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
*f*_{p}(*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 *b*^{D} and slope *a*^{D}.
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 .

where the term accounts for a volume correction. This unbiased distance estimator does not depend on the selection function in

The radial peculiar velocity

where

where

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