5 Individual estimates and properties of the sample

Figure 2: Histograms of the distributions of the differences of the observed color indices (obs) and the calculated (cal) - arbitrary scale - from estimated intrinsic luminosities for initial and final discriminations of stars into the groups.

   
5.1 Individual estimates

Once the parameter estimation and group discrimination is completed, each star in our initial sample is a posteriori attributed to one of the LPV groups identified in each bandpass, following the method described in Sect. 3. This allows us to estimate the most probable individual distance and absolute magnitude in each band according to the observed astrometric, kinematic and photometric data and attributed group.

Due to the probabilistic nature of the Bayesian procedure, some misclassification is unavoidable. To check and improve individual star assignations in each wavelength, we compare the calculated color indices ${\rm cal}=M_{\lambda_1}-M_{\lambda_2}$ (obtained from the estimated individual absolute magnitudes deduced by the Bayesian assignations in the $\lambda_1$ and/or $\lambda_2$ wavelengths) with the observed color indices ( ${\rm obs}=m_{\lambda_1}-m_{\lambda_2}$). 5% of the stars are re-assigned to groups reducing the differences $\rm cal-obs$ for their indices 25-12 and/or K-12.

Figure 2 shows the histograms of difference of $\rm cal-obs$for the indices 25-12 and K-12 of all the stars in the sample. These distributions are related to the errors of the individual estimated luminosities and of the observed magnitudes. We can deduce that the accuracies of our estimated individual luminosities are distributed according to a gaussian rule of standard error 0.3 mag. and 0.1 mag. respectively in K and IRAS bands. The lower accuracy in the IRAS bands is consistent with the fact that IRAS photometry is more homogeneous than the K photometry, and that the variability amplitude of LPVs is smaller in the IRAS bands than in K. As previously stated, the LM method has allowed us to take advantage of all the available information, leading to better estimations of the individual absolute magnitudes. Furthermore, the LM method has provided at the same time the statistical distribution of these individual magnitudes (see Sects. 3 and 4). The individual estimates of K, 12 and 25 luminosities are given in a table available in electronic form at the CDS and are included in the specialized ASTRID database.

   
5.2 Comparison of sample/population

The LM method gives unbiased calibrations for the base population. It also gives individual kinematic and photometric estimates for each star of the sample. The distribution of these individual estimates (Sect. 5.1) is, of course, biased by the sample selection criteria, contrary to the group characteristics derived in Sect. 4. A comparison of the statistical properties of the sample with the calibrated parameters for the population allows us to check the representativity or the bias of the sample with respect to the population.

   
5.2.1 Kinematic representativity

Let us analyze the representativity of our sample stars with respect to the kinematics. The observed proper motions and radial velocities, together with the estimated individual distances, allow us to compute the three velocity components (U,V,W) and the distance Z above the galactic plane for each star in our sample. The mean kinematical properties of our sample derived from these individual estimates are shown in Table 5 for each group of the K and 12 bands. They are very similar to the parameters describing the groups (Tables 2 and 3), showing that our sample is very representative of the LPVs population as far as the kinematics is concerned.

This conclusion was expected since there is a priori no selection affecting (directly or indirectly) the kinematics of our sample and thus no bias is introduced in the kinematics of the stars. We can also note that the proportion of the different groups in the sample is close to that found in the population.

   
5.2.2 Luminosity representativity

Figure 3: Distributions of individual luminosities in K, 12 and 25 from top to bottom, of each group (D,OD ED, or D,ODb,ODf,ED from left to right) of the sample compared to the distribution of the calibrated luminosity (in units normalized to the surface of each histogram) for the same group.

Although no kinematical bias is introduced when selecting a sample with a cut in apparent magnitude, it is well known that a bias in luminosity is introduced. Figure 3 shows the histograms of the individual absolute magnitudes of the stars in our sample in each group of the K and IRAS bandpasses, together with the normal unbiased distributions estimated by the LM method for the population. These distributions are in units normalized to the surface of each histogram - and not to the number of population stars in each sample -, thus only the relative shapes and the magnitude shifts of both histogram and unbiased distribution are relevant. The bias of our sample towards higher luminosities is very clear both in K and IRAS bands. In the IRAS bands, the under-representativity of faint stars in our sample is more pronounced for LPV stars in the disk group than in the other IRAS groups. This corresponds to the classical Malmquist bias (1936), increasing with increased $\sigma_M$value. In short, the under-representation of faint stars in our sample is important for the K or IRAS faint stars and even more for the disk population, specially in the case of IRAS bandpasses.

However, let us remark that the brightest stars in every group of the sample coincide with the brightest luminosity of the group base population.

 

 

Table 5: Mean kinematical parameters of the sample computed from individual velocities and positions of stars.

K
	D	OD		ED
V0	-13	-31		-121
$\sigma_U$	30	41		111
$\sigma_V$	18	27		62
$\sigma_W$	20	26		84
Z0	185	245		621
N	396	224		39
%	60	34		6
12
	D	ODb	ODf	ED
V0	-7	-28	-20	-105
$\sigma_U$	20	42	35	110
$\sigma_V$	14	27	22	66
$\sigma_W$	12	39	21	77
Z0	152	343	180	714
N	239	273	231	51
%	30	34	29	7

   
5.2.3 Envelope effects and representativity

The luminosity sampling bias is not independent of the existence, thickness and composition of a circumstellar envelope around LPVs. Figure 4, which shows the percentage of known LPVs measured by HIPPARCOS (LPVs:%HIP) as a function of the IRAS (25-12) color index, shows that the incompleteness depends on the IRAS color. This is not surprising because the thicker the envelope, the fainter the star in the visual wavelengths.

This is confirmed if instead of using the known LPVs we use the IRAS sources with a (25-12) color index compatible with the LPVs values of this index. In doing so, we include stars in the LPV region not necessarily classified as variables (IRAS sel.:%LPVs). The bias of the HIPPARCOS sample is more strongly dependent on the envelope thickness if we do the comparison with these selected IRAS sources. Thus the percentage of stars observed by HIPPARCOS (IRAS sel.:%HIP) strongly and abruptly increases up to 80 % for 25-12 decreasing to zero.

Finally, Fig. 5 shows how much the sample of carbon-rich stars measured by HIPPARCOS (C stars:%HIP) does not represents either the percentage of the C-rich stars among the known LPVs (LPVs:%C stars) or the percentage of stars known as LPVs measured by HIPPARCOS (LPVs:%HIP). Thus one should be careful about making any interpretation from the percentages of C-rich stars, as we will see in Sect. 6.4.

Figure 4: Percentages of LPVs observed by HIPPARCOS (full circles) as a function of the 25-12 IRAS color index. They are compared to the percentages of stars observed by HIPPARCOS (+) and of known LPVs (*) among the sample of IRAS sources selected as probable LPVs from their IRAS color indices.

Figure 5: Percentages of C stars observed by HIPPARCOS (empty circles) and among known LPVs (empty squares) as a function of the 25-12 IRAS color index compared with the percentages of LPVs observed by HIPPARCOS (full circles).