2 The pulsation modes of carbon LPVs

2.1 Introduction

The long period variables (LPVs) are cool giants whose pulsation properties are similar to those of their oxygen-rich analogues. A few Population II Cepheids (three stars with two confirmed carbon-rich Cepheids, are included in our sample: see Sect. 7.6 of Paper III). We have shown in Sect. 5 of Paper III that mean luminosities and photospheric radii are, on average, increasingly larger along a sequence of increasing variability. No discontinuity in the obtained data could be observed. Here, we concentrate on variables of the classes SRb and SRa (semi-regulars) and M (Miras) with carbon-rich atmospheres and known periods. Many SRb-variables or stars classified SR or SR:, are not included in our study in the absence of reliable periods. Conversely, new recent data of higher accuracy, mainly from HIPPARCOS, showed that a few stars previously classified as Lb (irregulars), actually display light curves of SRb (semi-regulars) variables.

As it is the case for classical Cepheids, period-luminosity relations, then refined as period-luminosity-radius relations (PLR), have been established. A period-luminosity relation was discovered for the Miras (essentially oxygen-rich) of the LMC (Glass & Lloyd Evans 1981). It was subsequently improved by Feast (1984) and Feast et al. (1989), who considered the relation as a relatively tight one, the dispersion being of only $\pm 0.15$ mag. The relation was slightly revised by Groenewegen & Whitelock (1996) for carbon Miras, with a zero-point based on the Galactic star UV Aur, a member of a double system (see also van Leeuwen et al. 1997).

A crucial question is that of the mode of pulsation. Concerning oxygen-rich LPVs, arguments in favor of the first overtone were given by Haniff et al. (1995), although the fundamental mode may often be seriously considered in some cases. Barthès (1998) modeled a sample of 22 Miras and found that a significant proportion of them pulsate in the fundamental mode. Bergeat et al. (1998) studied the period-luminosity relation of Galactic carbon variables from HIPPARCOS data, making use of $M_{\rm {K}},$ the absolute magnitude in the near infrared, and compared it with its analogue in the LMC. The period-luminosity relation they found for Galactic carbon variables ("Sample 1'') is very close to the relation of Reid et al. (1995) for their LMC analogues. In both systems however, brighter carbon variables are observed at short periods ("Sample 2'') and underluminous stars ("Sample 3'') are found at long periods. Following Wood & Sebo (1996), Bergeat et al. (1998) suggested that stars in Sample 2 are overtone pulsators while those in Sample 1 might be pulsating in the fundamental mode. The status of Sample 3 underluminous variables remained unclear, but they should be pulsating in the same (fundamental) mode as Sample 1 stars. Mode identification was provided in a period-radius diagram, for some oxygen-rich variables, by van Leeuwen et al. (1997). A similar approach is applied to the carbon-rich variables in Sect. 2.2.

Figure 1: The period-radius diagram of 155 carbon variables. The theoretical predictions for fundamental (F) and first-overtone (OV) pulsators are shown for masses 1 and $3.5~M_{\odot}.$ Linear calculations are labelled with l while (non-linear) dynamical ones are referred to as d. The proposed identification of pulsation modes is also shown (see text for details).

Figure 2: An enlarged portion of Fig. $\ref{pr1}$ including 12 carbon variables with confirmed bi-periodicity. The diagram is split into two zones for fundamental and first-overtone pulsators respectively. The adopted separation lines are given by Eqs. (1), (2) and (3). The two nearly horizontal loci over 200-500 $~R_{\odot}$ are interpreted in terms of increasing pulsation masses from $\left(0.8\pm 0.3\right)~M_{\odot}$ to $\left(3.5\pm 0.7\right)~M_{\odot}$ (see text for details).

Theoretical calculations of PLR relations were published for oxygen-rich LPVs (Fox & Wood 1982; Wood et al. 1983; Wood 1990, and references therein). Unfortunately, no calculation is available for carbon-rich opacities. Luminosities and photospheric radii predicted for overtone models are larger than those for fundamental models, for a given period of pulsation. The calculations of pulsations in stellar envelopes are usually linear ones, with or without the assumption of adiabaticity. In many cases, when the fundamental mode predominates, the model diverges. This mode was interpreted as a source of violent processes of strong mass loss (Wood 1974; Tuchman et al. 1979), and considered as not reproducing the actual observations which imply more steady mass loss. The modeled period of the first overtone and the velocity amplitude in the atmosphere often fall short of observed values. Ya'ari & Tuchman (1996, 1999) operated non-linear dynamical simulations which follow up on the development of oscillations due to initial perturbations chosen in various ways. The calculations which spread over 600 years, show that the models end up pulsating in a fundamental mode whose period is shorter than the period of the initial static model, after having pulsated e.g. for more than 150 years in the first overtone. This is the consequence of a thermal adjustment in the envelope (Ya'ari & Tuchman 1996). Models also exist which remain in the first overtone (Ya'ari 1999). Barthès (1998) however argued that dynamical models are not fully reliable at present, and presented satisfactory fits obtained from linear models.

