77m- AOB08 889 AIR FORCE INST OF TECH WRIGHT-PATTERSON AF OH SCHOO--ETC F/6 1/SPROBABILITY DISTRIBUTION OF IMPACT FOR A SATELLITE BOOSTEh AFTEETC(UMAR 79 S W BENSON

NCLASSIFIED AFIT/BA/AA/7BD-1

40

AFIT/GA/AA/78D- 1.

* iPROBABILITY DISTRIBUTION

OF IMPACT FOR A SATELLITE

BOOSTER AFTER LOSS OF CONTROL

THESISAFIT/GA/AA/78D-1 Scott 4. Benson

2nd Lt USAF

DTIC

A4d

f • Approved for public release; distribution unlimited.

A GA/AA/7 8D- 1

S,,PROBABILITY DISTRIBUTION OF IMPACT FOR

A SATELLITE BOOSTER AFTER LOSS OF CONTROL,

THESIS /

Presented to the Faculty of the School of Engineering1

of the Air Force Institute of Technology

Air Training Command

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

by

nier

Scott a.IBenson B.S. S. -€ U o': ,-'ced

2nd Lt -TSAP C..t-,

Graduate Astronautical Engineering '-.

' / arch 1979

Arrroved for -ublic release; distribution unlimited.

91/7'.

Acknowledgement

I I wish to thank my wife, Pam, for her understanding and

her persistence in keeping me at work. I also wish to thank

my advisor, Capt. Bill Wiesel, for his advice and patience

towards the completion of this study.

Scott W. Benson

ii

Contents

Acknowledgements ... .... ... ......... ii

List of Figures . ........ ......... iv

List of Symbols .............. ..... v

Abstract .9....................... vii

I. Introduction . . . . . . . . . . . . . . . . . . 1

Background . . . . . . . . . . . . . . . . 1Approach . . . . . . . . . . . . . . . . . .1Scope . . . . . . . . . . . . . . . . . . . . 3

II. Procedure . . . . . . . ........... . . . 4

Launch to Orbit Injection. . . . . . . . . 4Booster Failure ........... 7Flight Trajectory Propagation to'Imnact . . . 9Probability Determination . . . . . . . . . . 12

III. Results and Discussion . . . . . . . . . . . . . 16

Gravity-turn Trajectory ............ 16Impact Areas ......... . . . . . . . . 16Probability Distribution . . . . . . . . .. 18

IV. Recommendations . . . . . . . . . . . . . . 23

Bibliography . . . . . . . . . . . . . . . . . 24

Appendix A: Boundary Value Launch Computer Program . 25

Appendix Bt Impact Point and Probability PredictionComputer Program ............ 27

Vita .. 32

List of Figures

Figure Page

I Launch-to-Impact Trajectory . . . . . . . .. 2

2 Booster Launch..... . . . . . .. . ... 5

3 Booster Failure . . . . . . . 7... .. .

4 Residual Velocity Components . . . . . . . . 8

5 Vehicle Orbit to Impact ........... 10

6 Latitude and Longitude Traversed ....... . 12

7 Probability Propagation to Impact . . . ... 13

8 Impact Envelopes .. . . . . . . . ... . 17

9 Impact Envelopes with Changing Time of Failure 19

10 Probability Density along an Impact Envelope . 20

11 Regions of High Probability Along an

Impact Envelope ................ . . . . . . 22

iv

List of Symbols

' Symbol Definition

a Semi-major axis of conic section

e Eccentricity of conic section

E Probability density

EI Eccentric anomaly at impact

E0 Eccentric anomaly at failure

g Gravitational acceleration

h Altitude

hAngular momentum

m Mass

p Parameter of conic section

P Probability

r Position of vehicle

RpER Perigee height of conic section

RE Radius of earth

t Time

t B Rocket burn time

T Thrust

V Velocity of vehicle

VCIRC Velocity to obtain circular orbit

V Equivalent exit velocity of rocketEVR Residual velocity

x Range

a Azimuth

0 Mass flow rate

Y Flight rath inclination

v

Symbol Definition

a Elevation

0 Longitude

£ Angle of inclination

Earth gravitational constant

VC ,True anomaly at impact

10 True anomaly at failure

Latitude

Angle traversed by vehicle

ivi

vi

AFI T/GA/AI/7 8D- 1

Abstract

The impact area and regions of high impact probability

after booster failure were found for a samnle space system.

