3 Group identification procedure

  In early catalogs, galaxy groups were identified "by eye'' (e.g., de Vacouleurs 1975) and/or in projection (e.g., Turner & Gott 1976) from angular galaxy catalogs. Nowadays, groups are better identified by means of objective grouping procedures applied to galaxy redshift surveys. Several such prescriptions have been suggested in the literature (Turner & Gott 1976; Materne 1978, 1979; Paturel 1979; Tully 1980, 1987; HG82; NW87; N93; Pisani 1993, 1996).

3.1 Friends-Of-Friends algorithm

  We adopt the adaptive FOF algorithms introduced by HG82 for several reasons. First, most loose group catalogs extracted from galaxy redshift surveys (HG82; Geller & Huchra 1983; RGH89; Maia et al. 1989; N93; Garcia 1993; PGHR94; RPG97) are based on this technique, and we want to compare them with ours. Second, FOF algorithms are relatively faster and easier to implement than other objective grouping algorithms (Turner & Gott 1976; Materne 1978, 1979; Paturel 1979; Tully 1980, 1987; Pisani 1993, 1996), lead to a unique output catalog for given input parameters, and do not rely on any a priori assumption regard to the geometrical shape of galaxy groups. Third, FOF algorithms are well-studied tools, as they have been repeatedly applied to numerical simulations of galaxy surveys. These have been either (i) cosmological N-body simulations (NW87; MFW93; NKP94, NKP97; F95a, b), where all six space and velocity coordinates are known in advance and allow for a self-consistent matching of "real'' and FOF groups, or (ii) geometrical Monte-Carlo simulations (RGH89; RPG97) which accurately mimick the main LSS features of a given data set and allow to assess the impact of LSS on the group properties. All such simulations provided evidence that FOF-identified objects indeed mostly correspond to physically real galaxy groups, though poor groups with only $N_{\rm mem}=3$ or 4 members may be substantially contaminated (RGH89; F95a; RPG97). No similar extensive countercheck on the other group-finding techniques has been reported yet. Fourth, further and more direct evidence of the "reality'' of FOF groups was recently provided by direct observation of the neighbourhood of FOF groups by Ramella et al. (1995a,b, 1996), who also showed how the physical properties of FOF-identified loose groups are a reliable estimate of those of the "real'' underlying galaxy groups. To be fair, it should be noted that the HG82 recipe tends to include a high fraction of spurious members and/or groups (NW87; F95a, b). However, the compilation of a catalog requires, in the first place, a high degree of completeness; suspicious objects can still be discarded later on.

The operational definition of a group is a number density enhancement in (redshift) space (HG82; NW87 have a slightly different point of view). The algorithm can be thought of as a percolation technique, but truncated to a specified value R₀ of the connecting link $R_{\rm L}$:galaxies closer than R₀ are "friends'' of each other, friendship is transitive, and an isolated set of (at least 3) friends is what we call a galaxy group. In the ideal case of a luminosity-complete and volume-limited sample with N galaxies, volume V, average density $\bar n = N/V$, with no redshift distortions, groups would then be selected above a fixed number density threshold $\delta n/n$, given by:  

$$1 + {\delta n \over n} = {3\over 4 \pi R_0^3 \bar n} = {3 /... ...) \over \int_{-\infty}^{{M_{\rm lim}}} \phi (M) {\rm d}M } \ .$$ (1)

where $\phi(M)$ is the galaxy LF of the sample (Sect. 3.4, Eq. (7)) and all galaxies are brighter than the absolute magnitude completeness limit ${M_{\rm lim}}$.

3.2 Radial scaling of the links

  In practice, loose groups have usually been identified from apparent-magnitude-limited redshift surveys. This brings in two main complications: (i) strong radial redshift distortions due to peculiar motions, mainly induced by small-scale galaxy dynamics within the groups themselves, and (ii) distance selection effects due to the difficulty of observing fainter galaxies at larger distances. Neglecting the effect of strong spatial inhomogeneities (LSS), the expected number density of galaxies at a distance r from us, brighter than ${m_{\rm lim}}$ in apparent magnitude and brighter than ${M_{\rm lim}}(r)= {m_{\rm lim}}-2.5\log_{10}(hr/{\rm Mpc}) -25$ in absolute magnitude, is given by  

$$\bar n(r;{m_{\rm lim}}) = \int_{-\infty}^{{M_{\rm lim}}(r)} \phi (M) {\rm d}M \$$ (2)

which increases with ${m_{\rm lim}}$ and decreases with r.