We have collected the periods available for our pulsating variables, from the GCVS (Kholopov et al. 1985 and further extensions in the literature) and/or from HIPPARCOS (1997) data.

2.2 The period-radius diagram

With our goal of mode identification in mind, we established a period-radius diagram of carbon variables. The used photospheric radii were taken from the analysis of Sect. 3. They can be recovered from the absolute bolometric magnitudes and effective temperatures quoted in Table 2 of Paper III at CDS. The accuracy of individual values is rather low. They were essentially intended for statistical purposes. The period-radius (PR) diagram (Fig. $\ref{pr1}$) is preferred here instead of the period-luminosity plot since, contrary to the latter, model predictions in this diagram are practically independent of the value adopted for the mixing length $\alpha = l/H_{\rm {p}}$ of convection (e.g. Figs. 1 and 2 in Ya'ari & Tuchman 1999). The curves shown in Fig. $\ref{pr1}$ are those found from linear models of a $1~M_{\odot}$ mass pulsating either in the fundamental mode (Fl1) or in the first-overtone mode (OVl1). Non-linear, i.e. dynamic, results from Ya'ari & Tuchman (1996, 1999), fall on the Fd1-curve for the same mass. Applying the proportionalities at constant radius $P\propto M^{-0.84}$ and $P\propto M^{-0.5}$ respectively for fundamental and first-overtone models (Wood et al. 1983; Wood 1990), two curves were established for models of $3.5~M_{\odot},$ which are labelled Fl3.5 and OVl3.5 respectively.

It can be seen in Fig. $\ref{pr1}$ that most carbon LPVs fall in the region of the 1-3.5 $~M_{\odot}$ theoretical curves, even if both higher and lower masses are indicated for part of the sample. We denote those pulsation masses by $M_{\rm {p}}$ hereafter, with $M_{\rm {p}}\le M_{\rm {i}}$ due to mass loss. The range of $M_{\rm {p}}$ as implied by Fig. $\ref{pr1}$, is in good agreement with the $M_{\rm {i}}\simeq$ 1-4 $~M_{\odot}$ range found in Sect. 7.5.2 of Paper III. We emphasize that higher overtones (specially second and third) cannot be excluded, at least for some stars, since calculations suggest they have periods close to those of the first one.

The locus of observations for $R/R_{\odot}\le 800$ is roughly covered by curves for pulsation masses in the above range. The theoretical curves at constant mass shown in Fig. $\ref{pr1}$ exhibit curvatures which are oriented upward. The observed data show the opposite trend at small radii, and a flat portion for $R/R_{\odot}\ge 200{-}250.$In Fig. $\ref{pr1}$, two main loci are well-correlated with the sampling of Bergeat et al. (1998) in their $M_{\rm {K}}$-$\log {P}$ diagram: Sample 2 stars are found in the lower part of Fig. $\ref{pr1}$, while Sample 1 stars are located in its upper part. In addition, underluminous giants with long periods (Sample 3) are found on the left side of the Fl1 curve. Both Sample 1 and Sample 2 loci show flat portions. If those samples actually correspond to fundamental and first-overtone pulsators respectively, they should be indicative of a positive correlation of masses with respect to radii, possibly illustrating similar mass-radius relations for both modes. Making use of the evolutionary tracks of Fig. 8 in Paper III, one can find $M_{\rm {i}}\propto R^{1.5}.$ We shall see in the discussion of Sect. 6 that the exponent appears slightly larger (about R²) for pulsation masses. Thus, the loci of Fig. $\ref{pr1}$ are roughly consistent with the hierarchy of evolutionary tracks in the HR diagram. For both modes, the flat parts of period distributions from $200~R_{\odot}$ to $500~R_{\odot},$ can be the result of a 0.6- $4~M_{\odot}$mass distribution. We present in Sects. 2.3 and 2.4, arguments in favor of this interpretation (mode identifications and mass range).

