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Hazard Analysis for Uncontrolled Space Vehicle Reentry

Russell P. Patera∗

The Aerospace Corporation, Los Angeles, California 90009-2957

DOI: 10.2514/1.30173

Satellites in low Earth orbit ultimately reenter the Earth’s atmosphere at the end of the mission due to orbital

decay. Although most of the mass of a typical space vehicle is destroyed and rendered harmless, a signiﬁcant portion

survives to ground impact. As the number of reentry events has increased in recent years, some surviving

components have impacted near populated areas and drawn attention to the casualty risk. In response to this hazard,

the U.S. government developed guidelines to mitigate the danger posed by randomly reentering space objects that

survive to surface impact. If an upper bound for casualty expectation is exceeded, a controlled deorbit/reentry into a

sparsely populated region is recommended. This paper develops the mathematical methods needed to compute risk

to people and property from uncontrolled reentries. The impact probability density function for surviving debris

objects associated with an uncontrolled reentry is derived. The impact probability density function is used with the

population density function to compute the casualty expectation. Examples of casualty expectation and risk of

damage to property are provided.

Nomenclature

A= vehicle drag area

da =inﬁnitesimal surface area

g= gravitation acceleration near Earth’s surface

i= orbital inclination

L= latitude

M= debris mass

n= number of debris pieces per reentry event

P= probability of impact

pi= probability of the ith debris piece producing a casualty

R= Earth’s spherical radius

r1= effective radius of debris

r2= effective radius of a human

V= impact velocity

x= auxiliary parameter

= ballistic coefﬁcient

A= impact area

=inﬂuence function

"i= casualty area

= latitude probability density function

L= population density as a function of latitude

o= atmospheric density at the Earth’s surface

x= population probability density function

= orbit probability density function

= impact probability density function

= right ascension of the ascending node

Introduction

ORBITAL decay results in roughly 100 random reentries of

large space objects per year, and this number is expected to

increase in the future. Figure 1 illustrates the number and associated

mass of reentries in recent years that involved larger space objects,

not including service missions to the International Space Station or

space shuttle activity.†The decrease in the number of reentries per

year, shown in Fig. 1, is due in part to the decrease in solar activity

associated with the 11-year solar cycle that peaked in approximately

2001. The number of reentries per year is expected to increase as we

approach the next peak in the solar cycle, which is predicted to be

near 2012.

Some recent reentry events involving recovered upper stages have

demonstrated the danger of impacting debris [1–4]. Although the

severe reentry heating environment consumes a signiﬁcant fraction

of the reentering mass, 10–40% of the prereentry mass is expected to

survive to Earth surface impact [5].

Warnings are generally not given for randomly reentering space

debris because impact location cannot be accurately predicted.

Although all the space objects in Fig. 1 were tracked and had

cataloged state vectors updated periodically, the time and location of

each impact could not be accurately predicted due to unpredictable

variations in high-altitude atmospheric density that alter the drag and

orbital decay rate. Much of this variation is caused by unpredictable

solar activity. As a result, the reentry time has an uncertainty of

10% of the remaining orbit lifetime [6]. This uncertainty translates

into uncertainty in impact location. For long-term prediction, the

uncertainty is so large that it encompasses many orbital revolutions

that span all values of longitude. Thus, any impact longitude is

possible even for short-term predictions of about a day. For longer

term prediction the impact longitude probability distribution is

essentially uniform. Impact latitude distribution functions, however,

are nonuniform and are bounded by i, where iis the inclination of

the parent orbit of the reentering object.

It is now clear that reentering space debris poses a hazard to people

and property. How best to manage this risk has become an issue that

is being addressed by the space community [7]. The U.S. government

developed guidelines to limit the hazard from reentering space

vehicles [8]. If the casualty expectation of a reentering satellite

exceeds 1 in 10,000, a controlled deorbit is strongly recommended

[8]. A controlled reentry implies that the object can be made to impact

a desired location on the Earth’s surface. Controlled reentries usually

target a remote ocean area that is uninhabited. Thus, the risk

associated with a successful controlled reentry is essentially zero.

Controlled reentries were performed for both the Compton Gamma

Ray Observatory (CGRO) [3,9] and the Mir Space Station.

Controlled deorbit is also recommended for both the Hubble Space

Telescope and the International Space Station (ISS) at the end of their

respective missions. Less massive space vehicles are also subject to

controlled deorbit, if the casualty expectation exceeds U.S.

government guidelines. The recent controlled deorbit of the Delta IV

Second Stage [10] is such an example.

