On the scaling of crater dimensions: 2. Impact processes

Abstract

Existing theoretical and empirical scaling rules for cratering phenomena are commonly based upon implicit assumptions regarding the roles of the impactor conditions and the target properties. It is common, for example, to assume that only the kinetic energy of the impactor and the work against gravity in the target govern large impact craters. A more general case is considered here. A spherical impactor is characterized by its kinetic energy, momentum, and various material properties of both the impactor and the target. Under these general assumptions, scaling among both similar and nonsimilar events is studied. Similar events always satisfy the cube‐root mass scaling rule. The dependence on energy varies according to how the similarity is achieved. For constant mass or constant velocity tests, variations in gravity are required, and cube‐root energy scaling is obtained. For a strengthless material only, quarter‐root energy scaling is possible, but only for restricted variations in both impactor mass and velocity, which imply that the mass is proportional to the three‐quarters power of the energy. Scaling rules for the more common and more useful application to nonsimilar events (e.g., variable energy and/or velocity at constant gravity) are shown to be bounded by certain well defined limits. Dimensional analysis is used with plausible assumptions on the sign of the partial derivative of the crater volume with regard to each of the independent variables. Distinctions are made among different forms of cube‐root scaling obtained from strength‐dominated phenomena. Quarter‐root scaling is shown to be a distinct limit to gravity domination, occurring only when there is no dependence upon impactor velocity and target strength. Special cases of the general scaling obtained correspond to degenerate forms common in the literature. Of these, energy scaling, momentum scaling, strength scaling, and gravity scaling are shown to be the bounds for all permissible scaling rules. This result is shown to be in agreement with a comprehensive collection of data obtained from a survey of the literature. Included are experimental results for both geological materials and metals, analytical models, and finite difference code calculations.