6 Properties of crossed groups

   
6.1 Crossing K and IRAS

Due to the difficulties coming from the large uncertainties on m_V: large amplitudes and variations from one cycle to another, the Vresults will not be further used in our analysis and we will concentrate on the K and IRAS results.

The luminosity estimations in the K and IRAS bands complement each other in the sense that, in general, K fluxes characterize stellar properties while IRAS fluxes provide information on the circumstellar envelope. Thus, the most physically interesting results are obviously obtained by simultaneously considering K and IRAS luminosities. We have already seen that it is very difficult to calibrate these luminosities at the same time due to the incomplete knowledge and non-uniqueness of the relation between the different magnitudes (Sect. 3). Another possibility is to do a crossing of the groups from both the K and IRAS calibrations i.e. to examine the properties of the stars belonging to the same group in Kand IRAS.

The first remarkable result concerns the number of crossed groups: only 7 are not empty while 12 could, a priori, be expected. Interestingly, there is no mixture of the extended disk group (in either wavelength) with any other group, except two stars (O-rich SRa: RW Eri and O-rich Mira: SV And), which is compatible with the statistical classification errors. This is a nice confirmation of the power of the LM method to extract consistently distinct groups in biased samples of a given stellar population.

 

 

Table 6: Mean kinematical parameters for the crossing K(IRAS) groups computed from individual velocities and positions. They can be considered as representative for the LPVs population (see Sect. 6.1).

	Disk1		Disk2		Old Disk		Ext. Disk
	D(D)	OD(D)	D(ODf)	D(ODb)	OD(ODf)	OD(ODb)	ED(ED)
nb of stars	141	21	103	90	81	113	36
V0	-6	-7	-18	-19	-34	-32	-123
$\sigma_U$	23	28	35	36	42	40	114
$\sigma_V$	13	23	20	18	24	26	63
$\sigma_W$	11	11	34	16	21	29	80
Z0	166	191	208	231	160	310	620
age range	$1{-}4\times 10^9$ yr		$4{-}8\times10^9$ yr		8-gt $10 \times 10^9$ yr
lower mass limit	$2{-}1.4~{\cal M}_{\odot}$		$1.4{-}1.15~{\cal M}_{\odot}$		$1.15{-}lt1~{\cal M}_{\odot}$

Table 6 gives the number of stars in our sample which are assigned to every crossed group G(G') i.e. to group G in K and G' in IRAS. In Sect. 5.2.1 our LPVs sample is shown to be representative of the LPVs population as far as the kinematics is concerned. Thus the mean kinematics of the stars belonging to a crossing group K(IRAS) may be considered as representative of the mean kinematics of the LPVs population belonging to this group.

Obviously such a consideration does not apply to the luminosities (see Sect. 5.2.2). The assigned groups are given in annex A (electronic table).

   
6.2 Ages and initial masses

Table 6 gives the values of the axes of the velocity ellipsoids and the scale height of each of the 7 crossed K and IRAS groups. Given that our sample is representative of the population in terms of kinematics, as already seen in Sect. 5.2.1, we can use the kinematical values of Table 6 as representative in terms of galactic populations.

The relation between the mean kinematics of a galactic population and its age allows us to estimate the range of ages of the groups. Furthermore, classical statistical studies of stars known to belong to different galactic populations and of different metallicity abundances allow us to add an estimate of the range of metallicity. By comparing the values in Table 6 with the results on kinematics and metallicity of the galactic populations by Mihalas & Binney (1981) and by Stromgren (1987), we can deduce:

• D(D) and OD(D) populations are $1{-}4\times 10^9$ yr old with a solar metallicity. Part of the stars in D(D) are classified as belonging to the bright disk population (BD) by the V analysis, being younger and with a small over-abundance.

• The age of D(ODf) and D(ODb) populations can be estimated to be in the range $4{-}8\times10^9$ yr, with a metallicity from the solar one to [Fe/H] around -0.4 i.e. Z between 0.006 and 0.02.

• OD(ODf) and OD(ODb) populations are composed of stars older than $8\times 10^9$ yr, up to more than 10¹⁰ yr, with [Fe/H] from the solar value to -0.7 i.e. Z between 0.004 and 0.02
• Stars classified as belonging to ED are probably very old and deficient with [Fe/H] between -0.7 and -1.5 i.e. Z of the order of (0.001, 0.004).

