14
ISSN 00380946, Solar System Research, 2010, Vol. 44, No. 3, pp. 238–251. © Pleiades Publishing, Inc., 2010. Original Russian Text © T.V. Bordovitsyna, A.G. Aleksandrova, 2010, published in Astronomicheskii Vestnik, 2010, Vol. 44, No. 3, pp. 259–272. 238 INTRODUCTION A high proportion of usedup spacecraft, spent upper stages of rockets, and various spacecraft con struction units are known to become space debris (Rykhlova, 2003; Klinkrad, 2006). Additional debris are produced by deliberate or spontaneous explosions in orbits or during spacecraft collisions. According to NASA (http://www.nasa.gov), at the end of 2008 nearEarth space contained 12 850 large objects of artificial origin, out of which only 3190 were space vehicles, with the proportion of functional spacecraft being as low as 6%; all of the rest were space debris. These are the catalogued objects only. As compared to the data on January 1, 2008, the number of tracked objects grew by 395. The general assumption is that in space today there are about 19000 objects sized 10 cm or more, and only 4% of them are operational space craft. This entire population of uncontrolled objects has become part of the nearearth space environment evolving under the laws of celestial mechanics. How ever, the debris formation mechanism will be shown below to greatly affect the general pattern of their orbital evolution. The purpose of the present paper is to investigate the dynamic aspects of the development, orbital evo lution, and distribution of fragments of space debris in nearEarth space. For these research purposes, the authors have developed a complex of algorithms and software pro grams for the numerical modeling of all of the stages of the above process. In this paper, we describe the mathematical soft ware package that we have developed and discuss some research results on the dynamics of the formation and evolution of space debris which appear to be the most interesting. All of the catalogued objects are divided by their orbit types (Klinkrad, 2006) into the following classes or domains: LEO, objects in lowEarth orbits; MEO, mediumEarth orbits, i.e., object in orbits between LEO and GEO; GEO, objects in geostationary orbits; GTO, objects in GEO transfer orbits; and HEO, objects in highly eccentric orbits. We will adhere to this classification. The percent ages of all of the catalogued objects according domain are as follows: LEO (altitudes less than 2000 km), 69.2%; MEO, 3.9%; GEO 7.8%; and HEO/GTO, 9.7%. In addition, a small fraction of about 150 objects put into orbits far away from the Earth. Note that extensive software packages have been developed to track the catalogued objects; a review of this software is given by Nazarenko (2001). These application packages are designed to predict the haz ardous approaches of spacecraft to orbital debris on injection and functional trajectories and, as a rule, are not supposed to model the formation process or inves tigate the orbital evolution of the entire population of orbital debris resulting from spacecraft fragmentation. However, it is only the numerical modeling of the frag mentation and orbital evolution of all the fragments that allows one to find the spatial distribution of the debris and monitor their time dynamics. The mathe matical software we propose is designed for this kind of research. MATHEMATICAL MODELS OF SPACECRAFT FRAGMENTATION Since the launch of the first satellite, 188 explo sions and collisions of orbiting spacecraft have been Numerical Modeling of the Formation, Orbital Evolution, and Distribution of Fragments of Space Debris in NearEarth Space T. V. Bordovitsyna and A. G. Aleksandrova Tomsk State University, Tomsk, Russia Received July 13, 2009 Abstract—This paper describes the numerical models designed by the authors to investigate the formation, orbital evolution, and spatial distribution of fragments of space debris emerging in orbits as a result of space craft fragmentation. It cites the results of the testing of the models and the data of their use. DOI: 10.1134/S0038094610030068

Numerical modeling of the formation, orbital evolution, and distribution of fragments of space debris in near-Earth space

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ISSN 0038�0946, Solar System Research, 2010, Vol. 44, No. 3, pp. 238–251. © Pleiades Publishing, Inc., 2010.Original Russian Text © T.V. Bordovitsyna, A.G. Aleksandrova, 2010, published in Astronomicheskii Vestnik, 2010, Vol. 44, No. 3, pp. 259–272.

