Contribution of explosion and future collision fragments to the orbital debris environment

34, 1983 Ad~.~ No.2, Printed in p.~. Great Vol.3, Britain. All pp.25— rights reserved.

0273—1177/85 SO.0O

+ .50 Copyright ~ COSPAR

CONTRIBUTION OF EXPLOSION AND FUTURE COLLISION FRAGMENTS TO THE ORBITAL DEBRIS ENVIRONMENT S.-Y. SU5 and D. J. KessIer*~ *LockIIeed Engineering and Mana~eineniSer%’ices

Co~npain.

Inc., Houston, TX, U.S.A. ~

Space Center, Houston. TX. U.S.A.

ABSTRACT

The time evolution of the near—Earth man—made orbital debris environment modeled by numerical simulation Is presented in this paper. The model starts with a data base of orbital debris objects which are tracked by the NORAD ground radar system. The current untrackable small objects are assumed to result from explosions and are predicted from data collected from a ground explosion experiment. Future collisions between Earth orbiting objects are handled by the Monte Carlo method to simulate the range of collision possibilities that may occur In the real world. The collision fragmentation process between debris objects is calculated using an empirical formula derived from a laboratory spacecraft impact experiment to obtain the number versus size distribution of the newly generated debris population. The evolution of the future space debris environment is compared with the natural meteoroid background for the relative spacecraft penetration hazard. INTRODUCTION Recent studies by Kessler and Cour-Palais /1/ and Kessler /2/ compare the man—made orbital debris environment with the natural meteroid environment to determine the degree of hazard this new debris environment poses to the safety of spacecraft operations. It was then concluded that the existing space debris environment only poses a hazard to large (greater than 100 meters in diameter) spacecraft. However, their model reveals that because of the “self-regenerative~ nature and cascading property of the collisional fragmentation process. future man—made debris environment would become more dangerous than the natural meteoroid background to the safety of the spacecraft at all sizes. This paper will upgrade the previous model by using a Monte Carlo technique to numerically simulate the collision process instead of using the average collision rate used by the previous model. The mathematical foundation for describing the collision probabilities between orbiting debris objects has been published by Kessler /3/. Readers are referred to the original paper for the formulas. NORAD SATELLITE DATA BASE The data base of our model Is built from the satellite population tracked by the NORAD ground radar system. The information obtained from the NORAD radar system are the orbital elements and the radar cross section (RCS) of each orbiting Object. The only orbital elements used in this paper are the semi major axis, eccentricity and inclination. The orbital inclination was Important in calculating the relative velocities between the orbiting objects. However, the distribution of inclinations were found to be sufficiently close to random so that collision probability could be assumed equal at all latitudes. Other orbital elements such as the argument of perigee and the longitude of the ascending node are also assumed to vary randomly in this statistical analysis of the collision probability between debris objects. As of March 1983, 5138 orbital objects were cataloged by NORAD. All the relevant information on these objects is easily stored In the computer. However, a collision between any two large orbiting objects produces millions of fragments. It then becomes impractical to store all the orbital elements and sizes of these new debris objects in the computer for subsequent calculations. Thus, a three—dimensional mesh of storage bins, identified by d, a and e (object diameter, semi major axis and eccentricity, respectively) has been devised to store the debris objects. Table 1 lIsts the mesh sizes of different variables used in this model. It is noted that the values for semi major axis listed in the table are treated like the altitude height above the Earth. 25

2,

S.—?. Su and D.J. Kessler

TABLE 1

Data Base Arrangement for Debris

Parameter

Parameter Range

log d (m)

—3 to 3

1 to 30

0.2

a (km)

100 to 1000 1000 to 5000

1 to 180 180 to 260

50

0 to 0.1

1 to 20

0.005

0.1 to 1.0

21 to 38

0.05

100 to 2500

1 to 48

50

e h (km)

