Free Access
Volume 552, April 2013
Article Number A20
Number of page(s) 16
Section Stellar atmospheres
Published online 15 March 2013

Online material

Appendix A: Various comparative data

In Sect. 6.1 (and Appendix A) we use photometry of the 2MASS All-Sky Point Source Catalog (Cutri et al. 2003) for a section of the Galactic bulge, which is available from the NASA/IPAC Infrared Science Archive (IRSA)11. Photometric data were downloaded for objects of a field located in Baade’s window, we extracted photometry for an area of Δα × Δδ ≈ 0.5° × 1°. Only stars fulfilling σJHK< 0.05mag were chosen, the correction for interstellar reddening was carried out according to the extinction values given by Dutra et al. (2002). The 2MASS photometry had to be transformed to our Bessell system using the equations given in Carpenter (2001; see their Appendix A). To convert the K-magnitudes of 2MASS to bolometric magnitudes (Fig. 11) we made use of the bolometric corrections BCK as a function of (J – K) derived by Montegriffo et al. (1998). To this end we fitted the values tabulated in Montegriffo’s Table 3, which are labelled “metal-rich stars”. The resulting regression12 was then applied to the 2MASS data.

In addition, we used a sample of O-rich Miras in the Galactic bulge identified by Groenewegen & Blommaert (2005) while investigating the lightcurves produced by the OGLE-II survey. The authors cross-correlated their objects with the 2MASS catalogue leading to a subsample of 1619 LPVs which are plotted in Fig. A.1. Dereddening was done following the values for interstellar extinction for each OGLE-II field as listed in Matsunaga et al. (2005). For the conversion between 2MASS and Bessell filters we applied again the relations of Carpenter (2001).

The reddening-corrected photometric data (apparent magnitudes) for all bulge objects were shifted to absolute magnitudes by assuming a distance modulus of (m – M) = 14.7mag for the centre of our galaxy (e.g. Vanhollebeke et al. 2009).

For the comparison in Fig. 10 we made use of the data for variable stars in the LMC presented in Ita et al. (2004). The authors analysed OGLE lightcurves and provide estimates for the periods for a large number of objects, as well as the corresponding JHK photometry. We dereddened the data with the value for AK given by Feast et al. (1989) and transformed the near-IR photometry from the LCO to the Bessell system with the help of Carpenter (2001). We also adopted the P-K relations fitted by Ita et al. (2004; cf. their Table 3) and overplotted them in Fig. 10. As a distance modulus for the LMC we assumed (m – M) = 18.5mag (e.g. Gibson 2000; Freedman et al. 2001; Alves 2004; Laney et al. 2012).

The relation between mass-loss rates and (J – K) colours empirically found by Le Bertre (1997) and plotted in Fig. 7 is given in the ESO photometric system and was transformed to the Johnson-Glass system according to the equations given in Bessell & Brett (1988). The other relation given in Gullieuszik et al. (2012; their Eq. (2)) was transformed from the 2MASS system by applying the conversion of Carpenter (2001).

The magnitudes of the solar-like hydrostatic COMARCS models – a sub-grid is used in Fig. 9 – are of the same kind as the corresponding ones of the dynamic models (Paper I).

Finally, we compiled various data sets from different sources for the comparison of bolometric corrections (BCs) shown in Fig. 9. The observational results of Mendoza & Johnson (1965) are already given in the Johnson system, no conversion is needed therefore. The fit to the values of an extended sample of C-type giants provided by Bergeat et al. (2002; their Table A.1) is based on observed data collected by the authors from the literature. According to Knapik et al. (1997) these measurements were obtained in the “Arizona system or close to it” (i.e., Johnson) and can thus be compared directly (Bessell & Brett 1988, Table 1) to the models. The fit to the data of a sample of nearby field C stars provided by Kerschbaum et al. (2010) in their Eq. (1) is based on photometry in the new ESO system, which is very close to the Johnson system according to the authors (see also Bouchet et al. 1991). The fit for C-rich AGB candidates in the LMC given by Riebel et al. (2012) in their Table 7 was transformed from the 2MASS system to our standard system with the help of Carpenter (2001).

Appendix B: Loops in colour–magnitude diagrams

In the previous Sects. 5 and 6 we analysed the applied models on the basis of mean photometry. Here we want to outline how typical C-type Miras vary within CMDs constructed from NIR photometry.

