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
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.
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
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
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
R = The Reliability of the weapon.
The reliability of the weapon is the probability that the weapon will
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.
If you want more information on these calculations, I suggest you read
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