## Hierarchical clustering method (Zappala et al.) applied to the proper elements data set of 66,089 asteroids (analytic proper elements of Milani and Knezevic:  AstDys proper elements database)

The purpose of this is to update the asteroid families using now available-much larger databases of the proper elements. Currently, we can use: 1)  66,089 analytically computed (Milani and Knezevic), and 2)  31.502 numerically computed proper elements (Knezevic and Milani). Since 1) is a factor of 5-6 larger than the database used by Zappala et al (1995), we think this be a good idea. We use the hierachical clustering metod (HCM) of Zappala et al.. This criterion of family membership requires that all family members are connected by a chain, where each member is attached to its neigbour by less than a limiting distance (cut-off). There are two free parameters: i) How to define the distance ?  , and ii) How to choose the  cut-off?.

We adopt the distance metrics from Zappala et al. 1994, Eq. 1:   d_1 = n*a* \sqrt ( 1.25 (da/a)^2 + 2 de^2 + 2 (dsinI)^2, in usual notation. This is a metrics in the Eucledian space, where a,e,i, are weighted by certain coefficients. It is uncertain what weighting coefficients should be chosen here. Alternatively, Zapalla et al. used a  metrics where weighting was such that the net distance in inclination gets 4x more important, relatively to the semimajor axis. Such metrics should tend to create families more `compressed' in inclinations.

The main arbitrarity of HCM is in the choice of the cut-off. Zappala et al. defines so-called Quasi Random Level  (QRL), which is, if I understood well, ideally a typical minimum distance of background objects at the family place (the idea is that at this about level, the family separates from the background).  Practically, however, it is hard to determine what is QRL because one cannot distinguish among family members and background objects. For this reason, the practical calculation of the QRL is involved and the result not unique. The asteroid members are then usually determined with cut-off values similar to the QRL.

Our view of this is that there is almost no preference between different cut-off (within reasonable limits which usually accounts for an interval of a few 10 m/s), and that this choice mostly depends on your interpretation (on what you want to get from the data). We would oppose this subjectivity to the quest of an objective cut-off, which interests so much the family people. Anyway, we do not see the exact reason why we should compare the result of our family runs with the "established" families (where the cut-offs were fixed). May be that at the end there will not be much difference, but still, it is an interesting exercise to see how rigid are family sizes, shapes, size-distributions, etc. depending on the choice of the metrics and cut-off.

Here we go with the figures. Basically, we have run the HCM starting from the largest family member, determining all members at a cut-off, where cut-off ranges from 10 m/s to 150 m/s. This range is intentionally exaggerated (for example, Fernando found that the QRL of the Themis family with the current dataset is some 70-80 m/s) but shows the whole range of HCM efforts. It may be later interesting to see some families at special cut-offs. If you  see an interesting case, please tell me, the program runs a couple of seconds. Alternatively, you can download the  program  which identifies the family at one cut-off (here are the datafiles:  numb.ana  and  mult.ana ). This program can be used also for our family runs, identifing the simulated families on different backgrounds, and thus remove the subjective judging whether an object significantly drifting by Yarkovsky and/or by resonances would or would NOT be consider a family member. We have done some experiences with this with the Flora family. Basically, it removed all Mars-crossers (as anticipated) and shifted the family center to larger semimajor axis which leads to a better fit.

The effect of some resonances can clearly be seen in figures. Get you own impression and please tell me whatever interesting feature you noticed. May be that at some point we could debate issues like: At what asteroid's sizes the families dominate over the background  population? If there are some traces of the ejection velocities seen on largest family members (diagonality in (a,e) if the true anomaly=0 or180 deg???)? What else?

The figures are animations of the 10-150 m/s range and static plots at selected cut-offs.  We have tried to select the later at cut-offs, where interesting things happen).The size of animations is ~5Mb, which we hope, will be ok for viewing from Nice/Prague (if not we'll put them in an ftp directory). The static figures are much smaller in size.

One thing I regret  to realize late is that the size of points is a bit too large so that some figures become saturated. Starting with Maria, I used a smaller size of points.

Outer zone (2.8-3.3 AU)

Intermediate zone (2.5-2.8 AU)

Inner zone (2.1-2.5 AU)

... to be done ...

Last revised: July 14, 2001

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