7 The carbon and BaII giants in the HR diagram

7.1 The HR diagram

The coordinates in Fig. $~\ref{hr1},$ i.e. absolute bolometric magnitudes and effective temperatures, can be found in Table 2 (available at CDS) for nearly 370 objects, including

• about 290 carbon stars (HC and CV-groups, i.e. from early R to late N stars),
• 56 Ba II giant stars (some of them are also HC-stars),
• 15 RCB variables mostly classified in oxygen-rich groups (HdC stars are classified HC and found immediately on their right side),
• 4 SCV-stars (a group which actually coincide with the SC spectroscopic type),
• 6 non-BaII stars, which finally proved to be oxygen-rich objects classified in the corresponding groups.

The derived luminosity data is essentially intended for statistical use, determining the loci of various star categories and also locating photometric groups. Departing from this rule, we also consider a few individual stars with remarkable features

• 2 carbon stars with silicate-type excess at $10~\mu\rm {m}$ ( $\rm C1653=BM$ Gem and $\rm C4595=V1468$ Aql),
• 2 carbon cepheids ( $\rm C3533=V553$ Cen of CWB-type and period 2.0605 days, and $\rm C1622=RU$ Cam of CWA-type with period variable from 22.05 to 22.24 days), and the oxygen-rich cepheid RT TrA for comparison (CWB-type with period 1.9461 days),
• $\rm C3066=HD100764$, the HC1-star with strong infrared excess, shown without correction (brighter value) and with correction (fainter value),
• AC Her, a carbon-rich variable of RV Tau-type.

The mean values quoted in Table $~\ref{coef_gr}$ for the fourteen photometric groups of the carbon-rich atmospheres, are also shown as (heavy) starred symbols with dispersion bars. Spanning the 5800-2000 K range, from left to right, they are: HC0 (2 values: see caption), HC1, HC2, HC3 and HC4 ( $\left<M_{\rm {bol}}\right>\simeq -3$ at about the same $\log T_{\rm {eff}}\simeq 3.6$), HC5, CV1, CV2, CV3, SCV ( $\left<M_{\rm {bol}}\right>\simeq -5.8$) and CV4 (at about the same $\log T_{\rm {eff}}\simeq 3.445$), CV5, CV6 and CV7. Oxygen-rich and BaII stars are considered separately.

7.2 Theoretical evolutionary tracks and mass range

Detailed accounts of the evolution of red giants can be found in reviews like Iben & Renzini (1983) and more recently Busso et al. (1999). Sketches of evolutionary tracks in the two cases described hereafter, may be found in the latter reference (p. 243). For low- and intermediate-mass stars, the termination of evolution in the red giant portion of the HR diagram, is the asymptotic giant branch (AGB) prior to white dwarf and/or planetary nebulae stages. After ascending the red giant branch (RGB), the stars with initial mass $M_{\rm i}\ge M_{\rm {HeF}}$ start the $3~\alpha$ reaction in their central regions ( $^{4}\rm {He}$combustion) and a more or less pronounced loop is executed leftward in the HR diagram, depending on chemical composition and mass: the lower the metallicity, the larger the loop. After core helium exhaustion, the early-AGB (E-AGB) phase is reached, and a second dredge-up experienced. According to models, there is not enough carbon dragged up throughout the envelope, up to the atmosphere, to turn the star into a carbon one. Then higher on the AGB (TP-AGB), the star presents a double shell source surrounding a carbon-oxygen core. It undergoes thermal pulses (TPs; temporal He shell ignition), experiencing the third dredge-up (TDU). After a more or less large number of TPs, it becomes a carbon star, provided $M_{i}\la 4~M_{\odot}$ since, at larger masses, hot bottom burning (HBB) prevents $\rm C/O \ge 1$ to be reached (e.g. Marigo et al. 1999; their Figs. 6 and 7 p. 134). For $M_{\rm i}\ga 4~M_{\odot},$ $\rm C/O \ge 1$ could however be reached in a small luminosity range. Substantial mass loss may reduce the envelope mass and stop HBB while thermal pulses are still occurring (Frost et al. 1998).