Moreover, we can estimate initial masses from evolutionary tracks. From Binney & Merrfield (1998) we can estimate a lower limit of the initial mass of a given age star that has reached the AGB. Thus, values of 2, 1.4, 1.15, 1  ${\cal M}_{\odot}$ can be deduced as lower limits of ${\cal M}_{\rm ms}$ of the stars of solar metallicity of respectively 1, 4, 8, $12\times 10^9$ yr. This agrees with the results on the ages at the top of the early-AGB (Charbonnel et al. 1996).

All these results are in the same ranges as the ones given by Jura & Kleinmann (1992), but our classification is more refined because it is not based on the spectral types and periods which are now known to not really be discriminative parameters for LPVs.

In the rest of this paper the stars belonging to D(D) and OD(D), D(ODf) and D(ODb), OD(ODf) and OD(ODb), ED(ED) will be called disk1, disk2, old disk and extended disk LPVs respectively (see Tables 6 and 8).

   
6.3 Evolutionary tracks

Figure 6 shows the K magnitude as a function of the V-K color index for each of the disk1, disk2, old disk and extended disk groups. For comparison, evolutionary AGB model predictions are also shown for three different masses (1.5, 2.5 and ${{4\, M}_\odot}$) at solar metallicity, and at three different metallicities (Z=0.004, 0.008 and 0.02) for ${{2.5\, M}_\odot}$ stars. These models have been computed at Geneva, and are described in Mowlavi (1999) and Mowlavi & Meynet (2000). The conversion between model variables (effective temperature $T_{\rm eff}$and luminosity L) to observable quantities (V-K and M_K) was done by using the transformations given by Ridgway et al. (1980). Several uncertainties affect both model predictions and the color transformation relations for AGB stars (which are characterized by peculiar chemical compositions as a result of dredge-up episodes). They also affect the determination of the stellar V magnitudes as noted in Sect. 4.1. Thus, the comparison between the evolutionary tracks and the distributions of our sample stars in each group shown in Fig. 6 can only provide qualitative results.

Figure 6: Theoretical evolutionary AGB tracks for stars of 1.5, 2.5 and $4~{\cal M}_{\odot}$ with solar metallicity and for deficient (Z=0.008 and Z=0.004) stars of $2.5~{\cal M}_{\odot}$ compared to distribution of the individual estimated K luminosities as a function of the color index V-K according to the assigned kinematical groups.

Evolutionary tracks show that, at a given metallicity, stellar luminosities increase with initial stellar mass (at a given V-K). We thus conclude, at least qualitatively and due to the solar or slightly deficient abundance of the majority of HIPPARCOS stars, that our sample stars have lower mass limits which respectively decrease as we consider disk1, disk2 and old disk groups. This is in agreement with the conclusions drawn in Sect. 6.2. Stars of the extended disk group, on the other hand, are compatible with lower metallicities, given their higher K luminosities.

We note that the evolutionary tracks cannot, even if freed from any uncertainty, attribute a single (M,Z) set of parameters to a star because of the degeneracy of those two parameters. A higher luminosity in K at a given V-K could be either attributed to a higher initial mass or lower metallicity. Kinematics can help in distinguishing such ambiguous cases.

Finally, we can comment on the smaller V-K values for the bright disk LPVs. This confirms the strong circumstellar absorption in V for these massive stars.

   
6.4 Variability and spectral types

The composition of each crossed group K(IRAS) with respect to usual classifications of LPVs (variability and spectral types) is given by the contingency table of both assignations (Table 7). The associated attraction-repulsion indices (Tenenhaus 1994) - ratio of the observed frequency to the theoretical frequency in the case of independence of both modalities - are more significant in characterizing the correspondence analysis of types and of groups. These indices are given in Table 8 for oxygen and carbon-rich LPVs. The two modalities attract or repel each other if the attraction-repulsion index is larger or smaller than 1 respectively.

 

 

Table 7: Contingency table between crossing groups and L, SR and M variability and M and C spectral types.

