The improved proper elements of 4100 numbered asteroids have been searched for clusterings in a, e, i space using a computer technique based on the D-criterion. A list of 14 dynamical families each with more than 15 members is presented. Quantitative measurements of the density and dimensions in phase space of each family are presented

Lindblad, Bertil A.

English

NASA Technical Reports Server

co.e..Me,eo.=,.Z.pp 366 -- J O
Lunar and Planetary Institute, Houston, 1992 363
. t
N93-  fl'3
A COMPUTER SEARCH FOR ASTEROID F_ _"- ;
B.A. Lindblad, Lund Observatory, Box 43, S-221 00 Lund, Sweden
Abstract - The improved proper elements of 4100 numbered asteroids have been searched
for clusterings in a, e, i space using a computer technique based on the D-criterion. A list of
14 dynamical families each with more than 15 members is presented. Quantitative
measurements of the density and dimensions in phase space of each family are presented.
Introduction - Large scale clustering in the proper elements of the minor planets was first
noticed by Hirayama (1918), who coined the term 'families" to indicate such clusterings.
Hirayama (1928, 1933) added additional members to the previous families. Extensive lists of
asteroid families have subsequently been compiled by Brouwer (1951), Arnold (1969), van
Houten et al. (1970), Lindblad and Southworth (1971), Williams (1979), Kozai (1979a, 1983)
and others. Unfortunately the criterion for family membership is not always stated, and it
appears that the limits of a family in proper element space are largely a matter of personal
judgement. A strict criterion of family membership based on the D-criterion of Southworth
and Hawkins (1963) was introduced by Lindblad and Southworth (1971). The D-criterion has
been used in numerous meteor stream studies and subsequently also for studying clustering
amongst asteroid and comet orbits (Lindblad and Southworth 1971, Kres_ik 1982, Lindblad
1985). The D-criterion for orbital similarity in proper a, e, i space may be written
D(m, n) z = (e_-e=) 2 + (q.-qm) 2 + (2 • sin _ )22 (1)
where m and n represent two orbits to be compared. The search program computes D (m, n)
for all possible pairs in the sample. If D(m, n) is less than a stipulated value D_ the program
accepts these two orbits as belonging to a cluster. During the continued comparison process
more and more pairs are linked together to form a cluster. The program numbers the various
groups and lists their members, computes the mean orbit M of the group and also the
deviation D(M, n) of each individual member from the mean orbit. Finally it computes the
mean D(M, n) for all members in a group.
In a search in three dimensional space the discriminant D, should vary inversely as the cubic
root of the sample size, if the samples are otherwise similar. Lindblad and Southworth (1971)
noted that the best agreement with the classifications of Hirayama and Brouwer was obtained
using D, = 0.020 in a sample of 1697 numbered asteroids and D, = 0.013 in a mixed sample
of 2652 numbered asteroid and PLS orbits. Based on these results we may write
D, = 0.24 • N -1/3
Dm = 0.18. N -1/3
(2a)
(2b)
where equation (2a) is appropriate for asteroids with registration numbers less than about
2000 and equation (2b) for samples exhibiting a higher degree of concentration to the ecliptic
plane.
Improved proper dements - Early computations of proper elements involved only low-order
expansions in the eccentricities and inclinations. Subsequently Williams (1979), Yuaza (1973),
Kozai (1979b), Kn_zevi6 (1986) and Kn_zevi6 and Milani (1989) have improved the theory of
computing proper elements by developing methods to accurately handle high inclinations
and/or large excentricities. A set of proper elements for 4100 minor planets computed by
Kn_evi6 and Milani, forms the data base for the present study.
https://ntrs.nasa.gov/search.jsp?R=19930010010 2020-03-17T07:29:27+00:00Z
