research article resonant orbital dynamics in leo...

Research ArticleResonant Orbital Dynamics in LEO Region:Space Debris in Focus

J. C. Sampaio,1 E. Wnuk,2 R. Vilhena de Moraes,1 and S. S. Fernandes3

1 Universidade Federal de Sao Paulo (UNIFESP), 12231-280 Sao Jose dos Campos, SP, Brazil2 AMU, Astronomical Observatory, 60-286 Poznan, Poland3 Instituto Tecnologico de Aeronautica (ITA), 12228-900 Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to J. C. Sampaio; [email protected]

Received 21 December 2013; Accepted 17 February 2014; Published 13 April 2014

Academic Editor: Antonio F. Bertachini A. Prado

Copyright © 2014 J. C. Sampaio et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The increasing number of objects orbiting the earth justifies the great attention and interest in the observation, spacecraft protection,and collision avoidance. These studies involve different disturbances and resonances in the orbital motions of these objectsdistributed by the distinct altitudes. In this work, objects in resonant orbital motions are studied in low earth orbits. Using the two-line elements (TLE) of the NORAD, resonant angles and resonant periods associated with real motions are described, providingmore accurate information to develop an analytical model that describes a certain resonance. The time behaviors of the semimajoraxis, eccentricity, and inclination of some space debris are studied. Possible irregularmotions are observed by the frequency analysisand by the presence of different resonant angles describing the orbital dynamics of these objects.

1. Introduction

The objects orbiting the Earth are classified, basically, inlow earth orbit (LEO), medium earth orbit (MEO), andgeosynchronous orbit (GEO).TheLEO region has a big quan-tity of space debris and, consequently, most of the objectsare found in low earth orbits. Considering approximately10000 cataloged objects around the earth, one can verify thedistribution of the objects as 7% of operational spacecraft,22% of old spacecraft, 41% ofmiscellaneous fragments, 17% ofrocket bodies, and about 13% of mission-related objects. Theuncatalogued objects larger than 1 cm are estimated in somevalue between 50000 and 600000 [1, 2].

In the last years, the study about space debris mitigationin LEO region has been motivated by the increasing numberof this kind of object through the years. These aspectsconsidered in the studies englobe the observation, spacecraftprotection, and collision avoidance [3, 4].The space debris arecomposed of aluminum from spacecraft structures, aluminafrom solid rocket motor exhausts, zinc, and titanium oxidesfrom thermal control coatings and their sizes range fromseveral meters to a fraction of a micrometer in diameter [5].

Currently, the orbital motions of the cataloged objectscan be analyzed using the 2-line element set of the NorthAmerican Defense (NORAD) [6]. The TLE are composed ofseven parameters and epoch. These data can be compared,for example, with the mathematical model of the propagatorused to describe themotion of the artificial satellite. A similarstudy is done for the Brazilian satellite CBERS-1 in cooper-ation with China. In this case, orbital perturbations due togeopotential, atmospheric drag, solar radiation pressure, andgravitational effects of the sun and the moon are consideredin the numerical integration of the orbit, and the results arecompared with the TLE data [7–9].

The space between the earth and the moon has severalartificial satellites and distinct objects in some resonance.Synchronous satellites in circular or elliptical orbits havebeen extensively studied in literature, due to the study ofresonant orbits characterizing the dynamics of these satellitessince the 60’s (see [10–22] and references here in). In someof these works, resonant angles associated with the exactresonance are considered in the numerical integration ofthe equations of motion, with the purpose to describe theresonance defined by the commensurability between the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 929810, 12 pageshttp://dx.doi.org/10.1155/2014/929810

2 Mathematical Problems in Engineering

mean motion of the artificial satellite or space debris and theearth’s rotation angular velocity. However, tesseral harmonics𝐽𝑙𝑚 produce multiple resonances in the neighborhood of theexact resonance, and some variations in the orbital motionsof objects are not described.

