Carbon giants may span a more or less wide range, from $M_{\rm {bol}}\simeq 0$ or -1 to $M_{\rm {bol}}\simeq -6$ or -7 at most, with one (or two maximum), at $M_{\rm {bol}}\simeq -1$ or -2 (eventually), and $M_{\rm {bol}}\simeq -4$ or -5 respectively. Decreasing bolometric magnitudes (increasing luminosities) are correlated with decreasing effective temperatures in the 6000-2000 K range. Thus, the carbon giants populate inclined strips in the HR diagrams of those extragalactic systems. Those loci however shift toward higher effective temperatures and larger luminosities for lower initial metallicities (e.g. Westerlund et al. 1995 for SMC: Z=0.004 typically).

These observations are at least partly consistent with evolved models of LMS and IMS (e.g. the reviews of Iben & Renzini 1983 and Busso et al. 1999). The bright carbon giants are found on the asymptotic giant branch (AGB), especially the region where models experience thermal pulses (TPs), i.e. temporal He-shell ignitions (TP-AGB). After a more or less high number of TPs, they may become carbon-rich, provided $M_{\rm {i}}\la 4~M_{\odot},$ since at larger masses, hot bottom burning (HBB, i.e. transformation of ¹²C into ¹³C and then ¹⁴N) prevents C/O $\ge 1$ to be reached (e.g. Marigo et al. 1999). Mass loss can however reduce the envelope mass and stop HBB while thermal pulses are still ongoing (Frost et al. 1998). Stars with initial masses larger than $4~M_{\odot}$ could thus turn temporarily to carbon-rich objects embedded in thick circumstellar shells, a consequence of strong mass loss. The low-luminosity carbon giants ( $M_{\rm {bol}}\ga-3.5$ ) however fail to be explained by standard models (e.g. Marigo et al. 1996, 1999, and references therein).

An alternative explanation for anomalous surface abundances may be past mass exchange in a binary system, where a TP-AGB companion became a white dwarf (Han et al. 1995 and references therein). This is the currently-accepted model for BaII stars, many of them having proved to be binary members (Mc Clure et al. 1980; McClure 1984; Jorissen & Boffin 1992). The low-luminosity carbon stars classified into the R-types however fail to show any evidence of binarity, and Mc Clure (1997a) considered they could result from coalescence of components in a former binary system.

From the data collected in several star clusters in the LMC, Bessell et al. (1981, 1983) found a sharp transition from oxygen stars to carbon stars at $M_{\rm {bol}}\simeq -4.3$ and $T_{\rm {eff}}\simeq 3670~\rm {K}.$ The fainter and hotter carbon giants (i.e. counterparts of HC-stars in the Galaxy) were not found in those clusters, and other LMC-fields as well (see data from Costa & Frogel 1996). Such stars were observed in the SMC (e.g. Westerlund et al. 1995) and in the Galactic bulge (Rich 1989). Bessell et al. deduced that the minimum initial mass for obtaining carbon stars is about $0.9~M_{\odot}$ at $\rm {\left[Fe/H\right]}\simeq -1$ and $1.3~M_{\odot}$ at $\rm {\left[Fe/H\right]}\simeq 0.$ The cluster diagrams and specially the turn-off of NGC 121 aged 10.5 Gyr where no carbon star was observed, were used to produce those estimates.

The kinematics of carbon N stars we mostly classified into the CV-groups, being similar to that of F5 dwarfs (Dean 1976), the deduced equivalent mass is about $1.4~M_{\odot}.$ The carbon R stars we mostly classified into the HC-groups, correspond to G-K dwarfs (Mc Leod 1947), pointing to main sequence masses smaller than $1~M_{\odot}.$ The HC-stars which are CH stars on spectroscopic grounds, are halo tracers (Hartwick & Cowley 1985) and their initial masses should not exceed $0.8~M_{\odot}.$

