# 20 Computing Crater Diameters From Earth-Impacting Asteroids

Astrogeologist Gene Shoemaker proposes the following formula, based on
studies of cratering caused by nuclear tests. Units are MKS unless
otherwise noted; impact energy is sometimes expressed in nuclear bomb
terms (kilotons TNT equivalent) due to the origin of the model.

D = Sg Sp Kn W^(1/3.4)
Crater diameter, meters. On Earth, if D > 3 km, the crater is
assumed to collapse by a factor of 1.3 due to gravity.

Sg = (ge/gt)^(1/6)
Gravity correction factor cited for craters on the Moon. May hold
true for other bodies. ge = 9.8 m/s^2 is Earth gravity, gt is
gravity of the target body.

Sp = (pa/pt)^(1/3.4)
Density correction factor for target material relative to the Jangle
U nuclear crater site. pa = 1.8e3 kg/m^3 (1.8 gm/cm^3) for alluvium,
pt = density at the impact site. For reference, average rock on the
continental shields has a density of 2.6e3 kg/m^3 (2.6 gm/cm^3).

Kn = 74 m / (kiloton TNT equivalent)^(1/3.4)
Empirically determined scaling factor from bomb yield to crater
diameter at Jangle U.

W = Ke / (4.185e12 joules/KT)
Kinetic energy of asteroid, kilotons TNT equivalent.

Ke = 1/2 m v^2
Kinetic energy of asteroid, joules.

v = impact velocity of asteroid, m/s.
2e4 m/s (20 km/s) is common for an asteroid in an Earth-crossing
orbit.

m = 4/3 pi r^3 rho
Mass of asteroid, kg.

r = radius of asteroid, m

rho = density of asteroid, kg/m^3
3.3e3 kg/m^3 (3 gm/cm^3) is reasonable for a common S-type asteroid.

For an example, let's work the body which created the 1.1 km diameter
Barringer Meteor Crater in Arizona (in reality the model was run
backwards from the known crater size to estimate the meteor size, but
this is just to show how the math works):

```	r = 40 m	    Meteor radius
rho = 7.8e3 kg/m^3  Density of nickel-iron meteor
v = 2e4 m/s	    Impact velocity characteristic of asteroids
in Earth-crossing orbits
pt = 2.3e3 kg/m^3   Density of Arizona at impact site

Sg = 1		    No correction for impact on Earth
Sp = (1.8/2.3)^(1/3.4) = .93
m = 4/3 pi 40^3 7.8e3 = 2.61e8 kg
Ke = 1/2 * 2.61e8 kg * (2e4 m/s)^2
= 5.22e16 joules
W = 5.22e16 / 4.185e12 = 12,470 KT
D = 1 * .93 * 74 * 12470^(1/3.4) = 1100 meters
```

More generally, one can use (after Gehrels, 1985):

```    Asteroid	    Number of Impact probability  Impact energy as multiple
diameter (km)   Objects    (impacts/year)	    of Hiroshima bomb
-------------   --------- ------------------  -------------------------
10			10	 10e-8		    1e9 (1 billion)
1		       1e3	 10e-6		    1e6 (1 million)
0.1	       1e5	 10e-4		    1e3 (1 thousand)
```

The Hiroshima explosion is assumed to be 13 kilotons.

Finally, a back of the envelope rule is that an object moving at a speed
of 3 km/s has kinetic energy equal to the explosive energy of an equal
mass of TNT; thus a 10 ton asteroid moving at 30 km/sec would have an
impact energy of (10 ton) (30 km/sec / 3 km/sec)^2 = 1 KT.

References:

Clark Chapman and David Morrison, "Cosmic Catastrophes", Plenum Press
1989, ISBN 0-306-43163-7.

Gehrels, T. 1985 Asteroids and comets. "Physics Today" 38, 32-41. [an
excellent general overview of the subject for the layman]

Shoemaker, E.M. 1983 Asteroid and comet bombardment of the earth. "Ann.
Rev. Earth Planet. Sci." 11, 461-494. [very long and fairly
technical but a comprehensive examination of the
subject]

Shoemaker, E.M., J.G. Williams, E.F. Helin & R.F. Wolfe 1979
Earth-crossing asteroids: Orbital classes, collision rates with
Earth, and origin. In "Asteroids", T. Gehrels, ed., pp. 253-282,
University of Arizona Press, Tucson.

Cunningham, C.J. 1988 "Introduction to Asteroids: The Next Frontier"
(Richmond: Willman-Bell, Inc.) [covers all aspects of asteroid
studies and is an excellent introduction to the subject for people
of all experience levels. It also has a very extensive reference
list covering essentially all of the reference material in the
field.]

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