Anr Spe.Ra
vol ll, Nu 12,m (12)61(12)66,l99l
COLLISIONALCASCADING:TFTELIMITS OF
POPULATIONGROWTHIN LOW EARTHORBIT
DonaldJ. Kessler
NASAlJohNonStnceCe^Er,Haustan,TX77058,U.SA.
ABSTRACT
PrcdictionshaveIeen madeby severalaulhns lhal rudom collisionsbet*€€o ma&-made objectsin Eanh crbit
will lead to a signincant sourceof new orbibr debris,possiblywithin lhe next century. The aulhon haveal$
orcluded that thereaie a numberof $crrlainde.s in thesemodels,and ada[tioial analvsisand ilata arerequiftd
!o futly chaiacteriz the futu€ €nviroment. Howevq, the natureot theseun@ointies atEsuchthat while th€
future environmentis uncenaln,lhe fact that collisions will conEol thc future environmentis less unc€nain.
The rlatathal aL€adyexist is sufficient lo showlhal cascadingcollisions wil contmt the tuftrE debris
environmentwift tro, or very minor incr.asesin the cDrrentlow Eanh otbit population Two populalions
conml this process:ExplosionfnSmenls ard expendedmcket bodiesandpayloads. Ptelic€s arealEady
cllanging lo limit explosionsin low Eanh orbiq it is now ne.essaryto b€gin limilin8 the nutnb€rof expend€d
rocketbodicsand!6yload! in orbil
INTRODUCTION
In 198, colisionrl cascadlnas?s prcdrded !o be an imporiant souce of new o.bihl debris,p6sibly befo.e lhe
ye3r20m, if debrisconlrol t chniqu€sw€renot adopted/l/. Itese predictiorBwercre€nfotcedin 1986and 1990
/23,4,5/. In 1981,NASAado ed proceduteswhich would minimize le possibility ofexplosions in orbit. Bv
1989,mostolher iations andorganizalioosaplear !o haveadopledsilnilatlioctdures. Thes€proced|n€swill
resuhin a n€ar-lem reducbonin tlte poj€cled deb.is environmentlhowever,they do nol addresslhe lotr8 tern
!rcblem. In the long term, orc€ a criticlr populationdensity of obje.a is re*hed, the rale of t|agmenl
ptDductionfrom landoh colisions exc€€dsfE de of removalby amocpl|€ ic drag,and f'e proc€durEs1o
preventexplosionsbe.omesless imporlat Once$is $ilicar densityis rcshed, the deb.ispopulationwil
incr€asewithout placing any moreobi:cls into orbiL This increa!€will slop only ehen lhe popdanon of larSe
objectsis sufficiendy reduce4 eitlFr by rctive rcmoval,or by fmgmeniario. However,by lhe ttme
fmgmentatio. reduccsthe popDlationof large objats, lhe resuldngd€briservironment is f*ely 10be too
hosrile for fulure s!6ce use. Cdsequendy, n is inporlant to developopemtionalt€chniqueswhich mainlain a
popurationbetow the cridcal density. The obje.tive of ftis paF. is to &fne the conditionsproducinSthis
aidcal &nsily.
KEY OBSERVATIONS AND ASSUMPTIONS
By definition, a c.itical popuraior dersity il tEachedwhenthai popuiaiionwiU ptodEe hagmentsfrd rzndoln
collisio.s at an incrcasingmre ard al a mie that is gtcaterthanth€ rale ofrEmowl by aatural Focesses.Th€ mte
of fraFnent productionwil he coftoll€d in llie sizeregionswh€remassand dEa areconcenFated.More than
99% of d|e orbiting popolarionrnassand atta is conBinedin orbnal denrislar8e. than r0 cm y'y. The avense
payloador rocket body nass is rbout rom kg ,/, and lhe avemSel0 cm lo m cm mals is estmated !o t'e
about I kg /r/. At an avenge velocity of r0 hL/sec, an averasecolisidr beteeer any two orbidng object!
