6 Summary and discussion

Identification of pulsation modes (fundamental, overtone, presumably the first) of carbon LPVs was provided in a period-radius diagram. The comparison with theoretical tracks in Figs. $\ref{pr1}$ and $\ref{pr3}$ and a study of stars with bi-periodicity (Fig. $\ref{pr2}$), were of considerable help. Mean pulsation masses were derived from theoretical PMR relations. They are quoted in Table  $\ref{mass}$ together with mean densities and mean surface gravities. The range of pulsation masses was found to be 0.6- $4.0~M_{\odot}$ for the majority of carbon-rich giants. This is in good agreement with the previous studies made on various subsamples by Claussen et al. (1987), Thronson et al. (1987), Zuckerman & Dyck (1989), and Barnbaum et al. (1991). The agreement is also good with studies on individual sources like IRC+10216, from the comparison of theoretical nucleosynthesis models and measurements of abundances in circumstellar envelopes (specifically LMS-models favored and a $5~M_{\odot}$-model rejected from the isotopic ratios observed in this object: Kahane et al. 2000). A mass-luminosity diagram was derived (Fig. $\ref{m_l}$) and discussed. It should be kept in mind that it applies to mean values of both quantities, with large standard deviations on both of them. Intrinsic ranges are involved since parameters like abundances span large domains. Finally, the diagrams of C/O abundance ratios vs. effective temperatures were constructed (Figs. $\ref{co1}$, $\ref{co2}$ and $\ref{co3}$) for the three classes of carbon-rich giants in our sample (CH stars, HC not classified CH, and CV).

Having summarized the main results of the present paper, we discuss them in a wider perspective. The comparison of various LFs in Figs. 5, 6 and 7 of Paper III, and the study of the variations of the C/O abundance ratio vs. effective temperature (Figs. $\ref{co1}$, $\ref{co2}$ and $\ref{co3}$), confirmed the existence of three samples of carbon-rich giants in the Sun's vicinity, as shown in Paper II, on the grounds of space distributions and kinematics

• the CH stars (from spectroscopic criteria), classified in either HC-groups or oxygen-rich groups (members of the spheroidal component; $\left({\rm C/O}\right)\rm {max}\simeq8$ at $\left(T_{\rm {eff}}\right)\rm {max}\simeq4900~\rm {K}$),
• the HC-stars not classified CH stars (mainly members of the thick Galactic disk, with a LF similar to that of the MWB; $\left({\rm C/O}\right)\rm {max}\simeq3.3$ at $\left(T_{\rm {eff}}\right)\rm {max}\simeq3870~\rm {K}$),
• the CV-stars (mainly members of the old thin disk, with a LF similar to that of the LMC ( $\left({\rm C/O}\right)\rm {max}\simeq1.5-1.75$ at $\left(T_{\rm {eff}}\right)\rm {max}\simeq2500~\rm {K}$).

The metal-poor systems, like the SMC (Z= 0.004) and spheroidal dwarf galaxies, display faint carbon stars such as the Galactic HC-stars. The two maxima of the SMC-LF are however shifted from the Galactic and LMC ones (see Figs. 5, 6 and 7 of Paper III). This probably corresponds to evolutionary tracks shifted toward larger luminosities and higher effective temperatures, for decreasing (Z) metallicity. The 43 CH stars in the outer LMC halo (Hartwick & Cowley 1988) are much brighter (Feast & Whitelock 1992; $-4\ga M_{\rm {bol}}\ga -6$) than the Galactic ones.

Prochaska et al. (2000) found that the thick disk stars had a chemical enrichment history similar to the metal-rich halo stars $\left(\left[\rm {Fe/H}\right]\simeq-1\right).$They also concluded that the thick disk abundance patterns are in excellent agreement with the chemical abundances observed in the metal-poor bulge stars, suggesting the two populations formed from the same gas reservoir at a common epoch.

It was shown that, globally, the mean photospheric radius and mean luminosity increases with the mean effective temperature, along the sequence of the photometric groups (HC and CV, with the exception of the hottest one HC0 which is brighter than HC1, and oxygen-rich groups are still brighter at even higher temperatures). The results are quoted in Table 3 of Paper III and the corrections for the Malmquist bias, although small, were given as well. The mean radii and luminosities increase along the sequence "Constant, Lb, SRb, SRa and Miras'' of increasing variability, while mean effective temperature decreases. It corresponds to stars increasingly later on average, along the photometric sequence (see Fig. 8 in Paper II).

