2 The photometric data

2.1 Photometric groups and populations

The first classification of the carbon-rich giants in discrete photometric groups (13 groups named HCi and CVj with i=0 to 5 and j=1 to 7, and the fourteenth SCV as a transition to bright S stars), was established by Knapik & Bergeat (1997), Bergeat et al. (1999) and Knapik et al. (1999). It relies on the intrinsic spectral energy distributions (SEDs) of those stars from UV to IR (0.36 to $\rm {25~\mu m}$) which were gathered into discrete groups. A sequence of 13 groups was obtained from the earliest one to the latest one, completed by a fourtheenth one (SCV) corresponding to the spectroscopic SC variables which is a transition to S stars. The mean SEDs exhibit more or less regularly increasing color indices and the eighteen best-determined ones were selected and settled to a zero-magnitude at $\rm {1.08~\mu m}.$ The name "carbon variable'' (CV) was initially selected because the members of all those later seven groups (CV1 to CV7) are variable stars or "suspected new variables''. On the contrary, a majority of the members of the six earlier groups are constant in light, i.e. non-variable stars as confirmed by repeated high-accuracy HIPPARCOS photometric measurements (ESA 1997). A few irregulars and small-amplitude variables, carbon cepheids and RCB-variables are also present. Those earlier groups were named "hot carbon'' (HC, from HC0 to HC5).

The interstellar and circumstellar extinction on the line of sight to each star is simultaneously determined, and a mean coefficient $\left < k \right >^{1/2}$ calculated, which proved to be an angular diameter on a relative scale for a given effective temperature (or eventually for a given photometric group: see Paper I). The new effective temperatures proposed for carbon-rich giants and related objects in Paper I rely on spectral energy distributions (SEDs), model atmospheres and measured angular diameters. The classification and the homogeneous temperature scale obtained proved to be tightly correlated. It was concluded that the main parameter of this photometric classification is effective temperature (Paper I), which is not the case of classical spectral classifications. Amongst the remaining parameters, the C/O ratio has an important influence. For CV-stars, the mean values and dispersions increase along the sequence (CV2 to CV6) of photometric groups (Paper I, Sect. 10).

Bergeat et al. (Paper II) analyzed the space distribution and kinematics of Galactic carbon giants. They showed that the HC-sample (essentially early-R stars) is a component of the thick disk contaminated by the CH stars which are a spheroidal contribution (see also Hartwick & Cowley 1985). The HC-stars not classified as CH stars probably correspond to old $(\simeq$ $11~\rm {Gyr})$low initial-mass stars (say $\le$ $1.1~M_{\odot}$). It was also shown in Paper II that the CV-sample (mainly N stars) is a component of the old (thin) disk, dated $\left(3\pm 1\right)~\rm {Gyr}$on average, but with a large spread from a few $10^{2}~\rm {Myr}$ to nearly $10{-}12~\rm {Gyr}.$The corresponding initial masses should range from about $1~M_{\odot}$ to a few solar masses. Corrections to the (small) effects of the Malmquist bias on the HIPPARCOS sample of carbon giants were also obtained in Paper II. They are applied hereafter to quoted mean luminosities.

2.2 The calibration of bolometric corrections

Effective temperatures were derived for about 390 carbon-rich giants whose apparent bolometric magnitudes were deduced as well (Paper I). The used method makes use of both the whole integrated deredenned SED ("bolometric'' value) and calibrated color indices ("spectral'' value). About 220 carbon stars observed by the satellite HIPPARCOS (ESA 1997) were involved in this study. There remained 81 HIPPARCOS carbon stars (82 SEDs) with SEDs partially documented. The "spectral'' method was the only one available for effective temperature evaluation of those latter stars. Their apparent bolometric magnitudes were estimated from calibrations of various bolometric corrections $BC_{\rm {m}\left(\lambda\right)}$ in terms of several (intrinsic) color indices IC₀.

