifted to φbol≈0.775 in Fig. 3.10, suggesting a shift of ∆φ≈0.225between visual and bolometric phases for our models, such that φbol lags behind φv. Such a phaseshift is not found for the RV-curve of CN lines however, which resembles the observations of S Cepin Fig. 1.22 rather well. Line doubling appears for the same interval and at the same phases, as wellas zero-crossing. The synthetic RV-curve is symmetric around RV =0 (in contrast to CO lines),but the observed one even extends to more negative values. CN lines appear to originate slightly


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Figure 3.9: Upper panels: Lightcurve of the dynamic model atmosphere – representing a typicalMira and used for the line profile modelling (model S ) – for three different, separated periods. Thephases φbol for which synthetic spectra were calculated are marked (◦), including the referencephase (φbol=0.00 ). Lower panels: Compilation of the corresponding RVs derived from line profilesof different types of molecular lines (see text for details). Note that only the results from profilesat lower spectral resolutions (R=70.000) are adopted here for the CO ∆v=3 and CN lines.

further out10 than the CO second overtone lines, which can be inferred from plots of the radialgradient of the optical depth τ or from line doubling at later phases (the shock front propagatingoutwards reaches outer layers later, also demonstrated in Fig. 4 of Alvarez et al. 2000). Such acomparison would be interesting if done observationally in the future. Definitely, the amplitudes∆RV of both types of lines are too small compared to observations (see Sect. 3.5.2).

Although the line shapes and their variations of the synthetic CO ∆v=2 low-excitation lineprofiles are comparable to observed ones, absolute values of RVs are somewhat different. Theyshow a small blue-shift at all times and therefore sample the region where the stellar wind starts(≈3R∗). On the other hand, observed RVs are either constantly at ≈CMRV or vary little aroundthe CMRV. Comparing Fig. 1.12, this could be interpreted as either the lines should probe deeperlayers (≈2 or 1.5R∗) or pulsation should influence layers at larger radii.

Sampling the steady outflow, CO ∆v=1 lines appear clearly blue-shifted, more or less constantly

10Combining the difference of the median phases of line doubling for both type of lines (∆φbol≈0.975–0.775 ) withan estimation for the shock front propagation velocity from Fig. 2.1 this would correspond to a radial difference ∆R

of 0.176 R∗/86.8 R⊙/0.4AU.


over time. Compared to observed RVs from K I lines in SCep spectra (Fig. 1.22), the synthetic onesare ≈5 km s−1 too low (as already expected from the velocity structures of the model, see Fig. 1.12).

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Figure 3.10: Radial velocitiesfrom Fig. 3.9 (model S ) projectedonto one lightcycle. While CNand CO ∆v=3 lines probe deeppulsating layers, CO ∆v=2 linesoriginate in the dust-forming re-gion where the stellar wind is trig-gered and the outflow is sampledby CO ∆v=1 lines. To be com-pared with observational resultsof S Cep as shown in Fig. 1.22 (orother Miras, Figs. 1.17+1.19).

3.2.5 Resume

Using the dynamic model atmospheres for LPVs of Hofner et al. (2003a) we performed line profilemodelling for four different molecular features (known to be suitable from observational studies).Due to the large macroscopic motions (pulsation, wind) within AGB atmospheres, it is absolutelynecessary to include velocity effects in the radiative transfer to model the complex line profiles andtheir temporal variations. Based on the well-chosen model S we tried to simulate high-resolutionspectroscopy observations of the C-rich Mira S Cep and other typical or well-observed Miras.

The fundamental goal for the models is to resemble the global atmospheric structure [T(r), P(r)]of evolved red giants. We consider this to be fulfilled because of two reasons: (i) the line depthsof the absorption features (originating in distinctly different atmospheric depths) compare ratherwell to observations (as available); and (ii) the corresponding temperatures in the regions of lineformation have similar values as the ones derived from observations.

On top of this, the model atmospheres should be able to reproduce the typical global velocitystructure (as derived from Mira observations) from the pulsating atmospheric layers through thedust-forming region (where the stellar wind is accelerated) out to the outflow region (Fig. 1.12).As these velocity fields severly influence the synthetic line profiles, they can be examined by twoadditional characteristics: (iii) the line shapes and their temporal variations; as well as (iv) RVsderived from Doppler-shifts of (components of) the line profiles. Both appeared to qualitativelyresemble observations although quantitatively some differences are apparent.

