On de-Sitter geometry in crater statistics

Abstract

The cumulative size-frequency distributions of impact craters on planetary bodies in the solar system appear to approximate a universal inverse square power-law for small crater radii. In this article, we show how this distribution can be understood easily in terms of geometrical statistics, using a de-Sitter geometry of the configuration space of circles on the Euclidean plane and on the unit sphere. The effect of crater overlap is also considered.

Lie sphere geometry in lattice cosmology

“…As already stressed, this geometric method has-quite surprisingly and to the best of our knowledge-not been employed in the general-relativistic initial-value problem and hardly ever in astrophysics and cosmology. The only two notable exceptions we are aware of concern the statistics of craters on planetary bodies [24] and the statistics of cosmological voids [25] 3 . In our paper we will use it to systematically construct initial data for Einstein's field equations applied to lattice cosmology.…”

“…In this fashion the set of spherical caps in S n is not only put into bijective correspondence with points in dS n+1 , but is also endowed with the structure of a maximally symmetric P α cos α Lorentzian manifold with metric g dS (n+1) , the geometry of which turns out to be very useful indeed, with many and sometimes surprising applications. For example, in n = 2 and n = 3 dimensions, the volume form induced by this metric has been used for statistical discussions of distributions of planetary craters in [24] and cosmic voids in [25], respectively. In figure 2 we illustrate once more the geometric objects underlying this bijective correspondence between spherical caps of-or oriented hyperspheres in-the Möbius sphere S n and deSitter space dS (n+1) in the case n = 1.…”

On de Sitter geometry in cosmic void statistics

“…We begin by briefly recapitulating the notion of a configuration space of spheres in Euclidean space and its de-Sitter geometry, as discussed in some more detail in Gibbons & Werner (2013); see also Zee (2013), pp. 646-647.…”

“…Therefore, the space of unoriented spheres is in fact given by half of the full de-Sitter quadric, sometimes referred to as de-Sitter space modulo the antipodal map X µ → −X µ . Geometrically, the coordinates X µ of a sphere can be regarded as a form of Lie cycle coordinates for the Laguerre cycle representing the sphere (see, e.g., the appendix of Gibbons & Werner (2013) for more mathematical details). Taking y i = (R, x), 0 i n + 1, as coordinates of the de-Sitter configuration space, its metric g induced by the ambient Minkowski metric in the usual way can be read off from the line element…”

“…One can now study the size distribution of spherical voids in the Universe in terms of their distribution over this de-Sitter configuration space. The approach presented here has been inspired by other notions of configuration space measures in cosmology (e.g., Gibbons & Turok (2008)), and another application of this de-Sitter configuration space of spheres in a different astronomical context has been proposed recently (Gibbons & Werner (2013)). The outline of this work is as follows.…”