Meteors can be observed in the sky by anyone with the patience to watch for them. They are more common at certain times of the year during what are called meteor showers. Occasionally one of these meteors reaches the Earth's surface and is called a meteorite. People have always collected them when they could find them. In ancient times they were only source of metallic iron.

There is no record of a human being killed by a meteorite, although this may be a matter of the lack of records throughout so much of the world for the span of human history. There are cases of injuries and damage to property from meteorites. And probably there were deaths in the the cases of the recent impacts of large object listed below, but they were in such remote areas no one knew who had been in the area affected.

• The last and most famous was the Tunguska event in Siberia in 1908. It is said to have devastated an area half the size of Rhode Island. The object probably exploded in the atmosphere, because no metoric object has been located. Even without an actual explosion such objects will wreak havoc on the ground because in front of the object is a parcel of high pressure, high temperature gases that will cause devastation.
• In British Guiana, now Guyana, in 1935 there were reports of an area five to ten miles in one dimension and twenty miles in the other dimension in which the forest was flattened and fires were burning. This is the pattern that would be expected from an object impacting the surface obliquely.
• In 1930 observers in the Amazon region of Brazil reported three objects of a celestial nature crashed in to the jungle.
• In 540 AD various chronicles mentioned events concerning a comet that mean that a comet hit the Earth.
• In the Empty Quarter of souther Saudi Arabia there is an impact crater which Bedouin tradition says is where an ancient city was destroyed. The crater is called Wabar and was 12 meters deep when it was discovered in 1932.
• There is some evidence of an impact crater in southern Iraq which is about two miles across. It is suggested that this was due to a major impact of about 2300 BCE and that it resulted in the destruction of the Akkad civilization of Mesopotamia. Other civilization in the region, such as Egypt, would have been strong affected as well.

The Characteristics of In-coming Objects
That Determine Their Destructiveness

Meteors, asteroids and comets differ as to mass, velocity, shape and density. Most of the objects which fall into the atmosphere do not reach the Earth's surface, but even if a large object explodes in the atmosphere the resulting pressure wave can level forests or cities in the path of its trajectory. For example, in 1908 some object hit Earth's atmosphere over Tunguska, Siberia. It apparently exploded before reaching the Earth since no material remains of it have been found. It nevertheless leveled the forest of an area about half the size of Rhode Island. The felled trees were directed radially from a central point. Near the central point the pressure wave stripped the trees of their branches and bark so what was left standing looked like a forest of giant toothpicks.

Hills and Goda [1] carried out an analysis of the pressure wave destruction due to an incoming object such as small asteroids. In their analysis the area of destruction extends over a radius R where the excess pressure Δp exceeds an amount b sufficient to destroy buildings. The magnitude of b is roughly 2 pounds per square inch in excess of the normal pressure of about 15 pounds per square inch. In standard international (Systeme International d'Unites SI) units the value of b is about 14,000 Pascals (or 140 millibars).

Hills and Goda concluded from their analysis that the radius of distruction is given by:

R = bK^1/3

where K is the kinetic energy of the object and b is the critical excess pressure mentioned above. Thus the distribution which is important is the distribution of kinetic energies.

The kinetic energy of an object of mass m and velocity V is ½mV². The kinetic energy of an object is related to its equivalent spherical diameter D because its mass is given by

m = ρ[(4/3)π(D/2)³]

where ρ is the density of the object. As Gallileo noted long ago, the velocities of falling objects are not influenced by their masses. Therefore the kinetic energy distribution of objects is determined by their size and density. Densities are largely either that of stone or that of iron. This leaves size as the crucial variable.

The area A of destruction is proportional to the square of the radius R so

A = gD²

where the dimensionless constant of proportionality g is given by

g = b²(ρπV²/12)^1/3

The coefficient g depends upon the density of the objects and thus differs for stony versus metallic objects. This dependence will be expressed by writing the coefficient as g(ρ). Representive values of ρ for stony and metallic meteors would be 2500 kg/m³ and 8500 kg/m³, respectively. A typical value for V is 20 km/s, which is 2×10⁴ m/s. This means the value of g for a stony object is about 1.2×10¹². For a metallic object g would be about 1.8×10¹².

