5 Period--Luminosity--Colour relationship

5.1 PLC relations

The fitted distributions of period, magnitude and colour correspond to the following PLC relations (where the error bars correspond to $\pm 1\sigma$ deviations, as estimated using Monte Carlo simulations):

• Group 1:
  
  $$\begin{eqnarray*}M_K & = -1.07_{[\pm 0.50]} \log P & - 0.37_{[\pm 0.13]} (V-K)_0 \\ & & - 1.49_{[\pm 0.72]} \end{eqnarray*}$$
  $\sigma_M\, =\, 0.63_{[\pm 0.13]}$
  
• Group 2:
  $$\begin{eqnarray*}M_K & = -0.37_{[\pm 0.94]} \log P & -0.42_{[\pm 0.24]} (V-K)_0\\ & & -3.23_{[\pm 1.35]} \end{eqnarray*}$$
  $\sigma_M\, =\, 0.32_{[\pm 0.17]}$
  
• Group 3:
  $$\begin{eqnarray*}M_K & = -1.69_{[\pm 0.54]} \log P & -0.16_{[\pm 0.14]} (V-K)_0\\ & & -2.53_{[\pm 0.78]} \end{eqnarray*}$$
  $\sigma_M\, =\, 0.50_{[\pm 0.08]}$
  
• Group 4:
  $$\begin{eqnarray*}M_K & = -0.81_{[\pm 1.72]} \log P & -0.09_{[\pm 0.45]} (V-K)_0\\ & & -4.76_{[\pm 2.48]} \end{eqnarray*}$$
  $\sigma_M\, =\, 0.48_{[\pm 0.21]}$.
  

The coefficients of the relation found for Group 4 are obviously very uncertain. This is not surprising, in view of the small number of stars (14), their small dispersion, and the number of parameters to estimate. For Groups 1, 2 and 4, the error bars on the zero point are relatively large; this is due to the fact that the means of the three variables are far from zero, and thus any slope uncertainty rebounds magnified on the zero point.

5.2 Projection onto the {P, M_K} plane

The tridimensional model distributions may be projected onto the period-luminosity plane. The elliptic-looking lines shown in Fig. 5 are the projections of the isoprobability contours that, in the mean PLC plane, correspond to a $2\sigma$ deviation. The offset between the data and the model populations is due to the sampling bias, which is suppressed by the LM algorithm (see Appendix A for details). Then, Period-luminosity relations, liable to be compared to the ones observed in the Magellanic Clouds, are derived by means of a linear least-squares fit to the contours. Monte-Carlo simulations, as well as analytic computations, have shown that this is equivalent to a fit onto the projected population itself. Finally, the error bars of the coefficients are estimated, for each group, by applying the standard least-squares procedure to a simulated unbiased sample (thus they may be directly compared to the error bars usually given for the LMC stars).

The results are the following:

• Group 1:
  
  $$M_K = -5.04_{[\pm 0.72]} (\log P - 2.48) - 7.26_{[\pm 0.06]}$$
• Group 2:
  
  $$M_K = -1.13_{[\pm 0.16]} (\log P - 2.00) - 6.41_{[\pm 0.05]}$$
• Group 3:
  
  $$M_K = -2.11_{[\pm 0.13]} (\log P - 1.75) - 6.51_{[\pm 0.05]}$$
• Group 4:
  
  $$M_K = -1.37_{[\pm 2.32]} (\log P - 2.24) - 7.05_{[\pm 0.14]}.$$

It may be noticed that Groups 2 and 3 have similar mean magnitudes. The same holds for Groups 1 and 4.

Figure 5: PL calibrated distributions: individual data and projected model distributions ($2\sigma$ isoprobability contours in the mean PLC plane). The Mira strip of the LMC is also shown (thick lines)

From a sample of 29 O-rich Miras of the Large Magellanic Cloud with period $\le 420$ days, Feast et al. ([1989]) derived the following relation (were the error bars correspond to $1\sigma$ deviations):

$$M_K = - 3.47_{[\pm 0.19]} \log P + 0.93_{[\pm 0.45]}$$

$$\sigma_M=0.13$$

assuming a distance modulus of 18.55. Based on a sample of more than a hundred Oxygen-rich Miras of the LMC, the solution of Hughes & Wood ([1990]) is, under the same assumptions:

$$M_K = - 3.86_{[\pm 0.18]} (\log P - 2.4) -7.40_{[\pm 0.02]}$$

$$\sigma_M=0.26.$$

From a sample of 79 Miras, unfortunately including a significant number of Carbon stars, Hughes ([1993]) derived:

