Cratering Brainteaser

Ge151: Spring 2011

Due: May 19, 2010.

This problem is intended to get you to think about cratering, especially the distinct regimes dominated by strength or gravity and their relationship to the transition from simple to complex impact craters. Specifically, in the strength regime (where simple craters are formed), an object impacting a solid surface at hypervelocity will create an explosion leaving a big, quasi-hemispherical hole in that surface. The diameter of the hole will depend only on the impactor energy and the strength of the original surface.

In the gravity regime (where complex craters are formed) the crater diameter is much larger and the vast majority of the impacting energy must be used simply to move the material out of the crater. The crater size in this case will depend on the impactor energy and on both the weight of the material you need to move and how far you need to move it to get it out of the crater.

Use a simple dimensional argument to relate crater size and energy in the strength regime. Note that material strength is given in units of Pascals and assume a typical rock strength of 10⁸ Pa.
Use a simple dimensional argument to relate crater size and energy in the gravity regime.
Calculate, using simple assumptions, the transition diameter from the simple to complex regimes on Earth, Moon, Mercury and Mars. Use a typical density of 2700 kg m^-3.
If the diameter of an impact crater on the Moon were doubled from 1 to 2 km what would be the relative increase in impactor energy needed?
What would be the relative increase for doubling crater size from 30 to 60 km on the Moon?
Compare the expected transient cavity dimension to the empirical relation given in de Pater and Lissauer.
D=1.8 r_m^0.11 r_p^-0.33 g_p^-0.22 (2R)^0.13 E_k^0.22 (sin q)^1/3
Where p refers to the planet, m to the projectile, r is density, g is gravity, R is the projectile radius, E_k is the kinetic energy, and q the angle of impact. What is velocity radial dependence implied by these empirically derived exponents?

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