Transcript
Calhoun: The NPS Institutional Archive Theses and Dissertations
Thesis Collection
1965
Optimum ballistic missile trajectories and associated optimum intended height of burst of a warhead. Stillings, Thomas J. Monterey, California: U.S. Naval Postgraduate School http://hdl.handle.net/10945/12776
NPS ARCHIVE 1965 STILLINGS,
T.
OPTIMUM BALLISTIC MISSILE TRAJECTORIES AU-"> ASSOCIATED OPTIMUM INTENPEO HBGHT OF GU *tST OF A WARHEAD :
81 SHKir
liil«]^r_
I.
STILLINGS
DUDLEY KNOX LIBRARY STGRADUATE SCHOOL MONTEREY CA 93943-5101
yS
DUDLEY KNOX LI NAVAL POSH MONTEREY, CALIFOR..
?
OPTIMUM BALLISTIC MISSILE TRAJECTORIES
AND ASSOCIATED OPTIMUM INTENDED HEIGHT OF BURST OF A WARHEAD
* * * * *
Thomas
J.
Stillings
OPTIMUM BALLISTIC MISSILE TRAJECTORIES AND ASSOCIATED OPTIMUM INTENDED
HEIGHT OF BURST OF A WARHEAD
by
Thomas J. Stillings
Lieutenant, United States Navy
Submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE IN OPERATIONS RESEARCH
United States Naval Postgraduate School Monterey, California
19
6
5
Library S. Naval Postgraduate School Monterey, California
U.
OPTIMUM BALLISTIC MISSILE TRAJECTORIES
AND ASSOCIATED OPTIMUM INTENDED HEIGHT OF BURST OF A WARHEAD by
Thomas J. Stillings
This work is accepted as fulfilling the thesis requirements for the degree of
MASTER OF SCIENCE IN
OPERATIONS RESEARCH from the
United States Naval Postgraduate School
ABSTRACT A probabilistic model is developed for treating the
problem of optimizing the intended trajectory and associated height of burst of
a
missile-warhead.
is treated as a control variable.
The angle of re-entry
Computational techniques
which may be operationally acceptable are described.
11
TABLE OF CONTENTS Title
Section I
II
III IV
V
Page
Introduction
1
Construction of the Probabilistic Model; Mathematical Character of the Problem
9
Computational Problems
13
Simplified Approach to Obtaining Approximate Solution: Graphical Methods and Correction
18
Conclusion
33
in
LIST OF ILLUSTRATIONS Figure
Page
1.
Trajectory contours for 10 degree re-entry angle
2
2.
Peak overpressures on the ground for burst
4
3.
Destruction envelope
4.
Sketch of trajectory and co-ordinate system
17
5.
Sample of method employed in isolating optimum-intended burst point
19
6.
Graph of percentage error in predicting probability of destruction vs angle of re-entry
27
6A.
Percent correction to graphical approximation
30
6B.
Percent correction to graphical approximation
31
7.
Plot of actual destruct angle and polynomial fit
32
a
1-kiloton
6
IV
.
INTRODUCTION
I
Consider the problem of setting the fusing mechanism of
warhead aboard
a
a
ballistic missile in order to optimize the
probability of destroying
a
target.
The general problem as stated above becomes extremely
complicated when one considers such things as missile in-flight failures, warhead failures, back-up fusing mechanisms employed, type of warhead and type of burst, distance from launcher to
target (which will affect range and Azimuth errors), geodesic error,
(i.e.,
to what degree of accuracy do we know the
distance to target.)
and more frustrating, the problem of
,
partial kills, i.e., the partial destruction of the target or a
portion of the target for
a
period of time less than the
duration of hostilities. In order to reduce the problem to something that can be
handled, it will be convenient to cover briefly three topics: the concept of ballistic coefficient,
the reflected (mach)
wave phenomenon and the definite range law (cookie cutter) Ballistic Coefficient
:
For our purposes, we may think of the ballistic coefficient as
a
parameter which describes how much
a
re-entry
vehicle will be affected by the atmosphere (see Fig. 1).
It
is proportional to the ratio of the weight of the vehicle to
its area, i.e., the square of the maximum diameter (see Ref. 4);
e.g., for
a
given area, an increase in weight will tend to 1
*
30
40 Range
Fig.