The main goal of this work is to define resonant anglesand resonant periods associated with real motions, providingmore accurate information to develop an analytical modelthat describes a certain resonance. Amodel to detect resonantobjects and the subset of interesting resonant objects is thenpresented. The time behavior of orbital elements, resonantangles, and resonant periods of specified objects is analyzed.In the studies about irregular motions, a new way to deter-mine the frequency analysis is presented.

2. Resonant Objects Orbiting the Earth

In this section, the TLE data are used to verify if the objectsare in resonant motions and to define the resonance in whichthe most of objects are found [6].

The present distribution of objects indicates the commen-surability between the frequencies of the mean motion 𝑛 ofobject and the earth’s rotation motion. See the histogram inFigure 1.

In Figure 1, it is verified that most of the objects is in theregion 13 ≤ 𝑛 (rev/day) ≤ 15.

To study the resonant objects using the TLE data, acriterium is established for the resonant period 𝑃res, by thecondition 𝑃res > 300 days. Note that the resonant period isrelated to a resonant angle which can influence the orbitalmotion of a particular artificial satellite. The value of 𝑃reshelps to understand the influence of each resonant angle,and this knowledge can provide a more rigorous analyticalformulation of the problem. A minimum value is establishedfor the resonant period in order to define the main resonantangles that compose an orbital motion. The resonant anglesare represented by the symbol 𝜙𝑙𝑚𝑝𝑞. 𝑃res is obtained by therelation

𝑃res =2𝜋

𝜙𝑙𝑚𝑝𝑞

, (1)

and 𝜙𝑙𝑚𝑝𝑞 is calculated from [10]as follows:

𝜙𝑙𝑚𝑝𝑞 (𝑀, 𝜔,Ω,Θ)

= (𝑙 − 2𝑝 + 𝑞)𝑀 + (𝑙 − 2𝑝) 𝜔

+ 𝑚 (Ω − Θ − 𝜆𝑙𝑚) + (𝑙 − 𝑚)𝜋

2,

(2)

where 𝑎, 𝑒, 𝐼,Ω,𝜔, and𝑀 are the classical Keplerian elements:𝑎 is the semimajor axis, 𝑒 is the eccentricity, 𝐼 is the inclinationof the orbit plane with the equator, Ω is the longitude of theascending node, 𝜔 is the argument of pericentre and 𝑀 isthe mean anomaly, respectively; Θ is the Greenwich siderealtime and 𝜆𝑙𝑚 is the corresponding reference longitude alongthe equator. The term (𝑙 − 𝑚)(𝜋/2) in (2) is a correctionfactor used in some representations of the earth gravitationalpotential [23–26]. This term is constant and do not appear

0

100

200

300

400

500

600

700

800

900

0 2 4 6 8 10 12 14 16 18

Num

ber o

f obj

ects

n (rev/day)

Figure 1: Histogram of the mean motion of the cataloged objects.

in 𝑑𝜙𝑙𝑚𝑝𝑞/𝑑𝑡 used in the calculations of the present work. So,𝜙𝑙𝑚𝑝𝑞 is defined as

𝜙𝑙𝑚𝑝𝑞 = (𝑙 − 2𝑝 + 𝑞) �� + (𝑙 − 2𝑝) �� + 𝑚 (Ω − Θ) . (3)

Substituting 𝑘 = 𝑙 − 2𝑝 in (3), 𝜙𝑘𝑚𝑞 is obtained as follows:

𝜙𝑘𝑚𝑞 = (𝑘 + 𝑞) �� + 𝑘�� + 𝑚 (Ω − Θ) . (4)

The terms ��, Ω, and �� can be written considering thesecular effect of the zonal harmonic 𝐽2 as [7, 27]

�� = −3

4𝐽2𝑛𝑜(

𝑎𝑒

𝑎𝑜)

2 (1 − 5 cos2 (𝐼))

(1 − 𝑒2)2

,

Ω = −3

2𝐽2𝑛𝑜(

𝑎𝑒

𝑎𝑜)

2 cos (𝐼)(1 − 𝑒2)

2,

�� = 𝑛𝑜 −3

4𝐽2𝑛𝑜(

𝑎𝑒

𝑎𝑜)

2 (1 − 3 cos2 (𝐼))

(1 − 𝑒2)3/2

.