$$M_K = - 3.75_{[\pm 0.14]} (\log P - 2.4) -7.45 _{[\pm 0.02]}$$

$$\sigma_M=0.13.$$

Obviously, the slopes are significantly different from that of any Galactic PL relation found above. Such a difference was also observed between the LMC and Globular Clusters stars by Menzies & Whitelock ([1985]). It cannot be due to the well-known steepening of the PL relation at periods larger than 420 days, i.e. relatively high mass (Feast et al. [1989]), since only a very few sample stars may be concerned. The first possible explanation is that the so-called Miras of the LMC include, in fact, a significant number of SRb semiregulars, especially at "short'' periods. Indeed, for the outer galaxies, the observers use to call "Miras'' the LPVs with an amplitude larger than a given threshold (e.g. 0.9 mag in I), corresponding to the maximum amplitude of SRa stars according to the GCVS. This criterium is obviously not sufficient and the slope of the so-called LMC "Mira'' strip should then be intermediary between our Groups 1 and 2. A second explanation of the slope discrepancy is that the shorter-period "Miras'' in the LMC include a population more-or-less equivalent to our Group 4, i.e. metal-deficient with a mean mass similar to or lower than the one of the main population. This, too, would lead to a shallower global PL relation. The existence of such a population had been suggested by Wood et al. ([1985]) and Hughes et al. ([1991]). Of course, these two explanations do not exclude each other.

It is also worth noting that the PL slopes of Groups 2 and 3 are much smaller than the one of Group 1 (Miras), and similar to the one of the evolutionary tracks (-1.67), derived by Bedding & Zijlstra ([1998]) from the works of Whitelock ([1986]) and Vassiliadis & Wood ([1993]). Moreover, in Group 2 as well as in Group 3, the proportion of SRb's (as defined by the GCVS) decreases towards longer periods: at P> 200 days, they represent less than 25% of the stars of these groups, while Miras (GCVS) amount to 45% for Group 2 and 65% for Group 3. All of this indicates that, in each population, the sequence of SRb Semiregulars corresponds to an evolutionary sequence towards the Mira instability strip.

5.3 Projection onto the {P, V-K} plane

In the same way as in the preceding subsection, a linear fit to the projected model distributions (see Fig. 6) yields the following period-colour relations:

• Group 1:
  
  $$(V-K)_0 = 10.80_{[\pm 0.58]} (\log P-2.48) + 8.51_{[\pm 0.05]}$$
• Group 2:
  
  $$(V-K)_0 = 1.80_{[\pm 0.32]} (\log P-2.00) + 5.75_{[\pm 0.10]}$$
• Group 3:
  
  $$(V-K)_0 = 2.54_{[\pm 0.28]} (\log P-1.75) + 6.20_{[\pm 0.10]}$$
• Group 4:
  
  $$(V-K)_0 = 6.50_{[\pm 2.47]} (\log P-2.24) + 5.52_{[\pm 0.15]} .$$

The difference of slope between Groups 2 and 3 is (qualitatively) consistent with the differences of mass and metallicity expected from the kinematics. Indeed, the larger mass of Group 3 stars yields significantly higher temperatures and thus a larger { $T_{\rm eff},(V-K)$} slope, while the moderate metallicity difference has only a small influence (Bessell et al. [1989,1998]). This is due to the behaviour of the TiO lines in this temperature range.

On the other hand, the much larger slope of Group 1 may be explained by a difference of pulsation mode, consistently with the larger mean period. Indeed, the period of a lower-order mode must be more sensitive to the temperature (see, e.g., Barthès [1998]).

Figure 6: PC calibrated distributions: individual data and projected model distributions

5.4 Projection onto the {M_K, V-K} plane

The calibration results in the Luminosity-Colour plane are shown in Fig. 7. As explained in the Appendix A, the offset and the difference of width between the data distributions and the projected $2\sigma$ contours are effects of the projection and of the sampling bias. A linear fit to the contours yields the following luminosity-colour relations:

• Group 1:
  
  $$M_K = -0.42_{[\pm 0.05]} ((V-K)_0-8.47) -7.24_{[\pm 0.05]}$$
• Group 2:
  
  $$M_K = -0.45_{[\pm 0.03]} ((V-K)_0-5.75) -6.41_{[\pm 0.03]}$$
• Group 3:
  
  $$M_K = -0.34_{[\pm 0.04]} ((V-K)_0-6.19) -6.51_{[\pm 0.06]}$$
• Group 4:
  
  $$M_K = -0.10_{[\pm 0.17]} ((V-K)_0-5.56) -7.06_{[\pm 0.14]} .$$

As in the preceding subsection, the difference of slope between Groups 2 and 3 is easily explained by the temperature-dependence of the { $T_{\rm eff},(V-K)$} slope, which is little sensitive to moderate metallicity variations.