I
—
50 to
impact
60 (thousands
70
80
of ft)
Trajectory contours for 10-deg re-entry angle
"straighten" the trajectory whereas result in
a
decrease in weight will
slower re-entry and thereby increase the time spent
a
in the gravitational field.
The result will be
a
more curved
trajectory.
Mach Effect
When
:
warhead is burst in the air close to ground (e.g.,
a
300 feet for
a
1
KT warhead), there is
a
region of unusually
high overpressures due to the merging of the incident and reflected waves (see Ref. the Mach Region.
1
and Fig 2).
This region is known as
Because of this the curves in Fig.
2
are not
monotonic but have the unusual shape exhibited there. Definite Range Law
:
This is the "cookie-cutter" concept that is frequently used.
Because of its simplicity it seems to give useful
results for many objectives. A couple of examples should be sufficient for our purposes.
Consider
a
point target in space at which we are firing
jectile with
a
kill range of one mile.
be at the center of
a
a
pro-
Imagine the target to
sphere of radius one mile.
If we detonate
this weapon anywhere within the sphere we consider the target
destroyed; anywhere outside the sphere is considered course, no weapon behaves in such
a
manner.
a
Of
miss.
"Partial kills" are
generally the rule rather than the exception, and they are very difficult to quantify. convenience.
The definite range law is
a
mathematical
Presumably, this range can be regarded so that
a
controlled fraction of partial kills will be treated as misses 3
i
1.000
-3
r
«
900
>
> w
800
W H W u
700
o eoo
s m
w tr > 1
500
o K O w
s
400
< m
300
a w
o f
200
w
5 100
100
200
'
300
400
500
600
700
800'
900
1,000
1,100
1,200
1,300
1,400
1.500
DISTANCE FROM GROUND ZERO (FEET) Figure 3.67a.
Peak' overpressures on the ground
.
fotf
FIGURE
a 1-kiloton burst (high*pressure range).
2
-
T^g. 2~repYesents
a destruction -curve for various overpr'es'su^es/ .of a ^7' KT weapon. For a weapon of different;' yield, W ^the'/ording te, and 'abscissa are each multiplied by W-*.'. See Ref. 1. For example^'/a '600.' foot high burst of a 1 KT weapon, the 20 P.S.I, curve will', ex texid to about 940 feet' from ground zero. Similarly, a 1 MT weapon det'o'natedi at .'O.OOO'/feet will have a ,v P.S.I, 20 overpressure to about 9,400. feet. '
'
1
•
,
'
'
'
1
:
;,'
''
'
(
.
(and
a
controlled fraction will be scored as hits).
As another example consider Fig.
5.
If we imagine
at (0,0) and estimate that it takes about 20 P.S.I,
a
target
of over-
pressure for destruction, the 20 P.S.I, curve shown could be considered as
a
"cooker-cutter"; i.e.,
a
burst just inside
will destroy the target and one just outside will not. the following three
In light of the above discussion,
simplifying assumptions will serve as the basis of this paper: 1.
The re-entry vehicle has
a
sufficiently high ballistic
coefficient so that the trajectory may be assumed to be straight line in the neighborhood of the target.
a
Moreover,
the vehicle is launched from a sufficiently great distance so
that all possible paths may be assumed to be parallel in the
neighborhood of the target. 2.
The overpressure contours of Ref.
are valid and the target damage is
a
1
(and hence of Fig. 2)
function of overpressure
alone 3.
A definite range law holds and 20 P.S.I, is necessary for
destruction. The general objective is to select an optimum height of
burst.
Further examination of the problem reveals that some
other parameters enter into the picture.
The angle that the
vehicle's trajectory makes with the surface can be selected in advance.
There is an advantage to small angles since then
the missiles are more difficult to detect (and hence defend
against).
Larger angles generally lead to higher kill
FIGURE
3
DESTRUCTION ENVELOPE
.
The capability to vary this parameter will make
probabilities.
defense more difficult. burst point becomes
a
Given
a
trajectory angle, the optimum
function of height of burst and aim point
(intersection of the intended trajectory with the surface)
Locating the target at (0,0,0) and using the Cartesian system wi th )(,=
downrange distance
Yx =
crossrange distance
A3= vertical and introducing parameters
-0-= angle the trajectory makes with the surface
S
= distance from aim point
M
= intended height of burst
the problem now may be stated:
(see Fig. 3),
Given-^
to target
,
choose the pair
(
5
,
LK. )
so that the probability
of target destruction is maximized.
