JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. Ell, PAGES 23,195-23,210,NOVEMBER 25, 1994
Collisional
evolution
of the Earth's
orbital
debris cloud
A. Rossi, 1'2A. Cordelli, 3P. Farinella, 2 andL. Anselmo 1 Abstract. We have developeda numericalalgorithmto model the future collisional evolutionof the low-orbiting Earth debrispopulation,accountingfor both the wide
spectrumof masses(or sizes)of the orbitingobjects,and their differentaltitudes, whichresultin a variableefficiencyof the drag-induceddecay.The evolutionprocess has been assumedto be causedby a number of sourceand sink mechanisms,such as launches,explosions,atmosphericdrag, and mutual collisions.The collisional outcomeshave been describedthrough a semiempiricalmodel for the fragment massdistributions,consistentwith the availableexperimentalevidence.A runaway exponentialgrowthof collisionfragmentsis alwaysfoundin our model. Although its timingand paceare sensitive to somepoorlyknownparameters, fairly plausible parameter choicespredict that the runaway growth will occur within the next century,starting in the crowdedshellsbetween700 and 1000 km of altitude and, somewhatlater, between1400 and 1500 km. The runawaygrowth is delayeduntil a few centuriesin the future only if the catastrophicbreakup thresholdin specific impactenergyfor orbitingobjectsexceeds that for naturalrockybodiesby at leasta factor of 10. Our sim•ations showthat the sensitivityof the resultsto future launch and/or deorbitingand removalpoliciesis rather weak, so that drasticmeasures will need to be taken soonin order to significantlyavoid or delay a catastrophic outcome.
1. Introduction and Summary
the populations of objects of different mass and at different
altitudes.
We have then used the code to predict the future late in a semideterministic fashion the future evoluevolution of the debris population in a number of cases, tion of Earth-orbiting debris. In our model, the or- obtained by varying both the parametersdescribingthe biting objects can undergohypervelocitymutual colli- physicalpropertiesof the existingbodies,and the possisions,which affect their size distribution by generating ble future policy choiceson the launch and removalrates swarms of fragments. The outcomesof such impact of spaceobjects. We have found that a runaway expoevents have been modeled in a fairly realistic way, us- nential growth of collisionalfragmentsalwaysoccursin
We have developed a new numerical code to simu-
our model, although its timing and pace are very sensitive to some poorly known parameters related to the physicalpropertiesof the Earth-orbiting objects. However, we stressthat plausible parameter choicespredict that the runaway growth will occur within the next century, starting in the most crowded shellsbetween 700 sink terms have been included: launches, explosions, and 1000 km of altitude; somewhat later the processis collisional fragmentation and cratering, and orbit de- triggered also between 1400 and 1500 km. Our simulacay due to drag. The evolution processhas been sim- tions show that the sensitivity of the results to future ulated by integrating a set of 150 coupled, nonlinear, launchand/or deorbitingand removalpoliciesis rather first-order differential equations, having as unknowns weak, so that drastic steps will need to implemented soonif a catastrophicoutcomeis to be avoidedor at least delayedsignificantly. This is in line with previous
ing the available experimentalevidenceto derive suitable semiempiricalrelationshipsbetweenthe quantities involved. The spacearound the Earth is divided into a number of discrete altitude shells, with intrinsic collision probabilities at each altitude computed from a set of 2700 actual orbiting objects. Various sourceand
iCNUCE-CNR,Pisa,Italy.
2Gruppo di Meccanica Spaziale, Dipartimento di Matematica, Universitfi di Pisa, Pisa, Italy.
3Dipartimento di Fisica,Universitfi di Pisa,Pisa,Italy. Copyright1994by the AmericanGeophysical Union. Papernumber94JE02320. 0148-0227/94/94JE-02320505.00
recommendations [Fluvyand McKnigh•, 1992; Eichlev et al., 1993]. The short- and medium-term prospectsmay be less severeif the fragmentation thresholdin specificimpact energyfor orbiting objectsis substantiallyhigher than
our nominalvalueof 103J/kg, deduced fromexperiments on nearly homogeneous rocky targets. Actually, a higher breakup thresholdhas beenindicatedby some recent impact experimentsusingreal or mock-up space23,195
23,196
ROSSI ET AL.: EVOLUTION
OF EARTH'S ORBITAL DEBRIS CLOUD
craft as targets [Na#l et al., 1992]. Sincetheseexper- 400,000, while those larger than 1 mm could be more than 3,000,000[Klinkradand Jehn,1992]. resultsare in generalclassified (as theywerecarriedout In a sense,the low-orbiting Earth debrispopulation for military purposes),we believeit would be highly is similar to the asteroid belt, since it is subject to a iments are still very few and, moreover, their detailed
desirable to extend the experimental database in order to improve the reliability of predictionsbasedon longterm evolution algorithms suchas that discussedin this paper. Other areas where more experimental evidence
process of high-velocity mutual collisions that affects
the long-termevolutionof its sizedistribution[Kessler and Cour-Palais,1978;Davis e• al., 1979,1989].However, the situation is more complex than for the aster-
is neededare (1) the massand velocitydistributions of oidsfor at leastthreereasons:(1) launchesand explocollisionalfragments,(2) the correlationbetweenthese sionsprovidesources of material(partially)subjectto two quantities,and (3) the relationshipbetweenfrag- human control, adding to the fragmentsgeneratedby ment cross section and mass. As we will show in secimpacts;(2) the numberdensityof objectsis a sensition 5, the latter in particular may be a critical pieceof
tive function of altitude, and so is the sink mechanism
information.
due to drag;and (3) relativespeedsare dominatedby
Data from observationalsurveysof the existing Earth-orbiting population, as a function of both massand orbital parameters, are also neededto set more accurate initial conditionsfor modelssimulatingthe future evolution process. This holds in particular for objects in the 1- to 10-cm sizerange, for whichlimited information exists despitethe fact that they may alreadybe dangerous
mutual inclinations, which are much larger than typical
orbital eccentricities and unevenlydistributed(whereas among the asteroids,eccentricitiesand inclinationshave similar, fairly smoothdistributions,with averagevalues
•0.15). Accordingto a numberof recentofficialreports[EuropeanSpaceAgency(ESA), 1988;InteragencyGroup for breakingup (or at leastseverelydamaging)targets (Space),1989; Officeof Technology Assessment, 1990], in the 100-kg mass range, and thus may contribute to an early triggering of the fragment runaway growth. A more detailed knowledgeof the propertiesof operational and "dead" spacecraft, e.g., mass, shape, structure, potentiM for causing explosions,would also be valuable to better constrainthe propertiesof the possibletarget objects. Limited information of this type is currently available from the RAE Tables published by the U.K. Defence ResearchAgency, and also through the ad hoc
United Nations register set up followingthe 1975 Convention on the Registration of Objects Launched into Outer Space;this could be improvedif launchingstates or spaceagencieswere requiredto supplythe U.N. register with more systematic, detailed and updated information on the newly launched space objects. The remainder of this paper is organizedas follows. In section2, we provide somebackgroundmateriM and referenceson the subject of this work. In section 3, we describe in detail
the code and our "nominal
choices" for
the main model parameters and the initial conditions. Numerical results in the nominal case are given and commentedon in section4. In section 5, we explore the parameter sensitivity of our results and the effects of different policy options.
the proliferation of small fragmentsfrom the collisional processingof the existing debris is having a growing impact on space flight operations, spacecraft design and maintenance, and the overall costs of the space projects. Furthermore, for more than a decade it has been pointed out that the collisional breakup of orbiting objects can give rise to a sort of chain reaction, with a further increase of the probability of new catastrophic collisionsin the near future and a subsequent
exponentialgrowth of orbiting fragments[Kesslerand Cour-Palais, 1978; Eichler and Rez, 1990]. According to the ESA report cited above, "The self-sustained debris production by collisionsis a long-term concern. It is howeverthe most far-reaching threat which could terminate all space activities. This mechanismrequires further careful study." Contributing to such a study is the main purpose of the work describedin this paper.
Description of the Model To model the debris evolution process, we have numerically integrated a set of 150 coupled, nonlinear, first-order differential equations, with each equation giving the rate of changeof the population presentin a
discretesize bin and in a givenaltitude shell [Cordelli et al., 1993]. We haveused15 altitude shells(six 50-
2. Background
kin-thick shellsbetween 400 and 700 km, plus nine 100Since the beginning of the space age, several thou- km-thick shellsup to 1600km) and 10logarithmicmass sandsof artificial objects have been launchedinto space, bins (centeredat valuesrangingfrom I g to 6000 kg,
mostof themin low orbits(namely,at altitudessmaller and spanninga factor of 5.664each). than m2000 km). Intentionalor accidentalexplosions Initial conditionsare providedfrom the (limited) have added to this a growing population of fragments, larger and larger for decreasingsizes. About 7500 Earth-orbiting objects having sizeslarger than 20 cm are presently tracked and cataloged by the U.S. Space Command. Only about 350 of theseare active satellites, and recent studies indicate
that the number of nontrack-
able particles of 1 cm and greater is between 50,000 and
knowledgeof the existing population, which has to be extrapolated to the smaller size range of untrackable particles; the results of this exercise,which can be seen as plausible but by no means accurate estimates, are given in Table 1. The assumedinitial population consists of about 57,000 bodies, all exceedinga minimum massof 0.42 g: the vast majority of these objectshave
ROSSI ET AL.: EVOLUTION
OF EARTH'S ORBITAL DEBRIS CLOUD
23,197
Table 1. The AssumedCurrent Distributionof Objectsat Different Heightsh in the VariousMass Bins h, kin
1.00 g
5.66 g
32.1 g
182 g
1.03 kg
5.83 kg
33.0 kg
187 kg
1059 kg
6000 kg
425 475 525 575 625 675 750 850 950 1050 1150 1250 1350
160 250 600 1200 1330 1290 3200 4924 8689 5800 5000 3050 2475 2500 4250
20 70 100 140 105 120 520 493 869 650 280 310 248 310 370
15 30 45 60 70 50 240 197 348 200 150 llO 99 250 310
12 20 30 50 60 35 200 148 261 150 80 65 74 210 250
10 15 25 40 50 30 150 113 200 100 70 50 57
10 12 15 25 30 20 90 77 137 60 50 35 39
5 5 10 8 10 4 20 20 35 15 20 lO l0
50 25 65 12 25 15 70 79 282 15 30 30 25
12 12 12 10 12 0 0 0 0 0 0 0 0
200 160
150 95
5 8 8 10 20 10 80 29 52 30 40 20 15 100 60
80 35
60 40
0 0
1450 1550
Central valuesare listed in the first line. Thesenumbersare used as input initial valuesfor the integration of the equations of the model.
massessmaller than a few tens of grams,and only about
1650exceed10 kg [e.g.,EichlerandRe•, 1987];this ap-
about I kg), 10 in the second(5.7 g), and 30 in the first (1 g) are released.Finally, oneexplosionper year
pears as a realistic estimate, when one includesthe op- is assumedto occur on averagein eachof the two altierationaland the "dead" satellites,the orbiting upper tude shellsbetween700 and 900 km, involvingbodies stages,and the most massivefragmentsfrom past ex- of massMe - 1500kg and with the resultingfragment plosions. The altitude distribution of the small untrack- mass distribution modeled accordingto the empirical able particles is assumed to mimic that of the tracked exponential law proposed by Su andKessler[1985]'the population. cumulative number of fragmentsof masslarger than Apart from collisions,other sourcesof orbiting ob- m resulting from the explosion of an object of mass jects are of coursepresent: launchesof new satelhtes,re-
Me is 0.1708 Me exp(-0.65x/• for m > 1.936,and leaseof operationaldebris(pyrothecnics, yo-yomasses, 0.870Me exp(-1.82V/-•) for m < 1.936,all masses beetc.) duringlaunchesand orbit injections,and inten- ing given in kilograms. tional or accidental explosions.For every altitude, the
The frequency of collisionsis derived in each altitude
launchrate (in yr-•) is an inputparameter, givenin shell by a computation of the averageintrinsic colliTable 2. As far as the release of operational debris is concerned,we have assumedthat on average,for every
launch,two objectsin the fifth massbin (centeredat
sion probability for the populationof catalogedorbiting bodies(extractedfrom a total sampleof 2700 orbits,asdescribed by RossiandFarinella[1992]),whose
Table
perigeeand apogeedistanceslie within the shell boundaries. This has been achievedby modifinga program developedby Farinelid and Dav•s [1992]to calculate
2. Assumed Rate of Net Launches Per Year at
the Different Heightsh in the Last Four Bins of Mass
the collisional probabihties and velocities in the aster-
h, kin
puter to speed up the computations. We have used the
425 475 525 575 625 675 750 850 950 1050 1150 1250 1350 1450 1550
33.0 kg 0 0 0 0 0.5 0 3 0 0 0 0 0 0 12 0
187 kg 0 0 0.5 0 0 0 0 0 0 0 0 0 0 12 0
1059kg 3 1 3 0 1 0 3.5 2 9.5 0 0.1 0 0 3.5 1
6000kg 3 0 1 0 1 0 0 0 0.5 0 0 0 0 0 0
Central valuesare listed in the first line. Overall, we assume61.1 launches/yr.
oid belt, and by using a nCUBE 2 parallel multicom-
definition of intrinsic collisionprobabilityfor orbiting bodiesgivenby Wether•ll[1967],so that for eachaltitude shell, the colhsionrate is obtainedby multiplying the correspongingaverageintrinsic collisionprobability
(givenin Table3), timesthesquared sumof the (mean) radii of the consideredbodies(i.e., their impact cross sectiondivided by •r), times the productbetweenthe numbersof existingprojectilesand targets. It is worth noting that the valuesgiven in Table 3 are fully consistent with the resultsof Ross•and Farinelid[1992],who did not consider different altitude shells, but studied the whole orbiting population together. Of course,the
averageintrinsiccollisionprobabilityof 1.1 x 10-9 m-•
yr-• derivedin that paper(seealsotheirFigure8bfor the dependenceupon altitude) has to be scaledby a factor proportional to the ratio betweenthe total population and the population of each shell.
23,198
ROSSI ET AL.: EVOLUTION
OF EARTH'S
Table 3. AverageIntrinsic CollisionProbability at Different Heights h h, km
IntrinsicProbability,10-9 yr-ln1-2
425 475
5.8759 15.4701
525
31.6974
575 625 675 750 850 950 1050 1150 1250 1350 1450 1550
25.8042 41.0091 41.8788 18.9014 14.6353 15.7480 27.1655 17.4727 16.3149 15.1570 12.7350 12.7350
ORBITAL
DEBRIS
CLOUD
impact velocity V, the threshold projectfie-to-target
massratio for target breakupis just 2Q*/V 2. Since this is much lessthan unity, the massof the projectfie is neglectedin computing the resulting fragment mass distribution. In both the cratering and the breakup cases,this distribution is modeledas a truncated power law, in agreementwith the resultsof laboratory impact experiments[Fujiwaraet at., 1989;Petit and Farinella, In the cratering case, the characteristicexponent q of the incremental fragment mass distribution dN oc
m-qdm (where dN is the numberof objectshaving masses in the interval[m,m+dm]) hasbeenfixedto 1.8; massconservationimplies that in this casethe largest fragment comprises1/4 of the total excavatedmass. In the breakup case, the following empirical relationship betweenthe largest fragment to target massratio
ml/M and the specificimpactenergyE/M normalized
The number of collisionsper year between objects of mass mi
to Q* has been used:
and m j at a given height can be obtained by multiplying the
intrinsicprobabilitytimesthesquared sumoftheirradii0'i +•'j )2 and the number of objects residing in the mass bins centeredat
ml
mi andmi. Theaverage radius •'i isdefined asv/Ai/•r,Ai
M
being the cross-sectional area of the object of mass mi obtained
1(E)-1'24
(2)
[Fujiwara et at., 1977],whereasq can be derivedfrom
by inversionof (1) in the text.
the mass conservationrelationship and turns out to be
Foreachorbitingbody,the cross-sectional areaA (in squaremeters)wasassumedto be relatedto the mass m (in kilograms)by the simpleempiricalrelationship
m -- 62Al'la,
(2 + mllM)
q- (1+mxlM)
(3)
[Greenberget al., 1978], implyingthat q rangesfrom 5/3 in a barelycatastrophic collision(mx/M = 1/2)to (1) 2 in a supercatastrophic event(mx/M -• 0), the latter
introducedby Kesslerand Cour-Palais[1978].This relationship is physicallyplausible,as the exponent1.13 is intermediatebetweenthe valuesI (whichwouldapply to a hollowstructure)and 1.5 (homogeneous solid bodies).In reality,the averageexponentis likely to de-
case being such that equal logarithmic mass bins contain equal masses. We remark that while power law massdistributions have been widely chosenin the past to representthe outcomesof impacts against spaceob-
jects [Su and Kessler,1985; McKnight,1991],a vari-
pend on the massof the body, but we lack sufficientob- able exponent q, dependingupon the collisionenergy, servational or experimental evidence on this to devise has never been used before in this context. We have a reliable quantitative relationship; therefore, for the made this choicebecauseit appearedto us both physi-
time being,we use(1) (or the alternativerelationship callyplausibleand wellsupportedby experiments [e.g., betweenA and m discussed in section5). As for the Hartmann, 1969;Fujiwara et al., 1977]. Drag-induced orbital decay has been taken into acimpact speeds,Rossiand Farinella [1992]found that for the catalogedlow-orbiting objectsits distributionis
count by introducing, for each altitude shell, a charac-
characterized by an averagevalue(9.65 km/s) approxi- teristicresidence time, inverselyproportionalto the (avmately independentof altitude and a fairly small disper- erage)atmospheric densityat the corresponding height sion(0.88km/s standarddeviation);therefore, wehave andto the crosssectionof the considered object(seeTajust used a constant impact velocity value V - 9.65 ble 4). Up to 1000-kmaltitude,the valueslistedin Takm/s in computingthe collisionaloutcomes. ble 4 have beenobtainedusingthe averageatmospheric In our model, each collisioncan provokeeither local- densitiesof the 1976 U.S. StandardAtmosphere(for a ized target damage(with a fractionof target massM typicalthermospheric temperatureof 1000øK);beyond fragmented and ejected from a "crater" of mass pro- 1000km, we haveextrapolatedusingthe scaleheightat portional to the impact energy,up to a maximum of 1000km (235 km) and its derivative(0.4). Dragis not M/10), whenever the projectile's kineticenergyperunit assumed to determine the orbital lifetimes of massive of target massElM is smallerthan a giventhreshold objects at low heights, however,becausemost of these valueQ* (whichcanbecalledimpactstrength)assumed objects are active satellites, and for them the effect of
to be of the orderof 103-105J/kg;or complete target atmosphericdrag is actively compensatedby thrusters; breakup,with a largestfragmentincludinglessthan one thus, we have assumeda constantcharacteristicdecay half of the target mass,if the Q* breakup thresholdis time of 10 years for all the objectsin the three heaviest exceeded. It is easy to see that the crater massover im- massbins (centeredat about 187kg, 1.1 and 6 tons)
pactenergycoefficient is (10Q*)-1 andthat,fora fixed
and in the three altitude
shells below 550 km.
Other
ROSSI ET AL.: EVOLUTION
OF EARTH'S
ORBITAL
DEBRIS
CLOUD
23,199
Table 4. ResidenceTimes Due to AtmosphericDrag (seeTable 2), the releaseof small objectsand the exin the Altitude Shells Centered at Different Heights plosions, as specified above, whereas the second and for a Value of the Area-to-Mass Ratio A/m Equal to third terms account for the drag-induced orbit decay, 1 m2/kg assuminga characteristicresidencetime r dependingon both altitude and mass(computedfrom Table 4, using h, km Decay Time, years (1) to relatecrosssectionto mass),and the fourthterm 425 475
0.10 0.27
525 575
0.63 1.42
625 675 750 850 950 1050 1150 1250 1350 1450 1550
2.81 5.60 18.00 56.00 110.00 150.00 185.00 220.00 250.00 280.00 308.00
For different values, decay times are assuxnedto be simply pro-
portionalto (A/m) -1 .
types of active maneuvers for operational satellites have been neglected. Note that in our model drag provides the only mechanism causing an exchange of bodies between different shells. This means that we are making two different ap-
accounts
for the collisions.
Here the three-dimensional
arrayf(m•, mt, mi) represents the numberof objectsof massmi produced(or destroyed)by a collisionbetween twobodiesof massm• andmr;p(hj) is the intrinsiccollision probability for objects residing in shell j, as given
in Table 3; and rr(mk,mr) is the squaredsum of the radii of the same two bodies, derived from their cross
section,againrelatedto massthrough(1), i.e.,
a(m•,mt)- 8.2537 x 10-12(mø•'4424s + mtø'4424s) 2. (s) We stress that many areas of uncertainty remain in this modeling work. The most critical ones appear to
be (1)the relationship(1) betweenmass(m)and cross section(A) of the orbitingobjects;(2) the valueof the (average)impact strength Q* of the targets and the fragment massdistribution function, which may depend on the shape, structure and material properties of the
typicalspacecraft;(3) the existingEarth-orbitingpopulations as a function of mass and altitude, which are the
initial conditionsfor the future evolution;(4) the future
launch and explosion rates, the two dominant sources proximations:(1) we assumethat the evolutionprocess of new objects before disruptive impacts become fre(neglectedin is little affected by the fact that objects having eccen- quent enough;(5) the time dependence our model) of drag in the high atmosphere, whichis tricities larger than about 0.01 can actually move across mainly associated with the l 1-year solar activity cycle differentaltitude shells(in reality the existenceof these and may affect in a significant way the decay rate for objects probably somewhat "smearsup" the density dif-
ferencesbetweenneighboringshells);(2) collisionsand small, low-orbiting bodies;and (6) the effectsdue to explosionsare assumedto producefragment swarmsre- objectsin elliptic orbits, in particular, fragmentsgener-
maining in the same shellsas their parent bodies. This ated by high-intensity explosions. While we shall disis consistentwith the fact that typical ejection speeds cussin section5 the sensitivityof our resultsto changes for sizeablecollisionalfragmentsare m100 m/s, a fac- in some of our assumptions,on some of these issuesthe tor of 100 smallerthan orbital velocities[seeMcKnight, prospectsfor more reliable estimatesof the correspond1991];however,this assumptionis probablynot fulfilled ing parameters depend upon new observationaland labfor the small fragmentsgeneratedby high-intensity ex- oratory work, to be carried out for this specificpurpose.
plosions[Johnsonand McKnight,1987]. We shallshow in section 4 that the results do not depend strongly 4. opon these simplifying assumptions. In summary, we have obtained the following set of coupled, first-order differential equations for
Numerical
Simulations:
Nominal
Case
The integrations of our set of 150 differential equa-
N(mi, hj,t), the numberof objectsresidingat time t tions have covered a time span of several centuries in in the bin centeredat massrnj and in the shellcen- the future, starting from the initial population specitered at height hi' fied in Table 1. The net current rate of insertion into orbit of new massive objects is assumedto be about 60
dN(mi,hj,t)
dt
= •(mi,h•)
per year overall(seeTable 2); the actuallaunchrate is higher(some100per year), but we needto takeinto ac-
count that a significantfraction of the launchedobjects, e.g., U.S. space shuttles and many Russian reconnaissance satellites, reenter intentionally at the end of very + • f(mk,mt,mi)p(hj)rr(mk, ml) short-lived missions.The altitude distribution of newly k,I launched spacecraft in the future is assumedto always N(m•,hj)N(mt,hj) (4) resemble the current one; as specified earlier, for each where • is the matrix that accounts for the launches launch a few tens of small objects are inserted into the -
,
)
+
,
+ )
23,200
ROSSI ET AL.: EVOLUTION
OF EARTH'S
same altitude shell, to simulate the loss of small non-
operational objects and devicesduring the early orbital phases,and two explosionsper year, involving bodies of 1500 kg in mass, are assumedto occur between 700 and 900 km of height. In section5, we will describethe results correspondingto a number of variations on this scenariofor the future launch policy. As for the impact
ORBITAL
DEBRIS
CLOUD
bet is too small (at the extreme,it may evenbe one,as in FarineIIa and CordeIIi[1991]),then oneloses"resolution" on the way the population evolves at different
altitudes;if on the otherhand,this numberis high(like in the presentwork), oneneglectsthe "smearing"effect
of orbital eccentricities,which causea single object to move at different heights, and also, at the lowest altistrengthQ*, we haveadopteda nominalvalueof 10a tudes, decreasesignificantly the orbital lifetimes versus J/kg, similar to that found experimentallyfor natural drag. In order to test how much our results are sensistonytargets [Fujiwardet al., 1989]. A priori, it is not tive to this type of choice, we have rerun the nominal clear whether artificial bodies are weaker or stronger, casedescribed above, but with only three altitude shells, with respect to hypervelocity impacts, than the basalt obtained by merging together the previous ones. These or cement targets that have been used to simulate as- three "supershells," spanning 400 km each, were centeroidal collisions. However, a few recent experimental tered at 600, 1000 and 1400 km; starting populations results[Na9I et aI., 1992]providesomeindicationthat and launch rates were obtained by summing up the valhigher valuesof Q* are more likely; thus we have tested ues for the previous thinner bins; collision probabilities other values of Q*, up to 50 times the nominal value, were averaged,scalingeach value given in Table 3 with and we will describe the corresponding results in sec- the corresponding shell population; and drag-induced tion 5. decay times were assumed to be 5, 130, and 265 years, The qualitative evolution pattern in our nominal case respectively. The results are shownin Figure 4, for the is surprisinglysimilar to that predicted by a much sim- same three mass bins of Figures 1, 2 and 3: it is appler model,basedon two equationsonly [Farinelidand parent that the main features of the evolution process CordeIIi,1991]. Actually, this suggests that the results remain the same. The quantitative changeswith reare not sensitive to the adopted number of mass bins spect to the nominal case describedearlier are actually and altitude shells. We refer to the above-mentioned smaller than those resulting from reasonablechangesin paper for a simple analysis of the main mechanismsat parameters such as Q*. work in the different phases of the evolution. The typical features are shown in Figures 1, 2 and 3, which
refer to the second(m • 6 g), fifth (m • 1 kg)and ninth (m • 1 ton) massbins, respectively.After a period of slow and steady population growth ranging from decades to centuries, depending on the altitude, the generation of collisionalfragments exceedsthe insertion into orbit of noncollisionaldebris and significantly increases the frequency of catastrophic impacts. Since each breakup event generates a swarm of new potential "projectiles," thus increasingthe subsequentrate of catastrophic events,a kind of chain reaction is triggered. As a consequence,the growth of the small-size population becomes exponential, whereas the abundance
5.
Numerical
Simulations:
Parameter
Sensitivity and Future Scenarios We have explored the parameter sensitivity of the results of our model, in order to look for possibilities of preventing the occurrenceof the runaway fragment growth. The most important parameter appears to be
the averageimpact strength Q* of the targets (i.e., the threshold energy density resulting in catastrophic
breakup),whichwe haveassumedto be independent of sizeand altitude. In the nominalcasewith Q* - 103 J/kg, the exponentialfragment growth and the corre-
of larger objects(includingthe operationalsatellites) spondingrapid disruption of large objects starts about reaches a maximum and then rapidly drops. In this 50 yearsin the future,aswehaveseen.With Q* = 104 phase, the environmentis dominated by collisionalfragmentation, with more satellites being destroyed than launched. At the end, a quasi-steady state is reached, with all the material being launched rapidly converted into fragments. Of course,it is plausible to infer that before this phaseis reached,any spaceactivity will have ceasedbeing carried out in the correspondingaltitude shells. The most critical altitude range for the early onset of runaway fragment growth correspondsto the
and5 x 104J/kg, the "catastrophe" in the900-to 1000km shell is delayed until about 250 and 400 years in
the future, respectively(seeFigures5 and 6). While recent data (obtainedin experimentscarried out for military purposes;seeNagI et aI. [1992])wouldsuggest that the typical catastrophicbreakup thresholdis
about4 x 104J/kg, fairlycloseto ourhighestvalue,the
sensitivedependenceof the evolution on Q* highlights the need for further experimental work to obtain relicrowdedshells(7 to 9 in our labelingscheme)between able estimatesof this parameter for the existingorbiting 700 and 1000 km; here the runaway growth of fragments objects, rather than the potential utility of hardening starts about 50 years in the future. However, not much future satellites, as we shall see below, in the discussion later the processis triggeredalsobetween1400and 1500 on the consequencesof different future policy options. Another test that we made on our physical model was km. Only between 1000 and 1400 km and beyond 1500 km the fragment growthis delayeduntil 100 to 300 years that of eliminating altogether the effectsof small-scale, in the future. "cratering" impacts. Only limited changesto the nomA significantlimitation of modelslike that developed inal results have been observed in this case: the runin this paper is that they always require a compromise away growth of small fragments is somewhat delayed on the adopted number of altitude shells: if this hum- (by about 20 yearsin shell9, between900- and 1000-
ROSSI
ET AL..
6 x105
EVOLUTION
•
,
o o
50
tOO
OF EARTH'S
•
ORBITAL
DEBRIS
CLOUD
23,201
5.66gr.- h.shells 1-2-3 !
,
,
I
i
i
I
i
i
i
150
200
250
300
350
400
450
,
,
500
time(yr) xlO 6
5.6,6 gr.- h.,shells 455-6 ,
1 -
o o
50
100
150
i
i
i
I
i
i
200
250
300
350
400
450
500
time(yr)
xlO 7
,
0 0
50
100
150
5.6,6 gr.- h.,shells 7;8-9
i
I
l
i
200
250
300
350
!
400
I
-450
500
time(yr) x10 6
,
,
io
,;o
, 5.66•r.- h.sh, ellst0-!l-12
1 -
0'
o
t .......
150
&
200
250
i
i
I
300
350
400
450
500
300
350
400
450
500
time(yr) xlO 7
0
o
5o
100
150
200
250
time(yr)
Figure 1. Nominalcasefor the time evolutionover500 yearsin the future of the populationsof orbitingobjects residingin the massbin centeredat 5.66 g (and rangingbetween2.38 and 13.5 g), for eachof the 15 altitude shellsconsideredin our model. In every graph, three shellsare plotted together, with the lower, intermediate, and higheronesbeingrepresentedby the solid,dashed,and dotted curves,respectively.We recall that the shells
arecentered at 425 (number1),
(2), 525 (3), 575(4), 625 (5), 675(6), 750(7), 850(8), 950 (9), 1050(10), 1150 (11), 1250 (12), 1350 (13), 1447550 {14), and 1550 (15) km ofaltitude.
23,202
ROSSI ET AL ß EVOLUTION
8000
OF EARTH'S
,
,
,
50
100
150
ORBITAL
DEBRIS
CLOUD
1.0,3 kg-h.,shells 1-,2-3
m6000
0 0
i
i
200
250
i
i
300
350
•
I
450
500
time(yr) 3,
xlO 4
,
1.03 kg- h.shells 4-5-6
0 0
50
lOO
15o
,
,
i
I
i
i
i
i
200
250
300
350
400
450
500
time(yr)
xlO•
2.5
1.03kg - h. shells7-8-9
2 ....
1.5 1
/.. .
0.5 0 0
50
100
................................ -
i
150
200
2 0
300
i
350
•
i
450
500
time(yr)
2 xlOn ,
1.03•kg-h.sh, ells10-!1-12
1.5 ,, ,,
.......................... ;'•-:. .......................
1-
0.5-
0
' 50
0
i
i iJlll........... Ii a- .....
150
200
T•
250
i
300
350
400
450
,
,
500
time(yr) xlO •
,
, 1.03,kg-h.sh, ells13-!4-15 ,
, _----.4 ...... "• !
0 0
50
100
150
200
!
