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4 Loose groups in PPS2


4.1 Group catalog

  Here, we present the group catalog selected with $V_0=350 \ {\rm km \ s}^{-1}$ and $D_0=0.231 \ h^{-1}~{\rm Mpc}$,($\alpha = -1.15$, M*=-19.30, $\delta n/n = 173$ if $\phi_*=0.02 \ h^3 \ {\rm Mpc}^{-3}$), which will be made available in electronic form at the CDS (Centre de Données Astronomiques de Strasbourg,, The other catalogs, similarly selected with different parameter choices, may be obtained from the author upon request.

There are $N_{\rm G}=188$ groups with $N_{\rm mem}\geq 3$ members (105 with $N_{\rm mem} \geq 5$)corresponding to a total grouped fraction $f_{\rm gr}=35 \%$.There are 1406 singles ($47 \%$) and 283 binaries ($19 \%$). Group members and group centroids are shown in Figs. 3 and 4, respectively, using the "4+1-diagram'' described in Sect. 2.

Table 1: The catalog loose groups in PPS2: two example lines

{r r r r r r r r r r r r}
 & & & & & & & & & & & \\ $i_{...
 ... &.22 &9.97 &11.08 &1.11 &1.324 \\  & & & & & & & & & & & \\ \hline\end{tabular}

Coordinates and internal properties of each group are given in the electronic tables enclosed to the present paper. As an example, in Table 1 we show the first and the last line of the group catalog. For each group, we list a label, the coordinates of the group centroid, and group internal properties. All means are number-weighted, $\langle \ \ \rangle$ denotes averages over pairs with $i \not =j$. The Table columns are: (1) group identification number; (2) number of observed members $N_{\rm mem}$;(3) mean right ascension $\alpha_{1950}$; (4) mean declination $\delta_{1950}$; (5) mean redshift cz, MBR-corrected; (6) rms velocity dispersion $\sigma_{\rm v}$ (line-of-sight); (7) $R_{\rm p} = (cz /H_0) (4/\pi) 2 \sin (\langle \theta_{ij}\rangle/2)$,mean pairwise member separation; (8) $R_{\rm h}= (cz /H_0) (\pi/2) 2 \sin (\langle \theta_{ij}^{-1} \rangle^{-1}/2)$,mean harmonic radius; (9) $\log_{10}$ of the total blue luminosity $L_{\rm G} = \sum L_i$ of all observed members, extinction-corrected, assuming $M_\odot=+5.48$ (blue); (10) $\log_{10}$ of the virial mass ${{\cal M}_{\rm vir}}=6 G^{-1} \sigma_{\rm v}^2 R_{\rm h}$;(11) $\log_{10}$ of the virial-mass-to-observed-luminosity ratio ${{\cal M}_{\rm vir}}/L_{\rm G}$;(12) crossing time ${t_{\rm cr}}= 2 R_{\rm h} / \sqrt{3} \sigma_{\rm v}$.Definitions are the same as in RGH89, except ${t_{\rm cr}}$ as in NW87, whose numerical factor is a factor 4.30 larger than in Eq. (11) of RGH89. Celestial coordinates $\alpha$ and $\delta$(in hours, minutes and fractions and degrees, minutes and fractions, respectively), are given for the epoch 1950, as in the original PPS database and in most group catalogs. Velocities are in ${\rm km\ s}^{-1}$, distances in $h^{-1}~{\rm Mpc}$, masses in $h^{-1}{{\cal M}}_\odot$, luminosities in $h^{-2}L_\odot$.

For completeness, we provide the ratio ${{\cal M}_{\rm vir}}/L_{\rm G}$ as in other group catalogs, but we regard its physical interpretation with caution, for the following reasons. The physically interesting mass-to-light ratio involves the true group mass and its total luminosity L. The former is usually estimated assuming the virial theorem to hold for loose groups, which is probably not the case (Aarseth & Saslaw 1972; Giuricin et al. 1984, 1988; Heisler et al. 1985; Mamon 1993, 1996a; F95b; NKP97). The total luminosity L of a group could be estimated from the observed portion $L_{\rm G}$,the group richness $N_{\rm mem}$, and the galaxy LF (Gott & Turner 1977; Bahcall 1979; Mezzetti et al. 1985; NW87; Gourgoulhon et al. 1992; Moore et al. 1993). Unfortunately, this would introduce further uncertainties and a further sample-to-sample dependence of group properties on $\phi(L)$(other than that - physical - due to the different galaxy mixture, and the one - observational - due to the FOF identification procedure).

