An estimation of the Moon radius by counting craters: a generalization of Monte-Carlo calculation of $\pi $ to spherical geometry

TL;DR

This paper estimates the Moon's radius by extending Monte-Carlo methods to spherical geometry, using crater counts within a spherical square to derive the radius with quantifiable error.

Contribution

It generalizes Monte-Carlo calculation of π to spherical geometry and applies it to estimate the Moon's radius from crater distributions.

Findings

Derived new relations between spherical square and circle areas.

Computed theoretical deviations of π based on Moon radius.

Estimated Moon radius with quantifiable error from crater counts.

Abstract

By applying Monte-Carlo method, the Moon radius is obtained by counting craters in a spherical square over the surface of it. As it is well known, approximate values for $\pi $ can be obtained by counting random numbers in a square and in a quarter of circle inscribed in it in Euclidean geometry. This procedure can be extend it to spherical geometry, where new relations between the areas of a spherical square and the quarter of circle inscribed in it are obtained. When the radius of the sphere is larger than the radius of the quarter of circle, Euclidean geometry is recovered and the ratio of the areas tends to $\pi $. Using these results, theoretical deviations of $\pi $ due to the Moon radius $R$ are computed. In order to obtain this deviation, a spherical square is selected located in a great circle of the Moon. The random points over the spherical square are given by a specific zone of the Moon where craters are distributed almost randomly. Computing the ratio of the areas, the deviation of $\pi $ allows us to obtain the Moon radius with an intrinsic error given by the finite number of random craters.