To overcome effect (ii), the strategy is to "compensate'' the decrease in $\bar n(r; {m_{\rm lim}})$by allowing the links to be "more generous'' at larger r or fainter ${m_{\rm lim}}$.To deal with effect (i), the spatial link $R_{\rm L}$ is replaced by a transverse "sky-link'' $D_{\rm L}$ and a radial "redshift-link'' $V_{\rm L}$.Different authors proposed qualitatively different solutions to implement such a strategy (HG82; NW87; see also Gourgoulhon et al. 1992, N93; Garcia 1993; we refer to the original papers for details), involving also dynamical considerations about $V_{\rm L}$ because of effect (i). Advantages and shortcomings of one recipe over another are discussed in NW87, Garcia (1993) and F95a, b.

We adopt the scaling recipe of HG82, though with stricter normalizations. Both $D_{\rm L}$ and $V_{\rm L}$ are normalized by $D_0 \equiv D_{\rm L}(cz_0)$ and $V_0 \equiv V_{\rm L}(cz_0)$at some fiducial redshift cz₀=H₀ r₀. They are then scaled with cz as $\left[ \bar n(cz_{ij}/H_0;{m_{\rm lim}}) \right]^{-1/3}$,where cz_ij=(cz_i+cz_j)/2 is the median redshift of the ij-th pair of galaxies. Galaxies are linked if their transverse and radial separations $r_{ij}^{\perp}$ and $r_{ij}^{\parallel}$ satisfy $r_{ij}^{\perp} \leq D_{\rm L}(cz_{ij})$ and $r_{ij}^{\parallel} \leq V_{\rm L}(cz_{ij})/H_0$, respectively. The number density within groups at distance r is  

$$n_{\rm grp} (r;{m_{\rm lim}}) \geq {3\over 4 \pi D_{\rm L}^3... ...t (1 + {\delta n \over n} \right ) \bar n(r;{m_{\rm lim}}) \ ,$$ (3)

which with the HG82 scaling $D_{\rm L}(r) \propto \bar n(r)^{-1/3}$ yields  

$$1 + {\delta n \over n} = {3\over 4 \pi D_0^3 \bar n(r_0)} =... ...er \int_{-\infty}^{{M_{\rm lim}}(r_0)} \phi (M) {\rm d}M } \ .$$ (4)

We emphasize that the relation among $\delta n/n$ and D₀ depends on the galaxy LF. Also, the spherical symmetry of the "volume of friendship'' implicitly assumed in the idealized Eq. (1) is broken by the actual, anisotropic definition of friendship through the two links D₀ and V₀.

3.3 Normalization of the links

  As previously pointed out, group properties sensitively depend on the chosen algorithm. On the other hand, we want to define group samples which may be directly compared-to/combined-with those previously published. We discuss here how to face this problem. We emphasize that here we concentrate on the question of how to "match'' FOF algorithms for different data sets. The complementary problem of how to calibrate the "optimal'' (if any) FOF algorithm for a given data set has been extensively discussed by other studies (HG82; NW87; RGH89; MFW93; F95a, b; NKP94, NKP97), and it is out of our scope.