This paper must be viewed as
problem outlined above.
a
pilot study for solving the
It will be noted that
a
large amount of
detailed work remains to be done in order to provide good answers The methods herein can serve as outlines and guides. One of the more important results is the development of
graphical computation system using dividers, graphs of over-
pressure functions and correction curves.
Thus,
the optimum
solution may be obtained under operational conditions without the need of
a
digital computer.
a
The organization of the report is as follows:
The mathe-
matical model and nature of the problems of mathematical analysis are presented in section II. are described in section III.
The computational problems
The operational hand approxi-
mations and corrective curves are given in section IV.
The
The Monte Carlo technique
conclusions appear in section V.
employed, along with the computer program, appear in Appendices
A and B respectively.
Appendix
C
contains
a
tabulation of
the parameter values determined by the graphical method de-
veloped in section IV.
,
II
CONSTRUCTION OF THE PROBABILISTIC MODEL; MATHEMATICAL CHARACTER OF THE PROBLEM
We have already introduced the coordinate system; the angle the trajectory makes with the surface; the intended
height of burstM is actually
a
;
the aim point
(5
,
0,
0).
The trajectory
random phenomenon and by assumption
1
can be
characterized by the impact point and the incidence angle
-Q~
.
The actual height of burst is also subject to random errors. Thus, we let (Y^
,
Y2
)
,
be the random vector representing the impact
point (X\
,
X2
,
X3
)
be the random vector representing the burst
point
A sample situation is given in Fig. 4. It is assumed that:
Y]_,Y2has
!•
£
a
circular normal distribution with mean vector
and covariance matrix 0\, I
,
where
I
is the identity
matrix. 2.
X3 has
a
normal distribution with mean U
independent of Yi
,
,
variance^ 2 and
and \.
From Fig. 4 it is seen that
Xi = Yi
X2 = Y2
-
X3 cot
"6"
is
It follows that the probability density function of the
detonation point is
~
-y-,
,
fetfT*)** fc 4*j
(2.3)
77 *
fffftrf+XiCote-ffa^jJxM A
10
Examination of the character of equations (2.4) is facilitated if we make the change of variables (2.5)
y,
-
X
U.
yz -
,
_
,$-->£
and let
In this coordinate system the set A is transformed into
a
set B
the exact nature of which is difficult to visualize (see Fig.
2),
but we can say B is bounded since the transformation is linear.
In the new system, equations (2.4) may be expressed as
Geometrically, the problem now may be viewed as follows: a
set B, choose a centering point
(
J
,'-'•'
,/'0
Given
so that the
probability of B is maximized. All solutions of (2.6) are critical points which may be local maxima, minimum or saddle points.
The nature of these
points may be examined by means of second order derivatives.
LettinE
,,
,,*
we can write (2.8) (2.9)
-tffvi-dV.Vtf"?? i
^--JJTtOS-^-^^^.^^'^^ p/Mf f(f [(Wi'tyM*. \,ii) H**A* =
li
A critical point will be (2
-
n)
%
%a
'
%
a
local maximum if
and d
><*
It can generally be said that Daam
(Ref
a^ < ^
(and D,
reason) is negative since the variainee
-
-
,
3 p
-
232)
for the same
ffr(%-s>y(iw)Wii!
of Y3 is one and (2.8) compares a truncated (by B) version of
this variance with unity.
The remaining part of (2.11) is
a
difficult question, however, and may require more specific in-
formation about the set
B.
The set B plays a role in determining the number of critical
points.
For example, if B were
sufficiently long dumbell
a
shaped region, there would be at least two solutions to (2.6). It would be interesting to examine the question of number of
With this condition it may be
critical points if B were convex.
This contingency was not
possible to show there is only one.
examined since the set A (and hence B) is not convex. Since the set A is formed by rotation of Fig. X3
2
about the
axis, it can be shown that
(2. 12)
ttfx n
x -f
(*, t
^
*, )
dx dxx {
J/j -
The same must be true in the transformed coordinate system, and it follows from (2.6) and (2.12)
that
(J
,
0,iU) will be the
center of gravity of the conditional probability density given (Y]_,Y2 ,Y3
)
belong to B (if
(2.6)).
