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COMET GROUPS

Another observed phenomenon might be included in a survey of jetstreams; namely, what is often called comet groups. From a statistical point of view these are probably insignificant because very few members (2 to 4) are included in each group (except for the Sun-grazing group). The only reason for mentioning them here is that if comets are considered to accrete from jetstreams (meteor streams) one could as easily imagine a stream developing several condensations.

STATISTICAL REMARKS

An important problem as far as the statistics are concerned is to decide whether “observed streams” are real or not. Hence, we want to estimate the probability (risk) that a certain property of the observed distribution is a result of a Poisson process. This probability is the level of significance of our conclusions concerning, for example, jetstreams. The problem thus formulated is a very difficult one (see the appendix for a simple example), which has never been solved in an analytic way (with the exception of the example given in the appendix). For general references on this type of problem, see Kendall and Moran (1963, chs. 2 and 5) and Roach (1968, ch. 4). Analytical methods described in the first of these works could possibly be employed, but this would be quite difficult and it is not at all certain that the result would be useful. A remaining possibility is to test synthetic distributions for the property under consideration (Roach, 1968; Danielsson, 1969). This test has to be done, of course, on a substantial number of synthetic distributions because the significance of such a test only can be determined from the distribution of the studied property among these synthetic distributions. In the present case, even making synthetic distributions is a complicated task.

Thanks to the Palomar-Leiden survey (PLS) (van Houten et al., 1970), which represents an additional, independent sample of asteroids, we can get an indication concerning the reality of our jetstreams if we find them here also. The value of this test is limited because of the observational selection of the PLS; essentially the test has to be confined to streams of low inclination. Nine hundred and thirty-one well-determined orbits (class I) have been investigated. The three Flora streams do appear also in the PLS material; however, these clusters of orbits are much less noticeable here. Within a distance D = 0.10 AU of the mean orbits of Flora A, B, and C, there are four, five, and three objects in the new material. At the same time, the density in this region of the five-dimensional space is twice as large in the new material as in the old. (This fact is found by experiment.) Because the mean orbits of Flora A, B, and C can be regarded as random points in relation to the PLS sample, one would expect them to have two (experimentally found average) neighbors within 0.10 AU if the distributions were random. It is obvious that the significance of each individual stream tested in this way is not overwhelming. If the streams are tested together, one finds that the risk that they all are a result of a Poisson process is about 1 percent. are completely irrelevant in our case. (n and k are the uniform average and actually observed number of members in the cluster.) As earlier pointed out (see above and Danielsson, 1969) the problem of finding an analytical expression for the probability of coming across a certain cluster in a random distribution is in reality a very difficult one. It seems to have been solved only for a very special one-dimensional case (Ajne, 1968). The formulation of the problem should be as follows: Given a random distribution with n members, what is the probability of observing a cluster of k members in some volume of suitably chosen size and location (k being considerably larger than the uniform average)? The problem will be illustrated by two examples:

CONCLUSIONS

By means of the new definition of an average distance between celestial orbits (eq. (1)) asteroid streams can be defined. So far only the three streams in the Flora family, Flora A, B, and C (Alfvén, 1969), have been studied (and redefined) by this method. It is found that the orbits of the members in these streams are well collimated everywhere along their path in contrast to previously defined streams. Furthermore, two of the streams show marked focusing regions where a majority of the orbits come very close together and where the relative velocities are an order of magnitude smaller than between randomly coinciding asteroid orbits.

From the point of view of jetstream physics, the best definition of a jetstream might be connected more closely with regions where the density of orbits is high and at the same time the relative velocity is low. This argument is not quite in line with the one leading to the distance formula used here. Maybe a weight function, giving more weight to those parts of two orbits where the distance is smallest, should be included in the integration leading to equation (1). In view of this argument, the classical way to determine a meteor stream would be quite good. According to this, a meteor stream is defined by the geocentric quantities of radiant, velocity, and date.

The statistical significance of the studied streams, admittedly, is shown far from satisfactorily. More work is required on this problem.

APPENDIX—PROPERTIES OF A POISSON PROCESS

The need to estimate the probability that a certain property of an observed distribution can be expected to appear in one realization of a Poisson process arises frequently in works of the present type. Because this is a very difficult task and misconceptions concerning the fundamental character of this problem are not rare in the literature of nonspecialized disciplines, this comment is considered worthwhile.

Any of the above discussed methods for finding clusters of similar orbits among the asteroids can serve as an example. In some way, the number of neighbors to an orbit (a point in a five-dimensional space) is determined; and if this number is “large,” an orbit cluster is considered located. By “large” number is meant that the probability of finding the same cluster in a random distribution should be small. However, one has to be very careful as to what can be expected in a random (Poisson) distribution. It gives an entirely false result to regard an observation of a certain large cluster of this kind as a random observation. Thus probabilities according to the formula

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(1) Let five points be randomly distributed on the perimeter of a circle. The probability that all of them occur on one side of a given (in advance) diameter is of course 2-3 = 0.031. The probability that all of them can be located on one side of a suitably chosen diameter can be calculated according to a formula deduced by Ajne from straightforward combinatorial analysis: for 2k-n > 0,

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With k = n = 5, then P(X = 5) = 5 X 2-4 = 0.31; i.e., 10 times more likely than in the first case. (2) Consider the alleged asteroidal cluster Flora B as studied by Danielsson (1969). In a two-dimensional area, where only one point would be found on an average, seven were observed. If the area had been randomly located, the probability for this occurrence in a Poisson distribution would be (e 7!)-1 = 7 - 10-3.

