Hazard Analysis for Uncontrolled Space Vehicle Reentry
Russell P. Patera∗
The Aerospace Corporation, Los Angeles, California 90009-2957
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.
A= vehicle drag area
da =inﬁnitesimal surface area
g= gravitation acceleration near Earth’s surface
i= orbital inclination
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
"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
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
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 .
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 . 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 . The U.S. government
developed guidelines to limit the hazard from reentering space
vehicles . If the casualty expectation of a reentering satellite
exceeds 1 in 10,000, a controlled deorbit is strongly recommended
. 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  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
Reentry risk can also be reduced with only partial control of a
space vehicle via drag modulation . This was achieved during the
reentry of Skylab in 1979 by shifting the impact footprint thousands
of miles away from North America . Drag modulation has been
proposed as a method of reducing reentry risk for larger space
vehicles that would otherwise require expensive controlled deorbit
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
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  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
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.
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
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):
1999 2000 2001 2002 2003 2004 2005 2006
Ye a r
Number of Objects
Mass, kg Number of Large Objects
Fig. 1 Total mass and number of large object reentry for recent years.
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
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 .
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 . 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
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
The normalization condition is satisﬁed:
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:
Eliminating Rand taking the derivative of Eq. (4) yields
Thus, dis related to dLby
The impact location distribution function can be expressed in terms
of latitude by considering an inﬁnitesimal probability associated with
Using Eq. (6) in Eq. (7) yields
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
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
L; iL; idL
The normalization condition is given by
Notice that the range of Lis bounded by the orbital inclination. This
integral can be evaluated by a change of variable given by
The derivate of Eq. (14) is
Using Eqs. (14) and (15) in Eq. (13) gives the proper normalization as
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
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
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
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
. 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
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
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:
In Eq. (21), "iis an effective interaction area centered on xand is
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
-45 -30 -15 0 15 30 45 60 75
Cumulative Popu lation
1995 Population 2007 Population (Skewed)
Fig. 2 Population distribution function.
-1 -0.5 0 0.5 1
Sine of Latitude
Population Density, #/sq. m.
1995 Density 2007 Densit y (Skewed) 1995 Population 2007 Population (S kewed)
Fig. 3 Population density functions.
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
The integral can be broken into its southern and northern
It is convenient to change the integration variable from Lto Lin
the southern hemisphere integral, so that Eq. (25) becomes
Changing the integration variable from Lto iby noting that
sinLsinisin, one ﬁnds
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 . 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
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 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.
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 . 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:
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
Orbital Inclination, deg
Northern Hemisphere, #/sq. m.
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.
-200 -150 - 100 -50 0 50 100 150 200
Longitude of Ascending Node, deg
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.
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:
Thus, the effective interaction area for debris piece and human is
-200 -150 -100 -50 0 50 100 150 200
Longitude of Ascending Node, deg
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.
-200 -150 -100 -50 0 50 100 150 200
Longitude of Ascending Node, deg
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.
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:
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
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
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
 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
0 100 200 300 400 500
Ballistic Coefficient, lbs/ft
Vertical Acceleration, g
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.
-20 0 -15 0 -1 00 -50 0 50 100 1 50 200
Longitude of Ascending Node, deg
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.
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.
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 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Cumulative Probability Probability Density
Fig. 11 Statistical variation of fractional risk illustrated in Fig. 10.
0.00E+00 5.00E-04 1.00E-03 1. 50E-03 2.00E-03 2.50E-03 3.00E-03
Cumul ative Probability
Fig. 12 Cumulative distribution function for the casualty expectation for the Delta IV Upper Stage.
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
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.
The author would like to thank William Ailor, Charles Gray, and
Rolf Bohman for reviewing this paper and providing many insightful
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