12
J
and
M are
unique solutions of
Ill
COMPUTATIONAL PROBLEMS
Computer techniques are required for the solution of the system of equations (2.4)
,
the examination of the critical
points (2.11), and the determination of the optimum probability It was decided to use Monte Carlo
of target destruction (2.2).
techniques (see Appendix A) in this study, and the first step is to characterize the set A.
The curve in Fig.
2
may be viewed
as giving distance as a function.
This curve is not given in analytic form.
height, i.e., g(Xo
)
values for which g(Xo
that there is
a
small interval of X3
ignore the double valued feature and fit order
7
It was decided to
is double valued.
)
Also, note of
a
polynomial
by the method of least squares, using 23 values read
directly from the graph.
This fit appears in Fig. 8.
the polynomial as g(X 3 ),
the
f
A =
(3.1)
of
|
:
2
Xx+ X 2
2
£
g
(X 3 )l
(2.4) and (2.11) all are similar
The integrals in (2.2), in nature.
set A may be represented by
2
X 1 ,X 2 ,X 3
Using
A rapidly converging iterative technique may
generally be available for the solution of the system (2.3).
Letting
y
,
-
X LY * 2
the equations (2.4) may be written
°
=
tffarfUt A
cor
6-)
p(te±^*)0(£)fl&Jd)rMJ* '
13
Let functions gl(/U J
)
a nd
g2(/vv , /
/
)
be defined so that the above
equations are
Then, using an initial approximation of AA
f
we can use
the iterative scheme
(3-3)
/"*,--?,
(^,,
/J
,
fH ,*?*fa?*)
provided it converges. The examination of the question of convergence usually
goes along the following lines:
If the series
(3.4)
converges absolutely, then
AA^
and
f converge
numbers which will be the solutions of 2.4. place if
a
constant r
(
0
^
*
^%
.
?,fa.sj--£U[y
i
o 14
f(y
l
-f*)j
I
Proceeding as indicated, (3.7)
From the theorem of the mean
0(tl- 0(f)
i
U-t'l ^wf(f)
it is easily shown that
and this value may be used in 3.7. |
4
(/, -/*,)-
4
(/,
Thus we can write
-/*,.,)! * P
\h /V-<
\0(Y,-f,)-^Y.-f/ .,)lt(>\?
l
-
?ri
l
Sharper bounds may be available if the maximum is 3.8 and the
points
(
F
,0 >/^«) are constrained to be in the set
B.
second term of 3.7 can be estimated in a similar fashion:
which can be treated as above.
15
The
Collecting, we have
,
.
, ,
(3.10)
which will have the form (3.5) if it can be shown that the coefficients are less than one.
This is
a
difficult analytic
problem and is beyond the scope of this paper.
The technique
(3.3) was used in our computations and lend to answers that
were stable in the light of our capability to compute by Monte Carlo. The signs of the determinants (2.11) were computed at the solution values and all were found to be positive.
16
ACTUAL BURST
X,
FIGURE 4 SKETCH OF TRAJECTORY AND CO-ORDINATE SYSTEM
17
IV
A SIMPLIFIED APPROACH TO OBTAIN AN APPROXIMATE SOLUTION; GRAPHICAL METHODS AND CORRECTION
If the down range and cross range errors are assumed to be small, the problem can be reduced to a simple two-dimen-
sional model since we intuitively feel that
a
'best'
trajectory
is one that passes directly over target.
At this point we will assume we follow our intended trajectory with probability
1
and find values of AA and
>
that
optimize this conditional probability of destruction. If Fig. 2 were reflected about a vertical axis, Fig. 5
will result.
Again, assume the target to be at (0,0) and we
wish to detonate our weapon inside the 20 P.S.I, envelope. We will make the following assumption to supplement those in section III:
The warhead passes directly over target in
a
straight
line trajectory making an incidence angle of 0"with the surface.
The collection of possible trajectories will intersect the 20 P.S.I
envelope as shown in Fig. given trajectory) by
a
Denote the lower intercept(of
5.
and the upper by ©(
.
The maximization takes place in two stages.
0
First, holding
and J fixed, find the best value of A/( as a function of J
Second, vary
procedure (4.1)
f
,
)
until the maximum is obtained.
let
(M)
.
r
—'—
f
n 18
a
,
>
l
o CO S3
M Q
o
§
w CO
Taking the derivative and equating to zero
Making the substitution
t
£
±£
r
j^
iff
Y-u
=
T?
