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

Trident II D5

The Trident II (D5) is designed to carry either eight heavy RVs (the 475 kiloton Mk-5), for which it has a maximum range of roughly 7,400 kilometers, or up to 14 lighter RVs (the 100 kiloton Mk-4), With eight Mk-4 or four Mk-5 RVs the Trident II has a range of roughly 11,000 kilometers (Aldridge 1983),(Gronlund and Wright 1992).

Research goal

We try to optimize number of warheads of Trident D5 required to kill maximum of soft hardened targets (potential enemy land based bombers) given total number of SSBN (2) each with 24 SLBM. Each SLBM can carry 8 or 14 RV (warheads) with 0.475, 0.1MT correspondingly. The total yield of warheads must not exceed 100 MT.

Terminal kill probability

Here we apply formula from (G.Blair 1985) making assumption for depressed trajectory (DT) which gives \(420<CEP<980\) or mean value \(CEP=700\) meters and 6 minutes of flight by 1850 km distance from launch point. For soft targets we take 5 psi (pound per squared inch) overpressure (Gronlund and Wright 1992). For more about Trident D5 reliability see (Levakov 2016). 

OAR<-0.95 #Overall reliability of attacking SLBM
Y<-c(0.475,0.1)# Yield, MT
CEP<-c(0.700*1.86)#Circular error probable, miles
H<-5#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),2,2,byrow = F,dimnames = list(c("TKP","Yield,MT"),c("MK-5","MK-4"))),3)
##           MK-5  MK-4
## TKP      0.468 0.203
## Yield,MT 0.475 0.100

Linear optimization with simplex

library(lpSolve)

f.obj<-TKP
f.con <- matrix(c(1/8,1/14,0.475,0.1),2,2,byrow=T)
f.dir <- c("<=","<=")
f.rhs <- c(48,100)
lpans<-lp(direction = "max",objective.in = f.obj,
   const.mat = f.con,const.dir = f.dir,const.rhs = f.rhs)
lpans
## Success: the objective function is 148.6508
lpans$solution
## [1] 109.3333 480.6667

Linear equations for optimum solution

library(matlib)
## Warning: package 'matlib' was built under R version 3.3.3
A<-f.con
b <- f.rhs
showEqn(round(A,3),b)
## 0.125*x1 + 0.071*x2  =   48 
## 0.475*x1   + 0.1*x2  =  100
x<-as.integer(solve(A,b))
#Number of warheads by Yield required for maximum targets to be killed
matrix(rbind(x,Y),2,2,byrow = F,dimnames = list(c("Warheads","Yield,MT"),c("MK-5","MK-4")))
##             MK-5  MK-4
## Warheads 109.000 480.0
## Yield,MT   0.475   0.1
sum(x%*%Y) #Total yield of warheads required (MT)
## [1] 99.775
plotEqn(round(A,3),round(b,3),vars = c("MK-5","MK-4"),xlim=c(0,500),ylim=c(0,800))
## 0.125*MK-5 + 0.07MK-4  =   48 
## 0.475*MK-5   + 0.MK-4  =  100
grid()

Conclusions

  1. The optimum set of warheads to kill maximum of soft ground targets (148 bombers on the ground) with 2 SSBN (48 SLBM) yields 109 MK-5 and 481 MK-4.
  2. Depressed trajectory takes 6 minutes for RV delivery by 1850 km and leaves no chance for early warning while salvo launch of 48 SLBM would take more time.

References

Aldridge, Robert C. 1983. First Strike!: The Pentagon’s Strategy for Nuclear War. Book. South End Press. https://www.amazon.com/First-Strike-Pentagons-Strategy-Nuclear/dp/0896081540.

G.Blair, Bruice. 1985. Strategic Command and Control. Redefining the Nuclear Threat. Book. Brookings Institution. https://www.amazon.com/Strategic-Command-Control-Bruce-Blair/dp/0815709811.

Gronlund, Lisbeth, and David C. Wright. 1992. “Depressed Trajectory Slbms: A Technical Evaluation and Arms Control Possibilities.” Journal Article. Science & Global Security, no. 3. Gordon; Breach Science Publishers S.A.: 101:159. https://inis.iaea.org/search/search.aspx?orig_q=RN:24064890.

Levakov, Alexander. 2016. “Trident C4 and D5 Reliability by Binomial Test.” Paper. Academia. https://www.academia.edu/29635787/Trident_C4_and_D5_reliability_by_binomial_test.