After the formation of the Moon, the terrestrial planets were pummelled by impacts from planetesimals left over from terrestrial planet formation. Most lunar crater chronologies are fitted with an exponentially decreasing impact rate with an e-folding time of about 150 Myr. Dynamical simulations consisting of leftover planetesimals and the asteroid belt should reproduce this impact rate. This work attempts to reproduce the impact rates set by modern crater chronologies using leftover planetesimals from three different dynamical models of terrestrial planet formation. I ran dynamical simulations for 1 billion years using leftover planetesimals from the grand tack, depleted disc, and implantation models of terrestrial planet formation with the CPU version of the Gravitational ENcounters with GPU Acceleration (GENGA) N-body integrator. I fitted the cumulative impacts on the Earth and Mars as well as the fraction of remaining planetesimals using a function that is a sum of exponentials with different weighing factors and e-folding times. Most fits require three or four terms. The fitted timescales cluster around τ₁=10 million years (Myr), τ₂=35 Myr, τ₃=100 Myr, and τ₄>200 Myr. I attribute them to dynamical losses of planetesimals through different mechanisms: high-eccentricity Earth-crossers and the ν₆ secular resonance, Earth-crossers, Mars-crossers, and objects leaking onto Mars-crossing orbits from beyond Mars. I placed a constraint on the initial population using the known Archean terrestrial spherule beds, and I conclude that the Archean impacts were mostly created by leftover planetesimals. The inferred mass in leftover planetesimals at the time of the Moon's formation was about 0. 015 Earth masses. The third time constant, τ₃, is comparable to that of modern crater chronologies. As such, the crater chronologies are indicative of impacts by an ancient population of Mars-crossers. The initial perihelion distribution of the leftovers is a major factor in setting the rate of decline: to reproduce the current crater chronologies, the number of Earth-crossers at the time of the Moon's formation had to be at most half that of the Mars-crossers. These results together place constraints on dynamical models of terrestrial planet formation.
R. Brasser (Wed,) studied this question.