There have been sincere hopes that the inverse Tully-Fisher (TF) relation could overcome the distance dependent selection bias, the so-called Malmquist bias of the 2nd kind (Teerikorpi 1997), influencing the determination of the value of the Hubble constant. First, Schechter (1980) pointed out that the inverse relation of the form
where M is absolute magnitude (or linear diameter) and p is an observable parameter not affected by selection effects (e.g. the maximum rotation velocity in the TF relation), can be derived even from magnitude limited samples in an unbiased manner. Secondly, Teerikorpi (1984, or T84) showed explicitly that the inverse relation gives unbiased average distances for a sample of galaxies (if p is not affected by selection), a result later confirmed by Tully (1988) via simulations (see also Hendry & Simmons 1994). Thirdly, Ekholm & Teerikorpi (1997, or ET97) showed that with the inverse relation, one does not necessarily need a volume-limited calibrator sample, which on the contrary is quite critical for the use of the direct relation:
The direct TF relation has been succesfully applied for derivation of by the method of normalized distances that was first used by Bottinelli et al. (1986). Theureau et al. (1997b) have used an improved version of this method in an analysis of a sample of 5171 spiral galaxies that leads to the value of . In this method, which is rather similar to the Spaenhauer diagram method of Sandage (1994), one identifies the so-called unbiased plateau in the vs. normalized distance diagram for a magnitude or angular diameter limited sample, from which is determined without the need to make further model dependent correction due to selection. Normally, the unbiased plateau contains only 10-20 percent of the total sample. That one can use the whole sample without bias correction is a remarkable advantage of the inverse relation approach.
However, several problems balance the advantages of the inverse relation (Fouqué et al. 1990, Teerikorpi 1990, Sandage et al. 1995, ET97), and one has not yet been able to use it reliably for an independent determination of (see Theureau 1997, Ekholm et al. 1999). In the present paper, we study two important points which must be clarified before a successful use of the inverse TF relation:
As regards the first point, we emphasize that our approach follows the lines which we have adopted and found useful with the direct TF relation: to detect and overcome the bias without detailed modeling, by studying how the cutoffs would appear in kinematical applications (cf. the increase of as a function of true (or normalized) distance, when the direct TF relation is used). Another, more technical approach to the direct TF relation is that of Willick (1994), based on corrections to individual galaxies within an iterative scheme. In principle, a good knowledge of the selection function could also permit Willick's method to work for the inverse TF relation, in order to derive the slope as it would be without the selection.
Our approach necessarily cuts away a part of the sample. However, as shown by Ekholm et al. (1999) for the KLUN sample, the remaining subsample suitable for the determination of with the inverse TF method is still much larger than the unbiased subsample for the direct TF relation.
The observational motivation for studying question 2) comes from Theureau et al. (1998), who discuss the detection rate of 21cm line profiles at the Nançay radio telescope, in connection with the angular size limited KLUN galaxy sample. The detection rate is relatively high, 86 percent for galaxies with a previous redshift measurement and 61 percent for those with unknown z. However, more important than the average detection rate, is how detection depends onp, i.e. the selection function . Unfortunately this is difficult to derive from the raw data. Theureau (1997) notes that one expects loss of galaxies with either a narrow or broad 21 cm line width. Narrow profiles are difficult to detect among the noise, while broad profiles, being low, also tend to be missed.
Hence, as a first approximation we suppose that the distribution of is affected by sharp lower and upper cutoffs, and , although the selection more probably depends on various galactic parameters in a complicated manner. How such cutoffs influence, e.g. the value of as determined by the inverse relation method, has not been previously quantitatively discussed.
We study the above questions using normal distributions, which allow analytic expressions. This forms a natural sequel to our previous discussions of the TF relations, both direct and inverse, where the assumption of Gaussianity has been adopted (Teerikorpi 1984, 1987, 1990, 1993). It is worth noting that though we occasionally refer to the KLUN sample (see e.g. Theureau et al. 1997a), the theoretical ideas concerning the above mentioned points 1-3 have general validity. As an extensive sequel to the present work, Ekholm et al. (1999) have investigated whether the large value of , derived with a straightforward application of the inverse TF relation for the KLUN sample, could be due to the errors arising from cutoff and/or calibrator sample biases. They show that the calibrator sample bias is more important.
In this paper we also aim at a better general understanding of the inverse relation as a distance indicator, and present some arguments in a heuristic manner, since the method has been justly criticised as difficult to visualize, in comparison with the "more natural" direct relation approach.
This paper is structured as follows: In Sect. 2 we show qualitatively how cutoffs in influence the distance determination. In Sect. 3 the method of normalized distance is introduced for the inverse TF relation. In Sect. 4, the behaviour in the vs. diagram is calculated analytically. In Sect. 5, we derive an analytical expression for the average bias in when there is an upper or lower cutoff in , and give some examples in Sect. 6. The fundamental problems of slope and calibration are discussed in Sects. 7 and 8. Sect. 9 contains concluding remarks, emphasizing the essential points and consequences of the present study.
A note on the nomenclature: We use capital D's for the diameter of a galaxy, is the usual apparent 25th-mag isophotal diameter ( arcmin), D is the linear diameter ( kpc). In Sect. 4, to make the formulae more readable, we use for the 's. Small d's are used for inferred distances, is the normalized and the kinematical distance. It is assumed that kinematical distances have been calculated from a realistic velocity field model, giving reliable relative distances (in practice from a Virgo-centric infall model as e.g. in Theureau et al. 1997b). The true distance is denoted by r.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999