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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 with D the linear diameter) and the log line-width distance indicator p of galaxies ( for spirals and 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 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 ,

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 of zero mean and dispersion accounts for the intrinsic scatter about the DTF straight line of zero-point bD and slope aD. The distance modulus 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 is a selection function accounting for selection effects in observation on m and p, is the spatial distribution function of sources (along the line-of-sight) and is the normalisation factor warranting . Under the three following assumptions:

• 0) No measurement errors on m and p are present. Particularly, corrections on galactic extinction and on inclination effects are supposed valid.
• 1) The function is gaussian.
• ) Galaxies are homogeneously distributed in space, which implies that, whatever the line-of-sight direction, the distance modulus distribution function reads with .
a statistical estimator 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 accounts for a volume correction. This unbiased distance estimator does not depend on the selection function in m and p and on the specific shape of the distribution function fp(p). Its accuracy 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 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 . The accuracy of this radial peculiar velocity estimator is:

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