(5)

𝑎𝑒 is the earth mean equatorial radius and 𝑎𝑒 = 6378.140 km,𝐽2 is the second zonal harmonic, 𝐽2 = 1.0826 × 10

−3.The term Θ, in rad/day, is

Θ ≈ 1.00273790926 × 2𝜋. (6)

In order to use orbital elements compatible with the wayin which two-line elements were generated, some correctionsare done in the mean motion of the TLE data. Considering𝑛1 as the mean motion of the 2-line, the semimajor axis 𝑎1 iscalculated [7]

𝑎1 = (√𝜇

𝑛1)

2/3

, (7)

where 𝜇 is the earth gravitational parameter, 𝜇 = 3.986009 ×

1014m3/s2. Using 𝑎1, (7), the parameter 𝛿1 is calculated by (8)

[7]

𝛿1 =3

4𝐽2𝑎2

𝑒

𝑎21

(3 cos2 (𝐼) − 1)

(1 − 𝑒2)3/2

. (8)

Mathematical Problems in Engineering 3

Table 1: 2-line data of objects orbiting the earth.

File Number of data Number of objects𝑎𝑙𝑙𝑑𝑎𝑡𝑎 2011 045 15360 9745

Now, a new semimajor axis 𝑎𝑜, used in the calculations ofthe resonant period, is defined using 𝛿1 from (8) [7]

𝑎𝑜 = 𝑎1 [1 −1

3𝛿1 − 𝛿

2

1−134

81𝛿3

1] , (9)

and the new mean motion 𝑛𝑜, used in the calculations, isfound considering the semimajor axis corrected 𝑎𝑜

𝑛𝑜 = √𝜇

𝑎3𝑜

. (10)

Now, (1) to (10) can be used in a Fortran program tocorrect the observational data and to provide resonant anglesand resonant periods related to orbital motions. The fileanalyzed is “𝑎𝑙𝑙𝑑𝑎𝑡𝑎 2011 045” from the website Space Track[6], and it corresponds to February, 2011. In Table 1, thenumber of data and the number of objects are specified.

The simulation identified objects with resonant periodgreater than 300 days. Several values of the coefficients, 𝑘,𝑞, and 𝑚 are considered in (2) producing different resonantangles to be analyzed by (1). See Table 2 showing details aboutthe results of the simulation.

Note that Table 2 shows only the data satisfying theestablished criterium, 𝑃res > 300 days. The results show theresonant angles and resonant periods which compose theorbital motions of the resonant objects. Table 3 shows valuesof the coefficient𝑚 and the correspondent number of objects.It is possible to verify that most of the orbital motions arerelated to the coefficient 𝑚 = 14 considering the value of thesemimajor axis up to 15000 km.

Comparing the number of objects satisfying the condi-tion of 𝑃res > 300 days in the different regions, 𝑎 < 15000 kmand 𝑎 ≥ 15000 km, it is verified that about 62.37% of theresonant objects has 𝑎 < 15000 km. See this information inTable 4.

These studies allow to investigate the real influence ofthe resonance effect in the orbital dynamics of artificialsatellites and space debris.The number of resonant objects incomparison with the total number of objects in the TLE datashows the great influence of the commensurability betweenthemeanmotion of the object and the earth’s rotation angularvelocity in its orbits. In this way, a more detailed study aboutthe resonant period and the resonant angles is necessary.

In the next section, the orbital motions of some spacedebris in resonance are studied.