The impact footprints were found to be drop-shaDed. They

increased in size as the residual velocity vector elevation

apnroached the original fliht path, or when the booster

failure was assumed earlier in the boost trajectory. Proba-

bility densities at impact were found, and plotted against

the longitude of impact. Singularities in the probability

density were discovered. The singularities correspond to

areas of high imnact probability, and occur when a maximum

latitude or longitude is obtained at a fixed residual velo-

city vector azimuth or elevation. These singularities were

plotted along the impact envelopes. A sample failing booster

system was used to demonstrate the impact area and probabil-

ity characteristics.

vii

PROBABILITY DISTRIBUTION OF IMPACT FOR

A SATELLITE BOOSTER AFTER LOSS OF CONTROL

I. Introduction

Background

In light of the recent Soviet satellite reentry and

impact in Canada, it is becoming more important to determine

the impact areas and probabilities of failing space systems.

The main reasons for concern are the possibilities that

there are nuclear materials on board, in which personal

safety is endangered or that the system is highly classified,

in which national security is endangered. Of special interest

to the United States are the slightly retrograde orbits

launched from Vandenburg AFB, California. These type of

orbits pass over Moscow in the first revolution, and might

be cause for concern in the case of booster failure.

The objective of this study was to demonstrate a method

of predicting impact areas and probabilities for a failing

booster system. A booster is launched and, at some time be-

fore orbit injection, suffers total loss of control. The

vehicle follows a new trajectory to earth impact. The posi-

tion of impact and the probability is calculated for each

possible failure. The launch to impact trajectory is repre-

sented in Fig 1.

Approach

The booster equations of motion were first integrated

to a circular orbit. This was accomplished by setting final

--

Launch point Imact P~oint

Fig. 1. Launch-to-Trnract Trajectory

2

a '4A

orbit conditions and solving the boundary value problem.

Once a launch-to-orbit trajectory was obtained, the equa-

tions of motion were integrated to several times lower than

the time to orbit injection. At this point, the residual

velocity was applied in one of many failure directions. The

vehicle position and new vehicle velocity were obtained and

the orbit they define was found. From the orbital elements,

it was determined when and where the vehicle impacted the

earth. Knowing the impact point, the probability density at

impact was found by propagating the probability density at

failure, through the new trajectory, to the ground.

Scope

For this initial study, the scope of the problem was

limited. The booster was assumed to launch easterly along

the equator, through a vacuum, to an arbitrary circular or-

bit height. The booster was modeled as a simple single stage

booster, with a constant thrust profile. The number of pos-

sible failures was limited to a small, representative sam-

ple. There are several other less important assumptions made

throughout the body of this report.

The following chapter addresses the problem solving

procedure in detail.

II. Procedure

Launch to Orbit Injection

The beginning stategy was to find a launch trajectory

that terminated in a circular orbit. Obtaining this trajec-

tory requires the solution of a boundary value problem,

using the equations of motion for a gravity turn trajectory

(Ref 5:335-43).

- Vcos-Y = 0

- Vsin-Y = 0EVE+ gsin7 -- = 0

m

+ gcosY =oV

h+B =0

T = OVEiE

In these equations, x and h are the vehicle range and alti-

tude respectively, V is the velocity magnitude, 7 is the

flight path inclination with respect to the local horizontal,

m is the vehicle mass, 0 is the rocket mass flow rate, and VE

is the rocket equivalent exit velocity. The launch phase is

illustrated in Fig 2.

There are several underlying assumptions used in ob-

taining this set of equations. First, the trajectory is rel-

ative to a normal reference frame; therefore, the local earth

must be assumed to be flat. Second, flight is assumed to be

in a vacuum, negating the lift and drag effects. Thirdly,

the thrust is assumed to act parallel to the velocity.

Other assumptions -,,ere made to simplify the boundary

4

vh hV

mg mT

x

Fig. 2. Booster Launch

value launch problem. The mass flow rate, , and the equiva-

lent exit velocity, VE, were assumed to be constant, result-

ing in a constant thrust profile. The gravitational acceler-

ation, g, was also assumed to be constant.