Presented as Paper 6500 at the AIAA Atmospheric Flight Mechanics

Conference and Exhibit, Keystone, Colorado, 21–24 August 2006; received

31 January 2007; revision received 13 March 2008; accepted for publication

26 June 2008. Copyright © 2008 by The Aerospace Corporation. Published

by the American Institute of Aeronautics and Astronautics, Inc., with

permission. Copies of this paper may be made for personal or internal use, on

condition that the copier pay the $10.00 per-copy fee to the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include

the code 0022-4650/08 $10.00 in correspondence with the CCC.

∗Senior Engineering Specialist, Center for Orbital and Reentry Debris

Studies, Mail Stop M4-066.

†Data available online at http://www.aero.org/capabilities/cords/

reentry-stats.html [retrieved 1 August 2008].

JOURNAL OF SPACECRAFT AND ROCKETS

Vol. 45, No. 5, September–October 2008

1031

Reentry risk can also be reduced with only partial control of a

space vehicle via drag modulation [11]. This was achieved during the

reentry of Skylab in 1979 by shifting the impact footprint thousands

of miles away from North America [12]. Drag modulation has been

proposed as a method of reducing reentry risk for larger space

vehicles that would otherwise require expensive controlled deorbit

systems [13].

Controlled reentries are performed only if absolutely necessary

due to cost, mission impact, and difﬁculty in performing the

maneuver. Therefore, it is important to calculate the casualty

expectation to determine if a controlled deorbit is required.

Published methods to compute casualty expectation lack adequate

mathematical foundations. For example, the derivations of the

impact probability density function and casualty expectation for a

debris object randomly reentering from an orbit of inclination, i,do

not appear in the literature.

The goal of this work is to improve the mathematical methods

needed to quantify the hazard due to uncontrolled space reentry. This

is achieved by introducing a continuous function for impact

probability density, which is expressed using simple algebraic and

trigonometric functions. The impact probability density function is

used to determine the risk to a building structure, ship, or aircraft,

which depends on the location and associated area of the object

at risk.

The impact probability density function is also used with the

population density function to derive the integral expression for

casualty expectation. This integral enables one to compute the

casualty expectation per unit area for an object reentering from an

orbit with arbitrary inclination. An interaction function is included in

the formulation for mathematical rigor, as well as to quantify the

severity of the debris impact, which can result in greater or fewer

casualties depending on the impact energy. The casualty expectation

integral can be easily evaluated and results in the casualty

expectation per unit casualty area as a function of the orbital

inclination of the reentering space object. The resulting plot [1] has

been presented in a previous publication but the details of the

underlying computations have never been published in an archival

journal. The uncertainty in the casualty expectation is evaluated by

considering all possible longitudes of the ascending node of the ﬁnal

orbit revolution. The results are applied to a Delta IV Upper Stage

random reentry by calculating the casualty expectation distribution

function.

This paper is organized into sections involving reentry

survivability, impact probability density, global population density,

and casualty expectation calculation. The reentry survivability

section reviews information regarding reentry heating, vehicle

breakup, and key factors in survivability. The section on impact

probability density derives the analytic expression for the impact

probability density function as a function of impact latitude and

space vehicle orbital inclination. The section on global population

density derives the population density as a function of impact latitude

based on a gridded global population database. This is needed to

complement the impact probability density, which also depends on

latitude. The section on casualty expectation is the focus of the paper.

Here the impact probability density is used with the global

population density to derive the casualty expectation integral, which

is used to calculate the casualty expectation per unit casualty area as a

function of the orbital inclination. The variation in casualty risk as a

function of the longitude of the ascending node of the ﬁnal orbit

revolution is also presented.

Reentry Survivability

The subject of reentry heating and space vehicle survivability is

complex. The interested reader can consult numerous references on

the subject [5,9,14–21]. Only a high-level qualitative review of the

subject is presented here.

Over time, the orbits of space objects in low Earth orbit decay and

tend to circularize due to the higher drag at perigee altitude. The

enormous amount of orbital energy must be removed during the

reentry process. Most of the energy goes into heating the atmosphere

as the space object descends through the atmosphere. A fraction of

the energy goes into heating the space object itself. Aerodynamically

sleek shapes transfer more of the heat energy to the vehicle. In

contrast, blunt-shaped objects transfer more of the heat to the

atmosphere.

If the heating rate is sufﬁciently high, the space object skin or outer

components can melt or vaporize. The heating rate depends on the

speed at which the object descends through the atmosphere, which

depends on the ballistic coefﬁcient deﬁned in Eq. (1):

0.0E+ 00

5.0E+ 04

1.0E+ 05

1.5E+ 05

2.0E+ 05

2.5E+ 05

1999 2000 2001 2002 2003 2004 2005 2006

Ye a r

Mass, kg

0

30

60

90

120

150

Number of Objects

Mass, kg Number of Large Objects

Fig. 1 Total mass and number of large object reentry for recent years.