In the theoretical HR diagram, a majority of carbon-rich stars are found between the evolutionary tracks of initial masses $M=0.8~M_{\odot}$ and $M=4~M_{\odot}$ for Z= 0.02. The loci may also be fit with lower masses tracks, when lower metallicities are considered. The derived pulsation masses (0.5- $4~M_{\odot}$) are in good agreement with those initial masses. On a small sample, Alksnis et al. (1998) found a reasonable agreement with tracks of (0.7- $4~M_{\odot}$), making use of some observed parallaxes. For S stars, Van Eck et al. (1998) selected 1.5 to $5~M_{\odot}$ model tracks. The loci of C and S stars overlap in the HR diagram. The initial O/H ratio and mixing history determine whether a star of given mass and luminosity is now a C star or a S star. The Ba II giants $\left(\left<M_{\rm {bol}}\right>\simeq -0.3\pm1.3\right)$ are found fainter, on average, than the early HC-stars, and coincide with the clump observed in many clusters. Most RCB variables and HdC stars range from $M_{\rm {bol}}\simeq -1$ to -4 against -0.2 to -2.4 for those of the three population II Cepheids in the sample (mean pulsation mass $\simeq$ $0.6~{M}_{\odot}$ as expected).

The CV-stars are located at the ends of their TP-AGB tracks and those ends are located at increasingly higher luminosities (tip of TP-AGB), shifting toward lower effective temperatures due to increasing opacities occurring in the atmospheres. Increasingly larger C/O ratios responsible for that are actually observed (see Fig. 16 in Paper I and Fig. $\ref{co3}$ in the present paper). The apparent decrease at $T_{\rm {eff}}\le2500~\rm {K}$ is likely the effect of carbon atoms trapped in increasing numbers of SiC and carbon grains.

The observed general trend of increasing luminosities and radii along the HC-CV photometric sequence for decreasing mean effective temperatures is the consequence of stellar evolution along the RGB and then the AGB. Some peculiar objects, including hot oxygen-rich stars (RCB variables, HdC stars, carbon cepheids), were discussed and most objects previously classified in luminosity classes II or even I on spectroscopic grounds proved to be fainter than true supergiants or bright giants. The HC-giants were found to be, on average, brighter than the BaII giants whose main concentration coincides with the clump. A second version of the HR diagram (Fig. 9 in Paper III) was restricted to the carbon giants. The quasi-vertical boundary line between the HC-region (left) and the CV-region (right) is nearly identical to the Carbon Star Formation Line (CSFL; e.g. Scalo 1976) associated with the third dredge-up in stars of various masses reaching the TP-AGB phase, for a given metallicity (Iben & Renzini 1983; Busso et al. 1999; Marigo et al. 1999). The leftward evolution of this CSFL with decreasing metallicity is illustrated in Fig. 4 of Westerlund et al. (1995). The lower limit to the transition in luminosities intervenes at $M_{\rm {bol}}\simeq-3.6\pm0.4$which is in agreement with $M_{\rm {bol}}\le-4.0\pm0.4$obtained by Marigo et al. (1999) from evolutionary calculations. The positions of Tc-stars were also shown with a barycenter at $M_{\rm {bol}}\simeq-4.6~\left(-3.8;~-5.8\right),$about 1 mag brighter than the above-mentioned lower limit for TP-AGB carbon stars. The CV-giants and part of the HC5-objects, with 0.5- $4~M_{\odot}$ and aged 0.2 to 10-12 Gyr, members of the thin disk, can be identified with TP-AGB stars experiencing TDU. The situation is far less clear for the (thick disk members) HC-stars. Various models may be considered for those very old (11 Gyr?), low mass stars (initial value $\la$ $1.1~M_{\odot}$, present one $\simeq$0.5- $0.8~M_{\odot}$ due to mass loss).

