4 Mean data in terms of photometric groups

Mean photospheric radii and mean luminosities or equivalently mean absolute bolometric magnitudes, are summarized and briefly discussed.

4.1 The carbon stars

Mean values and dispersions of the three coefficients $C_{{T_{\rm eff}}},$ $C_{\rm R}$ and $C_{\rm L},$ are quoted in Table $~\ref{coef_gr},$ with normalization factors of 1, 10 and 100 respectively. Then mean luminosities $\left( L/L_{\odot} \right)$ are deduced from Eq. (12) and mean radii $\left< R/R_{\odot} \right>$ from Eq. (5). Finally mean absolute bolometric magnitudes were calculated from Eq. (17). The corresponding ranges in radii and absolute bolometric magnitudes are denoted as (inf;sup) in Cols. 8 and 9. Following our approach (Sects. 3.1 and 3.3), these mean values are in principle free of the bias described by Smith & Eichhorn (1996). They are however affected by the Malmquist bias whose influence was studied in Knapik et al. (1998) and in Sect. 2 of Paper II. Here, the corrections for the effect of the Malmquist bias on bolometric absolute magnitudes are applied as estimated in Sect. 4 of Paper II. The values (Sect. 2.3) are small when compared to dispersions in Table $~\ref{coef_gr}.$

After steady values close to $\left( 6.5 \pm 0.4 \right)\times 10^{-2}$ observed for the groups HC0 to HC2, C_L continuously decreases along the sequence HC3-HC5 and then along CV1-CV7. Since the averaged effective temperatures diminish continuously along that sequence, this is a consequence of $\left<L/L_{\odot}\right>$ increasing with decreasing $\left<T_{\rm {eff}}\right>$ (see Sect. 7 and the HR diagram in Figs. $~\ref{hr1}$ and $~\ref{hr2}$). This is in good agreement with the evolutionary tracks as predicted for stars of low and intermediate masses. At the same time, the $C_{\rm R}$-values decrease and $\left< R/R_{\odot} \right>$ increases.

 

 

Table 3: Mean data obtained for the fourteen photometric groups of carbon stars (HC, CV and transition SCV: Col. 1). The mean effective temperatures and reference relative angular diameters in mas from Paper I are quoted in Cols. 2 and 3 respectively, as well as the corresponding mean coefficients in Col. 4: $c_{{T_{\rm eff}}}~\pm~\Delta c_{{T_{\rm eff}}}= \langle C_{{T_{\rm eff}}}~\pm~\Delta C_{{T_{\rm eff}}} \rangle .$The reduced mean coefficients $c_{{\rm R}}\pm\Delta c_{{\rm R}}= 10~\langle C_{{\rm R}}\pm\Delta C_{{\rm R}}\rangle$ and $c_{{\rm L}}\pm\Delta c_{{\rm L}}= 10^{2}~\langle C_{{\rm L}}\pm\Delta C_{{\rm L}}\rangle$ are given in Cols. 5 and 6 respectively. The mean luminosities and radii in solar units, are shown in Cols. 7 and 9 together with the ranges (inf; sup) of the latter. The deduced mean absolute bolometric magnitudes and corresponding ranges can be found in Col. 8. Mean effective temperatures are quoted in Col. 10, which are very close to those of Paper I (Col. 2); $^{\rm {a}}$ without C4247 brighter than the other 3 stars; $^{\rm {b}}$ without the underluminous C2334; $^{\rm {c}}$ values corrected for the Malmquist bias (Paper II).