For $M_{\rm i}\le M_{\rm {HeF}},$ eccentric ignition of helium occurs in a degenerate core leading to a core flash which removes degeneracy, followed by a sudden luminosity decrease. According to current models, no dredge-up is found during those flashes. Such a mixing event could explain low-luminosity carbon stars if it actually occurred. After the flash, central helium burning operates in the core, and after exhaustion, the E-AGB and TP-AGB phases are subsequently experienced. The "helium flash mass'' is about $M_{\rm {HeF}}\simeq 2.1{-}2.2~M_{\odot}$ in models without convective overshoot, depending on the chemical composition.

The detailed evolution is quite complicated as can be seen from models evolved "all-the-way'' from main sequence to AGB (see e.g. Sackmann et al. 1993 for a $1~M_{\odot}$ model). Those models give nearly consistent results up to the E-AGB phase. On the contrary, they diverge in the TP-AGB phase, depending on parameters and codes used. A critical topic is that of the minimum luminosity at which a C/O value larger than unity is reached. Varying initial mass, metallicity, dredge-up efficiency, mixing length, eventually adding convective overshoot..., and using different codes with various time and space resolutions, the issue appears as seriously confused (see Marigo et al. 1999). The lowest transition obtained or carbon star formation line (CSFL), corresponds to $M_{\rm {bol}}\simeq -3.5$ but -4.0 at least is required in most models. More or less strong and episodic mass loss intervenes during those advanced stages as evidenced by the observations of circumstellar shells around many of those stars (Wallerstein & Knapp 1998; Schöier & Olofsson 2001). This phenomenon strongly influences stellar evolution and dissipating the stellar envelope may halt the ascent of the AGB (e.g. Sackmann et al. 1993). Detailed calculations of stellar evolution have been published which assume a fixed rate of mass loss (Steffen et al. 1998).

An alternative to fully-numerical models evolved all the way from the main sequence, are semi-analytical models which use relations calibrated from the former, like the core mass-luminosity relation and core mass-interpulse period relation. We have adopted the results of Marigo et al. (1996) coupled with the full calculations of Bressan et al. (Z=0.02, 1993) and those of Fagotto et al. (Z=0.008, 1994c). The former results are shown in Fig. $~\ref{hr1},$ as light continuous curves for initial masses 0.8, 1, 2 and $4~M_{\odot},$while the latter (heavy dashed curves) are displayed for 1 and $1.9~M_{\odot}.$ Those tracks illustrate the shift due to metallicity differences. They cannot describe the full complexity of those evolution phases, but are merely guidelines we adopt here. Taking into account the large dispersions of the observational results ( $\pm 0.9~ \rm {mag}$ on average), they delineate fairly well the loci of the carbon and BaII giants (we estimated intrinsic dispersion on mean values for photometric groups to be about $0.75~\rm {mag}$). We conclude that the $0.8~M_{\odot}\le M_{\rm i}\le 4.0~M_{\odot}$ range, with probably a large spread in chemical composition, reproduces fairly well the observations. A comparable result $\left(1.0~M_{\odot}\le M_{\rm i}\le 4.0~M_{\odot}\right)$ was already reached by Alksnis et al. (1998), making use of about forty observed HIPPARCOS parallaxes. Their sample is however limited in number and accuracy $\left(0.16 \le \sigma _{\varpi _{0}}/\varpi _{0} \le 0.98\right),$ and mean values would then be biased. This kind of range in initial masses is also what is excepted putting together the various subsamples in the pre-HIPPARCOS studies, as obtained from different techniques and data bases (Claussen et al. 1987; Thronson et al. 1987; Zuckerman & Dyck 1989; Barnbaum et al. 1991). The agreement is also good with studies on individual sources like IRC+10216. Isotopic ratios observed in the circumstellar envelope point to a low stellar mass, excluding a $5~M_{\odot}$-model, making use of theoretical nucleosynthesis (Kahane et al. 2000).

