This article is from the Space FAQ, by Jon Leech leech@cs.unc.edu and Mark Bradford tla@surly.org with numerous contributions by others.

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.]

Continue to: