According to Fig. 9 most of the variables in our sample are on the AGB. Therefore, we can use our results to discuss the variability during the AGB phase. Our classification system for the type of variability aims to measure the regularity of the light change. Even not taking into account variations in the amplitude of the light change, we show that most stars have light curves that cannot be fitted by the simple combination of one or two excited periods. Regular variations are found with a wide range in period, while semiregular variability typically occurs mainly on time scales below 150 days (see Fig. 8). In Fig. 14, we compare the period distribution of the semiregular variables in our sample with the Milky Way SRVs listed in the GCVS. While in both cases the maximum of the distribution is at short periods, the GCVS distribution shows a significantly larger fraction of stars with periods longer than 150 days. These long periods may have been missed by our rather short time window. One would also expect a bias of the GCVS sample towards large amplitude variables as most of the data used there are based on photographic measurements. Furthermore, the period distribution from the GCVS given in Fig. 8 includes only one (main) period per object, while for the AGAPEROS data we give also secondary periods found for these stars.

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{lebzfig14.eps}\end{figure}$

Figure 14: Period distribution of semiregular variables in our sample and in the GCVS.

Due to the separation of amplitude and regularity in our classification system, we can explore the relation between these two quantities. We find that large amplitude variation occurs almost exclusively among the regular variables (see Fig. 7). However, there exist regular pulsators with small amplitudes. It is therefore not correct to classify all red variables below a certain amplitude limit as semiregular. A division into large and small amplitude variables seems to be more meaningful. Large and small amplitude variables are both found all along the AGB. This is illustrated in Fig. 15 where the $R_{\rm EROS}$ light amplitude is plotted against the DENIS K band measurement. Towards the tip of the AGB the fraction of regular as well as large amplitude variables increases. Below the RGB-tip, amplitudes become on the average smaller. The occurrence of regular and semiregular as well as small and large amplitude variables on the AGB indicates that AGB stars have to be seen as a highly inhomogeneous group. One reason for this may be a difference in stellar mass as noted above.

Summarizing, large amplitudes are well correlated with regular pulsations, but we find no correlation between large amplitude and stellar luminosity nor between small amplitude variability and semiregularity of the light change. This result is in agreement with Wood et al. (1999).

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{lebzfig15.ps}\end{figure}$

Figure 15: $R_{\rm EROS}$ amplitude versus $K_{\rm S}$ . Open boxes denote regular variables, filled triangles semiregular stars.

7.2 PL-relation

In the literature, the observed PL-relation of long-period variables is considered to be the same in different environments such as the LMC, the Galactic Bulge or globular clusters (see e.g. Glass et al. 1995; Feast et al. 2002). It is therefore independent of metallicity, contrary to the predictions of pulsation theory (see e.g. Wood & Sebo 1996). However, previous studies were restricted to Mira variables mainly due to limitation of sensitivity. Thanks to the microlensing surveys such as EROS, MACHO or OGLE, we can study systematically small amplitude variations over a (still rather small) time interval. Wood (2000) found for the SRVs in the LMC different PL-relations for different pulsational modes. However, these separations cannot be reproduced in the Galactic Bulge (Schultheis & Glass 2001). In addition, the PL-relation of the solar neighborhood (Bedding & Zijlstra 1998) looks different. Why is the PL-relation the same for Miras in different galactic environments, but not for SRVs?