1032 PATERA

mg=CdA(1)

where gis the gravitational acceleration at the Earth’s surface, mis

the mass of the debris piece, Ais the drag area, and Cdis the drag

coefﬁcient. Although some analysts deﬁne the ballistic coefﬁcient as

CdA=M, the deﬁnition in Eq. (1) is used in this work.

High ballistic coefﬁcient objects descend at faster rates and have

higher peak heating rates. Lower ballistic coefﬁcient objects descend

at lower rates through the atmosphere and have a longer amount of

time to radiate the thermal energy away. By the time an object

reaches the denser portions of the atmosphere, its velocity is lower;

therefore, the peak heating is lower.

There are several processes that serve to reduce the predicted peak

temperature during a reentry: heat capacity, ablation, and radiation.

The heat capacity of the space object’s skin serves, to some extent, as

a heat sink to reduce the peak temperature. Because a heat sink

absorbs heat throughout the reentry, its effectiveness is determined

by the total heat load and not the peak heating rate. Thus, a heat sink is

more effective against a rapid reentry that has a high peak heating rate

but a lower total heat load.

Ablation is the vaporization of the space object’s skin. Ablation

can absorb a great deal of energy per pound of surface material. Thus,

the heat energy that would go into raising the temperature goes into

changing the phase of the skin material. In addition, the release of

ablation products tends to impede the heat ﬂow from the heated

atmosphere to the object’s skin.

Radiation serves to reduce the peak temperature by emitting the

heat energy as electromagnetic radiation. Because the radiated power

is proportional to the fourth power of temperature, it is more effective

at high temperatures. The radiated power is proportional to the area of

the emitting surface. Radiation helps reduce the total heat load. It is

more effective for lower ballistic coefﬁcient objects that are heated

higher in the atmosphere, thus experiencing a lower heating rate

spread over a longer time. In this manner, more heat is radiated away

earlier in the descent and the velocity is reduced when the object

enters the denser regions of the atmosphere.

Vehicle survivability prediction is complicated by vehicle

breakup, which exposes shielded components to direct atmospheric

heating. Thus, internal components of various ballistic coefﬁcients

are released at breakup and follow unique trajectories and heating

proﬁles. Secondary breakups of released components are possible,

and a cascade of secondary breakups can occur. Objects with the

greatest likelihood of survival are lower ballistic coefﬁcient pieces of

blunt shape made of materials with a high melting temperature and

high heat capacity. Spherical pressure vessels and propellant tanks

have been recovered with relatively little physical damage caused by

the reentry heating environment. Very large space vehicles tend to

have more surviving components due to shielding and heat capacity

effects. Skylab, Cosmos 954, and Salyut-7/Kosmos-1686 are

examples of massive space vehicles that had a signiﬁcant amount of

surviving debris.

It should be noted that the on-orbit ballistic coefﬁcient is typically

signiﬁcantly different from the reentry ballistic coefﬁcient because

the space vehicle breaks up due to aerodynamic and heating loads

and because the value of Cdchanges based on changes in the

aerodynamic ﬂow regime. Large solar panels usually break off ﬁrst.

As a result, there is a range of ballistic coefﬁcients for space vehicle

fragments after the main breakup event. An analysis of the Hubble

Space Telescope indicated that 98 different components (some with

multiple quantities) will survive the reentry environment. Variation

in particulars, including ballistic coefﬁcients, results in an impact

footprint 1220 km long. The analysis estimated that 2055 kg of mass

survives to impact, which represents 17.4% of the on-orbit mass [22].

An analysis of a Delta II Second Stage indicated that at least 331 kg

(one propellant tank and four pressurization spheres) of the 920 kg

Second Stage survived reentry [5]. That amounts to at least 36%.

These two examples fall within the expected 10–40% of surviving

mass stated earlier.

Space vehicles can be designed to reduce reentry survivability,

and thereby reduce casualty risk. The easiest method is to choose

low-melting-point materials for major components. It might be

possible to construct the space vehicle in a manner that will assure a

particular breakup scenario. For example, if the total vehicle is

predicted to survive, it may be possible to design the vehicle to

separate into two or more less-survivable segments during reentry.

On the other hand, if more-survivable components tend to separate

from the core vehicle early in the reentry, thereby avoiding peak

heating, it might be possible to prevent this early separation. Keeping

these more-survivable components attached to the core vehicle could

subject them to higher heating loads sufﬁcient to destroy these more

resilient components. As more is learned about the reentry heating

environment, more approaches to minimizing reentry survivability

can be applied.