The boundary value launch was executed as follows. An

initial mass, and anprorriate values of thrust, mass flow

rate, and equivalent exit velocity were chosen. The equa-

tions of motion were integrated through a number of differ-

ent burn times, and the resulting flight path inclinations

5

were obtained. The integration routine used was ODE, an

ordinary differential equation integrator found in the IMSL

subroutine package. The burn time to zero path inclination,

i, when the booster is parallel to the earth, was found using

the Aitken method of interpolation. The Aitken subroutine,

INTERP, is also found in the IMSL library. Integrating again

to this burn time gives values for path inclination, velocity

and altitude. To achieve successful circular orbit, the final

velocity must be equal to the circular orbit velocity at

that altitude. The initial value of mass was changed to ob-

tain this required boundary condition.

The method used in changing the initial condition, mass,

is discussed here. Rearranging the equations of motion gives

an equation for instantaneous change in velocity for a

change in mass.

dV -T sinYdm F3M 9 0

The required change in velocity is

AV = V - VCIRC

where

VCIRC = RE +h

The first change in mass becomes

AM AVAm - dV

dm

This change in mass was added a number of times; the

equations of motion being integrated to a horizontal flight

path for each mass increment. Again, the Aitken inter-,ola-

6

tion scheme was used to find the initial value of the total

mass which satisfied the boundary conditions.

The interpolated values of mass and burn time were then

used to integrate the equations of motion to the final set

of parameters.

Booster Failure

Booster failure was modeled by an instantaneous change

in direction of the residual velocity at a given time in the

booster trajectory.

A time decrement was chosen and subtracted from the

circular orbit trajectory burn time. The equations of motion

were integrated to the resultant time, which gave a set of

parameters before orbit injection. The magnitude of the re-

sidual velocity was found by subtracting the final velocity

from the circular orbit velocity. , U•~~ = VIR , V

~R CIRC )(. r

It was assumed that the direction of the residual ve-

locity could be towards any point on a sphere surrounding

the vehicle. For this study, the direction was incremented

by 30" in azimuth and elevation, to give a total of 62 ros-

sible directions. The increment was decreased later in the

study to achieve a better understanding of the probabilities

of impact. The sphere of points was defined with the polar

axis along the local vertical, as seen in Fig 3.

The components of the residual velocity can easily be

found, as shown in Fig 4.

7

4,z

Fig. 3.Booster Failure

rV

VRz

*. Fig. L'. Residual Velocity Com

onents

V V

VRx =VRCOSaCOS

VRy = VRsi/aCOS6

VRz = VRsin.6

where

the z axis is aligned along the local verticalthe x axis is in the plane of the original flight path

These components were then added to the flight velocity

components, x and fi, at burnout to give a resultant velocity

after booster failure, herein referred to simply as V. This

velocity and the vehicle position, as found through the equa-

tion integration, gives the information needed to propagate

the trajectory to imract.

8

Flight Trajectory Frovaration to Impact

[U 4The requirements for this part of the study were to see

'if the booster impacts the earth, and if so, where it impacts

V and how long a time interval it covers before impact.

The method used in finding these answers starts with

the conversion of the vehicle velocity and position after

failure to the orbital elements. Using the orbital elements,

it is simple to check if the vehicle impacts by checking the

orbit perigee height. If it does impact, the angle traversed

and the angle of deviation from the original flight path can

be found. Knowing the angle traversed, the time of flight

can be calculated, and the impact latitude and longitude are

easily obtained.

The vehicle velocity and position after failure were

converted to the necessary orbital elements(Ref 3,17-26).

h2

h 2-A

a P21 - e

Using the calculated parameter and eccentricity, the orbit

perigee height was found and checked against the earth radius,

which is assumed to be constant.

RPER = 1 + e

check ER 5 R

If the perigee height was greater than the earth radius, the

9

t

Ve

UP

Fig. 5. Vehicle Orbit to Impact

vehicle would enter earth orbit, and those cases were ignored.

If the perigee check indicated that impact did occur, the

angle the vehicle traverses before impact, 4, can be found by

#= 2w-v0 -

where

P is the true anomaly at failurePt is the true anomaly at impact, as seen in Fig 5

These true anomalies can be calculated using the equation of

a conic section. -

Yo = cos-1 1

e

The angle of inclination, &, after failure is found by

10

j =COS. j h

where k is the unit vector alone the z axis.a

To obtain the time of flight from failure to impact,

the eccentric anomaly, E, at each point of interest must be

found.