would resuhin the calastrophicbreahp of bolh objects. Cons€qucndy,ll|e maelhat couision fragmentsarc
genemredwill be controll€dby $e couirion nte in this 3i2eran8e,and dle condrtronsdescribirg the clilical
dcnsity can b€ definedin tlnru of d|e densityof obje.ts 10 cn and larger. Folo\.,in8 a collision, the amountof
tlE mlss thai go into smaller $an l0 cm sizesdep€ndson a larg€nurnb€tof faclon, dd is signilicandy
urcertain. Howev€(. mosttagments largt' than 10 cm to 20 cm at! obcrv.d by US SpaceCommand
following a bratup, and $e numberis not asuncenain. ThereforE,the curretudatais €apableofdescribhg 0t€
conditionsof critical &osity wilh more cartaintythsn the time dep€ndantconsequerceof d|e sna er than l0 cm
popuialion. A fundame al asunptiol| of this analysisis that lhc currcnt popularionof la4e, inlact obj€cts,a
maintain€dby repla.emenlwh€nthey deray, Educitg ilte analysisto de&rmininglhe sourceand sin* rataqrtr
Bre€kupmodelshaveb€€ndcaekf€d io {kscribe tle numb.r tnd velocity ol tEsmerts relative io the cene( of
mass. Thesemodelsareba*d on observarionsof salellitebrEakupsin $ce a'|d h the lab, ard attemptto mver
a largerangeof sizes/8,9/. The r€locity of 10 cm fragmentslanges fron 0 lo 2!0 rvse4. The velocity
distribution given in /9/ was usedard new orbi6 carcuhl€dfor l0 cn fi'agments. An atrnosphericmodel,
assuminga solar activity of I l0 F10.7 units, wrs usedto f'ropagatetheseorbits as a funclion of time. Spadal
densily was calculat€dat variou! tim€ irlervals usinglhe techniquesin /l0l It *?s assumedthat tle breakupof
a 1000kg space.mftwould poduce 3m tmgmentslarg€r$an 10 cm, ard thal lh€ fragmentareato massmios
are given by the expressiongiven h /l/. The resultsfo. a break-upat 1000tm aldtudeis show in figure l Ar
!i8.1. Slteuit trc.bp { 1000 bn. Top cwe i.
irilid sparial deNily dnrihrio
*irh rlritDde. Tine
henal berw.m lowq cw6 is 40O ydr.
Fi8. 2. Dett in Spltid D.roiry folowine ! br.dop.
SFriil daDity d.qys .4ondrially
util it rceh.3
sho*n in figure 2, fte d€€ayin spaaialdeosiryat rhebreak,opalritudecan b€ scu..rely expressedas rhesM of
rwo erponcndals
conkininga 'me.n UIe-.ju$ a, in naturatradiertiviry. Tharis, Sparirtdensiryal rhe
b€:r up altrude asa runctjoi ot |i me follo* ing a breakupc.n beexpr€ssedas
(tl
SB G) = S0 (e-ttl + e-t'/rln
wnere S0 is the initial sparialdensiryfollowin8 rhe bre€kup,t is time and 11 and 12 are consrrntsand can be
thoughtof as lhe m€3nlife of two componentsof rhe ilsbrjs cloud. Thee $/o c.mponenrr are rhe hatf of rlle
cloud that go into higher orbirs, and |,hehar fiar go inlo lower otbirs. The m€anlife of rhe half tlat go into
higher orbirs is conEouedby tl|e amocphericder;iry ar the brea*W altitud., while rhe m€anlife of die ha|f $ar
80 nto lower orbils is conrolled by rheaEnos?h€ricdensrryar sometowEratdnie. For 3m hagments,
So=2.2XI0 v^nr. and at lom km tl = 174yeals and 12 =700 yetrs. Ar 5m kn, q =.72 years ajdr2-7-2
y€ars. Eeh meanlife is apprcrihately inyerselypropdtional !o lhe arnoaphedcderjiry aarhebreakup
altrtude,
If the $tellile br€akuplare is give! by dB/d! andbreakupdare€qDatlylitely lo occur within somesmall attitl&
band(small enoughnot lo sgnifi@rly changEthe menl Ufe),$en by in@gErirSover rime, lhe equitibrium
spaualdensity ot the bMl-lp fmgmerts is Sivenby
.JB= d B ! . -
?irol=
dB \?i7u 1
el
whereSOis lhe inilial s?atial densiryaruag€d over breakupswftnir dE aftiro& band. The varueof r is
simply lhe averageof.r1and12.Ir is convenienrro wrire S0 in @ms of rhe avengc numbs of fragmenB,frc
wnich are initialy in th€ volune clemenr AU defineatby rhealli0ldc tand. IIe valueof fro asymploticalty
approaches&e numberof fragDeds gcn€$redFr avs"ge brs*up asAU iffrcasas. Th6 varueof AU shoutdbe
targe,10accoNt for all of dE fi-aSmenlshowever,if ioo larg., rhc valu€ o{ r will significanrty ch3rg€ wirhin
$e volme el€ment. A vdue cones?onding!o an altitudebandofabod lm bn a..outr for aboutde rhird of
the t'aSment!and is suffrci€rdy stnaUthar mear life faidy accuraElyn?Gsanrse averagefor the altitudeband.
