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The numerical value of D, is known to be a function of sample size and the precision of the orbital elements. In a family search, D, should vary inversely as the cube root of the sample size, if the samples are otherwise similar. If D, = 0.020 is accepted as the rejection level in the numbered asteroid population, a rejection level of
would have been appropriate for the somewhat smaller PLS sample. However, the stronger concentration to the ecliptic plane suggests a lower D, value because the density of observed orbits corresponds to a larger sample. Because of this truncation effect, we are inclined to use about the same D, value as in the numbered asteroid sample. In order not to prejudice the choice, computer searches were made at four rejection levels, D, = 0.017, 0.018, 0.019, and 0.020. Our study showed that at the stricter rejection levels the recognized families often were split into two or three subgroups. The searches at D, = 0.019 and D, = 0.020 provided reasonable agreement with the classification of van Houten et al. The finally adopted rejection level was D, = 0.019. This level is, perhaps, slightly on the conservative side.
Previously Known Families
Table II compares the classification of van Houten et al. with that of our search at D, = 0.019. Table headings are similar to those of table I. It is seen that 214 out of 386 orbits are classified in the same manner, whereas 110 orbits are assigned to satellite families or to other groups. Inspection of families 1 through 3 suggests that the classification boundaries used by van Houten et al. are slightly wider than those of Brouwer.
The newly introduced families 30, 33, and 34 were readily identified by our search, although about half of their members were assigned to smaller satellite families. Families 4 and 12, with very few members in the PLS sample, were not found in our study at D, = 0.019. Nor were they found in the search at D. = 0.020. Family 33 was split into several groups, two of which were related to Brouwer 19 and 20 and a third which appeared to be related to the lo family introduced by us (table III). These results again indicate that the limits of the van Houten families in general are wider than those of the Brouwer study.
In the classification of the PLS orbits, the newly found Michela-Nysa group (van Houten families 31 and 32) presented some problems. The Michela and Nysa families represent two fairly close groupings, which our search could not disentangle. The division line between the two families appears to be one of inclination much in the same way as in the Flora group. Our investigation TABLE II.-Classification of Family Members in the PLS Data
van Houten Number of Asteroids Total Number of
1 . . . . . . . . . . . 40 16 56 l . . . . . . . . . . . . . . . . . . . . . . . . . 63
produced more members in the Michela-Nysa group than found by van Houten et al. The search at D, = 0.019 listed 181 members in the combined Michela-Nysa group. Of these, 97 were included among the 103 members ascribed to the Michela-Nysa families by van Houten et al. The properties of the Michela-Nysa families deserve further detailed study.
The search at D, = 0.019 produced a number of previously unknown families or groups. These have not yet been fully investigated and therefore are not detailed here. The more important families were detected in a subsequent search in the total asteroid sample (table III).
FAMILY SEARCHES IN TOTAL SAMPLE
The number of orbits available for study in the combined numbered/PLS asteroid sample was 2674. Of the PLS orbits, 22 of type 1 were excluded because they are already included in the 1697 numbered asteroid sample. The total number of orbits used in the study was 2652. As previously, proper elements were used.
The appropriate rejection level to use in the family search was estimated as follows. From the relations
a rejection level slightly smaller than 0.014 or 0.017 may be estimated. Searches were made at four different levels, D, = 0.011, 0.012, 0.013, and 0.014. The investigations at D, = 0.011 and 0.012 separated and identified the low-inclination families but gave a far too severe rejection level for the moderate- and high-inclination groups. Family 2 Eos was split into two groups. The search at D, = 0.013 gave adequate separation of the Brouwer families and this rejection level therefore was adopted. The adopted D, value again was chosen conservatively.
Previously Known Families
The results of the search at D, = 0.013 were compared with the classifications of Brouwer and van Houten. Families identified were 1 through 9, 16, 18 through 22, 25, 27, and 30 through 34. Family 24 was incorporated into the Nysa group. Families 11 through 15, 17, 23, and 26 were not detected in the total sample; i.e., at the chosen, stricter, rejection level their members were classified as nonfamily objects.
TABLE III.-New Families
No. Family Number of Mean” Members Remarks”
1499, 2802, 6523
separate from wh93
*Computed from orbital elements a, e', and i'. bA, L, and whistand for Arnold, Lindblad, and van Houten, respectively. New Families
Table III lists those new families that had six or more members detected by the search. The first and second columns give the numbers and names suggested for these families, the third and fourth columns give the number of members and the mean value of D(M, N) at the rejection level D. = 0.013. The individual members are given in column 5. Numbers above 2000 refer to the PLS asteroids. Parentheses indicate additional members obtained at the rejection level D. = 0.014. Column 6 lists the family number given by Arnold (1969) to these groupings.
The new families have been named after bright asteroids occurring within their boundaries, and have been numbered 35 through 44 in continuation of the numbering of van Houten et al. It is seen that the new families have members in both data samples studied by us. Further, it should be mentioned that the new families (table III) were also detected in the separate searches in these two samples.
The mean D(M, N) listed in table III is a measure of the concentration within an asteroid family, a low D(M, N) value implying a high degree of concentration. For most families, D(M, N) is of the order of 0.010. This value is lower than that found in our study for the majority of the Brouwer families.
In addition to the families reported in table III, a large number of minor groups were detected by the search. It is believed that a number of the new four- and five-member families are significant, but there is an increasing probability as we proceed to small-sized groups that the associations are due to chance. Subsequent studies may very well give reasons to include some of the smaller groups in an extended listing of families. It is also possible that a future study will revise upward the rejection level D, thus allowing more members in the families reported in table III.
An asteroidal stream (jetstream) is defined as an assembly of orbits showing similarity in all five orbital elements a, e, i, co, and Q. For the purpose of the study of streams, the present-day elements will be used. From a geometrical point of view, the asteroidal streams are analogous to the meteor streams. Southworth and Hawkins' D criterion can thus be used without modification to search for similar orbits once the rejection level D, is determined.
Very little information is at present available as to the size and number of streams in the asteroidal population. Alfvén (1969) found three separate streams, denoted A, B, and C, among the members of the Flora family. The statistical significance of these groupings has been discussed by Danielsson (1969). Some additional streams have been listed by Arnold (1969).
Lacking more detailed information, the investigator is faced with the problem of setting the rejection level D, more or less arbitrarily. Test runs in the numbered asteroid population indicated that D, values in the range 0.050