Impact cratering: The effect of crustal strength and planetary gravity

Upon impact of a meteorite with a planetary surface the resulting shock wave both ‘processes’ the material in the vicinity of the impact and sets a larger volume of material than was subjected to high pressure into motion. Most of the volume which is excavated by the impact leaves the crater after the shock wave has decayed. The kinetic energy which has been deposited in the planetary surface is converted into reversible and irreversible work, carried out against the planetary gravity field and against the strength of the impacted material, respectively. By using the results of compressible flow calculations prescribing the initial stages of the impact interaction (obtained with finite difference techniques) the final stages of cratering flow along the symmetry axis are described, using the incompressible flow formalism proposed by Maxwell. The fundamental assumption in this description is that the amplitude of the particle velocity field decreases with time as kinetic energy is converted into heat and gravitational potential energy. At a given time in a spherical coordinate system the radial velocity is proportional to R^(−z), where R is the radius (normalized by projectile velocity) and z is a constant shape factor for the duration of flow and a weak function of angle. The azimuthal velocity, as well as the streamlines, is prescribed by the incompressibility condition. The final crater depth (for fixed strength Y) is found to be proportional to R_0[2(z + 1)u_(or)²/g]^(1/(z+1)), where u_(or) is the initial radial particle velocity at (projectile normalized) radius R_0, g is planetary gravity, and z (which varied from 2 to 3) is the shape factor. The final crater depth (for fixed gravity) is also found to be proportional to [ρu_(or)^2/Yz]^(1/(z+1)), where ρ and Y are planetary density and yield strength, respectively. By using a Mohr-Coulomb yield criterion the effect of varying strength on transient crater depth and on crater formation time in the gravity field of the moon is investigated for 5-km/s impactors with radii in the 10- to 10^7-cm range. Comparison of crater formation time and maximum transient crater depth as a function of gravity yields dependencies proportional to g^(−0.58) and g^(−0.19), respectively, compared to g^(−0.618) and g^(−0.165) observed by Gault and Wedekind for hypervelocity impact craters in the 16- to 26-cm-diameter range in a quartz sand (with Mohr-Coulomb type behavior) carried out over an effective gravity range of 72–980 cm/s²