According to HG82, groups are selected above a density contrast $\delta n/n$ given a priori. The physical justification is the hypothesis-requirement that galaxy groups correspond to dynamically bound matter overdensities, whose dynamical state is dictated by their density contrast $\delta \rho/\rho$,in turn related to the number density contrast $\delta n/n$,maybe through some mechanism of biased galaxy formation (e.g., Kaiser 1984). Unfortunately, one does not know exactly the value of the normalization of $\phi(M)$ in Eq. (1). Due to the presence of LSS, this may vary by a factor of 2 from survey to survey, and fluctuates strongly even within the same galaxy survey (e.g., de Lapparent et al. 1988, 1989; see also Sect. 3.4). Fluctuations and uncertainties about the shape of the LF also have a (slight) effect on the D₀-$\delta n/n$ relation (4) (TB96; Sect. 4.3). Moreover, the relation among the (physically well-motivated) mass density-contrast $\delta \rho/\rho$ and the observatio galaxy number density-contrast $\delta n/n$,is still very uncertain (e.g., Bower et al. 1993). Actually, the very existence of a universal value of $\bar n$, valid at any sufficiently large spatial scale, has been repeatedly questioned (e.g., Coleman & Pietronero 1992; Baryshev et al. 1994) and the debate on this point is still alive (Davis 1996; Pietronero et al. 1997).

What normalizations should we adopt? For the sake of clarity, let us first consider only data samples of the same depth, but different galaxy LF. (We will discuss the effect of different depth, and of further sample-to-sample variations due to local LSS features, later on.) The RGH89 group catalog in the CfA2 Slices is normalized by $V_0=350 \ {\rm km \ s}^{-1}$ and $D_0 = 0.270 \ h^{-1}~{\rm Mpc}$ at $cz_0=1000 \ {\rm km \ s}^{-1}$.With the galaxy LF adopted by RGH89, D₀ translates into $\delta n/n =80$.This very value may be regarded as an update of $\delta n/n =20$used by HG82 and based on theoretical considerations, enhanced by a factor 4 to account for the locally high density of the the Great Wall region within CfA2 Slice where most of the groups reside (Ramella et al. 1992; Ramella, private communication). Subsequent observational (Ramella et al. 1995a-b, 1996) and theoretical (F95a,b) studies confirmed such values of V₀ and D₀ as "optimal'' for the compilation of a group catalog, though different normalizations may be more appropriate in different contexts (NW87; MFW93; N93; F95a, b; NKP94, NKP97). We also note that the specific conclusions of most such works should be taken with caution, as (i) they do not always use the precise HG82 scaling for $D_{\rm L}$ and $V_{\rm L}$,and (ii) their observational sample is often the CfA1 survey, brighter and shallower than CfA2 and PPS2. (For a discussion of the effect of depth, see below.) Later, PGHR94 and RPG97 used a slightly different $D_0=0.231 \ h^{-1}~{\rm Mpc}$, due to the higher density of CfA2 North and to their requirement $\delta n/n =80$.The galaxy LF of PPS2 is still different, yielding further combinations of D₀ and $\delta n/n$.With our choice of $\phi(M)$ (Sect. 3.4), D₀=0.231 and $0.270 \ h^{-1}~{\rm Mpc}$ yield $\delta n/n = 173$ and 108, respectively, instead of the desired $\delta n/n =80$.The latter is only recovered if we adopt $D_0=0.300 \ h^{-1}~{\rm Mpc}$,substantially larger than D₀ used by RPG97. This is consistent with the different normalizations suitable for PPS2, CfA2 Slice, and CfA2 North ($\phi_*=0.02 \pm 0.1 \ h^3 \ {\rm Mpc}^{-3}$,$\phi_*=0.025 \ h^3 \ {\rm Mpc}^{-3}$, and $\phi_*=0.05 \pm 0.2 \ h^3 \ {\rm Mpc}^{-3}$, respectively) and well within the LF normalization uncertainty.