3. Study of Objects in 14:1 Resonance

Considering the results of the simulations shown in thesecond section, some cataloged space debris is studied withrespect to the time behavior of the semi-major axis, eccen-tricity, inclination, resonant period, resonant angle, and the

0

Sem

imaj

or ax

is (k

m)

Resonant period (days)

9

8

7

6

5

4

3

2

1

0 9 1087654321

×104

×104

a ≥ 40000 km15000 km ≤ a < 40000 kma < 15000 km1:1 resonance2:1 resonance

14:1 resonanceObject 325Object 546Object 2986Object 4855

Figure 2: Semimajor axis versus resonant period of objects satisfy-ing the condition 𝑃res > 300 days.The objects studied in LEO regionare identified by the numbers 325, 546, 2986, and 4855.

frequency analysis of the orbital elements. These data areanalyzed in this section observing the possible regular orirregular orbital motions.

Figure 2 shows the semimajor axis versus resonant periodof objects satisfying the condition 𝑃res > 300 days. Theobjects are distributed by the value of semimajor axis in threedifferent regions: (1) 𝑎 < 15000 km, (2) 15000 km ≤ 𝑎 <

40000 km, and (3) 𝑎 ≥ 40000 km.Figure 2 shows some objects around the 1:1, 2:1 and

14:1 resonances. The orbital motions of these objects areinfluenced by the resonance for several years and possibleirregular motions can be confined in a region delimited forresonant angles with biggest resonant periods. In order tostudy the orbital motions of these objects, four catalogedspace debris are analyzed in the LEO region; see Figure 2.From the TLE data, objects are identified by the numbers 325,546, 2986, and 4855.These objects have been chosen becausethey show resonant period 𝑃res > 10000 days.

Figures 3, 4, 5, and 6 show the time behavior of thesemimajor axis, eccentricity, and inclination of the objects325, 546, 2986, and 4855, and the frequency analysis of theseorbital elements.

Usually, in the studies of the dynamical systems withthe purpose to identify chaos, the frequency analysis isconsidered for a long time. But, in low earth orbits, a bignumber of cataloged objects have theirmotions influenced bydifferent resonant periods and unknown objects can collidewith each other providing more unpredictability in theirmotions. In this case, the frequency analysis can be moreappropriate for a short time.

Several authors, [28–33], also use the frequency analysisto study the possible regular or irregular orbits in different


7220.9

7221

7221.1

7221.2

7221.3

7221.4

7221.5

7221.6

0 50 100 150 200 250

Sem

imaj

or ax

is (k

m)

Time (days), January–September, 2011

(a)

0.1

1

10

100

1000

10000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Inte

nsity

Frequency

(b)

0.0061

0.0062

0.0063

0.0064

0.0065

0.0066

0.0067

0.0068

0.0069

Ecce

ntric

ity

0 50 100 150 200 250Time (days), January–September, 2011

(c)

0.0001

0.001

0.01In

tens

ity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Frequency

1e − 006

1e − 005

(d)

66.91

66.912

66.914

66.916

66.918

66.92

66.922

66.924

Incli

natio

n (d

eg)


(e)

0.001

0.01

0.1

1

10

100

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Frequency

(f)

Figure 3: Orbital motion of object 325 corresponding to January–September, 2011: (a) time behavior of the semimajor axis, (b) powerspectrum of the semimajor axis, (c) time behavior of the eccentricity, (d) power spectrum of the eccentricity, (e) time behavior of theinclination, and (f) power spectrum of the inclination.


