tokens: g is the gravitational acceleration. We can use the velocity profile of the shock to calculate the surface density profile of the ejecta blanket Energy and momentum conservation yield limits on the slope of the surface density distribution −2.5 > dln σ/dln r > −3. If we assume a constant density for the settled ejecta, then the thickness of the ejecta blanket scales the same way as the surface density. Most ejecta blankets around craters on the Moon are within this range (Zhu et al. 2015 , 2017 ; Raggio et al. 2016 ). In contrast, ejecta blankets from chemical and nuclear explosions on Earth are slightly steeper −3 > dln σ/dln r > −4, due to air drag (Carlson Jones 1965 ). Using the same shock trajectory we can make a theoretical prediction for the density slope we numerically calculated in the previous section. We saw that ∝ m −β . At distances much larger than the radius of the impactor, the velocity profile is roughly homologous ≈ d / t , where d is the distance from the impact site and t is the time since the impact. Combining these relations with the definition of density m ≈ ρ d 3 we find a scaling relation for the density profile ρ ∝ r −3 − 1/β . For β = 2/3 we obtain dln ρ/dln r = −4.5, which is close to the value obtained in the numeric simulation dln ρ/dln r = −4.1. Eventually, the shock stops and the crater reaches its final size. There are two criteria that can determine the final size of the crater (which will always be the smaller of the two). One possibility is that the shock stops when the shock pressure drops below the elastic modulus of the soil Y , so the final shock radius is where R i ≈ ( m i /ρ 0 ) 1/3 is the radius of the impactor. The second possibility is that the shock stops when the shock pressure drops below the hydrostatic pressure g ρ R , and so In the case of the lunar regolith, this is the deciding condition. 3.2 Craters from PBHs The density of PBHs is much higher than that of meteors, so instead of transferring all of their energy to the target at the impact site, the impactor just passes through and deposits roughly the same amount of energy per unit length along its trajectory inside the target. For this reason, above some critical threshold, the evolution of the shock wave is independent of the density of the impactor. To understand how the shock wave will behave in this case, one can think about this problem as a series of point explosions, where the energy of each explosion scales with its depth E ∝ x . If the mass of the BH is m i , then the effective cross-section is determined by the Bondi radius , and so . For a uniform density soil, the mass in a sphere of radius x around the hotspot is m ≈ ρ 0 x 3 . Hence, . We thus obtain a new power-law relation between the shock velocity and the swept up mass, which is shallower than that obtained for classical craters, i.e. β = 1/3. Using the same reasoning as in the previous section, we find the slopes of ejecta blanket surface density profile As was done
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