The calibrations were acquired from the sample of 390 stars (Paper I). Several couples $\left(BC; IC\right)$ were finally selected by trial and error amongst our data. They represent our "best choices'' on the $V\left(0.55~\mu \rm {m}\right)$ to $K\left(2.2~\mu \rm {m}\right)$ range. The classical CB_K vs. $\left [J-K\right ]_{0}$ plot is shown in Fig. $~\ref{CBK_J-K}.$ The earlier diagram of Frogel et al. (1980; their Fig. 2, p. 498) is confirmed and extended at both ends.

Except for about 20-30 objects, the stars populate a well-defined locus from (0.0, 0.8) to (2.5, 3.4). The discrepant objects are located below this locus due to thermal emission from circumstellar (CS) grains in the infrared, usually starting from the H-bandpass at $1.65~\mu\rm {m}$ and increasing longwards. The J-bandpass at $1.25~\mu\rm {m}$ is much less affected while the K-bandpass at $2.2~\mu\rm {m}$ is much more affected. Finally the CS vector in the diagram is oriented toward lower CB_K and larger $\left [J-K\right ]_{0}$ as seen in Fig. $~\ref{CBK_J-K}.$ The extreme ("infrared'') carbon stars such as $\rm C2619=IRC+10216=CW$ Leo and $\rm C2724=CIT6=RW~LMi$, are so strongly shifted that they fall outside the frame. In this diagram, the Ba II stars do not differ appreciably from the hot carbon (HC) stars, which allows one to adopt the same mean relation. With the exception of a few stars, the RCB variables do exhibit a substantial IR excess. They populate a strip located well below the relation for carbon and Ba II stars. This is also the case of C2011, a star with a silicate-type IR excess. In such cases the CB_K vs. $\left [J-K\right ]_{0}$ couple cannot be used to estimate bolometric corrections. For $\left[J-K\right]_{0}\ge 2.1,$ that is dealing with carbon variables of the CV6-CV7 groups, the proportion of stars with an appreciable IR excess increases. The locus widens in a leek-shaped structure for increasing color indices. Following a wide maximum, a decrease of the bolometric correction is observed for increasing indices. Here again, the mean relation cannot safely be used. Thus, we calibrated data at shorter wavelengths. The assumption of constant total radiated power when absorbed light is re-radiated in the IR by CS grains deserves discussion, depending on shell geometry and/or dust properties. Bergeat et al. (2002, Paper I, Sect. 13) found that for a few hot carbon (HC) stars and RCB variables, corrections need to be applied due to non-spherical symmetry. On the contrary, they estimated that reasonable values can be obtained from dereddened SEDs without IR excesses or from observed SEDs including IR excesses in the case of cool carbon variables (CV6-CV7) with strong thermal emission.

Finally, we made use of eight $BC_{\rm {m}\left(\lambda\right)}$ vs. IC₀ mean relations, namely $CB_{\left[0.78\right]},$ $CB_{\left[1.08\right]},$ CB_H and CB_K vs. $\left[V-\left[1.08\right]\right]_{0},$ CB_H and CB_K vs. $\left[\left[1.08\right]-K\right]_{0},$ and finally CB_Hand CB_K vs. $\left[J-K\right]_{0}.$ The magnitudes $\left[0.78\right]$ and $\left[1.08\right]$ are such as defined by Knapik & Bergeat (1997) from the data obtained in narrow bandpasses by Baumert (1972). The adopted mean relations are given in Appendix. Departures between values deduced from short wavelengths on one hand and from long wavelengths on the other hand, if any, are estimated to be less than 0.1 mag. Finally, the estimates are averaged and the errors on the resulting bolometric correction should not exceed 0.05 mag for the 81 carbon stars studied in this way. The absolute bolometric magnitudes in forthcoming sections are of course affected, but the effect can be neglected when true luminosity dispersions and uncertainties on true parallaxes are taken into account.

2.3 The three encountered biases

A detailed study of the influence of three encountered biases on averaged quantities as derived from the HIPPARCOS parallaxes was given by Knapik (1999). No appreciable bias is expected on mean HIPPARCOS proper motions. Essential parts of that work can be found in Knapik et al. (1998) and Paper II. Here, we present a summary of the effects of biases on averaged quantities.