G	$\langle T_{\rm {eff}}\rangle$ n	$\langle \Phi_{0}\rangle$	$c_{{T_{\rm eff}}}~\pm~\Delta c_{{T_{\rm eff}}}$	$c_{{\rm R}}\pm % \Delta c_{{\rm R}}$	$c_{{\rm L}}\pm\Delta c_{{\rm L}}$	$\langle L/L_{\odot} \rangle$	$\langle M_{\rm {bol}} \rangle$ (inf;sup)	$\langle R/R_{\odot} \rangle$ (inf;sup)	$\langle T_{\rm {eff}}\rangle$
HCO	5620 4	5.94	$2.843\:\pm\:0.145$	$1.47\:\pm\:0.65$	$5.15\:\pm\:2.28$	380	-1.71 -0.92;-2.98	20 14;36	5645
HCO(a)		5.94	$2.843\:\pm\:0.145$	$1.77\:\pm\:0.31$	$6.17\:\pm\:1.24$	380	-1.32 -0.92;-1.81	17 14;20	5675
HC1	4890 25	6.72	$2.416\:\pm\:0.045$	$1.46\:\pm\:0.38$	$6.96\:\pm\:2.68$	205	-1.06 -0.35;-2.11	23 18;31	4895
HC1(c)				$2.0 \:\pm\:0.64$	$8.05\:\pm\:3.10$	154	-0.74 0.0;-1.8	17 13;25
HC2	4290 27	8.38	$2.305\:\pm\:0.105$	$1.48\:\pm\:0.51$	$6.44\:\pm\:2.24$	240	-1.22 -0.58;-2.15	28 21;43	4280
HC2(c)				$1.64\:\pm\:0.56$	$7.09\:\pm\:2.5$	200	-1.02 -0.3;-2.0	26 19;39
HC3	4005 19	9.41	$2.256\:\pm\:0.056$	$1.09\:\pm\:0.44$	$4.54\:\pm\:2.16$	485	-1.99 -1.15;-3.36	43 31;73	3995
HC4	3965 15	9.57	$2.275\:\pm\:0.065$	$0.650\:\pm\:0.033$	$2.85\:\pm\:1.45$	1235	-3.00 -2.11;-4.54	74 49;150	3980
HC5	3480 20	11.9	$2.151\:\pm\:0.093$	$0.517\:\pm\:0.153$	$2.40\:\pm\:0.70$	1735	-3.37 -2.81;-4.12	115 89;164	3465
CV1	3285 36	13.6	$2.197\:\pm\:0.069$	$0.438\:\pm\:0.171$	$2.04\:\pm\:0.79$	2400	-3.72 -3.01;-4.79	156 112;256	3275
CV1(c)				$0.48\:\pm\: 0.18$	$2.14\:\pm\:0.87$	2190	-3.62 -3.00;-4.70	142 102;233
CV2	3035 46	16.5	$2.293\:\pm\:0.044$	$0.359\:\pm\:0.149$	$1.53\:\pm\:0.57$	4255	-4.34 -3.65;-5.35	230 162;394	3040
CV2(c)				$0.38\:\pm\: 0.17$	$1.61\:\pm\:0.63$	3870	-4.24 -3.50;-5.30	216 152;370
CV3	2915 43	18.2	$2.337\:\pm\:0.047$	$0.349\:\pm\:0.119$	$1.50\:\pm\:0.52$	4450	-4.39 -3.74;-5.32	261 195;395	2920
CV3(c)				$0.36 \:\pm\:0.12$	$1.57\:\pm\:0.57$	4055	-4.29 -3.70;-5.20	253 189;383
CV4	2775 32	20.5	$2.370\:\pm\:0.033$	$0.305\:\pm\:0.129$	$1.32\:\pm\:0.54$	5700	-4.66 -3.91;-5.80	336 236;583	2770
CV4(c)				$0.33 \:\pm\:0.13$	$1.39\:\pm\:0.59$	5200	-4.56 -3.80;-5.70	313 220;543
CV5	2645 44	23.5	$2.434\:\pm\:0.045$	$0.299\:\pm\:0.092$	$1.23\:\pm\:0.37$	6590	-4.82 -4.24;-5.60	392 299;567	2630
CV5(c)				$0.30 \:\pm\: 0.11$	$1.29\:\pm\:0.41$	6030	-4.72 -4.20;-5.50	394 299;567
CV6	2445 49	29.4	$2.610\:\pm\:0.061$	$0.307\:\pm\:0.078$	$1.11\:\pm\:0.32$	8130	-5.05 -4.49;-5.79	479 382;644	2430
CV6(c)				$0.31 \:\pm\:0.08$	$1.16\:\pm\:0.35$	7450	-4.95 -4.40;-5.70	467 372;628
CV7	1955 16	69.3	$3.932\:\pm\:0.506$	$0.358\:\pm\:0.14$	$0.93\:\pm\:0.40$	11580	-5.43 -4.6;-6.8	969 700;1580	1945
CV7(b)		69.3	$3.932\:\pm\:0.506$	$0.311\:\pm\:0.09$	$0.78\:\pm\:0.27$	16400	-5.81 -5.2;-6.7	1112 870;1560	1945
SCV	2775: 11	20.5	$2.395\:\pm\:0.080$	$0.188\:\pm\:0.028$	$0.79\:\pm\:0.09$	15835	-5.77 -5.51;-6.06	545 475;641	2790

