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TABLE D-III.-Comparison of MDS Extrapolations

and PLS values P0 Equation (6) Equation (7) PLS 17. . . . . . . . . - 1990 3700 3240 18. . . . . . . . . . 3240 8300 11500 19. . . . . . . . . . 4940 18600 28.200

NOTE.-The mean photographic opposition magnitude is defined in the MDS.

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7 8 9 10 11 12 13 1. 19

Figure D-1.-MDS frequency distribution of absolute magnitudes g for three distance zones and their sum. This was originally figure 5 of the MDS; revised by I. van Houten-Groeneveld.

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asteroids at a given opposition fall outside the MDS region. Therefore the comparison between MDS and PLS indicates that the correction factor for the inclination cutoff, integrated over the three distance groups, should be 1.10 times 1.90, or 2.09.

Using the approximation of circular orbits, the correction factors for the inclination cutoff for the three distance intervals separately were found to be 1.94, 2.42, and 2.45, respectively. The numbers of objects in the three distance groups are 52, 29, and 19 percent of the total (first-class orbits used only). This results in an integrated correction factor for the inclination cutoff of 2.18. The two numbers differ by only 4 percent. This shows that the correction factors for the inclination cutoff, as used in the PLS, are completely satisfactory.

The correction factor used to extend the PLS field to the whole sky depends on the size of the PLS field, which is accurately known. This correction factor cannot give rise to any inaccuracy.

In short, I do not see any reason to suppose that systematic errors are present in the PLS results. The accuracy of the PLS material is probably better than the MDS because the photometric material was larger, every asteroid being measured about six times, and because accurate reductions to absolute magnitude were available. Therefore I do not share Dohnanyi's reluctance to combine the two surveys. According to me, such a combination is completely justified.


Nairn, F. 1966, Spatial Distribution and Motion of the Known Asteroids. J. Spacecr.
Rockets 3, 1438–1440.
Watson, F. G. 1956, Between the Planets. Harvard Univ. Press. Cambridge.

*For additional information on the PLS, the reader is referred to van Houten's paper, p. 183.


University of Hawaii

This paper is concerned with some aspects of determining the evolution of the size distribution of a finite number of mutually colliding and fragmenting particles such as the asteroids or interplanetary dust. If n(m, t) is the number of particles per unit volume per mass interval at time t, then n = dn/dt is the rate at which that number changes with time. This rate can be calculated if the laws are known according to which the colliding bodies erode one another and fragment and if the influence of collisions on the motion of the particles is known. To reduce the complexity of the problem, one assumes that the speed of approach between the bodies is always the same veoli and that they, as well as the debris, occupy a fixed volume (“particles in a box”). Only collisions between two bodies are considered, and the way in which erosion and fragmentation occurs at a given value of veoll depends only on their masses. The particles are assumed to be spherical. One is particularly interested in stationary states (i.e., cases where n can be factored into independent functions of t and m):

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and in steady states (i.e., where dT/dt = 0). Steady states can of course be reached only over limited ranges of m because no particles are supplied from outside to the system. Even for very simple assumed fragmentation laws, the equation for n is extremely complicated, being of a multiple integrodifferential type, and analytical solutions can be found (sometimes) for very restricted mass ranges and even then only by making some rather drastic approximations. A simpler problem—namely, where the probability of destruction of a particle is independent of the total number and the mass distribution of the other particles in the system—was solved by Filippov (1961). Assuming that the probability is proportional to a power of the mass and that the size distribution of the fragments of a particle is given by a power law, he derived a formula by which the asymptotic (t + Co) solution for n can be calculated. In the collision problem, the probability of the collision depends on the relative numbers of other particles, and the size distribution of the fragments and erosion products depend on the masses of the colliding components. By neglecting the role of debris in further collisions, Piotrowski (1953) found a stationary solution with 6 or m-3/3. Dohnanyi (1969), who did consider the role of debris in the evolution of the distribution, derived analytically a steady-state solution with 0 or m-111°, applicable to particles with intermediate sizes. Hellyer (1970) subsequently concluded that Piotrowski's law applies only to large masses when one considers the role of debris; i.e., to masses not covered by Dohnanyi's law. More recently, Dohnanyi (1970) has investigated the evolution of the large particles in a mass distribution, in particular, those particles that are not created by collisions of others; and he has concluded that their distribution function approaches asymptotically the m-111° law. To appreciate the significance of these results, let us look at the equation for n and at the fragmentation laws themselves.

There are two types of collisions. In erosive collisions the target mass M is very much greater than the projectile mass pl. (We assume that M > 1.) As a result of the collision, a small amount of matter Me is eroded from the target mass M. For hypervelocity collisions that occur between asteroids as well as between interplanetary dust particles, M. P. u. Because Me or u (as experiments confirm (Gault, Shoemaker, and Moore, 1963)), for sufficiently large values of p/M, the mass M will be disrupted. Those are explosive collisions.

The m-111° law is determined almost exclusively by explosive collisions. The threshold projectile mass is equal to M/T', where the parameter T' depends on veoli. A first approximation of the mass distribution of collisional debris derived from either type of collision is

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where n < 2. For erosive collisions, n = 1.8, Me = Tu, and the largest debris has a mass Mb = Au; T, T', and A are much greater than unity. These relations complete the erosive fragmentation law. In the case of explosive collisions, the corresponding relations are much less well known, except Me = M + u in this case; n is perhaps somewhat less than 1.8 (Gault, Shoemaker, and Moore, 1963; Gault and Wedekind, 1969). A relation for M, is unknown. Dohnanyi (1969) initially assumed M, to be proportional to u. This leads to absurd consequences, however; and he more recently suggested (Dohnanyi, 1970) that Mb = AM, where A* 1. It may be worthwhile at this time to consider qualitatively how the value of Mb is affected by increasing or decreasing the masses M and u. The kinetic energy in the center-of-mass rest frame is, per unit mass of the colliding

particles, -1 E -(# + 2 + #)

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