2 Calibration method

This work is based on the LM method, which has been designed to fully exploit the HIPPARCOS data to obtain luminosity calibrations. The mathematical foundation of this method was presented in Luri ([1995]) and Luri et al. ([1996a]). Its main characteristics are:

• It is based on a maximum-likelihood algorithm;
• It is able to use all the available information on the stars: apparent magnitude, galactic coordinates, trigonometric parallax, proper motions, radial velocity and other relevant parameters (photometry, metallicity, period, etc.), and takes into account, as an additional constraint, the existence of mean relations between, e.g., period, luminosity and colour, whose analytical form is given a priori;
• It allows a detailed modelling of the kinematics, the spatial distribution, and also the distribution of luminosity, period and colour of the sample.

  In the implementation presented in this paper, the stars are assumed to be exponentially distributed about the galactic plane and their velocities to follow a Schwarzschild ellipsoid. The period and colour are assumed to follow a bivariate normal distribution, including a correlation between the two variables. This generates elliptic iso-probability contours in the period-colour plane. For each given combination of period and colour, the individual absolute magnitudes of the stars are assumed to be normal-distributed about the mean value given by a period-luminosity-colour relation, e.g. $M_K = A \, \log P \, + \, B \, (V-K)_0 \, + \, C$. The resulting 3D distribution looks like a flattened ellipsoid whose main symmetry plane is the PLC relation. All parameters of the model are determined by maximum-likelihood estimation;

• The method takes into account the observational selection criteria that were used when making the sample -- this is very important for obtaining unbiased results (Brown et al. [1997]);
• It takes into account the effects of the observational errors; the results are not biased by them and even low-accuracy data (which would otherwise be useless) can be included;
• The galactic rotation is taken into account by introducing in the model an Oort-Lindblad first-order differential rotation with A₀=14.4, $B_0=-12.8\,\, {\rm km}~{\rm s}^{-1}~{\rm kpc}^{-1}$ and $R_{\rm sun} = 8.5$ kpc;
• The interstellar absorption is taken into account, using the 3D model of Arenou et al. ([1992]).

A further important feature of the LM method is its capability to separate and characterize, in the sample, groups of stars with different properties (e.g. luminosity, kinematics, spatial distribution...). The number of groups has to be fixed beforehand (see Sect. 4 for criteria). Then, separate results are obtained for each group, and this provides a much more meaningful information than a global result for the mixture of all of them would.
For the population corresponding to each identified group, the LM method provides unbiased estimates of the model parameters, i.e. for the version used in this study:

• The parameters of the absolute magnitude distribution, i.e. the coefficients of the mean period-luminosity-colour relation, and the dispersion around it ( $\sigma_{M}$);
• The velocity distribution: mean velocities (U₀,V₀,W₀) and dispersions $(\sigma_{U},\sigma_{V},\sigma_{W})$;
• The spatial distribution: the scale heigth Z₀;
• The period-colour index distribution: mean of the logarithm of the period $\overline{\log P}$, mean de-reddened colour index, e.g. $\overline{(V-K)_{0}}$, the associated dispersions $\sigma_{\log P}$ and, e.g., $\sigma_{(V-K)_{0}}$, and the correlation between log period and colour;
• The percentage of the sample in each group: %.

In addition, the parameters of the selection function generating the sample are obtained for each group.

The LM method also yields improved individual distance estimates (and thus improved absolute magnitude estimates) which take into account all the available information on each star: the trigonometric parallax $\pi _{\rm t}$ and other measurements (magnitude, $\alpha$, $\delta$, $\mu _{\alpha }$, $\mu _{\delta }$, $v_{\rm r}$, P, colour). This estimation is free of any bias due to observational selection or observational errors, because both are taken into account by the method.