Although the amount of surviving debris can be reduced, it is

likely that there will always be some debris objects that survive to

surface impact. It is the surviving debris that impacts with sufﬁcient

energy to injure people or damage property that is the subject of

this work.

Impact Probability Density

Objects that reenter the Earth’s atmosphere via drag-induced

decay tend to circularize before reentry. This is because drag at

perigee lowers apogee and drag at apogee lowers perigee. Because

drag at perigee is higher due to denser atmosphere, apogee decreases

faster than perigee. Even if reentry occurs at a nonzero eccentricity,

uncertainty in long-term prediction results in a uniform distribution

of the reentry position within the orbit plane. That is, the angular

position of the reentry point from the ascending node (argument of

perigee plus true anomaly) is uniformly distributed. At low Earth

orbit, the equatorial bulge causes the orbit plane to precess about the

Earth’s spin axis, thus changing the right ascension of the ascending

node, . The uncertainty in atmospheric drag over several days,

which involve tens of orbits, results in a uniform distribution of the

longitude of the ascending node, . In addition, the uncertainty in

reentry time randomizes the longitude range of the reentry footprint.

Only for short-term predictions on the order of a few days is it

possible to predict the longitude of the reentry footprint. Therefore,

we assume that the impact longitude is uniformly distributed.

Let be the angle between the ascending node and the reentry

position in the orbit plane. The probability density function for is

given by

1=2(2)

The normalization condition is satisﬁed:

Z2

0

dZ2

0

d

21(3)

For each value of , a latitude angle can be computed. Here we

neglect the slight oblate shape of the Earth and assume a spherical

shape for simplicity. There are two values of for each value of

latitude, one for ascending motion and the other for descending

motion. The distribution of latitude for a random reentry is not

uniform. The latitude distribution can be computed by relating to

latitude via the intermediate Cartesian coordinate parameter z:

zRsinLRsinsini(4)

Eliminating Rand taking the derivative of Eq. (4) yields

cosLdLcossinidsin2isin2L1=2d(5)

Thus, dis related to dLby

dcosLdL

sin2isin2L1=2(6)

The impact location distribution function can be expressed in terms

of latitude by considering an inﬁnitesimal probability associated with

dLand d:

PATERA 1033

dd

2L; idL(7)

Using Eq. (6) in Eq. (7) yields

d

2L; idLcosLdL

sin2isin2L1=2(8)

where a factor of 2 was included to account for the fact that the range

of Lis , whereas the range of is 2. The latitude probability

density function is therefore

L; i cosL

sin2isin2L1=2(9)

Notice that, if the inclination is =2radians, the latitude distribution

becomes uniform as expected:

L; =2 1= (10)

The impact probability density on the Earth’s surface is the

probability per unit area. It can be obtained from Eq. (9) by simply

dividing by the surface area associated with an inﬁnitesimal

latitude, dL:

da2R2cosLdL(11)

That is

L; iL; idL

daL; idL

2R2cosLdL

1

22R2sin2isin2L1=2(12)

The normalization condition is given by

Zi

i

L; i2R2cosLdLZi

i

cosLdL

sin2isin2L1=2(13)

Notice that the range of Lis bounded by the orbital inclination. This

integral can be evaluated by a change of variable given by

sinLsinisinx(14)

The derivate of Eq. (14) is

cosLdLsinicosxdx(15)

Using Eqs. (14) and (15) in Eq. (13) gives the proper normalization as

expected:

Z=2

=2

sinicosxdx

sin2isin2isin2x1=2

Z=2

=2

dx

1

2

21(16)

One can use Eq. (12) to compute the probability that a piece of

reentering debris will strike an area on the Earth’s surface. If a

building is located at latitude 30 deg north or south and has an area of

1000 ft2and a piece of debris reenters the Earth’s atmosphere from

an orbit having an inclination of 45 deg, the impact probability is

given by

PAL; i1000=6;=4

10002:314 10162:314 1013 (d17)

Notice that the longitude of the building is not important for this

random reentry. This result indicates that the probability of any one

single building being struck by space debris is quite small.