7101

7102

7103

7104

7105

7106

7107

7108

7109

0 50 100 150 200 250

Sem

imaj

or ax

is (k

m)


(a)

0.1

1

10

100

1000

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Frequency

(b)

0.0028

0.003

0.0032

0.0034

0.0036

0.0038

0.004

Ecce

ntric

ity


(c)

0.0001

0.001

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Frequency

1e − 006

1e − 005

(d)

67.207

67.208

67.209

67.21

67.211

67.212

67.213

67.214

Incli

natio

n (d

eg)


(e)

0.0001

0.001

0.01

0.1

1

10

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Frequency

(f)



7257.1

7257.15

7257.2

7257.25

7257.3

7257.35

7257.4

7257.45Se

mim

ajor

axis

(km

)


(a)

0.01

0.1

1

10

100

1000

10000

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Frequency

(b)

0.0053

0.0054

0.0055

0.0056

0.0057

0.0058

0.0059

0.006

Ecce

ntric

ity


(c)

0.0001

0.001

0.01

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Frequency

1e − 006

1e − 005

(d)

90.508

90.51

90.512

90.514

90.516

90.518

90.52

90.522

90.524

90.526

90.528

Incli

natio

n (d

eg)


(e)

0.001

0.01

0.1

1

10

100

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Frequency

(f)

Figure 5: Orbital motion of object 2986 corresponding to January–September, 2011:(a) time behavior of the semimajor axis (b) powerspectrum of the semimajor axis, (c) time behavior of the eccentricity, (d) power spectrum of the eccentricity, (e) time behavior of theinclination, and (f) power spectrum of the inclination.


7381.3

7381.4

7381.5

7381.6

7381.7

7381.8

7381.9Se

mim

ajor

axis

(km

)


(a)

0.1

1

10

100

1000

10000

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Frequency

(b)

0.00145

0.0015

0.00155

0.0016

0.00165

0.0017

0.00175

0.0018

0.00185

0.0019

Ecce

ntric

ity


(c)

0.0001

0.001

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Frequency

1e − 006

1e − 005

(d)

100.01

100.015

100.02

100.025

100.03

100.035

100.04

Incli

natio

n (d

eg)


(e)

0.001

0.01

0.1

1

10

100

Inte

nsity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Frequency

(f)



Table 2: Results of the simulation: objects in resonant orbital motions.

Number of data Number ofdifferent objects Coefficient 𝑘 Coefficient 𝑞 Coefficient𝑚

3180 1276 −50 ≤ 𝑘 ≤ 50 −5 ≤ 𝑞 ≤ 5 1 ≤ 𝑚 ≤ 50

Table 3: Number of objects according to the number of thecoefficient 𝑚, considering the value of the semimajor axis up to15000 km.

𝑚 Number of objects7 28 19 310 111 312 2013 3114 35115 6516 418 320 121 222 223 224 1025 9326 1927 6628 17929 9730 3131 1332 533 234 135 436 537 4238 1439 1440 2441 18642 11643 11444 5245 2546 847 448 649 2250 33

dynamical systems. A brief description of the frequencyanalysis involving the Fast Fourier Transforms (FFT) ispresented in what follows. See this method with more detailsin [28, 29, 33].

Generally, the frequency analysis is applied in the outputof the numerical integration in the study of dynamicalsystems. However, in the present work, this analysis is used

Table 4: Number and percentage of objects satisfying the condition𝑃res > 300 days.

𝑎 < 15000 km 15000 km ≤ 𝑎 <

40000 km 𝑎 ≥ 40000 km

1276 331 43962.37% 16.18% 21.46%

in the real data of the orbital motions of the objects 325, 546,2986, and 4855, from the TLE data of the NORAD [6].

For regular motions, the orbital elements oe(𝑡) show adependence on time as follows [28, 29, 33]:

oe (𝑡) = ∑

ℎ

𝐴ℎ𝑒2𝜋𝑖hf𝑡

, (11)

where 𝐴ℎ represent the amplitudes, h ∈ Z𝑁 and f is a fre-quency vector.The components of f compose the fundamen-tal frequencies of motion, and the spectral decomposition ofthe orbital motion is obtained from the Fourier transformwhen the independent frequencies are constant in the courseof time [33]. It is possible to observe that f depends on thekind of trajectory [28].