I
I
I
250
300
350
400
450
500
time(yr)
Figure 2. The sameas Figure1, but for the massbin centeredat 1.03kg (and zangingbetween0.43 and 2.45
ROSSI ET AL.: EVOLUTION
100
,
,
,
OF EARTH'S ORBITAL DEBRIS CLOUD
23,203
105,9 kg- h.,sh½ils 1•2-3 ,
• 80 o
60
20
0
I
i
i
50
100
150
200
250
3
3 0
450
500
450
500
time(yr) 50
•
30
o
20
!
!
i
1059kg - h. shells4-5-6
10
øo
150
200
250
300
350
400
time(yr)
800
,
,
,
100
150
105,9 kg- h.,shells 7•8-9 ,
,
i
2oo 0
0
50
200
250
300
350
400
450
500
400
450
500
time(yr)
60
, ,
,
, I059,kg-h.s,hcUs I0-,11-12,
2O
O' 0
I , 50
i l O0
i 150
i 200
i 250
3
350
time(yO
3OO 200
_
ss ss
.
ss
ss
...
.
• ' ?= =•""'-"•=
0
0
50
100
150
200
.... '•........ 250
',... -.....
r ...... 300
:::i:....... 350
'r....... 400
-r ....... 450
500
time(yr)
Figure 3. The sameas Figure1, but for the massbin centered at 1059kg (andrangingbetween445 and 2521
23,204
ROSSI ET AL.. EVOLUTION
lO
x lO?
OF EARTH'S ORBITAL DEBRIS CLOUD 5.66 gr. - h. shells1-3
,
i
i
i
i
i
i
ß
ß
=. .1 _
..... •'
50
x 10s
i
,,"'f'
i
100
200
150
I
I
I
,I
I
250
300
350
400
450
-
500
tirne(yr) 1.03 kg - h. shells 1-3 I
I
•)4 •'
I
/
i
'
"""
i
"- --
•
i
i
i
i
- .-...-........
.
02 -'
e-
•
O0
ß __ --. -"
50
/
/
...' .... i
.,,
....
1''
i
I
I
I
1O0 150 200 250 300
i
i
350
i
400
450
500
tirne(yr) 1059 kg - h. shells1-3 1500
• 1000
-
500 O0
..... 5I0
1O0 I - '-150 1'------I;-' 200----"• 250 .......300 -i-.....350 t.......400 -.i-.....450 I.... 500 , tirne(yr)
laigure4. Thetimeevolution in thenominal caseforthesamethreemassbinsshown in Figures 1, 2 and3, but merging togetherthealtitudeshells,asdiscussed in thetext. Thesolid,dashed anddottedcurves correspond to the threenew"supershells"centered at 600,1000,and 1400km, respectively. km altitude) and their final abundances are 20 to 40% The atmospheric density profile assumed to comlower; as a consequence, the disruptionof large bod- pute the rate of drag-induced orbit decay has some ies is somewhatless effective,but the differenceis not influenceon the resultsonly in the lowestfew shells, quantitatively relevant. where the abundanceof fragments,albeit reduced,is xlO 8
5.6,6 gr.- h.,shells 7;8-9
.
I
!
o
50
lOO
150
-
"" '"'" ;
200
250
300
I
I
I
350
400
450
500
time(yr)
.
2OOO
105,9 kg- h.,shclls 7•8-9 ,
1500 500 0
50
100
I
I
I
I
I
I
I
150
200
250
300
350
400
450
500
time(yr)
laigure5. Evolution in shells 7, 8, 9 forthemass binscentered at 5.66g and1059kg,withQ* - 104J/kg.
ROSSI ET AL.. EVOLUTION x107
1.5
OF EARTH'S ORBITAL DEBRIS CLOUD
5.6,6 gr.-h.,shells 7-;8-9 ,
i
23,205
i
_
.... .-..-..'..'..'-:" 0
50
0
100
150
200
250
300
350
400
450
500
time(yr)
,
30O0
,
,
105,9 kg-h.,shells 7•8-9 .................... .........
•2000-
... .... ....' ....
.....
.....................
....................
..........
.......
.......'"
u 1000
0
50
0
1O0
150
200
I
I
I
I
I
250
300
350
400
450
500
time(yr)
Figure 6. The sameas Figure5, but with Q* - 5 x 104J/kg. not prevented from reaching the exponential growth The explosionrate affects the abundanceof fragments stage. Here, on the other hand, bodiesin eccentricor- in the most dangerousshellsin the immediate future, bits would tend to have shorter lifetimes, thus further before collisionalfragmentation takes over, and theredepressing the populationsof small particles(seeFig- fore plays somerole in its timing; however,this role is in ure 4); thus,a morerefinedmodelis probablyneededto most casesmarginal, and typically muchlessimportant obtain reliable quantitative predictionsat low altitudes. than that of Q*. xlO 6
5
,
5.6,6 gr.-h.,shells 7;8-9
i
i
i
_ :
:'
3-
i
2-
...!
,,xx"..........'
1I
0
......... .. 100
0
150
200
250
I
I
I
I
300
350
400
450
,
,
500
time(yr)
105,9 kg- h.,shells 7•8-9
1500
,-
•1000
o
I 50
I lOO
I 150
i 200
I
250
I
I
I
I
300
350
400
450
500
time(yr)
l•ignre 7. The sameas Figure5, but adoptingthe massversuscross-section relationshipderivedby Badhwar and Anz-Meador[1989].
23,206
ROSSI ET AL- EVOLUTION OF EARTH'S ORBITAL DEBRIS CLOUD xlO ?
5
5.6,6 gr.- h.shells 7-•8-9 !
i
i
_
321-
0
0
50
100
!
I
150
200
I
,
250
I
I
300
350
I,,
400
I
450
5O0
time(yO 800
105,9 kg-h.,shell. s.7.:8-9 ,
'
O0 ' 50 100 150 200 250 300 350 400 450 500 time(yr)
Figure 8. The sameas Figure8, but assuming a currentpopulation of 100(insteadof 282)in altitudeshell9 (between900 and 1000kin) andmassbin 9 (between44• and2521kg).
(m/kg)- 37.07(A/m2)TM,
On the other hand, we havefound that the paceof the evolution is fairly sensitiveto the assumedrelationship
(6)
between the mass and the cross section of the orbitderivedby Badhwarand Anz-Meador[1989].This reing objects.For instance,we havederivedFigure7 by lationship was derived for relatively large objectsonly substituting(1) with the relationship (more than about 20 cm in diameter),so this is in-
4x10? ,
,
,
5.6,6 gr.- h.,shells 7•8-9
,
. . , .....
,.
.