One can compute the probability of an unsheltered person being

struck by a piece of space debris in a similar fashion. If the area of an

average person is 4ft

2, then the probability of impact for 30 deg

latitude is

PAL; i4=6;=4 42:314 1016

9:255 1016 (18)

If the entire global population of about 6 billion people was located

at latitude 30 deg north or south and distributed in longitude to

prevent occupying the same location, the probability of a single

person being struck by a debris piece is given by

P6109AL; i24 109=6;=4

24 1092:314 10165:553 106(19)

This calculation assumes that the population is unsheltered and

neglects the area of the debris piece itself. It is meant to simply

illustrate the use of the impact probability density in Eq. (12). To

compute the casualty expectation associated with reentering debris,

one must properly treat the global population density. In addition, the

interaction between the debris and population must be included in the

formulation.

Global Population Density

Unlike impact probability density, the global population density

cannot be represented by a simple analytic function. Instead, the

population density can be computed numerically using a database

containing population as a function of latitude and longitude [23,24].

The longitude dependence only matters if the location of the reentry

impact footprint is fairly well known. Because we are only interested

in random reentry for which the location of impact is unknown, we

assume the longitude of the impact footprint is uniformly distributed.

Therefore, we only need the population density as a function of

latitude for casualty expectation analysis.

The most recent available global population database was

processed to obtain the population density as a function of latitude

[23]. Figure 2 illustrates the population distribution function as a

function of latitude. Figure 3 contains the population density as a

function of the sine of the latitude, which is needed later for

computing casualty expectation. Also illustrated is the distribution

function, which is essentially the integral of the probability density

starting in the southern hemisphere and integrated northward

through the northern hemisphere. One can account for population

density in years later than 1995 by simply assuming a constant 1%

growth rate. If the growth rate is asymmetric, with the southern

hemisphere having a 2% per year growth rate and the northern

hemisphere having a 0.5% per year growth rate, the distribution does

not change signiﬁcantly when extrapolated to the year 2007, as

illustrated in Figs. 2 and 3. The higher population density in the

northern hemisphere is clearly visible in both Figs. 2 and 3.

Although the population density as a function of latitude cannot be

expressed using elementary functions, it can be computed and

tabulated for related computations, such as casualty expectation.

Casualty Expectation Calculation

The casualty expectation calculation requires that the reentering

debris impact probability density be coupled to the population

density function. Impacting debris can only affect population in the

vicinity of the impact point. The closer the debris impact point is to a

person, the greater the chance of casualty. This coupling between the

debris impact point and population density function can be

represented mathematically by an inﬂuence function, x, where xis

the separation distance between a small area with a population

density and debris impact point. The inﬂuence function relates the

magnitude of the casualty to the separation distance between a person

and a debris object. The inﬂuence function includes the effective area

of the debris piece, the average area of a human, and the energy of the

impacting debris, as well as a structure that may protect the human

population. In this analysis, we do not require the detailed form of the

inﬂuence function, but just its integral over the region of interest.

This is because both the impact probability density and population

1034 PATERA

density vary very little over the impact region. Thus, we assume a

step function centered on the debris impact point.

The casualty expectation depends on the probability density of a

debris object, x; the probability density of the population, x;

and the inﬂuence function, x. The probability that a debris object

impacts the inﬁnitesimal area, da0, is given by x0da0. The number

of people residing in the inﬁnitesimal area, da, is given by xda0.

The amount that the impacting debris at position x0will affect the

population at xis given by xx0. The casualty expectation is

computed as the product of these functions integrated over the

respective areas. Expressed mathematically, one ﬁnds for the ith

debris object

piZZa;a0

xx0xx0dada0(20)

One can perform the daintegration ﬁrst. Because the impact

probability density does not vary much over the integration region,

a0,x0equals xand it can be brought outside of the integral:

"ixxZa0

xx0da0Za0

x0xx0da0(21)

In Eq. (21), "iis an effective interaction area centered on xand is

given by

"iZa0

xx0da0(22)

The functional form of xis not important, only its integral over

region a0. Therefore, one can assume a step function for simplicity

and interpret "ias the effective casualty area for the ith debris object.

The parameter "iis also used to quantify the severity of the

inﬂuence of the debris impact. If the size, mass, and energy of the

impacting debris are large, then "iwill be large as well. On the other

hand, if the population is sheltered by strong housing units, "iwill be

smaller. A 100% sheltering of the population implies that "iis 0.

Using Eq. (21) in Eq. (20) yields

0.E+00

1.E+09

2.E+09

3.E+09

4.E+09

5.E+09

6.E+09

7.E+09

-45 -30 -15 0 15 30 45 60 75

Latitude, deg

Cumulative Popu lation

1995 Population 2007 Population (Skewed)

Fig. 2 Population distribution function.

0.E+00

1.E-05

2.E-05

3.E-05

4.E-05

5.E-05

6.E-05

-1 -0.5 0 0.5 1

Sine of Latitude

Population Density, #/sq. m.