238

INTRODUCTION

A high proportion of used�up spacecraft, spentupper stages of rockets, and various spacecraft con�struction units are known to become space debris(Rykhlova, 2003; Klinkrad, 2006). Additional debrisare produced by deliberate or spontaneous explosionsin orbits or during spacecraft collisions. According toNASA (http://www.nasa.gov), at the end of 2008near�Earth space contained 12850 large objects ofartificial origin, out of which only 3190 were spacevehicles, with the proportion of functional spacecraftbeing as low as 6%; all of the rest were space debris.These are the catalogued objects only. As compared tothe data on January 1, 2008, the number of trackedobjects grew by 395. The general assumption is that inspace today there are about 19000 objects sized 10 cmor more, and only 4% of them are operational space�craft. This entire population of uncontrolled objectshas become part of the near�earth space environmentevolving under the laws of celestial mechanics. How�ever, the debris formation mechanism will be shownbelow to greatly affect the general pattern of theirorbital evolution.

The purpose of the present paper is to investigatethe dynamic aspects of the development, orbital evo�lution, and distribution of fragments of space debris innear�Earth space.

For these research purposes, the authors havedeveloped a complex of algorithms and software pro�grams for the numerical modeling of all of the stages ofthe above process.

In this paper, we describe the mathematical soft�ware package that we have developed and discuss someresearch results on the dynamics of the formation andevolution of space debris which appear to be the mostinteresting.

All of the catalogued objects are divided by theirorbit types (Klinkrad, 2006) into the following classesor domains:

LEO, objects in low�Earth orbits;MEO, medium�Earth orbits, i.e., object in orbits

between LEO and GEO;GEO, objects in geostationary orbits;GTO, objects in GEO transfer orbits; andHEO, objects in highly eccentric orbits.We will adhere to this classification. The percent�

ages of all of the catalogued objects according domainare as follows: LEO (altitudes less than 2000 km),69.2%; MEO, 3.9%; GEO 7.8%; and HEO/GTO,9.7%. In addition, a small fraction of about 150 objectsput into orbits far away from the Earth.

Note that extensive software packages have beendeveloped to track the catalogued objects; a review ofthis software is given by Nazarenko (2001). Theseapplication packages are designed to predict the haz�ardous approaches of spacecraft to orbital debris oninjection and functional trajectories and, as a rule, arenot supposed to model the formation process or inves�tigate the orbital evolution of the entire population oforbital debris resulting from spacecraft fragmentation.However, it is only the numerical modeling of the frag�mentation and orbital evolution of all the fragmentsthat allows one to find the spatial distribution of thedebris and monitor their time dynamics. The mathe�matical software we propose is designed for this kind ofresearch.

MATHEMATICAL MODELS OF SPACECRAFT FRAGMENTATION

Since the launch of the first satellite, 188 explo�sions and collisions of orbiting spacecraft have been

Numerical Modeling of the Formation, Orbital Evolution, and Distribution of Fragments of Space Debris

in Near�Earth SpaceT. V. Bordovitsyna and A. G. Aleksandrova

Tomsk State University, Tomsk, RussiaReceived July 13, 2009

Abstract—This paper describes the numerical models designed by the authors to investigate the formation,orbital evolution, and spatial distribution of fragments of space debris emerging in orbits as a result of space�craft fragmentation. It cites the results of the testing of the models and the data of their use.

DOI: 10.1134/S0038094610030068

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

NUMERICAL MODELING OF THE FORMATION, ORBITAL EVOLUTION 239

registered; 173 of them are included into the NASAcatalogue History of In�Orbit Satellite Fragmentation,which can be found at http://orbitaldebris.jsc.nasa.gov. The data from this catalogue will be used inour model calculations.

We assume that at the moment of spacecraft frag�mentation caused by an isotropic explosion, the coor�dinates of each fragment are identical to those of theparent body, and the fragment’s velocity componentsare determined by the formulae (Bordovitsyna andDruzhinina, 1998)

(1)

where are the parent object’s velocity compo�nents in the geocentric coordinate frame. The param�eters , τ, and ϕ specify the magnitude and directionof the fragment’s velocity vector with respect to theparent body and are treated as random variables deter�mined by the inverse function method from the givenfunctions of the distribution density.

