Calculations relevant to Nuclear
Exchanges

The Nukes We Need: Preserving the American Deterrent by Keir A. Lieber and Daryl G. Press (PDF Link)
MS Excel 972003 Worksheet with preworked out equations based on this page available HERE.
The Circular Error Probable is actually the radius in which 50% of all weapons fired would land. The 95% Radius (R95) is the radius in which pretty much all of the weapons would land. To convert from CEP to R95; use the following formula:
R95 = CEP * 2.1
Where:
P = Probability of a Hit
R = Radius of Target
CEP = Circular Error Probable of Weapon
Example:
Given a CEP of 300 feet and a target radius of 100 feet, what is the Probability of a Hit?
1 – exp( 0.6931 * [ 100^{2} / 300^{2}] ) = 0.0741 or 7.41%
Reference:
Missile Accuracy (CEP) Excerpt from
Statistical Snacks (HTML
Link)
Explain the Circular Error Probable Formula mentioned
in this Article (HTML
Link)
SSPS: 0.5 ^{(LR/CEP)^2}
Where:
LR = Lethal Radius
CEP = Circular Error Probability
Example:
A missile with a CEP of 1,000 meters is attacking a hardened target. The lethal radius of the warhead against that type of target is about 250 meters. What is the SSPS?
0.5 ^{(250/1000)^2} = 0.957603 or 95%
LR = 2.62 * Y^{(1/3)} / H^{(1/3)}
Where:
Y = Yield in Megatons
H = Hardness of target in PSI
LR = Lethal Radius expressed in nautical miles.
Example:
A nuclear warhead with a yield of 1.2 megatons is attacking a target with a hardness of 10 PSI. What is the Lethal Radius?
Y: 1.2^{(1/3)} = 1.062659
H: 10^{(1/3)} = 2.154435
2.62 * (1.062659 / 2.154435) = 1.29 nautical miles
SSPK: 1 – 0.5 ^{(LR/CEP)^2}
Where:
CEP: Circular Error Probable
LR: Lethal Radius
Note: The measurements can be in Feet/Meters/Nautical miles; but the measurement system must be consistent for both variables!
Example:
A Nuclear missile with a CEP of 1.39 nautical miles and a Lethal Radius of 1.29 nautical miles is attacking a point target. What is the SSPK?
1 – 0.5 ^{(1.29/1.39)^2} = 0.4495 or 44.95%
TKP = R * SSPK
Where
R = Probability of the Delivery System AND warhead functioning correctly.
SSPK: Single Shot Probability of Kill
p(kill)_{n} = 1 – (1TKP)^{n}
Where:
TKP = Terminal Kill Probability
n = Amount of weapons targeted on target.
Example:
Five warheads are fired at a target for which the TKP of each individual warhead is 25%. What is the P(kill)_{n}?
1(10.45)^{5} = 0.7627 or 76.27%
EMT = N x Y^{2/3}
Where:
EMT: Equivalent Megatonnage against Soft Targets
N: Number of ReEntry Vehicles or Bombs carried
Y: Yield of each bomb or reentry vehicle in megatons
Example:
The SS90 SYPHILLIS missile carries 30 x 550 kiloton warheads. What is it's EMT?
30 x 0.55^{2/3} = 20.14 Megatons of EMT