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

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2. Distance estimation and its pdf

The uncertainties of parallaxes determined by Hipparcos obey a Gaussian law of the form

[EQUATION]

Now, if one transform (2) by changing the independent variable from [FORMULA] to [FORMULA], the law of distribution of r is proportional to

[EQUATION]

where [FORMULA] is, as in (2), the standard deviation of the observed parallaxes and [FORMULA]. For lack of denomination, we shall refer to it as classical . The problem is that although [FORMULA] is a pdf, the calculation of the second moments does not converge, a problem dealt in particular by Smith and Eichhorn (1996). This is even more true for the transform of Lutz and Kelker (LK) representation given by formula (1) and which can be inverted directly since no a priori knowledge is assumed, and becomes with the same notations:

[EQUATION]

To analyse these expressions, we introduce a new variable

[EQUATION]

which represents the relative deviation from the true (hence unknown) distance defined as [FORMULA], and a parameter

[EQUATION]

in which the uncertainty of the observed parallax is introduced. We obtain respectively for [FORMULA] and [FORMULA]

[EQUATION]

[EQUATION]

where one may chose [FORMULA] and [FORMULA] in such a way that the maximums of the functions are scaled to 1.

The maximums are obtained by differentiating (5) or (6) with respect to x and writing that the derivative is equal to zero. One obtains, for the classical [FORMULA], the relation

[EQUATION]

The maximum corresponds to the larger solution of this equation:

[EQUATION]

In the case of the LK distribution, one has

[EQUATION]

Replacing x in (5) and (6) respectively by (7) and (8), and imposing the value 1, one gets [FORMULA] and [FORMULA]. The scaled (in this way) functions, [FORMULA] and [FORMULA] are represented in Fig. 1. One sees that while in the LK solution, the maximum corresponds to a distance larger than [FORMULA], the [FORMULA] distribution gives a maximum smaller than [FORMULA] (see Fig. 1). Within the assumptions behind each of these distributions, the maximum corresponds to the most probable value. It is tempting to use it as an estimate of the distance. This has been frequently done for the LK distribution and I believe this is dangerous. If [FORMULA] is small, both converge to a unique value, equal to zero (distance [FORMULA]). But for large values of [FORMULA], there are increasing differences which depend upon the underlying assumptions.

[FIGURE] Fig. 1. Comparison of the scaled functions [FORMULA] and [FORMULA] for [FORMULA]. Note the position of the maximums in comparison with [FORMULA], corresponding to [FORMULA]

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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