Jf-o
which is true for all
(
} Q*
*(
The value of F at the point Aimax which is the probability of destruction given a particular line is given by
nwZ*r €*»*> ^
Le tting
FfW
y-
_
«^£fl
= tC
,/rr >
*
^5f
Oi
A
1(T or 1
.x
(*•»
FW* M ^fee^f-i.o ^ ..
So far we have the optimum height of the IBP, but still
must find out where to plan the detonation, i.e., to select f
.
Since we are interested in maximizing the probability of
destruction (4.5), we see this is clearly equivalent to
maximizing (
o(
(
d^
-a)
since the function is monotonic.
Maximizing
-a) is in turn equivalent to finding the longest line that
can be drawn through Fig.
5
at an angle
&
.
In Fig. 5, a series of parallel lines were drawn roughly every 20 feet at an angle-©- to the horizontal and the longest
21
.
line determined by the use of dividers, and choosing the one
C
with max
y
o^
-
7 C(
^ If
,
[
a
maximum occurred between lines it
was deemed to be at the mid-point of the two which gives an
"accuracy" of 10 feet for
a
1
KT weapon or 100 feet for
a
MT
1
weapon By assuming the measurements were accurate in the sense that one can differentiate between a line being longer than an
adjacent line (which is reasonable since the vast majority
were easily discernable) we can say with probability one, the
maximum error is 10 W
'/J
feet for a W KT weapon.
Measurements were taken in this manner for several values of
-O"
and the resulting values of
ilmax
were plotted against S~
•
Readings were concentrated at points where the curve appeared
A similar procedure was used for values
to be rapidly changing.
of
©
vs
/
and
of degree 1,
0- vs
2,
,
\
/
V
.
in order to interpolate polynomials
20 were fit in a least-squares sense to
The degree chosen from the one with the
the data points.
smallest
f
-
Q
I
where
>
and Oi = computed value.
/
^
=
emperical value
The above criterion was chosen since
it was planned to interpolate values of •0-=5,
6,
7,
,
90
and the measurements were felt to be fairly accurate to start
with and large deviations in the fit would only tend to aggrevate the original data points. The polynomials were fed into the computer and evaluated at the points "©"= 5, 6,
90.
22
The results were tabulated
.
in table
1
where:
Jimax = planned burst point
measured vertically along HOB axis
XCOORD = planned burst point measured horizontally along "DISTANCE FROM GROUND ZERO" AXIS -0*= Angle trajectory intersects ground measured from the
horizontal J =
Aim point measured positively to the right along "DISTANCE FROM GROUND ZERO" AXIS
REMARKS CONCERNING THE GRAPHICAL APPROXIMATION The most interesting discovery is that the 'best' IBP is
never over target; for low angles of re-entry, the burst is planned prior to target; for steep angles of re-entry, the IBP is past target.
The sharp discontinuity in XCOORD at about 28° is caused by
entry into the envelope in the region (850, 700) of Fig.
5
where
the slope of the envelope and the slope of the trajectory are
nearly equal We will next apply the correction discussed in section III. The method developed above was used as the first approximation for the iterative technique developed in section III.
details follow.
23
The
,
L
«
1Z P
J
-
i
ff
r-
Solving for AA
let
>CH
J
l*it)*C
>
10°
= 10°.
Data was collected for
-©• =
sets of the standard deviation
percent of error in
f or
M
(
10°, 20°,
-----80° for two
(H
Table
and f and Fig.
(JT
)
.
contains
1
a
contains the plot of $- vs
percent of error. Thus, operational procedures can be developed that do not
require a high speed computer.
24
The first approximation is
f>;
lA
II
c>
»
tf
few I
o
00
o
* -4
o
o m H M PQ
o
O fa
o o
U D M Q fa W O
CO
v£3
o
Oh W s o H S
M fa
w g o W M
/ C_>
'?.
CO
w W Q CM Pi
-V
•
•
1
\
1
1
1
1
p c*oor^.v£>m^cooj _| ^^ _l _J _j
CM
i-H
i—
r~|
«5>
1
\
1
1
f
,
,
,
,
ONOOr^^oLTivtrooM
O U u
'
f
w
B*S
27
_m
t
t-l
.
o
fa
fa O O
.
obtained by the use of dividers applied to graphs like Fig.
5.
The results are corrected by curves like Fig. 6.
As was expected the method of section III gives a more accurate first approximation when the ballistic sigma is small.
