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Astron. Astrophys. 321, 84-104 (1997)

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7. Non-equilibrium and orbits of ELG

For the 75 systems with [FORMULA] we have computed the virial and projected masses (see, e.g. Heisler et al. 1985); we have done this for the datasets that include all members as well as for the subsets of non-ELG members. For both mass estimators we find that the estimate based on all the galaxies is 8 % larger than that based on non-ELG only. The distribution of masses computed using only non-ELG is significantly different (at the [FORMULA] 0.999 conf.level) from that computed from the combination of non-ELG and ELG.

We have estimated the average ratio of the masses we would have derived separately for ELG and non-ELG. Using an average ELG fraction of 0.15 for the 75 systems used here, we estimate that cluster mass estimates based solely on ELG must, on average, be [FORMULA] 50 % larger than the estimates based on the non-ELG only. Note that this result involves the assumption that ELG and non-ELG have the same type of orbital distribution, so that the same velocity projection factor applies. If the orbital characteristics of ELG and non-ELG are not the same, the difference in the mass estimates may in reality be larger or smaller.

In deriving the difference of 50 % in estimated mass, it has also been tacitly assumed that ELG and non-ELG are both pure categories. However, one must realize (see also Sect.  3.2) that the non-ELG class may harbour a significant contribution of late-type galaxies (about two-thirds of all spirals were not detected as ELG). If the latter share the kinematics of the ELG, there might be an even larger difference between mass estimates based on spiral and non-spiral galaxies.

To forge consistency between the mass estimates based on ELG and non-ELG, the orbits of the non-ELG should be more radial than those of the ELG in order to counteract the lower value of [FORMULA] by a larger velocity projection factor. However, this is very unlikely in view of the more centrally concentrated distribution of the non-ELG. Another, more probable solution to the apparent inconsistency between the mass estimates based on ELG and non-ELG is to assume that the ELG are not in equilibrium with the non-ELG. As both classes are in the same potential (to which both probably contribute in a limited way), it would only seem possible for them not to be in equilibrium if they had not adjusted to the potential in the same manner. This could happen if their relaxation times were very different, which in turn could be a natural consequence of the differences in their spatial distribution. As the non-ELG are significantly more concentrated and find themselves in a denser environment, they are more likely to have reached equilibrium than are the ELG.

We have tried to obtain more information on the orbits of ELG and non-ELG by analyzing the dependence on projected radius of the distribution of the line-of-sight component of their velocities. If the statistics of the orbital parameters of ELG and non-ELG are different, their velocity distributions must depend on position in different ways, and that would manifest itself in the distribution of line-of-sight velocities (see e.g. Kent & Gunn 1982, and Merrit 1987).

We have determined the radial dependence of the dispersion of the line-of-sight velocity, using the synthetic cluster formed by adding the 75 systems with at least 20 members. This has the clear advantage of statistical weight but the equally clear disadvantage of producing an 'average' cluster that may bear little resemblance to any of the real clusters that it is made up of. After clipping of the [FORMULA] outliers this cluster contains 3699 galaxies of which 549 are ELG. In Table 7 we show the values of [FORMULA] for two radial bins as well as the overall value, for ELG and non-ELG. The bins have been chosen as a compromise between optimizing the detection probability for orbital anisotropy, if it exists, and the statistical weight for its detection. E.g., decreasing the size of the inner bin increases the discrimating power for orbital anisotropy, but decreases the statistical weight for its detection.


Table 7. The observed radial dependence of [FORMULA]

In adding the data for the 75 systems, the values of [FORMULA] have not been scaled but the velocities have been scaled with the global value of [FORMULA] of each cluster. The values of [FORMULA] in Table 7 are thus in units of the overall [FORMULA] of the synthetic cluster. One might wonder why [FORMULA] of the non-ELG is larger, instead of smaller, than 1.00 (after all, the combination of ELG and non-ELG should give [FORMULA] exactly equal to 1.00). The reason is that, in adding many normalized guassians, the errors in the means of the individual guassians produce an overall sigma that is slightly larger than 1.00.

The Table confirms the large difference between ELG and non-ELG as regards overall [FORMULA] and, at the same time, shows that the ratios of the [FORMULA] 's in the inner and outer bin are remarkably similar (viz. 1.14 and 1.13) for ELG and non-ELG. The first impression could be that this indicates similar orbits for ELG and non-ELG. However, that cannot be the case, as the density distributions of ELG and non-ELG are significantly different.