364 Asteroids, Comets, Meteors 1991
Results of computer search - In the Kn_evi6-Milani sample of 4100 orbits the asteroid
population with serial numbers above 2000 showed a much stronger concentration to the
ecliptic plane than the first 2000 numbered asteroids. It follows that a family search in this
sample must be made at a rather strict rejection level. Eq. (2b) suggests a Ds value in the
range 0.011 - 0.012. In order to take a conservative approach the search was made at Ds =
0.011. The search produced a list of 316 "families" totaling 2163 members, i.e. slightly more
than 50 % of the asteroid population was grouped into families. However, 211 families had
only 2 members, 54 had 3, 18 had 4, 6 had 5 members each, etc. Pending a detailed study of
the possible significance of small families the present report=focuses on the fourteen largest
families only. A more detailed study will be published elsewh-_re.
Table 1 lists all asteroid families with 15 or more members together with their mean proper
elements and mean D(M, n) value. Families are named after the asteroid of lowest serial
number. In a few cases a double name has been introduced so as to facilitate a comparison
with the results of other workers. All the Hirayama families, except Phocaea, are detected. In
addition nine "new" families with from 15 to 100 members appear. _e _three families:
Eunomia, Nysa and Vesta have been previously mentioned by other investigators. The six
new families in Table 1 represent small, but very compact groups.
=_nsion and &ns_d i_:in phase-_:]:As_a Check on ihe :c/Snsistency of
the data we have for each family in Table 1 computed the product of Lhe three standard
deviations o(a), o(e) and o(i). This product is a measure of the extent of each family in phase
space. In order to obtain suitable numerical values we define
cr = o(a) • o(e). o(i). 10s (3)
The product o is listed in column seven of Table 1. In Fig. 1 we have plotted log o versus the
equivalent radius D(M, n). A high correlation between D(M, n) and o is evident, (correlation
coefficient r = 0.89), i.e. both parameters are measures of the extent of an asteroid family in
phase space. In column eight of Table 1 we list the number of family members per unit
volume of phase space. Table 1 shows that Koronis, Dora and Adeona are extremely dense
groups, whereas the other eleven families all have about the same density.
Statistical significance of asteroid families - The statistical significance of the families listed in
asteroid studies is a matter of some concern. In most studies dynamical families are
identified as groupings in proper a, e, i space and the boundaries are drawn rather
subjectively. Since the agreement between the various lists of families is only partial, one is
inclined to doubt the correctness of the used statistical significant tests. A different approach
to the statistical significance problem was used by Lindblad and Southworth (1971). These
authors constructed random samples of asteroid orbits and searched them at the same
rejection level d, as was used in the real sample. The random samples were created by
scrambling the proper elements a and e of the asteroids under study. This procedure
preserves the frequency distribution of each orbital element, but does not preserve any
correlations which may exist between the proper elements a and e. Fortunately, this
correlation is very low. In the present study a number of random samples are being searched
at the rejection level D, = 0.011. When these searches are completed one can directly derive
g
the probability that a family of N members is due to a chance grouping in proper a, e, i
space. In the previous study (Lindblad and Southworth 1971) it was found that in a sample of
a_u_=!70Q orbjt_bou_ 50% of the two-and three-member families and 30% of the
four-member families were random groupings in th_ d_ta,