4.2 Group properties

  External properties of members and groups (e.g., spatial position, clustering, grouped fraction...) are easily visualized in Figs. 3 and 4. They should be compared with the parent galaxy sample (Fig. 1). Our catalogs are built using all galaxies in PPS2. Previous works (RGH89; F95; RPG97) cut their subsamples at $cz \le 12000-15000 \ {\rm km\ s}^{-1}$,in order to exclude unreasonably elongated groups. Tests with similarly cut subsamples in PPS show negligible differences. In practice, the two procedures are equivalent, as in the faraway regions there are too few galaxies to be grouped (compare Figs. 1, 3, and 4). We further compare the radial distribution of galaxies, members, and groups in Fig. 5. The ratio among the observed number densities $\tilde n_{\rm G}$ of FOF-identified groups and $\tilde n_{\rm g}$ of galaxies in PPS2 is rather constant and independent of redshift, $\tilde n_{\rm G}(r)/ \tilde n_{\rm g}(r) \sim 1/15$ with our chosen FOF. Consistent with this result, the total number ratio in CfA2 North is $N_{\rm G}/N_{\rm g} \sim 6\%$ (RGH89; PGHR94; RPG97).

Internal properties ($X=N_{\rm mem}$, $L_{\rm G}$, $\sigma_{\rm v}$, $R_{\rm h}$, ${{\cal M}_{\rm vir}}$, ${{\cal M}_{\rm vir}}/L_{\rm G}$, ${t_{\rm cr}}$), are shown in Figs. 6, 7, 8, 9, as distribution histograms ${\rm d}N(X)$ and scatter plots cz-X against redshift. Table 2 lists typical values and range of variability (average, median; minimum and maximum, $1^{\rm st}$ and $3^{\rm rd}$ quartile; half interquartile range, rms deviation, rms deviation$\sqrt{N_{\rm G}}$)of group properties.

Table 2: Global properties of loose groups in PPS2

{l r r r r r r r r r}
 & & & & & & & & & \\ 
 ...&0.02 &0.02 &0.03 &0.03 &0.06 &0.02 \\  & & & & & & & & & \\ \hline\end{tabular}

Table 3: Group properties (medians) for different combinations of LF, D0, and ${\delta n \over n}$

{l l l r r r r r r r r r}
 & & & & & & & & & & &\\ $\phi(...
 ...&1.81$\ 10^{13}$\space &520 &0.29\\  & & & & & & & & & & &\\ \hline\end{tabular}

Table 4: Group properties (medians) for different redshift corrections

{l r r r r r r r r}
 & & & & & & & & \\ Rest Frame
 ...0}$\space &1.64$\ 10^{13}$\space &0.27\\  & & & & & & & & \\ \hline\end{tabular}

4.3 Variability of group properties

  Group catalogs selected from different galaxy samples might be significantly inhomogeneous with each other even when the parent galaxy samples are homogeneously selected. The primary source of discrepancy would be, of course, an inconsistent matching of FOF parameters among different catalogs, i.e. (i) a different link normalization for a given sample depth, or (ii) a different sample depth for a given link normalization (see Sect. 3.3). Further, subtler sources of discrepancies could be: (iii) a different galaxy LF, which depends on physical differences among samples, but also plays an active role in the FOF algorithm itself; (iv) large scale flows of peculiar motions; (v) sample-to-sample variations, e.g. due to local LSS features within the samples.

In Table 3, we test for variations of internal properties due to different assumptions about the galaxy LF. We consider several group catalogs with several combinations of LF, D0, and $\delta n/n$,but the same $V_0=350 \ {\rm km \ s}^{-1}$.(We normalize the LF of TB96 with $\phi_*=0.02 \ h^3 \ {\rm Mpc}^{-3}$.) The main differences are connected with the different D0. For given V0 and D0, the residual net effect of the galaxy LF on group internal properties is generally rather small, $\delta X/X \lower.5ex\hbox{$\;\buildrel < \over \sim \;$}5-10 \%$.Similar results hold also for group positions and clustering properties (TB96; TBIB97).

In Table 4, we test the effect of different cz corrections for the motion of the Sun (none, MBR, local group centroid) described in Sect. 2. As expected, we find negligible differences in group properties and membership in the three cases (except, of course, an overall modulation of cz of group members and centroids). In particular, no-correction or MBR-correction yield almost indistinguishable results, as the direction of $v_{\odot {\rm MBR}}$ ($\alpha=11.2^{\rm h}$; $\delta=-7^{\rm o}$)is almost orthogonal to (the bulk of) PPS.

Table 5 is a preliminary comparison of our group catalog with similarly selected groups in previous studies. We note explicitly that all these samples have the same depth, given by ${m_{\rm lim}}=15.5$.A more thorough investigation (Trasarti-Battistoni 1997) is beyond the scope of this paper. We list global properties for our groups in PPS2 and for groups in the CfA2 survey as given in RGH89, F95b, PGHR94, and RPG97. Their results and ours are in good agreement. Comparing our Table 2 with Table 6 of RGH89 and Table 1 of PGHR94 shows that also the ranges of variability of group properties are in very good agreement in the three cases.

