Online material

Appendix A: Details of the representative cases

In Sect. 3 we defined different classes of time-dependent behaviour and illustrated them with the help of selected models, as shown in Fig. 1. While we kept the stellar mass (0.75 M_⊙) and the f_L parameter (2) fixed for this group of models, the other parameters varied as listed in Table A.1. The models are labelled as T_⋆/log (L_⋆)/M_⋆/log (C–O) + 12/Δu_p/f_L with L_⋆ and M_⋆ in solar units.

The photometric amplitudes and other properties for the illustrative cases can also be found in Table A.1. The column gives the extinction in the V band (at 0.55 μm) at the phase 0.25 (or, more precisely, the mean of these values for all computed cycles); as is further discussed in Paper III, this is a measure of the lower limit of the dust extinction in the model.

pp: 3000/3.55/0.75/8.50/2/2.0

In this particular case, the largest extension is 1.5 stellar radii. No dust is formed in this model, the light curves show a very regular and repeated variation from cycle to cycle; the amplitudes (maximum-to-minimum) are 0.38^mag, 0.42^mag and 0.72^mag for the bolometric, K and V magnitudes. It has the position (0.53, 0.92) in the (J − H) vs. (H − K) two-colour-diagram.

pm: 2800/3.55/0.75/8.50/2/2.0

This model has the same parameters as the pp one except that it is 200 K cooler. It is classified as pm and shows a different dynamic behaviour where the outer boundary varies from smallest to largest extension and back in almost precisely two luminosity periods. The largest extension of the atmosphere is about 2R_⋆; it extends to ≳2R_⋆ every fourth (luminosity) cycle interleaved with an expansion to ≲2R_⋆ (also every fourth cycle). This behaviour is coupled to a higher dust condensation every fourth cycle when the stellar material reaches greater heights. We can note that although the condensation degree is lower than 10^-3, the influence of the dust on the photometric behaviour is clearly visible. The amplitudes are 0.34^mag, 0.40^mag and 0.84^mag for the bolometric, K and V magnitudes, respectively, where we note that the amplitude in V has increased (from the previous pp model) because of the dust. The minima (and maxima) are fainter in the presence of dust a distance, r, ~1.5–2R_⋆. [It might be classified as a SRa type variable star.] Its position in the (J − H) vs. (H − K) diagram is at (0.58, 0.91), that is, slightly redder in (H − K) than the previous one.

pn: 2600/3.70/0.75/8.20/4/2.0

Compared with the pm model, it has a lower T_⋆ by 200 K and an increased luminosity, and also an increased piston velocity amplitude but a lower carbon excess. Here the pulsations cause the atmosphere to expand to between 2 and 4R_⋆ in a rather irregular way. The condensation degree varies from a few percent up to about 25% without any obvious correlation to the pulsation behaviour of the atmosphere. Now we see a stronger influence of dust on the photometric properties. The amplitudes of the variation in bolometric, K and V magnitudes are 0.69^mag, 0.68^mag and 1.33^mag (if we take the mean over all computed cycles; the maximum-to-minimum variation in V over the computed range is 2.34^mag). [It might be classified as a mira variable.] This model is situated around (0.6–0.9, 0.7–1.2) in a (J − H) vs (H − K) plot, that is, redder than the pm model, especially in (H − K).

we: 2800/3.85/0.75/8.20/6/2.0

This model is 200 K warmer than the pn one, but with increased luminosity and piston velocity amplitude. This model rarely has an atmospheric extension smaller than 5R_⋆. The condensation degree varies between 5 and 30%, which implies quite a strong effect on the photometric properties. The amplitudes for the bolometric, K and V magnitudes are 1.42^mag, 1.48^mag and 2.37^mag (the total range in V over all the computed cycles is 4.67^mag). (It might be classified as a mira variable during its wind phase.) In the two-colour (J − H) vs (H − K) diagram the model occupies the range 0.9–1.4 in (H − K) and 0.9–1.8 in (J − H); compared with the pn model, it has moved along the blackbody line towards cooler temperatures.

ws: 2800/3.85/0.75/8.50/2/2.0

This model has the same temperature and luminosity as the we model, but a higher carbon excess, so it develops a wind even at the low piston amplitude of Δu_p = 2 km s^-1. It has a dust condensation degree that is almost constant in time, at 15±1%. The amplitudes in the bolometric, K and V magnitudes are 0.45^mag, 0.43^mag and 0.69^mag, respectively, (the total range in V over all the computed cycles is 1.00^mag). Its position in the (J − H) vs. (H − K) diagram is at (0.91, 1.25) with a small variation since the amount of dust is almost constant.