The Size Distribution of NEO's

Gregory H. Canavan of the Los Alamos National Laboratory gives some rough estimates of the frequency distribution of NEO's. He defines size in terms of diameter and divides the range of sizes into three classes: 1. Less than 200 meters in diameter, 2. 200 meters to 2 kilometers in diameter, 3. Greater than 2 kilometers in diameter.

The functional form of the frequency distribution which Canavan finds satisfactory is

n_i(D) = L_iD^−β_i

The values for the parameters for the three classes (from Canavan [2]) are shown below:

Below are the computed frequencies for these parameters as a function of object diameter.

The graph below shows the logarithm of frequency as function of the logarithm of object diameter:

One crucial bit of information is the composition of the population of objects. The objects can be of three types: 1. Stoney, 2. Combination stone and metal, 3. Metal, primarily iron with a small proportion nickel. The proportion have been estimated to be roughly 7:3:1; i.e., 7/11 stoney, 3/11 a combination and 1/11 metallic. The frequency of metallic meteors is generally overestimated because of the much proportion of metallic meteors that survive the passage through the atmosphere and are identified and recovered.

Hills and Goda [1] concluded that stony objects with diameters of less than 50 meters do not penetrate the atmosphere deep enough to cause damage. Metallic objects, almost always iron, do have greater penetrating power but only those with diameters of six meters or more will create the pressure wave destruction they were considering.

Economic Costs

Let γ be the spatial density of economic value so the economic loss from the distruction of an area A is γA. The value of γ takes into account the length of time that an area would be an economic loss. Expressed in terms of characteristics of the in-coming object the economic cost c for one object of diameter D would be:

c = γg(ρ)D²
and multiplied by the frequency n(D)
to give the annual cost C(D)
this is
C(D) = γg(ρ)D²n(D) = γg(ρ)D²LD^−β
= γg(ρ)LD^2−β

To get the annual cost for objects of diameters in the range [h, k] the above must be integrated; i.e.,

E = ∫_h^kC(D)dD
= ∫_h^kγg(ρ)LD^2−βdD
= γg(ρ)L[D^3−β/(3−β)]_h^k
= γg(ρ)L[k^3−β−h^3−β]/(3−β)

The integration is to be limited to a range for which the L and β parameters are constant.

The Values of the Parameters

The Gross Domestic Product and Gross National Product of the Earth are equal and have a value in 2005 of about $55.5 trillion on a purchasing powere parity (PPP) basis. The area of the Earth's surface is 5.09×10⁸ km². Thus the annual economic production per square kilometer of Earth's surface, land and water, is $109 thousand. An area that is devastated would be taken out of production for a period of time, perhaps forever. Even if an area is taken out of production forever the present value of the future loss would not be infinite. The present value of a perpetuity is the reciprocal of the interest rate. Thus, if the interest rate is 5 percent, the present value of an annual loss of L is (1/0.05)=20 times L. If the annual loss is growing at a constant real rate per year the multiplying factor is the reciprocal of the difference between the real interest rate and the real rate of growth. Thus, if the real interest rate is 5 percent and the rate of real growth 3 percent the multiplying factor for the current annual loss is 1/(0.05-0.3)=50.

In general the multiplying factor for converting the current annual loss to the present value of all current and future losses is a function of the period of time the loss is sustained, the real interest rate and the real growth rate.

The time period that a area would be out of production probably depends upon the size of the area. This time period would reflect the amount of resources required for recovery and, for very large losses, the loss of world resources available to affect the recovery. There are some impacts so large that human civilization would never recovery from them.

The multiplying factor, denoted as γ above, for converting current annual loss into the present value of all present and future loss is probably in the range of 10 to 50. This means that the loss per square kilometer is in the range of $1.1 million to $5.5 million. For the calculations below the value of γ will be taken to be $2.4 million.

For the calculations the combined stone and iron objects are treated as equivalent to the stone objects. The value of the parameter g for stone object is 1.2×10¹². The first range of integration for the stone objects is from 50 meters to 200 meters.

(To be continued.)

Sources:
• 1. J.G. Hills and M.P. Goda, "The fragmentation of small asteroids in the atmosphere," Astronomy Journal, vol. 105 (1993), pp. 1114-1144.
• 2. G.H. Canavan, "Cost and Benefit of NEO Detection and Interception", pp. 1157-1189 in Hazards Due to Comets and Asteroids, edited by Tom Gehrels, (University of Arizona Press, 1994).

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