Returning to Fig. $~\ref{hr1},$ we observe that the RCB variables strongly depart from the above loci and a $\left(3.0~M_{\odot} \le M_{\rm i}\le 6.0~M_{\odot}\right)$range for Z=0.02 might be considered, or alternatively less massive models with lower metallicities and thus larger leftward loops. The chemical peculiarities of at least some RCB variables and cooler non-variable hydrogen-deficient analogues, i.e. carbon stars named HdC, are best reproduced by "born-again'' low-mass giants (Schönberner 1979; Herwig et al. 1999 and references therein). Born-again stars experience a last thermal pulse while approaching the white dwarf zone in the HR diagram, and thus return into the red-giant region, with a tiny envelope.

7.3 The BaII giants

The BaII giants populate the whole domain $1.1\ge M_{\rm {bol}}\ge-2.9$ for $3.6\le \log\left(T_{\rm {eff}}\right)\le 3.72,$ but the largest concentration is found between 1.2 and -0.6, centered at about $M_{\rm {bol}}\simeq 0.3.$ With $BC_{V}\simeq -0.3$ to -0.8, depending on the effective temperature, it corresponds to $0.6\le M_{V}\le 1.1.$ Those values point to the luminosity class IIIb as revised by Keenan & Barnbaum (1999), henceforth centered on the clump, an obvious concentration in the HR diagrams of some old open clusters. The attributing to classes III (or IIIab) and then to class IIIa is somewhat more difficult. A second concentration close to $M_{\rm {bol}}\simeq -2.5,$ i.e. $M_{\rm {V}}\simeq -2.0,$ is corresponding to class II (bright giants).

It can be seen from Fig. $~\ref{hr1},$ that the BaII giants of class III are located between the tracks for stars of initial masses in the range 1 to $4{-}5~M_{\odot},$ in the RGB region and/or leftward loops of central He-burning. The brighter BaII giants of class II seem to be in the E-AGB phase. For Z=0.02, many BaII stars could have initial masses in the $1.5{-}2.5~M_{\odot}$range. From radial velocity variations, it has been shown that those stars are members of binary stars, whose former primary then on the AGB, is now a white dwarf (McClure et al. 1980; Jorissen & Boffin 1992). Heavy elements enrichment (Sr, Ba) is thus the consequence of mass transfer in the binary system. This is also the case of "extrinsic'' S stars (Van Eck et al. 1998) and possibly of the CH stars (McClure & Woodsworth 1990).

7.4 The HC-giants

Considering the evolutionary tracks of Fig. $~\ref{hr1},$ it appears that the hot carbon stars (HC-groups, mostly R stars) are located in the RGB and/or He burning loops, and also the E-AGB, like the BaII stars. The former are however brighter by about 1 mag than the latter, on average. The cool carbon variables (CV-groups, mostly N variables: Sect. 7.5) seem essentially located in the TP-AGB region. The case of the HC5-group, located at the junction $\left(M_{\rm {bol}}\simeq -3.4\right)$ is not clear as already noticed on the grounds of space distributions and velocities (Paper II). The latter studies have shown that most HC-stars are members of the thick disk population, while the CV-stars are old (thin) disk members. Amongst the former stars, are found the CH stars which are members of the spheroid. Despite the aspect of Fig. $~\ref{hr1},$most of the HC-giants are not the progenitors of the CV-ones. The age of the thick disk certainly lies between 9 and 12 Gyr, with 11 Gyr usually adopted. From calculations of Fagotto et al. (1994a, 1994b) the corresponding initial masses range from 0.88 to $0.98~M_{\odot}$ for Z=0.0004 to Z=0.008.

The low luminosity of HC-stars ( $\left<M_{\rm {bol}}\ga -3.4\right>$) makes their carbon enrichment not explained by TDU on TP-AGB. It occurs, at least in models, at higher luminosities. Alternative possibilities will be discussed in Sect. 6 of Paper IV.