In Fig. A.1 we show the same combined population of galactic objects as in Fig. 11, but with absolute K magnitudes on the ordinate instead of bolometric ones. This enables us to overplot also the Miras (most likely O-rich9) found by Groenewegen & Blommaert (2005) in the 49 OGLE fields across the Galactic bulge. For a description of the necessary post-processing of the 2MASS photometry we refer to Appendix A. The fact that the 2MASS survey provides “only” single-epoch data (instead of a time series that could be averaged) introduces some scatter in for these variable stars. Still it is clear that the latter objects are all located on the extension of the pronounced red giant branch towards even brighter MK in Fig. A.1. The C-type Miras of the W06 sample form again the noticeable sequence from the bright end of the red giant branch towards redder colours with (J – K)0 up to ≈6mag. Also plotted in Fig. A.1 is the synthetic photometry for model S, illustrating the difference between the dust-free hydrostatic initial model and the developed dynamic model in a CMD.

thumbnail Fig. B.1

Colour–magnitude diagram containing observational data compiled from different sources: (i) 2MASS photometry of stars in Baade’s window (black dots); (ii) 2MASS photometry of Mira variables in the Galactic bulge (turquoise dots) identified by Groenewegen & Blommaert (2005); and (iii) reddening corrected mean magnitudes and colours of field C-Miras (grey filled circles, blue diamond for the individual RU Vir data) from Whitelock et al. (2006, Table 6) shifted to an absolute scale MK according to the distances given by the same authors. Overplotted are the modelling results for model S, i.e. the location of the hydrostatic initial model as well as the average of several phases of the different cycles in Fig. A.2. The colour-code is the same as in Fig. 16 of Paper II. The box (dotted lines) marks the range covered in Fig. A.2.

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thumbnail Fig. B.2

Colour–magnitude diagram containing individual observations of RU Vir adopted from Table 2 of Whitelock et al. (2006) and shifted to an absolute scale MK by applying the distance modulus derived by the same authors (cf. their Table 6). The dashed line marks the average variation of RU Vir during the light cycle as derived from the Fourier fits in Fig. 8 of Paper II. Overplotted are the corresponding synthetic photometric data of model S. Three different pulsation cycles are plotted with the same symbols, colour-coding, and labels as in Fig. 13 of Paper II. The arrows mark the directions of how the objects pass through the loops during a light cycle, selected phases are labelled (convention for φbol / φv as in Paper II).

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Figure A.2 (showing a close-up of Fig. A.1) illustrates the temporal variations, with model S and the C-type Mira RU Vir as examples. Both occupy the same range of the CMD and show variations on the order of 1mag in K as well as (J – K)0. Apart from the recognisable cycle-to-cycle variations the model shows for every pulsation period similar broad loops throughout the light cycle. It appears bright and blue at maximum light (phase φbol ≈ 0.0), while it is faint and red around minimum light (φbol ≈ 0.5). As described in Paper II, we cannot plot such loops for RU Vir directly as the observed data points were obtained with an insufficient sampling rate over several periods. Instead, we merged the measurements from different periods into one combined light cycle. Based on the sine fits (one Fourier component) to the resulting lightcurves (see Fig. 8 in Paper II) we can derive simulated variations of RU Vir, which are shown with a dashed line in Fig. A.2. The range of this simulated loop is narrower than the range covered by the individual data points. As noted in Paper II (cf. the comparison of original (J – K) data vs. the corresponding simulated colour curve in Fig. 12 there), this can be explained by the occuring cycle-to-cycle variations in the lightcurves. Nevertheless, the general behaviour of looping through the CMD, with extreme phases in the same order (φbol ≈ 0.0 at the upper left end, φbol ≈ 0.5 at the lower right end), is similar to the model. Again we find that the model loops counter-clockwise in the CMD of Fig. A.2, while RU Vir follows a clockwise movement. We demonstrated in the appendix of Paper II how a (relatively small) phase shift between the lightcurves obtained in the different filters can lead to a change in the sense of rotation. Interestingly, this behaviour has only scarcely been investigated directly by observational studies. Payne-Gaposchkin & Whitney (1976; their Figs. 5a−c) studied a large sample of LPVs and found them – averaged over certain groups – to vary in CMDs very similarly to our results. Unfortunately, they show only loops for stars of spectral types M and S (clockwise), while the corresponding data for C-rich objects was according to the authors afflicted with large scatter. Eggen (1975; see his Figs. 18, 24, and 30) monitored large-amplitude variables (mostly M-type) and plotted their variations in the Mbol-(R – I)-plane. Although the sampling of these time series is limited, the stars seem to perform loops with the sense of rotation being mostly counter-clockwise. A slightly different approach was pursued by Wing (1967), who obtained a spectro-photometric time series over ≈2.5 years for about 25 Miras from which he could derive oxide band strengths and temperature estimates. According to Wing (priv. comm.) the objects showed quite a variety of behaviours. The results for only five of these objects were then published in Spinrad & Wing (1969; their Fig. 9), showing clockwise as well as counter-clockwise variations.

