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5 Mean data in terms of variability classes

According to the classification of Kholopov et al. (1985), we consider a sequence of increasing variability, namely "Cst, Lb, SRb, SRa and M'' from constant stars to Miras including irregular variables (Lb) and semi-regulars with either small amplitude of variations (SRb) or large amplitude (SRa). We used the same classification in Paper II, in the study of the reflex solar velocity in terms of variability classes. The approach used here is similar to that of Sect. 4, except for $\left<T_{\rm {eff}}\right>$ which is a "mean'' value deduced from Eq. (13) where $\left<C_{{T_{\rm eff}}}\right>$ itself is derived from  $\left<C_{{\rm R}}\right>$ and $\left<C_{{\rm L}}\right>$ replaced into Eq. (14) which is assumed to hold for mean values. The relation $\left<\Phi _{0}\left(T_{\rm {eff}}\right)\right>$ of Paper I was adopted. The corresponding $\left<\Phi _{0}\right>$ allows the determination of an "equivalent'' group $\rm\left <G\right >$for each variability class. The results are given in Table % latex2html id marker 2841
$~\ref{coef_cl}.$


  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{fmw.eps} \end{figure} Figure 5: The luminosity function (LF) of Galactic carbon giants in the Sun vicinity (dashed lines). The LF for stars with $\varpi\ge0.9~\rm{mas}$ presumably free from effects of the Malmquist bias is shown with full lines ("MW'' hereafter).

The luminosity and the photospheric radius increase and the effective temperature decreases, on average, while following the sequence of increasing variability: Cst, Lb, SRb, SRa and M. The stars classified as constant (i.e. for which no photometric variations larger than a few hundredths of a magnitude could be evidenced) are essentially early HC-giants from HC0 to HC3, many HC4 and HC5-stars being variables. We obtained HC2 as their equivalent group with good consistency for mean luminosity, photospheric radius and effective temperature. Most variable carbon stars belong to a CV-group as initially intended and a strong discontinuity in properties appears when skipping from Cst to Lb. There is then a slow evolution along the sequence from Lb to M, with increasing mean luminosities and photospheric radii, and decreasing mean effective temperatures. Considering dispersions, the regular evolution along the sequence is more convincing than the differences found between two successive classes. The mean $\left<M_{\rm {bol}}\right>\simeq -4.7,$ i.e. $\left<L/L_{\odot}\right>\simeq 4900,$ value obtained for the full sample M(1) of carbon Miras, is similar to that quoted for SRa variables. From HIPPARCOS data, a few Miras proved to be strongly underluminous (e.g. Bergeat et al. 1998): this is the case of $\rm C2165=T$ Lyn and C5570=RZ Peg which we excluded from sample M(2). Then we obtained $\left<M_{\rm {bol}}\right>\simeq -4.9,$ i.e. $\left<L/L_{\odot}\right>\simeq 7100.$ On average, the amplitudes of variations in carbon Miras are lower than those of their oxygen-rich counterparts. There is no gap between the former and large-amplitude carbon semiregulars classified SRa. The results of Table % latex2html id marker 2855
$~\ref{coef_cl}$ can be interpreted in terms of increasing variability, on average, along the evolutive tracks in the TP-AGB region (Sect. 7).


  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{flmc.eps} \end{figure} Figure 6: The MW-LF of Fig. % latex2html id marker 2857
$~\ref{lf1},$ confronted to the LF of the Galactic bulge (MWB, 33 stars; Rich 1989) and to the LF of the LMC (LMC, 895 stars; Costa & Frogel 1996).


  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{fsmc.eps} \end{figure} Figure 7: The MW-LF of Fig. % latex2html id marker 2859
$~\ref{lf1},$ confronted to the SMC-LFs of Azzopardi (1993; "SMC-Azzo.''), Westerlund et al. (1995; "SMC-West.''), and Groenewegen (1998; "SMC-Groe.'').


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