0.E+00

1.E+09

2.E+09

3.E+09

4.E+09

5.E+09

6.E+09

7.E+09

Population

1995 Density 2007 Densit y (Skewed) 1995 Population 2007 Population (S kewed)

Fig. 3 Population density functions.

PATERA 1035

pi"iZa

xxda(23)

The problem is now reduced to calculating the densities, and ,

appropriate for a given problem and evaluating the integral.

Equation (23) is valid for both random and controlled reentry. This

paper is concerned with random reentry, and so the impact

probability density is a function of latitude and orbital inclination

only, as derived earlier. The global population density as a function

of latitude can be computed from a population database and is

illustrated in Fig. 3.

Using Eqs. (11) and (12) in Eq. (23), we ﬁnd

pi"i

Zi

i

sinL cosL

sin2i sin2L1=2dL(24)

The integral can be broken into its southern and northern

hemisphere contributions.

pi"i

Z0

i

sinL cosLdL

sin2isin2L1=2"i

Zi

0

sinL cosLdL

sin2isin2L1=2

(25)

It is convenient to change the integration variable from Lto Lin

the southern hemisphere integral, so that Eq. (25) becomes

pi"i

Zi

0

sinL cosLdL

sin2isin2L1=2"i

Zi

0

sinL cosLdL

sin2isin2L1=2

(26)

Changing the integration variable from Lto iby noting that

sinLsinisin, one ﬁnds

pi"i

Z=2

0

sinisin d"i

Z=2

0

sinisin d

(27)

Because the integrands are simply the population density as a

function of the sine of latitude, as illustrated in Fig. 3, these integrals

are easily evaluated. Figure 4 illustrates pias a function of inclination

for the sum of populations in both the northern and southern

hemispheres. Figure 5 shows the contribution of each hemisphere to

the casualty expectation.

The peak in Fig. 4 occurs near a 35 deg inclination and indicates a

high population density in the northern hemisphere near 35 deg

latitude. The contribution from the southern hemisphere, is small as

illustrated in Fig. 5. The large peak associated with the southern

hemisphere in Fig. 5 is due to the large population in Indonesia.

It may be possible to predict the longitude of the ascending node of

the ﬁnal orbital revolution closer to the time of reentry. For a given

inclination and longitude of the ascending node of the ﬁnal orbital

revolution, the ground track can be computed. Using the population

near the ground track, one can compute the casualty risk relative to

that of the random reentry casualty risk [13]. It can be greater or less

than the random risk by a factor termed the “fractional risk.”The

average of all fractional risk values associated with all ascending

node values is 1. Figure 6 illustrates the fractional risk as a function of

the longitude of the ascending node for the ﬁnal orbital revolution of

an orbit inclined at 10 deg. Variation in the fractional risk is due to the

nonuniform distribution of the Earth’s population in both latitude and

longitude. If the longitude of the ascending node is 150 deg, the

fractional risk is 75% of that obtained from Fig. 4. The casualty

expectation in this case is 0:75 8:51066:4106.Ifitis

known with certainty that the reentry will be in the southern

hemisphere, one can use Fig. 5 to obtain the casualty expectation for

an orbit inclined at 10 deg. In this case, the casualty expectation

becomes 0:75 3:31062:5106. If the longitude of the

ascending node is 105 deg, the fractional risk is about 3 times

larger than that of Fig. 4. Thus, the casualty risk for an ascending

node of 105 deg is 4 times greater than that of a 150 deg

ascending node. If it is possible to change the orbital decay rate to

change the longitude of the ascending node, the casualty risk can be

reduced.

Figures 6–8 illustrate the fractional risk vs longitude of the

ascending node for orbital inclinations of 28.5, 35, 51, 82, and

98 deg, respectively. The ﬁgures indicate that the fractional risk can

vary by an order of magnitude, whereas the average fractional risk is

equal to 1 in each case. The values of fractional risk can be used with

the values from Fig. 4 to compute the casualty expectation per square

meter of casualty area.

The damage caused by impacting debris can vary depending on

the size, mass, and drag coefﬁcient. Atmospheric drag slows most

debris objects to their free fall state before surface impact. A

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

1.00E-05

1.20E-05

1.40E-05

1.60E-05

1.80E-05

0 10 203040 50607080 90100

Orbital Inclination, deg

Casualty Expectation, #/sq. m.

Fig. 4 Casualty expectation for a random reentry as a function of orbital inclination.

