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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 260830 11 pageshttpdxdoiorg1011552013260830

Research ArticleA Study of Single- and Double-Averaged Second-OrderModels to Evaluate Third-Body Perturbation ConsideringElliptic Orbits for the Perturbing Body

R C Domingos1 A F Bertachini de Almeida Prado1 and R Vilhena de Moraes2

1 Instituto Nacional de Pesquisas Espaciais (INPE) 12227-010 Sao Jose dos Campos SP Brazil2 Universidade Federal de Sao Paulo (UNIFESP) 12231-280 Sao Jose dos Campos SP Brazil

Correspondence should be addressed to A F Bertachini de Almeida Prado bertachinipradobolcombr

Received 6 November 2012 Accepted 29 April 2013

Academic Editor Maria Zanardi

Copyright copy 2013 R C Domingos et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The equations for the variations of the Keplerian elements of the orbit of a spacecraft perturbed by a third body are developedusing a single average over the motion of the spacecraft considering an elliptic orbit for the disturbing body A comparison ismade between this approach and the more used double averaged technique as well as with the full elliptic restricted three-bodyproblem The disturbing function is expanded in Legendre polynomials up to the second order in both cases The equations ofmotion are obtained from the planetary equations and several numerical simulations are made to show the evolution of the orbitof the spacecraft Some characteristics known from the circular perturbing body are studied circular elliptic equatorial and frozenorbits Different initial eccentricities for the perturbed body are considered since the effect of this variable is one of the goals ofthe present study The results show the impact of this parameter as well as the differences between both models compared to thefull elliptic restricted three-body problem Regions below near and above the critical angle of the third-body perturbation areconsidered as well as different altitudes for the orbit of the spacecraft

1 Introduction

Most of the papers on this topic consider the third-body per-turbation due to the Sun and due to the Moon in a satellitearound the Earth This is the most immediate applicationof the third-body perturbation One of the first studies wasmade by Kozai [1] that developed the most important long-period and secular terms of the perturbing potential of thelunisolar perturbations written as a function of the orbitalelements of the Sun the Moon and the satellite Moe [2]Musen [3] and Cook [4] studied long-period effects of theSun and the Moon on artificial satellites of the Earth Theyapplied Lagrangersquos planetary equations to study the variationof the orbital elements of the satellite and its rate of variationThis idea was expanded by Musen et al [5] that included theparallactic term in the perturbing potential Kozai [6] studiedthe secular perturbations in asteroids that are in orbits withhigh inclination and eccentricity assuming that the bodiesare perturbed by Jupiter Blitzer [7] made estimates for the

lunisolar perturbation for the secular terms Around the sametime Kaula [8] obtained the disturbing function consideringthe perturbations of the Sun and the Moon

Later Giacaglia [9] calculated the disturbing function dueto the Moon using equatorial elements for the satellite andecliptic elements for the Moon All terms were expressed inclosed forms Kozai [10] worked again on that problem andexpressed the perturbing function as a function of the polargeocentric coordinates of the Moon the Sun and the orbitalelements of the satelliteThe short-period terms are shown inan analytical form and the secular and long-period terms areobtained by numerical integration

Hough [11] considered the effects of the perturbation ofthe Sun and the Moon in orbits near the inclinations of 634∘and 1166∘ (critical inclinations when considering the geopo-tential of the Earth) and showed that those effects areimportant only in high altitudes Ash [12] also studied thisproblem using the double-averaged technique for a high-alti-tude satellite around the Earth Collins and Cefola [13] used

2 Mathematical Problems in Engineering

the same technique for the prediction of orbits for a long timespan

In the last years several researches based on both ana-lytical and numerical approaches studied the third-bodyperturbationThemajority of them concentrated on studyingthe effects of a perturbation caused by a third body usinga double-averaged technique like those of Sidlichovskyy [14]Kwok [15 16] Broucke [17] and Prado [18] The disturbingfunction is expanded in Legendrersquos polynomials and theaverage is made over the periods of the perturbing and theperturbed body using an approximated model expanded tosome order Some others studied this problem based on asingle-averaged analytical model where the disturbing func-tion is expanded in Legendrersquos polynomials but the averageis made over the short period of the perturbed body onlysee for example [19] and references therein In all of theresearches cited before the perturbing body was in a circularorbit There are also several papers considering the third-body perturbation in constellations of satellites (see [20 21])and in other planetary systems not involving the Earthlike those of Carvalho (see [22ndash25]) Folta and Quinn [26]Kinochita and Nakai [27] Lara [28] Paskowitz and Scheers(see [29 30]) Russel and Brinckerhoff [31] and Scheers etal [32] In particular Roscoe et al [33] used the analyticalequations derived in Prado [20] to study the dynamics of aconstellation of satellites perturbed by a third body