E0 = Cos- ( + cosyvo cOS\ + eco s~

=~~~ eco(~s2rP

The time of flight is found by solving Kepler's equation for

each eccentric anomaly, and getting the difference in time.

At [(E esinE,)- (Eo - esinEo)]

"* This change in time represents the time from failure to im-

pact. This time is added to the burn time to failure to ob-

tain the total time of flight.

Using spherical trigonometry, the latitude, 0, and

longitude, 8, are easily found from# and t, as shown in Pig

6. The original simolifying assumition of launching along

the equator takes effect here.

0 = sin-t(sintsinM)

a = sin -t tan'tan&

In the cases when the change in velocity occurs in the

Dlane of the original orbit, the latitude and longitude are

simply

,--0

It

EarthcenterI

I-I

Equator,'-'

Fig. 6. Latitude and Longitude Traversed

The longitude can be added to the longitude covered

prior to failure, Of, to obtain a total longitude. The fail-

ure longitude is found with the original flat earth assump-

tion.

0f= sin -1 x

f ~ R EOt = 0 + f

At this point, a position and time of imnact has been

found for a vehicle which had been subjected to a velocity

change before injection. The probability of imract at this

position will now be determined.

Probability Determination

To determine the probability of imract at a specific

latitude and longitude, the probability density at failure

for the corresnonding velocity change is found, then propa-

gated along the trajectory to the ground, as seen in Fig 7.

12

Fig. 7. Probability Propagation to Impact

13

For a small change in azimuth and elevation, the proba-

bility density at failure iscos5

E(a ,6) =

Therefore, the probability of the vehicle rassing through an

area of the sphere is

cowsa+,) f U

When integrated over the whole sphere (from 0- 2n in azimuth

and -2-- in elevation), the Probability becomes unity.

Propagation of the Drobability to a latitude and longi-

tude at impact results in

P(0,0) =f dO~ do -4,r C)-(O'6

The Jacobian is defined as a determinant.

As the changes in latitude and longitude become infinitesmal,

the -robability density becomes the integrand.

4r r a 3

5T 56___

In this study, the partial derivatives are estimated by

numerical methods. Very small changes in azimuth and eleva-

tion are chosen and added. When the computer program is run,

there are resulting small changes in latitude and longitude.

The ratio of these changes becomes the numerical partial de-

rivative.

14

• ~~Cosa -

This probability density represents probability in relative

magnitude. 'When the density is high, or approaching a singu-

larity, the probability is high for that impact noint.

The following chapter will discuss the resulting impact

envelopes and probability density behavior for this study.

15

III. Results and Discussion

Gravity-turn Trajectory

To initiate the gravity-turn launch, the following rara-

meters were chosen.

m = 21,000 lbm/g

= .466 lbm/sec

T = 58,800 lbf

The launch-to-orbit trajectory had these final values.

tB = 276.54996 sec

x = 3078159.92872 ft

h = 526417.72579 ft

V = 25616.04328 ft/sec

= .00000036 rad0

m = 398.14796 lbm

Impact Areas

Impact points for each possible failure in each time

step were found. The nlot in Fig 8 is a representative rlot

using one time decrement and elevations from - radians

in 7 radian increments. Each curve of ooints'rerresents a

constant elevation of the residual velocity vector, with azi-

muth rotating through .1 radians. It is seen that minimum

latitude and longitude disrersions occur for elevations ar-

proaching the poles of the srhere of failure. The maximum

dispersions occur for elevations equal to zero.

From the oroximity of the imract roints for each rosi-

tive and negative elevatio trajectory (i.e. +30 deg), it is

I 2.00

00

Latitude

Fig. 8. Impact Envelopes

17

assumed that there are high and low trajectories to each

impact noint, excepting the case when the elevation is zero.

This point is important in finding a probability for any im-

pact point, since the probability density is composed of two

probability densities summed.

In Fig 9, the maximum impact envelopes are plotted for

five time decrements from orbit injection time. These repre-

sent each time step with the residual velocity vector eleva-

tion equal to zero. For the launch trajectory utilized in

this study, failure results in a large dispersion in longi-

tude and a relatively small dispersion in latitude. As the

time decrements are subtracted, these dispersions grow.