It can be seen that the whole domain of the brightest normal giants (luminosity class III), bright giants (II) and finally faint supergiants (Ib), is populated, with no gap left, from $\left<M_{\rm {bol}}\right>\simeq -1.1$ to -5.8. The BaII stars, which are fainter on average (Sect. 4.2), are observed down to $\left<M_{\rm {bol}}\right>\simeq 1.0.$A peculiar case is represented by the SCV-group. Knapik et al. (1999) found it to stand as a transition between S and CV-stars. We adopted the $\Phi_{0}$-value of group CV4 since it leads to $\left<T_{\rm {eff}}\right>$ close to $2775~\rm {K},$ the mean value of the CV4-group. The SCV-group does exhibit a luminosity of $16~ 000~L_{\odot}$that is $\left<M_{\rm {bol}}\right>\simeq -5.8,$ close to the values for CV7. Both groups are however poorly-documented. For instance, including the underluminous CV7-star C2334 (n=16) leads to $\left<M_{\rm {bol}}\right>\simeq -5.4$ instead of -5.8 (Table $~\ref{coef_gr}$). At the faint end, the three groups HC0, HC1 and HC2 directly yield $\left<M_{\rm {bol}}\right>\simeq \left(-1.2 \pm 0.13\right)$ i.e. about $\left(240~L_{\odot}\right)$ with uncorrected values, and $\left<M_{\rm {bol}}\right>\simeq \left(-1.0 \pm 0.2\right)$ when our correction for the Malmquist bias is applied. Their locus is found at the junction between classes II and IIIa, well above the clump of classes IIIb-IIIab. A practically continuous luminosity distribution is observed for carbon giants in Table $~\ref{coef_gr},$ and delineating discrete luminosity classes is hardly justified. This is also clear for HIPPARCOS oxygen-rich giants in the HR diagrams published by Perryman et al. (1995). From mean absolute magnitudes of Table $~\ref{coef_gr}$ and bolometric corrections of Sect. 2, and mean color indices of Table 11 in Paper I, we found $\left<M_{{K}}\right>\simeq -2.5$ to -3.3 for early HC-stars.

Cooler R-stars, e.g. HC3, are still brighter with $\left<M_{{K}}\right>\simeq -4.4.$ Since HC0 stars are very few, the obtained mean value $\left<M_{{K}}\right>\simeq -3.0$ is dominated by HC1 and HC2 stars. This is 1 mag brighter than found by Knapp et al. (2001) for corresponding R0-R3 stars. They adopted a Gaussian distribution of $\left<M_{{K}}\right>\simeq -2.0$ and standard deviation 1.0 mag, very close to the $M_{{K}}\simeq -1.6$ value for clump red giants of Alves (2000). We actually found this is the case for many BaII stars that are on average fainter than early R stars (Sect. 4.2).

Our method (Knapik et al. 1998, Paper II) and that of Knapp et al. (2001) are statistical in nature (see also Pourbaix & Jorissen 2000), but they differ in several aspects. The samples used (HC1 to HC3-stars, or equivalently spectral types R0 to R3) include nearly the same HIPPARCOS stars. Pourbaix & Jorissen however re-analyzed the HIPPARCOS Intermediate Astrometric Data, while the published observed parallaxes and proper motions (ESA 1997) were used in our work. The biases as described in Sect. 2.3 are not accounted for in the same way. An additional difference is that Knapp et al. (2001) assumed that the absolute magnitudes are distributed about a single mean absolute magnitude in the infrared ( $\left< M_{{K}}\right>\simeq -2$) with a 1 mag intrinsic standard deviation, as consistent with their analysis. On the contrary, we find a gradient on $\left< M_{{K}}\right>$ from HC1 to HC3, as mentioned above. Knapp et al. (2001) mention a possible shift of their results, but toward $\left< M_{{K}}\right>\simeq -1.$ We have simulated possible systematic effects on our mean absolute magnitudes. Varying the q-slope of Sect. 2.3 from 2.35 to 3, up to 0.3 mag shifts are deduced (possible underestimates of the Lutz-Kelker bias). A value in excess of 4 would be required to reach a 1 mag-shift, which we exclude. The corrections for the Malmquist bias are known to within 0.1-0.15 mag. We consider the combined effect of both uncertainties should not exceed 0.4 mag in the worst case.

