The present work is motivated by the longstanding problem in observational cosmology, concerning the structure of galaxy distribution: is the Universe homogeneous or fractal on large scales, and what is the scale where homogeneity begins? Paturel et al. (1994) and Witasse & Paturel (1997) left open the possibility that the apparent partial incompleteness in some galaxy samples could actually reflect inhomogeneity or fractal distribution.
The completeness of the KLUN-sample of spiral galaxies was studied by Paturel et al. (1994) from the distribution of the angular sizes (, with isophotal 25 mag/arcsec2 diameters expressed in units of ). Separating the sample into parts with and without the plane of the Local Supercluster, it was concluded (on the assumption of homogeneity outside of the LS) that the sample is almost complete down to (or ). In fact, it is a general problem, how to determine the completeness of a sample, if there is no a priori guarantee that the galaxy counts in a complete sample should follow, say, the distribution predicted by a homogeneous distribution of galaxies.
We continue here the study of incompleteness (the knowledge of which is crucial for our applications of KLUN), and attempt simultaneously to determine the all-sky average radial distribution of galaxies. This is done by a method which utilizes the TF distance moduli available for every KLUN galaxy. As the question of the radial distribution is closely connected with the problem of the scale and dimension of fractal distribution, we give below a short resumé of the state of this field.
The early history of determination of the fractal dimension and the maximum scale of fractality is described by Peebles (1980): the main result was that 1.2 and Mpc 1. The first evidence for a universal fractal distribution of galaxies with fractal dimension 2 was given by Baryshev (1981) from number counts and virial mass distribution arguments. The same 2 inside the Virgo supercluster was indicated by Klypin et al. (1989) from analysis of fractal properties of superclusters, with Mpc. Later in several works a double power law behaviour was found so that at distances Mpc 1.2, but for 6 Mpc Mpc there is 2.2 (Einasto 1991, Guzzo et al. 1991, Calzetti et al. 1992, Einasto 1992, Guzzo 1997).
In recent years several extensive galaxy redshift surveys have become available (CfA1, CfA2, SSRS1, SSRS2, Perseus-Pisces, LEDA, LCRS, IRAS, etc., see the review by Sylos Labini et al. 1998a). Detailed analyses of the surveys have been made by different quantitative statistical methods which have revealed the fractal structure of the 3-dimensional galaxy distribution at the scales corresponding to the surveys. The fractality implies that around any galaxy the number density behaves with radius r as for .
However, considerable debate has arisen on the value of fractal dimension and the maximum correlation length (see e.g. Guzzo 1997, Sylos Labini et al. 1998b). One may even speak about two schools. According to the first one (see Davis 1997), using the standard correlation function analysis, fractal structure exists within the scale interval 0.02 Mpc 20 Mpc with dimension 1.2, while on the scales larger than 40 Mpc the Universe becomes homogeneous. According to the second school (see Pietronero et al. 1997, Sylos Labini et al. 1998a), using a more general statistical approach which includes the correlation function method as a particular case, the fractal structure exists at least up to 200 Mpc and has dimension 2. The debate shows the need for complementary approaches using other data and methods.
Recently, Di Nella et al. (1996) analyzed the correlation properties of the galaxy space distribution using the LEDA all-sky data base containing over 36000 galaxies with known redshifts. They concluded that the distribution has a fractal character up to the scales of 300 Mpc with the fractal dimension 2.2. In a recent discussion, Guzzo (1997) concluded that the galaxies are clustered in a fractal way on small and intermediate scales, and the turnover to homogeneity may occur somewhere between 200 and 400 Mpc. Taken at face value, such results predict that in an all-sky sample like KLUN, one should see a radial galaxy number density behaviour which deviates from the uniform distribution. In terms of the distance modulus µ, uniformity corresponds to the differential law for the number N of galaxies in the distance modulus range , when µ has been derived by the "standard candle" method. In a distribution with fractal dimension , the expected density law around any galaxy is , or with 2, we have the " -law".
There are a few noteworthy aspects in the present study. We concentrate on investigating how the radial density behaves around us, while the analyses mentioned above are usually based on the average behaviour around the different galaxies in the sample, which reveals the actual fractal law. However, there is special importance in seeing directly the behaviour around our individual vantage point. One thing is that we are naturally eager to see and check in concrete manner the "strange" prediction of fractality, or the thinning of average density, if our Galaxy occupies an average position in the fractal structure. On the other hand, Sylos Labini et al. (1996) have pointed out that in order to see the fractal behaviour around one point, one needs an all-sky sample with a sufficiently faint magnitude limit () in order to avoid so-called finite-volume effect inside a few Voronoi-lengths. The KLUN sample can thus be used as a kind of first firm step for deriving the radial density law, which may be later straightforwardly extended deeper when the sample is gradually increased.
In order to derive the density law, we use a new method based on photometric (TF) distances. It has two special advantages. First, we can use almost the whole sample for constructing the density law. Second, unlike the methods based on redshift distances, we do not require a local velocity field model. In view of the recent suggestion (Praton et al. 1997) that working in the redshift space may enhance or even produce structures of the "Great Wall" type (and hence influence the correlation studies and presumably, the derived value of ), it is especially important to use photometric distances in complementary studies.
© European Southern Observatory (ESO) 1998
Online publication: May 15, 1998