We are then led to the following question. In order to build group catalogs physically as similar as possible to each other, should they be compiled using (the same link V₀ and) the same link D₀, or the same density threshold $\delta n/n$? We emphasize that the parameter $\delta n/n$ defined in Eq. (4) is customarily used only in order to label a given FOF catalog. In fact, no such assumption is needed to identify the groups, and one could as well use as label the D₀ parameter effectively used by the FOF algorithm itself. Moreover, the theoretically motivated mass overdensity $\delta \rho/\rho$is ideally referred to the mean mass density $\bar \rho_0$ of the whole Universe, and not $\bar \rho_S$ averaged only over the considered sample. However, no matter whether the universal value of $\bar \rho_0$ coincides with $\bar \rho_S$ and whether it is known or not, for a given $\bar \rho_0$ identical D₀'s will correspond to identical $\delta \rho/\rho$.So, though strong theoretical motivations suggest $\delta \rho/\rho$ to be the relevant physical quantity, and suggest $\delta n/n$ as its observational counterpart, on practical grounds it is more justified to relate the link normalization directly to D₀. As a matter of fact, the issue of D₀ vs. $\delta n/n$ was considered also by Maia et al. (1989). They compared FOF groups in SSRS1 and in CfA1, identified either with the same D₀ or with the same $\delta n/n$, and found that the median physical properties of the groups were more similar in the former case than in the latter. However, it is not clear how to interpret such result, since the two galaxy samples SSRS1 and CfA1 differ in depth and selection criteria.

We can now discuss the case of data samples of different depth, for which things are still slightly more complicated. For simplicity, we assume them to have the same parent LF. We adopt the Schechter (1976) functional form $\varphi(L/L_*){\rm d}(L/L_*)=\phi_* (L/L_*)^{-\alpha} \exp(-L/L_*){\rm d}(L/L_*)$.Then from Eq. (4) above, constant $\delta n/n$ requires

$$\begin{eqnarray} D_0 &&\propto \left [\int_{(r/r_*)^2}^\infty \varphi \left({L\o... ...\!+\!\!\alpha, \left({r\over r_*}\right)^2\right ] \right )^{-1/3}\end{eqnarray}$$ (5)

$\phantom{}$
where r_* is the maximum distance where a galaxy of luminosity L_* (absolute magnitude M_*) is still observable. Adopting $r_0 = 10 \ h^{-1}~{\rm Mpc}$ ($cz_0=1000 \ {\rm km \ s}^{-1}$), $M_* = -19.3 + 5 \log h$, and $\alpha = -1.15$,going from a catalog limited at ${m_{\rm lim}}=14.5$ to one limited at ${m_{\rm lim}}=15.5$, r_* goes from 58 to $91 \ h^{-1}~{\rm Mpc}$, the minimum L/L_* from (r₀/r_*)²=0.030 to 0.012, so to keep $\delta n/n$ constant at the normalization location r₀ the link normalizations should be related by  

$${D_0 ({m_{\rm lim}}=14.5) \over D_0 ({m_{\rm lim}}=15.5)}= \... ...0.030) \over \Gamma (-0.15, 0.012) } \right ]^{-1/3} = 1.12 \ .$$ (6)

$\phantom{}$
In other words, at any given distance r the deeper sample contains more galaxies and has a higher average density $\bar n(r; {m_{\rm lim}})$, so fixing the same $\delta n/n$ at a given r₀ yields a smaller D₀ for fainter ${m_{\rm lim}}$, albeit only by $12 \%$.One way out of this technical difficulty (TB96) is to choose a normalization location $\tilde r_0({m_{\rm lim}})$ variable from sample to sample, and equal to (i) a constant fraction of the characteristic sample depth r_*, or (ii) zero distance. The problem with (i) is to introduce a dependence on L_* (which also varies from sample to sample), while (ii) is technically delicate as Eq. (2) with $\phi$ given by Eq. (7) formally diverges for $r \rightarrow 0$, but suitable limits can still be defined (TB96). In practice, given the depth of the galaxy samples from which galaxy groups can be meaningfully identified, $cz_0=1000 \ {\rm km \ s}^{-1}$ is already close enough to zero that simply keeping the same normalization location r₀ for samples of different ${m_{\rm lim}}$introduces only a small inconsistency in the values D₀ and V₀ (Eq. (6)), largely overwhelmed by the other sources of uncertainty.