These characteristics would be expected in any launch con-

figuration.

Probability Distribution

In an attempt to uncover curves of equal probability,

the probability density was olotted versus the impact longi-

tude. The result, for a specific time decrement and residual

velocity vector elevation, is shown in Fig 10. A number of

singularities were discovered in the probability density.

These singularities represent areas of high probability.

The end singularities occur when the residual velocity

vector azimuth is 0 and r radians. Close to these azimuths,

a small change in residual velocity vector elevation results

in a very small change in impact latitude, so there is a

higher prrobability of imractinf, in those areas.

The middle singularity correslponds to an azimuth of

18

100

roo - 3&rTo- 3h-r 101

1*10

1'40

+30

0

jo

610

df3 2I0

Latitude

-ig. 9. Imact Thnvelores ,-.ith ChangirTime of Failure

19

0)

400

200

radians. Two phenomena occur near this point. A small change

in latitude, and a small change in azimuth result in a neg-

ligible change in longitude. These occurences give this area

a high probability of impact.

The remaining singularities occur at maximum latitudes

at set conditions. The singularity at 2181.5 degrees corre-

sponds to a maximum latitude for a constant azimuth of

radians. The singularity at ; 169.4 degrees corresponds to

the maximum latitude for the elevation used for this plot.

In these areas, a small change in elevation and azimuth, re-

spectively, results in a negligible change in latitude, in-

creasing the probability of impact. The position of the sing-

ularities along the impact curves, for the case in Fig 10,

can be seen in Fig 11.

The number of singularities occuring can now be real-

ized. Taking each burn time separately, for each set azimuth

or elevation, there is a maximum latitude and longitude at-

tainable, and a singularity corresponding to each. It would

also be expected to have singularities corresponding to max-

imum latitudes and longitudes for a fixed residual velocity

magnitude. All these singularities could be plotted, as in

Fig 11, to obtain curves of high probability of impact for

each time decrement. This is an area where further study

would be enlightening, along with those mentioned in the

next chapter on reccomendations.

21

I SO

07

Latitude

Fig. 11. Regions of High Probability Alongan Impact Envelope

22

IV. Recommendations

There are many candidate areas for further study in

this thesis. The sco-pe of the study could be expanded to

work with specific launch vehicles or launch sites. The

launch equations of motion could be changed to incorporate a

spherical earth, a changing gravitational acceleration, a

non-constant thrust profile, a multi-stage vehicle, or flight

through an atmosr-here. The boundary value conditions could

be changed to achieve an elliotical orbit.

As mentioned in the results chapter, the study could be

taken deeper, without changing the scope, to find curves of

high probability. Another aoproach to finding probability

distributions would be to start at imoact and work back to

failure. This method would eliminate the problem of summing

probabilities from near coincedent high and low trajectories.

The depth in which the interested reader wishes to follow

the study is certainly dependant on his desired results.

23

. L 23

Bibliograuh

1. Baeker, James B., et al. "Launch Risk Analysis," Journalof Spacecraft and Rockets,L_ (12)t733-8 (December 1977)

t2. Baker, Robert, Jr., et al. "Range Safety Debris PatternAnalysis," Journal of the Astronauti~cal Sciences,fl 14)t287-323 (Octoe-December 3T

3. Bate, Roger R., W.'ueller, Donald D. and Jerry E. Whffite.Fundamentals of Astrodynamics. New Yorki Dover Publica-tions, Inc., 1971.

4. Burrus, John M. Generalized Random Reentry Hazard jaLjtioris. Contract Report. S-oace and 'isile Systems Organ-ization, Los Angeles AFS, California, October 1969. (AD862205).

5. M~iele, Angelo. Flight r.echanics, Volume 1., Theory ofFlight Paths. Reading, D.,assachusetTs: -Addison- I-esleyPublishing Corncrany, Inc., 1962.