Increment Between Bins

Bin Number

For parameters a and e, it is noted that the increments are smaller at their lower values and change to larger increments at their higher values. Uneven increments are taken to assure an optimal mapping between the true orbital elements and the discrete storage bins since most orbital objects have small eccentricities and are in orbits lower than 1000 km in altitude. It also increases the accuracy of the output of the atmospheric drag program where the orbital debris changes bin positions. All altitude bins are fixed at a 50—km increment. The present model will describe the debris environment below an altitude of 2500 km. To assure that orbital debris with high eccentricity are included in the data base, the value of semi major axis is extended to 5000 km as shown in the array in Table 1. However, the geosynchronous transfer orbits are not included in this analysis because of limited storage bins for semi major axis. EXPLOSION FRAGMENTS Of the 5138 NORAD—cataloged objects in the data base, almost half come from known explosion events. Since small fragments from in—space explosions at high altitudes cannot he detected by the ground radar system, the observed data in each explosion event is extrapolated to obtain the number of untrackable small fragments. This is accomplished by comparing the tracked data with an empirical formula derived from the ground explosion experiment. The empirical formula, which was obtained by examining the ground explosion data reported by Bess /4/ and is listed as 1”2), for M > 1936 g; (la) 1.71 x ~ Mt exp(—0.02O56M 8.69 x l0

Mt exp(—O.O5756M

), for M

<

1936 g.

(1b)

Here N is the cumulative number of fragments with mass greater than M measured in grams. M is the total mass of the pre—explosion object in grams. In order to apply Equations (la) a~ld Mb) to the observed data, the fragment mass is first obtained from the fragment size in RCS using a formula relating the mass and RCS of a space object given in the paper by Kessler and Cour—Palais /1/. Next, as shown in Figure 1, the trackable explosion data is plotted as the logarithm of cumulative number versus the square root of the f’agment mass in kilograms. The data plotted in Figure 1 is from the Delta second stage for NOAA 3 satellite (international designation number 1973—86B). It is seen that the data points in Figure 1 do not fall on a straight line as predicted by Equation (la), except for the middle portion of the curve. The falloff from the straight line for the larger object size could be due to the statistical fluctuation in the maximum size, whereas the falloff in the cumulative number for the small size objects is assumed to be caused by detectability limitations in the ground radar system. Thus the extrapolated total cumulative number of explosion fragments is obtained from the straight line, and a weighting factor is assigned to the original data from the ratio of the extrapolated number to the observed number. For fragments smaller than 1936 g (13 cm in diameter), the number is further increased by the same ratio between Equations (la) and (lb). Thus, when the spatial density of the fragments from 1973—86B is plotted versus altitude as shown in Figure 2, it is noted that the corrected cumulative spatial density for fragments down to 4 cm in size (upper curve) is several times higher than the original trackable population (lower curve). A total of 23 known explosion events with at least seven fragments being tracked in each event have been analyzed. The results are plotted in Figure 3 as three curves in cumulative spatial density versus altitude, representing three different minimum object sizes in the debris population. It is noted that more than one order of magnitude in spatial density is contributed by the small fragments in explosion events. For objects larger than 4 cm, the population is predicted to be a factor of 2 to 3 larger than the tracked population. This compares to a factor of 3 given by Kessler /2/, and comparable to the results given by

Explosion and Collision ContributionS

27

~~4 INTE9N~T~ONALO$S~GN~TOR ~973960

MA$S~~

Fig. 1. Cumulative number of explosion fragments versus the square root the fragment mass. The international designator of the exploded spac~craft is 1973—86B, the second stage of the Delta rocket for satellite NOAA The empirical formula for the number—size distribution is fitted to the

observed data.

NTEMNATIC)’.AL DESiG’~A’CIR 973

39$

~o7

0

500

1100

2400

32f~O

4000

ALTITUDO ~

Fig. 2. The spatial density of the observed explosion fragments from 1973-86B versus altitude. The lower curve is from the observed data only. The upper curve is from the fitted curve of empirical formula for explosion data in Fig. 1.

28

S.—Y. Su and D.J. Kessler

Lennertz /5/. The small fragments obtained from the explosion analysis are not given the orbital elements of their parents. They are kept in a separate data base with fictitious circular orbit elements that can reproduce the spatial density shown in Figure 3. These ‘average~’ orbit elements allow atmospheric drag to change their altitude positions.

