A stochastic model of the formation and survival of lunar craters: IV. On the nonrandomness of crater centers

Abstract

It is shown that even if the underlying distribution of craters is completely random, the distribution of centers of observable craters is not. In particular, in finite regions in which a very large crater is found, we would expect to find a dearth of small craters, since the large crater would have obliterated older small craters in its neighborhood. The resulting distribution of the number of crater centers in equiareal regions of the lunar surface is then a mixture of Poisson distributions, and tests of randomness based on goodness of fit of this distribution to the Poisson distribution will not be accurate. Under the meteoroidal impact hypothesis, with disappearance of craters due to filling as well as obliteration, the coefficient of variation of density of small craters in regions of small size can be as large as 30%. This shows that large deviations from the Poisson distribution can be expected, even in homogeneous regions. Another source of variation in crater counts is the grouping together of inhomogeneous regions, for example, maria whose crater densities differ by a factor of 2 or 3. The resulting distribution of numbers of crater centers in equiareal sectors will again be a mixture of Poisson distributions. Empirical crater counts clearly show the negative correlation between density of small craters and the presence of large craters. Simulated plots of randomly formed craters are used to study the probability levels to be expected on applying the test of goodness to fit of the Poisson distribution. It is concluded that there is no reason to reject the hypothesis of randomness of formation of craters of large and moderate size.