Astron. Astrophys. 342, 1-14 (1999)

3. ESP geometry and the measure of velocity dispersions

To all practical purposes, the projection of the ESP survey on the sky consists of two rows of adjacent circular OPTOPUS fields of radius 15 arcmin and a separation of 30 arcmin between adjacent centers. The angular extent of groups and clusters at the typical depth of the survey () are comparable, or even larger, than the size of the OPTOPUS fields. Therefore, most systems falling into the survey's volume are only partially surveyed.

The main effect of the "mask" of OPTOPUS fields is to hide a fraction of group members that lie within or close to the strip containing the mask (the OPTOPUS fields cover 78% of the area of the "un-masked" strip). Because of the hidden members, several poor groups may not appear at all in our catalog. On the contrary, our catalog might include parts of groups that are centered outside the ESP strip. These problems notwithstanding, we expect to derive useful information on the most important physical parameter of groups, the velocity dispersion, .

Our estimate of the parent velocity dispersion, , is based upon the sample standard deviation . The sample standard deviation defined as is a nearly unbiased estimate of the velocity dispersion (Ledermann, 1984), independent of the size of the sample. We make the standard assumptions that a) barycentric velocities of members are not correlated with their real 3D positions within groups, and that b) in each group the distribution of barycentric velocities is approximately gaussian. Because the position on the sky of the OPTOPUS mask is not related to the positions of groups, its only effect is to reduce at random . Therefore, using an unbiased estimate of the velocity dispersion, the mask has no effect on our determination of the average velocity dispersions of groups.

The effect of the mask is to broaden the distribution of the sample standard deviations. The variance of the distribution of sample standard deviations varies with approximately as (Ledermann, 1984). This distribution, proportional to the distribution, is skewed: even if the mean of the distribution is unbiased, is more frequently underestimated than overestimated.

While it is easy to predict the effect of the mask on the determination of the velocity dispersion of a single group, it is rather difficult to predict the effect of the mask on the observed distribution of velocity dispersions of a sample of groups with different "true" velocity dispersions and different number of members. In order to estimate qualitatively the effect of the mask on the shape of the distribution of velocity dispersions, we perform a simple Monte Carlo simulation.

We simulate a group by placing uniformly at random points within a circle of angular radius corresponding, at the redshift of the group, to the linear projected radius = 0.5 Mpc. This radius is the typical size of groups observed in shallow surveys (e.g. RPG). We select the redshift of the group, , by random sampling the observed distribution of ESP galaxy redshifts.

In order to start from reasonably realistic distributions, we set and the velocity dispersion, , by random sampling the relative histograms obtained from our ESP catalog. We limit the range of to 3 18 and the range of to 0 1000 km s^-1 .

We lay down at random the center of the simulated group within the region of the sky defined by extending 15 arcmin northward and southward the "un-masked" limits of the ESP survey. We then assign to each of the points a barycentric velocity randomly sampled from a gaussian with dispersion centered on . We compute the velocity dispersion, , of the velocities. Finally, we apply the mask and discard the points that fall outside the mask. We discard the whole group if there are fewer than 3 points left within the mask (). If the group "survives" the mask, we compute the dispersion of the members. On average, 78% of the groups survive the mask (this fraction corresponds to the ratio between the area covered by the mask and the area of the "un-masked" strip). The percentage of surviving groups depends on the exact limits of the region where we lay down at random groups and on the projected distribution of members within . For the purpose of the simulation, the fraction of surviving groups is not critical.

We repeat the procedure 100 times for = 231 simulated groups (the number of groups identified within ESP). At each run we compute the histograms N() and N().

In Fig. 1 we plot the input distribution N() -thin line- together with the average output distribution, N() - thick line-. Errorbars represent one standard deviation derived in each bin from the distribution of the 100 histograms N(); for clarity we omit the similar errorbars of N(. The factor normalizes the output distribution to the number of input groups. The two histograms in Fig. 1 are clearly very similar since N() is within one sigma from N() for all values of . We point out here that the similarity between the input and output histograms does not mean that the surviving groups have not changed. In fact only about 63% of the triplets survive the mask while, for example, 88% of the groups with 5 members and 98% of those with 10 members "survive" the mask.

Fig. 1. Effect of the "OPTOPUS mask" on N(): the thin histogram is the "true" distribution, the thick histogram is the average distribution "observed" through the "OPTOPUS mask", normalized to the number of input groups. Errorbars represent one standard deviation.

Fig. 2 shows the results of our simple simulation for the velocity dispersion. The thin histogram is the input "true" distribution N(). The dotted histogram is the average "observed" distribution obtained without dropping galaxies that lie outside the OPTOPUS mask, N(). This is the distribution we would observe if the geometry of the survey would be a simple strip. The third histogram (thick line) is the average output distribution in presence of the OPTOPUS mask, N() (errorbars are one-sigma). The input distribution N(), the distribution N(), and the distribution N() are all within one-sigma from each other. In particular the two distributions we observe with and without OPTOPUS mask are undistinguishable (at the 99.9% confidence level, according to the KS test). The low velocity dispersion bins are slightly more populated in the "observed" histograms because the estimate of the "true" is based on small . Note that in the case of real observations, some groups in the lowest bin will be shifted again to the next higher bin because of measurement errors.

Fig. 2. Effect of the "OPTOPUS mask" on N(). The thin histogram is the "real" distribution, the dotted histogram shows the effect of sampling on the input distribution, and the thick histogram is the average distribution "observed" through the "OPTOPUS mask", normalized to the number of input groups. Errorbars represent one standard deviation.

Our results do not change if we take into account the slight dependence of from observed within our ESP catalog: also in this case the effect of the mask is negligible.

In conclusion, the simulation confirms our expectation that the OPTOPUS mask has no significant effect on the shape of the distribution of velocity dispersions.

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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