To estimate the actual probability under the proper formulation of the problem, 100 synthetic random distributions were made to simulate the observed population. Seven points were observed in the given area, suitably located, 26 times. Thus the probability was estimated to be 0.26. More than seven points were observed three times so that the probability of finding seven or more points was 0.29.

It is clear that formula (A-1) can be wrong by very many orders of magnitude when the number of points is large. For example, the probability 10-199 mentioned by Arnold (1969, p. 1236) may very well be wrong by a factor of 1090 or more.

ACKNOWLEDGMENTS

The author is indebted to Prof. Hannes Alfvén for initiating this project and to Lynne Love for help with the computer work.

This work has been supported by grants from NASA (NASA NGR-05-009-110) and The Swedish Council for Atomic Research (AFR 14-89).

REFERENCES

Ajne, B. 1968, A Simple Test for Uniformity of a Circular Distribution. Biometrika 55,
343.
Alfvén, H. 1969, Asteroidal Jet Streams. Astrophys. Space Sci. 4, 84.
Alfvén, H., and Arrhenius, G. 1970, Origin and Evolution of the Solar System, I.
Astrophys. Space Sci. 8, 338.
Arnold, J. 1969, Asteroid Families and Jet Streams. Astron. J. 74, 1235.
Danielsson, L. 1969, Statistical Arguments for Asteroidal Jet Streams. Astrophys. Space
Sci. 5, 53.
Houten, C. J. van, Houten-Groeneveld, I. van, Herget, P., and Gehrels, T. 1970,
Palomar-Leiden Survey of Faint Minor Planets. Astron. Astrophys. Suppl. 2, 339-448.
Kendall, M. G., and Moran, P. A. P. 1963, Geometrical Probability. Griffin & Co. London.
Roach, S.A. 1968, The Theory of Random Clumping. Methuen & Co. London.
Southworth, R., and Hawkins, G. 1963, Statistics of Meteor Streams. Smithson. Contrib.
Astrophys. 7, 261-285.

DISCUSSION

WILLIAMS: Were observational selection effects considered in judging the significance of jetstreams?

DANIELSSON: The problem of observational selection certainly needs to be investigated very carefully to determine whether the asteroid streams are real or not. One can probably assume that asteroids of absolute magnitude (visual) g × 12 are unbiased with respect to observational selection. In a paper examining the Flora family (Danielsson, 1969) I have shown that if the asteroids with g » 12 are excluded, one of the streams (Flora C) remains statistically significant. Selecting the largest asteroids of the family in this way, of course, meant a substantial reduction of the number of members.

UREY: Are the jetstream particles the result of a collision in which the components that were produced remained in neighboring orbits?

DANIELSSON: The appearance of focusing points could possibly be the result of a collision, but this must then have been a very recent (104 to 105 yr) event because the phases of these orbits are very quickly spread out.

UREY: Do the geometrical properties you describe support a model based on fragmentation?

DANIELSSON: The geometrical properties that I have described do not tell you anything directly about accretion or fragmentation. However, as far as I can see, the well-collimated streams with focusing regions would have a very short lifetime unless there were some viscous force in the stream producing and maintaining these properties. Thus, if these geometric characteristics are found to be common for most of the streams, it would indicate the existence of such a force. This in turn would probably favor an accretion model.

DISCUSSION REFERENCE

Danielsson, L. 1969, Statistical Arguments for Asteroidal Jet Streams. Astrophys. Space Sci. 5, 53.

SPATIAL DISTRIBUTION OF INTERPLANETARY DUST

ROBERT G. ROOSEW
WASA Goddard Space Flight Center

Interplanetary dust can be defined as solid particles outside Earth's atmosphere in the size range larger than a molecule and smaller than an asteroid. It is studied by a number of quite different techniques. For Earth-based observers, these techniques include measurement of the brightness and polarization of the interplanetary light,' optical radar studies of particles entering the upper atmosphere, photographic and radar meteor observations, study of meteorites, and various methods of collecting dust particles in the atmosphere, in ice cores, and in deep sea sediments. - de from spacecraft include some interplanetary light observations and measurements of individual particles by means of microphones, penetration sensors, and collection experiments. These observational techniques are described by Millman (1969) and Bandermann (1969).

EARTH-ASSOCIATED DUST?

At the beginning of the last decade it was generally considered probable—if not certain—that interplanetary dust was concentrated at a number of preferred locations in the near-Earth environment. In particular, Whipple (1961) reported evidence for a high concentration of dust near Earth with a maximum concentration with respect to the average interplanetary medium perhaps as high as 10° (the so-called geocentric dust cloud (GDC)). Kordylewski (1961) reported that he had observed concentrations of dust (the so-called libration clouds) associated with the quasi-stable triangular EarthMoon libration points L4 and L5 (fig. 1). He further stated, “The surface intensity of the libration clouds is a little less in their opposition than that of the Gegenschein” [counterglow].” Also, there was a widespread belief that the

*Interplanetary light” has been suggested by Roosen (1971a) as a general term to describe all light scattered (or emitted) by interplanetary material. It includes the zodiacal light, which by definition is concentrated toward the plane of the ecliptic, the counterglow, which is a weak brightening in the antisolar direction, and also the light known to come from high ecliptic latitudes, up to and including the ecliptic poles.

*Editorial note: The responsibility for replacing “Gegenschein” with,"counterglow” is entirely mine; I thank Dr. Roosen for accepting this change, which he did reluctantly and only because it had already been made when he received galley proofs.-T. Gehrels.

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