The distribution densities for τ and ϕ are given bythe following formula:

(2)

The fragments’ velocity with respect to the parentbody Δv is determined by the formula (Pardini et al.,1998)

(3)

where y is a random variable from 0 to 1 with a uniformdistribution density, and is the average velocityfound from the following relations:

(a) explosion

(4)

(b) collision

(5)

where is the relative velocity andkinetic energy of the striker; are the followingconstants:

10 1

20 2

30 3

cos ,

sin cos ,

sin sin ,

x

x

x

= + Δ τ

= + Δ τ φ

= + Δ τ φ

v v

v v

v v

1 2 3, ,v v v

Δv

τϕ = τ =

π

1 sin( ) , ( ) .2 2

p p

(0.1 0.6 3 ) 0.00 0.75,

(1.3 0.6 1 ) 0.75 1.00,

y y

y y

⎧Δ + ≤ ≤Δ = ⎨

Δ − − ≤ ≤⎩

v

v

v

Δv

2log 0.0676(log ) 0.804 log 1.514,d dΔ = − − −v

( ) [ ]c c m m

c m

/

V

log( ) ,log

,

A B d d d d

A d d

+ ≥⎧Δ = ⎨<⎩

v

m c/ ,13 ,Vd E C E=

c c c, ,A B C

c

c

c8

0.125,

0.0676,

8.01 10 ,

A

B

C

= −

= −

= ×

and d is the fragment’s diameter (m). If we assume thatthe fragment has a spherical shape, the diameter is cal�

culated by the formula: , where is thecross�section area of the fragment (m2) found from thelog�normal distribution:

(6)

(7)

where m is the fragment’s mass (kg) and is the aver�age cross�section area of the fragment (m2). The frag�ments’ masses are determined from the following rela�tions:

(a) high�intensity explosion

(8)

where N0 and c are the parameters determining theexplosion power and are varied so that the mass distri�bution would remain continuous (Pensa et al., 1996);

(b) medium�intensity explosion

(9)

(c) low�intensity explosion

(10)

where is the spacecraft mass (kg) and N is the num�ber of fragments whose mass is bigger than m (kg); and

(d) collision

(11)

where is the total ejection mass (kg). In the case of

catastrophic fragmentation, , where

and are the masses of the target and striker (kg).

/2d A= π A

2ln ln1

2 0.81 1( ) ,0.8 2

A A

p A eA

−⎛ ⎞− ⎜ ⎟⎝ ⎠

= ×

π

1.13 5

1.5 5

62.013 8.04 10 ,

2030.33 8.04 10 ,

A Am

A A

⎧ ≥ ×⎪= ⎨⎪ ≤ ×⎩

A

⎧ ≥⎪

= ⎨ ⎛ ⎞<⎪ ⎜ ⎟×⎩ ⎝ ⎠t

0

0.75

0.05,

( )0.439 0.05,

0.1

c mN e m

N m m mM

t

t

t

/3

1.8202

1.8202

( )

9.4561 10 0.015,

0.7901 0.015 1.936,

0.1555 1.936,

m

m

N m

M m m

M e m

M e m

⎧ × <⎪⎪

= ≤ <⎨⎪

<⎪⎩

t

t

0.6202

1.8202

0.171 1.936,( )

0.869 1.936,

m

m

M e mN m

M e m

⎧ ≥⎪= ⎨⎪ <⎩

tM

( )e0.7496

( ) 0.4396 ,MN mm

=

eM

e t pM M m= + tM

pm

240

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

BORDOVITSYNA, ALEKSANDROVA

Formula (5) is true for catastrophic fragmentationonly. The following condition determines whether thecollision is catastrophic:

TESTING SPACECRAFT FRAGMENTATION MODELS

The fragmentation models were tested on LEOobjects using the data from the aforementioned NASAcatalogue. In regards to GEO, so far only two explo�sions have been registered in this zone. The first onewas the break�up of the American rocket 68081ETranstage 13 into three pieces; the second one was thesolar battery explosion aboard the Russian satelliteEkran�2. The latter did not result in the fragmentationof the satellite, but we are going to use the data on

p

t

p

t

craterization mode,

catastrophic fragmentation.

2

2

1.15 V0.1

1.15 V0.1

m

M

m

M

⎧ × ×<⎪

⎪⎨

× ×⎪ ≥⎪⎩

Ekran�2’s orbit and mass in the modeling of the space�craft fragmentation in GEO and the exploration of theorbital evolution of the fragments. The main parame�ters of the spacecraft are given in Table 1.

