ON-ORBIT COLLISION HAZARD ANALYSIS IN LOW EARTH ORBIT

USING THE POISSON PROBABILITY DISTRIBUTION

Version 1.0

Prepared for:

Office of Commercial Space Transportation

Licensing and Safety Division

Prepared by:

Kaman Sciences Corporation

Engineering Sciences Division

2560 Huntington Ave., Suite 200

Alexandria, Virginia 22303-1410

August 26, 1992
INTRODUCTION

 Any object placed in orbit around the Earth has a risk of colliding with other orbiting objects. For

mission planning and safety reasons, it is useful to understand and assess the level of collision risk by

determining the probability that a given object will collide with currently orbiting objects. Several

techniques can be used to calculate the collision hazard for orbiting spacecraft. This primer introduces

one method for analyzing collision risks that utilizes the Poisson distribution and principles of the

kinetic theory of gases. Also, this primer addresses its application to low Earth orbit (LEO) only.

THE PROBABILITY OF COLLISION

 The Poisson probability distribution function is given by the equation:

(1)

where: Pk = Probability of k events

k = Number of events, and

R = Rate of occurrence parameter.

The probability of no events occurring (k = 0) can be found:

(2)

In general, a Poisson distribution is used in those situations when the probability of two or

more events occurring is very small. If an assumption is made that the probability of more than one

event is negligible, the probability of one collision can be represented by:

(3)

The rate of occurrence parameter, R, is obtained from the kinetic theory of gasses. It is defined

as the number of collisions during a given length of time between one molecule of gas with others in a
confined volume, and is given by the product of the molecule's size,1

 density of the gas, relative speed,

and the length of time considered.

 By applying the same kinetic gas assumptions to a confined volume in space, the collision rate

between a specified object and all other objects in a given volume can be found for a specified length

of time. For an object in orbit, the rate of occurrence parameter would then be the product of that

object's cross-sectional area (AC), average spatial density of all other objects (SPD), average relative

velocity (VR) between the object under consideration and all other objects, and the time (T), which is

the length of time the object is exposed to the given population density, SPD. Substituting the analytic

form of the rate of occurrence parameter (AC*SPD*VR*T) for orbital collisions into Equation (3)

yields Equation (4); the probability that there will be a collision between a given object and any other

object in a given volume.

(4)

where: PC = Probability of collision for the duration of time, T

AC = Cross-sectional area, km2

SPD = Spatial density, objects/km3

VR = Relative velocity, km/s, and

T = Time at risk, seconds.

CALCULATING THE PROBABILITY OF COLLISION

 Equation (4) shows there are four parameters to be considered when calculating the probability of a

collision. Each of these terms are discussed below.

Spatial Density (SPD): Spatial density is the number of objects that reside in a given volume of space.

Figure 1 shows the object spatial density up to 2000 km altitude for 50 km altitude "shells." For

example, on the average2

, 40 objects reside in the 1700 - 1750 km region. The volume of the "shell,"

with an inner radius of 8078 135 km (1700 km altitude) and an outer radius of 8128.135 km (1750 km

altitude), is 3.69E+10 km3

. The spatial density is then the number of objects divided by the volume or

1.06E-9 objects/km3

. Appendix A provides a table of the spatial density values from Figure 1.

 

 

1

It is assumed that all molecules in a gas have the same diameter, d. Therefore, when deriving the

mean free path from the kinetic theory of gases, the simplification is made that the diameter of the molecule

under consideration is 2d, and the impacting molecules are point masses.

 

2

In reality, the number of objects in a region at any given time changes as objects in elliptical orbits

pass between regions. For this reason, an average number of objects is used to describe the population

in time invariant terms.
Time (T): The time at risk is the length of time the object poses a risk to the population. If the object

is to remain in orbit for a predetermined length of time, the time at risk is this value. If an object is in

an orbit which is subject to decay by atmospheric drag, the lifetime of the object must be estimated.