24

Aorendix A

Dgnarv Value Launich Comuter Frogram

,T~~~~ A~t() (?)2 I

w1 0

flEI (11 20 3

PE=. C l 7 P

G§ " I.ItG ; A*6 ' -

P I 7 .it ,,s 5

y-(3) .1'G

Y (4) =6,1 .0P /8Y (3) .,210 0 0/ trR I' +0EM(

o-= I !,G4 x(51 /Y .a

TH=2 . 5' X (5)GPRI147T'9 DARA,X(;),PM7SB9ETA,-A

4 EL 3 I . C :-8IVES= I.*CE-'.

TOUT= 0 . 0flFLT =T 61/". 0

± TTCTOIJT+IrELT

TF (x(,L).'3T a. .)G -') T,) L

TF(L.Er .I)GU TOt

jF(rJV.LT,0o0)SO T3 It

,(J ='~OJT

25

(T UJT r) T3) GO TO Dr-0 T~ 1

F TF (Pzi (J) LT 0o~O,;o rn 2GO TO 16

2 K=A,CALL iNTcRP(PS I. IT JvKPXOYOTE14D9E~

GO T' 107 TCL'T=TF

CALL OEE,~IL,-d W

' V IM =- 1 Hf (TA () G, S INx Yi 'Er T

T(4U, 1.) GO TO 1

f)FL V (I) =X ()-JF(I.Frn.?5)Go To t

!'F '=- LJ( T) / rVI)

CGO TO' 151'R COt I M

CALL TNER' ' V ~-4 OY)T7D F

0fpitJ14-, D;LTI lL--S=",PlELM;:

I" ~~N 4 P~ SI DOFS 'jOrT REAC{H 0~

C ND

I~' (5 XE

P~EIUF N.

26

Aorendix B

t Impact Foint and Frobability Prediction Commnuter Frovram

PROGRAM FAIL (INPJT175O, OU'p)UT)

COMMON GBETAtTHPI,01,REREAL LAT,LATDttD-pLTPOG93 ALCG~LNGpOGEXIIEFNAL LEQ

K=OG=32.17'405RE=2.*09?- 673E?GM=i .40764688E i5P1=3. 14159 26536REAOl ,T'3,WEPR1N1I ,

PRINT*, 9 B Tq MASS=",WEBETA=15.0OftTH=2 .8*WE*GPRINT#," BETA= ", BETA T4fRUST=*,riTOUT =T8IF(KE0.O3GO To 20

24 K=K4I20 IFLA6=i

X(210G.0X(3)=O.

XC.) =89.8*PI/i80 .X(5)=WENEQN=5AB=1 * E-8REL=AB

DELT=T8/f 0 0.0IF(K.EO.0)Go TO 21TOUT=T OUT-DELT

21 CALL OOE(LEOIEl4,X, TTOUTRELA3,1?-LAGWTW)

PRINT',9 RA'4GF OLT VEL PSI+amA33 rIME*

PRINT' ,X(1) XC 2) ,X( X,)(4tX (5), tiJrXP(l)mX(3)*COS (XC.))XP(2)=X(3)'SI4(X C,))

IF(K.GT.0)GO To ?5VCIRC:X(3)GO TC 24.

25 CONTINUEOELV=VCIRC-K (3)FL (1)=-I/2s0AZ(1 0.00

27

DEL=FI/6 . 0DAZ=LELDELA 7=i. 0 E-5OELEL=1. OE-500 23 1=1,600 22 J1,1i21=0

26 CONTINUEDV (1) DELV4100 7( j)#OS (EL (1)OV (2) =DEL V*SI.4 (A7 (J))rOS (EL (1))DV (3) DFLV4 SI4V: ( 1.()IF (L.aGT.a0) GO TO 2?'PRINT4,'----------------------------------------------------PRINT4 9" AZ=",AZ (J), ELm.0EL([1)

27 CONTINUECALL IMPACT(DV ,X,XPLAT,LONGLO~l8?,)ELT.,ORRIU)IF(I.EO.t)GO TO 32

EPS=.O0iZING=(A?(J)-PI)*17-IFf71NGLT.EP3)G) rJ 32IF(OPBIT.GT.a'GO 17,) 31IF(L.EQ*1)rGO TO SIF(L.EQ,2)GO rO ?31=1LATI=LATLONGI=LONG2AZI=tZ(J) OLZ

