7 Application examples

  The examples presented in this section are intended to show how the methods presented in the present paper can be incorporated in various studies. They do NOT intend to be exhaustive presentations of these studies. The emphasis is always on how the Hipparcos intermediate astrometric data can be used, or in some cases, should be used.

7.1 The absolute magnitude of RR Lyrae stars

  The Hipparcos variability annex gives 38 RRc and 146 RRab stars. Of the latter there is one which was newly defined as RRab, and which now can be rejected as such on the basis of its parallax: HIP 100859, which has a parallax of 26 mas, giving it an absolute magnitude of 2.9. Furthermore, RRc types with periods below 0.3 days were found to show often fainter absolute magnitude (see also VL98 on the classification of A and F type periodic variables). Leaving out those cases, 155 RRab and RRc stars were left. These stars provide 9345 individual abscissae measurements, from which, without taking into consideration the effects of reddening, a parallax based absolute magnitude of $M_{\rm Hp}=0.94\pm0.19$ was obtained. This is not significantly different from kinematically derived parallax estimates by Fernley et al. (1997), who note, however, that some of the stars in the selection used here are not confirmed independently as RR Lyrae stars. The unit weight standard deviation of the solution is 1.021, which, given the number of observations, is close to significantly different from the expected value of 1.00. The consistency between the magnitude based parallaxes and the parallaxes obtained from the individual solutions is shown in Fig. 5.

 

Figure 5:  The relation between the photometric parallaxes, calibrated using Hipparcos abscissa data, and the observed individual parallaxes with their estimated errors for RRab and RRc stars. The solid line represents the expected one-to-one relation

7.2 The PL relation of Mira stars

The Mira stars in the Magellanic Clouds show in the K band a period luminosity relation. The slope of this relation was used by van Leeuwen et al. (1997b) in combination with Hipparcos parallaxes to derive a zero point for this relation. A major problem with the Mira stars is the effect of the very large brightness and colour variations on the astrometric measurements. Most of the brighter Miras received so-called ``V'' type solutions, implementing a brightness related correction to the abscissae residuals.

We started off with the same selection of 16 stars as used by van Leeuwen et al. (1997b), and removed from this list two stars suspected of being fundamental mode pulsators, a symbiotic, a C-type and a double mode Mira, and tested all solutions for a standard 5-parameter solution. Three stars showed very bad fits according to the unit weight variances of their residuals, and were removed. There remained 8 stars, for which the reddening corrected ``K'' magnitudes were fitted using:  

$$M_K = -3.47 \log P + \beta_1.$$ (37)

Predicted parallaxes were calculated for assumed values of $\beta_1$.Corrections to this assumed value reflected in a scaling parameter for the parallaxes, which was solved for all 8 stars simultaneously using their 465 abscissae records. No corrections for brightness variations were introduced. The value thus obtained is $\beta_1=0.65\pm 0.16$, slightly smaller than what was given by van Leeuwen et al. (1997b), but this may be related to ignoring the above mentioned corrections. The need for these corrections was reflected in the unit weight standard error, which measured 1.35.

7.3 The Magellanic Clouds

A preliminary calculation of the proper motions of the Large and Small Magellanic Clouds was carried out for the paper by Kroupa & Bastian (1997). The covariance values have since been recalculated in more detail (see Sect. 2.2), and were in most cases found to be smaller than shown in Figs. 17.11 and 17.12 of Volume 3, Chapter 17.

The situation for the LMC and SMC is simple. The Hipparcos Catalogue contains 31 members of the LMC and 8 members of the SMC for which a single star solution of type ``5'' or ``7'' was obtained. Given the distances of the Magellanic Cloud stars, it was assumed that solutions of type ``7'' were spurious. The astrometric solutions for these stars were obtained from in total 2229 abscissa measurements in the LMC and 514 in the SMC. Each star had a parallax and proper motion derived in its individual solution. The first task was to remove the parallax and proper motions and to obtain abscissae relative to a zero proper motion and parallax (using Eq. 10). The abscissae were sorted on orbit number and for every orbit a set of de-correlated observation equations was obtained (corrected for the consortia correlations and the correlations described in Sect. 2.2). The observation equations described corrections to individual star positions and to a common LMC or SMC proper motion, but did not allow for a parallax solution. These de-correlated observation equations were accumulated in a single least squares solution, (using Eq. (31) for the proper motions) providing the results shown in Table 3. From the unit weight standard errors it is clear that the condition of a single common proper motion over the LMC may not be entirely satisfied, producing a $\chi^2$significantly above the expected value. The situation is better for the SMC. The differences between the values presented here and those presented earlier by Kroupa & Bastian have no significant effect on the discussions presented in that paper.

 

Table 3:   Preliminary results for common proper motion solutions for stars in the LMC and SMC. The correlation coefficients $\rho$ between the proper motion in declination and right ascension are also given

7.4 The Pleiades and Praesepe

The determination of the Pleiades parallax and proper motion will be dealt with in detail by van Leeuwen and Hansen (in preparation). Here we present the preliminary results of that study and a similar application to stars in the Praesepe cluster.

The Hipparcos catalogue contains 60 members of the Pleiades cluster. Of these, 54 were selected as single stars with solution type ``5'', providing a total of 2182 abscissae (see van Leeuwen & Hansen-Ruiz 1997). All abscissae were incorporated, including those rejected in the individual star solutions. In an iteration over the combined solution a total of 4 abscissa residuals were rejected. All abscissae were corrected to a reference parallax and proper motion, and sorted on orbit number. For each orbit a set of de-correlated observation equations was created, which were combined in one least-squares solution. The observation equations described positional corrections for all 54 stars and one parallax and proper motion for the cluster centre, using Eq. (31). Thus, the degrees of freedom were reduced from $54\times 5 = 270$ to $3 + 54\times 2 = 111$. The results are summarized in Table 4. From the unit weight standard error and its uncertainty it appears that there remained unmodelled effects in the parallax and proper motion. This is most likely the internal proper motion dispersion in the cluster, which has a dispersion in the projected centre of the cluster of around 1 mas s^-1. The correlation coefficient of $\rho^{\mu_\delta}_{\mu_{\alpha\cos\delta}}= 0.244$ reflects the proximity of the Pleiades to the ecliptic and limited range of scan-directions resulting from this situation.

Table 4 also provides similar results for the Praesepe cluster. Here there is no significant contribution from an internal proper motion dispersion.

 

Table 4:   Preliminary results for common proper motion and parallax solution for stars in the Pleiades and Praesepe clusters. The various correlation coefficients $\rho$ are also given