 

Figure 2:  Dependence of group properties (medians) from the FOF normalizations. a) the line-of-sight velocity dispersion $\sigma_{\rm v}$ vs. the redshift link V0 ($D_0=0.231 \ h^{-1}~{\rm Mpc}$); b) the harmonic mean radius $R_{\rm h}$, and the pairwise mean separation $R_{\rm p}$ vs. the spatial link D0 ($V_0=350 \ {\rm km \ s}^{-1}$). Symbols are explained in the figure. The straight lines are linearly fitted to the data of the PPS2 groups only. The bars are $\pm 1$ standard deviation divided by $\sqrt{N_{\rm G}}$.Such bars have not been used in the linear fit to the medians. They are only shown in order to give an idea of the typical dispersions of group properties around their central values. We also show data for the CfA2N groups of Ramella et al. (1997; their Fig. 1), and for PPS2 groups selected with $D_0=0.270~h^{-1}~{\rm Mpc}$ and variable V0

 

Figure 3:  Members of loose groups in PPS2: 4+1 diagram. Each dot is a group member galaxy. Everything else as in Fig. 1

 

Figure 4:   Loose Groups in PPS2: 4+1 diagram. Each dot is a group (number-weighted centroid, $N_{\rm mem}\geq 3$ member galaxies). The group magnitude is computed by adding up the luminosity of all observed group members, and the lower envelope corresponds to 3 member galaxies of magnitude ${m_{\rm lim}}=15.5$. Everything else as in Fig. 1

 

Figure 5:  Radial distribution of groups and galaxies in PPS2. Redshift distribution for groups (thick histogram), group members (thin dashed histogram), and all galaxies (thin histogram). For comparison, all curves are normalized to the same total area

 

Figure 6:  Internal properties X of loose groups in PPS2. Left panels: distribution histogram ${\rm d}N(X)$.Right panels: variation with distance. Here we have: a,b) $X=N_{\rm mem}$;c,d) $X=\log_{10}(L_{\rm G}/L_\odot)$.The curves are not fitted to the data. Here, the horizontal solid line corresponds to $N_{\rm mem}=3$and the smooth curve is the magnitude limit for groups, corresponding to $3 L_{\rm lim}(cz;{m_{\rm lim}}=15.5)$ for galaxies. We assume h=1.0 and $M_\odot=+5.48$ (blue)

 

Figure 7:  As for Fig. 6, but here a,b) $X=\sigma_{\rm v}$, the smooth curve is $V_{\rm L}$;c,d) $X=R_{\rm h}$, the smooth curve is $D_{\rm L}$.The curves are not fitted to the data. They are directly obtained from the FOF links and the definition of the X's (see text), by replacing the harmonic radius $R_{\rm h}$ and the line-of-sight velocity dispersion $\sigma_{\rm v}$ with the transverse spatial link $D_{\rm L}$ and the radial velocity link $V_{\rm L}$, respectively

 

Figure 8:  As for Fig. 7, but here a,b) $X=\log_{10}({{\cal M}_{\rm vir}}/{{\cal M}}_\odot)$, the smooth curve is $6 \times G^{-1} \sigma_{\rm v}^2 R_{\rm h}$;c,d) $X=\log_{10}({{\cal M}_{\rm vir}}/L_{\rm G})$, the smooth curve is $2 \times G^{-1} \sigma_{\rm v}^2 R_{\rm h} /L_{\rm lim}$

 

Figure 9:  As for Figs. 6 and 7, but here $X=\log_{10}(H_0 {t_{\rm cr}})$,the smooth curve is $2/ \sqrt{3} \times D_{\rm L} / V_{\rm L}$

All these complications, due to the nature of the FOF grouping algorithm, would be avoided if group properties turned out to change little for reasonable variations of FOF parameters. Actually, RPG97 report that group properties are non-significantly sensitive to the choice of linking parameters over a wide range of $\delta n/n$.On the other hand, the symmetry and the "orthogonality'' of $D_{\rm L}$ and $V_{\rm L}$ in the HG82 algorithm suggests D₀ to be directly related to the group size ($R_{\rm h}$ or $R_{\rm p}$), and V₀ to be directly related to the group velocity dispersion $\sigma_{\rm v}$,possibly with some residual "non-orthogonal'' dependence. Since $(1+\delta n/n) \propto D_0^{-3}$, any dependence of a given group property X on D₀ or $\delta n/n$is equally recovered if one plots X against $\log D_0$ or $\log(1+\delta n/n)$.In summary, even a substantial dependence of group properties on the links V₀ and D₀, though not removed, might be missed or hidden by focusing attention only on the customary $\delta n/n$ parametrization.