SATELLITE DREAKUP RATE
The salellile br€akupral€ drE lo rddoh collisionsis a funcrionof sFdar density,ard canbe calculateddE
n
n_I
i=t
.t=i+t
::=+s-vo^U= >
E
..o,;dU
J S. ,r s .'V
, ,,
,ohrrc
t]l
Th€ firsr er!.€ssion assumess"hericarsrnmcEy abourrheEaih, atrd soDc arcnge ste ard velociry. Thc
value of S is rhe averagesparialdensityar someatriNde' and is calcutaledusiDgtie objectsin the U3 Spac€
CommandcahloSueby the l@hniquesgiven in /10t Valuesfor oreretialiveeeiocity v ana comion crosssecLnno. can eirherbe arsum€d,or cahuraledby seuinglhe trst erpr€ssotr equallo rhe moreexacq2nd
e^pression. In rhis erpEssior, Si is 0E s?ariatdensity of obi:ct i ar sonE alritudead tatitude n is $e numb€r
of cailoguen objecti in orbn, ard Vii and 6ii arethe calcutakn rdadve velocitie! ard colision crcss-s€criotrs.
respective,betweenobjeclsi andj in volun; eteme dU.
Saielite physicaldinensions ard mas *ere obrainedfrom ,/. lvhen no pbysicaldimensionswereavaitable,
ine radmcross-s€ciion!?s assumedro be a n€asur of fie physicd dim.rsions as describedn lU u1n/9/. A
nunerical progran which Ds.da seriesof t0% rafllom samplesro perfdm fie inEgradonsand sumnatioosrrl
LmrB of PoI'LltLionCotrth
(3). A collision rate for calatoguedobjccts of 0 05 collisions per yeat b€tw€en3m Lm and 18m lm altjlude
was obtained. By requiring thc first eipressiotr!o equalthe samecollision mle, lhe propcrlv weightedavcrage
vclocity and cross-sectionar€ v = 7.5 knvs€cand o = | 0 sq m. In addition,$e amountof massinvolved wilh
eachdlision wd us€dlo detemirc the arnountofmass thal would b€.ome fragne.6. It wa assumedthat if
the rarroof fte mas of thc large. satellile times lhe relatrvevelocity squaftd [o dE massof $e sDaUersalelli|e
rimes l0 krvsec sqoaredw6 lcasftan l0m, fsn aI of the massof both satelliteswent itto fragments lfthis
ra.io was Erster than 1000,then nm of the masswent into fiagments- This c€lcutaron detemined thal,an
avenge c6lision bctwe.n 300 tm and l8m ].m produced1600kg of fraSments-Thjs meaJEthar a signilicant
fradi6n of the collisions are betee4nlargerobje.b, so ftal an averagecollision would produceabout480
band or
lBsmentslargerthar |0cm.*rth lnof them'nioallywithin a lm krn altrode
N^=160.
C;mbining eqDations(2) and (3), ar|d assuminsthat one-ha[ of he catalogueis irEt sarellitesand tne othe.
c.itical densily of casloguedobjeca is give$ bv
half is breatup fragnenls, the ave!_age
s = v, i_^ro-
6)
Using the valuespreviously discusse4fi8ure 3 comparesihe resultinga!€ra8ecritical densitv wilh -thespatul
densiryof iie cunenl calatoSue'Thc resulE indicatethat much of he curr€ntpopulalion above900 km is in
"avenge" cnrh.al densitybecausen assunesdtat the sarellitesize andorbial
uE unsrableregion. This is;
independeni
of altilude. We know this is nol exacdy tnE. For exmple, larger
;Elinalion disEibutionsare
spac.cmf! aregercra y launch€dto lowet altitr es,and orbilal inclin tionsdo varv with altilude, producing
v_aryingcollision velociries. As$mtug fiat theselocal size and inctinariqutEfeftnces willalwavsexi$ ften
- uoj,it o criri.l a"nSty."" bc c3idh€d. Eff@tive valuestor v, o ad No se'e obui'ed asa funclion of
aturualeby equiring rhe lwo explNioll in (3) lo be equarto one atnth€t tt w3s conclod€dlhal the inclination
eff€ts do increaseGe couisionrat€sov€. the aveEgearound9m kn and r 5m tn bv abot 30%' and decms€d
ir slishdr at other altiLud€s.Ho*evet, theelfeds of size weremorediatnalic...dighdv incr€astrg fte average
tagieni generarionmrs ar tow alutldes, and siSnif'cantlydeceasingdle mleat hiShq aldtudes fhese
adjustrnentsare shownin fignre 4.