It was demonstrated that the behaviour of lines sampling different regions within AGB atmo-spheres can be modelled simultaneously by only one consistently computed atmospheric model andsuccessive radiative transfer as summarised in Fig. 1.12:

3.3. A closer look at CO ∆v=3 lines 97

• CO ∆v=3 lines, CO ∆v=2 high-excitation lines and CN lines probe the deep photosphere,which is dominated by pulsation. At lower spectral resolution (70 000), time series of syn-thetic spectra showed line profile variations similar to observations (Fig. 1.13). The typicaldiscontinuous, S-shaped RV-curves and the line-doubling around phases of maximum lightcan be reproduced. However, the splitting of the two components during phases of line dou-bling is significantly too weak and the amplitude ∆RV is too small by roughly a factor oftwo. At highest resolution (300 000), the profiles appear more complex.11

• CO ∆v=2 low-excitation lines sample the dust-forming region, where the stellar wind istriggered. They show complex, broadened shapes with irregular temporal variations andappear slightly blue-shifted (reminiscent of a static layer).

• CO ∆v=1 lines probe the layers of steady outflow and show typical P Cygni-type shapes atany instance of time.

Thus, it can be ascertained that the dynamic model atmosphere used here shows fundamentalagreement with dynamic processes (pulsation, dust formation, mass loss mechanism) occurring inAGB atmospheres (Miras).

3.3 A closer look at CO ∆v=3 lines

3.3.1 Line formation in model S

In the right panel of Fig. 3.11 we present synthetic line profiles for second overtone CO linescomputed by using atmospheric structures of the C-rich model S at certain phases φbol. Thesespectra were published in Nowotny et al. (2005a) and are rather similar to the results presentedin Sect. 3.2.1.1. The main difference is that no pseudo-continuous opacity contribution by C2H2

was taken into account here. This facilitates a direct comparison with observed line profiles of theS-type Mira χ Cyg as shown in the left panel of Fig. 3.11.

As already discussed in Sect. 3.2.1.1, the very characteristic behaviour of CO ∆v=3 lines inMiras (; discontinuous, S-shaped velocity curve; cf. also Sect. 1.3.2.1) can be reproduced reason-ably by our synthetic line profiles at the lower spectral resolution. However, at the highest resolutionof R=300000 we find somewhat more complex profiles not observed so far. Although some sub-structures may be present in observations as well (but suppressed by the limited resolution of theFTS spectra), there are probably some shortcomings of our model atmosphere responsible for thedifferences. Eye-catching is the fact that we do not find one line component proceeding from blue-to red-shifts during phases of φbol≈0.2–0.5 but rather a complex evolution of a multi-componentline profile. For example at phase φbol=0.30, the synthetic profile exhibits some ”line splitting”not present in any observed spectra.

For investigating the process of line formation in more detail we used the method to cut theatmospheric structures at certain radial points12 and calculate synthetic spectra based on theseshortened model structures. This method may be applicable only in some restricted cases13 as(translating the 1D atmospheric structure into 3D spherical geometry by the spherical radiativetransfer) we use spheres of a constant radius instead of constant projected velocity. Still, it may beuseful for deriving some information concerning our line profile modelling of CO ∆v=3 lines (comingfrom the pulsating layers of the inner atmosphere; cf. Fig. 1.12), as demonstrated in Figure 3.12.

The first case shown there is phase φbol=0.06 of model S. The line profile changes only weaklywhen going from the cutted model (B) to the full model. In general, the formation of the spectrum

11It will be interesting to see if future instruments (with higher resolutions than the FTS spectrograph used byHinkle and collaborators) will allow us to verify whether there are such substructures in the profiles or if this is apeculiarity of the model atmospheres used.

12meaning to let the model end there or setting all quantities to =0 for larger radii13i.e. compact objects with thin regions of line formation and monotonic gas velocities


Figure 3.11: Left: Time series of average line profiles of CO ∆v=3 lines in FTS spectra of χ Cyg(cf. Fig. 1.13). Taken from Lebzelter et al. (2001) who analysed the radial velocities, the spectrawere kindly provided by T. Lebzelter. Observed heliocentric velocities were converted to systemicvelocities RV ∗ of the star by assuming a CMRV heliocent of –7.5 km·s−1 (HHR82).Right: Synthetic second overtone CO line profiles (5–2 P30) for selected phases during one lightcy-cle, calculated on the basis of model S (similar to Fig. 3.1; see text for details about the differences).For the lower resolution of 80.000, spline fits through the spectral points are plotted.


Figure 3.12: Atmospheric structures (gas densities, gas velocities; cf. Fig. 2.4) of model S for threedifferent phases φbol as given in the plots. Also shown are the resulting CO ∆v=3 line profiles(black) if the model structures are cutted at certain depth points (A, B) together with the finalspectra for the full, un-cutted model (grey). Finally (rightmost panels of spectra), the resulting lineprofiles for the full models (black) are plotted together with the corresponding profiles if velocityeffects are not taken into account (green).


is completed just behind the shock front (B) at ≈1.2R∗. In front of the propagating shock wave,the gas density drops sharply down to log ρgas < –11.5. There is not enough material further outresulting in a distinct amount of optical depth. Thus, only minor contributions from these layersto the spectrum can be recognised.