2.3 Multiperiodicity and mode identification

A detailed period analysis for 93 red semiregular variables by means of Fourier and wavelet analysis of long-term visual observations was published by Kiss et al. (1999).

A majority of the variables show multiperiodic behavior, typically two or three periods. Kiss et al. concluded that their data can be explained as the direct consequence of different pulsational modes taking place. Their stars are mostly oxygen-rich objects. Percy et al. (2001) reported long-term VRI photometry of a sample of small-amplitude red variables, including four carbon stars. Three of them display two periods. We studied this phenomenon in our sample of carbon-rich variables, concentrating here on bi-periodicity. Amongst the 33 possible cases of multiperiodicity we studied, the 12 best-documented cases are shown in Fig. $\ref{pr2}$as connected symbols. This figure is an enlarged version of Fig. $\ref{pr1}$, restricted to $R/R_{\odot}\le 700.$With no exception, the longer periods point to the upper "Sample 1'' locus while the shorter ones correspond to the lower "Sample 2'' locus. These additional data contribute in delineating more confidently the two regions in the diagram. Two segmented lines are shown in Fig. $\ref{pr2}$ which verify respectively

$$% P_{\rm {l}}=1.203~\left( R/R_{\odot} \right)-34$$ (1)

for $R/R_{\odot}\la 230,$ and

$$% P_{\rm {l}}=0.090~\left( R/R_{\odot} \right)+221$$ (2)

for $R/R_{\odot}\ga 230.$ A tentative third half-line is shown for $R/R_{\odot}\ga 440,$namely

$$% P_{\rm {l}}=1.203~\left( R/R_{\odot} \right)-269$$ (3)

assuming an end of distribution at $\left(3.5\pm0.7\right)~M_{\odot}.$ For $R/R_{\odot}\ga 800,$ there is some indication of a new mass increase (4- $10~M_{\odot}$?) among those rare extreme objects. This is the third part of the distribution which is denoted by "First Overtone?'' in Fig. $\ref{pr1}$, for $R/R_{\odot}\ga 700.$

Identification of modes can then be operated with reasonable confidence. The main problem is probably the possible existence of higher-order overtones (second and third) whose periods are presumably close to those of the first one. The longer periods (Samples 1 and 3) populate the range 270-560 days with a mean $\left<P_{0}\right>\simeq 382$ days to be compared to 70-270 days and $\left<P_{1}\right>\simeq 166$ days for the shorter periods (Sample 2). We thus obtain a ratio of $2.3\pm0.7$ or directly from a larger sample

$$% \left< P_{0} / P_{1}\right> \simeq 2.24\pm 0.7.$$ (4)

This is close to the mean ratio theoretically calculated $\left(\ge2.0 \right)$ for fundamental and first-overtone modes respectively (e.g. Fox & Wood 1982, their Fig. 3, p. 204). Amongst 10 stars in common with Kiss et al., we found good agreement for 8 carbon variables and discrepancies in two cases (C36=VX And and C2803=U Hya).

Figure 3: The distribution of carbon LPVs with respect to the ratio $P/\left (R/R_{\odot }\right )$ of period in days to photospheric radius in solar unit. Two extremas are noticed which correspond to the fundamental (F) and first-overtone (OV) modes, respectively. The ranges found from theoretical predictions ( $M\simeq 1~M_{\odot}$) of Ya'ari & Tuchman (1996, 1999) are shown with l for linear calculations and d for (non-linear) dynamical ones.

2.4 The distribution in P/R

To confirm the results of Sect. 8.3, we show in Fig. $\ref{pr3}$, the frequency distribution of the carbon variables with respect to the ratio

$$% \beta = P/\left(R/R_{\odot}\right)$$ (5)

that is the slope calculated from origin in the former diagrams. Two maxima are noticed that correspond well to the values of $\beta$ of the OV and F models respectively. Mean masses larger than $1~M_{\odot}$ are however indicated in both cases (see Sect. 8.2).

There are about 10 objects with very large radii $R/R_{\odot}\simeq 700$-1500. They belong to the CV6 and CV7 groups, and they show evidence for substantial circumstellar shells. We tentatively ascribed pulsation in the first overtone to those stars. The estimated masses in the case of fundamental pulsators would be much too high. Unlikely values (20-30 $~M_{\odot}$) would be attributed to those extreme objects, in strong disagreement with the theoretical tracks in the HR diagram. Alternatively, their radii could have been overestimated due to undetected circumstellar contributions. This is precisely the sample of extreme stars whose low effective temperatures were derived with some difficulty (Paper I).