Using the double-averaged analyticalmodel inDomingoset al [34] the analytical expansion to study the third-bodyperturbation was extended to the case where the perturbingbody is in an elliptical orbit Liu et al [35] included the incli-nation of the perturbing bodyrsquos equatorial plane with respectto its orbital plane

The main idea behind the single-averaged technique isthat it eliminates only the terms due to the short-time peri-odic motion of the perturbed body The results are expectedto show smooth curves for the evolution of the mean orbitalelements for a long-time period when compared with thefull restricted three-body problem In other words a betterunderstanding of the physical phenomenon can be obtainedand it allows the study of long-term stability of the orbits inthe presence of disturbances that cause slow changes in theorbital elements

So an interesting point would be a study to show thedifferences between those two averaged models when com-pared with the full elliptic restricted three-body problemThe idea of the present paper is to study this problem butassuming that the perturbing body is in an elliptical orbitStudies under this assumption are available in Domingos etal (see [34 36 37]) In particular in the present paper a studyof the effects of the eccentricity of the perturbing body inthe orbit of the perturbed body is made for several differentconditions regarding the orbit of the primaries as well as theorbit of the spacecraft

So our goal is to make a more complete study of thisproblem and to perform some tests to verify the differencesin the results obtained by those two approximated techniquesA comparative investigation is made to verify the differencesbetween the single-averaged analytical model with the model

based on the double-averaged technique as well as a compar-ison with the full restricted elliptic three-body problem thatcan provide some insights about their applications in celestialmechanics

The assumptions used to develop the single-averagedanalytical model are the same ones of the restricted ellipticthree-body problem (planet-satellite-spacecraft) Our anal-ysis for the evolution of the mean orbital elements will bebased only on gravitational forces The equations of motionare obtained from Lagrangersquos planetary equations and thenwe numerically integrated those equations Different initialeccentricities for the perturbing body are considered

The set of results obtained in this research performs ananalysis of several well-known characteristics of the third-body perturbation like (i) the stability of near-circular orbitsto investigate under which conditions this orbit remains nearcircular (ii) the behavior of elliptic equatorial orbits and (iii)the existence of frozen orbits which are orbits that keep theeccentricity inclination and argument of periapsis constants

A detailed study considering the ldquocritical anglerdquo of thethird-body perturbation which is an inclination that makesa near circular orbit that has an inclination below this valueto remain near-circular is made This work is structured asfollows In Section 2 we present the equations of motionused for the numerical simulations Section 3 is devoted tothe analysis of the numerical results for near circular orbitsThe theory developed here is used to study the behaviorof a spacecraft around the Earth where the Moon is thedisturbing body The choice of the Moon as the disturbingbody is made based on the fact that its effect on the spacecraftis larger than the effect of the Sun Several plots show the timehistories of the Keplerian elements of the orbits involved Ourconclusions are presented in Section 4

2 Dynamical Model

For the determination of the equations of motion we startedby assuming the existence of a central body with mass 119898

0

that is fixed in the center of the reference system 119909-119910-119911 Theperturbing body with mass 1198981015840 is in an elliptic orbit withsemimajor axis 1198861015840 eccentricity 1198901015840 inclination 119894

1015840 argumentof periapsis 1205961015840 longitude of the ascending node Ω1015840 andmean motion 1198991015840 The spacecraft with negligible mass 119898 is ina generic orbit with orbital elements 119886 (semimajor axis) 119890(eccentricity) 119894 (inclination) 120596 (argument of periapsis) Ω(longitude of the ascending node) and it has mean motion119899 This system is shown in Figure 1

Using the traditional expansion in Legendrersquos polynomi-als (assuming that 1199031015840 ≫ 119903) the second-order term of thedisturbing potential averaged over the eccentric anomaly ofthe spacecraft is given by the following (see [17 18 34])

⟨1198772⟩ =

1205831015840

1198862

11989910158402

2

(

1198861015840

1199031015840

)

3

(1 +

3

2

1198902

) [

3

2

(1205722

+ 1205732

) minus 1]

+

15

4

(1205722

minus 1205732

) 1198902

(1)

Mathematical Problems in Engineering 3

Z

Y

X

m0

m

r

r998400

m998400

Figure 1 Illustration of the dynamical system

where (Murray and Dermott [38])

1199031015840

=

1198861015840

(1 minus 11989010158402

)