Table 5: Group properties in PPS2 and in CfA2 (medians, and global values)

{l l l l l l l}
 & & & & & & \\ Galaxy sample &$\sigma_{\...
 ... 10^{13}$\space &3.0$\ 10^{2}$\space &0.44\\  & & & & & & \\ \hline\end{tabular}

This seems to contradict the results of RGH89. They found a significant difference among groups in different samples, namely $\sigma_{\rm v, {\rm med}}=131 \ {\rm km\ s}^{-1}$ in the CfA1 survey and $\sigma_{\rm v, {\rm med}}=192 \ {\rm km\ s}^{-1}$ in the CfA2 Slice. From their Fig. 9, one sees that groups in the CfA1 survey and the CfA2 Slice are located preferentially around $cz_1 \approx 1000 \ {\rm km\ s}^{-1}$ and $cz_2 \approx 8000\ {\rm km\ s}^{-1}$,respectively. RGH89 correctly notice that this different location of the LSS within the samples induces a significant physical difference among the groups in the two samples, those in the shallower sample being typically nearer to us and brighter. Then, they argue that these sample-to-sample variations might also be responsible for the discrepancy in $\sigma_{v, {\rm med}}$.However, RGH89 do not give an explanation why $\sigma_{v, {\rm med}}$ should be higher in one sample than in the other, and they reject the possibility that the discrepancy be induced by the FOF grouping algorithm. (See also the discussion in Maia et al. 1989). In fact, the difference between the groups in CfA1 and CfA2 Slices could be due to a combination of the different LSS features present in the two galaxy samples and of the different radial scaling of the FOF links adopted for the two samples. We recall that the links increase with r and decrease with ${m_{\rm lim}}$ (Sect. 3.2). Therefore, normalizing the links with the same D0 and V0 for both samples, but scaling them proportionally to $\left[ \bar n(r; {m_{\rm lim}}) \right]^{-1/3}$ (different for the two samples!) as in RGH89, at any given r the links $D_{\rm L}$ and $V_{\rm L}$ will still be always more generous in the shallower sample than in the deeper one. But, if group properties are so directly related to $D_{\rm L}$ and $V_{\rm L}$ as suggested by Sect. 3.3, then the relative location of LSS features within the sample boundaries must also be taken into account. E.g., if the LSS features lie at comparable distance in both samples, one would expect a higher $\sigma_{v, {\rm med}}$ in the shallower sample (more generous links). But, if the LSS features are very differently distributed in the two samples, they could be differently "weighted'' by the adopted links. So, if LSS features lie at a much greater distance (more generous links) in the deeper than in the shallower sample - as in the case for CfA1 and CfA2 Slices - scaling up the links with r might even overtake the effect of ${m_{\rm lim}}$ in $\bar n(r; {m_{\rm lim}})$.One would then expect a higher $\sigma_{v, {\rm med}}$ in the deeper sample - once more, as in the case for CfA1 and CfA2 Slices. Interestingly, one finds a link ratio $V_{\rm L}(cz_1;{\rm CfA1})/V_{\rm L}(cz_2;{\rm CfA2}) \approx 2.5$,to be compared with $\sigma_{v, {\rm med}}({\rm CfA1})/\sigma_{v, {\rm med}}({\rm CfA2})\approx 1.5$,taking into account that groups are not only located precisely at cz1 and cz2. In a sense, by modulating the properties of the groups according to distance, the FOF links may either amplify or deamplify the sample-to-sample variations according to how the galaxy LSS is arranged within the samples. On the other hand, such effects will be reduced when samples of the same ${m_{\rm lim}}$ are compared using the same $D_{\rm L}(r)$ and $V_{\rm L}(r)$, as we do here. In this case, all discrepancies would be purely due to the intrinsic sample-to-sample variations, i.e. a different amount and/or location of LSS within the survey limits, but not further modulated by a different radial scaling of the links. Considering medians or averages over the whole group distributions would further reduce the sample-to-sample discrepancies.

Related to the previous point, note the smooth curves in the cz-X scatter plots (Figs. 6-9). They were not obtained by fitting the observed distribution on the diagrams. They were obtained simply by replacing $\sigma_{\rm v}$ and $R_{\rm h}$ by $D_{\rm L}(cz)$ and $V_{\rm L}(cz)$, respectively, in all formulae defining internal properties. However, there is often a clear similarity of redshift dependence between the smooth, FOF-induced curves and the (upper envelopes, or median values of) group internal properties.

Note also how the peaks in the ${\rm d}N(X)$ histograms often correspond to denser region in the cz-X plane (projected onto the X axis), in turn related with dense concentrations in redshift space (e.g., the peak at $cz\sim 5000 \ {\rm km\ s}^{-1}$ in Figs. 1, 3 and 4).

In summary, in PPS2 as well as in other samples (Maia et al. 1989; RGH89), there seem to be a complex interplay among LSS features, sample depth, FOF algorithms, and group properties. Disentangling these effects is a subtle matter, out of the scope of the present paper, and it is left for a future work (Trasarti-Battistoni 1998, in preparation).

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