With the exception of CH stars, no evidence for binarity was found in any of those stars (McClure 1997a), and coalescence of components in a former binary system was invoked. It seems unlikely on the basis of low pulsation masses $\left(0.5~M_{\odot}\right)$ found for HC3 to HC5-stars, and the high frequency of those objects (at least 24% of carbon stars, including 6% for CH stars: Paper II). The "extrinsic'' models with mass transfer in a binary system, on a dwarf or giant, should not be considered any longer for HC-stars that are not CH stars. We are left with the following possibilities

• (1) unpredicted extra mixing on RGB or E-AGB (in a low mass star or around a almost stripped-off core), or sufficient dredge-up of carbon occurring at He-flash, as proposed by Westerlund et al (1991a) for the faint MWB carbon stars and Westerlund et al. (1995) for the faint SMC carbon giants,
• (2) low-mass stars near minimum in an interpulse phase or post last pulse (e.g. $1~M_{\odot}$ models of Sackman et al. 1993),
• (3) RGB or E-AGB carbon giants evolved from dwarfs born with $\rm {C/H\ga O/H}$ at low O/H (no carbon dragged up needed); in extremely metal-poor stars, Norris et al. (2001) confirmed large supersolar values of $\left[\rm {C/Fe}\right]$ that can hardly be attributed to internal mixing effects.

Some extra mixing like cool bottom processing (CBP) is needed to explain the isotopic abundance ratios (specially $^{12}\rm {C}/^{13}\rm {C}$) in low-mass AGB stars (Busso et al. 1999). As shown by Schlattl et al. (2001), in initially metal-free LMSs, the low entropy barrier between the helium- and hydrogen-rich layers enables a penetration of the helium-flash-driven convective zone into the inner tail of the extinguishing H-burning shell. As a consequence, protons are mixed into high temperature regions, triggering an H-burning runaway. The subsequent dredge-up of matter processed by He and H burning enriches the surface with large amounts of helium, carbon and nitrogen. Large $\rm {\left[C/Fe\right]}$ and $\rm {\left[N/Fe\right]}$ abundance ratios are precisely observed in very metal-poor stars (e.g. $\rm {\left[C/Fe\right]}$ enhanced by up to 2.0 dex for $\rm {\left[Fe/H\right]} < -~2.5;$ Rossi et al. 1999; see also Chieffi et al. 2001). The possibility (1) is however the more simple and direct explanation for HC-stars in the thick disk.

A very recent result may highlight the issue in favor of possibility (3): new stellar models with masses ranging between 4 and $8~M_{\odot},$ Z=0 and Y=0.23 were published by Chieffi et al. (2001). In models with a mass larger than $6~M_{\odot},$ the second dredge-up is able to raise the CNO abundance in the envelope enough to allow a normal AGB evolution with TPs and TDU. In models of lower mass, the authors find efficient convection associated successively with a He-flash and a H-flash, resulting in carbon abundance in the envelope rising to a level high enough to lead to further evolution similar to that of more metal-rich AGB stars. These population III stars now became white dwarfs provide an important source of primary carbon and nitrogen, which would imply a major revision of the history of chemical evolution in the early Galaxy. Our HC-giants, members of the thick disk, may have received part of this material. Enrichment in s-process elements also takes place in population III AGB stars (Goriely & Siess 2001).

In possibility (2), the HC-giants could have been halted in their ascent of the AGB by mass loss, leaving only a tiny envelope around a 0.5- $0.6~{M}_{\odot}$-core. This scenario can actually be invoked for faint early HC-stars $\left(-3.5 \la M_{\rm {bol}} \la -1\right),$ as descendants of low initial mass stars. Brighter $\left(-3.0 \la M_{\rm {bol}} \la -4\right)$ HC5-giants $\left(M\simeq 0.55~M_{\odot}\right)$ and similar CV1-stars could be the remnants of part of higher initial mass stars ($\ga$ $1.1~M_{\odot}$) of the thin disk population. The effect of mass loss is still observed in low mass giants $\left(M\simeq 0.5~M_{\odot}\right)$like the underluminous CV5-CV6 objects $\left(T_{\rm {eff}}\le2700~\rm {K}\right)$ of Sample 3 in Bergeat et al. (1998), which are about 1.4 mag fainter than the other cool CV-stars (such underluminous stars are also observed in the LMC).