NORAD MARCH 1983 DATA BASE 10.6

—

DEBRIS OBJECT ~ 1 mm

—

DEBRIS OBJECT ~ 1 cm MARCH 1983 NORAD CATALOGUED DATA

I.;-.

10•~

S r’ ~ I(2 14

I

~

0 0

~L...

14 00

.--.--~

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

ALTITUDE 1km)

Fig. 3. The spatial density of the orbital debris as function of altitude. The bottom curve is taken from the NORAD cataloged space objects as of March 1983. The two upper curves include the explosion fragments too small to be tracked by NORAD radar system. The top curve is for objects greater than 1 m, while the middle curve is for objects greater than 1 cm.

Explosion and Collision Contributions

29

COLLISIONS BETWEEN DEBRIS OBJECTS The average collision rate between the debris objects within a volume element tU is given by

c1~it= s~s~ vo ij

(2)

~.

where S1 and S. are the latitude-averaged spatial densities of the debris objects allocated In size bins d~and d4, respectively. v Is the average relative velocity which Is obtained by dividing the colli~ion rate of a random sample of 200 debris objects calculated from the equations in Kessler’s paper /3/ by the latitude—averaged spatial density of these 200 objects in the same volume element. The average velocity is found to be about 8.7 km/s for all altitudes and is assumed to remain unch~ngedwith time. The collision cross—section can be approximately given byij~d4 + d.) /4. The average collision rate in one volume el~ment for objects between different siz~bins and f~rdifferent objects in the same size bin calculated by using Equation (2). When i j, S. in Equation (2) is subtracted by the sum of the square of the spatial density from each Ir~dividual object in the same size bin and then multiplied by 1/2.

By adding up all the size bins and then all the volume elements, the average collision rate, C, is obtained for all the debris objects in the volume under consideration. The average collision rate is found to be 0.26 collIsion per year for all the objects down to millimeter in size object below 2500—km altitude. For objects larger than 4 cm, the collision rate Is 0.04/yr. This compares to 0.06/year given by Kessler /2/. The following procedure has been adopted to simulate the collision process.

After the

number of collisions that could occur in a given number of years has been determined, the altitude where the collision took place and then a pair of size bins (or within one single particle size bin) which were involved in the collision will be decided. Finally, the identities of collision objects and the collision velocity are picked. Setting t

=

1 year, and knowing the average collision rate for all the objects in the whole

volume, the Poisson distribution is needed to determine the probability ~k for exactly k collisions to occur in each year, i.e., — exp(—C) ce/k! (3) A collision probability line Is set up as shown in Figure 4, with the length of the line segment representing the probability of collisions. For example, the first, second, and third line segments in this figure (denoted by P , P , and P ) represent no collision, one .

collision, and two collisions in a year, respect?vel~. The ~otal length of the probability line in this case should equal to unity, representing all the possible outcomes of the event.

I

P0

I

P1

I

I

0

lii 1

Fig. 4.

CollIsion probability line.

A random number generator is used to obtain a number between 0 and 1 to determine the line segment on which the random number will fall. The line segment ~k’on which the random number falls indicates that k collisions have taken place. Each time the random number Is generated, the time increment is Increased by one year. Since P turns out to be a larger number than other P.s in the probability line, there is a good chance that no collision will occur In a year. If this is the case, another random number will be generated and the time step is increased by one year. This procedure is repeated until a non—zero collision takes place. Once it Is determined that one or more colflslons have occurred, the next step is to determine the altitude bin at which the collision(s) took place. This is accomplished by constructing another probability line composed of the line segments proportional to the collision probability at each individual altitude bin. The line segments containing P ‘S are not included here since it has already been determined that the collision has occuPred. The total length of the probability line in this case has to be normalized to unity. Once again the random number is picked to find the line segment on which the random number falls. The altitude bin number corresponding to the chosen line segment Is assigned as the collision altitude. Then the collision probabilities from all different size bins in that

