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Astron. Astrophys. 348, L33-L36 (1999)

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2. The "reduced parallax" method

Let us consider a relation of the form:

[EQUATION]

If V is the intensity-mean visual magnitude and [FORMULA] its reddening corrected value, then one can write:

[EQUATION]

which defines the quantity RHS and where [FORMULA] is the parallax in milli-arcseconds. A weighted-mean of the quantity [FORMULA] is calculated, with the weight (weight = [FORMULA]) for the individual stars derived from:

[EQUATION]

with [FORMULA] the standard error in the parallax. This follows from the propagation-of-errors in Eq. (2). We have adopted the slope [FORMULA] (see the discussion in Fernley et al. 1998b), which is the one used by F98 and which is in agreement with the results from Baade-Wesselink methods (see, e.g., Clementini et al. 1995), Main Sequence fitting (Gratton et al. 1997) and theoretical models (see, e.g., Salaris & Weiss 1998, Cassisi et al. 1999).

The sample we consider is identical to that of F98, that is 144 stars out of a total of 180 stars in the HIPPARCOS catalogue. F98 discuss the reasons for discarding the 36 stars. Arguments include the fact that these stars do not have reddening determinations, are not RR Lyrae variables, or have poor quality HIPPARCOS solutions. Table 1 of F98 (retrievable from the CDS) lists all necessary data to perform the above analysis: periods, intensity-mean V and K magnitudes, colour-excesses [FORMULA], and metallicities [Fe/H]. The extinction is calculated from [FORMULA] (as done by F98).

An important requirement when applying this method is that the value of [FORMULA] is small compared to the errors on the parallax. If the dispersion [FORMULA] of the exponent in the factor RHS is large, the distribution of errors on the right-hand term in Eq. 2 is asymmetrical and a bias towards brighter magnitudes is introduced (Feast & Catchpole 1997, Pont 1999). The adopted value of [FORMULA] has been computed by considering four different contributions: errors on the intensity-mean V values of the RR Lyrae stars (as given in Table 1 of F98), on the extinction (as derived from the errors on E(B-V) given in Table 1 of F98), on [Fe/H] (again, from Table 1 of F98), and the intrinsic scatter due to evolutionary effects in the instability strip. This last term is the most important one, and we have adopted for it a 1[FORMULA] value by 0.12 mag (as in Fernley et al. 1998b), following the results of the exhaustive observational analysis by Sandage (1990). The final value is [FORMULA] = 0.15, a quantity small enough in comparison with the parallax errors so that no substantial bias is introduced on the right-hand term of Eq. 2, as we have verified by means of numerical simulations. Even a [FORMULA] of 0.20 mag. would lead to a bias by at most 0.02 mag.

Table 1 lists the values of the zero point with error we obtain with different sample selections for the [FORMULA]-[Fe/H] relation. Solution 1 corresponds to the case of the whole sample; the zero point of 0.67 [FORMULA] 0.24 mag is about 0.4 mag brighter than the value derived by F98, and consistent with the value listed in Koen & Laney (1998) using the same method with slightly different values for [FORMULA]. The sample with [Fe/H] [FORMULA] (Solution 2) corresponds to a sample constituted entirely (according to the discussion in F98) by Halo RR Lyrae stars, with a negligible contamination from the Disk population. In this case the zero point is equal to 0.77 [FORMULA] 0.26 mag; it is slightly fainter than Solution 1, but well in agreement within the statistical errors. We also re-derived the zero point for Solution 2 in the case of [FORMULA] = 0.0, and we found a change by only 0.04 mag. A systematic change in the metallicity scale (Solution 4) by 0.15 dex does not affect appreciably the zero point determination, while the result is more sensitive to a systematic variation of the adopted reddenings (Solution 5).


[TABLE]

Table 1. Values for the zero point of the [FORMULA]-[Fe/H] and [FORMULA]-[FORMULA] relations from the RP method


The RP method has also been used to derive the zero point of the [FORMULA]-[FORMULA] relation. This relation appears to be insensitive to the metallicity (Fernley et al. 1987, Carney et al. 1995) and is also very weakly affected by reddening uncertainties, since [FORMULA] (Rieke & Lebofsky 1985). Moreover, the intrinsic scatter around this relation is smaller than in the case of the [FORMULA]-[Fe/H] relation (Fernley et al. 1987). In the sample considered here there are 108 RR Lyrae stars with an observed intensity-mean K magnitude. The procedure is the same as described before, the only difference is that now, instead of Eq. 1, we use the expression [FORMULA] where [FORMULA] is the fundamental pulsation period. For the first-overtone RRc variables we have derived the fundamental periods using the relation [FORMULA] = +0.120 (Carney et al. 1995). We adopt a slope [FORMULA] following Carney et al. (1995); for the value of [FORMULA] we have considered the same contributions previously described (with the exception, of course, of the contribution due to the error on [Fe/H]). In this case the observational estimate of the intrinsic scatter due to the width of the instability strip comes from Carney et al. (1995), and the final value results to be [FORMULA]=0.10.

In Table 1 the values of the zero point for the [FORMULA]-[FORMULA] relation are listed. When considering the entire sample we obtain a zero point of [FORMULA] mag, [FORMULA] 0.4 mag brighter than the value from the Baade-Wesselink method (see, e.g., Carney et al. 1995). In the case of a pure Halo RR Lyrae sample ([Fe/H][FORMULA]) we obtain [FORMULA] mag, slightly dimmer but again in agreement with the value derived for the whole sample. The influence of [FORMULA] is even less than for the [FORMULA]-[Fe/H] relation.

As the sample of the RR Lyrae stars is not volume complete it may be subject to Malmquist type bias. If the space distribution of RR Lyrae is spherical it implies that the true zero points of the [FORMULA]-[Fe/H] and [FORMULA]-[FORMULA] relations may be fainter by up to 0.03 and 0.01 mag, respectively, for the adopted values of [FORMULA]. This applies when average absolute magnitudes of a volume and brightness limited sample are compared. Oudmaijer et al. (1999) showed empirically that when the averaging is done over [FORMULA] the effect of Malmquist bias is less.

In Fig. 1 we compare, for the 62 HIPPARCOS RR Lyrae stars with [Fe/H] [FORMULA] and both observed K and V magnitudes, the true distance moduli derived from the [FORMULA]-[Fe/H] and [FORMULA]-[FORMULA] relations, using zero points of 0.77 and -1.16 mag, respectively. Each data point has an error bar of 0.26 mag in [FORMULA] and 0.27 mag in the [FORMULA]direction. The comparison of the two photometric distances can in principle give us an independent indication for possible biases in the determination of the zero points of the two relations with the RP method. As it is evident from the figure, the distance moduli from both relations agree very well. A linear fit to the data is consistent with a slope of unity, and the dispersion around the 1:1 relation is equal to 0.098 mag. A dispersion of this order is what is expected from the dispersions in the observed [FORMULA]-[Fe/H] and [FORMULA] relations for the RR Lyrae sample.

[FIGURE] Fig. 1. A comparison of the true distance moduli to the 62 metal-poor RR Lyrae with [Fe/H] [FORMULA] from the [FORMULA][Fe/H] and [FORMULA] relations. Each point has an error bar of about 0.26 mag in both [FORMULA] and [FORMULA]direction. The solid line is the 1:1 relation. The dispersion is less than 0.10 mag.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
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