The same numerical procedure is done for irregular andregular motions. The difference in behavior of these orbitalmotions is used to identify the two kinds of trajectories.The power spectrum of regular motion can be distinguishedbecause it generally has a small number of frequency com-ponents produced by the Fourier transform. The irregulartrajectories are not conditionally periodic and they do notcompose an invariant tori. In this way, the Fourier transformof an orbital element, for example, is not a sum over Dirac𝛿-functions, and consequently the power spectrum is notdiscrete for irregular motions [28, 33].

Observing the time behavior of the orbital elements ofthe objects 325, 546, 2986, and 4855 in Figures 3, 4, 5, and 6,one can verify possible regular and irregular motions in thetrajectories of these space debris. The time behavior of theinclination shows irregularities. Analyzing object 325, notethat, in 200 days, a fast increase in the inclination occurs,but, this variation is about 0.01∘ and it may be related to somedisturbance added to the motion.

The power spectrumof the orbital elements is not discreteand they have a big number of frequency components inFigures 3 to 6. So, it is also important to observe if the orbitalmotion is influenced by different resonant angles.

Table 5 shows the resonant angles related to the orbitalmotions of objects 325, 546, 2986, and 4855 corresponding tothe period January–September, 2011. Observing Table 5, onecan verify that the orbital dynamics of objects 2986 and 4855are influenced by some resonant angles, while for the objects


0

20000

40000

60000

80000

100000

0 50 100 150 200 250

Reso

nant

per

iod

(day

s)


d𝜙2 14 −1/dt

d𝜙3 14 −2/dtd𝜙4 14 −3/dt

d𝜙4 28 −2/dtd𝜙5 28 −3/dt

d𝜙6 28 −4/dt

d𝜙7 28 −5/dtd𝜙6 42 −3/dt

d𝜙7 42 −4/dt

d𝜙8 42 −5/dt

d𝜙1 14 0/dt

Figure 7: Time behavior of the resonant period corresponding tothe orbital motion of object 325.

10000

20000

30000

40000

50000

60000

70000

80000

0 10 20 30 40 50 60 70

Reso

nant

per

iod

(day

s)


d𝜙4 43 −1/dt

d𝜙5 43 −2/dt

d𝜙6 43 −3/dt

d𝜙7 43 −4/dt

d𝜙8 43 −5/dt

d𝜙3 43 0/dt


325 and 546, several resonant angles influence their orbitssimultaneously.

If the commensurability between the orbital motions ofthe object and the earth is defined by the parameter 𝛼 andby the condition 𝛼 = (𝑘 + 𝑞)/𝑚, one can say that the exact14:1 resonance is defined by the condition 𝛼 = 1/14. This way,by analyzing Table 5, it is verified that the motions of objects325 and 2986 are influenced by the exact 14:1 resonance,𝛼 = 1/14, while the objects 546 and 4855 are influenced byresonant angles in the neighborhood of the exact resonance.

10000

20000

30000

40000

50000

60000

70000

80000

0 50 100 150 200 250

Reso

nant

per

iod

(day

s)


d𝜙0 14 1/dt

d𝜙−1 28 3/dt

d𝜙−1 42 4/dt


10000

20000

30000

40000

50000

60000

70000

80000

0 50 100 150 200 250

Reso

nant

per

iod

(day

s)


d𝜙7 41 −4/dt

d𝜙8 41 −5/dt


The condition 𝛼 for object 546 is 𝛼 = 3/43 and for object 4855is 𝛼 = 3/41.

Figures 7, 8, 9, and 10 show the time behavior ofthe resonant period corresponding to the resonant anglespresented in Table 5.

