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2. Distance and velocity estimates

  The Tully-Fisher like relations are based on an observed linear correlation between the absolute magnitude M (or similar quantity such as tex2html_wrap_inline2685 with D the linear diameter) and the log line-width distance indicator p of galaxies (tex2html_wrap_inline2691 for spirals and tex2html_wrap_inline2693 for ellipticals). They allow to estimate distance and radial peculiar velocity of an individual galaxy from its measured apparent magnitude m (or similar quantity such as tex2html_wrap_inline2697 with d the apparent diameter), parameter p and redshift z. In this section, we recall in mind the basic statistical model describing these relations (see Triay et al. 1994 or Rauzy & Triay 1996 for details).

Regardless of the distance of the galaxy, selection effects in observation and measurement errors, the theoretical probability density (pd) in the M-p plane reads:
The Direct (i.e. Forward) Tully-Fisher (DTF) relation assumes that it exists a random variable tex2html_wrap_inline2707,
statistically independent of p such as the theoretical pd of Eq. (1 (click here)) rewrites:
where fp(p) is the distribution function of the variable p in the M-p plane. The random variable tex2html_wrap_inline2707 of zero mean and dispersion tex2html_wrap_inline2719 accounts for the intrinsic scatter about the DTF straight line tex2html_wrap_inline2721 of zero-point bD and slope aD. The distance modulus tex2html_wrap_inline2727 of an object reads:
where m is the apparent magnitude of the galaxy and r its distance in Mpc. Regardless of measurement errors, the observed probability density takes the following form:
where tex2html_wrap_inline2733 is a selection function accounting for selection effects in observation on m and p, tex2html_wrap_inline2739 is the spatial distribution function of sources (along the line-of-sight) and tex2html_wrap_inline2741 is the normalisation factor warranting tex2html_wrap_inline2743. Under the three following assumptions:

a statistical estimator tex2html_wrap_inline2761 generally adopted for the distance of a galaxy with measured m and p reads as follows (see Appendix A for details, or Lynden-Bell et al. 1988; Landy & Szalay 1992; Triay et al. 1994):
where the term tex2html_wrap_inline2767 accounts for a volume correction. This unbiasedgif distance estimator does not depend on the selection function tex2html_wrap_inline2733 in m and p and on the specific shape of the distribution function fp(p). Its accuracy tex2html_wrap_inline2801 is proportional to the distance estimate of the galaxy (see Appendix A):
The radial peculiar velocity v of a galaxy, expressed in km s-1 with respect to the velocity frame in which the redshift z is measured, reads:
where H0 is the Hubble constant and z is expressed in km s-1 units. Assuming that the above hypotheses hold, the estimator tex2html_wrap_inline2815 of the radial peculiar velocity of a galaxy with measured m, p and z reads thus as follows:
where H0 and bD have been merged into a single parameter tex2html_wrap_inline2827. The accuracy tex2html_wrap_inline2829 of this radial peculiar velocity estimator tex2html_wrap_inline2815 is:

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