3 The C-coefficients, luminosities and radii

3.1 The C $\mathsfsl{_R}$-coefficient

In Paper I was established (Eq. (33)) the relation

$$\left< k \right> = \left(\Phi~/~\Phi_{0}\right)^{2}$$ (3)

where $\Phi$ is the true (observed) angular diameter of the star and $\Phi_{0}$ its diameter would its magnitude be $\left[1.08\right]_{0}=0,$ other thinks being equal.

The $\left< k \right>$-coefficient is a mean value over the wavelengths used, that is derived directly from photometry (Knapik et al. 1999, Sect. 2.4). It is thus an angular diameter on a relative scale as confirmed in Paper I (Sect. 8). It correlates with astrometric data, namely estimated true parallaxes $\varpi$ from Knapik et al. (1998; Sect. 4), in the expected way. The reference angular diameters $\Phi_{0}$ were calibrated as a function of effective temperatures (Paper I, Sect. 11), viz. $\Phi_{0}\left(T_{\rm {eff}}\right)$. The coefficient may still be written as

$$\left< k \right>^{1/2} \simeq 2.063\times 10^{8} \frac{R}{R_{... ..._{\odot}}{D}\:/\: \left(\frac{\Phi_{0}}{2}\right)_{\rm {mas}},$$ (4)

where R is the photospheric radius of the star. Replacing $\varpi=1/D,$ we introduce a new coefficient

$$C_{R}\:=\frac{\Phi_{0}/2}{R/R_{\odot}} \simeq \frac{\varpi _{\rm {mas}}}{214.94\: \left< k \right>^{1/2}}\cdot$$ (5)

The latter quantity is a linear and homogeneous function of $\varpi.$ Parallax errors predominate over photometric errors on $\left< k \right>^{1/2}.$ Within a restricted sample, say in a given photometric group, the distribution of true parallaxes may be assumed Gaussian, which is also the case of the quantity in Eq. (5). The $C_{\rm R}$-coefficient can then be averaged without being affected by the bias described by Smith & Eichhorn (1996). This is the angular radius (in mas) shown by a star of photometric coefficient $\left< k \right>$ and parallax $\varpi$ if its magnitude would have been $\left[1.08\right]_{0}=0$and its radius have been $R=R_{\odot}.$ In practice, the third member of Eq. (5) will be used to evaluate C_R, to be averaged for a given group (or class) of stars. The mean values $\langle R/R_{\odot} \rangle$ are then evaluated from $\Phi_{0},$ a quantity that correlates with groups and effective temperatures, and not with $\left < k \right >^{1/2}$ (Paper I).

Individual estimates of $R/R_{\odot}$ may be derived from individual $\Phi_{0}$'s as quoted in Paper I or from $\Phi_{0}\left(T_{\rm {eff}}\right)$ relations, to be replaced in Eq. (5). They should not be averaged since the corresponding means could be biased.

3.2 The $\mathsfsl{\left< k \right> ^{1/2}}$ vs. $\mathsfsl{\varpi}$ diagrams

The above Eq. (5) can be written as

$$\left< k \right>^{1/2} \simeq \alpha \: \varpi$$ (6)

where $\alpha \simeq \left( 214.94\: C_{R}\right)^{-1}.$ That coefficient is expected to span a range delineated by those in R and $\Phi_{0}$ for the considered group. The corresponding diagram of $\left < k \right >^{1/2}$ vs. $\varpi$ should exhibit a populated interior of an angle with vertex at origin. This is the result shown by Knapik et al. (1998) for 34 CV2-stars (their Fig. 3), whereas the diagram with observed parallaxes $\varpi_{0}$ as abscissae did not have this property. We present here an updated version of the former diagram including 43 CV2-stars and 46 SEDs (Fig. $~\ref{k_par1}$) that confirms the previous one. The diagrams for the groups HC0+HC1, HC2 to HC5, CV1, CV3 and CV4 are displayed in Fig. $~\ref{k_par2},$ and CV5 and CV6+CV7 in Fig. $~\ref{k_par3}.$ The poorly-documented SCV-group is not shown.

Figure 2: The photometric relative angular radii $\left < k \right >^{1/2}$ as a function of true parallaxes $\varpi$ in mas, for the CV2-group.

 

 

Table 1: Minimum and maximum slopes estimated as described in Sect. 3.2. A strong increase is noticed from HC3 to CV3 followed by a flat portion (CV3-CV6). It illustrates the increase of photospheric radii along the sequence of groups.

