Crater scaling-laws

[Widdekind]

According to Ivanov (2008),

 

The Schmidt-Housen scaling law expresses the crater diameter, in terms of dimensionless (scaled) [math]\pi[/math]-value for transient crater diameter, [math]D_{at}[/math], named [math]\pi_{D} = D_{at} \left( \frac{\rho}{m} \right)^{1/3}[/math] ([math]\rho[/ma th] is the target material density, and m is the projectile mass), and the other [math]\pi[/math]-value for the projectile, [math]\pi_{2} = 1.61\frac{g D_p}{v^2}[/math] ([math]g[/math] is the gravity acceleration, [math]D_p[/math] is the projectile diameter, and [math]v[/math] is the impact velocity). For target materials without an appreciate cohesion (e.g., dry sand), the scaling law is well approximated with the exponential relation:

 

[math]\pi_D = k_D \pi_D^\beta[/math]

 

where [math]K_D[/math] and [math]\beta[/math] are experimentally derived coefficients. For nonporous rocks, [math]K_D= 1.6[/math], [math]\beta = 0.22[/math], whereas for porous rocks (e.g., dry sand) [math]K_D= 1.68[/math], [math]\beta = 0.17[/math]

 

Note the apparent typo, in the (second?) subscript, of the above equation. Assuming, to begin with, that the authors meant to say:

 

[math]\pi_D = k_D \pi_2^\beta[/math]

then please ponder the application, of the above equation, to planetary impacts, by simplifying the notation:

 

[math]\pi_{D} = D_{at} \left( \frac{\rho}{m} \right)^{1/3}[/math]

 

[math]\equiv D_{at} \left( \frac{\rho}{\frac{\pi}{6} D_p^3 \rho_p} \right)^{1/3}[/math]

 

[math]\equiv \frac{D_{at}}{D_p} \left(\frac{6}{\pi} \frac{\rho}{\rho_p} \right)^{1/3}[/math]

 

[math]\approx 1.24 \left( \frac{D_{at}}{D_p} \right) \left( \frac{\rho}{\rho_p} \right)^{1/3} \rightarrow 1.24 \left( \frac{W}{d} \right) \left( \frac{R}{\rho} \right)^{1/3}[/math]

where [math]W[/math] is the crater width; lower-case (Greek) letters denote the impactor; and upper-case (Greek) letters denote the planet being impacted.

 

Along like lines, let us simplify further, by assuming a 'natural' impact, [math]g = \frac{G M}{\left( \frac{D}{2} \right)^2}[/math], and [math]\frac{1}{2} v^2 \approx \frac{G M}{\left( \frac{D}{2} \right) }[/math]:

 

[math]\pi_{2} = 1.61\frac{\left( \frac{4 G M}{D^2} \right) d}{\frac{4 G M}{D}}[/math]

 

[math]\approx 1.61 \left( \frac{d}{D} \right)[/math]

where we keep the same 'case-conventions' as per previous.

 

Now, 'putting all the pieces together', w.h.t.:

 

[math]\pi_D = k_D \pi_2^\beta[/math]

 

[math]1.24 \left( \frac{W}{d} \right) \left( \frac{R}{\rho} \right)^{1/3} \approx K_D \left( 1.61 \frac{d}{D} \right)^{\beta}[/math]

 

[math]\left( \frac{W}{d} \right) \left( \frac{R}{\rho} \right)^{1/3} \approx 1.3 \left( 1.61 \frac{d}{D} \right)^{0.2}[/math]

 

[math]\left( \frac{W}{d} \right) \left( \frac{R}{\rho} \right)^{1/3} \approx 1.4 \left( \frac{d}{D} \right)^{0.2}[/math]

Now, for 'natural' impacts, of stony asteroids, into a rocky planet's surface crust, the above density ratio is roughly unity (or, icy comets, into an ice-crust). So, in such situations, w.h.t.:

 

[math]\frac{W}{d} \approx 1.4 \left( \frac{d}{D} \right)^{0.2}[/math]

This formula fails, for the LHS > 1, whereas the RHS < 1. Such could be corrected, if, in fact, the LHS is 'up-side-down':

 

 

[math]\pi_D = k_D \pi_D^{-\beta}[/math]

 

[math]\rightarrow[/math]

 

[math]\frac{W}{d} \approx 1.2 \left( \frac{D}{d} \right)^{0.2}[/math]

That formula would 'say', that, for a given impactor, of size [math]d[/math], impacting into the cooled crust, of a rocky planet, impacts upon larger worlds, produce bigger craters. And, such seems plausible, as larger planets would 'pull harder' upon the impacting projectile. For a ~10 km impactor (e.g. Chicxulub), this formula forecasts a ~100 km crater, in close accord with actual observations (e.g. Chicxulub). (Note, that, in-so-far as the accuracy of the above formula can be relied upon, then, observations, of actual craters, anomalously deviant, from the the formula's forecasts, could indicate 'non-natural' impacts, which, for some, surely inferable, reason, did not impact at 'natural' near-escape-velocities [extra-solar high-speed asteroids?? impactors of anomalous density?? small impactors slowed by atmospheric drag??].)

 

Moreover, the previous formula can be re-written, as:

 

[math]\frac{W}{D} \approx \left( \frac{d}{D} \right)^{0.8}[/math]

If so, then, for a given world of size [math]D[/math], larger-and-larger impactors, would produce larger-and-larger 'world-relative' craters.

 

Note, that 'world-wrecking' collisions ([math]W \approx D[/math]), require 'same-size' collisions ([math]d \approx D[/math]). Indeed, to completely crack apart a particular planet,

 

[math]\frac{1}{2}m v^2 \approx \frac{G M^2}{D}[/math]

with a 'natural' collision,

 

[math]\frac{1}{2}v^2 \approx \frac{G M}{D/2}[/math]

w.h.t.:

 

[math]m \approx \frac{M}{4}[/math]

Such suggests, that impactors must be almost as massive, as the planet that they impinge upon, in order to completely crack the 'space stone', sundering the same.

 

Note, too, that central concentration of planet mass, into a dense metallic core, increases (the magnitude of) the world's gravitational binding energy, in the first formula immediately above; whilst adding nothing to the impactor's impact speed, in the second formula preceding this paragraph. So, a 'fluffy crust', surrounding a 'mass-centralized' core, functions as a 'flak vest', for that core, requiring more massive impactors, to completely crack apart the planet. Earth's moon, having possibly formed from a near-world-wrecking caliber collision, may be mostly 'flak vest', without much metal in the mixture (excepting metals, from impactors, which created the craters seen on the surface of our moon, after that surface had solidified).

Edited May 6, 2011 by Widdekind