Carbon-rich giants in the HR diagram and their luminosity function

The luminosity function (LF) of nearly 300 Galactic carbon giants is derived. Adding BaII giants and various related objects, about 370 objects are located in the RGB and AGB portions of the theoretical HR diagram. As intermediate steps, (1) bolometric corrections are calibrated against selected intrinsic color indices; (2) the diagram of photometric coefficients [k](1/2) vs. astrometric true parallaxes (omega) over bar are interpreted in terms of ranges of photospheric radii for every photometric group; (3) coefficients C-R and CL for bias-free evaluation of mean photospheric radii and mean luminosities are computed. The LF of Galactic carbon giants exhibits two maxima corresponding to the HC-stars of the thick disk and to the CV-stars of the old thin disk respectively. It is discussed and compared to those of carbon stars in the Magellanic Clouds and Galactic bulge. The HC-part is similar to the LF of the Galactic bulge, reinforcing the idea that the Bulge and the thick disk are part of the same dynamical component. The CV-part looks similar to the LF of the Large Magellanic Cloud (LMC), but the former is wider due to the substantial errors on HIPPARCOS parallaxes. The obtained mean luminosities increase with increasing radii and decreasing effective temperatures, along the HC-CV sequence of photometric groups, except for HC0, the earliest one. This trend illustrates the RGB- and AGB-tracks of low- and intermediate-mass stars for a range in metallicities. From a comparison with theoretical tracks in the HR diagram, the initial masses M-i range from about 0.8 to 4.0 M. for carbon giants, with possibly larger masses for a few extreme objects. A large range of metallicities is likely, from metal-poor HC-stars classified as CH stars on the grounds of their spectra (a spheroidal component), to near-solar compositions of many CV-stars. Technetium-rich carbon giants are brighter than the lower limit M-bol similar or equal to -3.6 +/- 0.4 and centered at [M-bol] similar or equal to -4.7(-0.9)(+0.6) at about [T-eff] similar or equal to (2935 +/- 200) K or CV3-CV4 in our classification. Much like the results of Van Eck et al. (1998) for S stars, this confirms the TDU-model of those TP-AGB stars. This is not the case of the HC-stars in the thick disk, with [T-eff] greater than or similar to 3400 K and [M-bol] greater than or similar to -3.4. The faint HC1 and HC2-stars ([M-bol] similar or equal to1.1(-1.0)(+0).(7)) are found slightly brighter than the BaII giants ([M-bol] similar or equal to -0.3 +/- 1.3) on average. Most RCB variables and HdC stars range from M-bol similar or equal to -1 to -4 against -0.2 to -2.4 for those of the three population II Cepheids in the sample. The former stars show the largest luminosities [M-bol] less than or similar to -4 at the highest effective temperatures (6500-7500 K), close to the M-bol similar or equal to -5 value for the hot LMC RCB-stars (W Men and HV 5637). A full discussion of the results is postponed to a companion paper on pulsation modes and pulsation masses of carbon-rich long period variables (LPVs; Paper IV, present issue).