G	$(\alpha \pm \Delta\: \alpha)_{\rm min}$	$(\alpha \pm \Delta\: \alpha)_{\rm max}$
HC1	$(1.48\: \pm\:0.10)\:\times\:10^{-2}$	$(3.80\: \pm\: 0.20)\:\times\:10^{-2}$
HC2	$(1.95\: \pm\:0.13)\:\times\:10^{-2}$	$(4.48\: \pm\: 0.15)\:\times\:10^{-2}$
HC3	$(2.27\: \pm\:0.15)\:\times\:10^{-2}$	$(7.84\: \pm\: 0.40)\:\times\:10^{-2}$
HC4	$(4.36\: \pm\:0.63)\:\times\:10^{-2}$	$(1.18\: \pm\: 0.18)\:\times\:10^{-1}$
HC5	$(6.41\: \pm\:0.51)\:\times\:10^{-2}$	$(1.55\: \pm\: 0.28)\:\times\:10^{-1}$
CV1	$(4.93\: \pm\:0.85)\:\times\:10^{-2}$	$(1.87\: \pm\: 0.13)\:\times\:10^{-1}$
CV2	$(6.81\: \pm\:0.98)\:\times\:10^{-2}$	$(2.22\: \pm\: 0.11)\:\times\:10^{-1}$
CV3	$(8.43\: \pm\:1.22)\:\times\:10^{-2}$	$(2.04\: \pm\: 0.16)\:\times\:10^{-1}$
CV4	$(1.16\: \pm\:0.12)\:\times\:10^{-1}$	$(2.47\: \pm\: 0.24)\:\times\:10^{-1}$
CV5	$(9.41\: \pm\:1.17)\:\times\:10^{-2}$	$(2.27\: \pm\: 0.20)\:\times\:10^{-1}$
CV6	$(1.10\: \pm\:0.15)\:\times\:10^{-1}$	$(2.44\: \pm\: 0.30)\:\times\:10^{-1}$

Unfortunately, the number of stars per diagram is rather limited. The group CV2 was studied in detail. The Gaussian distribution

$$N\left(\alpha\right)~{\rm d} \alpha = \frac{N}{\left(2~\pi\ri... ...-\left<\alpha\right>\right)^{2}}{2~\sigma ^{2}}\;\rm {d}\alpha$$ (7)

could be fit around the arithmetic mean

$$\left<\alpha\right> \simeq 0.148 \pm 0.052.$$ (8)

Despite a large dispersion, a satisfactory solution was found with $N \simeq 0.859$ and $\sigma \simeq 0.057,$ that is

$$N\left(\alpha\right)~\rm {d} \alpha \simeq 6.050 \: \exp \left[ -156~\left(\alpha - 0.148 \right)^{2}\right]\;\rm {d}\alpha.$$ (9)

Two linear relations shown as full lines in Fig. $~\ref{k_par1}$ were derived which correspond roughly to

$$\alpha _{\rm {min}} \simeq \left<\alpha\right> - 1.36 \: \sigma$$ (10)

and

$$\alpha _{\rm {max}} \simeq \left<\alpha\right> + 1.36 \: \sigma$$ (11)

which results in 83% of data falling inside the materialized sector $\left(\alpha _{\rm {min}}, \alpha _{\rm {max}} \right).$ There is no significant departure except perhaps for $\rm C5418=V460$ Cyg.

Fits to the other diagrams for HC and CV-groups were also derived, which proved frequently more difficult to obtain. The obtained estimates are quoted in Table $~\ref{pmin_max}$ for the sequence of groups in our classification. They roughly correspond to inferior and superior limits in relative radii  $R/R_{\odot},$ for a given range in $\Phi_{0}.$ A general increase of both limits is noticed along the sequence of photometric groups which is moderate till HC3 and then from CV3 to CV6, where steady extremas at nearly $0.10\pm 0.01$ and $0.23\pm 0.02$ are reached. Such flattenings at both ends of the sequence will be found again while deriving luminosities and studying the locus of carbon giants in the HR diagram (Sects. 5 to 7).

Figure 3: Same diagram as Fig. $~\ref{k_par1}$ for the groups HC0+HC1, HC2 to HC5, and CV1, CV3 and CV4.

Figure 4: Same diagram as Figs. $~\ref{k_par1}$ and $~\ref{k_par2}$ for the groups CV5 and CV6+CV7.