Probabilities of destruction can be predicted more accurately for higher values of 6".
In general, the method of section IV
gives values of AAand J that are low and nearer to target for small -6>"(©-
—
40°)
;
high and further from target for higher
-Q-
This has high intuitive appeal since if re-entry is at small angle we tend to shoot for the point in Fig.
destructive envelope intersects the ground.
a
where the
5
But if ballistic
errors are allowed, we run a high risk of having the vehicle impact with the earth before ever reaching target.
Similarly,
for steep angles of re-entry we tend to aim our errorless
vehicle to pass thru the highest point on the envelope since the vertical distance traversed is a maximum.
But if we allow
errors, we run a high risk of passing to the left of the
envelope of Fig.
Appendix f or
£* =5,
5.
contains
C
6,
7,
probabilities.
list of starting points (AA
a
,
,
f. )
90 along with associated destruction
If one chooses to use the curves given in
Fig. 5, the destruction probabilities given may serve as a
check on the final probabilities, the percent of difference
decreasing as
(ji
decreases.
28
.
Outside of Appendix C, all probabilities were computed
using the polynomial fit to the curve since the curves were not believed to be accurate in the first place and we were
interested primarily in technique. Graphs of the polynomial used and the actual curves are found in Fig.
7
29
X O
o o
o o
t
V,
o 1-1
X o oi Pm p->
<
M
<
CM
o
O M Pn
cT
o H
13
O M H U H O H W w Cm
>e>
s
f r^
CSo
fise
Us
c^
~
O
o S-i
to
L
— 13
o M
H o pd Cu
< <:
a M a Pm
PQ
vD
0^ A
K
=k A
X O
o o
w
o o CM
o M En
\S
W
o H M H W
C_>
O
H S3
W 3 w PL,
o
or I
__o
4 -1
5^
o
8 o J-l
w
*S
5"*»-
O
O
•
"^^^1
-x^^
i
i
>
^r>-^ ** e
'X,, x)
X
m/
'
!
J '
/*
r
i
.
,
\ v>
i
)
/
-
'
> -
-
0
~~^
382
0S2;
m
323
*
036
g(x3 )
-H R-l
FIGURE
7
PLOT 0? ACTUAL DESTRUCT CURVE AND FIT K-SCfiLZ -
(both scales in hundreds of feet) UNITS']
Y-SCflLE - 2.08E+02 UNITS/INCH.
STILLINGS,ToJ
pia'
V
CONCLUSIONS
The optimum probabilities of target destruction, and the
optimum values of Table
2
A*\
and
for parameter
appear as a function of
j
(
*
— 1
be
f^W^
The required integral is therefore
in our case. 2
FfAix)]'
"^*
^_
n
"h
(X i )
USE OF PROGRAM The following procedure is recommended for finding 1.
M.f and
PKILL:
Function X GOF must be rewritten to fit destruction curves
employed. 2.
Values of(pand(T)
must be changed in the main program to
suit the weapon employed.
36
.
3.
Since only curves for
1KT weapon are generally available,
a
the problem must first be solved for this case using as 1
approximation the value of section IV
W
((J*
(M
,
f
)
a
first
found by the method of
and the parameters/^ f and (f must be multiplied by
is not
a
function of the yield since it is inherent
in the fusing mechanism)
Conditional probability of destruction given the "best" trajectory when
THETA
=
angle of re-entry
A
=
lower intercept of trajectory with destruction envelope
ALPHA
=
upper intercept of trajectory with destruction envelope
Umax
=
XCOORD
-
down range horizontal coordinate of IBP
J
=
down range distance from target the trajectory
vertical coordinate of IBP
intercepts the ground
37
APPENDIX
B
PROGRAM PROB TYPE REAL MU
DIMENSION MU(50),SI(50) PRINT 11 FORMAT( 1H1 PRINT 14 14 F0RHAT(4Xi HI, 5X.5HMU ,5X, 5HSI ,10HEXPOFXl 1.10HPKILL = 670. $ MU(1) = 425. $1 = 1 SI 1 A=.(2.*3. 1415926535)»*1.5 B=l./A 152 THETA * U0./57. 2957795131 153 SIGMA =200. $ G=l./F $ F =SIGMA*SIGMA CC=TANF(THETA) $ C = l./CC S1GMA1 = 500. 11
)
1
,10HEXPOFX3
)
(
)
SIGMA1»SIGMA1
D = E =
l./O DO 777 LL =
1,70
N =
VS1 = VS2 = VS3 = 0. 160 CONTINUE Zl = RANF(-l) $ Z2 = RANF(-2) $ Z3 = RANF (-3) XI = 2000.MZ1-.5) $ X2 = 20 00.* Z2-.5) $ X3 = 1000. *Z3 (
X1*X1 $ Y2 = X2»X2 1023 W=XG0F(X3) ww=w*w IF(Y1+Y2.LE.WW) 101,102 Yl
=
102 V = VI 101
V2
=
=
0.