Figure 13 shows the PL-relation for the AGAPEROS sample. On the one hand, the Mira variables, classically defined as long period and large amplitude stars, concentrate along Wood's sequence C. Regular variables at shorter periods would not have been classified as Miras. On the other hand, the semiregular variables (both according to the classical and to our definition), are spread all over the K- $\log P$ -plane. Making one fit with all semiregular stars would not result in a K vs. $\log P$ relation. In the solar neighborhood, Bedding & Zijlstra (1998) note that the SRVs are actually found on two sequences: the first one corresponds to the LMC Mira PL-relation (Wood's sequence C); the second one is located close to a PL-relation derived from Galactic globular cluster LPVs shifted 0.8 mag from the Whitelock globular cluster sequence (Whitelock 1986), as shown in Fig. 13. The Bedding & Zijlstra sequence, defined for SRVs, obviously mixes objects from Wood's sequence B and C, as shown in Fig. 13. The increase towards longer periods is consistent with the larger fraction of long period SRVs in the GCVS (Fig. 14) assuming that the detection of long periodic small amplitude variations is biased towards bright objects. Therefore, three PL-sequences seem to be more appropriate for semiregular variables. Multiperiodic stars are found on all three sequences A, B and C (see Fig. 13). Sequence D is almost exclusively occupied by stars with two periods in agreement with the suggestion from Wood (2000) that these long periodic variations are either due to binarity or a pulsation mode resulting from an interaction of pulsation and convection. However, there are also a few regular pulsating variables on this sequence with only one period. These stars would be definitely worth further investigation.

Schultheis & Glass (2001) showed that the interpretation of the PL-relation of Bulge SRVs is rather complex due to the depth of the Bulge ( $\sim$ $\rm\pm 0.35^{mag}$ , see Glass et al. 1995) and the variable interstellar extinction. There is no clear separation of the four sequences. We also showed that the LMC variables are much more homogeneous in their metallicity than the Bulge AGB stars (Fig. 11). This would explain part of the scatter in the K vs. $\log P$ plot for the Bulge.

7.3 Number densities

The number of semiregular variables in comparison to the regular variables is about a factor of 3. If we use the selection criterion of Cioni et al. (2001), i.e. all stars with $R_{\rm EROS}$ amplitudes smaller than 0.9 mag are SRVs, we end up with a ratio of almost 37 between SRVs and Miras in our sample. This value is much higher than what was found by Cioni et al. ( $\sim$ 5), so we assume that our sample is more complete at smaller amplitudes. In the Galactic Bulge, Alard et al. (2000) found that the proportion of SRVs with respect to Miras is about a factor of 20. Most recently, Derue et al. (2002) found a similarly large ratio between semiregulars and miras in the Galactic spiral arms. However, this ratio is of course very sensitive to the classification of SRVs (see above). For the Galactic disk, Kerschbaum & Hron (1992) found equal number densities for Miras and semiregular variables. However, they note that their sample of semiregular variables is probably not complete due to the difficulties in detecting small amplitude variables.

Do we see in different environments the same ratio of SRVs to Mira variables or does it depend on metallicity? Vassiliadis & Wood (1993) calculated lifetimes of the major evolutionary phases for different initial masses and different metallicities. They found that higher metallicity will increase the lifetime of the early-AGB but decrease the lifetime on the TP-AGB. Miras stars populate the TP-AGB, therefore in environments with higher metallicities, such as the Galactic Bulge the lifetime of the TP-AGB is shorter and thus the number densities should decrease. This might explain the correlation between the ratio of SRVs to Miras and metallicity. However, while a large fraction of our variables on the TP-AGB are regular variables also semiregular variables are found. Lebzelter & Hron (1999) have shown that for stars in the solar neighborhood stellar evolution goes from SRVs to Miras. The semiregular stars found at a similar luminosity as the Miras (see Fig. 13) are therefore probably not in the same evolutionary state or they have different masses. Comparison of the number densities with expected lifetime is therefore problematic.

A lower metallicity leads also to a shift of the AGB towards higher temperatures in the HR diagram. The visual light change of these cool variables is dominated by highly temperature sensitive molecules like TiO (e.g. Reid & Goldston 2002). If the stellar temperature is higher, these molecules will play a minor role. Lower metallicity will also make the TiO bands weaker. Therefore one would expect that the visual amplitudes will in general be smaller for lower metallicity. This would favour small amplitude variability in metal poor environments and would explain the smaller fraction of large amplitude objects in the LMC compared to the Bulge. It would also be consistent with the complete lack of Miras in metal poor globular clusters (Frogel & Whitelock 1998).

However, one has to be extremely careful concerning possible selection effects, in particular for small amplitude variables. A homogeneous survey of variable stars in different Galactic environments is therefore needed.

7 Discussion

7.1 Variability on the AGB

7.2 PL-relation

7.3 Number densities