The Hipparcos Space Astrometry Mission: FAQ's supplement: Biases

Biases

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From Hans Schrijver (98-02-02):

The answer to the question whether trigonometric parallaxes are biased is in some sense a matter of perspective. The astrometrist asks the question: given a real parallax, what is the expectation of the measured parallaxes? The astrophysicist, on the other hand, poses the question: given a measured parallax, what is the expectation of the real parallax? Perhaps not surprisingly, this will lead to some different answers.

I think one can distinguish three areas where bias can play a role:

(1) The observational area. Given a star with real parallax pi_0, what will the distribution of measured parallaxes be for a given sigma_pi? Most people agree that the Hipparcos measurements show a nice, almost Gaussian distribution around pi_0. This in strong contrast with ground based measurements where the problem of determination of the zero point (correction from relative to absolute parallaxes) easily leads to a bias.

(2) Bias related to transformation to other physical quantities (distances, absolute luminosities). This is caused by the transformation of the distribution of observed parallaxes to the distribution of the desired quantity by applying the Jacobian of the transformation. This is illustrated in Figures 1 and 2 of Luri & Arenou ("Utilisation of Hipparcos data for distance determinations: error, bias and estimation", Hipparcos-Venice '97 Proceedings, ESA-SP 402, 449; available at this site). It is my strong impression that many people think that this effect is the Lutz-Kelker one (for example, van Leeuwen, Space Science Reviews, 81, 361); it is not, however: see (3).

(3) Biases related to the question: given a measured parallax pi, what is the distribution of real parallaxes pi_0? The answer to this question (i.e., the expectation of pi_0) depends critically on the distribution (in distance, luminosity, etc.) of the sample of stars that may contribute to the measurement. In the Lutz & Kelker paper, building on the ideas of `Statistical Astronomy' (Trumpler & Weaver 1953), the authors show that for a uniform distribution of stars, without any constraint on luminosities, the measured trigonometric parallaxes are strongly biased towards the observer (i.e., too large), depending on the relative measurement error. Although their results are certainly much too pessimistic for the general case, we must assume that parallaxes will, in principle, always have a bias in this sense, the magnitude of which can be very different from case to case.

I think the general message to the users must be (and I think we all agree about that): use the parallaxes, especially those with lower precision with great care. The method proposed by Luri & Arenou (in their Section 5) represents a valid approach to this problem (taking into account all available information about the star sample). Although from their introduction one gets the impression that only the bias of category (2) is treated, in the end their method seems quite general.

It is mainly the bias described under (3) that is addressed in the paper by Oudmaijer et al. It is the effect that is to be taken into account by the astrophysicists making use of the data. And in this area, the trigonometric parallax is as much a `derived physical quantity' as all the others.

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