Eci

Pi(

Aci

Ai

)Ni...........(1)

Ect

n

i

Eci.............(2)

1

INTRODUCTION

An object being returned from orbit will exhibit a dispersio n

pattern at the landing area th at is caused by uncertainties in the

orbital parameters, retro-maneuver variations and other factors .

Failures of the reentry systems can shift the impact area eithe r

up-range or down-range of the intended landing area. This prime r

provides two methods that may be used in computing impac t

probability and casualty expectancy for such failures.

CASUALTY EXPECTANCY

Casualty expectancy (Eci) for an area (Ai ) is defined by the

following equation as the expected number of casualties occurring

during the event:

The total casualty expectancy (Ec t) for all areas is:

Where:

The populated area exposed is A i

The object's impact probability in A i is Pi

The casualty area of the object to people in A i is Aci

The number of people in A i is Ni

Casualty expectancy is highly dependent on the probability o f

failure, computation of the probability of impact (P i) and the

numbers of people exposed in a given area. In general, use of the

variables cited above requires judgment as to when approximations

can be used without over or underestimating the risks that ca n

occur.
P

1

2 x y

[ 1/2(

x

2

2

x

y

2

2

y

)]dxd

y

..........(3)

2

PROBABILITY OF IMPACT

Most impact dispersion areas exhibit a bivariate norma l

distribution. The general equation for this distribution is shown

below.

Two methods used for computing Pi are illustrated in Figures 1 and

2. The first method uses the basic equation for a bivariat e

distribution and the second illustrates a method used when th e

exposure is time dependent. This computation can take many form s

other than the examples shown, however, the general equations are

applicable.

Figure 1 gives an illustration of a failure that produces a finite

displacement of the impact dispersion area in the down-rang e

direction and Figure 2 illustrates a time variant displacemen t

based on the time of failure. A finite displacement of the impact

dispersion can be caused by various systems failures such as timing

system failure. A time dependent shift can be caused if th e

retro-motor, for example, should fail at various times during its

burn period.

A problem always addressed by risk analysts is when approxim ations,

such as averaging of the numbe rs of people exposed, can be applied

without biasing the results. I f the dispersion area of exposure is

very large but contains highly concentrated populated areas wel l

away from the mean point of im pact, then use of average population

densities may tend to overestimate the risk. If the dispersi on area

of exposure is small and does not contain any significant po pulated

areas, using average population density for a larger region ma y

tend to underestimate the risks. If there are significan t

population centers near the mean point of impact, averaging wil l

usually underestimate the risks. Hence, one must examine th e

exposed areas and make reasoned judgments on when approximation s

can be used.
3

Shown in Figure 1 is a planned landing site showing the normal 3

Sigma dispersion area of the reentry vehicle. Also shown is th e

landing zones tolerance to variations of the mean point of impact

that the site can contain. This tolerance is necessary t o

accommodate the variations in de-orbit opportunities which occur.

The 3 Sigma dispersion area is increased by including the landing

site tolerance which also defi nes an exposure corridor both up and

downrange of the landing site.

The methodology depicted in Figure 1 can be used for determinin g

the Probability of Impact (P i) when the shifted dispersion are a

contains significant population centers. For other rural area s

exposed, the population density can be approximated by remov ing the

city populations and computing the average population density for

the remaining population in the region. If the dispersion ar ea does

not contain any major population centers, the average populatio n

density can be derived as described above or the region's average

population density including cities can be used as a conservative
4

Figure 1. DISPLACED DISPERSION PROBABILITY

 OF IMPACT

estimate for computing Ec.
5

EXAMPLE 1

This example will use the methodology depicted in Figure 1 an d

Equations (1) and (2). The following conditions are establis hed for

the example:

1. The probability of a failure causing a down-range shift

is = 0.001

2. The dispersion area is defined as a bivariate norma l

distribution centered on the mean point of impact with 

x = 10 s.mi. and y = 4 s.mi.( s.mi. is statute miles )

3. There are three cities in the dispersion area as follows :

a. City 1:

Population N1 = 200,000

City Area A1 = dx (4 s.mi.) x dy (3 s.mi.) =

12 square miles.

Distance to centroid of A1 is Xdr (+ 4 s.mi.)

and Ycr (+ 5 s.mi.)

b. City 2:

Population N2 = 50,000

City Area A2 = dx (2 s.mi.) x dy (2 s.mi.) = 4

sq.mi.

Distance to centroid of A2 is Xdr(-5 s.mi. )

and Ycr (-7 s.mi.)

c. City 3:

Population N3 = 30,000

City Area A3 = dx ( 2.5 s.mi.) x dy ( 1 s.mi.)

= 2.5 sq.mi.

Distance to centroid of A3 is Xdr (+ 15 s.mi.)

and Ycr (- 1 s.mi.)

4. The total 3 area exposed is 60 x 24 miles ( 1,440 sq.

mi.) and contains 14,400 peopl e in addition to the three

cities above.