1 + 1198901015840 cos1198911015840

cos1198911015840 = cos1198721015840 + 1198901015840 (cos 21198721015840 minus 1)

+

9

8

11989010158402

(cos 31198721015840 minus cos1198721015840) + sdot sdot sdot

sin1198911015840 = sin1198721015840 + 1198901015840 sin 21198721015840

+ 11989010158402

(

9

8

sin 31198721015840 minus 78

sin1198721015840) + sdot sdot sdot

(2)

For the case of elliptic orbits of the perturbing body theparameters 120572 and 120573 are written as follows (see [34])

120572 = cos120596 cos119863 minus cos 119894 sin120596 sin119863

120573 = minus sin120596 cos119863 minus cos 119894 cos120596 sin119863(3)

where119863 = Ω minus 1198911015840

minus 1205961015840

The mean anomaly1198721015840 of the perturbing body is given as

1198721015840

= 1198721015840

119900

+ 1198991015840

119905 (4)

Thus the variations in the orbital elements of the per-turbed body are obtained To do this we derived Lagrangersquosplanetary equations that describe the variations of the meanorbital elements of the spacecraft The semimajor axis isconstant since themean anomaly119872was eliminated from theperturbing functionThose equations can bewritten as shownbelow

119889119890

119889119905

= 119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899

times [sin 2120596 (cos2119863 minus cos2119894sin2119863)

minus cos 119894 cos 2120596 sin 2119863]

119889119894

119889119905

= 119870

1

sin 119894[120583119886 (1 minus 1198902)]123

4

1205831015840

11989910158402

1198862

times 1198902

[ minus 5 cos 2120596 sin 2119863cos2119894

+

3

2

(minus sin 2119863 + sin 2119863cos2119894)

minus

5

2

sin119863 cos 2120596 (1 + cos2119894)]

+ (sin119863 minus sin 2119863cos2119894)

minus5 cos 2119863 sin 2120596 cos 119894

119889Ω

119889119905

= 119870

3

4

1205831015840

11989910158402 119886

2

sin 119894[120583119886 (1 minus 1198902)]12

times (1 +

3

2

1198902

) (minussin2119863 sin 2119894)

+ [cos 2119894 cos 2120596 sin2119863

+ sin 119894 sin 2119863 sin 2120596] (52

1198902

)

119889120596

119889119905

= 119870

3

4

1205831015840

11989910158402

minus

1198862

sin 119894[120583119886 (1 minus 1198902)]12

times [(1 +

3

2

1198902

) (minussin2119863 sin 2119894)

+ (

5

2

1198902

) (cos 2119894sin2119863 cos 2120596)

+

1

2

sin 2119894 sin 2119863 sin 2120596] +radic1 minus 119890

2

119899

times [5 (cos2119863 minus cos2119894 sin2119863) cos 2120596

minus 5 cos 119894 sin 2119863 sin 2120596

+

6

2

(cos2119863 + cos2119894 sin2119863)]

(5)

where

119870 =

1 + 31198901015840 cos1198911015840 + 311989010158402cos21198911015840

1 minus 61198901015840

+ 1511989010158402

(6)

Those equations show some characteristics of this systemcompared with similar researches (see [18 34]) which arelisted as follows

(i) Circular orbits exist but they are not planar It isimmediate from the inspections of the time deriva-tives of the Keplerian elements If the initial eccentric-ity is zero then the time derivative of the eccentricityis also zero and the orbit remains circular but the time


derivative of the inclination is not zero so the orbitdoes not stay planar

(ii) Elliptic equatorial orbits are not stableThe equatorialcase (119894 = 0) has a singularity in the equation for thetime derivative of the inclination so this model isnot able to make prediction for the behavior of theinclination Regarding the eccentricity it is visible thatthis case has a nonzero value for the time derivativeof the eccentricity so those orbits do not keep theeccentricity constant The only exception is when theinitial orbit is circular as shown above where theeccentricity remains zero In fact the equation for thetime derivative of the eccentricity for equatorial orbitscan be simplified to the following

119889119890

119889119905

= minus119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899

sin 2 (119863 minus 120596) (7)

(iii) Those facts explained above also show that there areno frozen orbits under this analytical model whichwould be orbitswhere the time derivatives of the incli-nation eccentricity and argument of periapsis are allzero

Domingos et al (see [34]) made a similar study but usingthe double-averaged technique and showed that circular andequatorial orbits exist in the double-averaged model sinceorbits with zero initial inclination or zero initial eccentricityremainwith zero eccentricity and zero inclinationThe differ-ence in the equations ofmotionwhen considering the circularand the elliptic motion of the primaries under the double-averaged models is the existence of the term (1 + (32)119890