From some fairly simple modeling, we have predicted values of [FORMULA] for ELG and non-ELG in the two radial bins defined in Table 7. We used a model with 3 - D density profiles described by [FORMULA] -models with a [FORMULA] of -0.71, and [FORMULA] 's of 0.15 and 0.42 h-1 Mpc for non-ELG and ELG respectively, i.e the values we found from the fits to the surface density profiles (see Sect.  5). In addition, we specify a velocity distribution at each position that is described by a value for the velocity dispersion in the radial direction, [FORMULA] (r), and a constant anisotropy parameter [FORMULA] (= 1- [FORMULA]). The value of [FORMULA] (r) was assumed to depend linearly on radius, viz. [FORMULA] (r) = [FORMULA] (0) - [FORMULA] r.

From the 3-D density profile, we randomly extract 105 points. Since our clusters are sampled to different limiting radii, the total sample including all clusters is not complete at large distances from the center. To mimic this incompleteness in our simulated sample, we did not include the simulated points with [FORMULA] 1.2 h-1 Mpc . From Table 1 is can be seen that the median of the largest [FORMULA] in the clusters is essentially 1.2 h-1 Mpc . We included all points with [FORMULA] 1.2 h-1 Mpc as long as the 3 - D clustercentric distance was less than 10 h-1 Mpc . In this manner, we reproduced to within a few percent the observed fractions of galaxies in the inner and outer bins, for ELG as well as non-ELG.

For each of the simulated points, we randomly extracted three velocity components from three gaussian velocity distributions that follow from the velocity dispersion profile and the anisotropy. We assumed the two components in the tangential directions to have the same dispersion, which follows from [FORMULA] and [FORMULA]. We then projected the velocity vector along the line-of-sight, and added noise to the resulting line-of-sight velocity by adding two random deviates. The first simulates the errors in the individual velocity measurements (assumed to be 0.1 [FORMULA] ), the other the error in the average cluster velocity (assumed to be 0.2 [FORMULA] ). Finally, as in the observations, we rejected all simulated (line-of-sight) velocities outside [FORMULA].

The model parameter were optimized as follows. First, we estimated the best values of [FORMULA] (0) and [FORMULA] for the non-ELG, assuming the orbits to be isotropic, i.e. [FORMULA]  0.0. In order to reproduce the observed values of [FORMULA] for the non-ELG we need [FORMULA] (0) = 1.13 and [FORMULA] = -0.14. This implies that [FORMULA] decreases to zero at a radial distance of 8.1 Mpc, which is quite acceptable in view of the expected turn-around radius of the 'synthetic' cluster. This model produces inner and outer [FORMULA] 's for the non-ELG of 1.05 and 0.92 respectively, i.e. exactly as observed.

Next, we tried to model the observed [FORMULA] values of the ELG, again assuming isotropic orbits. We have no prescribed relation between the [FORMULA] (0)'s of ELG and non-ELG, except that it is very hard, if not impossible, to imagine that the ratio of the [FORMULA] (0)'s for ELG and non-ELG could exceed [FORMULA]. The best value of [FORMULA] (0) for the ELG was found to be 1.50, which then implies a value [FORMULA] = -0.19 (because we assume that the radius at which [FORMULA] decreases to zero is the same for ELG and non-ELG). This model does not do a very bad job, but it does not fully reproduce the decrease of [FORMULA] from the inner to the outer bin. One way to improve the agreement between model and observations would be to assume a larger value for [FORMULA] for the ELG than for the non-ELG. Although we cannot totally exclude the possibility that the ELG have a steeper velocity dispersion profile than the non-ELG, the data that we have for the inner part of the 'synthetic' cluster do not indicate this (see also below).

Another way to improve the agreement between observed and predicted [FORMULA] 's of the ELG is to assume that the velocity distribution of the ELG is anisotropic; in other words: to assume that the anisotropy parameter [FORMULA]  0. In that case there are two free parameters: [FORMULA] (0) and [FORMULA] ; the value of [FORMULA] will follow directly from [FORMULA] (0) and the maximum radius of 8.1 Mpc found for the non-ELG. The best solution has to be determined by iteration because the observed global value of [FORMULA] does not, for a given value of [FORMULA] (assumed to be independent of radial distance), immediately yield the value of [FORMULA] (0). If [FORMULA] had been derived in an aperture with sufficiently large projected radius, and if [FORMULA] = 0, [FORMULA] (0) would have followed directly as [FORMULA]. As neither of these two conditions is fulfilled, we had to iteratively find the best combination of [FORMULA] (0) and [FORMULA] for each value of [FORMULA] that we assumed.

