Mathematics of Counterforce Targeting

Mathematics of Counterforce Targeting


The explosion of a nuclear weapon causes a number of effects. There is a blast wave (about 50 percent of the released energy), heat (about 35 percent of the released energy), and radiation (about 15 percent of the released energy). When we are dealing with a small target that is built to withstand a significant nuclear explosion (such as a missile silo), we ignore both the heat and radiation, and assume that it is only the blast that can destroy the target.

The Variables

There are several critical variables to be examined in assessing whether a nuclear weapons can destroy a hardened target. Briefly, we need to know whether the weapon can (a) create an area of blast sufficient to destroy the target, and (b) the chances that the blast area sufficient to destroy the target will land close enough to target (i.e., "cover" the target). The critical variables in this assessment are:

  • H = The Hardness of the target. The hardness of the target is the number of pounds per square inch of overpressure that the target can survive. In our calculations, it will be assumed that if the blast of the weapon over the target is less than H, the target will survive intact, otherwise it will be completely destroyed.

  • Y = The Yield of the weapon. The yield of the weapon is expressed in megatons (MT), the number of millions of pounds of TNT that are equivalent to the blast produced by the nuclear weapon.

  • LR = The Lethal Radius of the weapon. Given Y, the yield of the weapon, and H, the hardness of the target, it is possible to calculate LR, the lethal radius of the weapon against the target. This is simply the radius of the circle produced by the weapon in which the overpressure is greater than H. If the target is "covered" by this circle, it is considered destroyed. The larger the value of Y relative to H, the larger the circle. The lethal radius is measured in nautical miles.

  • CEP = The Circular Error Probable of the weapon. Once we have determined the size of the lethal radius, we turn to the question of accuracy. That is, we determine whether the circle of the lethal radius will land close enough to the target so that it will be over the target. The standard measure of warhead accuracy is the CEP, the circular error probable. This is defined as follows. Suppose we fire a large number of warheads at a target and mark where the warheads land. The CEP is the radius of the circle that can be drawn to include 1/2 of the warheads. The tighter the pattern of the warheads, the smaller the CEP, and the greater the assumed accuracy of the warheads. CEP is measured in nautical miles.

  • B = The Bias of the weapon. The bias of the weapon is the distance between the center of the CEP and the target. That is, suppose the warheads land close together, but at some distance from the target. The distance of the miss is the bias. In all calculations of nuclear attacks that appear in "the open literature" (i.e., the stuff you and I can read), bias is assumed to be 0.

  • R = The Reliability of the weapon. The reliability of the weapon is the probability that the weapon will work.

    If the weapon is perfectly reliable (i.e., R = 1.0), the chances of a weapon with yield Y destroying a target of hardness H is

    Pk = 1 - .5((LR/CEP)2)

    Pk is called the kill probability of the warhead.

    If the reliability of the warhead is less than 1.0, then the probability of the warhead destroying the target must take this into account. The probability of the warhead destroying the target is:


    If the resulting probability of destroying the target is not high enough for you, then you may aim more than one warhead against the target. But there are very difficult issues of timing involved. If one warhead arrives somewhat earlier than another, the effects of the first explosion (the release of neutrons, the blast, the debris) may disable the next warheads or "push" them away from the target. The destruction of following warheads by those that arrive earlier is called fratricide. It is commonly assumed that two -- but no more than two -- warheads can be detonated close enough in time so that fratricide can be avoided.

    The KP program will calculate two different versions of the probability that 2 warheads will destroy the target. One value for Pk2 assumes that both warheads aimed at the target come from the same missile. The other value assumes that each of the pair of warheads comes from a different missile. This is known as cross-targeting. Cross-targeting is better because if both warheads come from the same missile, and that missile fails, the target will not be destroyed. If the warheads come from different missiles, and one missile fails, there is still a chance that the other warhead will work and destroy the target. The two values will differ, but at some times the difference is so small that is does not show up in the calculations.

    Additional Reading

    If you want more information on these calculations, I suggest you read the following:

    Davis, Lynn and Warner Schilling. 1973. All You Ever Wanted to Know About MIRV and ICBM Calculations But Were Not Cleared To Ask. Journal of Conflict Resolution 17: 207-242.

    Speed, Roger D. 1979. Strategic Deterrence in the 1980's. Stanford, CA. Hoover Institution Press.

    Steinbruner, John D., and Thomas M. Garwin. 1976. Strategic Vulnerability: The Balance between Prudence and Paranoia. International Security 1,1: 138-181.

    Tsipis, Kosta. 1983. Arsenal: Understanding Weapons in the Nuclear Age. New York: Simon and Schuster.

    Last updated - 2/17/10