1036 PATERA

computer simulation was used to illustrate the situation at impact by

placing objects with a range of ballistic coefﬁcients in an 80 n mile

circular orbit and allowing them to decay to Earth impact. Figure 9

illustrates the acceleration and ﬂight-path angle at impact, when free

fall is equivalent to a vertical acceleration of 0g. Because most space

vehicle components that survive reentry have ballistic coefﬁcients of

less than 200, they impact nearly vertically at essentially their

respective terminal velocities. A component with a very high ballistic

coefﬁcient of 500 would experience an acceleration of less than 20%

of 1gat impact.

Because some pieces of debris are not very massive and may have

a low impact velocity, they may not have sufﬁcient energy to pose a

hazard. Therefore, debris pieces should be screened based on impact

energy before they are included in the casualty expectation (CE)

calculation. Hazards to people from falling debris have been studied

in depth [25]. To simplify modeling, one can exclude objects with

impact energies of less than a chosen threshold in the range of

15–35 ft lbs16. The impact velocity and energy can be expressed in

terms of object mass and ballistic coefﬁcient:

VI

2

o

s(28)

EIm=o(29)

where mis the mass of the debris piece and rois the atmospheric

density at the Earth’s surface. As an example, a 1 lb object with a

ballistic coefﬁcient of 12 lbs=ft2impacts the ground at 104 ft=swith

167 ft lbs of energy. From Fig. 9, it is clear that an object with a

0.0E+ 00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

1.6E-05

0102030405060708090

Orbital Inclination, deg

Northern Hemisphere, #/sq. m.

0.0E+ 00

1.5E-06

3.0E-06

4.5E-06

6.0E-06

Southern Hemisphere, #/sq. m.

Northern Hem isphere Southern Hem isphere

Fig. 5 Casualty expectation for a random reentry for both the northern and southern hemispheres.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-200 -150 - 100 -50 0 50 100 150 200

Longitude of Ascending Node, deg

Fractional Risk

10 Deg Inc l. 28.5 Deg Inc l.

Fig. 6 Fractional risk as a function of longitude of ascending node for orbital inclinations of 10 and 28.5 deg.

PATERA 1037

ballistic coefﬁcient of 12 lbs=ft2is in a state of free fall with a

90 deg ﬂight path, which corresponds to vertical descent.

The effective area used for each piece of debris in the CE

calculation must be increased due to the ﬁnite size of a typical person.

Each dimension should be increased by a man border in the range of

0.5–1.0 ft. For example, a piece of debris with an impact cross-

sectional area of 3 ft by 2 ft is increased to 4 ft by 3 ft when given a

0.5 ft man border. A debris object with a circular cross-sectional

radius of 2 ft is increased to 2.5 ft when given a 0.5 ft man border.

If the debris is approximately circular, let its radius by represented

by r1. Let the average area of a human be represented by radius r2.

For a debris object of area A, one ﬁnds its effective radius:

r1

A=

p(30)

Thus, the effective interaction area for debris piece and human is

given by

"ir1r22(31)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-200 -150 -100 -50 0 50 100 150 200

Longitude of Ascending Node, deg

Fractional Risk

35 Deg Incl. 51 Deg Incl.

Fig. 7 Fractional risk as a function of longitude of ascending node for orbital inclinations of 35 and 51 deg.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-200 -150 -100 -50 0 50 100 150 200

Longitude of Ascending Node, deg

Fractional Risk

82 Deg Incl. 98 Deg Incl.

Fig. 8 Fractional risk as a function of longitude of ascending node for orbital inclinations of 82 and 98 deg.

1038 PATERA

If the debris object is shaped more like a rectangle than a circle, the

effective interaction area should be computed by increasing its length

and width appropriately:

"il2r2w2r2(32)

The effective interaction area is computed for each debris object

and used in Eq. (27) to calculate the respective casualty expectations.

The total casualty expectation is obtained by summing over all n

debris objects:

CE X

n

i1

pi(33)

Because the population is expected to continue growing at 1.099%

per year, the casualty expectation will increase at the same rate.

Because the population data in Fig. 3 is from 1995, the casualty

expectation in years beyond 1995 is obtained from

CE tCE199510:01099t1995(34)

Equation (34) should be updated when a revised global population

database becomes available.

An example of the use of the casualty expectation calculation is

provided in the decision to deorbit a Delta IV Medium Upper Stage

[10] on 4 November 2006. The Upper Stage was used to place

DMSP-17 into its sun synchronous orbit inclined at 98.7 deg. A

reentry survivability analysis for the Upper Stage determined that the

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

Ballistic Coefficient, lbs/ft

2

Vertical Acceleration, g

-95

-90

-85

-80

-75

-70

Flight Path Angle, deg

Vertic al Ac celeration Flight -Path Angle

Fig. 9 Vertical acceleration and ﬂight-path angle at impact for various values of ballistic coefﬁcient.

