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Astron. Astrophys. 340, L35-L38 (1998)

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1. Introduction

A major contribution of the Hipparcos mission was to provide an unprecedented large number of independently determined accurate trigonometric parallaxes (ESA, 1997). For about 20 000 stars, the parallax is determined to better than 10% and for 30 000 others, to better than 20%. Various comparisons with otherwise estimated parallaxes of distant stars or clusters have shown no hint of regional or global systematic errors exceeding 0.1 mas. The possible effect of correlations in small fields, not exceeding a few percents within 2-3 degrees field as demonstrated by Lindegren (1989), makes no significant exception. However for small regions of the sky (like the case of clusters) may need some special consideration.

This large amount of data opens the use of star distances in large statistical investigations in stellar kinematics or astrophysical applications without calling for additional assumptions on the probability distribution function of the parameters to be determined. Indeed, in the absence of a sufficient number of independent determinations, and particularly in using photometric or spectroscopic parallaxes such a Bayesian approach has been widely used in the past. Let me mention the pioneering work by Lutz and Kelker (1973), who assumed that stars are uniformly distributed in space so that the number of stars within an interval of distances between r and [FORMULA] is


This leads to a law of distribution of the possible true values of the parallax [FORMULA] given the observed value [FORMULA]


where [FORMULA] is the standard deviation of the trigonometric parallax [FORMULA]. It is not a probability density function (pdf), because the integral does not converge for infinity.

Other constraints on the pdf have been used. To take an example, Hanson (1979) uses an a priori assumption that the number of stars per unit magnitude interval around M inside unit volume is


so that the distribution of absolute magnitudes for stars of a given apparent magnitude m is


This led to a distribution law where the exponent of [FORMULA] differs from 4 as in Lutz and Kelker, and depends on the values of A and B retained.

This remark is just to empharise the fact that Bayesian approaches provide distance evaluations that depend upon the a priori assumptions that one makes on the distribution of some parameters derived from the distance.

In what follows, it is intended to provide a working pdf for the distance that can be used, with the restriction of the end of the first paragraph but without any other a priori assumption of a Bayesian type. In addition, it can be used simply to compute uncertainties of derived quantities, in particular in dynamical investigations in the Galaxy, in which proper motions and radial velocities with their Gaussian pdf are associated in the evaluation of the space velocities. This is also the case for absolute magnitudes, provided that the photometric uncertainty is also Gaussian.

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998