7.5 The CV-giants

7.5.1 The TDU on TP-AGB

We consider now the luminous CV-stars. They are essentially N variables and late R stars which can be explained by the third dredge-up (TDU). This phenomenon does intervene when the convective envelope is able to penetrate the region located between the two shells (He-burning inside, H-burning outside, alternatively ignited) where lies the stellar material which just experienced He-burning during the flash (Iben & Renzini 1983). Busso et al. (1999) summarize recent versions of the scenario, with a better knowledge of the fate of $^{13}\rm {C}$-pockets formed by diffusive and semiconvective mixing at dredge-up. These pockets are the site where the s-elements observed in many CV-stars are produced. A fraction $\Delta M_{\rm {dredge}}$of the intershell material is convected to the surface, which is polluted in He, C and O, and traces of heavy elements from the s-process (Lattanzio & Boothroyd 1997; Busso et al. 2001). Through subsequent mass loss, low- and intermediate-mass stars contribute to the chemical enrichment of the interstellar medium (e.g. Marigo 2001). However, most numerical models evolved all the way from the main sequence up to the TP-AGB, either neither reach $\rm C/O \ge 1$ or become carbon stars at very high luminosities corresponding to $M_{\rm {bol}}\simeq -5$ to -6.5 (see Marigo et al. 1999, their Table 1 p. 125, for a summary). This was the reason for developing semi-analytical models whose parameters are adjusted to fit the observations (Marigo et al. 1996; Marigo et al. 1999), like

• the minimum core mass $M_{\rm c}^{\rm {min}}$ allowing TDU appearance,
• the TDU efficiency $\lambda=\Delta M_{\rm {dredge}}/\Delta M_{\rm {c}},$ where $\Delta M_{\rm {c}}$ is the downward convective penetration,
• the chemical composition of the intershell convective zone.

They include analytical laws deduced from numerical models evolved all the way from the main sequence, and initial conditions taken at the onset of the first pulse from those models. The TP-AGB portions of the tracks of Fig. $~\ref{hr1}$ were obtained from this method (Marigo et al. 1996). Actually, the used $M_{\rm {c}}{-}L$ relation is only representative of the bright terminal portion of the inter-pulse phase. It explains the steady high luminosity of those tracks, as opposed to the sudden drops of complete numerical models such as those of Sackmann et al. (1993).

The concept of $M_{\rm c}^{\rm {min}},$ a function of mass and metallicity (say Z), is abandoned in the improved version of Marigo et al. (1999). They claim they satisfactorily reproduce the LFs of LMC (Z=0.008) and SMC (Z=0.004). Their minimum luminosity for carbon star production is at about $M_{\rm {bol}}\simeq -4$ in both cases (their Figs. 6 and 7 p. 134). However, due to two quiescence phases intervened in star formation history of the LMC clusters, one could not establish the lower and upper limit to carbon star masses in the LMC (Marigo et al. 1996).

7.5.2 The inferred mass range

Taking into account the errors on parallaxes, the locus of CV-stars in Fig. $~\ref{hr1}$ is nearly bounded downward by the $M=0.8{-}1~M_{\odot}$ tracks at Z=0.02, which is not surprising. This is an old (thin) disk population and the evolution time needed by such low mass stars to reach the red giant region, is at least $\tau \simeq 10.3\times 10^{9}$ years (Bressan et al. 1993). This is about the age of the Galactic old disk ( $10{-}11\times 10^{9}$ years). The tracks and evolution time are however dependent on the chemical composition (X, Y, Z). For Z=0.008, the $M=1~M_{\odot}$ track is shifted toward higher luminosities and temperatures. The evolution time shortens to $\tau \simeq 8.5\times 10^{9}$ years, and to $\tau \simeq 7.1\times 10^{9}$ years for Z=0.004. There is probably a large scatter around a mean age-metallicity relationship, due to local inhomogeneities. We conclude that the distribution of Fig. $~\ref{hr1}$ corresponds to an inferior limit for initial masses of $0.85{-}1.1~M_{\odot}$ with various metallicities and an evolution time to the red giant region of $8{-}11~\rm {Gyr},$ depending on Z-values.