The second case presented in Fig. 3.12 is phase φbol=0.75 with pronounced line doubling forthe CO absorption feature (see also Fig. 3.15 and the corresponding text in Sect. 3.3.2). Again,there is a blue-shifted component mainly originating in the layers below the shock front (A) at≈ 0.9R∗, which is slightly modified though when going to the model cutted at B. Complementaryto the first case above, the densities in the infalling layers in front of the shock wave (between theradial points A and B) are larger here. Leading to relevant optical depths, a second red-shiftedcomponent arises therefore. Note, that there is almost no change in the line profile from the modelcutted at B (log ρgas = –11.3) to the full model. Thus, the spectrum containing CO ∆v=3 linesappears to be formed in atmospheric layers with approximately log ρgas > –11.5 g·cm−3.

The third example in Fig. 3.12 shows the line profile for phase φbol=0.41 of model S. Consideringthe density range stated just above, we find that the spectrum is formed over a large radial range inthis case. Namely from the innermost layers out to the position of the shock front at ≈ 1.5R∗. Allkinds of velocities – outflow as well as infall – can be found in this region. Together with sphericityeffects, this leads to the final rather complex line profile.

From these examinations we can conclude that – at least for the model we used here – thepicture of one component that arises, developes and disappears again (as derived from observations;Sect. 1.3.2.1 + Fig. 1.13) appears to be too simple. Depending on the phase, the absorption line canbe formed over a whole range in radius with quite different velocity fields leading to complex lineprofiles. Also we do not necessarily sample the same layer by studying a certain line throughout thelight cycle and the derived RVs do not trace the movement of a specific mass shell (Fig. 2.1). In thissense, the scenario sketched in Fig. 1.14 may be too simple but still helpful for examplifying thefundamental behaviour. Future spectrographs with spectral resolutions larger than 100000 mayopen the road for investigations of such fine structures as found in our synthetic line profiles withR=300000.

As the aim is having dynamic model atmospheres as realistic as possible (Sect. 3.5.2), a modelwith a more pronounced pulsation would be needed for even better CO ∆v=3 line profiles. A moreuniform velocity field throughout the region of line formation (log ρgas > –11.5) may result in onespectral component appearing blue-shifted and moving to red-shifts during the lightcycle.

3.3.2 Miras and semi-regular variables (SRVs)

In Sect. 1.3.2.1, the temporal behaviour of second overtone CO lines as found in observed spectrawas discussed in detail. The observational results showed that there is a fundamental differencebetween Miras and SRVs in the line profile variations and the resulting RV values. The first typeof LPVs exhibits a discontinuous RV curve with pronounced line doubling at phases of maximumlight (cf. Fig. 1.13). On the other hand, SRVs only show velocity variations with the general trend ofmaximum RVs at minimum phases (φv≈0.5) and minimum RVs around maximum phases (φv≈0.0).The amplitudes ∆RV are significantly smaller (only a few km·s−1). A clear distinction comparedto Miras is the fact that SRVs do not show the phenomenon of line doubling. Special emphasiswas put on the semi-regular WHya in Sect. 1.3.2.1. High-resolution spectroscopy at several phases(Lebzelter et al. 2005a) provides us with a well-sampled RV curve for CO ∆v=3 lines which is shownin the right panel of Fig. 1.15. Velocity variations appear rather sinusoidally with an amplitude(∆RV ≈15km·s−1) rather large for the group of SRVs. Still, the characteristic of no line doublingat any phase was found.

During our studies of different dynamic model atmospheres, we found that model W (describedin detail in Sect. 2.1.2.2) is able to reproduce reasonably well the behaviour of CO ∆v=3 linesin spectra of SRVs – particularly of WHya. It is a model with a rather low piston amplitude of


Figure 3.13: Left: Time series of average line profiles of CO ∆v=3 lines in FTS spectra of the semi-regular variable WHya. Taken from Lebzelter et al. (2005a) who analysed the radial velocities(Fig. 1.15), the spectra were kindly provided by T. Lebzelter. Observed heliocentric velocities wereconverted to systemic velocities RV ∗ of the star by assuming a CMRV heliocent of +38km·s−1

(Hinkle et al. 1997).Right: Synthetic second overtone CO line profiles (5–2 P30) at two different spectral resolutions,calculated on the basis of model W for selected phases φbol.


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Figure 3.14: Left: Radial velocities derived from Doppler-shifted CO ∆v=3 lines in FTS spectra ofWHya (left panel of Fig. 3.13; cf. also Fig. 1.15). Data adopted from Lebzelter et al. (2005a).Right: RVs for φbol=0.0–1.0 as derived from the corresponding synthetic CO profiles based onmodel W (right panel of Fig. 3.13).

