1 Introduction

Most of the methods used for extragalactic distance determination are based on a linear relationship between the absolute magnitude M and a given observable parameter P.

$$M = a \log P + b.$$ (1)

The distance modulus is thus derived from the classical relation

$$\mu = m^{\rm c} - M$$ (2)

where $m^{\rm c}$ is the apparent magnitude corrected for perturbing effects (inclination, redshift, galactic extinction...) Similarly, the parameter P must also be corrected for such effects. Without considering the galactic extinction, these corrections depend essentially on axis ratio and morphological type. For spiral galaxies, one of the best parameters P is the 21-cm line width (Tully & Fisher 1977).

The principle of the method of sosie galaxies (Paturel 1981; Sandage 1996) consists in selecting galaxies having similar observational properties as calibrating galaxies. For instance, one can select galaxies having the same morphological type, the same axis ratio and the same 21-cm line width, as a given calibrating galaxy (e.g., M 31). From Rel. 1, these galaxies have the same absolute magnitude as the considered calibrating galaxy. This method has revealed to be a powerful tool in obtaining accurate galaxy distances because it does not assume anything about the expression of, e.g. the Tully-Fisher relationship. For instance, the value of a or b does not intervene in the final result. Any morphological type dependences are removed etc.

We started a new observational program to search for new sosie candidates. This means that we are observing galaxies with at least the same morphological type and the same axis ratio as calibrators, but, obviously not necessarily the same 21-cm line width. Among detected galaxies, some will be pure sosies of the calibrators (i.e. galaxies with the same 21-cm line width), others not. This last class will be called "sosie candidates". Nevertheless, sosie candidates can also be used through the conventional TF relation with the still valid advantage that the result does not depend on morphological type and inclination effects. Only, uncertainty about the value of the slope a may affect the final result which anyway will be more secure.

Another problem still plagues the use of any distance determination. It has been shown that Malmquist bias appears at very small distances. For instance, for galaxies as luminous as M 31, the bias starts for radial velocities as small as 2000 km s^-1. To push this limit deeper we will enlarge our sample and observe galaxies up to fainter apparent magnitude.

In order to check the effect of the bias it is compulsory to work with a sample complete up to a well determined apparent magnitude. We present in Fig. 1 the completeness curve for all sosie candidates of M 31 extracted from the LEDA database. This sample contains 1434 galaxies. The completeness curve (filled circles) is drawn. On the same figure the completeness curve is given for those 351 galaxies of the sample having known 21-cm line width (open circles). This shows that more than 1083 galaxies have still to be observed to get all sosie candidates of M 31 (hence, all sosies of M 31) up to an apparent B-magnitude 15. Fortunately, among these 1083 galaxies, 367 already have an optical radial velocity and will be easy to observe in HI without having to use the search mode. They will constitute our first priority target.

  

Figure 1: Completeness curve for all sosie candidates of M 31 (filled circles) extracted from LEDA database. The completeness is fullfilled up to $\approx$15 Bmag (vertical line on the right). Today, only a part of this sample is observed in HI (open circles) and the completeness of this subsample is fullfilled only up to an apparent magnitude of about 12.5 Bmag. (vertical line on the left). The dashed line represents the idealized completeness curve

In order to show the benefit of using a deeper sample, we plot in Fig. 2 the apparent Hubble constant H₀ vs. the radial velocity for M 31 sosie galaxies and four different completeness limits ($B_{\rm lim}=12$, 13, 14 and 15). These curves show the effect of the Malmquist bias under different conditions. They are calculated according to Teerikorpi (1975, Eq. (5)) using a dispersion of the Tully-Fisher relation of $\sigma=0.6$ and a Hubble constant of 50 km s^-1 Mpc^-1. It appears clearly that the bias would be negligible up to a radial velocity of 6000 km s^-1 if a completeness limit of $B_{\rm lim}=15$ is reached. The Hubble constant derived in this way would not be influenced very much by large scale flows, because these flows are generally assumed to be smaller than about 600 km s^-1. Note that, the bias effect would be much stronger if sosies of a less luminous calibrator were used.

  

Figure 2: Bias curves of M 31-sosies for four completeness limits: $B_{\rm lim}=12$, 13, 14 and 15. It appears that the bias is negligibly small up to a velocity of 6000 km s-1 when the sample is complete up to a limit of $B_{\rm lim}=15$

For instance, we compare the bias for M 33 and M 31 (Fig. 3) using the same bias curve as above but a completeness limit of $B_{\rm lim}=13$. It is obvious that it is better to use a luminous calibrator.

  

Figure 3: Bias curves of M 31-sosies and M 33-sosies for a completeness limits $B_{\rm lim}=13$. It appears that the bias is much stronger for M 33 which is less luminous than M 31

A preliminary study from pure sosies of M 31 and M 81 is presented in a companion paper (Paturel et al. 1998) where distances to the calibrators come from Cepheid Period-Luminosity relation calibrated with geometrical parallaxes. Another study is in preparation with all available sosie candidates. This program will be pursued by systematic HI observations of sosie candidates of different calibrators. In this paper we present a first step of observations obtained for this program. We started it by searching sosie candidates of two intrinsically luminous galaxies (M 31 and M 81), one low luminosity galaxy (NGC 253) and one face-on galaxy (NGC 5457).

The look-alike galaxies are far more distant than the calibrating galaxies. Hence morphological classification and axis ratio measurements are less certain. Further, the morphological types and axis ratios of galaxies of our program are revised from time to time due to inclusion of new data. The morphological type is coded numerically with T according to de Vaucouleurs et al. (1976). The axis ratio is noted R₂₅ and corresponds to the ratio of the major to the minor axis of the external isophote at the limiting surface brightness of 25 Bmag arcsec^-2. In Figs. 4 and 5 we present the histograms of $\Delta T = T({\rm look}-{\rm alike}) - T({\rm calibrator})$ and $\Delta\log R_{25}=\log R_{25}({\rm look}-{\rm alike})-\log R_{25}({\rm calibrator})$in order to show the actual uncertainty in the definition of a look-alike candidate. The standart deviations of these quantities, $\sigma(\Delta T)=1.0$ and $\sigma(\Delta\log R_{25})=0.07$, correspond to the typical error of T and $\log R_{25}$, respectively but two galaxies are rejected afterwards (PGC 06289 and PGC 68360).

  

Figure 4: Histogram of deviations in morphological types $\Delta T = T({\rm look}-{\rm alike}) - T({\rm calibrator})$. The standard deviation is $\pm~1.0$

  

Figure 5: Histogram of deviations in axis ratios $\Delta\log R_{25}=\log R_{25}({\rm look}-{\rm alike})-\log R_{25}({\rm calibrator})$.The standard deviation is $\pm~0.07$