Acknowledgements - This investigation was supported by grant no. 204-301 from the Swedish
Natural Science Research CounciL Travel grants were received from Kgl. Fysiografiska
S/illskapet in Lund and from the Swedish B0ard for Space Activities. The sample of 4100
proper elements was made available through the courtesy of Drs. Z. Kn_zevi_ and A. Milani.
Asteroids, Comets, Meteors 1991 365
Family name N a e i D(M, n) o N]o Comments
Themis 212 3.144 0.156 1.4 0.027 10.9 19.5
Eos 215 3.014 0.077 10.1 0.024 27.0 8.0
Koronis 141 2.874 0.049 2.1 0.021 2.08 67.8 Compact
Maria 36 2.555 0.091 15.0 0.021 2.85 12.6 Compact
Flora 440 2.231 0.137 4.5 0.036 62.5 7.0 Loose assoc.
Eunomia 100 2.626 0.149 13.2 0.031 10.8 9.3 Brouwer 19-20
Nysa-Hertha 56 2.392 0.163 3.0 0.024 17.7 3.2
Vesta 41 2.354 0.103 6.5 0.019 11.5 3.6
Amalasuntha 39 2.446 0.153 2.9 0.017 5.04 7.7 Rel. to Flora
Mildred-Beer 17 2.385 0.190 2.2 0.015 1.37 12.4 Rel. to Flora
Oppavia-Gefion 17 2.789 0.136 9.1 0.013 1.68 10.1
Dora 16 2.787 0.196 7.8 0.007 0.19 83.3 Compact
Adeona 16 2.674 0.167 11.6 0.007 0.24 66.7 Compact
Goberta 15 3.174 0.135 1.1 0.010 1.45 10.3
Table 1. Mean proper elements, mean discriminant D(M,n), volume cr and spatial density
N/a of large families detected in search amongst 4100 asteroids at rejection level D, -- 0.011.
I.,q0
er
I.UO
O
Fig. 1. Log a versus t_
equivalent radius
D(M, n). _ u,
_m
w
• _M, t)
| t | t
UIO _ Ua0 0.040
MEAN VALUE OF DISC_IMIN.d_T D(M. n)
366 Asteroids, Comets, Meteors 1991
References
Arnold, J., 1969, Asteroid Families and Jet Streams. Astron. J. 74, 1235-1242.
Brouwer, D., 1951, Secular Variations of the Orbital Elements of Minor Planets. Astron. J.
56, 9-32.
Hirayama, K., 1918, Groups of Asteroids Probably of Common Origin. Astron. J. 31,
185-188.
Hirayama, K., 1928, Families of Asteroids, Jap. J. Astron. Geophys. 5, 137-162.
Hirayama, K., 1933, Present State of the Families of Asteroids, Proc. Imp. Acad. Japan, 9,
482-485.
van Houten, C., van Houten-Groeneveld, I., Herget, P. and Gehrels, T., 1970, Astron.
Astrophys. Suppl. 2, 339-448. _: --- _ _ _ _ _:_ _ :_ _ ......
Kn_zevi6, Z., 1986, Comparison of the Asteroid Proper__Elements Obtained from Various
Theories. In Asteroids Comets Meteors II (C.-I. Lagerkvist, B.A. Lindbladl H. Lundstedt
and H. Rickman, Eds.), 129-134. Uppsala Universitet, Uppsala.
Kn_zevid, Z. and Milani, A., 1989, Asteroid Proper Elements from an Analytical Second
Order Theory, in Asteroids II (Binzel, R., T. Gehrels and Mathews, M., Eds.), i073-1089,
Univ of Arizona Press, Tucson. :......
Kozai, Y., 1979a, Families of Asteroids by a New Criterion. Ann. Tokyo Astron. Obs. Second
Ser. 17, 194-220.
Kozai, Y., 1979b, The Dynamical Evolution of the Hirayama Family, in Asteroids
(T. Gehrels, Ed., 334-358, Univ. of Arizona Press, Tucson. _ _ _
Kozai, Y., 1983, Families of Asteroids. In Dynamical Trapping and Evolution in the Solar
System (V.V. MarkeUos and Y. Kozai, Eds.), 117-122, Reidel, Dordrecht (LAU Coll. 74).
Kres_ik, L., 1982, Bull. Astron. Inst. Czechosl., 33, 150.
Lindblad, B.A., 1985, DO Comet Groups exist? In A. Carusi and G. Valsecchi (eds),
Dynamics of Comets. Their Origin and Evolution, D. Reidel, Dordrecht.
Lindblad, B.A. and Southworth, R, 1971, A Study of Asteroid Families and Streams by
Computer Techniques, in Physical Studies of Minor Planets (T. Gehrels, ed.), NASA
SP-267.
Southworth, R. and Hawkins, G., 1963, ;tatistics of Meteor Streams. Smithson. Contrib.
., AStrop_ys. 7, 261-285.
Williams, J.G., 1979, Proper Elements and Family Memberships of the Asteroids. In
Asteroids (T. Gehrels, Ed.), pp. 1040-1063. University of Arizona Press, Tucson.
Yuasa, M., 1973, Theory of Secular Pertubations of Asteroids, including terms of higher
orders and higher degrees, Publ. Astron. Soc. Japan, 25, 399-445.
=


A computer search for asteroid families

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server