2 Data

2.1 Solar neighbourhood

Paper I has provided us with the mean absolute K₀ magnitudes and (V-K)₀indices of about 350 M-type LPVs of the solar neighbourhood. For about 250 stars, (J-K)₀ is available. The K magnitude is known to closely mimic the behaviour of bolometric magnitude. Moreover, as mentioned in Paper I, simple simulations of light curves have shown that the mean K magnitude and the magnitude corresponding to the mean K flux agree with one another to within a few percent (0.06 mag for a full amplitude of 1 mag). The adopted magnitude is thus a good representative of the actual mean luminosity of each star. We must recall however, that, because of the paucity of the data, which were taken at arbitrary phases, the observational error on K is $\sigma=0.25{-}0.50$ mag.

The mean V-K index was defined in Paper I as the difference between the midpoint V magnitude and the mean K magnitude. Simulations have shown that the midpoint value of V is systematically larger than the mean by, usually, only a few hundredths of mag, or at the very most a few tenths. On the other hand, the magnitudes at minimum brightness tend to be underestimated by visual observers. However, this systematic error (a few tenths of mag; the half for the mean) depends altogether on the distance, the mean absolute magnitude and the amplitude (which is correlated to the brightness, but with an important scatter). Thus, globally over the sample of stars and the subsamples defined in Paper I, the systematic error on <V> should not exceed -0.1 mag and thus may be neglected. Besides, the random error is about 0.3 mag. Summarizing, the V-K data used in Paper I are subject to a random error of about 0.5 mag.

The luminosity calibrations performed in that paper led us to identify four groups, differing by their kinematics and/or luminosity:

Group 1: mainly composed of Mira stars (84%), with a kinematics corresponding to old disk stars;

Group 2: mainly composed of SRb stars (56%), also with old disk kinematics;

Group 3: mainly composed by SRb stars (82%), with a younger kinematics;

Group 4: a small group (14 stars, including 13 Miras) with a kinematics corresponding to extended-disk or halo stars.

The calibration procedure also provided us with the de-biased distributions of the populations corresponding to these groups. As explained in Paper I, the overall error bars of their parameters were computed by means of many Monte-Carlo simulations, taking into account the above-mentioned photometric errors, the astrometric ones and those of the radial velocities. Concerning the barycenters of these distributions (i.e. the population means), the uncertainties are: about 0.1 mag (Groups 1, 2 and 3) or 0.25 (Group 4) for (V-K)₀; about 0.05 (Groups 1 and 2), 0.08 (Group 3) or 0.1 (Group 4) for M_K.

Last, it must also be mentioned that, for a minor but significant proportion of semi-regular stars, the periods may by erroneous by as much as a factor of 2, either because there are few data and/or we could not check them (see Paper I), or because the light curves and/or Fourier spectra are ambiguous (two or three large-amplitude pseudo-periodicities liable to correspond to a mode - see Mattei et al. 1997). This means that the period scattering in the groups including a large proportion of semi-regulars (i.e. Groups 2 and 3) is probably overestimated. Here again, the population mean should remain nearly unaffected.

2.2 Globular cluster stars

For oxygen-rich LPV stars belonging to globular clusters (GC) of our Galaxy, periods, mean absolute bolometric magnitudes (derived from blackbody fits to mean, de-reddened JHKL data) and mean (J-K)₀ index values have been found in Whitelock (1986) and Feast (1996). The $<m_{\rm bol}>$ precision is about 0.1-0.3 mag according to the amplitude and data sampling. That for J-K is 0.03-0.15 as explained above, but most often $\simeq 0.05$.

2.3 LMC stars

As hundreds of O-rich Mira-like stars (i.e. LPVs with $\Delta I \le 0.9$ mag) have been observed in the Large Magellanic Cloud, we will handle them in a synthetic way, by considering their mean Period-Luminosity ( $M_{\rm bol}$) and Period-Colour (J-K) relations, the $1\sigma$ scattering about it, and the barycenter of this population. The latter is defined by the mean period, already computed by Reid et al. (1995) and the corresponding mean bolometric magnitude and colour. The PL and PC relations were taken from Feast et al. (1989) and Hughes & Wood (1990) and hold as long as days. Due to the large number of stars, the error bars of the barycenter may be neglected.

We have also derived from Wood & Sebo (1996) mean K₀ magnitudes and (J-K)₀ colours of O-rich LPVs found near a few clusters of the LMC (2 data points per star, taken at arbitrary phases). The magnitudes were then converted into bolometric ones by applying the empirical bolometric correction $BC_K\,=\,f((J-K)_0)$ given by Bessell & Wood (1984). The obtained precision, including intrinsic variability effects, should be roughly 0.3 mag for <K> and $\le 0.10$ for <J-K>. Two thirds of these stars are obviously pulsating on a higher-order mode than the Mira-like population, since they form a second, parallel strip in the PL plane. That is why we included this sample among the LMC data.

Last, K magnitudes and periods of hundreds of pseudoperiodic red variables belonging to the MACHO sample have been found in Wood (1999). Being single-phase observations, these data represent the mean magnitude within about 0.3-0.5 mag. On the other hand, the periods of these stars are secure, since they were derived from MACHO light curves spanning years.