The observational data on the objects produced byin�orbit spacecraft fragmentation are presented in theNASA catalogue in the form of graphs showing thedependence of the fragments’ apogee and perigee alti�tudes on the rotation period.

We obtained analogous dependences by means ofsimulation using the algorithms above; the respectivegraphs and those from the NASA catalogue are givenin Figs. 1–6, with the data in Figs. 1–3 depicting thedistribution of the fragments after the explosions andthose in Figs. 4–6 depicting the distribution after thecollisions. A comparison of the graphs shows that thesimulation results agree well with the data from theNASA catalogue. Figures 7–9 present the resultsshowing that in questionable cases, when we do notknow exactly what event caused the spacecraft to breakapart, modeling may help us to obtain the truth. Fur�thermore, the closer the observations are to the event,

Table 1. Data on the objects

Spacecraft Fragmenta�tion date m, kg h, km e i, degree Ω, degree ω, degree M

Cosmos�1275 26.07.81 800 980 0.0036 82.963 119.824 139.033 221.356

Cosmos�1375 06.06.82 650 995 0.0005 65.839 350.281 26.567 333.500

Nimbus�6 R/B 01.05.91 840 1090 0.0006 99.5801 326.211 148.399 211.752

P�78 (SOLWIND) 24.02.79 850 525 0.0022 97.634 82.502 99.408 260.964

USA�19 05.09.86 930 220 0.0391 39.066 28.1524 26.707 335.326

USA�19 R/B 05.09.86 1455 220 0.0288 22.783 10.4654 54.777 307.938

Ekran�2 23.06.78 1750 35790 0.0001 0.1137 78.3897 325.277 78.3897

Cosmos�375 30.10.70 1400 535 0.1022 62.805 96.408 56.086 313.310

Cosmos�397 25.02.71 1400 585 0.1016 65.762 352.867 50.306 318.552

Cosmos�1405 04.09.82 3000 330 0.0021 65.006 126.126 318.093 42.037

GEMINI�9 01.06.66 3400 250 0.0025 28.844 223.906 135.251 221.977

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

NUMERICAL MODELING OF THE FORMATION, ORBITAL EVOLUTION 241

1400

1200

1000

800

110108106104102100600

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)1400

1200

1000

800

110108106104102100600

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

Fig. 1. Distribution of debris from the fragmentation of the satellite Cosmos�1275 a week after the explosion: (a) by the data fromthe NASA catalogue and (b) simulation data.

1300

1200

1100

1000

900

800

108107106105104103

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

700

1300

1200

1100

1000

900

800

108107106105104103700

Fig. 2. Distribution of debris from the fragmentation of the satellite Cosmos�1375 several hours after the explosion: (a) by the datafrom the NASA catalogue and (b) simulation data.

242

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

BORDOVITSYNA, ALEKSANDROVA

the surer our choice is. Figure 10 presents a graphicalanalysis of the fragmentation of the spacecraft GEM�INI 9 ATDA R/B. The results show that during thefragmentation, there must have been an explosion fol�lowed by collisions of the fragments.

The explosion model was also tested using theobservation data on the fragments of spacecraft

68081E Transtage 13. Some orbital data on this space�craft were available after its explosion on February 21,1992 (Vershkov et al., 2001); in addition, Pensa et al.(1996) made a rather comprehensive study of thisevent.

The constructed spacecraft fragmentation modelwas used to determine the orbits of the spacecraft and

Table 2. Orbital elements of Transtage�13 and its fragments before and after the explosion obtained from the observation and modeling data

68081E Transtage�13 before the explosion

68081E Transtage�13 after the explosion 68081G fragment 68081H fragment

observations model observations model observations model

i, degree 11.887 11.9055 11.917 11.9177 11.890 11.9261 11.857

Ω, degree 21.744 21.7657 21.3226 21.4662 21.7015 21.4775 22.1738

ω, degree 45.840 71.4630 71.436 73.0072 74.0772 132.4364 125.8261

v, degree 33.081 37.7610 38.001 36.7980 35.7611 345.0407 352.9804

u, degree 109.2272 109.2371 109.437 109.8052 109.8383 117.4771 118.8065

e 0.009486 0.009486 0.00839 0.01330 0.01331 0.006757 0.00709

a, km 41837.44 41820.05 41847.37 41951.56 41991.69 41766.12 41683.32

4000

3500

3000

2500

2000

1500

1000

500

13612611610696

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

0

4000

3500

3000

2500

2000

1500

1000

500

136126116106960

Fig. 3. Distribution of debris from the fragmentation of the satellite Nimbus 6 R/B a week after the explosion: (a) by the data fromthe NASA catalogue and (b) simulation data.

