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Figure 3.—Ratio of the three pressure components over density along a radius vector in the symmetry plane of the stream. (a) Case I. (b) Case II. (c) Case III. (d) Case IV. The same arbitrary units are used for all cases.
accordance with qualitative arguments. The degree of inelasticity, however, will have a varying degree of importance depending on details of the distribution function. For certain situations this parameter will determine whether the stream will expand or contract. Case II is such an example.
Figure 4.—Collision-induced mass flux at two points in a cross section of the stream for the four different cases studied. The broken line indicates the symmetry plane; the x's indicate the density maximum in the stream. The flux vectors are drawn for the three values 1.0, 1.5, and 2.0 of the restitution parameter 3. The vectors belonging to the two extreme cases are indicated by 1 and 2, respectively.
To estimate the time interval for which this type of analysis is valid, the ratio of the average mean free time between collisions to the average orbital period for the particles in the stream must be evaluated. This ratio is not a simple relation between particle diameter, mass, and volume of the stream, for instance. To a large degree it depends on the details of the distribution function. It is clear that this ratio increases rapidly as the particle diameter increases if the mass in the stream and the form of the distribution function are kept constant. According to the arguments presented above, this ratio will also increase with time, reducing the importance of collisions.
Results from an initial rate of change study should be interpreted with care because transient effects from a specific choice of initial state could easily mask important properties of the system. For our case, the above pattern was repeated for all distribution functions studied. Numerous questions, however, were not treated. Can instabilities develop in the stream? Is there a preferred profile the stream will try to reach? To what degree will the final state depend on initial state and on the degree of inelasticity? Answers to several of these and similar questions will hopefully be obtained from numerical simulations of jetstreams in the near future.
Alfvén, H., and Arrhenius, G. 1970, Origin and Evolution of the Solar System, I. Astrophys. Space Sci. 8, 338.
Chapman, S., and Cowling, T. G. 1960, The Mathematical Theory of Non-Uniform Gases. Cambridge Univ. Press. London.
Danby, J. M. A. 1962, Fundamentals of Celestial Mechanics. Macmillan Co. New York.
A STUDY OF ASTEROID FAMILIES AND STREAMS
B. A. L/WDBLAD
A study of asteroid orbits is made to determine if there exist groupings of similar orbits in the asteroid population. A computer program based on Southworth's D criterion for similarity in meteor orbits is used. The program successfully sorted out the asteroid families listed by Hirayama, Brouwer, and van Houten et al. A number of new families were detected, several of which appear to be more significant than the minor Brouwer families. Asteroidal streams (jetstreams) are studied and a list of such streams is presented.
It is well known that the distribution of orbital elements among the minor planets is nonrandom. The frequency distribution of the semimajor axis a exhibits gaps corresponding to commensurabilities with Jupiter. Hirayama (1918, 1928) has shown the existence of families; i.e., groups of asteroids with almost equal values of the orbital elements a, e, and i. Brouwer (1951), who restudied this problem using the proper elements, has added several new families. Arnold (1969) introduced computer methods in the classification of families and found additional families among the numbered asteroids. Alfvén (1969, 1970) has drawn attention to the fact that within the Flora family there exist groups of orbits that exhibit similarity also in the orbital elements co and Sl.
The orbital elements of about 1700 numbered minor planets are published in the standard asteroid Ephemeris (1970). Orbital elements of some further 2000 minor planets from the Palomar-Leiden survey (PLS) have recently been reported by van Houten et al. (1970). In the PLS material, van Houten et al. have verified a number of the Hirayama and Brouwer families. Four new families were discovered.
The PLS considerably increased the number of orbits available for study, and a comprehensive search in the data will probably reveal additional families and streams. The purpose of the present paper is to make such a search with the use of computer techniques.
STREAM DETECTION PROGRAM D Criterion
The problem of classification based on orbital similarity is well known in meteor astronomy, where the study of meteor streams has necessitated the use of sophisticated computer techniques for the detection and classification of streams. The basis for our stream detection program is Southworth and Hawkins' (1963) criteria for orbital similarity, which for low-inclination orbits may be written
where M and N represent two orbits to be compared and a, e, i, o, and Q are the customary notations for the orbital elements. The stream detection program computes D(M, N) for all possible pairs in the sample under study. If D(M, N) is below a certain stipulated value D, the program considers these two orbits as forming a stream. In the continued comparison process, more and more orbits are grouped into the stream. The program finally lists the meteor streams and their members, their mean orbit, and the deviation of each stream member from the mean stream orbit. An extensive survey of photographic meteor orbits using this stream detection program has been made by Lindblad (1971). The Southworth D criterion is an objective method of classification on the basis of the orbital elements; i.e., it selects concentrations in five-dimensional (q, e, i, co, Q) space. The reason for using the perihelion distance q instead of the semimajor axis a is that the perihelion distance q for meteor orbits is better defined than a. In adapting the method to asteroid orbits, we did not consider it necessary to modify the original program. The main problem encountered in our study was how to determine the appropriate rejection level D.
Data Sample and Data Preparation
Present and proper orbital elements for 1697 numbered asteroids and for 1232 PLS asteroids were available on cards. The 1697 numbered asteroids have well-defined orbits, and the entire data sample was used in our study. Of the PLS orbits, 28 were excluded because they are already included in the 1697 numbered asteroid sample. In the PLS, the investigators assigned each individual orbit a quality class. The 977 orbits of highest quality (type 1) were used by van Houten et al. in their study of asteroid families. The same data