30

S.—Y. Su and D.J. Kessler

altitude are used to construct the third probability line to determine the pair of size bins (or within one single size bin) in which the collision will take place. After it has been decided that the collision should occur in the particular object size bin(s), the identities of the colliding objects within that given size bin(s) have yet to be determined. This can be accomplished by returning to the data base to generate a list of all the objects in the given size bin(s) that pass through the collision altitude. For each object in the list, the spatial density at that altitude is calculated as the weighting factor for use in constructing the fourth probability line. The spatial density is needed because the average collision rate given in Equation (2) is proportional to the spatial density of the colliding objects. With the probability line laid out, it is easy to identify the objects that have collided. The collision velocity of these two objects is determined in the same manner with the fifth probability line composed of line segments proportional to the magnitude of the relative velocity between the objects. Incidentally, the orbital inclination of the two colliding objects could be usee to determine the collision velocity and the collision latitude. In the present paper we shall neglect this and keep the results as functions of altitude alone. After the collision veloclty between the two colliding objects has been obtained, the outcome of a collision can be evaluated. The output of the collision process is new debris objects with a given size and velocity distribution to be incorporated into the data base for use In calculation of future collisions. COLLISION FRAGMENTATION When two debris objects are about to collide, the most difficult thing is to determine how they will collide. If the two objects differ greatly in size, the larger object is considered to be the target and the smaller one to be the projectile during collision. Although the location of the impact area in the target is important in considering the momentum transfer between the two objects, the effect of momentum transfer will be ignored as the first approximation in the calculation of the final orbits of the two objects. In a simple impact model derived from the laboratory experiment the ejecta mass N , is set equal to the product of the projectile mass, M and the square of the collision vefocity v in km/s normalized by a constant factor. In the~present case, the constant factor is set to equal unity. Thus 2M~ (4) v The projectile is assumed to be destroyed in every collision, and its mass is added to the ejecta mass for calculating the number versus size distribution. The cumulative number versus size distribution for the ejecta fragments, which was obtained by reanalyzing Bess’s test data /4/, is given as follows: N

N

=

( ~e)_0~7496

0.4478

=

(5)

Here N is the cumulative number of ejecta with mass M or greater. Equation (5) differs slightly from the formula used by Kessler and Cour—Palais /1/ in that the slope for the new curve is slightly flatter. Thus a few more pieces of larger fragments are available. When the ejecta mass exceeds certain amount of mass relative to the target, it is assumed that the remaining target would be broken up. The critical ejecta mass, M~,is assumed as Mc

=

lOMe

=

1Ov2M~

(6)

Thus, when the ejecta mass is larger than 10 percent of the target, a catastrophic collision has occurred such that the target is destroyed. If this is not the case, the collision is non—catastrophic and the projectile only excavates out of the target. The larger fragments from the destroyed target will be treated as a low—intensity explosion event (i.e., using Equation (1)), while the smaller ‘ragments are obtained using Equation (5). Once the collision and explosion fragments spread out from the parent body, their orbital

elements must be determined in order to return them to the data base for use in future calculations of the collision process.

In the current model, the fragments are assumed to The ejecta velocity from non—catastrophic impact is assumed to vary randomly from 10 to 30 rn/s regardless of ejecta size, whereas the large fragments from the catastrophic breakup process are assumed to have

be ejected isotropically outward with respect to the target. velocities varying randomly from 5 to 15 m/s.

If the target is much larger than the projectile, the transfer of momentum to the target is small, and the center of mass of the target will remain in Its old orbit. The velocity of the fragments will be the vector sum of the target velocity and the ejecta or breakup velocity. If the target and projectile are about the same size, and the collision is not a