	L-C	SRb-C	SRa-C	M-C	L-O	SRb-O	SRa-O	M-O
D(D)	29	25	7	9	20	29	4	31
OD(D)	1	3	0	1	1	4	1	9
D(ODb)	7	3	1	4	6	33	7	28
D(ODf)	2	4	0	2	42	39	3	2
OD(ODf)	3	5	0	1	34	22	2	0
OD(ODb)	0	3	0	0	14	28	13	54
ED(ED)	1	1	0	1	4	7	3	18

 

 

Table 8: Attraction-repulsion indices between crossing groups and L, SR and M variability and M and C spectral types.

		L-C	SRb-C	SRa-C	M-C	L-O	SRb-O	SRa-O	M-O
Disk1	D(D)	2.7	2.3	3.5	2.0	0.6	0.7	0.4	0.9
Disk1	OD(D)	0.7	1.4	0	1.6	0	0.9	0.9	1.9
Disk2	D(ODb)	0.3	0.5	0.8	1.5	0.3	1.4	1.4	1.4
Disk2	D(ODf)	0.3	0.6	0	0.7	2.3	1.5	0.6	0.1
Old Disk	OD(ODf)	0.6	1.1	0	0.5	2.5	1.2	0.4	0
Old Disk	OD(ODb)	0	0.4	0	0	0.6	0.9	2.2	2.1
Ext. Disk	ED(ED)	0.4	0.4	0	0.9	0.6	0.7	1.6	2.2

From Table 8 we can deduce that:

• The groups corresponding to initially less massive stars are essentially attractive for O-rich LPVs and the ones corresponding to initially massive stars are essentially attractive for C-rich LPVs;

• The strong attraction of C-rich stars by groups corresponding to more massive initial stars is clear. It agrees with the results already given by V calibration (Mennessier et al. 1999) and with both observations and model predictions according to which dredge-up of carbon from core to the surface of AGB stars is more efficient in massive stars (Dopita et al. 1997). However, we must be very cautions. Indeed, the biases described before (Sect. 5.2.3) show the over-representativity of C-rich stars in the sample and the large number of missing LPVs. To which galactic population do the not-HIPPARCOS-observed LPVs belong? Are they O or C-rich? The ratio of carbon to oxygen-rich LPVs that goes up to 78% for the D(D) sample could be due to the sampling. Thus deduced implications for the dependence on the mass of the efficiency of the dredge-up are to be taken with caution;

• O-rich SRa's are distributed close to O-rich Miras;

• O-rich SRb's are significantly present in the two disk population groups. The first one is composed of stars without a shell or with a thin one (cf. Sect. 4.3). This probably corresponds to early AGB stars with a relatively high initial mass. The second one contains stars with a shell and corresponds to stars in the same stage as SRa and Miras. This agrees with results by Kerschbaum & Hron (1992). O-rich irregular LPVs are close to the SRb's lacking a shell and are probably early AGB stars;

• The difference between L irregular variable stars according to their spectral type is evident. C-rich L stars correspond to massive TP-AGB's with a thick envelope. On the contrary, O-rich L stars are close to early AGB O-rich SRb's but their initial mass range seems to be more extended to lower masses;

• One CH star (V Ari) is assigned to the Extended Disk population and it is the faintest star in all the luminosities. This completely agrees with the usual hypothesis considering this type of star as an old giant star. Its amplitude of variability is less than 1 magnitude, a very small value even for a semi-regular. The C character of this type of star is however difficult to explain. HIPPARCOS observations suggest that V Ari could be a suspected multiple star but this is not conclusive.

   
6.5 Upper limit of the AGB

In Sect. 5.2.2 we remarked that the brightest stars in the sample agree with the brightest luminosity for each group population (see Fig. 3). Thus we can consider our sample as representative of the LPVs population as far as the brightest luminosities are concerned. Our calibrations show that the upper limit in K luminosity of the OD population ( $(K_0-3\sigma_K) = -8.1$ mag) is fainter than that of the D population ( $(K_0-3\sigma_K) = -9.4$ mag) as seen in Table 2. This confirms the dependence of the upper limit of the AGB on ${\cal M}_{\rm ms}$. Willson (1980) has described a schematic evolution on the AGB related to the mass-loss rate, its acceleration by the pulsations and probably the induced dust formation. She found a difference in solar luminosities of ${\sim}0.3 L/L_{\odot}$ where stars of solar abundance and ${\cal M}_{\rm ms}$ equal to 1.5 and 1 ${\cal M}_{\odot}$ leave the AGB. Our result is of the same order.