2 km·s−1, no dust-formation and thus no stellar wind. In comparison, WHya shows only a lowmass loss rate of M≈2·10−8 M⊙ yr−1, too. It shall however explicitly be noted, that the stellarparameters of (the C-rich) model W were not chosen to resemble (the O-rich star) WHya.14 Still,a comparison of this one certain aspect – the RV variations of CO lines – shall be made, as themodel reproduces the behaviour characteristic for all SRVs observed so far.15

For the line profile modelling we followed the procedure of Sect. 3.2.1.1. The resulting syntheticprofiles for a representative number of phases during the lightcycle are plotted in the right panelof Fig. 3.13. Whenever the corresponding profiles of the observed FTS spectra were available, theyare also plotted for comparison. The figure demonstrates that the CO ∆v=3 lines only show blue-or red-shifts as well as rather asymmetric line profiles due to the non-monotonic velocity field ofthe atmosphere (the extreme case of phase φbol=0.48 was investigated in detail in Fig. 2.8).

RVs derived from the deepest point of the line profiles in highest-resolution (R=300 000) aredrawn in the right panel of Fig. 3.14. For a direct comparison, the velocity data of WHya (same asFig. 1.15, as a function of visual phase here though) are also shown in the left panel of this figure.The plot shows, that our model W is able to reproduce the principal behaviour of RV curves ofsemi-regular variables as described in Sect. 1.3.2.1 – except for the asymmetry w.r.t. RV =0. Onlya small difference in phase may be suspected such that the bolometric phases of the model lagbehind the visual phases of the observations by 0.1–0.2. The amplitude ∆RV ≈8 km·s−1 of thesynthetic velocity curve is similar to SRV observation (left panel of Fig. 1.15), the amplitude ofWHya is probably at the upper end of the found range.

A very interesting feature of this comparison is the fact that the line profiles of second overtoneCO lines based on our model W – just as observed ones for SRVs – do not show doubled lines forany phase during the light cycle. This behaviour (already discussed in Sect. 1.3.2.1) is astonishingas one would expect shock waves in any pulsating atmosphere and also Balmer emission lines wereobserved in spectra of SRVs without line doubling of CO lines. The results of our modelling – asshown in Fig. 3.15 – may provide a possible answer to this puzzling fact. For two selected phases ofmodel W and model S, we present the atmospheric structure (gas velocities and the correspondinggas densities) together with the resulting CO line profiles and the radial optical depth at certainwavelength points (ABC) of the profiles. For the model without mass loss, a shock front can be seen

14as it was done for model S and the C-rich Mira S Cep15to be published in the future (Nowotny et al., in prep.)


Figure 3.15: Illustration of the fact that the occurrence of a shock wave within the atmospheredoes not necessarily lead to line doubling in spectra containing CO ∆v=3 lines. Demonstrated byrepresentative phases of model W (dust-free, no M) and model S (with M). See text for details.


in the velocity structure around 1 R∗. Nevertheless, only a blue-shifted component from behind thefront shows up in the line profile. No red-shifted component is formed (the very weak one whichcan be seen in the synthetic profile at approx. 10 km·s−1 would certainly not be recognisable inobserved spectra). The reason for this is the sharp drop in density in front of the shock wave. Theoptical depth is simply not large enough at these wavelengths so that a component becomes visiblein the spectrum (although model W shows gas velocities of up to 15 km·s−1, these are not translatedinto the spectra as the outer layers with that high velocities do not contribute to the formationof the spectra). As stated in Sect. 3.3.1, the densities needed to form an individual component ofCO ∆v=3 lines have to be larger than log ρgas ≈ –11.5, which is not met for model W. However,this requirement is fulfilled in the case of model S which is shown in the right panel of Fig. 3.15.Due to the formation of dust and the resulting wind, the atmospheric structure differs somewhatin the layers around 1R∗ which are relevant for the line formation of CO ∆v=3 lines. The layers infront of the shock show smaller velocities, but the densities are enhanced by roughly one order ofmagnitude compared to model W. Thus, a clear increase of optical depth can be recognised in thelower right panel for the radial plot corresponding to wavelength point C (red-shifted) for whichvelocities are taken into account. Two separated components can clearly be seen in the line profile.The phenomenon of line doubling is therefore not a matter of the occurrence of shock waves butrather depending on the question if there is enough material (large enough densities) with infallingvelocities to produce large enough optical depth and thus the second component in the line profiles.

