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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:
kλ
( ) ( )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
NUMERICAL MODELING OF THE FORMATION, ORBITAL EVOLUTION 247
600
500
400
300
100
200
9392919089880
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
600
500
400
300
100
200
9392919089880
Apo
gee/
Per
igee
alt
itud
e, k
m
Period, min
600
500
400
300
100
200
9392919089880
Apo
gee/
Per
igee
alt
itud
e, k
m
Period, min
600
500
400
300
100
200
9392919089880
(c) (d)
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.
248
SOLAR SYSTEM RESEARCH Vol. 44 No. 3 2010
BORDOVITSYNA, ALEKSANDROVA
Δϕ
Δλ
ri Δrϕj
Fig. 11. Space cell in spherical coordinates.
42220
42200
42180
42160
42140
36027018090042120
λ, deg
43000
42500
42000
41500
36027018090041000
λ, deg
43000
42500
42000
41500
36027018090041000
λ, deg
r, k
m
43000
42500
42000
41500
36027018090041000
λ, deg
r, k
m
43000
42500
42000
41500
36027018090041000
λ, deg
r, k
m
I
II
III
IV
V
r, k
mr,
km
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
ϕ
r
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NUMERICAL MODELING OF THE FORMATION, ORBITAL EVOLUTION 249
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
2E�008
4E�008
6E�008
8E�0088E�008
6E�008
4E�008
2E�008
1050–5360300
–10
λ, deg
I
II
III
IV
V
ϕ, deg
240180
12060
00
0
ρ, km–3
λ, degϕ, deg
ρ, km–3
8E�0101.2E�009
4E�010
1.6E�009
360300
240180
12060
00
λ, deg
ρ, km–3
8E�0096E�0094E�0092E�009
360300
240180
12060
00
λ, deg
ρ, km–3
6E�009
4E�009
2E�009
360300
240180
12060
00
λ, deg
ρ, km–3
1.6E�009
1.2E�009
8E�010
4E�010
360300
240180
12060
00
ρ, km–3
2E�009
4E�010
1.2E�0098E�010
1050–5–100
ρ, km–3
ϕ, deg
2E�0094E�0096E�0098E�009
1050–5–100
ρ, km–3
ϕ, deg
2E�009
4E�009
6E�009
1050–5–100
ρ, km–3
ϕ, deg
4E�010
8E�010
1.2E�009
1.6E�009
1050–5–100
ρ, km–3
1E�008 1E�008
1.6E�0092E�009
Fig. 13. Distribution density of debris from the fragmentation of a spacecraft like Ekran�2 10 years after the event.
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BORDOVITSYNA, ALEKSANDROVA
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
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a, k
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