Figure 7: LC calibrated distributions: individual data and projected model distributions

5.5 {P, J-K} distribution

Once the distances have been calibrated, it is possible to check the distribution of the sample stars with respect to the de-reddened J-K index. The results are shown in Fig. 8, together with the raw data. The J-K Period-Colour distribution appears similar to the V-K one. The scattering, also existing in the raw data, makes the PC relation more difficult to see. It is due to the smaller number of stars in the J data set, and probably also to the peculiar sensitivity of this colour index to the surface gravity and extension of the envelope (Bessell et al. [1989,1998]).

A linear least-squares fit to the de-reddened data (excluding a few obviously misclassified stars, namely two having (J-K)₀ > 2 and two having (J-K)₀ < 0.6) yields:

• Group 1 (86 stars):
  
  $$(J-K)_0 = 1.01_{[\pm 0.22]} \log P - 1.27_{[\pm 0.56]}$$
  
  
  $$\sigma_{J-K}=0.02$$
• Group 2 (63 stars):
  
  $$(J-K)_0 = 0.16_{[\pm 0.07]} \log P + 0.83_{[\pm 0.14]}$$
  
  
  $$\sigma_{J-K}=0.02$$
• Group 3 (91 stars):
  
  $$(J-K)_0 = 0.17_{[\pm 0.04]} \log P + 0.85_{[\pm 0.08]}$$
  
  
  $$\sigma_{J-K}=0.02$$
• Group 4 (12 stars):
  
  $$(J-K)_0 = 1.02_{[\pm 0.67]} \log P - 1.25_{[\pm 1.51]}$$
  
  
  $$\sigma_{J-K}=0.01.$$

These fit relations are probably slightly biased, and thus should be shifted by a certain amount so as to represent the whole populations.

Contrary to what was found with V-K, the relations of Groups 2 and 3 cannot be reliably distinguished here. This may be due to the crossing-over of the iso-metallicity curves in the { $T_{\rm eff}, J-K$} diagram: the effects of the differences of temperature and metallicity between the two groups tend to compensate each-other (Bessell et al. [1989,1998]).

Figure 8: J-K versus Period distribution of sample stars: raw ( top) and dereddened ( bottom) indices. Two stars with J-K > 2 are not shown

Based on 29 Oxygen Miras, the relation found by Feast et al. ([1989]) for the LMC is:

$$(J-K)_0 = 0.56_{[\pm 0.12]} \log P - 0.12_{[\pm 0.29]}$$

$$\sigma_{J-K}=0.08.$$

From a sample of 21 stars, Hughes ([1993]) derived:

$$(J-K)_0 = 0.37_{[\pm 0.05]} (\log P - 2.4) + 1.215_{[\pm 0.014]}$$

$$\sigma_{J-K}=0.06.$$

As for the PL relation, we find a significant discrepancy between the Miras in the LMC and the ones in the solar neighbourhood. Since, in J-K as well as in V-K, Group 4 is approximately aligned with Group 1, and thus metal-deficient Miras of the LMC should not significantly influence its PC relation, it seems that, as suggested in Sect. 5.2, we are actually encountering a problem of misclassification of the LPVs in the outer galaxies. This, of course, does not preclude the existence of a metal-deficient population which would further influence the PL relation.

5.6 {M_K, J-K} distribution

The LC relations yielded by a linear least-squares fit to the calibrated and de-reddened data are:

• Group 1:
  
  $$M_K = -1.22_{[\pm 0.39]} (J-K)_0 -6.32_{[\pm 0.49]}$$
  
  
  $$\sigma_M=0.33$$
• Group 2:
  
  $$M_K = -1.37_{[\pm 0.37]} (J-K)_0 -5.24_{[\pm 0.44]}$$
  
  
  $$\sigma_M=0.22$$
• Group 3:
  
  $$M_K = -2.39_{[\pm 0.59]} (J-K)_0 -4.41_{[\pm 0.70]}$$
  
  
  $$\sigma_M=0.69$$
• Group 4:
  
  $$M_K = -0.18_{[\pm 0.65]} (J-K)_0 -7.25_{[\pm 0.68]}$$
  
  
  $$\sigma_M=0.06.$$

We remind that these fit relations are subject to sampling bias, and thus should be significantly shifted downwards, so as to represent the whole population (see Appendix A). In view of the error bars, the slope of Group 3 may be the same as the one of Groups 1 and 2, which is what is expected from AGB evolutionary models. This supports our interpretation of the slope differences found in V-K (Sect. 5.4). The slope of Group 4 is, once again, not reliable.

Figure 9: Magnitude versus (J-K)0 distribution of the sample stars, deduced from the luminosity calibration