Assuming average solar activity, a first order estimate can be obtained from Appendix B for circular

orbits below 925 km (525 nm).

Cross-sectional Area (AC): Cross-sectional area is the projected area of the object along the trajectory.

Since this object could actually be tumbling through space (i.e., changing the projected cross-sectional

area), calculating and using the largest possible cross-sectional area will provide the most conservative

collision estimate. The largest cross-sectional area can be determined from the dimensions of the

object. For example, if the object left in orbit is a cylindrical rocket stage, the largest cross-sectional

area will be the length of the stage times the width or diameter.

 For accuracy, the average collision cross-sectional area3

 between the specified object and all other

 

 

3

 The collision cross-sectional area between two objects comes from the impact parameter,

b, in the kinetic theory of gasses. The impact parameter is the sum of the radii of the two

molecules (i.e., b = rl

 + r2) in a collision. The collision cross-sectional area between molecules

is ∏b

2

. Since radar cross-sectional areas of space objects are known, it is more convenient to
objects with which it could collide should be used in Equation (4) instead of the physical

cross-sectional area. The average collision cross-sectional area is found by:

(5)

where: XC = Average collision cross-section

AC = Cross-sectional area of the object

ACi = Cross-sectional area of objecti

 in the population, and

n = Number of objects.

However, calculation of average collision cross-section can require significant effort. To simplify

Equation (5), an assumption is made that all objects in the population are small compared to the object

under consideration Thus, the collision cross-section can be substituted by the cross-sectional area of

the object for a reasonable first order approximation.

Relative Velocity (VR): Relative velocity is a function of the relative angle of encounter of the object

with another object and the velocity of the colliding objects. The relative angle of encounter varies

markedly with the orbital inclination, right ascension, and eccentricity of both objects being

considered. Consequently, many possible combinations of these orbital parameters make up the

distribution of angles of encounter that any object may have during the time period considered. Thus,

determination of relative velocity is very difficult to discretely calculate.

 Current methods for obtaining accurate relative velocity values involve orbital simulations, or

interpolation and extrapolation of empirical data. Relative velocity can theoretically range from 0 m/s

to twice the orbital velocity. However, for simplification purposes, average relative velocity in LEO is

commonly estimated at 2 times the orbital velocity or (approximately 10 km/s). This comes from the

assumption that the average relative angle is 90 degrees

EXAMPLE PROBLEM

 The following examples demonstrate the calculations that are required for: 1. an object of

cross-sectional area 11 m2

 and a mass of 121 kg in a 525 km circular orbit left to decay by

atmospheric drag; and 2. an object with cross-sectional area of 11 m2

 in an elliptical orbit with an

apogee at 1185 km, and a perigee at 835 km that will remain in the same orbit for 10 years and then

be removed.

For Object 1:

 To calculate the lifetime of object 1, the ballistic coefficient of this object is needed. The ballistic

coefficient is found by:

 

work with area dimensions rather than radii. It can be shown that ∏b

2

 is equivalent to ([∏r12]0.5

+ [∏r2

2

]

0.5)2

 This assumes that all objects are spheres with circular physical cross-sectional areas

[Ref 6]
(6)

where: BC = Ballistic coefficient,m2/kg

Cd = Drag coefficient,2.0 to 2.2 [2.2]

A = Cross-sectional area, m2

, and

m = Mass, kg.

· SPD: For a circular orbit at 525 km, the spatial density is found using the altitude shell from

500-550 km in Appendix A to be 2 53E-9 objects/km3

.

· T: Given that the orbit is circular (e = 0) at an altitude of 525 km, and the ballistic coefficient

is found from Equation (6) to be 0.2 m2

/kg, the lifetime is approximated from Appendix B to

be 65 days.

· VR: Since the object is in LEO, the average relative velocity is assumed to be about 10 km/s.

· AC: The object's physical cross-sectional area is 11 m2

.