0
I
0
time(yr)
., .
500
.-
,
,
100
150
105,9 kg- h.,shells 7:8-9
,.
,
,.
.
..
300
.-
;
:. ß
.:
:.'
200 100 0 0
50
200
250
300
350
400
450
500
time(yr)
Figure 9. The sameas Figure5, but assuming a currentpopulationof 182(insteadof 282)and a launchrate of
5.5yr-1 (instead of 9.õyr-1) in altitudeshell9 (between 900and1000kin)andmassbin9 (between 44õand is).
ROSSI ET AL.: EVOLUTION xlO 6
5
,
,
OF EARTH'S ORBITAL DEBRIS CLOUD
5.6,6 gr.- h.,sh½11s 7;8-9
,
,
23,207
,
.
32.'
1-
,,,.....
ß
.: .."
ß
0
50
0
100
150
!
I
!
!
I
I
200
250
300
350
400
450
500
time(yr)
,
,
105,9 kg-h.,sh½11s 7•8-9 ,
,
300
200
100
0
I 50
I 100
I"'"t•,,,, 150 200
250
I 300
I
I
I
350
400
450
500
fime(yr)
Figure 10. The sameasFigure5, but assuming a complete stopof all newlaunches andexplosions afterthe year 2000.
tended to be only a sensitivitytest with respectto the roleoftargets (abouta factor2 at m = 10s kg)andeven assumedrelationship between A and m, and not a re- more to the oppositeeffect for the small "projectiles" alistic case. At any rate, Figure 7 showsthat the expo- (the crosssectionat m - 1 g is largerby a factor 60 nential fragmentgrowthis significantlydelayedin this with the Badhwar and Anz-Meador formula, enhancing case. This is due to the smaller collisional cross section both collisionaland drag-inducedelimination of small
difference (hencelongerlifetime) of the largebodies,playingthe fragments).On the otherhand,nosignificant xlO 7
1.5
!
I
ß
5.66 gr. - h. shells7-8-9 I
!
!
I
........... .I................ .i............-.,:a.
..
ß
ß ß
I _
..
.
..
/
/
I
o
o
50
lOO
150
200
I
250 3{•0 350 &
4•0
500
time(yr)
,
,
lOO
15o
105,9 kg-h.,shells 758-9 ,
,
,
400
450
200
o
o
50
200
250
300
350
500
time(yr)
Figure 11. ThesameasFigure5, butassuming nomoreexplosions andthathalfofthesatellites aredeorbited after 10 years sincelaunch, starting at present.
23,208
ROSSI ET AL.. EVOLUTION
2x106
i
i
i
OF EARTH'S
ORBITAL
DEBRIS
CLOUD
5.66gr.- h.,shells 7-8-9 i
i
1.5
ß
.
ß
... ..
0.5-
,
o o
50
o-e'"""' 100
•11 150
I 250
200
I 300
I 350
I 400
I 450
500
time(yr)
05,9 kg- h.shells 7-8-9 i
i
!
!
I 300
[ 350
I 400
i
3OO
200
100
0
50
100
150
I 200
I 250
I 450
500
time(yr)
Figure12. Thesame asFigure 5,butassuming nomore explosions andthat,starting 10years inthefuture, for everynewlaunch in mass bins8, 9 and10(i.e.,those involving bodies exceeding about80kg)twooldsatellites of the same mass were deorbited.
with respect to the nominal caseis obtained by chang-
ing (1) only for subcentimetricobjects,as proposedby Kessleret al. [1989].
9, where the earliest fragment runaway growth is observed in the nominal case. First, we have reduced the
assumedpopulationof the ninth massbin (1-ton class
We have made a few tests concerning our assump- spacecraft) at thisaltitudefrom282(seeTable1) to 182 tions on the current population and launch rate in shell and then to 100. Even in the latter case, only a 20-year 1.5
xlO 6
5.6,6 gr.- h.,shells 7;8-9
,
I _
I
o o
50
100 150 21•0 250
300
350
I
i
400
450
,
,
500
tim½(yr)
,
40O
105,9 kg- h.,shells 778-9 ,
300
200
100
0
50
100
150
200
250
I
I
I
I
300
350
400
450
500
tim½(yr)
Figure 13. The same as Figure 12, but assumingin addition that 1000 objectsper year are removedfrom each of the shells 8 and 9 and the mass bins 1 and 2.
ROSSI ET AL.: EVOLUTION OF EARTH'S ORBITAL DEBRIS CLOUD
delay in the runaway growth is observed(Figure 8). Then
we reduced
at the
same
time
the
current
1-ton
population in shell 9 to 182 and the assumedlauch rate
from9.5yr-1 (seeTable2) to 5.5yr-1, but again,only a very limited improvementis observed(Figure 9).
23,209
ris, Collisional evolution of asteroids: Populations, rotations, and velocities, in Asteroids, edited by T. Gehrels, pp. 528-557, University of Arizona Press, Tucson, 1979. Davis, D.R., P. Farinella, P. Paolicchi, S.J. Weidenscbilllng, and R.P. Binzel, Asteroid collisional history: Effects on sizesand spins, in Asteroids II, edited by R.P. Binzel, T.
Concerningfuture launch policies,we havealsotested more radical options. Stopping altogether the launch-
Gehrels, and M.S. Matthews, pp. 805-826, University of Arizona Press, Tucson, 1989. ing activity 50 yearsin the future (or, better, assum- Etchlet, P., and D. Rex, Evolution of the number of space objects and debris in various altitudes with regard to fuing that sincethat time an old satellite is deorbitedfor ture collisional risks in space, Rep. 8718, Tec. Univ. any newly launchedone) doesnot delaythe "catastroBraunschweig, 1987. phe" in any significant way, but .just changesthe sub- Etchlet, P., and D. Rex, Debris chain reactions, paper pre-
sequenttrendsin fragmentabundances(whichreacha peak and then drop as all the large objects are elimi-
nated by collisionalbreakup). Even in the scenarioof
sentedat AIAA/NASA/DOD Orbital Debris Conference: Technical Issuesand Future Directions, Baltimore, Md., April 16-19, 1990. Etchlet, P., H. Sdunnus,and J. Zhang, Reliability of space debris modelting and the impact on current and future space flight activities, Adv. Space Res., 13, 225-228, 8,
zero net launch rate and no further explosionssince the year 2000, the ongoingcollisionalprocesswill trigger an exponentialfragment growth phaseand a corresponding 1993. rapid decreaseof the spacecraft population within the European SpaceAgency,SpaceDebris Working Group, Space
next century(seeFigure 10). A morerealisticoptionis
Debris, European Space Agency Spec. Publ. 1109, Nov.
possiblythat of deorbiting half of the satellites after 10 years since their launches. If this practice were started
Farinella, P., and A. Cordelli, The proliferation of orbiting
now (and preventingalsoall explosions),the catastrophe would be delayed only about 70 years in the future
(Figure 11); the delaywith respectto the nominalcase would be even smaller if the deorbiting of old satellites were started only in the year 2000. The runaway fragment growth occurseven if, starting 10 yearsfrom
1988.
fragments: A simple mathematical model, Sc. Global Security, 2, 365-378, 1991. Farinella, P., and D.R. Davis, Collision rates and impact velocities in the main asteroid belt, Icarus, 97, 111-123, 1992.