Analyzing the time behavior of the resonant period inFigures 7, 8, 9, and 10, it is verified that the resonant anglesremain confined for a few days.The term confinedmeans thatthe orbital motion is inside a region delimited for resonantangles with biggest resonant periods. The orbital motion ofobject 546 has resonant angles which satisfy the establishedcriterium 𝑃res > 300 days for 70 days and, consequently, theorbital dynamics of this object may be losing the influenceof the 14:1 resonance. Figures 9 and 10, corresponding to the


0

0.005

0.01

0.015

0.02

0.025

0 50 100 150 200 250 300 350Time (days), January–September, 2011

−0.005

−0.01

−0.015

−0.02

−0.025

d𝜙kmq/dt

(rad

/day

)

d𝜙6 42 −3/dt

d𝜙4 28 −2/dt

d𝜙2 14 −1/dt

d𝜙7 42 −4/dt

d𝜙5 28 −3/dt

d𝜙3 14 −2/dt

d𝜙4 14 −3/dt

d𝜙8 42 −5/dt

d𝜙6 28 −4/dt

d𝜙7 28 −5/dt

d𝜙1 14 0/dt

Figure 11: Time behavior of 𝜙𝑘𝑚𝑞 corresponding to the orbitalmotion of object 325.

0


0.005

0.01

0.015

0.02

0.025

−0.005

−0.01

−0.015

−0.02

−0.025

d𝜙kmq/dt

(rad

/day

)

d𝜙4 43 −1/dt

d𝜙5 43 −2/dt

d𝜙6 43 −3/dt

d𝜙7 43 −4/dt

d𝜙8 43 −5/dt

d𝜙3 43 0/dt


objects 2986 and 4855, show a tendency to not remain inresonance, because the resonant period decreases fromvaluesgreater than 10000 days to values smaller than 500 days. Theorbital motion of object 325 is more influenced by the 14:1resonance and this fact can be observed by the time behaviorof 𝜙𝑘𝑚𝑞, Figure 11.

To continue the analysis about the irregular orbitalmotions, the time behavior of the 𝜙𝑘𝑚𝑞 is studied verifyingif different resonant angles describe the orbital dynamics ofthese objects at the same moment. This fact can indicate

0

0.01

0.02

0.03


−0.01

−0.02

d𝜙kmq/dt

(rad

/day

)

d𝜙0 14 1/dt

d𝜙−1 28 3/dt

d𝜙−1 42 4/dt


0 50 100 150 200 250 300

0

0.005

0.01

0.015

0.02

0.025

−0.005

−0.01

−0.015

−0.02

−0.025


d𝜙kmq/dt

(rad

/day

)

d𝜙7 41 −4/dt

d𝜙8 41 −5/dt


irregular orbits requiring a full systemwith all resonant anglesas described in Table 5. Figures 11, 12, 13, and 14 show the timebehavior of the 𝜙𝑘𝑚𝑞 corresponding to the resonant anglespresented in Table 5.

Analyzing the time behavior of the resonant angles 𝜙𝑘𝑚𝑞in Figures 11, 12, 13, and 14, it is possible to observe resonantangles in the exact 14:1 resonance: 𝜙1 14 0, 𝜙6 42 −3, 𝜙4 28 −2,𝜙2 14 −1, 𝜙7 42 −4, 𝜙5 28 −3, 𝜙3 14 −2, 𝜙8 42 −5, 𝜙6 28 −4,and 𝜙4 14 −3 and𝜙7 28 −5 (object 325), 𝜙−1 28 3, and 𝜙−1 42 4 (object 2986) andresonant angles in the neighborhood of the 14:1 resonance:𝜙3 43 0, 𝜙4 43 −1, 𝜙5 43 −2, 𝜙6 43 −3, and 𝜙7 43 −4 and 𝜙8 43 −5 (object546), 𝜙7 41 −4, and 𝜙8 41 −5 (object 4855). In Figure 11, one canverify that all resonant angles have the same 𝛼, but differentvalues for (𝑘+𝑞) and𝑚 appear in the conditions𝛼 = (𝑘+𝑞)/𝑚,


Table 5: Resonant angles 𝜙𝑘𝑚𝑞 related to the orbital motions ofobjects 325, 546, 2986, and 4855.