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

NUMERICAL MODELING OF THE FORMATION, ORBITAL EVOLUTION 243

its fragments after the explosion. The observation andmodeling results given in Table 2 demonstrate thegood agreement of the observed and modeled orbits.

NUMERICAL MODELING OF THE ORBITAL EVOLUTION OF SPACECRAFT FRAGMENTS

The population dynamics of GEO objects was sim�ulated using a modified version of the NumericalModel of Artificial Earth Satellite Motion softwarepackage (Bordovitsyna et al., 2007) developed at theResearch Institute of Applied Mathematics andMechanics at Tomsk State University. The source codewas written in Fortran�90 and translated into an exe�

cutable code with the Salford Fortran 5.0 free com�piler with 19 digits of precision. Therefore, all of thecalculations in the numerical model were performedwith 19 significant digits. This allowed us to abandonEncke’s method when writing motion equations andthus simplify and unify the software package. The inte�gration procedures in the new version use motion andvariational equations written in rectangular coordi�nates. The equations are integrated using the Everharthigh�order method (up to the 29th order, inclusive).The integration procedures takes into account pertur�bations from the geopotential harmonics of arbitraryorder as well as those from the Moon, Sun, and solidtides in the Earth’s body and light pressure.

Table 3. Forecasting error for the position Δr, major axis Δa, and longitude of the subsatellite point Δλ

λ0, degreesError estimate by direct and inverse solutions Estimate of the error of the method

Δr, m Δa, m Δλ, arcsec Δr, m Δa, m Δλ, arcsec

75 9.9 0.003 0.049 10.8 0.003 0.053

120 16.6 0.085 0.081 9.5 0.323 0.046

344 705.6 0.549 3.449 1568 1.253 7.617

164 3645 3.151 19.37 2190 1.958 10.66

1800

1600

1400

1200

1000

800

600

400

200

109104999489

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

0

1800

1600

1400

1200

1000

800

600

400

200

1091049994890

Fig. 4. Distribution of debris from the fragmentation of P�78 (SOLWIND) 11 h after the collision: (a) by the data from the NASAcatalogue and (b) simulation data.

244

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

BORDOVITSYNA, ALEKSANDROVA

For the applied version of the model, we obtainedprecision estimates for geosynchronous satellitemotion forecasts over a 10�year period.

Note that a detailed analysis of the application pos�sibilities of a numerical model with the Encke motionequation to investigate the orbital evolution of GEOobjects over a 200�year period was done by Kuznetsov(2008).

Like Kuznetsov and Kudryavtsev (2008), to esti�mate the effect of the geostationary satellite librationmotion parameters on the forecast precision, weselected four objects for which the initial longitudes ofthe subsatellite point λ0 (column 1 in Table 3) enableone to account for all types of libration motion.

The obtained estimates confirm the conclusion inthe work (Kuznetsov and Kudryavtsev, 2008) that theforecast precision depends on the amplitude of libra�tion motion. The lower the amplitude, the higher theprecision is. The least precise forecasts are made forquasi�random trajectories with λ0 = 164°. Table 3 givestwo estimates: using direct and inverse integration andfrom the comparison of the 19th and 23rd order solu�tions.

CALCULATING THE SPATIAL DENSITY OF THE FRAGMENTS

To construct a spatial density distribution for thefragments, the debris area is divided into cells. Thisdivision is performed in steps by three parameters: dis�

Table 4. Scatter boundaries

No. аmin, km аmax, km emin emax imin, degree imax, degree vmin, km/s vmax, km/s

1 32691 63006 0.0015 0.332 0.0025 9.261 2.59 3.54

2 38953 45609 0.0012 0.103 0.0029 3.701 2.96 3.24

3 37475 52675 0.0003 0.202 0.0003 4.006 2.87 3.26

4 40352 44963 0.0010 0.056 0.0015 2.670 2.99 3.20

5000

4000

3000

2000

1000

0

1351159575

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

–1000

5000

4000

3000

2000

1000

0

1351159575–1000

Fig. 5. Distribution of debris from the fragmentation of USA 19 a day after the collision: (a) by the data from the NASA catalogueand (b) simulation data.