The Malmquist bias (1924, 1936) affects any sample delineated on the grounds of a limited apparent magnitude. We found $V_{\rm {l}}\simeq 10$ for the HIPPARCOS sample of carbon stars (Knapik et al. 1998), refined as $V_{\rm {l}}\simeq 10.4$for the HC-stars and $V_{\rm {l}}\simeq 9.6$ for the CV-stars (Paper II). Mean absolute magnitudes are found too bright since increasing selection occurs amongst more and more distant stars, favoring the intrinsically brighter objects. In Sect. 4 of Paper II, corrections to mean absolute bolometric magnitudes were derived which amounted to a few tenths of a magnitude at most (+0.32 mag at HC1 and +0.2 mag at HC2 are the largest values, against 0.1 mag or less for the other groups). A second aspect is the increasing deficits at increasing distances from Sun, in stellar number densities (the missing objects in Fig. 1 of Paper II) and in stellar surface densities as projected on the Plane (Sect. 5.1 of Paper II). This latter effect lowers the q-exponent of the HIPPARCOS distribution of true (as opposed to observed) parallaxes

$$N\left(\varpi\right){\rm d}\varpi \propto \varpi^{-q} ~ \rm {d}\varpi$$ (1)

from about 3 for stars of $\varpi \ga 1~ \rm {mas},$ to 2.35 on average (Knapik et al. 1998 and Sect. 2.3 of Paper II). The influence is still larger at even larger distances, but fewer HIPPARCOS carbon stars are involved.

The bias studied by Smith & Eichhorn (1996) occurs every time a mean value is calculated for a quantity which is not a linear function of true parallaxes. Even if the distribution according to true parallaxes is a Gaussian, that of distances or absolute magnitudes is not. Smith & Eichhorn derived the mathematical expectation of distances from parallaxes affected by arbitrarily high errors ($\sigma$, their Fig. 1, p. 213). An overestimate is found for $\alpha = \sigma / \varpi < 0.74$ which changes into an underestimate for $\alpha > 0.74.$ The consequence of this effect and the methods used to avoid it in various circumstances can be found in Bergeat et al. (1998), Knapik et al. (1998), Sects. 3.1 and 3.2 of Paper II, Sects. 3.1 and 3.3 of Paper III, and in Sect. 2.1 of the present paper.

The third bias results from the effect of errors on a distribution non-uniform in parallaxes. Let us assume that the probability distribution function (pdf) of repeated individual observations of a single true parallax $\left(\varpi \right)$ is a Gaussian centered on the $\varpi$-value, with $\sigma$ as a standard deviation. Now we consider an other problem which is to derive the pdf of true parallaxes $\left(\varpi \right)$ for a given sample of stars whose observed parallaxes are the $\varpi_{0}$'s from the HIPPARCOS catalogue. This pdf is the same Gaussian as mentioned above, provided the stars are uniformly distributed in true parallaxes (Lutz & Kelker 1973). This latter condition is completely unrealistic and Lutz & Kelker adopted a uniform space density in their pioneering work, that is

$$N\left(\varpi\right) \rm {d}\varpi \propto \varpi^{-4} ~ \rm {d}\varpi.$$ (2)

The resulting pdf weighted by $\varpi^{-4}$ is no longer a Gaussian. The net effect when exploiting observed parallaxes is that stars are seen, on average, as closer to Sun than they are indeed. The larger the errors on observed parallaxes, the larger the bias for a given pdf. Coupled with the Smith & Eichhorn bias on absolute magnitudes, this is the so-called Lutz-Kelker bias (Lutz & Kelker 1973). The effect on mean values for samples of carbon giants is lower than it is in their original paper since Eq. (1) prevails on Eq. (2) for that Malmquist-biased sample whose distribution is flattened toward the Galactic plane. A flattened sample uniform in volumic density, free of the Malmquist bias, would have $q\simeq 3.$ Knapik et al. (1998) proposed a calculation of the mathematical expectation of the difference $\varpi$- $\varpi_{0}.$ They were able to model the distribution of errors on observed HIPPARCOS parallaxes of carbon stars.

It is interesting to note that future astrometric missions like GAIA (ESA 1995) will be almost free from those biases, due to their very high accuracies and extensive sampling, reaching the edges of the galactic disk. The main difficulty will then result from non-uniform brightness distributions on extended stellar discs.