Explosion and Collision Contributions

3)

direct, “head—on’ collisiàn, then debris from both objects will orbit the Earth with reduced orbital angular momentum. The probability is very low for two nearly equal mass to experience a “head—on” collision such that all of the fragments would have insufficient velocities to remain in orbits. Therefore it was assumed that all collision fragments would maintain the average momentum of the larger object. MODEL RESULTS The collisional fragmentation process is assumed to be the most significant source for producing small debris objects in the future. The larger objects are assumed to come from launches of future spacecraft. In the present analysis, it has been assumed that the 1982 launch schedule, with the associated launch debris, would be representative of all future years. This schedule contained 121 launches, of which about 100 were USSR. The sink for all debris is reentry into the Earth’s atmosphere, due to atmospheric drag. Atmospheric density is calculated by using a modified Jacchia model which averages daily and seasonal variations, but maintains the variation based on solar cycles. Average solar cycles were assumed for dates past the current cycle. Several runs of the evolution model have been completed for a 50—year period between the years 1983 and 2033. In general, it is noticed that although the first collision occurs shortly after the model is executed, almost all collisions involve centimeter or millimeter sized objects. The collision generates many more millimeter sized objects, which in turn push up the subsequent collision rate many folds for the millimeter sized objects. This

cascading effect keeps producing large amounts of millimeter sized objects without an end unless checked by atmospheric drag. Thus, noticeable fluctuations in the millimeter sized objects were observed from year to year. On the other hand, the larger sized objects behaved more steadily. atmospheric drag.

They were created less frequently, and took longer to decay by

The results from one of the sample runs are plotted in Figure 5, in which two separate sets of figures are grouped in two separate columns. The figures in the left—hand column show the evolution of man—made debris environment, while figures in the right—hand column show the man-made debris superposed on the curve for the natural meteoroid background. The natural meteoroid background flux level, which is taken from NASA 1969 meteoroid model prepared by Cour—Palais /6/ has been converted to the equivalent energy of the Earth orbiting debris object for comparison with the man—made debris flux intensity. There are four curves in each panel of figures shown in Figure 5. These curves, from top to bottom, represent the latitude averaged cumulative number flux intensity with minimum object sizes of 1 mm, 4 mm, 1 cm, and 4 cm, respectively. Each panel represents a snapshot of the debris environment for the year indicated. In the beginning year of 1983 shown in the first panel of Figure 5 in the right column, the man—made debris flux level slightly exceeds the natural meteoroid background flux level for objects with the minimum size of 4 mm. Examining the second panel for the year 1991 reveals that only small differences exist between this panel and the first one in lower altitudes. This could be interpreted to mean that a dynamical semi-equilibrium seems to be reaching between the generation of small objects from collisional fragmentation process, the decay of objects by atmospheric decay, and the input of newly launched spacecraft from the traffic input model. In the panel for the year 2000, a drastic change in flux level occurs at around 900 km in height. The flux of man—made debris objects down to 1 mm in size actually exceeds the meteoroid flux background at an altitude around 900 km. In order to explain what has happened during this period, the major collision events that account for the large production of debris objects as listed in Table 2. It is seen that between the years 1991 and 2000, three major collisions have taken place around the 900—km altitude. The first major collision occurs in the year 1992. It involves two large sized objects, one in size bin 13 and the other in bin 19. A collision velocity ~f 14 km/s has been chosen. The collision generates a total ejecta mass Me = 6.89 4x 10 g and causes the larger object to break up with a total breakup mass M = 1.12 x 10 g. The ejecta mass, the breakup mass, and the breakup process were all ca1cu1a~edusing Equations (la), (lb), and (4) through (6). The two subsequent collisions listed in Table 2 do not seem as catastrophic as the first one. Only a moderate number of ejecta are released. Over the next 30 years, three additional major collisions take place, and release a very large amount of debris.

At this point, a check of the results plotted in Figure 5 is very desirable. The very high level of man—made debris flux that resulting from major collisions is reflected in the panels for the year 2010 and later. One particularly disturbing fact is that the low—altitude collisions occurring in the year 2016 resulted in an extremely high debris flux level being observed at around 500 km for next several years. This is evident from examining the plot of panel 5 in which the effect of the high flux level in the year 2020 is still being observed. Since the altitude band between 300 km and 500 km is the most active

32

S.—Y~ Su and D.J. Kessler

~:

~il

u~ 4-10(2

—

9

C 5

— ,_T

—

0

-

0

—

‘

— -.

. •

--

$ ~

H

-..-~.

-~

---

..—

4-’

C.0

•-

C

.--

j’—

40 4.1 0-do

__1.

II~~

C

SO C

C i— 0 L LW

5

rr

— .• . .-~:-.-

.~

4-00

~

.