Using these values in Equation (4) provides a probability of collision of 1.56E-6 for the object's entire

lifetime.

 Note, however, that the object will not be at 525 km altitude for the entire lifetime; rather, its

altitude will decrease as the orbit decays. In this case, the result can be considered the worst case

probability of collision by assuming the object remains in this altitude for the entire time duration,

rather than decaying into and through the lower altitude regions which are less populated, because the

initial orbit's altitude has a higher spatial density than altitudes below it.
For Object 2:

 This example is complicated by the fact that object 2 is an elliptical orbit that crosses several

regions with different spatial densities.

⋅ SPD: A more complex method for calculating a weighted spatial density that is good for all

eccentricities involves solving Kepler's equation and accounting for other factors to determine

the time spent in each region. For simplification purposes, an average spatial density is

calculated as shown in Table 1 for a first order approximation.

TOTAL AVERAGE SPATIAL DENSITY = 5.51 E-08 / 6.6 = 8.35E-09 .

The spatial density contributions from the regions of apogee and perigee are weighted based on

the approximate distance the object will travel into the region. For example, an object with a

perigee at 835 km will travel 15 km into the 800-850 km region, thereby creating a factor of

15/50 or 0.3 to be multiplied by the spatial density to determine the contribution to the total

density. Thus, given an elliptical orbit 835 x 1165 km, the average spatial density was found to

be 8.35-09 objects/km3

. This method is good for low eccentricity, elliptical orbits in LEO, and

will result in values within 10% of the more complex method that directly calculated the

weighing factor for each region based on the time spent crossing the region.

⋅ T: The object has a scheduled lifetime of 10 years; at which time it will be removed from orbit.

It should be noted that by calculating the probability of collision over 10 years, an assumption

is made that the population (SPD) does not change significantly over that time.

⋅ VR: Since the object is in LEO, the average relative velocity is assumed to be about 10 km/s

⋅ AC: The object's physical cross-sectional area is 11 m2

 Applying these values to Equation (4) gives a probability of collision of 2.90E-4 for the object's entire

duration in this orbit.
CONCLUSION

 This document provides the basic information needed to estimate a general probability of collision in

LEO. Although the method described in this primer is a first order approximation, its results are

reasonable. Furthermore, the methods described above can be the basis for developing more rigorous

techniques for calculating collision probabilities through the use of the Poisson probability distribution

REFERENCES AND BIBLIOGRAPHY

1. Vedder, J.D. and Tabor, J.L., "New Method for Estimating Low Earth Orbit

Collision Probabilities," JOURNAL OF SPACECRAFT AND ROCKETS, Vol. 28, No.2, Mar-Apr 1991,

pp.210-215.

2. Rossi, A. and Farinella, P., "Collision Rates and Impact Velocities for Bodies in

Low Earth Orbit," submitted to ESA JOURNAL, April 1992.

3 Resnick R. and Halliday D., PHYSICS FOR STUDENTS OF SCIENCE AND

ENGINEERING, John Wiley & Sons, Inc., New York, NY,1962.

4. HAZARD ANALYSIS OF COMMERCIAL SPACE TRANSPORTATION, VOL II:

HAZARDS, U.S. Department of Transportation, Washington, D.C., May 1988.

5. Johnson, N.L. and McKnight, D.S., ARTIFIClAL SPACE DEBRIS, Orbit, Malabar, FL,

 1987

6. Kessler, D. J. and Cour-Palais, B. G., "Collision Frequency of Artificial Satellites:

 Creation of a Debris Belt," published by the American Geophysical Union,1978.
APPENDIX A

SPATIAL DENSITY OF EARTH ORBIT UP TO 2000 km CALCUALTED

FROM DATA IN THE MARCH 1,1992 SATELLITE CATALOG.
APPENDIX B

CALCULATING AN OBJECT LIFETIME.

[Source: Reference 4]