This article is from the Space FAQ, by Jon Leech firstname.lastname@example.org and Mark Bradford email@example.com 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
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
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)