—

-

[.. T

-. ~1

SOW

I

._--

:_

W

4 — •

.5—I ~ I/IC

SO

LOOL

-

.-

. •

-~

.

O O

~r.—

..

.

Cs-I . 0

—

-

•::

. — -— .

LC) . - .

C

-

-

—

-~

—

— .

-

.0 SO E —‘

0 .0 0<00

.4.2

•~

~WI.fl

1.~..

— —

—

-•

I.~C.C

~‘-~

--

SO ~00

—~ Jn •

.,IA,,SA7,flI,I.flhj

•

,nl*tk),.a~~.fl1,

.

~Wl*,f$j3Y4$II

..I U’.— 4.1 4-1 SO

C L

In,,

XL

~ 4-

E

0)0

.C 4.1 .5.04.I

0

L

_________________________

_______

SO C.0 .0 E 0

___________________________ 0

_________________ -

,.

~

C.- 0)0.

IIS

-

00-CO

.

4010.4-I .

• -. -I

=

.-

~

.•

-

0)0)0

.. .

.-

..~

-

-••--

-I —

—

9

9 . --

.

0

~ ——

— .

SS

..-..~ -

—

—-. -~

.

~. .

~1~

.2o-.-510) .4-ICC

C040

E40 0 00 C

-~ 0

-

-

. -

COOC

•--

.

—— 00.-U,

____________I

L~__________

—-.—~-

~ .0-—

0..

~

(2

0).0

___________________

________________

II’’

.~-~ -.

-

.~: -.• -~

.

.8 0

•

:~ :~ • . —- =

•—-. .

— S

_---. •~ • .

•

S

—.-•-•--~-~..

0.. “—4--

0

I

.:_ --

-

—.-.• ~3.

.

~—--

•

-~,

—

•

r’~’’•n~~

•fl3* 1**1L~dvII, Iflhl

~01* ~.1*3V4S1~ XflhI

C 10. I/I SOs.’ 0) 51.0 .0 L 4.100 E

~05OSO0 •.L.C

—

-T .01*

0)

.0 OEL> C 0000)4-I ~. L .0 ~ 0

-

~

-s-ISO-.-- -~~ ~

:

~

0) .CE~

i-.-~ ...~

•

L 0

LI.. ‘0 4.440.

Explosion and Collision Contributions

0)00 ~.

—

E

40

• .0<

.‘

0(0 .-~

0(0 ..4

40

~

.-l 00

L)>.

L 10

a)

>0

4.1

4..’ ‘I,

10 0<

c’J Oh

0(0

C-’.’ Oh

0(0 I—

Qh

Oh =4

oc

.41 (0.’

0(0 1..-

0’

~

0.)

‘0

It, •

0-

C, C 4.) •0~~~~

—

.CE C -—

0.. .0< C—

() 40

40. 00 ~

~ 00 Oh

00

4.10-

00*00 (0) .4

—

0’ ~ 00

5) U)

C

Oh

L

O

10

-— 00

I—~

O

C Oh

‘0

L 0.)

Oh

-

C C —

40~

0’)

40

0’)

0(1

(00 40’ 0’

0 —5

Oh

L 0 0

0 5) >

•

—0

15

IOZ 0.

Oh

C C’.)

Oh

—

0’ —

~ —

C

Oh

I~

~

•

—

O

•C

‘5

O~

C’) .‘

L

5)

—

—

C (‘4

—

—

.0 .0 0 .0

C..) 4-

00

O

4-’ SO SO

C

0

4-’

C SO

E

U) —

Oh Oh 15

>, L 40

E E If)

L

00 C

00 C

.1

..4

0<

U) 00 ‘0 ‘0

00

00 C

00 C —

00 C —

00 C

0<

0<

0<

>0

or,

00 Oh

0 0 — —

—

~ —

0

—

—

0’)

I—

E 0

L 4--

CO.’ SO

-w

—

In

SO

.0 ‘0 I—

(0

4)

15 E

00 C

0-Oh

‘VZ 0)

.0

It) C

— 0< C0J

—

C

C

C

0..