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

NUMERICAL MODELING OF THE FORMATION, ORBITAL EVOLUTION 245

6000

5000

4000

3000

2000

1000

0

–1000

1501301109070

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

–2000

6000

5000

4000

3000

2000

1000

0

–1000

1501301109070–2000

Fig. 6. Distribution of debris from the USA 19 R/B fragmentation a day after the collision: (a) by the data from the NASA cata�logue and (b) simulation data.

2800

2400

2000

1600

1200

800

400

109 111 1171151130

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

107

2800

2400

2000

1600

1200

800

400

109 111 1171151130107

2800

2400

2000

1600

1200

800

400

109 111 1171151130107

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(c)

Fig. 7. Distribution of debris from the fragmentation of Cosmos�375 4 months after the event: (a) by the data from the NASAcatalogue, (b) by the simulation data for fragmentation from a collision, and (c) by the simulation data for fragmentation from anexplosion.

tance from the center of the Earth r (km), longitude λ(degrees), and latitude (degrees).

The spatial density of the fragments at a given pointin time is determined as the ratio of the number offragments in the cell to its volume.

ϕ

Let the geocentric distance ri, latitude ϕj, and lon�gitude determine the cell center (Fig. 11); then, itsvolume can be found from the formula:

( ) ( )2 2, ,

2 13 ( ) cos sin .3 4 2

i j k i jV r r rΔφ= + Δ φ ΔλΔ

246

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

BORDOVITSYNA, ALEKSANDROVA

2800

2400

2000

1600

1200

800

400

111 112 116115114113110

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

(c)

0

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

2800

2400

2000

1600

1200

800

400

111 112 1161151141131100

2800

2400

2000

1600

1200

800

400

111 112 1161151141131100

Fig. 8. Distribution of debris from the Cosmos�397 fragmentation 7 weeks after the event: (a) by the data from the NASA cata�logue, (b) by the simulation data for fragmentation from a collision, and (c) by the simulation data for fragmentation from anexplosion.

1050

900

750

600

450

300

150

0

100959085

ApogeePerigee

Apo

gee/

Per

igee

alt

itud

e, k

m

Period, min

(а) (b)

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

(c)

–150

1050

900

750

600

450

300

150

0

100959085–150

Period, min

Apo

gee/

Per

igee

alt

itud

e, k

m

1050

900

750

600

450

300

150

0

100959085–150

Fig. 9. Distribution of debris from the Cosmos�1405 fragmentation an hour after the explosion: (a) by the data from the NASAcatalogue, (b) by the simulation data for fragmentation from a collision, and (c) by the simulation data for fragmentation from anexplosion.

DEPENDENCE OF THE DISTRIBUTION AND ORBITAL EVOLUTION OF GEO

FRAGMENTS ON THEIR FORMATION MECHANISM

Orbital debris dynamics as a result of spacecraftfragmentation in GEO was investigated by the exam�ple of satellites like Ekran�2.

We modeled several explosions of various intensityand catastrophic fragmentation as a result of a colli�sion (Aleksandrova, 2008). Table 4 cites the scatter

boundaries for the fragments’ orbits by the major axisa, eccentricity e, inclination i, and velocity v of thefragments in a decreasing order of explosion intensity(1)–(3) and for the collision (4). Overall, each cloudcontained about 800 fragments with masses from0.05 g to 1 kg. The evolution of the fragmentation wassimulated over a 10�year period.

Furthermore, we simulated satellite fragmentationfrom ageing. This kind of fragmentation was modeledby varying the mean anomalies from 0° to 360° with an

SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010

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Fig. 10. Distribution of debris from the GEMINI 9 ATDA R/B fragmentation after the explosion on June 21–24, 2006: (a) by thedata from the NASA catalogue, (b) by the simulation data for fragmentation from a collision, (c) by the simulation data for frag�mentation from an explosion, and (d) by the simulation data for fragmentation from an explosion and collision.