The underluminous CV5-giants might be objects observed at $M_{\rm {bol}}\simeq -3.5\pm0.3$ at their luminosity minimum in the interpulse phase (e.g. the evolutionary tracks from: Lattanzio 1987; Sackmann et al. 1993; Steffen et al. 1998; and Ford & Neufeld 2001).The interpulse period may reach up to $\simeq$ $2 \times 10^{5}$ Yr for a $M_{\rm {c}}\simeq0.5$ core (e.g. Marigo et al. 1996). The time interval spent by the star close to the luminosity minimum is only a small fraction of this period, which could explain why underluminous CV5 and CV6-stars are so few (2% of our CV-sample). The low (but physically acceptable) pulsation masses derived ($\simeq$0.5- $0.6~{M}_{\odot}$, similar to those obtained for the HC5- and CV1-giants) may alternatively suggest that we are observing nearly stripped-off cores with only a tiny envelope of less than $0.1~M_{\odot}.$ In such a case, having suffered their last thermal pulse, they may be transiting leftward in the HR diagram, on their track toward the white dwarf region. With further envelope thinning, increasing effective temperatures at nearly constant luminosity are predicted (e.g. the above references). At least part of the HC5- and CV1-giants could thus have the underluminous CV5-stars as progenitors (evolution from $T_{\rm {eff}}\simeq 2650~\rm {K}$ up to 3300-3500 K at nearly constant luminosity $-3.3\ge M_{\rm {bol}}\ge-3.7$). In the mass-luminosity diagram (Fig. $\ref{m_l}$), both objects do populate the lower left corner with $M/M_{\odot}<1.$

Girardi et al (2000) published a diagram showing the relation between the final masses (white dwarfs) and the corresponding initial masses, from both empirical and theoretical origin (their Fig. 4, p. 380). Final masses in the $M\simeq 0.5$- $0.7~M_{\odot}$ range have progenitors of less than $3~M_{\odot}$ while more massive white dwarfs ( $M\simeq0.8$- $1.1~M_{\odot}$) should be produced by stars initially in the $\simeq 3.5$- $5.5~M_{\odot}$ interval. For objects having become carbon giants, our results in Table  $\ref{mass}$ suggest that the boundary between both categories might lie at CV5 with $\left<M/M_{\odot}\right>\simeq\left(2.7\pm0.5\right).$

The mean stellar density is continuously decreasing along the photometric sequence from HC5 to CV6 at least, from a few 10^-7 to slightly less than 10^-8, in solar units. The surface gravity is nearly constant along the photometric sequence (i.e. along evolutionary tracks) with a mean value of about $g=\left(5.1\pm0.7\right) \times 10^{-3}$ SI or $\simeq 0.5$ CGS $\left(\log~g\simeq-0.3\pm0.1\right).$ For stars earlier than HC5, an increase to about $\log~g\simeq 0$ (or -0.1) at 4000 K is indicated. Both values are in good agreement with the $\log~g$ vs. $T_{\rm {eff}}$ diagram of Hill et al. (2000, their Fig. 6) for carbon-enriched metal-poor stars. The LPV data roughly obey $P\propto \left(R/R_{\odot}\right)^{1/2}$ for both modes, like simple oscillators. With theoretical PMR like $P= Q\left(R/R_{\odot}\right)^{1.5}\left(M/M_{\odot}\right)^{-0.5}$ and Q constant or slowly variable, one obtains $M\propto R~^{2}$ or nearly so. We found about $\left(L/L_{\odot}\right)\propto \left(M/M_{\odot}\right)^{0.5},$ and thus $\left(L/L_{\odot}\right)\propto \left(R/R_{\odot}\right)$ or nearly so with an exponent slightly larger than unity. The uniform surface gravity in Table  $\ref{mass}$ is thus a direct consequence of internal structure and shape of the evolutionary tracks in the HR diagram.

We have attempted to build a global scheme including all of the derived data and many studied objects. Some speculative considerations were of course needed in this discussion, but the authors feel comforted by the consistency and overall agreement of the masses and luminosities, with what can be learnt from other sources (both theory and observations). The lowest effective temperatures $\left(T_{\rm {eff}}\le2400~\rm {K}\right)$ are not reproduced by evolutionary models. This is not surprising since detailed model atmospheres with extensive opacities (gas + dust) are needed to achieve that, but luminosities are practically not affected. For Milky Way carbon giants, the accuracy of luminosities, photospheric radii and inferred pulsation masses will certainly be strongly improved by future astrometric space missions. Meanwhile, many additional angular diameters (and time variations) will become available from the Very Large Telescope Interferometer (VLTI).

Acknowledgements

Valuable suggestions from the referee Dr. Maurizio Busso are gratefully acknowledged.