X ~-

—‘ .

.

(0 C

—s

‘5 0.. SO C C)

0<

Oh

.~

10 .sJ

Oh •

5)51

E4-’ 400

Oh 10 00

00

r

—.

0) 40.—

44

0 a)

•0’)

0<

C — 0<

C

00 C

It) C

00 C

_,

—

—

—

00

0<

00

0<

—

14-.

L

1*.. ,5

051 >, 00

0) 0)

0) 40

C~) C’.)

C

0) 00

C.4

00

0’)

—

(0)

—

-.

—0

-

4.’

L

C’.)

or,

00

Oh 0’ —

Oh Oh —

0(1 C’-) C 0%)

00 0%) C C’.J

00-

‘V SO

00 Oh Oh —

~

5)

0(1 C —

LU

00 SO ‘S.C ‘04-’

JAfl 5:2—C

-

C (0)

33

34

S.Y.

Su and D.J. Kessler

region for manned spacecraft operations, the high level of debris flux could pose an unacceptable level of hazard on the safety of Spacecraft operations. The high flux level at low altitudes apparently drops slightly in the last panel shown for the year 2030. However, this level may still be unacceptably high. CONCLUDING REMARKS The present model offers a powerful tool in examining the dynamical evolution of the future man—made debris environment. Because of the nature of the statistical approach adopted for the model, several runs cannot cover all the possible outcomes. The results shown in Figure 5 are for only one of the sample runs and may not be typical. However, a general trend of evolution described in the previous section is observed in the rest of sample runs. Many additional runs are required to obtain the extremum condition of the debris environment for use in assessing future spacecraft operations. The most severe problem in the present model, and in any other model in the future, is the lack of better understanding of the collisional fragmentation process. More specifically, the ejecta size versus velocity distribution in the collision process and the effect of momentum transfer between the colliding space objects must be evaluated in order to perfect the model. Unfortunately, no known data currently exists in the open literature. There are some data on the ejecta size versus velocity distribution published from the study of the geological impact cratering process; but these may not be suitable for application to a collision process between metallic spacecraft. The greatest uncertainty about the accuracy of our prediction depends directly upon the accuracy of the collisional fragmentation process used in the model. In conclusion, our dynamical model of debris environment evolution produces useful information concerning future spacecraft operations. The central part of our model implements the Monte Carlo technique to simulate the collision process statistically. This method is capable not only of imitating the random fluctuation of the real physical process. but also of easily implementing the physical process into the computer coding. The computer coding can be easily modified as soon as the collision process becomes better understood. ACKNOWLEDGEMENT We would like to express our appreciation to Mary A. Tarlton for her assistance in the analysis of explosion fragments. Herb A. Zook’s critical review and coments on the paper are very helpful. The atmospheric drag program is provided by Alan C. Mueller. REFERENCES 1.

Kessler, 0. J. and Cour—Palais, B. G., Collision frequency of artificial satellite: creation of a debris belt, Journal of Geophy. Res., Vol. 83, June 1, 1978, pp. 2637—2646.

2.

Kessler, D. J., Sources of orbital debris and the projected environment for future spacecraft, Journal of Spacecraft and Rocket, Vol. 18, July—August 1981, pp. 357—360.

3.

Kessler, D. J.,-Derivation of the collision probability between orbiting objects: lifetimes of Jupiter’s outer moons, Icarus, Vol. 48, 1981, pp. 39—48.

4.

Bess, T. D., Mass distribution of orbiting man—made space debris, NASA Technical Note TND—8108, 1975.

5.

Lennertz, T. (Major USAF), The number of objects in space: Modifying the perimeter acquisition radar (PAR) for unknown track experiment for probability of detection (U), Office of the Secretary of the Air Force Space Systems (OSAF/SS), Pentagon, Va., classified by HQ NORAD/ADCOM, Tech Memo 81-4, June 1982 (Declassify 31 December 2005)

6.

Cour—Palais, B. G., Meteoroid environment model NASA Sp-8013, 1969.

—

The

The

1969 (near Earth to lunar surface),