 

 

Table 4: Mean data obtained for the five variability classes of carbon stars: M(1) denotes the full sample of carbon Miras, whereas the underluminous Miras T Lyn and RZ Peg were removed from the M(2)-sample. The meaning of symbols is the same as in Table  $\ref{coef_gr},$ except for the "equivalent'' group $\rm\left <G\right >$ (see text for details).

Cl	$c_{{\rm\rm R}}\pm\Delta c_{{\rm R}}$	$c_{{\rm L}}\pm\Delta c_{{\rm L}}$	$\langle L/L_{\odot} \rangle$	$\langle M_{\rm {bol}} \rangle$	(inf;sup)	$\langle T_{\rm {eff}}\rangle$	$\langle \Phi_{0}\rangle$	$\langle R_{\rm {p}}/R_{\odot} \rangle$	(inf;sup)	$\langle \textrm{G} \rangle$	n
Cst	$1.40\:\pm\:0.58$	$6.02\:\pm\:2.37$	275	-1.37	(-0.65;-2.46)	$4375\:\pm\:475$	8.15	30	(20;50)	HC2	73
Lb	$0.409\:\pm\:0.179$	$1.80\:\pm\:0.86$	3075	-3.99	(-3.14;-5.39)	$2950\:\pm\:330$	18.2	225	(155;400)	CV3	111
SRb	$0.337\:\pm\:0.147$	$1.50\:\pm\:0.65$	4475	-4.40	(-3.61;-5.64)	$2895\:\pm\:325$	18.2	270	(190;480)	CV3+	60
SRa	$0.315\:\pm\:0.130$	$1.31\:\pm\:0.58$	5860	-4.69	(-3.90;-5.96)	$2730\:\pm\:270$	20.5	325	(230;555)	CV4	20
M(1)	$0.332\:\pm\:0.117$	$1.30\:\pm\:0.57$	5965	-4.71	(-3.92;-5.97)	$2550\:\pm\:540$	26.5	400	(295;615)	CV5-6	53
M(2)	$0.312\:\pm\:0.081$	$1.19\:\pm\:0.40$	7015	-4.89	(-4.26;-5.77)	$2535\:\pm\:540$	26.5	425	(340;570)	CV5-6	51

4.2 The BaII stars

Bergeat & Knapik (1997) studied a sample of 52 BaII stars they locate in the observational HR diagram, i.e. absolute visual magnitudes vs. spectral types, or almost equivalently oxygen-rich photometric groups. HIPPARCOS data of a much larger sample were used, together with radial velocities, by Mennessier et al. (1997). Both studies conclude this spectroscopic group is an inhomogeneous one, spreading upon luminosity classes V to Ib. Preparing the theoretical HR diagram $\left(M_{\rm {bol}},~T_{\rm {eff}}\right)$ of Sect. 7, we have integrated the 44 best-documented dereddened SEDs and derived the values of the apparent bolometric magnitudes $m_{\rm {bol}}.$ For 29 additional stars, mean bolometric corrections

$$\left<BC_{{V}}\right>=\left<m_{\rm {bol}}-V_{0}\right>=\left<M_{\rm {bol}}-M_{{V}}\right>$$ (18)

were used. The reader is referred to Bergeat & Knapik (1997) for the use of observed parallaxes. We mention here that HD199394 is classified as IIIb instead of IIIa as was erroneously quoted in their Table 1. In the present study, we exclude the supergiants (class Ib or photometric groups sg) and the dwarfs above the main sequence and subgiants (classes V to IV or photometric groups d and g), and the dwarfs on the main sequence (class V or groups d). We only keep the BaII giants either bright (class II) or normal (class III subdivided into IIIa, IIIab and IIIb, or photometric group g). Revised (post-HIPPARCOS) definitions of classes III can be found in Keenan & Barnbaum (1999). For the corresponding 56 stars, we obtained

$$\left<M_{\rm {bol}}\right>\simeq -0.27\pm 1.26.$$ (19)

This is fainter than the faintest value deduced for HC carbon stars (about -1), even if both samples substantially overlap. Apart from a concentration around revised class IIIb (the "clump''), no marked structure is observed.

The effective temperatures were derived from Table B.7 of Knapik (1999), a calibration of mean values for photometric groups adapted from Perrin et al. (1998) and Richichi et al. (1999). Individual values of $M_{\rm {bol}}$ and $T_{\rm {eff}}$can be found at the end of Table 2 for 62 BaII stars and  $T_{\rm {eff}}$ for 5 additional stars.