Flury, W., and D.S. McKnight, Policy aspects of orbital debris mitigation, paper presented at 43rd Congressof now (1993), for everynewlaunchin the threeheaviest the International Astronautteal Federation, Washington, D.C., Aug. 28-Sept. 5, 1992. massbins (i.e., thoseinvolvingbodiesexceeding about Fujiwara, A., G. Kamimoto, and A. Tsukamoto, Destruc80 kg) two old satellitesof similarmassweredeorbited; tion of basaltic bodies by high-velocity impact, Icarus,
as shown by Figure 12, the decline in the population 31, 277-288, 1977. of sizeable objects is not fast enough to prevent the Fujiwara, A., P. Cerroni, D.R. Davis, E. Ryan, M. Di Marfragment population explosion. Pinally, we have tested tino, K. Holsapple, and K. Housen, Experiments and scalthe possibility of actively removingfrom spacesmall oring laws on catastrophic collisions,in Asteroids II, edited by R.P. Binzel, T. Gehrels,and M.S. Matthews, pp. 240bital debris, possibly by using high-power laser sweep265, University of Arizona Press, Tucson, 1989. ers [Schall, 1991, 1993]. Figure 13 illustratesthe case Greenberg, R., J.F. Wacker, W.K. Hartmann, and C.R. when, assumingno more explosionsand the two-forChapman, Planetesimals to planets: Numerical simulaone deorbiting policy describedearlier, 1000 objectsper tion of collisional evolution, Icarus 35, 1-26, 1978. year are removedfrom eachof shells8 and 9 (i.e., be- Hartmann, W.K., Terrestrial, lunar, and interplanetary rock
tween700 and 900 km) and massbins1 and 2 (masses fragmentation, Icarus, 10, 201-213, 1969. smallerthan 13.5 g). Although the fragmentpopula- InteragencyGroup (Space), Report on Orbital Debris for National Security Council, Washington, D.C., Feb. 1989. Johnson,N.L., and D.S. McKnight, Artificial SpaceDebris, Orbit Books, Malabar, Fla., 1987. Kessler,D.J., and B.C. Cour-Palais, Collisionfrequencyof artificial satellites: The creation of a debris belt, J. GeoAcknowledgments. We are grateful to B. Bertotti, D. phys. Res., 83, 2637-2646, 1978. McKnight, C. Pardini, and T. Parrinello for valuable discus- Kessler, D.J., R.C. Reynolds, and P.D. Anz-Meador, Orbital sionsand comments. Constructive remarks by C. Chapman, debris environment for spacecraft designedto operate in F. Vilas, and an anonymous referee were also useful in relow earth orbit, NASA Tech. Memo., 100471, April 1989. vising the paper. P.F. worked on this project in 1993-1994 Klinkrad, H., and R. Jehn, The space-debris environment while at the Nice Observatory courtesy of the "G. Colombo" of the Earth, ESA J., 16, 1-11, 1992. fellowship of the European Space Agency. McKnight, D.S., Determination of breakup initial conditions, J. Spacecraft Rockets, 28, 470-477, 1991. Nagl, L., D.S. McKnight, and R. Maher, Review of data used to support breakup modelling, paper presented at References tions in the relevant
shells are much reduced
with
re-
spect to the nominal case, the onset of the exponential growth processis not prevented.
AIAA/AAS AstrodynamicsConference,Hilton Head IsBadhwar, G.D., and P.D. Anz-Meador, Determination of the area and massdistribution of orbital debrisfragments, Earth Moon Planets, ,15, 29, 1989. Cordelli, A., P. Farindia, L. Anselmo, C. Pardini and A.
Rossi, Future collisionalevolution of Earth-orbiting de-
bris, Adv. SpaceRes., 13, (8), 215-219, 1993. Davis, D.R., C.R. Chapman, R. Greenberg,and A.W. Hat-
land, S.C., Aug. 10-12, 1992. Office of Technology Assessment,Orbital Debris: A Space Environmental Problem,US GovernmentPrinting Office, Washington, D.C., 1990. Petit, J.-M., and P. Farindia, Modelling the outcomes of high-velocity impacts between small solar systembodies, Celestial Mech., 57, 1-28, 1993. Rossi, A., and P. Farindia, Collisionrates and impact veloc-
23,210
ROSSIET AL.: EVOLUTION OF EARTH'SORBITALDEBRISCLOUD
ities for bodies in low Earth orbit, BSA J., 16, 339-348, 1992.
Schall, W.O., Removingsmall debrisfrom earth orbit, Z.
A. Rossi and L. Anselmo, CNUCE-CNR, Via S. Maria 36,
56126Pisa, Italy. (e-mail:
[email protected];
Flugwiss. Weltraura[orsch.,15, 333-341, 1991.
[email protected] ) Schall, W.O., Active shieldingand reduction of the numA. Cordelli, Dipartimento di Fisica, Universith di Pisa, ber of small debriswith high-power lasers,in Proceedings PiazzaTorricelli2, 56126Pisa,Italy. (e-mail: cordelli@ of the First European 6'onferenceon SpaceDebris, pp. ipifidpt.difi.unipi.it) 465-470, European SpaceOperation Center, Darmstadt, P. Farinella,Dipartimentodi Matematica,Universithdi Germany, 1993.
Pisa, Via Buonarroti2, 56127Pisa,Italy.
Su, S.-Y., and D.J. Kessler,Contributionof explosionsand
[email protected]) future collisionfragmentsto the orbital debris environ-
(e-mail:
ment, Adv. SpaceRes., 5, 25-34, 1985.
Wetherill, G.W., Collisionsin the asteroidbelt, J. Geoph•ts. (ReceivedSeptember27, 1993;revisedJune23, 1994; acceptedSeptember6, 1994.) Res., 7œ,2429-2444, 1967.