The upper edge of the locus of CV-stars in Fig. $~\ref{hr1}$ is approximately the $M=4~M_{\odot}$ track at Z=0.02. It corresponds to a lower limit of $1.5\times 10^{8}$ years for the evolution time of those objects. This upper limit in initial masses decreases if lower metallicities are assumed. Apart from RCB variables and HdC stars classified HC0 or HC1 with $T_{\rm {eff}}\simeq 4500{-}7000~\rm {K}$ and $-4\le M_{\rm {bol}} \le -2,$ a few stars are located on the left of this limit, namely

• SC stars classified SCV and oxygen-rich red giants,
• some very cool and luminous carbon stars classified CV6 or CV7.

Those stars may correspond to $M=4{-}6~M_{\odot}$ evolutionary tracks (Z=0.02) not shown in Fig. $~\ref{hr1}.$ Recently, very bright carbon stars $\left(M_{\rm {bol}}\simeq-6.8\right)$ surrounded by thick circumstellar shells were discovered in the LMC (van Loon et al. 1998, 1999). They correspond to our brightest Galactic CV7-stars (up to $M_{\rm {bol}}\simeq -7$), but unfortunately, very few parallaxes are available for them. They probably represent a minority amongst CV-stars. According to Figs. 6 and 7 of Marigo et al. (1999), they may correspond to "obscured C-stars'' with $M_{\rm {i}} \ge 4{-}4.5~M_{\odot},$ which skip from oxygen-rich to carbon-rich objects, and then backwards.

Figure 9: The HR diagram of about 300 carbon giants. The nearly vertical limit between HC-stars on the left, and CV-stars on the right, is clearly seen. The Tc-rich stars are located in the bright locus of CV-stars which are TP-AGB objects.

The six oxygen-rich giants (neither RCB variables nor BaII stars) and four SC stars (classified SCV) in our sample fall either above the $M=4~M_{\odot}$ track or slightly below it, but within $\pm 0.9~ \rm {mag}.$ It is confirmed that the upper part of the HR diagram is populated with oxygen-rich giants or supergiants, and OH-IR LPVs. This is also the region of intrinsic S stars (see Van Eck et al. 1998), which are Tc-enriched TP-AGB stars (see Sect. 7.5.3 for Tc-rich carbon stars). The admitted explanation is hot bottom burning (HBB) in stars with $M_{\rm {i}}\ge4{-}5~M_{\odot}$: the convective envelope spreads in depth until high temperature regions are reached where hydrogen combustion via the CNO cycle is going on (e.g. Marigo et al. 1998). Then  $^{12}\rm {C}$ precedently synthesized by the $3~\alpha$ reaction is burnt into  $^{14}\rm {N}.$ The main consequences are a decrease of the C/O ratio which can become again less than unity, and a lowering of the $^{12}\rm {C}/^{13}\rm {C}$-ratio. Making use of the $M_{\rm {c}}{-}L$ relationship of Wagenhuber & Groenewegen (1998), the calculations have been refined (Marigo 1998; Figs. 6 and 7 of Marigo et al. 1999 may also be used). Strong mass loss might promptly reduce the envelope mass in such a way that HBB could stop while TDU is still going on. Frost et al. (1998) thus obtained models of carbon stars with optically thick wind at the tip of TP-AGB, and luminosities comparable to those of the obscured LMC C-stars of van Loon et al. (1998, 1999), and of our brightest CV7-stars.

We note that the carbon stars with strong silicate-type excesses in the infrared (BM Gem and V1468 Aql) fall exactly in the middle of the loci at HC5-CV1 $\left(T_{\rm {eff}}\simeq 3500{-}3300~\rm {K}\right),$ far below the oxygen-rich limit roughly identified with the $M=4~M_{\odot}$ evolutionary track.