3.4 Gas velocities vs. measured RVs

To infer actual gas velocities from radial velocities of Doppler-shifted spectral features, a correctionfactor p – with ugas=p·RV – should be considered, caused by geometric aspects and center-to-limbeffects. A first theoretical estimation of 24/17≈1.41 for any radial pulsating star was given byGetting (1935). Parsons (1972) later found smaller values from synthetic lines calculated with planeparallel Cepheid model atmospheres and a dependence on chosen parameters (his Fig. 2). His ap-proximate value of 1.31 was then used by Hinkle (1978) to analyse the spectroscopic observationsof R Leo. As the very extended Mira atmospheres have more complicated velocity structures (e.g.Fig. 2.4) and the fact that lines from different wavelengths originate in different depths, this correc-tion becomes more difficult. Willson et al. (1982) continued Parsons’ calculations; focussing on thepeculiarities of LPVs, they present a detailled list of conversion factors. As stated in Scholz &Wood(2000) a factor of ≈1.4 was often used to interpret the observations. They themselves derive factorsof 1.25/1.4/1.6 for different lines (CO, OH) in their model calculations by comparing RVs derivedfrom the two components of splitted lines to gas velocities in front of and behind the shock frontsof their atmospheric models.

We applied the same method to synthetic CO ∆v=3 lines at phases 0.75/0.85 of model S(Sect. 3.2.1.1, Figs. 3.1+3.11), which show clear line doubling. A correction factor of p≈ 1.36 canbe derived from our model results. Recognisable in the right panel of Fig. 3.15 is the fact thatthe blue-shifted component is formed immediately behind the shock front (marked with the bluediamond) while the red-shifted component is formed (derived from the increase in optical depth)over a larger region in front of the shocked (marked with the red quadrangle). The lower velocitiesof the latter radial region (e.g. middle of the quadrangle) may decrease the factor p slightly.

We also derived values for p for the line profiles shown in Fig. 2.7 and described in the text ofSect. 2.2.5. For the computation of these lines, constant velocities ugas for every depth point of theatmospheric model were assumed, and RV were measured afterwards. The results are summarisedin Fig. 3.16. The values range between 1.2 and 1.5 which seems to be consistent with the findingsfrom the literature listed above.

3.5. Steps towards realistic models 105

Figure 3.16: Conversion factors p as derived from the line profiles shown in Fig. 2.7. For two differentatmospheric models uniform velocity fields with constant ugas were assumed and the values for RVmeasured from the Doppler-shifted lines.

3.5 Steps towards realistic models

As pointed out in Sect. 3.2.4, our model S can qualitatively reproduce the velocities found forS Cep, but some differences remain concerning the absolute values. Therefore we tried to vary theinput parameters of the model (Tab. 2.1) to get even more realistic atmospheric structures. Theresults16 of this preliminary parameter study shall be presented in this section.

3.5.1 Fitting models to observations of selected targets (e.g. S Cep)

For example a model where only the stellar mass is changed from 1 to 1.5M⊙ was investigatedwhich is named model S1 in Tab. 2.1. This seems comparably plausible, as we only have rough massestimates for S Cep (e.g. 2.5–4M⊙ from Barnbaum et al. 1991). The differences are immediatelyvisible from the the moving mass shells and atmospheric structures plotted in Fig. 3.17 (compareFigs. 2.1+2.4 for the original model S). Dust shells and density enhancements are more stronglypronounced. The global velocity structure looks markedly different. The very regular motions dueto pulsation only reach out to ≈1.4R∗. From there on the model structures are not necessarilysimilar for the same phases of different lightcycles any longer, because the dust formation cyclespans two pulsational periods. Throughout the dust-forming region strong variations in velocitiescan be seen; a steady outflow and a smooth distribution of the degree of dust condensation is notreached until ≈10R∗.

These differences are also reflected in line profiles and RVs (synthesised in the same way as theresults presented in Sect. 3.2). The latter are shown in Fig. 3.18, which is immediately comparable toFigs. 3.9+3.10. While the RV-curve for CO ∆v=3 lines is almost the same as for the original model,line strengths change considerably from one period to the next (duplicating only every secondperiod). CN ∆v=–2 lines are more affected by the double periodicity. Not only the line shapes(variable line strengths; line doubling much clearer), but also RV-curves (irregular with a larger∆RV of ≈16 km s−1 for cross-correlation) evidently deviate. However, synthetic velocity amplitudesfor lines from the inner regions could not be increased enough with this changed model to reproduceobserved amplitudes ∆RV . While the faster outflow leads to more realistic RVs measured fromCO ∆v=1 lines (≈18km s−1), the largest differences by far compared to the original model are

16partly published in Nowotny et al. (2005c, 2006), partly to be published in the future (Nowotny et al., in prep.)