0

1

2

3

4

5

-20 0 -15 0 -1 00 -50 0 50 100 1 50 200

Longitude of Ascending Node, deg

Fractional Risk

Fig. 10 Fractional variation in casualty expectation as a function of longitude of ascending node of the ﬁnal orbit revolution for the Delta IV Upper

Stage, which has an orbital inclination of 98.7 deg.

PATERA 1039

casualty area of surviving debris would be about 70 m2. Initial plans

involved a perigee lowering maneuver that would have resulted in a

random reentry. The casualty expectation per unit area for a random

reentry was obtained from Fig. 4 by noting that risk from an orbit

inclined at 98.7 deg is equivalent to an orbit inclined at 81.3 deg due

to symmetry considerations. From Fig. 4, 8:51 106casualties per

square meter are expected based on 1995 population data. This

results in 5:96 104for the expected 70 m2of casualty area.

Updating this number based on population growth expressed in

Eq. (34) results in a casualty expectation of 6:72 104casualties.

The uncertainty in the casualty expectation can be obtained by

analyzing the fractional risk data for an orbit inclined at 98.7 deg, as

shown in Fig. 10. The probability distribution and cumulative

probability for the fractional risk is illustrated in Fig. 11. It indicates

that there is a 20% chance of the fractional probability being above

1.6, which corresponds to a casualty expectation of 1:08 103. The

cumulative distribution for the total casualty expectation is shown in

Fig. 12. It indicates that there is a 50% chance of the casualty

expectation being above 4:93 104. Because both of these

numbers are signiﬁcantly greater than the 1104guideline, a

decision was made to perform a controlled deorbit. It should be noted

that this deorbit was only possible because of the signiﬁcant

performance reserve on the Upper Stage. Not all upper stages have

sufﬁcient performance reserve to achieve deorbit.

Conclusions

Some debris from objects that reenter the Earth’s atmosphere can

survive the reentry heating environment and impact the Earth’s

surface. This impacting debris can injure people and damage

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Fractional Probability

Cumulative Probability

0

0.3

0.6

0.9

1.2

1.5

Probability Density

Cumulative Probability Probability Density

Fig. 11 Statistical variation of fractional risk illustrated in Fig. 10.

Cumulative Probability

0

0.2

0.4

0.6

0.8

1

0.00E+00 5.00E-04 1.00E-03 1. 50E-03 2.00E-03 2.50E-03 3.00E-03

Casualty Expectation

Cumul ative Probability

Cumulative Probability

Fig. 12 Cumulative distribution function for the casualty expectation for the Delta IV Upper Stage.

1040 PATERA

property. An analytical expression for the impact probability density

of a surviving debris object can be used to quantify the risk to a

structure on the Earth’s surface. The impact probability density was

found to depend on the inclination of the space vehicle’s orbit and the

latitude of impact. Impact is not possible for latitudes with greater

magnitudes than the orbital inclination. The highest impact

probability density is at a latitude equal to the inclination of the space

object orbit. The impact probability density is a symmetric function

of latitude; therefore, 20 deg north latitude has the same impact

probability density as 20 deg south latitude. Orbits with smaller

inclinations have less potential area to impact. As a result, smaller

inclination orbits have higher associated impact probability

densities.

The impact probability density multiplied by the exposed area of a

property in question equals the probability of impacting the property.

The probability of impacting a particular building was found to be

very small due to the large surface area of the Earth. The damage

sustained at impact depends on the kinetic energy of the debris.

The casualty expectation associated with a space vehicle

undergoing a random reentry is needed to determine if a controlled

deorbit is necessary to keep the risk below U.S. government

recommended guidelines. The population density can be used with

the impact probability density to determine the casualty expectation

per unit area of surviving debris. Although the population density has

a longitudinal dependence, only the latitude dependence is

signiﬁcant due to the uniform longitude distribution of the impact

probability density of the debris object. A global population database

was used with the impact probability density to compute the casualty

expectation per unit area as a function of the orbital inclination of the

reentering debris. One needs only the casualty area of the surviving

debris pieces and the precomputed casualty expectation per unit area

as a function of the orbital inclination to compute casualty

expectation. Although methods to reduce reentry survivability are

being developed for future space vehicles, the need to perform

casualty expectation calculations will remain for years to come.

Acknowledgments

The author would like to thank William Ailor, Charles Gray, and

Rolf Bohman for reviewing this paper and providing many insightful

suggestions to improve it.

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O. de Weck

Associate Editor

PATERA 1041