7.5.3 Tc and $\mathsf{^{12}{C}/^{13}{C}}$-ratio

A relation was established between the CV-classification, and thus effective temperature, and the C/O ratio (Paper I; Sect. 17 and Fig. 16): mean ratios and dispersions increase along the CV2-CV6 sequence, i.e. for decreasing effective temperatures and increasing luminosities. Then we checked for possible correlation of our data with the $^{12}\rm {C}/^{13}\rm {C}$-ratio: the carbon stars with low ratios prove to be rather uniformly distributed in the carbon star locus of Fig. $~\ref{hr1}$ in both the HC- and CV-portions. The so-called J-stars (¹³C-rich) show no preference concerning effective temperatures and luminosities. The ¹³C producing neutrons must be created inside He-rich, partly He-burning regions (He shell), not in the envelope. Thus the J-stars cannot be explained by an extra-mixing, i.e. cool bottom processing (CBP) on the AGB (Nollett et al. 2001, 2002) because this does not account for other abundances, and would invariably lead to s-element rich stars, while the J-stars are not s-process rich. The ¹³C observed in J-stars must be produced by some extra-mixing at relatively low temperatures, in phases that never experienced conditions suitable for producing neutrons.

We finally show in Fig. $~\ref{hr2},$ a new version of the same HR diagram, but restricted to the carbon giants. The practically vertical separation line between the loci of HC- and CV-stars, is located at $\log T_{\rm {eff}}\simeq 3.52$ i.e. $T_{\rm {eff}}\simeq 3300~\rm {K}.$ It is very close to the theoretical curve produced by Scalo (1976) for the onset of helium shell flashes for a Galactic disk composition. Following Westerlund et al. (1995; Fig. 4), this is also the carbon star formation line (CSFL) for the CV-stars which are old (thin) disk stars on TP-AGB. Taking into account the errors on parallaxes and the mean values of Table $~\ref{coef_gr},$ we set the lower limit

$$M_{\rm {bol}}\simeq -3.6\pm 0.4$$ (20)

for the majority of the CV-stars which is in satisfactory agreement with $M_{\rm {bol}}\le -4.0$ of Marigo et al. (1999). A few CV-giants (including Miras), appear as underluminous ( $\left<M_{\rm {bol}}\right>\simeq-3.3$ to -3.6) as always noted by Bergeat et al. (1998) in a period-luminosity diagram (their Sample 3; similar objects are observed in the LMC). Pulsation masses for those underluminous CV-stars, are systematically found in the $0.5{-}0.6~M_{\odot}$ range (Sect. 3.2 of paper IV). The bright CV-stars are expected to be enriched in heavy elements as predicted from TDU. A probing case is that of tecnetium ( $^{99}\rm {Tc}$ an unstable isotope with half life $\tau \simeq 2.14\times 10^{5}$ years). A Tc-rich atmosphere has thus necessarily experienced a recent dredge-up (a few 10⁵ years). The Tc-stars among carbon giants are shown as (heavy) starred symbols in Fig. $~\ref{hr2}.$ We note that they all are CV-stars. No one is observed in the HC-region of the diagram, leftward of the separation boundary or CSFL for CV-stars. The statement was however made of Tc-lines absent from the spectra of several CV-stars like UU Aur, X Cnc, Y CVn and probably SS Vir, the former two being enriched in s-process elements (Little et al. 1987). On the contrary, Tc seems present in every bright (intrinsic) S star (Van Eck et al. 1998, their Fig. 2 p. 977 to be compared to Fig. $~\ref{hr2}$). However, technetium is best seen from its blue resonance line at $\lambda \simeq 426.0~\rm {nm},$in a region of the spectrum generally unreachable in very red C-stars, due to a strong depression at $440~\rm {nm}.$ Other features like the inter-combination Tc I line at $\lambda \simeq 592.447~\rm {nm},$ are extremely blended and much weaker. It often cannot be seen, which cannot be directly interpreted as inferring that it is not here (Abia et al. 2001). Further investigations are clearly needed before this matter can be settled once and for all. We have determined for Tc-rich carbon giants (Abia et al. 2001, and references therein), a kind of a barycenter (n=22 stars) at $\left<C_{\rm L}\right>\simeq 1.30\pm 0.45$ i.e. $\left<L/L_{\odot}\right>\simeq 5900,$ or alternatively