Note that in Fig. 3 of Nowotny et al. (2011b) we plotted a colour–magnitude diagram MK vs. (V – K)0, showing the combined contents of Figs. A.1 and A.2 from here. Although this differing filter combination provides a valuable colour index because of the broad basis in wavelength, it is generally harder to obtain simultaneous measurements in these two filters. For most of the objects in the W06 sample there is no information concerning their brightnesses or even light variations in the V-band and they are not included in this plot. For a quite limited sub-sample of the C-type Miras we could estimate a mean visual magnitude from the photometric time series available in the AAVSO database, providing a hint on a mean (V – K)0. This is only possible for stars that are bright in the visual and not strongly influenced by circumstellar reddening. Thus, only objects with  ≤ 2 × 10-6M yr-1 could be studied. The C-rich Mira RU Vir is the only target for which simulated variations in the MK-(V – K)0-plane could be derived from the Fourier fits of the lightcurves (see Fig. 8 in Paper II). Both RU Vir and model S show a relatively similar behaviour as found here (loops), with the averaged colour of the latter being significantly redder, though.

Appendix C: Extreme dust formation

thumbnail Fig. C.1

Upper panel: movement of the outer layers of a specific dynamic model atmosphere with dust formation occuring every second pulsation period (Appendix A). Colour-coded is the degree of condensation of carbon atoms into amorphous carbon dust particles. Lower panel: synthetic lightcurves for the broad-band filters V and K (filled circles), together with the corresponding photometry for which the dust opacities were neglected (open circles). Note that the K-lightcurves were shifted by 3mag to better fit into one panel.

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thumbnail Fig. C.2

Radial atmospheric structures for the dynamic model used for Fig. A.1 and discussed in Appendix A. Shown are two characeristic phases (marked with dotted lines in Fig. A.1) where dust formation takes place (φbol = 2.573) or not (φbol = 1.585).

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In Sect. 4.2 we illustrated how the time-dependent dust formation in the outer layers of a C-type Mira effects the resulting lightcurves of such an object. This was based on a dynamic model atmosphere with relatively moderate dust formation processes. Here we show a different model13 adopted from the recent grid of Mattsson et al. (2010) which represents the quite rare group of strictly “double-periodic” models (cf. the discussion in Höfner et al. 2003). As it can be seen in the upper panel of Fig. A.1, only every second pulsation period a new dust shell emerges and propagates outwards due to the radiation pressure acting upon the dust grains. Figure A.2 shows that the radial structure of the model replicates in the inner, dust-free layers from cycle to cycle for a given phase φbol. However, the layers outwards from the dust-forming region at ≈2–3 R can differ significantly because of the extreme dust formation. The emergence of the quite narrow dust shell is also reflected in the photometric variations plotted in the lower panel of Fig. A.1 (see also the discussion in Sect. 4.2). While the K-lightcurve follows approximately the sinusoidally changing luminosity input, the model exhibits especially in the visual a characteristic lightcurve

which is reminiscent of “Miras with secondary maxima” as discussed e.g. by Lebzelter et al. (2005). It would be interesting to make a more detailed comparison with observed targets showing a similar behaviour, as for example the cases presented in Wood (1999; object 7308.113 in their Fig. 2), Groenewegen et al. (2004; object 050709.50-685849.4 in their Fig. 9), Lebzelter et al. (2005; R Nor in their Fig. 8), Lebzelter et al. (2005; LW10 in their Fig. 2) or Lebzelter (2011; ASAS 201445-4659.0 in their Fig. 1), in the future.

© ESO, 2013

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