## Abstract

Mazrouei *et al*. (Reports, 18 January 2019, p. 253) found a nonuniform distribution of crater ages on Earth and the Moon, concluding that the impact flux increased about 290 million years ago. We show that the apparent increase on Earth can be explained by erosion, whereas that on the Moon may be an artifact of their calibration method.

Mazrouei *et al*. (*1*) analyzed the age distribution of lunar and terrestrial impact craters and found a nonuniform distribution. An apparent kink in the cumulative distributions at the same age of about 290 million years (Ma) led to the conclusion that the impact flux increased at this time.

Their analysis involves 38 terrestrial craters with diameters *D* ≥ 20 km and requires that the crater inventory in this diameter range has not been affected by erosion. To support this hypothesis, they adopted a low estimate of long-term erosion rates (*2*). We argue that the vast majority of Earth’s surface has been exposed to erosion rates much higher than the adopted 2.5 m Ma^{–1}. The record of the terrestrial craters with *D* < 90 km is more affected by erosion than by the age of the underlying crust (*3*).

Figure 1 illustrates the effect of erosion on the age-frequency distribution of the craters with *D* ≥ 20 km exposed at the surface. The theoretical distribution was obtained by simulating 10^{7} hypothetical impacts following the most recent estimate of the terrestrial crater production rate (*4*). We assumed that crater production and erosion were constant over the entire age of Earth, and that the entire crust was as old as Earth. The spatial distribution of erosion rates was inferred from present-day erosion rates of 30 large drainage basins (*5*). Lifetimes of craters at a given erosion rate were estimated from their depth as a function of the diameter (*3*).

Although our simulation only predicts a statistical distribution and not absolute numbers of craters, it is visible from the shapes of the curves alone that constant crater production in combination with constant (over time, but spatially variable) erosion can reproduce the observed age-frequency distribution. We conclude that the curvature in the age-frequency distribution can be explained by degradation of the crater inventory by erosion and does not necessarily require an increase in impact flux.

The age distribution of the lunar craters is sensitive to the relationship between the rock abundance (RA_{95/5}) values and the crater ages. Mazrouei *et al*. assume a power-law relationship derived in a previous study (*6*) from a calibration dataset consisting of nine index craters of known ages.

Figure 2 shows the cumulative distribution of the RA_{95/5} values from which ages were derived for 111 craters by Mazrouei *et al*. The distributions for diameter ranges *D* ≥ 10 km and *D* ≥ 20 km can be approximated by exponential distributions with almost the same decay constants (λ = 59.7 ± 5.7 for *D* ≥ 10 km, λ = 58.3 ± 12.1 for *D* ≥ 20 km, obtained from a maximum likelihood method).

This exponential distribution suggests an alternative model for the RA_{95/5}-age relation. If it was an exponential function of the form_{95/5} values would be consistent with a uniform age distribution, so there would be no requirement for any change in impact flux through time.

We fitted the nine index craters and found τ = 1213 ± 330 Ma. The resulting age distribution in both diameter ranges is shown in the inset of Fig. 2. Compared to figure 3 of Mazrouei *et al*., the concave curvature indicating an increase in impact flux has vanished. The value of τ only defines the absolute time scale, not the shape of the curves.

It is unclear whether the nine index craters used for calibration (*6*) are sufficient to refute the exponential hypothesis. Our exponential fit yields *R*^{2} = 0.862 and a root mean square deviation (RMSD) of 0.012 in the RA_{95/5} values, while the power law suggested by Mazrouei *et al*. even has a slightly higher RMSD of 0.014. In the original data, the craters Jackson and Aristarchus differ in their RA_{95/5} values by more than 0.01, and this difference is opposite to the trend with age. Thus, the variation of the RA_{95/5} values at given crater age should be larger than 0.01, and the RMSDs of both fits are in this order of magnitude.

There are two alternative models that are both consistent with the data, but the exponential model would be consistent with a constant impact flux. We do not claim that the exponential model is better than the power law suggested by Mazrouei *et al*., but given the uncertainties in the data, we are not convinced that the calibration dataset consisting of only nine craters can refute the exponential model.

In summary, we find that the apparent increase in the terrestrial impact flux is consistent with the removal of craters by erosion. The apparent increase in the lunar impact flux is very sensitive to the relationship between the values of RA_{95/5} and the ages of craters, and other fitting functions are also consistent with the data under a constant impact flux.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**

**Author contributions:**S.H. performed the calculations and wrote the paper; G.W. and T.K. contributed knowledge about terrestrial and lunar impact craters.

**Competing interests:**The authors have no competing interests.

**Data and materials availability:**The MATLAB codes for reproducing our results are provided in data S1.