In order to test such effects on group properties, we built several arrays of group catalogs with different normalizations, and the same LF adopted for PPS2. The first array has given $V_0=350 \ {\rm km \ s}^{-1}$ and variable D₀= 0.184, 0.192, 0.202, 0.215, 0.231, 0.254, 0.270, 0.290, 0.300, 0.330, $0.363 \ h^{-1}~{\rm Mpc}$, including the D₀'s used by RPG97 (Ramella, private communication) and yielding $\delta n/n=$342, 301, 259, 214, 173, 129, 108, 87, 79, 59, 45, respectively, with our LF. The second array has given $D_0=0.231 \ h^{-1}~{\rm Mpc}$ and variable V₀= 150, 250, 350, 450, 600, $750 \ {\rm km\ s}^{-1}$.The third array is like the second one, but for $D_0 = 0.270 \ h^{-1}~{\rm Mpc}$.In Fig. 2a, we plot the line-of-sight velocity dispersion $\sigma_{\rm v}$ (medians) against the redshift link V₀. Analogously, in Fig. 2b, we plot the mean harmonic radius $R_{\rm h}$ and mean pairwise member separation $R_{\rm p}$ (medians) against the spatial link D₀. In all cases, we find strong, approximately linear correlations: $\sigma_{\rm v, med} \simeq 0.23 V_0 + 95 \ {\rm km\ s}^{-1}$,$R_{\rm h, med} \simeq 1.32 D_0 +0.04 \ h^{-1}~{\rm Mpc}$,$R_{\rm p, med} \simeq 2.15 D_0 -0.08 \ h^{-1}~{\rm Mpc}$.Similar results also hold for groups in CfA2 North (Fig. 2, RPG97's data), in CfA1 (NKP94; NKP97), and in SSRS1 (Maia et al. 1989, their Table 5). Although RPG97 produce a figure with a decreasing trend of $R_{\rm h}$versus $\delta n/n$, they do not mention that this trend simply arises from the proportionality between median $R_{\rm h}$ and D₀ (which would have been obvious had they made a log-log plot).

For all the above mentioned reasons, several of which are mainly practical, we prefer to parametrize a given FOF algorithm by its D₀ and V₀. In particular, we specify the threshold spatial separation used by the FOF algorithm directly in terms of D₀, instead of through the density contrast (HG82) or as a fraction of the mean inter galaxy separation (NW87). We then compute $\delta n/n$ a posteriori through Eq. (4), and interpret its value mainly as an approximate measure, rather than a precise parametrization, of density contrast.

3.4 Luminosity function

  The galaxy LF is parametrized with the Schechter (1976) form:  

$$\phi(M)= {\rm const}\cdot \left[10^{0.4(M_* - M)}\right]^{1+\alpha} \exp\left[-10^{0.4(M_* - M)}\right]$$ (7)

$\phantom{}$
where ${\rm const}= \phi_* \cdot 0.4 \cdot \ln 10$.We assume distance proportional to cz, extinction-corrected $m_{\rm Z}$,and no K-corrections (very small for this low redshift sample),

Recently, Marzke et al. (1994; MHG94 hereafter) estimated the LF of the whole CfA2 Survey, and several subsamples therein. For their CfA2 South subsample, very similar to PPS2, they get: $\alpha = -0.9 \pm 0.2$, $M_* = -18.9 \pm 0.1$, $\phi_*= 0.02 \pm 0.01 \ h^3 \ {\rm Mpc}^{-3}$.Note that they explicitly corrected for (a form of) Malmquist bias, the Eddington (1913) bias: Zwicky magnitudes have an uncertainty $\sigma_m \simeq 0.3$-0.4 mag, which causes a "random diffusion'' of the more numerous fainter galaxies towards brighter magnitudes, and in turn modifies the overall shape of the LF. This effect induces an artificially bright M_*, and a correspondingly too negative $\alpha$.