GO TO 2628 CONTINUE

LAT(74=LATLONGDAzLONG2DADLA=OELA7/,(.ATJD -L TT)0A0LC'=DELAZf(LO4:J)A-LONGI)1=2AZ (J) =AZIEL (I)=ELIIDELELGO TO ?6

29 CONTINUELAl DE=LkTLONGCE=LONG2CEOLt=DELEL/CL ATDF-LATI)DEOLO)DELEL/ (LON'F)E-LONGI)

P=AB' (DAOLA'0EDO-OD.DLO*OEDLA) *E<'EL (I)=ELI

32 CONTINUEIF(L.GTo0)GO TO !3LATI=O.O

*=~c

13 CONTINUELATI=LATI' 180. 0/nt

23

TIME=TOUT.DELTPRIN17,fPRIN14,~ LATITLJ)E LONGITUDE PROB43ILITY TIME"PRINT',LATIL9N; ,' TIMFIF(I.EO.1)GO TO ?3

31 CONTINUE22 A7(J+1)=AZ(J)+ona23 EL(I+l)=EL(Ifl!JE

IF(K.LT.5)GO'TO 24.PRINT#, DONE"STOPEND

SUBROUTINE LE(I(T,x,x)(pDIMENSION X(5) ,X:)(.;)COMMON G,BETA,THTh,PIXP(l)=X(3)*COs (XC~w

XP(4)=-G*COScX (4)) /XC3)XP (5) -BETARET U FNEND

SUBROUTINE IMPACT ( D~ X IXP PLATgLO%43,.TG2j OELTL ORBIT)COMMON G98ETATN,34,P1,REDIMENSION V(3) 9R ( 3H (3) E0 3)1 X(IT) r XP(15) ,)V (3)REAL MAGE,MAG:Z,MAGH,M1ArVREAL LONGoLONGi,.-ON;32PLAT

V(i) :DV(l) 4)P( j)V(2)=DV( 2)V(3)=DV(3) +XP( 2)R(I)=X(i)R(2) :0.0R(3)=X(2) +REH=R'V

H()R(2)*V3) -R( 3)'V(C2)

H2=0R2=0V2=0COEFV=0DO 11 1:1,3H2=H(I)**2IN2R2=R( I) 42+R2V2=V(I)*" 2+V2

GET ECCENTRI-ZITY ElMAGH=St)RT (12)MAGRrS!)RT CR2)MAGV=SORT CV?)COEF9 :MAGV4* 2-Ut 44GR

29

E2 =00O 1.2 1=1,3

12 E2=EC(I),-2+E2MAGE=SORT (E2)TO FIND IF IT T4PACrS

RPER=P/(1..0+MkGE)IF (RFER.GT.RE) GO TO 15~

A=P/ (i.0-VAGE*4 7)COEFTJU= ( (P/MA GR) -1. 0) IMAGFTANOM=AC0S (COE FT4J)IF (CCEFV*GT* 0.0) 3 TO 16TANOP,=2. 0* PI-TAN014

16 COEFENU=(UPfRE)-l.O)/MAGEEANOP-ACOS (COE FENJ)ANG=2.* PI-TA4O4E~4Of'TANG=2*04 PI-EANOIE=ACOS((MAGE+140S(TA))f(I.0,MAGE---,'S(TO4})))Eo=ACOS( (MAGE+COS ( TANOm))/i .04;E COS(T NNOM))IF(TANG.LT.PI) GO TO 1.7E=2. 0PI-E

17 IF(TAN0M.LToPI) 3 T3 iAEO:2.* P1-EU

18 CONTINUE0ELT=SQRT(A*'3J)l((F-MAGEvSIN(F)) -(EO-MAE'lSI'4(FO)))

IF(V(2).GT.0.0)GO TO 8AINC=-AINC

8 CONTINUE

A16O=PZIF(AINCEO*0.0)S) TO 9IF(AINC.EQ.Al3O);O TO 9LAT=ASsIN(WIN:l 1Nf(ANG))LONG2=ASIN (TA~4 (L!&T) 'rid'(AI4C))CALL REDUCE (A?4G9LING2)GO TO 19

9 CONTINUELONG 2ANGLAT=C00

19 CONTINUELONG= (LONGI1.LON ,2) "i 0 .0/PIANGIING* 18 0.Of PIAINC=AINC4I80. Of'!IF(L.GT.D)GO TO 10PRIN7*9" LAT=**LATvLONJG="LONG2PRINT#,PRINT*169 ANGLE TIME INCLINATION"PRINTS ,ANGDELT, &1N'