Figure 3.17: Movement of mass shells (left) and the corresponding atmospheric structures (right)at selected phases for the dynamic model atmosphere denoted by model S1 in Tab. 2.1. Note thata dust shell is formed only every second pulsation period and the atmospheric structure (especiallythe gas velocities) is affected by this multi-periodic effect from ≈1.4R∗ on outwards.


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Figure 3.18: Radial velocities derived from Doppler-shifts of all molecular lines used for the spectralsynthesis based on model S1. Same as Figs. 3.9+3.10.


found for CO ∆v=2 low-excitation lines. As suspected from the large variations of gas velocitieswithin the dust-forming region at ≈2–3R∗ in Fig. 3.17, these lines show various components. In partwith even more pronounced variability than second overtone lines which has never been observedso far, however.

The fine-tuning to achieve a dedicated fit for the velocities of one particular star (like S Cep)throughout the whole atmosphere is not a straightforward task (compare also the discussion inSect. 2.1.4). Not only are the stellar parameters of S Cep poorly known (L, M , etc.), but a mul-tidimensional grid of dynamic models would also be needed. Too many parameters interact andinfluence the velocity structures in subtle ways. Especially the transition region between regularpulsation and steady outflow (and thus the behaviour of first overtone low-excitation CO lines)seems to depend sensitively on the chosen set of stellar parameters. This dependency was discussedalso by e.g. Hofner & Dorfi (1997) or Winters et al. (2000b).

3.5.2 Larger velocity amplitudes in the pulsating layers

A first and fundamental aim for the small parameter study we carried out was a dynamic modelatmosphere with a more realistic amplitude of the gas velocities (and therefore ∆RV ) within thepulsating layers. As stated in Sect. 1.3.2.1, the RV-curves of CO ∆v=3 lines show a rather uniformpicture for all Miras studied so far, independent of their properties. Figure 1 of Lebzelter & Hinkle(2002a; reproduced in the left panel of Fig. 3.20) and the results in HSH84 demonstrate thatamplitudes of ∆RV ≈20–30kms−1 are observed. This common feature seems to be a fundamentalcharacteristic of Miras and should be reproduced by a realistic model atmosphere. The ∆RV valuesgiven in Sect. 4 of Scholz & Wood (2000) could serve as a guideline for modelling. They state thata typical Mira appears to have a true (observed RVs corrected by their factor p to get real gasvelocities; Sect. 3.4) post-shock outflow velocity of ≈–14kms−1 and a pre-shock infall velocity of≈20km s−1, resulting in a full velocity amplitude of ≈34km s−1. Comparing this with gas velocitiesin the inner regions of model S used here (Fig. 2.4), it can be deduced that the dynamic modelshould have an amplitude larger by a factor of ≈2. With these more extreme velocity gradientsacross the shock front, line doubling should be more pronounced for CO ∆v=3 and CN linesand easier to recognise even in low-resolution spectra (Sect. 3.2.1). From our studies so far weknow that a larger shock amplitude cannot be achieved simply by applying a piston with a highervelocity amplitude ∆up. This is due to the self-regulating interdependence of the density structure(levitation) and the maximum shock strength.

We studied several models with changed stellar parameters and investigated the influence ofthe parameters on the velocity structure, the line profiles as well as on the derivable RVs. Oneoutcome of this limited study was model M (parameters also given in Tab. 2.1), which showsvelocity variations in deep atmospheric layers that are close to the values discussed above. Themost promising results of the line profile modelling based on this model shall be summarised inthe following.

For the spectral synthesis we proceeded in the same way as described in Sect. 3.2.1. Figure 3.19shows the resulting synthetic CO ∆v=3 line profiles for representative phases during the lightcyclein comparison with observed FTS spectra of the S-type Mira χ Cyg (due to the lack of observationalmaterial for C-type Miras). The typical behaviour of second overtone CO lines (cf. Sect. 1.3.2.1and Fig. 1.13) can clearly be recognised. The difference in RV between the two splitted componentsis more pronounced than for model S, reflecting the larger drop in velocity in the atmosphericstructure at the location of the shock front. As discussed in Sects. 3.2.1.1+3.3.1, the syntheticprofiles of model S do not show a smooth transition of one component from blue- to red-shift,but more complex profiles and even line splitting around phases of φbol≈0.3 (not observed withthe resolutions of spectrographs available today; disappearing by rebinning the synthetic profilesto R=70000). For the new model M, this shortcoming is somewhat diminshed as can be seen inFig. 3.19. The RV curve also exhibits the typical discontinuous S-shape and shows similar variationswhen going from spectral resolutions of 300 000 to 70 000 (Fig. 3.19). In Fig. 3.20 we compared