We use an estimate of $\phi(M)$ based directly on our data (TB96). First, though PPS2 and CfA2 South are similar samples, they are still slightly different. Their galaxy LF's might be different, and this might effect group identification through the $\phi(M)$ ingredient of the FOF algorithm. Second, group identification requires a Malmquist-uncorrected $\phi(M)$. Malmquist corrections to $\phi(M)$ are global, and they do not apply to each single galaxy. In fact, only the study of intrinsic physical properties of the galaxies themselves would require such (unknown) corrections (and others, e.g. internal extinction). Our sample PPS2 is spatially inhomogeneous (Fig. 1). This requires a density-inhomogeneity-independent technique (e.g., Efstathiou et al. 1988; de Lapparent et al. 1989; see also the review of Binggeli et al. 1988).

Using the inhomogeneity-independent STY method (Sandage et al. 1979) and Zwicky magnitudes (corrected for Milky-Way extinction), TB96 found $\alpha = -1.15 \pm 0.15$ and $M_* = -19.3 \pm 0.1$.By construction, the STY technique does not provide an estimate of the density normalization $\phi_*$.Simple tests with the non-parametric inhomogeneity-independent C-method (Lynden-Bell 1971; Choloniewsky 1986, 1987) or, on the other hand, a countercheck with the more naïve inhomogeneity dependent $1/V_{\rm MAX}$ technique (e.g., Felten 1977), all seem to suggest $\phi_* \sim 0.01 \ h^3 \ {\rm Mpc}^{-3}$ for PPS2 (TB96). However, matching the absolute magnitude counts ${\rm d}N/{\rm d}M$ or the radial counts ${\rm d}N/{\rm d}r$is better accomplished using $\phi_* \sim 0.02$ (TB96). Both values are consistent with the typical uncertainty $\Delta \phi_*/\phi_* \approx 0.5$,and with the CfA2 South value $\phi_*= 0.02 \pm 0.01 \ h^3 \ {\rm Mpc}^{-3}$ given by MHG94. Since their analysis is superior TB96's on this point, here we also adopt $\phi_*= 0.02 \pm 0.01 \ h^3 \ {\rm Mpc}^{-3}$.(Note that this only matters when we insist on translating a given D₀ in the grouping algorithm into an approximate density threshold $\delta n/n$.)

Regarding the other two Schechter parameters, estimating errorbars as in Marshall (1985) the same tests also suggested an uncertainty $\Delta \alpha \approx \pm 0.15$ and $\Delta M_* \approx \pm 0.1$ (TB96). This is close to the typical observational uncertainties (and/or the scatter among different techniques and/or data sets) $\Delta M_* \sim 0.1-0.2$ and $\Delta \alpha \sim 0.1-0.2$reported by MHG94, and the similar earlier studies quoted above. The results of TB96 (not Malmquist-corrected) and of MHG94 (Malmquist-corrected by $\delta \alpha \simeq 0.1-0.2$, $\delta M_* \simeq 0.3-0.4$, both positive) are in good agreement, once we take into account the different details between the two analyses. In fact, by using inhomogeneity-independent techniques, several authors estimated uncorrected Schechter parameters and their corresponding additive corrections $\delta M_*$ and $\delta \alpha$.In the first two Northern CfA2 Slices, de Lapparent et al. (1989) found $\delta M_* \simeq 0.3$ and $\delta \alpha \simeq 0.1$.For the whole CfA1, similar values were found by Efstathiou et al. (1988; $\delta M_*\simeq 0.39$ and $\delta \alpha \simeq 0.18$), and by N93 ($\delta M_*\simeq 0.45$ and $\delta \alpha \simeq 0.24$). Our LF and those of de Lapparent et al. (1988, 1989) for the CfA2 Slice(s) ($\alpha=-1.2$-$1.1 \pm 0.1$, $M_*=-19.15-19.2 \pm 0.1$, not Malmquist-corrected), used in previous group catalogs, are also rather similar. Let us outline that, concerning group properties, such residual small differences have however a rather small effect (Sect. 4.3; TB96).