3.3 The C $\mathsfsl{_L}$-coefficient

In order to again avoid the bias described by Smith & Eichhorn (1996), we introduce now the coefficient

$$C_{\rm L}\:=\:\left(L\:/\:L_{\odot}\right)^{-1/2}$$ (12)

from luminosity in solar units. We used the coefficient

$$C_{{T}_{\rm {eff}}}\:=\:\left(\frac{\Phi _{0}}{2}\right)\left(\frac{T_{\rm {eff}}} {T_{\rm {eff}\odot}} \right)^{2}$$ (13)

in Paper I (Eq. 36). Substituting Eq. (5), we obtain

$$C_{{\rm L}}\:=\:C_{{\rm R}}\:/\:C_{{T}_{\rm {eff}}}.$$ (14)

From Eq. (43) of Paper I, viz.

$$C_{T_{{\rm eff}}}=\frac{10^{-0.2\left(m_{{\rm bol}}-M_{{\rm bol \odot}}\right) +2}}{214.94~\left< k\right> ^{1/2}}$$ (15)

and the above Eq. (5), we finally transform Eq. (12) into

$$C_{{\rm L}}=\frac{\varpi _{\rm {mas}}} {10^{-0.2\left(m_{{\rm bol}}-M_{{\rm bol \odot}}\right)+2}}\cdot$$ (16)

This latter relation shows that $C_{\rm L},$ like $C_{\rm R},$ is a linear and homogeneous function of the true parallax $\varpi.$ Thus it can be averaged without being affected by the above-mentioned bias. The coefficient can be calculated from Eq. (16) with Eq. (14) used as a checking relation. We adopted $M_{{\rm bol} \odot} \simeq 4.73.$The coefficients thus obtained can be averaged and estimates of mean luminosities derived from Eq. (12) for a group or class of carbon giants. Finally we adopt mean bolometric luminosities as obtained from

$$\left< M_{\rm {bol}} \right>\:-\:M_{\rm {bol}~\odot} \simeq-~2.5 \log \left<L\:/\:L_{\odot}\right>\cdot$$ (17)

Individual values of $L/L_{\odot}$ can be derived from Eqs. (16) and (12) i.e. from individual values of the $C_{\rm L}$ coefficients. They should not be averaged due to the mentioned bias. The same statement applies to individual absolute bolometric magnitudes. We shall use the latter to locate every carbon star in the HR diagram (see Sect. 7).

3.4 The data for HIPPARCOS carbon stars

The values of $C_{\rm R}$ and $C_{\rm L}$ were calculated for about 300 HIPPARCOS carbon stars with usable astrometric data. The corresponding values of $C_{T_{\rm eff}}$ from Paper I were completed with the 81 additional stars of Sect. 2.2. Unfortunately, a few stars from initial proposals were missed. In addition to C3236 = SS Vir and C5496 = RX Peg, which were not correctly inserted in the entry catalogue (INCA), erroneous identifications affected C1616 = BK CMi, C1787 = BE CMi, C3614 = NSV7110, C4241 = U Lyr, C5371 = LU Cep and C5970 = V532 Cas. In addition HIC 22767 is not C808 = IRAS 04505+2241. No astrometric solution is given for C5721 = V363 Lac, and SU Tau = HD247925, a RCB variable, was replaced by HD247901.

The values of $C_{\rm R}$ were calculated from Eq. (5), making use of true parallaxes $\varpi$ derived from HIPPARCOS observations, and the mean relative angular diameter $\left < k \right >^{1/2}$ from multicolor photometry (see Sect. 3). The parallaxes were then used in Eq. (16) together with apparent bolometric magnitudes $m_{\rm {bol}}$ from Paper I, assuming $M_{\rm {bol}\odot} \simeq 4.73$ (see Sect. 5). Individual radii and luminosities were also derived as described at the ends of Sects. 3.1 and 3.3. The deduced individual absolute bolometric magnitudes are quoted in Table 2, which is only available at CDS. This table is a modified, augmented and updated version of Table 10 of Paper I available at CDS. It contains 508 entries corresponding to 508 analyzed SEDs. Effective temperatures, interstellar (and eventually circumstellar) extinctions, classification in our photometric groups, bolometric magnitudes and comments are quoted there. For about 370 objects, absolute bolometric magnitudes are given as derived from true parallaxes.