$
CONTINUE H =SIGMA*D P=1./H$Q=B*P
Y3
$
RR =
X3*X3
GO TO 575
Xl-SK )+X3*C 167 U=EXPF(-.5*UE*( S + R) )+T«G) 161
=
I
R=RR*RR$S=X2*X2$TT=X3-MU( I)$T=TT»TT
V
=
Q*U
$
575 CONTINUE VS1 = VS1+V N
=
=
VI $
Xl«V
$
V2
=
VS2 = VS2+V1
)
X3*V $
VS3 = VS3+V2
N+l
IFIN.LT. 10000) 160,500 XN=N PKILL = 40000000C0.*VS1/XN El = 4000000000. »VS2/XN E3 = 4000000000. *VS3/XN PRINT 15, I,MU(I) ,SI( I),E1,E3,PKILL 15 FORMAT 1X,I4,F10.5,F10.5,F1D.5,F10.5,F10.5>
500
|
(
I
=
1
MUU) SKI)
+
1
= =
777 CONTINUE END
E3/PKILL (El+MUtI )*PKILL*C)/»KILL
FUNCTION XGOF (Y) 1022 Bl= 71.1229667E CI
B2=-31.7485590E-C1
B3= 66.9311997E-C3
B4=-46.2518018E-C5 B5= 14.8951096E-C7 B6=-24.1298172E-10 B7= 19.0555101E-13 B8=-58.4969877E-17 ((((((( B8*Y)+B7)»Y+B6)»Y+B5)*Y+B4)»Y+B3)*Y+B2)*Y+B1 XGOF = END
APPENDIX C
J;
THETA
ALPHA
J
XCOORD
max
.
I.
I 9 10 ii
12 13 14 15_
lo 17 18 19 20_ "21
22 23
24 25
499.999 424.907 189.185 369.360 351.793 329.999 269.999 235.552 201.599 169.999 114.999 89.067 61.310 2 9_l2_9JL ""•00*t)0u
-00*000 -00*000 -00*000 QQ.QQO
631.038 630.126 629.931 630.912 637.916 644.412 652.994 663.553 TT5T8-59-
34 -35 37
38 39 40
ET
47 48 49
V
494 487 479
60. 080
4899
46.210 1.486) 47.129 77.037
•3865 .3178 .2712
(
•
"4o8~
"~90."5crar"
454 438 420
92.987
1W
91.176 95.059
TGZTrzer 111.540 117.810 118.230 111.320
36.9
356 346
235L
-1096
-964 -849 -753
3y-
9oT4'4U"~"
333 334 336
84.139
JUL
Tiy
821.032 828.780 835.665 841.917
421 427 432 437
853.488 859.259 865.257 871.596
446 450 454 457
"F7373T8"
~%o(T
885.401 892.?48 900.205 907.564
462 463 465 466
39
2332
-1923 •1738 -1564 -1399
91. 381
401
80_2.227
W777B-
42 43 44
505
"8127226"'
764.841 778.468 790.967
30.023
587 531
3^0 352 359 366 374 "3BT 388 395 402 408
689.581 704.321 719.641 735.102
r
50
634.545 634.304 633.962 633.26I
"14*1737^
32 33
N
660.099 653.707 646.386 640.607 636.991
302T32T
2'0 • 0^3 103.780 1
•
i
-
T5ST -5?6 -596 -620
-
ro7o"
.117.540 .131.330 ;i_44.830_
•687 •704 .716
725
"i~5'/TBY0
729"
•170.220 182. 160
730 •728 •724
193.820 •205.430 -2177220"
-229.370 -241.960 -254.920 -263.120 =28TT25Cr -293.940 -305.720 -316.090 -324.550
.
719
-7w -709 •705 .702 .700 ~98
%s
12
>99
1.000
S 15 16 17 18 19
.