Object Coefficient 𝑘 Coefficient𝑚 Coefficient 𝑞325 1 14 0325 6 42 −3325 4 28 −2325 2 14 −1325 7 42 −4325 5 28 −3325 3 14 −2325 8 42 −5325 6 28 −4325 4 14 −3325 7 28 −5546 3 43 0546 4 43 −1546 5 43 −2546 6 43 −3546 7 43 −4546 8 43 −52986 −1 28 32986 −1 42 42986 0 14 14855 7 41 −44855 8 41 −5

𝛼 = 1/14, 𝛼 = 2/28, and 𝛼 = 3/42. The time behaviorof curves with 𝛼 = 2/28 are similar and different of curveswith 𝛼 = 3/42 or 𝛼 = 1/14. The orbital motion of object546 is governed by distinct resonant angles, but, after seventydays, the influence of resonant effects decreases or disappears.Objects 325 and 2986 have their orbital motions influencedby the exact 14:1 resonance and they need a full systemwith different resonant angles which compose their motions.Otherwise, the orbital motion of object 4855 is defined forresonant angles separately, which can be defined as a regularorbit.

The results and discussions show the complexity in theorbital dynamics of these objects caused by the resonanceeffects. Furthermore, the increasing number of space debrisand collisions between them can cause big problems forartificial satellites missions.

4. Conclusions

In this work, the orbital dynamics of synchronous spacedebris are studied. From the TLE data of the NORAD,objects orbiting the earth in resonant orbital motions areinvestigated.

Results show that some objects around the exact 1:1,2:1, and 14:1 resonances remain in resonance for a longtime. In other words, the orbital motions will be influencedfor resonant angles for several years and possible irregularmotions can be confined in a region delimited for resonantangles with biggest resonant periods.

Analyzing the cataloged objects satisfying the establishedcriterium of the resonant period greater than 300 days, fourspace debris (325, 546, 2986, and 4855) are studied observingthe irregular characteristics in their orbits.The time behaviorof the orbital elements and the respective frequency analysisare used to verify the orbits of these objects.

Several resonant angles influence, simultaneously, theorbital motions of objects 325 and 2986. The time behaviorof the inclination and the frequency analysis confirm theirregularities in their orbits. The orbital dynamics of object546 is firstly influenced by several resonant angles in theneighborhood of the 14:1 resonance, but, after some days, theregular characteristic in its orbit seems to dominate.The timebehavior of the 𝜙𝑘𝑚𝑞 of object 4855 shows resonant anglesseparately indicating a regular orbit.

The resonant orbital dynamics of artificial satellites andspace debris can be described using observational data. Res-onant angles which compose the orbital motions are definedbefore the development of analytical model, providing moreaccuracy. Using the same procedure shown in the presentwork, the resonant periods associated with resonant anglescan be determined and the study about any resonance,involving the commensurability between themeanmotion ofobject and the earth’s rotation motion, can be described withmore details.

Using real data, the time behavior of orbital elementsallows to validate by comparison several analytical modelsdescribing perturbations in orbital motions, like atmosphericdrag, solar radiation pressure, gravitational effects of the sunand the moon, for example. The frequency analysis fromobservational data is a new way to verify unstable regions.

In a future work, the method presented in this paperwill be used to determine all resonant angles that influ-ence a determined orbital motion and the analytical modelpresented in a previous paper [19]; it will be necessaryto propagate the orbit, using each resonant angle and thecorrespondent resonant period previously determined. Thedescription of the resonance problem is tending to a moreunderstandable way.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was accomplished with support of the FAPESPunder the Contracts nos. 2012/24369-0 and 2012/21023-6, SP,Brazil, CNPQ (Contract 303070/2011-0), and CAPES.

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