From an investigation of a flux-limited sample of Galactic carbon stars taken from the Two Micron Sky Survey (TMSS 1969), Claussen et al. (1987) found that their main-sequence progenitors have masses between 1.2 and $1.6~M_{\odot},$ i.e. should be F-type dwarfs. Thronson et al. (1987) used the IRAS Point Source Catalogue (IRAS 1988) to study a flux-limited sample of highly-evolved carbon-rich and oxygen-rich stars in the Milky Way. They found a local birthrate for their carbon stars equivalent to the deathrate for stars that leave the main sequence with masses in the $3{-}5~M_{\odot}$ range. This is consistent with the wavelength ranges of the used catalogues, resp. TMSS $\left(2.2~\mu \rm {m}\right)$ and IRAS $\left(12,~25,~60,~100~\mu \rm {m}\right).$ Zuckerman et al. (1986) presented evidence that carbon-rich stars which have circumstellar envelopes with large "terminal'' outflow velocities $\left(V_{\infty}\right),$ are preferentially located close to the galactic plane (low |b| used instead of |z|, which may generate some confusion). This class of carbon stars which lies at $\vert b\vert\le 10\degr,$ has outflow velocities $V_{\infty}\ga 18$ $\rm {km~s^{-1}}$ (Zuckerman & Dyck 1989). Large masses $\left(M_{\rm {ms}}\ge 3~M_{\odot}\right)$ were proposed for those objects, while a range of $1\le M \le 3~M_{\odot}$ was suggested for $-9\la V_{\infty} \la 18~\rm {km~s^{-1}}.$ A 107 pc scale height was obtained for the stars with high outflow velocities, a value typical of main-sequence stars with masses larger than $2.5{-}4~M_{\odot}$ (Barnbaum et al. 1991). These results imply that the range of initial main-sequence masses, is fairly large for carbon giants. In their investigations of TMSS and IRAS carbon stars, Claussen et al. (1987) and Thronson et al. (1987) actually delineated different subsamples of CV-stars.

Making use of the true parallaxes as estimated by Knapik el al. (1998) from HIPPARCOS data, we investigate in the present paper, the LF and loci of about 370 carbon and BaII giants of the Galactic disk, in the HR diagram. The Lutz-Kelker bias being taken into account, those parallaxes are intended for statistical purposes. The effective temperatures and bolometric apparent magnitudes of Paper I extended to a larger sample, will be used hereafter. The evaluation of the pulsation masses of carbon long period variables (LPVs) and of the mass-luminosity diagram is postponed to a companion paper (Bergeat et al. 2002b, hereafter Paper IV).

The data of Bergeat et al. (2001, hereafter Paper I) and references therein, and Bergeat et al. (2002a, hereafter Paper II), are summarized and extended in Sect. 2 through the calibration of bolometric corrections (Table A.1 in appendix) to be applied to additional stars from the HIPPARCOS sample (not considered in Paper I because of incomplete SEDs). Then the coefficients $C_{\rm R}$ and $C_{\rm L}$ are introduced in Sect. 3, to allow derivation of unbiased mean photospheric radii and luminosities respectively. The corresponding data for individual stars are given in Table 2 (only available in electronic form at CDS), specifically effective temperatures and absolute bolometric magnitudes.

The diagrams of relative angular diameters $\left < k \right >^{1/2}$ from photometry vs. estimated true parallaxes $\varpi$ from HIPPARCOS astrometry, are presented for the different photometric groups. They are interpreted in terms of ranges in photospheric radii for each photometric group. Mean values and ranges (effective temperatures, photospheric radii and luminosities) were computed for the fourteen photometric groups of carbon giants (Sect. 4). The mean bolometric magnitudes of BaII giants were computed for comparison purposes from data of Bergeat & Knapik (1997).

The mean values according to variability classes amongst carbon giants, were also calculated (Sect. 5). The luminosity function (LF) of Galactic carbon giants in the Sun vicinity (Sect. 6), is compared to those of the Galactic bulge and of the Magellanic Clouds. The loci of carbon and BaII giants in the HR diagram are then presented (Sect. 7), and confronted to the predictions of theoretical models of stellar evolution. The various star categories are discussed (BaII, HC- and CV-giants, RCB variables, HdC giants and carbon-rich cepheids). The presence of Technetium is discussed, and the ranges in initial masses $\left(M_{\rm {i}}\right)$ inferred. The results are briefly summarized in Sect. 8, and a full discussion is postponed to Paper IV mainly devoted to pulsation modes and pulsation masses of carbon-rich long period variables (LPVs).

$\begin{figure} \par\includegraphics[width=10cm,clip]{fi1.eps} \end{figure}$

Figure 1: The CB_K vs. $\left [J-K\right ]_{0}$ diagram for 375 red giants: 315 carbon stars of the SCV, HC and CV-groups (diamond-shaped symbols; 1 with "silicate'' IR excess: X-symbol), 15 RCB variables (ox.-groups; crosses), 45 Ba II stars (ox.-groups; squares).

1 Introduction