Tbe resulb leavethe 9m tm to 1m0 tm altibde region fimly in $e urutableEgion. while tEducingthe
Egion of in$ability betwen rmo hn and r4m kn. This latler region of s?6cccontatusmosllv breakup
f;gmenl! and few laige salelttes. [email protected] the low numberdensityin $ese rcgiots, il hasb€ensuggested
tl)ar they might be Ned as the region !o disc2rdnDctearpayloads sEh an act on woold chrnge 6e cunent size
distnbuion, ds@sing thecritical dssiry line ftom that shown in ligulP 4. as wU as incrE$e tne s?alal
dersity: cons€quentlya mor€linil€d amountof matedalcould be plac€diDlo tle region thar suggest€dbl
aiive, if we could b€ sur. dur dE e *erc no other facbrs 0oconsider,
figurc 4...figurE3 woutd t'e morer"ree
suchas a significant uncabloguedpopulatior
RecentElescopic meau€menls, discussedin /11/ indiczle thar the US Spdc Co(nmandcalrlo8l]e is onlv abour
50% completedown b a liniting size of l0 cm. Arftough this mesnsthat $e cDrlml populalion densilv
shouldbe doubleot€r that shovn in tigurEs3 and4, the cone$onding cntical &-nsity lines woold also itcr€3se.
By placirS snaller objeds inlo lhe populalion,the effecti\€ vatuesfor both 6 and N0 would decrpase,increasing
dle sllowable oitical density- Cons4uendy, alftough this un€talogucd pop'iation wiu havean elIe€i, it will
be less thu a factor of two. In addirion,it is difficult to makemearingful stalen|entsabout lhe numberand size
capab ides,csl€cially when the
of objecrswhenlheseobjectsarEn.lr $e thr€sholdof th€ me3sl.[ements
quantily of intcrcst is mass.and $e limits of &e sensorsarc expressedir tems of ndar cross'seclionor limitiog
magrilude. A full analysism!s! imlde all of lhe disEibutionswhich effect lhe rettrmedsigtar srenSlh, strch
asobjet shape,alb€do,and a numberofother parameters/5, Similar di{ficultias occur in /12l which show $e
avcragefragmentmassdecreasirsno!! r"pidly lt|an in /l/ (which is b3s€dor groud explosiondau) as 6e 10
cm thiesholdis appoached.
CONCLUDINC REMARKS
This analysisindicabs 6ar cenain rcgionsof low Earthorbil are alEady unstable. Most of the larger debris
Eeneded in theserEgionswill be contiDed,o thc unstableregions..$js is de of Oe rcasonsthey aE unstable.
Ho*€ver, smallerdebriswill be e.jectedlo grEaterdisbncesthanlargpd€bris,atd lheseuslable regionswill act
asan incr€asin8souiceof small debris in aI of low E8fth orbit for cettuti:s. If noding moreis addedto the
Nstable regions,rhe mle thal lhe debris qvimnment wil gmw is slow...aboulone collisional breahp every 10
to 20 yeds, dep.nding on |e natu.eofde uncalaloguedFpularion, wilh abou!halfof thes€breakupsoccning
witin the unsrableregions. Therefore,if objectsare removedf.om thet€ altitdas, it may slill b€ possibleto
reversefte inslability. Howevq, ifobjeds areconlinu€dio beplacedinto the unstableand surmundingregions,
I
:
!
Fb, 3. Avatle Crili..l D€ciq. A.sm6 eize ed
o.bi&l irclincrioB de indeFnddr of .llitud..
Fi!.4, Adjut <l Cnricd Dcreiry. A.s@a
.nd orbid i&lin.riom.
l@d siz.
the bRal.upralf, vould be siSnficandy higlrr and rhercgion of inirbitity woutd expand. In rhelasr 9 yw\
ttrelopulation densnybelw€€n9OOkm and t 0m km hasdoubt.d. U recur€trr Dqutarion abov. 800 iJn re
lo dorble, the reSionof insbbilily coutd expand. ro in€tude8m kn, anl tlle brE€iruirare wouLt incrcascrcotE
evtry 2.5 yeals to 5 years. Uder rhesec.ndirions. il would be moredilEc,rtato re!€rse the irstabiliry.
Thereis someuncenainryiD rlE exrr rhrcsholdof insubiliryi howevcr,thi! uccnainry is small comDarlqto
'
the c.nainty rharacrionsmusrb€gr trow in order ro be ettecove. Sone acdo s coutd ;tnosr belin
inmedia.ely. Fulu.e upfrl $aaeslhat haw reshn capabiliry,suchaslhc Delra2rd $ag€, could & reenterea
afier delivering d|en payloorl O0Er elions, suchEstheconEoU.dreenEyar fie cnd.of_tifeof Dayl@ds,o, the
retie!"j of payloadt wi[ rcquiE ar t€asrl0 yqrs ot engin€sins devetcDnenl rrur engineerini develoimenr
snoukl b€gmnow, sinc! lhere i! I k uncenaintyrhaLirwillbe €quir€dREFERENCES
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