Analytical formulation of lunar cratering asymmetries

School of Astronomy and Space Science and Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, 210046 Nanjing, PR China
e-mail: zhoujl@nju.edu.cn

Received: 28 March 2016
Accepted: 3 July 2016

Abstract

Context. The cratering asymmetry of a bombarded satellite is related to both its orbit and impactors. The inner solar system impactor populations, that is, the main-belt asteroids (MBAs) and the near-Earth objects (NEOs), have dominated during the late heavy bombardment (LHB) and ever since, respectively.

Aims. We formulate the lunar cratering distribution and verify the cratering asymmetries generated by the MBAs as well as the NEOs.

Methods. Based on a planar model that excludes the terrestrial and lunar gravitations on the impactors and assuming the impactor encounter speed with Earth v_enc is higher than the lunar orbital speed v_M, we rigorously integrated the lunar cratering distribution, and derived its approximation to the first order of v_M/v_enc. Numerical simulations of lunar bombardment by the MBAs during the LHB were performed with an Earth–Moon distance a_M = 20−60 Earth radii in five cases.

Results. The analytical model directly proves the existence of a leading/trailing asymmetry and the absence of near/far asymmetry. The approximate form of the leading/trailing asymmetry is (1 + A₁cosβ), which decreases as the apex distance β increases. The numerical simulations show evidence of a pole/equator asymmetry as well as the leading/trailing asymmetry, and the former is empirically described as (1 + A₂cos2ϕ), which decreases as the latitude modulus | ϕ | increases. The amplitudes A_1,2 are reliable measurements of asymmetries. Our analysis explicitly indicates the quantitative relations between cratering distribution and bombardment conditions (impactor properties and the lunar orbital status) like A₁ ∝ v_M/v_enc, resulting in a method for reproducing the bombardment conditions through measuring the asymmetry. Mutual confirmation between analytical model and numerical simulations is found in terms of the cratering distribution and its variation with a_M. Estimates of A₁ for crater density distributions generated by the MBAs and the NEOs are 0.101−0.159 and 0.117, respectively.