wp: 2800/3.70/0.75/8.80/4/2.0

A model with a lower luminosity than our we case but a higher carbon excess and piston velocity amplitude, it has a wind in which dust condenses at r ~ 1.5R_⋆ every cycle around phase 0.5. The photometric amplitudes are 0.81^mag, 0.84^mag and 3.78^mag for the bolometric, K and V magnitudes (the total range in V over all the computed cycles is 4.46^mag). (It might certainly be classified as a mira variable.) The dust condensing around phase 0.5 makes the minima in V quite deep. In the two colour diagram the model varies (loops), along the blackbody line, between 1.1−1.4 in (H − K) and between 1.3–1.8 in (J − H).

wn: 2800/3.70/0.75/8.80/6/2.0

It has the same parameters as the wp case, but a higher piston velocity amplitude. Here the wind is more irregular, the dust is still formed around r ~ 1.5R_⋆ but in a more irregular fashion. The amplitudes in the bolometric, K and V magnitudes are 1.26^mag, 1.53^mag and 6.71^mag, respectively, (the total range in V over all the computed cycles is 8.94^mag). In the two-colour diagram the model moves during the pulsations along the blackbody line between 1.2–1.8 in (H − K) and between 1.8–2.4 in (J − H).

Appendix B: Model overview

In the following pages we present a table with photometric and dynamic properties of the models in the present grid. The models are arranged in increasing effective temperature, luminosity, and stellar mass. For each such combination the data are ordered by increasing carbon excess, piston velocity amplitude, and f_L.

In each line, after the model parameters we list the assigned class, log g (surface gravity in cgs units). Then come dynamic quantities evaluated at the outer boundary: mass-loss rate (in solar masses per year), the wind velocity (km s^-1), the carbon condensation degree, and the dust-to-gas ratio. Note that all given values are temporal means, see Sect. 2.3. Then follow the photometric properties: the (full) amplitude of the bolometric magnitude, the mean V magnitude, the range of V magnitudes, the mean K magnitude and its range, and finally the colours (V − I), (V − K), (J − H) and (H − K).

The luminosities and stellar masses are given in solar units. The carbon excesses, log (C–O) + 12 are given on the scale where log N_H ≡ 12.00. The piston velocity amplitudes, Δu_p, are given in km s^-1. The f_L and class designations are described in Sect. 2.2 and Sect. 3 respectively. All the photometric quantities are given in magnitudes.

Appendix C: Material at the CDS

The resulting spectra and synthetic photometric magnitudes in various filters are available for downloading, some photometric data are also summarized in Table B.1 along with dynamic data, such as mass-loss rates and wind velocities.

Among the downloadable material there is also, for each individual model, a two-page fact sheet that looks like Figs. C.1 and C.2 summarizing the dynamic and photometric behaviour. It should be noted that the values for mass-loss rates, gas velocities, and carbon condensation degrees plotted on the fact sheets are values for each snapshot, whereas the values for these quantities given in the figures and tables in this article are computed as averages over pulsation periods as described in Sect. 2.3.

Fig. C.1 Example of the first page of the fact sheets. The top label gives T⋆, log L⋆, M⋆, log (C–O) + 12, Δup, and fL for a quick overview. The upper leftmost diagram gives the K, J and V magnitudes as a function of phase (with maximum luminosity at phase 0.0). The ranges are identical on all fact sheets to facilitate comparisons of different models; this is true for the three upper diagrams. The different colours correspond to the different epochs selected for the spectral synthesis. The middle and right upper panels show two colour–colour diagrams, the blackbody line is also shown. The three middle panels show, as a function of time, the mass-loss rate, the gas velocity, and the condensation degree for carbon, all three at the outer boundary (25R⋆); here the vertical scales are identical on all fact sheets, the time intervals do vary from model to model. For the models without a wind, the position of the outer boundary is shown instead of the mass-loss rate. For the episodic models, we show both of them. The horizontal dashed and dotted lines denote the mean value, and the mean value ± the standard deviation. The bottom part contains the light-curves in K, J and V; here the dotted vertical lines denote the positions of the luminosity maxima, the maximum range in magnitudes is also given, for the V magnitude also the mean amplitude over a (luminosity) cycle. Enlarged versions of the three middle plots, now with automatic scaling, are shown for the selected epochs. Finally, a summary of parameters, dynamic and photometric properties are listed.

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Fig. C.2 Example of the second page of the fact sheets. The colours in the plots correspond to the selected epochs as given on the first page. In each column we show, from top to bottom, the gas velocity, density, carbon condensation degree, and gas temperature, all as a function of distance from the stellar centre. In each panel, individual curves correspond to snapshots in time, and time increases downwards. The distance between two curves is approximately 0.05 in phase. The curves closest in time to phase 0.0 are drawn with thick black lines. The ordinate scales apply to the bottom (last) curve. The vertical short-dashed lines in the panels denote the distances 1, 2, 5, 10, and 20 stellar radii. Note that time increases downwards. Scales OK for last curve (bottom black).

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© ESO, 2014