4 The mass-luminosity diagram

The values of Table  $\ref{mass}$ are mean values obtained over an unknown mass distribution. Intrinsic dispersion and errors from parallaxes both contribute. The main feature is the increase of mean masses along the sequence of photometric groups, at least from HC5 ( $T_{\rm {eff}}\simeq 3500~\rm {K}$) to CV7 (about $T_{\rm {eff}}\simeq 2000~\rm {K}$). This is also a sequence of increasing luminosities from about $1700~L_{\odot}$ to $17~000~L_{\odot}$ (see Table 3 of Paper III). It was shown in Paper I that the mean C/O ratio and its dispersion increase along this sequence (at least from CV2 to CV6). The SEDs are obviously affected by increasing opacities (gas + grains) along the same sequence. This is also a sequence of increasing mass loss and increasing CS dust emission of SiC at $11~\mu\rm {m}$ (Bergeat 2002c).

Our mean masses in Table  $\ref{mass}$ are shown in Fig. $\ref{m_l}$, which is a mass-luminosity diagram, making use of the absolute bolometric magnitudes. Mean luminosities increase with mean pulsation masses as predictable from initial masses of theoretical tracks. It was expected since mean pulsation mass increases along the sequence of groups (Table  $\ref{mass}$), which is a sequence of increasing mean luminosities (Table 3 of Paper III). The locations of the onset and tip of TP-AGB for Z= 0.008 are adapted from Fig. 6 of Marigo et al. (1999). A good agreement is observed between our results and those of predictions. Typical dispersion bars are shown as double arrows. Taking them into account, it can be seen that most of the results are close to the tip (solid line) of the TP-AGB region. It is also likely that, on average, the metallicity is somewhat larger than Z= 0.008 and less than Z= 0.02 (solar value). The tip-line shifts slightly downward in Fig. $\ref{m_l}$ for such a value (say Z= 0.015). This conclusion concerning location with respect to TP-AGB tip and the above-mentioned sequence of increasing mean luminosity vs. mean pulsation mass, suggests that, on average, the carbon giants are located close to their track end in the HR diagram. In addition, this end shifts toward lower effective temperatures for increasing masses, i.e. increasing C/O ratios and opacities (Paper I). This trend is seen for Z= 0.02 in Fig. 8 of Paper III, and when comparing the track for $M=1~M_{\odot}$ to that for $M=4~M_{\odot}$ (but not clearly for $M=2~M_{\odot}$). It is worth noting that tracks ending at too high effective temperatures may result from opacities missing in the models.

Considering $l=\left< L/L_{\odot} \right>$ and $m=\left<M/M_{\odot}\right>,$ we found that the unweighted data satisfy the mean relation

$$% \log l \simeq \left(0.72\pm0.07\right)\log m +\left(3.58\pm0.12\right)$$ (10)

with a correlation coefficient of $\rho^{2}\simeq0.87.$ Restricting the fit to the best-documented results from CV2 to CV5 (over 97 stars, eight mean masses averaged in the 1- $3~M_{\odot}$ range), we obtained

$$% \log l \simeq \left(0.497\pm0.075\right)\log m +\left(3.658\pm0.034\right)$$ (11)

with $\rho^{2}\simeq0.88.$ The intermediate part of the mass-luminosity relation is thus very nearly

$$% l \propto m^{1/2}$$ (12)

a result we comment on in Sect. 6.

There is also a reasonable correspondence with the carbon star formation line (CSFL) for Z= 0.008 from Marigo et al., specially that of their intermediate case B ( $\log~T^{\rm {dred}}_{\rm {b}}=6.4$ and $\lambda =0.5;$ dashed line in Fig. $\ref{m_l}$). The observed stars actually span a large range in metallicities.

Estimates of mean stellar densities are given in Table  $\ref{mass}$ (Col. 9). These data can be compared to the result of internal structure calculations. It is seen that they continuously decrease from HC5 to CV7 (a factor of about 22). In addition, estimates of surface accelerations or gravities are quoted in Table  $\ref{mass}$ (Col. 10). These data, useful for atmosphere modeling, complement the effective temperatures provided in Paper I. The values are around $g=\left(5.1\pm0.7\right)~10^{-3}$ SI or $\simeq$0.5 CGS $\left(\log~g\simeq-0.3\right).$ They only diminish by a factor of 2 at CV6-CV7, for the extreme objects with the overtone assumption (O$\arcmin$), but this result is only indicative.

In summary, it appears that the CV-giants are close to the tip of the TP-AGB region, and close to the end of their track towards the right in the HR diagram. With larger masses, this end shifts toward lower effective temperatures and higher luminosities.