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Astron. Astrophys. 343, 713-719 (1999)

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10. Concluding remarks

In the present study we have searched for answers to a few basic questions concerning the inverse TF relation as a distance indicator. First , how to see the signature of observational incompleteness in the TF parameter [FORMULA], analogously to the well-known increase in [FORMULA] at increasing (normalized) distance, in the case of the direct TF relation? Secondly , in which sense and how significant is the influence of the [FORMULA] incompleteness alone on the derived value of [FORMULA] (supposing one knows the correct inverse TF slope)? Thirdly , how to work with the situation when the inverse slope obeyed by the calibration sample differs from the relevant slope for the field sample (or for a cluster), and how important is in this regard the nature of the calibration sample?

We have generalized the "fine-tuning" scheme discussed by ET97, and have pointed out that there is a useful [FORMULA] vs. [FORMULA] representation for the inverse relation, revealing both the relevant inverse slope and cutoffs in [FORMULA]. Here [FORMULA] is the normalized distance applicable to the inverse TF relation, actually the (log) linear diameter within an unknown constant.

We have derived simple analytical formulae which give the influence of the [FORMULA] cutoffs on derived average distance (hence, on [FORMULA]). The treatment is based on Gaussian distributions, and, in the case of the average bias, on the assumption that the space density is proportional to [FORMULA]. The probable case that there is an upper observational cutoff in [FORMULA], would cause an overestimated value of [FORMULA]. However, this effect is not at all dramatic - even if the upper cutoff excludes half of the galaxies, [FORMULA] would be increased typically by only 14 percent. It may be concluded that a cutoff in [FORMULA] cannot alone explain derived large values of [FORMULA].

The problems of the relevant inverse TF slope and the nature of the calibrator sample lead to serious consequences on the use of the inverse TF relation as a distance indicator. It has been previously realized that the inverse slope [FORMULA] applicable to the field sample is not necessarily the same as the slope [FORMULA] obeyed by the calibrators, which may result in a biased value of [FORMULA] (Teerikorpi 1990). In the present paper, such situations were systematically studied.

If the two slopes are the same (in practice, when accuracies of calibrator distances are high and the photometric measurements for the field sample are not less accurate than those for the calibrators), then there is no bias, and it is also true that the nature of the calibrator sample (volume-limited, magnitude-limited, etc.) is not important (ET97).

If the sample inverse slope [FORMULA] is shallower than the calibrator slope (as normally expected), then matters come to depend critically on the nature of the calibrator sample. First, there is the possibility that the calibrator sample is a true, volume-limited representation of the Gaussian cosmic distribution function of [FORMULA] from which the field sample has been taken (the latter subject only to Malmquist bias). In this ideal case, it is permitted to use the field sample slope, with the zero-point obtained by a force-fit through the calibrators. This was the solution suggested by Teerikorpi (1990), and in this case it is not necessary to know the value of the calibrator slope.

However, generally the calibrator sample's [FORMULA] is shifted from the cosmic [FORMULA]. If one knows the shift [FORMULA] (or [FORMULA]) and the slopes [FORMULA] and [FORMULA], then one may calculate the resulting error in [FORMULA], when the field sample's [FORMULA] is used. The systematic error is given by Eq. (25).

Another, more convenient route, which bypasses the explicit need for [FORMULA], is to use the calibrators' slope [FORMULA] (and corresponding zero-point) and to calculate analytically the resulting systematic error in terms of the Malmquist bias of the 1st kind. This assumes that the average space number density around us is proportional to [FORMULA]. The systematic error is given by Eq. (27).

We repeat that in the general case both [FORMULA] and [FORMULA] (with corresponding zero-points from force-fits through calibrators) result in erroneous average distance estimates. Fortunately, we have now some quantitative control on this error.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999