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Figure 3.19: Left: Time series of average line profiles of CO ∆v=3 lines in FTS spectra of χ Cyg(cf. Fig. 1.13). Taken from Lebzelter et al. (2001) who analysed the radial velocities, the spectrawere kindly provided by T. Lebzelter. Observed heliocentric velocities were converted to systemicvelocities RV ∗ of the star by assuming a CMRV heliocent of –7.5 km·s−1 (HHR82; cf. the caption ofFig. 1.17). Also shown are the corresponding synthetic line profiles (5–2 P30) for selected phases,calculated on the basis of model M (same as Fig. 3.1).Right: RVs for φbol=0.0–1.0 as derived from the synthetic profiles, plotted repeatedly for the twodifferent spectral resolutions (+ for 300 000, squares for 70 000).

the synthetic velocity variations of our models S and M (only for R=70 000) to observationalvelocity data adopted from Lebzelter & Hinkle (2002a). The latter represents a significant largesample of Miras and shows that this RV curve is a fundamental characteristic of this type of LPVs(compare the discussion in Sect. 1.3.2.1). The figure demonstrates that model M is able to reproducethe observations rather well, especially the amplitude of ∆RV =21.6 km s−1 is very realistic. Thisbehaviour could be reproduced for the first time by dynamic model atmospheres and subsequentline profile modelling. We needed to apply a shift in phase of 0.3 to the synthetic RV curve forachieving agreement in phase with observations. This value may be suspected to be a measure forthe difference between φv and φbol (cf. Sect. 2.2.6).

We also modelled line profiles for CN ∆v=–2 lines based on model M, for the spectral synthesiswe followed the approach described in Sect. 3.2.1.2. The results of the modelling for selected phasesare shown in Fig. 3.21, together with similar lines as observable in high-resolution spectra of theC-type Mira S Cep. This synthetic CN line profiles appear to be the most realistic ones whichwe were able to produce by our modelling efforts so far. Although similar substructures as formodel S (cf. Fig. 3.2) may be identified at the highest resolution of 300 000, the characteristicbehaviour of lines sampling the pulsating layers (Fig. 1.13) is much more pronounced here. Thecongruence between observed and synthetic profiles is rather high, especially when rebinned downto the lower spectral resolution of 70 000. The transition from blue- to red-shift during phasesof φbol≈0.0;0.6 is much smoother compared to model S. The resulting discontinuous RV curve(right panel of Fig. 3.21) shows the same S-shape as known from CO ∆v=3 lines. Interesting isthe fact that this is not clearly recognisable in observed CN velocities of S Cep as shown in thecomparison of Fig. 3.22. There the RV curve appears more like a straight line. One may suspectthat the cross-correlation technique, applied to obtain these velocities from the FTS spectra, is not


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Figure 3.20: Left: Composite RV curve of second overtone CO lines in FTS spectra of a large sampleof Miras, demonstrating the rather universal behaviour (cf. Sect. 1.3.2.1). Observed heliocentric RVsare converted to systemic velocities by the respective CMRV of each object. Data adopted fromLebzelter & Hinkle (2002a). The grey line was fit to guide the eye. Right: RVs derived from thecorresponding synthetic line profiles (only the values for lower spectral resolutions of R=70000)of model S (grey; Fig. 3.1) as well as model M (black; Fig. 3.19). An artificial shift of 0.3 in phasewas applied for easier comparison.

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Figure 3.21: CN ∆v=–2 line profiles (1–3 P238.5) as observed in FTS spectra of S Cep by HB96(spectra kindly provided by K. Hinkle, priv. comm.). Observed heliocentric velocities were con-verted to systemic velocities RV ∗ of the star by assuming a CMRV heliocent of –26.7 km·s−1 (HB96;cf. the caption of Fig. 1.22). Also shown are the corresponding synthetic line profiles (1–3 Q24.5)for selected phases, calculated by using model M (same as Fig. 3.2).Right: RVs for φbol=0.0–1.0 as derived from the synthetic profiles, plotted repeatedly for the twodifferent spectral resolutions (+ for 300 000, squares for 70 000).


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Figure 3.22: Left: RV curve of CN ∆v=–2 lines in FTS spectra of S Cep. Data adopted from HB96(compare also Sect. 1.3.5.3, Fig. 1.22). Right: RVs derived from the corresponding synthetic lineprofiles (only the values for lower spectral resolutions of R=70 000) of model S (grey; Fig. 3.3) aswell as model M (yellow; Fig. 3.21).

able to resolve the weak components as they appear slightly before light maximum and disappearslightly after the next maximum. Two facts of our modelling are worth noticing as they are muchcloser to observational results: the splitting into two well separated components during phasesof line doubling (e.g. φbol=0.72 ) is very well pronounced and the amplitude of the RV curve of∆RV≈22km s−1 is exactly what was found for S Cep (cf. Fig. 1.22). The above mentioned phaseshift of 0.3 – needed to align the synthetic RV curve with the observed one – appears plausiblehere, too.