$$\left<M_{\rm {bol}}\right>\simeq -4.7~\left(-4.05; -5.6\right)$$ (21)

which is 1 mag above the lower limit, and

$$\left<T_{\rm {eff}}\right>\simeq\left( 2935\pm200\right)~\rm {K}$$ (22)

which correspond to the groups CV3-CV4 in Table $~\ref{coef_gr}.$ The absence of Tc detection in cooler stars (CV7- and most CV6-giants) may be due to insufficient coverage and/or technical difficulties while investigating those overcrowded spectra with many blends. The identification of the cool carbon variables of the CV-groups with TP-AGB objects having recently experienced dredge-up of nuclearly-processed material, makes no doubt on the grounds of the above arguments, even if the interpretation of detailed chemical peculiarities remains difficult.

7.6 The RCB variables, HdC giants and carbon-rich Cepheids

About fifteen RCB variables and hydrogen deficient carbon (HdC) stars are shown in Fig. $~\ref{hr1}$ where they form a leftward protuberance. The plot in Fig. $~\ref{hr2}$is restricted to stars classified in carbon-rich (HC) groups. Many of those stars are close to the $M=4~M_{\odot}$ track or above it. It is however surprising to find faint RCB variables of $M_{\rm {bol}}\ge -2,$ since they were believed to be bright giants or even supergiants on the grounds of their spectra. The latter are probably more typical of atmospheric extension and/or mass loss, than of truly large luminosities. We shall see in Sect. 3 of Paper IV that there is no correlation between surface gravity and luminosity in the case of the CV-giants.

 

 

Table 5: The results obtained for 11 peculiar stars: 7 RCB variables, 3 cepheids and AC Her a RV Tau-type variable. They are selected from Table 2 available at CDS. The photometric group and corresponding effective temperature (Cols. 3 and 5), the extinction $A_{{J}}\simeq0.87~E\left({B-V}\right)$ (Col. 4), were derived from our SED analyses (the star-symbol denotes a circumstellar contribution to the extinction). The obtained absolute bolometric magnitudes are quoted in Col. 6 and comments in Col. 7 refer to periods in days, phases and eventual label Hd or HdC for the hydrogen-deficient stars.

Name	Type	PhG	AJ	$T_{\rm {eff}}$	$M_{\rm {bol}}$	Comments
R CrB	RCB	F8sg	0.0	6100	-3.7	Hd; max
RY Sgr	RCB	F5sg	0.07	6900	-4.4	Hd; max
V CrA	RCB	F9g	0.11	5800	-2.3	max
V CrA	RCB	F9g	0.20*	5800	-1.7	int
RS Tel	RCB	G0sg	0.07	5800	-1.4	max
RS Tel	RCB	G0sg	0.62*	5800	+0.5	min
RT Nor	RCB	HC0	0.28	5615	-1.8	HdC
S Aps	RCB	HC1	0.20	5115	-2.5	HdC
SV Sge	RCB	HC3	0.40	4010	-2.8	HdC
RT TrA	CWB	F6.5sg	0.075	6500	-1.0	1.9461 d; 0.0
RT TrA	CWB	G0sg	0.070	5850	-0.8	0.2
RT TrA	CWB	G2sg	0.090	5200	-0.6	0.6 (min)
RT TrA	CWB	F8sg	0.097	6100	-0.75	0.8
V553 Cen	CWB	F8g	0.07	6150	-1.3	2.0605 d; max
V553 Cen	CWB	G1g	0.07	5650	-1.0	min
RU Cam	CWA	HC1	0.03	5215	-1.8	22.05 d; HdC
AC Her	RVa	G0g	0.17	5850	-2.3	75.01 d; 0.66