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Fig. 11. Space cell in spherical coordinates.

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Fig. 12. Dependence of the change in the radius vector onthe longitude of the subsatellite point.

interval of 1°, with the Keplerian elements keptunchanged. Thus, we obtained 360 objects evenly dis�tributed along the orbit.

Figure 12 shows the distribution of satellite pieces10 years after the fragmentation (Graphs I–III showthe breaking up of spacecraft from an explosion; theyare arranged in a decreasing order of explosion inten�sity; IV shows the breaking up from catastrophic colli�sion; and V shows the breaking up from spacecraft age�ing). As the fragmentation intensity decreases, evenmore fragments are emerging in resonance orbits witha maximum concentration of objects near the stablelibration points. This is most noticeable if we considerthe example of fragmentation from ageing. In the caseof collision fragmentation, we can also clearly see theconcentration maxima near the stable libration points.

For each fragmentation case, we found the spatialdensity of the fragments 10 years after the event.

To construct the spatial density, the debris area wasdivided into cells in the following steps: 15° by longi�tude λ, 5° by latitude , and 50–2000 km by geocen�tric distance (depending on the fragmentation inten�sity). The spatial density of the fragments at a givenpoint in time is determined as the ratio of the numberof fragments in the cell to its volume.

Figure 13 shows the spatial density distribution bylongitude and latitude 10 years after the fragmenta�tion. The sequence of graphs is the same as in Fig. 12.

An analysis of the results allows us to conclude thatthe cells with maximal density are located in the libra�tion motion domain, with the concentration maximabecoming more pronounced as the fragmentationintensity decreases.

RESEARCH ON THE LONG�TERM ORBITAL EVOLUTION OF GEO OBJECTS

Starting in 2006, the dynamic evolution of realGEO objects was studied for all of the uncontrolledobjects in this zone that are included in the catalogueof the European Space Agency (http://linkinghub.elsevier.com). To study the orbital evolution of theselected objects, their motion equations were integratedby the Everhart 19th order method over a 10�year intervalfrom January 1, 2007 to January 1, 2017. The follow�ing perturbing factors were considered: the Earth’s

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nonspherical shape, solar and lunar gravitation, tidesin the Earth’s body, and light pressure.

A general picture of the objects’ evolution and dis�tribution for 10 years is given in Fig. 14. The main dis�

tribution tendency of the objects gathering near thelibration points is retained. While performing theinvestigation, we discovered that two spacecraft Pro�ton�K (82044F) and Cosmos�2224 (92088A) passed

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Fig. 13. Distribution density of debris from the fragmentation of a spacecraft like Ekran�2 10 years after the event.

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from the libration mode near the stable point 75° intothe libration mode near two stable points. We alsoobtained estimates for the approaches of GEO objectsto the stable libration point of 75°. This was done byforecasting their motion over a 10�year period in stepsof 0.1 days. The data on all of the approaches are givenin Table 5. Column 1 cites the name of the spacecraft,column 2 gives its minimal distance to the librationpoint, and column 3 gives the date of every approach.

CONCLUSIONS

Thus, the present paper describes the numericalmodels developed by the authors to investigate the for�mation, orbital evolution, and spatial distribution offragments of space debris emerging in orbits as a resultof spacecraft fragmentation. The main focus is on test�ing the fragmentation models by observations andstudying space debris in GEO.

The models are shown to be in good agreementwith observations, and in questionable cases, when thecause of the fragmentation is unknown, they can beused to find the cause and understand the fragmenta�tion pattern.

The dynamics of fragments in GEO was simulatedfor various formation mechanisms; a dynamic pictureof evolution over a 10�year period was constructed forall of the uncontrolled objects in this domain of thenear�Earth space environment as of January 1, 2007.

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42500

42400

42300

42200

42100

42000

36027018090041900

a, k

m

λ, deg

Fig. 14. Dynamic evolution pattern for the GEO object population in rotating coordinates for 10 years from January 1, 2007 toJanuary 1, 2017.

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Satellite Δr, km Date

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SIRIO�1 7 15.05.2014

Luch 1�1 12 21.04.2008

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