5. The casualty area of two objects is 30 sq. ft. for each

object.

The results for Example 1 are shown below in Table 1.

Pi x ( 10 ) E x ( 10 )

-5

c

-7

CITY 1 4.0 7.2

CITY 2 0.61 0.82

CITY 3 0.63 0.81

REMAINING AREA 194.8 0.21
Pi x ( 10 ) E x ( 10 )

-5

c

-7

6

TOTAL 200 9.1

AVERAGE 200 4.4

TABLE 1. EXAMPLE 1 PROBABILITY OF IMPACT AND CASUALTY EXPECTANCY

Shown in Table 1 is the total casualty expectancy determined using

the equations cited. The casualty expectancy was also compute d

using the total population averaged over the entire exposed are a

(AVERAGE). This illustrates th e tendency to underestimate the risk

by averaging the population over the area exposed.

The methodology depicted in Figure 2 can be used when the tim e

dependent exposure area sweeps over various populated region s. When

the distance exposed as a function of time is small (50-20 0

miles/sec.) and the exposure corridor narrow, any significan t

population centers exposed should be considered independently and

the average populations of other areas included as described above.

If the distance exposed as a f unction of failure time is large and

no major cities or metropolitan areas are near mean of the e xposure

corridor, average population densities may be acceptable. I n

general, the analyst should perform sample calculations t o

determine the effect of cities before concluding that use o f

average population densities is appropriate.
Figure 2. TIME DEPENDENT PROBABILITY OF IMPACT 7
8

EXAMPLE 2

This example will use the methodology depicted in Figure 2 and by

Equations (1) and (2). This methodology is applicable to th e

failure of a retro-motor during its thrusting period. The fo llowing

conditions are established for this example.

1. The probability of failure of the retro-motor is 0.01 an d

the burn time of the motor is 16 seconds. The failur e

rate probability (Pf) assuming a uniform failure distributi

o n i s

then: 

 

 

 0.01 x

1/16 =

6.25 x

10 /sec. -4

2. The acceleration caused by the retro-motor during burn i s

40 ft./sec. and the range shift sensitivity near th e

2

landing site is 6 s.miles/ft/sec. If the retro-moto r

failed during the last second of burn, the area exposed

would be (1sec. x 40ft./sec. x 6s.mi./ft./sec. = 240

2

miles) long. 

3. The cross range dispersion is a normal distribution that

is centered on the ground track center line (C/L) with y

= 4 s.miles.

4. There are three cities contain ed in the exposure area as

follows:

a. City 1:

Population = 50,000

dy = 2, dx = 2 and Ycr = 1 s.miles

b. City 2:

Population = 200,000

dy = 3, dx = 3 and Ycr = -10 s.miles

c. City 3:

Population = 800,000

dy = 3, dx = 4 and Ycr = 3 s.miles

d. The total area exposed for this 1-secon d
9

period is 240 s.miles x 24 s.m iles = 5,760 sq.

s.miles. This area contains a population o f

100,000 people in addition to the cities.

e. The casualty area of the re-en try object is 30

sq.ft.
10

The results from this example are shown in Table 2 below. Shown in

Table 2 is the total casualty expectancy determined using th e

equations cited. The casualty expectancy was also computed usin g

the total population averaged over the entire exposed are a

(AVERAGE). This example also illustrates the tendency t o

underestimate the risk by averaging the population over the are a

exposed.

Pi x ( 10 ) E x ( 10 )

-6

c

-7

CITY 1 1.0 0.13

CITY 2 0.1 0.24

CITY 3 2.4 1.7

REMAINING AREA 621.5 0.12

TOTAL 625 3.1

AVERAGE 625 1.3

TABLE 2. EXAMPLE 2 PROBABILITY AND CASUALTY EXPECTANCY

This type of failure during the retro-motor burn will cause th e

re-entry vehicle (RV) to remai n in orbit for failures early in the

burn period and cause impacts down range of the landing site fo r

failures later during the motor burn period. When the RV has been

decelerated sufficiently to re-enter and impact, the initial impact

distance will be approximately half way around the world dow n range

from the landing site. As deceleration continues, the impac t

distance down range moves rapidly back toward the landing sit e

until burn-out of the motor occurs. To determine the total ris k

from such a failure requires the process above be completed fo r

each time interval and the risks for each summed.
11

SUMMARY

Presented above are two methods that may be used in computin g risks

for reentry vehicles. Obviousl y, there may be a number of failures

which can also produce risk to areas outside the landing are a. Once

all the failures and their ris ks have been determined, they can be

summed to arrive at the total mission risk for all areas. 

The effects on risk of averaging various population data ove r large

or small areas can lead to over or underestimating the risks a s

described and illustrated in the examples provided. The exposur e

area should be carefully examined and the risk sensitivity fro m

major population centers tested before concluding that averag e

population densities can be used to estimate the risk.