The results for a few representative stars as collected from Table 2 available at CDS, are given in Table $~\ref{coef_sp}.$ A correlation is obvious from Fig. $~\ref{hr1}$ and Table $~\ref{coef_sp}$: the RCB variables classified into oxygen-type groups, with relatively high effective temperatures (about 6000-7000 K and spectral type F), reach $M_{\rm {bol}}\simeq -3.5$ to -4.8, i.e. luminosities similar to those of the brightest HdC stars like C4247 $\left(M_{\rm {bol}}\simeq-3.7\right)$ and C3606 $\left(M_{\rm {bol}}\simeq-3.5\right)$ which are labelled in Fig. $~\ref{hr1}.$ It is the case of R CrB and RY Sgr. This is also close to $M_{\rm {bol}}\simeq -5.0$ found for the early RCB-variables in the LMC (W Men and HV 5637; a distance modulus of 18.5 was assumed). Then, from $M_{\rm {bol}}\simeq-1.5$ to -3.0, we found some RCB variables which have been included by Stephenson in his catalogue (1989), e.g. V  $\rm CrA=C4098$ and RS  $\rm Tel=C3982$, as carbon stars, but to whom we attributed oxygen-rich groups in our analyses. At nearly the same luminosities, but with slightly lower effective temperatures, we find a few RCB variables classified as carbon stars by Stephenson, and HC0 to HC3 by us. It is the case of RT  $\rm Nor=C3687$, S  $\rm Aps=C3562$ and SV  $\rm Sge=C4181$.

The transition between the oxygen-type groups and the HC-ones is thus confirmed at $T_{\rm {eff}}\simeq\left(5600\pm200\right)~\rm {K},$ as is the case for non-variable stars. The variations of the RCB stars seem to be not intrinsically photospheric in origin, but rather atmospheric and/or circumstellar, as shown by the increase of extinction at phases outside maxima (see Bergeat et al. 1999 and references therein). The existence of two classes of RCB variables with respectively $\left<M_{\rm {bol}}\right>\simeq -3.8$ and $\left<M_{\rm {bol}}\right>\simeq -2.0,$ or alternatively of a continuous distribution from -4 to -1, can be invoked. Those results need to be considered with caution since they are based on very few stars, with data of limited accuracy.

Three cepheids are included in our sample, two of which (RU  $\rm Cam=C1622$ and V553  $\rm Cen=C3533$) were catalogued by Stephenson (1989), and not the third, namely RT TrA. They all are classified CW, i.e. they are members of a Population II group including the pulsating cepheid-like variables of the W Vir and BL Her types. The mean absolute bolometric magnitudes of the CW-stars spread from -0.15 to -2.45 with effective temperatures ranging from 6600 down to 5250 K (Hall 2000). The results of Table $~\ref{coef_sp}$ are in good agreement with those values. Making use of the pulsation constant Q as quoted by Hall (2000), a mean pulsation mass of $M_{\rm {p}}\simeq0.57~M_{\odot}$ is found for those three stars, in good agreement with the $0.6~M_{\odot}$ quoted in Hall's table. We find here the same sequence for decreasing effective temperatures, as that found for RCB variables: early oxygen-types not in Stephenson's catalogue, oxygen-types in the catalogue, and carbon HC-types. The data is compatible with a second transition still located at about 5600 K. As is well-known, the variations of those cepheids are clearly photospheric in origin, with pulsations characterized by large changes in effective temperature and photometric group. The associated change in luminosity amounts typically to $\Delta M_{\rm {bol}}\simeq0.3$ to 0.4.

The carbon-rich star AC Her, classified as a variable of the RV Tauri type, was studied in some detail by Bergeat et al. (1999 and references therein). The luminosity found here $\left(M_{\rm {bol}}\simeq -2.3\right)$ places it in the RCB-range, but it is brighter than the above CW-variables.