Fractal Radiometry of Lunar Surfaces Suggests a Significant Underestimation of Near-Earth Object Impact Rates

Abstract

We present an experimental method to infer the mass distribution of meteoritic impacts on the Moon by confronting Lambert's theoretical model (the distribution of brightness reflected by a sphere) with the empirical measurement. In the theoretical model, surface brightness should decrease with the cosine of the distance from the centre of the disc, becoming completely dark at the edges, which would not be well defined (they would form a smooth gradient). Observation shows edges that are much brighter and more sharply defined. This occurs because the lunar "surface" is a multifractal, forming quasi-retroreflectors at different scales. The ratio between theoretical and observed brightness, compared across different regions, makes it possible to estimate the mean fractal dimension of the lunar "surface". Assuming that the Moon is "smooth" at large scale (Euclidean with D~2) and that the anomaly in visible brightness results from the combination of effects at all scales from 532 nm to 3500 km, we formulate the variation in fractal dimension as a function of scale over a wide range of orders of magnitude, yielding a predictive model more comprehensive than is possible via direct counts. Given the distribution of crater sizes, we use an adjusted version of the Imke formula of Imke de Pater & Jack J. Lissauer (UCLA/NASA) to estimate the sizes of the objects that produced these craters. The results obtained are consistent with "manual" counts performed by Robbins, S. J. (Journal of Geophysical Research) for diameters between 2 km and 1,200 km. For craters smaller than 1 km, our model yields results that differ from manual counts, suggesting undercounting due to operational difficulties. These results show that there may be a need for a broad revision of collision estimates with NEOs.

Keywords: Astrophysics, Astronomy, Space Science, Fractal Analysis, Near-Earth Objects, Mathematical Modelling, Remote Sensing, Risk Assessment, Global Catastrophic Risks, Space Safety, Statistics, Risk Management