10 CONTINUEORf3IT=0,oGO TC 14.

15 PRINT*, OR91T 4A-, FrCENT RICIT f'I AGF

30

~*1 ORSIT~lo 0

14 RETUPN9 END

SUORCUTrNE RE0UF(AMNLoNG2JREAL LONG2

- Pl,=3.14l5926535A90=PI/2.0

j ~AIBOPI

A360=2.0-*PlIV(ANG.LT.A36O) ) TI. 6ANI=ANG/A360IAN=INT(ANI)ANG= (AN4I-IAN) '2. O*PT

*8 CONT I NUEIF(AtQGLT*AcJO)GO TO 7IF(AVG.LT.AI8o)S0 T) 6IF(AVG.LTeA270)'5O TO 6LONG 2=A360 +LO4G2GO TO 4

7 LONG2=LONG2GO TO 4

6 LONG2=AI80-LO4G24 RETIJPN

END

31

Vita

t Scott W. Benson was born on 8 November 1955 in Geneva,

Illinois. He graduated from Geneva Community High School in

1973. In May 1977, he received the degree of Bachelor of

Science in Aeronautical and Astronautical Engineering from

Purdue University. He received his commission through Air

Force ROTC in rlay 1977. He entered the Air Force Institute of

Technology in September 1977 as his first active duty assign-

ment. Upon completion of his studies, he was assigned to the

Foreign Technology Division, FTD/SDSY, at Wright-Patterson

Air Force Base.

Permanent Address: 1022 James Street

* Geneva, Illinois 60134

32

UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE (When Data Fntered)

READ INSTRUCTIONSREPORT DOCUMENTA:TION PAGE 3EFORE COMPLETING FORM

t. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

ArIT/GA/AA/78D-74. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

PROBABILITY DISTRIBUTION OF IMPACT P4S ThesisFOR A SATELLITE BOOSTER AFTER LOSSOF CONTROL 6. PERFORMING ORG. REPORT NUMBER

7, AUTHOR(s) S. CONTRACT OR GRANT NUMBER(s)

Scott W. Benson, B.S.2nd Lt USAF

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

Air Force Institute of Technology AREA& WORK UNIT NUMBERS

AF T/1

Wright-Patterson AFB, Ohio L 5433It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

i;arch 197913. NUMBER OF PAGES

4O14. MONITORING AGENCY NAME & ADDRESS(If different from Controlling Office) 15. SECURITY CLASS. (of this report)

Unclassified15sa. DECL ASSIFIC ATION'OOWN GRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for r ublic release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, ii different from Report)

IS. SUPPLEMENTARY NOTES Arnroved for ,ublic release, IAd AFR 190-17

JOSEI F.'HI , ':ajor, USAFDirector of Information

19. KEY WORDS (Continue an reverse side if necessary and identify by block number)

Booster FailureImpact FredictionImpact P-robabilityScacecraft Launch

Z0.,JBSTRACT (Continue on reverse side If necessary and Identily ty block number)

The imnact area and reFions of high im-act rrobability afterbooster failure vere found for a samle space system. The imr-actfootrrints were found to be droo-shared. They increased in size asthe residual velocity vector elevation anproached the originalflight rath, or %,.hen the booster failure -.as assumed eariier inthe boost tra,ectory. lrobabilitv densities at im-act ..erc 2o_and ,.!otted against the lor7itude of iw,;act.. Singularities in ,.!herrobability dens-ity ;.-ere discovered. The singularities corresrond

DD I 1473 EDITION OF I NOV 65 IS OBSOLETE UnclissifiedSECURITY CLASSIFICATION OF lHIS PAGE (li'en Pata Entered)

Unclassif ied(;URITY CLASSIFICATION OF THIS PAGE(Wihen Data En tered)

to areas of hig;h imact probh.bility, and occur ',hen a maximumlatitude or longitude is obtained at a fixed residual velocityvector azimuth or elevation. These singularities ,were ilottedalong the imr.act envelores. A samnle failing booster system wasused to demonstrate the impact area and probability character-istics.

tN

Unclassified

SECURITY CLASSIFICATION OF r, PAGE(When Do(& Fntered)