SIOP

The Single Integrated Operational Plan (SIOP) was the United States’ general plan for nuclear war from 1961 to 2003. The SIOP gave the President of the United States a range of targeting options, and described launch procedures and target sets against which nuclear weapons would be launched. The plan integrated the capabilities of the nuclear triad of strategic bombers, land-based intercontinental ballistic missiles (ICBM), and sea-based submarine-launched ballistic missiles (SLBM). The SIOP was a highly classified document, and was one of the most secret and sensitive issues in U.S. national security policy. https://en.wikipedia.org/wiki/Single_Integrated_Operational_Plan

Research goal

We try to estimate number of warheads of Trident D5 required to kill 1380 hardened targets (potential enemy land based missiles in silos) given total number of SSBN (14) each with 24 SLBM. Each SLBM can carry 3, 8 or 12 RV (warheads) with 0.5, 0.3 and 0.15 MT correspondingly (initial plan for Trident D5 armament in the end of Cold War).

Terminal kill probability

Here we apply formula from Bruice G.Blair “Strategic Command and Control. Redefining the Nuclear Threat”, page 305

OAR<-0.95 #Overall reliability of attacking SLBM
Y<-c(0.5,0.3,0.15)# Yield, MT
CEP<-c(0.090*1.86)#Circular error probable, miles
H<-300#Hardness of target, psi
TKP<-OAR*(1-0.5^(8.41*Y^(2/3)/((H^0.7)*(CEP^2))))#Terminal kill probability of target

round(matrix(rbind(TKP,Y),2,3,byrow = F,dimnames = list(c("TKP","Yield,MT"),c("3","8","12"))),3)
##              3    8    12
## TKP      0.865 0.78 0.629
## Yield,MT 0.500 0.30 0.150

Linear equations

library(matlib)
## Warning: package 'matlib' was built under R version 3.3.3
A<-matrix(c(TKP,1/3,1/8,1/12,Y),3,3,byrow=T)
det(A)
## [1] 0.01166236
b<-c(1380,14*24,600)# Targets, SLBM, Total Yield
showEqn(round(A,3),b)
## 0.865*x1  + 0.78*x2 + 0.629*x3  =  1380 
## 0.333*x1 + 0.125*x2 + 0.083*x3  =   336 
##   0.5*x1   + 0.3*x2  + 0.15*x3  =   600
x<-as.integer(solve(A,b))
#Number of warheads by Yield required for 1380 targets to be killed
matrix(rbind(x,Y),2,3,byrow = F,dimnames = list(c("Warheads","Yield,MT"),c("x1","x2","x3")))
##             x1    x2     x3
## Warheads 624.0 768.0 383.00
## Yield,MT   0.5   0.3   0.15
sum(x) #Total number of warheads required
## [1] 1775

Gausian elimination

echelon(A, b, verbose=TRUE, fractions=FALSE)
## 
## Initial matrix:
##           [,1]      [,2]       [,3] [,4]
## [1,] 0.8653472 0.7798948 0.62853089 1380
## [2,] 0.3333333 0.1250000 0.08333333  336
## [3,] 0.5000000 0.3000000 0.15000000  600
## 
## row: 1 
## 
##  multiply row 1 by 1.155606 
##           [,1]      [,2]       [,3]     [,4]
## [1,] 1.0000000 0.9012508 0.72633377 1594.736
## [2,] 0.3333333 0.1250000 0.08333333  336.000
## [3,] 0.5000000 0.3000000 0.15000000  600.000
## 
##  multiply row 1 by 0.3333333 and subtract from row 2 
##      [,1]       [,2]       [,3]      [,4]
## [1,]  1.0  0.9012508  0.7263338 1594.7356
## [2,]  0.0 -0.1754169 -0.1587779 -195.5785
## [3,]  0.5  0.3000000  0.1500000  600.0000
## 
##  multiply row 1 by 0.5 and subtract from row 3 
##      [,1]       [,2]       [,3]      [,4]
## [1,]    1  0.9012508  0.7263338 1594.7356
## [2,]    0 -0.1754169 -0.1587779 -195.5785
## [3,]    0 -0.1506254 -0.2131669 -197.3678
## 
## row: 2 
## 
##  multiply row 2 by -5.700704 
##      [,1]       [,2]       [,3]      [,4]
## [1,]    1  0.9012508  0.7263338 1594.7356
## [2,]    0  1.0000000  0.9051460 1114.9354
## [3,]    0 -0.1506254 -0.2131669 -197.3678
## 
##  multiply row 2 by 0.9012508 and subtract from row 1 
##      [,1]       [,2]        [,3]      [,4]
## [1,]    1  0.0000000 -0.08942974  589.8992
## [2,]    0  1.0000000  0.90514596 1114.9354
## [3,]    0 -0.1506254 -0.21316689 -197.3678
## 
##  multiply row 2 by 0.1506254 and add to row 3 
##      [,1] [,2]        [,3]       [,4]
## [1,]    1    0 -0.08942974  589.89923
## [2,]    0    1  0.90514596 1114.93540
## [3,]    0    0 -0.07682892  -29.43023
## 
## row: 3 
## 
##  multiply row 3 by -13.01593 
##      [,1] [,2]        [,3]      [,4]
## [1,]    1    0 -0.08942974  589.8992
## [2,]    0    1  0.90514596 1114.9354
## [3,]    0    0  1.00000000  383.0619
## 
##  multiply row 3 by 0.08942974 and add to row 1 
##      [,1] [,2]     [,3]      [,4]
## [1,]    1    0 0.000000  624.1564
## [2,]    0    1 0.905146 1114.9354
## [3,]    0    0 1.000000  383.0619
## 
##  multiply row 3 by 0.905146 and subtract from row 2 
##      [,1] [,2] [,3]     [,4]
## [1,]    1    0    0 624.1564
## [2,]    0    1    0 768.2085
## [3,]    0    0    1 383.0619

Matrix form equation

\[\begin{array}{lllllll} 0.865 \cdot x_1 &+& 0.78 \cdot x_2 &+& 0.629 \cdot x_3 &=& 1380 \\ 0.333 \cdot x_1 &+& 0.125 \cdot x_2 &+& 0.083 \cdot x_3 &=& 336 \\ 0.5 \cdot x_1 &+& 0.3 \cdot x_2 &+& 0.15 \cdot x_3 &=& 600 \\ \end{array}\]

Conclusions

  1. This example is a fake story though equations are real.
  2. SIOP for SLBM could be planned using linear equations.
  3. Real values of variables for TKP should be verified with care.
  4. Total yield of 600 MT by 1775 warheads would bring catastrophe for our planet with no winner in nuclear war.
  5. Nuclear weapons must be banned forever!