20 21 22
u 25 26 27 28 29 30 31 32
100 .576 .760 .814 .834 .851 J? 75 .9 04 .932 .95^ .969 .Q7Q .986 .990 .993 .996 .QQ7 .999 .999 1.000 1.000 .999 .999 .999 1.000
3 35 37 38
40
42 m
44 ^5 47 48 49 A 50
200
500
1000
.311 .443 .492 .511 ^530
.127 .186 .209 .218 .227
.064 .09^ .105
X
W7
.9«tt
.780 .803 .824 .845 .86? .889 .909 .923 .925 .90? .909 .915 .922 .928 /q?4 .939 .9^.948 .952 .655 .958 .960 .962 .963 .Q65 .966 .967 .968 .969 .071
.
if
yf
\
40
'
' 1
9 AU
•
.595 .633 .681 .720
-
36X
1,
'
.972 .973 .974 .976 .977
...
,
•
.261 .285 .310 .335 .357 .376 .39^ .412 .430 .452 .477 .502 .521 .524 .4Q3 .501 ,510 .519 .528
.110 .115 \
..
'
1
??
.132 .145 .158 .171 4j-a3__ 1,1 ^^^^ .194 .204 '.213 .224 .??6 .250 .265 .277 .278 .7^0 .265 .270 .275 .281
—
.
<^R
.OR?
.5^7 .556 .564 .571
.583 .588. .593 .5977
.292 .298 .303 .308 119 .315 .319 .321 .324
.603 .607 .610 .613
.328 .330 .323 • 335
.
.
<7fi
^nn
.A17 .620
.624 .628 .632 .636
.
•
•>oA
• «a«a-7
.339 .3^2 .3^5 .3^7 .350
APPENDIX C (Con't)
THETA
A
ALPHA
00.000
51 52 53
909.663 911.76? 921.521 930.684 932.232
/Jmax
XCOORD
466 467 467 467 468
-330 -333
4&8 468 467 467 467
-315 -303 -289 -273 -256
62 63 64 65
466 465 465 464
-223 -207 -194 -I83
£5
464 464 464 465 467
-174 -169 -165 -164 -164
ZJ70
474 ^79 484 490
Ti64 -165 -164 -160 -154
494-
TT2+4
495 493 484 468
-130 -113 -937 -732
TO
I53J -374 -269 -241 -297
5^ 55
3?
. •
933-000 933.000
57 58 59 60
67 68 69 70
i
71 72 73 74 75 75 77 78 79 80
Si
!
>.
!
82 83 84 85
401 350 288 221
H5
135 116 124 221
87 88 89 90
468
41
^ 695 689 681 669 653
.33^ -331 -324
1530 -588 -682 -538 -105
633"
•
611 585 356 526
466 438 412 389
•
.
•
'
371 356 345 338 333 330"
326 322 314 301
283 259 230 196 160
iz6 986'
801 756
865
rrr 142 161 136
165
APPENDIX C (Con't)
THETA
PROBABILITY OF DESTRUCTION
SIGMA
10
100
50
200
500
1000
.978 .979 .979 .980
.640 .643 .646 .648 .649 .649
.353 .355 .357 .358
1.000 1.000 1.000
51
52
is 55 56 57
.
/ •
.3*59
-
*
V
58 59 60
61 62 63
1
l
*
64 65 00 67 68 69 70 71 72 73
'
\
-
1
•
»...
.
74 75 76/.:
77 78 79
'
•
80
a
1
82 83 84 85 gb 87 88 89 90
*
;
/ •
• 1
' I 1
1
. \
s
p
1 \ /-
\
'
V
42
\
(
'
f
/
BIBLIOGRAPHY
1.
"The Effects of Nuclear Weapons", April 1962.
2.
Bliss, G. A. Mathematics for Exterior Ballistics. Wiley and Sons, 1944.
3.
Meyer, Hebert A. Symposium on Monte Carlo Methods. Wiley and Sons, 1956.
4.
Taylor, A. F.
Advanced Calculus.
43
Grimm and Co., 1955
Distribution Limited to U. S. Government Agencies Only: (Test and Evaluation): Other requests for (29 February 1972). this document must be referred to the Naval Postgraduate School, Monterey, California 93941. Code 023.
thesS7255
Optimum
a ballistic missile trajectories
3 2768 001 00859 2 DUDLEY KNOX LIBRARY
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