In Table 8 we summarize the results of the modeling, for non-ELG as well as ELG. It can be seen that models with [FORMULA] 1.0 (i.e [FORMULA] 0) predict that the observed dispersion of the line-of-sight velocity component does not decrease sufficiently with [FORMULA]. This was to be expected since we needed the anisotropy to increase the ratio between the inner and outer values of [FORMULA] of the ELG, but that can only be accomplished with radially elongated orbits.


Table 8. The modeled radial dependence of [FORMULA]

The most probable value of [FORMULA] for the ELG is not very well determined from our data, because the uncertainties in the observed inner and outer values of [FORMULA] for the ELG are estimated to be 4-5%. However, taking the data at face-value we conclude that the best fit is obtained for [FORMULA] 0.9, but a value of 0.3, or even 0.0 cannot be formally exluded. On the contrary, we think that negative values of [FORMULA] can be fairly safely ruled out. From Table 8 we therefore conclude that [FORMULA] 0.5 [FORMULA] 0.5, which thus provides some evidence for anisotropy of the ELG orbits but not very strong evidence.

We think that the evidence for anisotropy of the ELG orbits is, in fact, quite a bit stronger if one considers not just the inner and outer values of [FORMULA] but includes the total velocity dispersion profile. In Fig. 13 we show the observed velocity dispersion profiles for the non-ELG (upper panel) and the ELG (solid lines in the lower panel) in the synthetic cluster. The [FORMULA] profiles were derived with the LOWESS method (Gebhardt et al. 1994). The heavy lines represent the observed [FORMULA] , the two thin lines on either side indicate the 95% confidence bands, obtained from 1000 Monte Carlo simulations of the dataset (for details, see e.g. Gebhardt et al.) The increase of the uncertainty in the estimate of the ELG [FORMULA] for [FORMULA] 0.3h-1 Mpc is due to the flat ELG surface-density profile. It must be realized that the values in Table 8 are weighted averages over the indicated intervals, where the weights follow from the observed surface densities shown in Fig. 10. In the lower panel we also show the predicted velocity dispersion profiles of ELG for three models, viz. those with [FORMULA] = -0.6, 0.0 and 0.6

[FIGURE] Fig. 13. The velocity-dispersion profiles for non-ELG (upper panel) and ELG (lower panel) in the synthetic cluster. The full-drawn curves refer to the data, with heavy lines representing the observed profiles while the thin lines indicate the 95% confidence limits. In the lower panel, three model predictions for different values of the anisotropy parameter [FORMULA] are given, viz. [FORMULA] = -0.6 (dotted), 0.0 (short dashes) and 0.6(long dashes).

On the basis of the steep gradient of the ELG [FORMULA] within 1.0 h-1 Mpc , we consider it quite unlikely that for the ELG the value of [FORMULA] is practically 0.0 (i.e. that the ELG orbits are isotropic). Instead, the sharp increase of the ELG [FORMULA] within 1.0 h-1 Mpc is produced by galaxies that mostly are at quite some distance from the centre where, in our model, [FORMULA] has already decreased considerably from its central value [FORMULA] (0). To produce the observed values of the line-of-sight velocity dispersion would seem to require inevitably that the line-of-sight velocity component is significanty 'boosted' by radial anisotropy. Combining the data in Table 8 and the information in Fig. 13, we conclude that the best estimate for [FORMULA] is probably [FORMULA] 0.6 [FORMULA] 0.3 which implies a value for [FORMULA] between [FORMULA] 0.3 and 0.8. Note that we have not given much weight to the data within 0.3h-1 Mpc due to their large uncertainty. Had we given those more weight, we would have arrived at a lower value of [FORMULA], probably between 0.0 and 0.3. However, for those values of [FORMULA] the predicted velocity dispersion profile seems to give a rather bad representation of the observed slope between 0.3 and 1.0h-1 Mpc .

As mentioned before, the shape of the full distribution of line-of-sight velocities (rather than only the dispersion) at different [FORMULA] could, in principle, provide information on the orbits. However, we have calculated those distributions for the models summarized in Table 8 and found that with our statistics the shapes are not a sufficiently sensitive discriminant.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998