Note that the stronger piston of the new model M (∆up=6kms−1 instead of 4 km s−1 formodel S; cf. Tab. 2.1) results only in slightly larger outflow velocities while the maximum infallvelocities are increased approximately by a factor of 2. This can be recognised in Fig. 3.20 for COlines as well as in Fig. 3.22 for CN lines. The reason for this effect is the larger log g∗ of model Mcompared to model S. Thus, the increase in total velocity amplitude ∆RV is mostly due to thefaster infalling material.

Model M proved to be rather realistic concerning the velocity variations in the deep atmosphericlayers (governed by periodic pulsations) and the resulting line profile variations of molecular fea-tures originating there. This is the first step towards model atmospheres with dynamics close toreality. The next step would be to fit in addition the velocities in regions where other lines originate(simultaneously with one model for a certain object).


3.6 Quasi-static, warm molecular envelopes and dynamicmodel atmospheres

Different velocity components have been found by studies of line profiles in observed high-resolutionspectra of pulsating AGB stars in the past (HHR82). Among these, a layer with almost zero velocityrelative to the stellar systemic velocity and an excitation temperature of about 800 – 1000K couldbe identified from low-excitation lines of CO (∆v=2) and OH. Being present in a number of Miras[e.g. in R Leo (Hinkle 1978) or χ Cyg (HHR82)], this layer is variable in intensity. The variationsdo not follow the lightcycle and sometimes the static layer vanishes completely. HHR82 interpretedthis layer as a reservoir for mass loss driven by radiation pressure on the dust, which can condenseat these low temperatures.

Tsuji (1988) found anomalies in low-excitation CO ∆v=2 lines in high-resolution spectra of var-ious M giants (with slightly higher temperatures and smaller light amplitudes). These could not beexplained by photospheric absorption alone but seemed to be composed of two components. Sub-tracting synthetic spectra (based on hydrostatic model atmospheres) from observed ones revealedexcess absorption with small blue- or red-shifts. Tsuji also interpreted this as being caused by a”quasi-static layer” in the outer atmosphere and derived excitation temperatures of 1000– 2000Kas well as mass estimates for it. Such a molecular formation zone could be well separated fromthe warm photosphere but not yet in the expanding regions of the cool stellar wind. Applying thesame method to ISO SWS spectra, Tsuji et al. (1997) found excess absorption/emission for otherspectral features (molecular bands of H2O, CO, SiO, CO2), too. They claimed that such a ”warmmolecular layer” could be located at about 2 R∗ and could be common not only in Miras, but inevolved red giants in general. Still it was not clear how such an extra component within a red giantatmosphere could develop and persist.

The atmospheric model S used here shows a similar velocity behaviour to that described above(compare Figs. 1.12+2.4). Around 2R∗ we find a region where gas velocities are rather small atall times. In this transition zone between the pulsating photosphere and the cool outflow, dustis being formed (cf. Fig. 2.4d) and the stellar wind is triggered. Low-excitation CO ∆v=2 lines,which sample these layers, show only small blue-shifts and RVs would be observable close to thestellar systemic velocity. This means that dynamic model atmospheres of Hofner et al. (2003a) canproduce a phenomenon resembling a quasi-static layer in a consistent way, and it is not necessaryto introduce it artificially. This is at least true for some combinations of stellar parameters. Asit was shown in Sect. 3.5.1, a relatively small change in the chosen parameters (e.g. 1→1.5M⊙)can strongly influence velocities in the mentioned layers, due to the non-linear response of dustformation to the thermodynamic conditions in the upper atmosphere.

It will be interesting to perform a parameter study of various models with regard to this phe-nomenon. On the other hand observational evidence of a static layer for a statistically meaningfulnumber of stars should be possible with the upcoming IR spectrographs. There is already some ef-fort being spent to investigate existing high-resolution spectroscopic data (Lebzelter et al. 2003). Inaddition, interferometric measurements of Miras will help to tackle this question (e.g. Perrin et al.2004).

The possible formation mechanism of ”warm molecular layers” was investigated by Woitke et al.(1999) for O-rich Miras and by Helling & Winters (2001a, 2001b) for C-rich ones. They foundenhanced molecular column densities in the layered structures of dynamic models of AGB outeratmospheres (compared with predictions of hydrostatic models), caused by stellar pulsation andpropagating shock waves. Similar results regarding the partial pressures of selected molecules weredescribed by Hron et al. (1998) and Loidl et al. (1999), also discussing the effects on the resultingsynthetic spectra of C-rich models.

synthetic line profiles · 2007-05-09 · chapter 3 synthetic line profiles 3.1 modelling line...

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