10158402

+

(158)11989010158404

) which is an extra term that depends on theeccentricity of the primaries 1198901015840 and this term does not destroythose properties The same is true for the frozen orbits sincethe appearance of a multiplicative nonzero constant does notchange this property

3 Numerical Results

In this section we show the effects of the nonzero eccen-tricities of the perturbing body in the orbit of the perturbedbodyWe numerically investigate the evolutions of the eccen-tricity and the inclination for a spacecraft within the ellipticrestricted three-body problem Earth-Moon-spacecraft aswell as using the single- and the double-averaged modelsup to the second order The spacecraft is in an ellipticthree-dimensional orbit around the Earth and its motion isperturbed by the Moon To justify our study and to makeit a representative sample of the range of possibilities weused three values for the semimajor axis of the spacecraft8000 km 26000 km and 42000 km They represent a Low-Altitude Earth Orbit aMedium-Altitude Earth Orbit (the oneused by the GPS satellites) and a High-Altitude Earth Orbit(the one used by geosynchronous satellites) respectivelyThespacecraft is assumed to be in an orbit with the followinginitial Keplerian elements 119890

0= 001 120596

0= Ω0= 0

We focus our attention On the stability of near-circularorbits Results of numerical integrations showed that this

stability depends on 1198940(the initial inclination between the

perturbed and perturbing bodies) When 1198940is above the

critical value (119894119888) the orbit becomes very elliptic If 119894

0lt 119894119888

then the orbit remains near circular (see [17 18 34]) Thevalue of 119894

119888is 0684719203 radians or 3923152048 degrees

Thus for our numerical integrations the initial inclina-tions for the orbit of the spacecraft received values near thecritical inclination (in the interval 119894

0= 38∘ and 41∘) below this

critical value (1198940= 30∘ 20∘ 10∘ and zero) and then above this

critical value (1198940= 40∘ 50 119900 60∘ 70∘ and 80∘)The eccentricity

of the perturbing body is assumed to be in the range 0 le

1198901015840

le 05 in general but some results are omitted when theapproximations are found to be very poor It means that wegeneralized the Earth-Moon system in order to measure theeffects of the eccentricity of the orbit of the primaries

All of the cases considering the Low-Altitude Earth Orbitwere simulated for 80000 canonical units of time whichcorrespond to 12800 orbits of the disturbing body This timeshows the characteristics of the system since the third-bodyperturbations are weaker at 119886

0= 8000 km The simulations

for the other values of the semimajor axis were made onlyfor 20000 canonical units of time which correspond to 3200orbits of the disturbing body because it appeared to be a goodnumber to evidence the characteristics of the system

Our numerical results are summarized in Figures 2ndash6Such figures refer to the orbital evolution of the eccentricityand inclination as a function of time for the initial inclinationscited above and a direct comparison between the models

31 Low-Inclination Orbits Figure 2 shows the results of theevolution of the eccentricity as a function of time for the casewhere the initial inclination assumes the values 119894

0= 30∘ (red

line) 20∘(blue line) 10∘ (pink line) and zero (orange line)The first second and third columns show the results for thecases ofLow-Medium- andHigh-altitude orbits respectivelyFigure 2 uses a solid line to represent the full elliptic restrictedthree-body problem a dashed line to represent the single-averaged model and a dashed-dotted line to represent thedouble-averaged model Regarding the overall behavior theresults are according to the expected from the literature (see[17 18]) and the increase of the initial inclination causesoscillationswith larger amplitudes in the eccentricity It is alsoclear that the averaged models are very accurate for the casewhere the perturbing body is in a circular orbitThis accuracydecreaseswith the increase of the eccentricity of the primariesfor all values of the initial inclinations

The use of second-order averaged models is not recom-mended for eccentricities of the primaries of 03 or larger Alarger expansion is required in this situation In general bothaveraged models have results that are much closer to eachother than closer to those of the full model It means that bothaverages tend to give the same errors at least for eccentricitiesof the primaries up to 02 For eccentricity of 03 and abovethe averaged models begin to show different behaviors andthose differences increase with this eccentricity Regardingthe comparison between both averaged models the resultsdepend on the value of the initial inclination

The increase of the semimajor axis makes the spacecraftto stay more time at shorter distances of the Moon so the


0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

0005

001

0015

002

Ecce

ntric

ity

e998400= 02

e998400= 03

e998400= 0

e998400= 01

Time (canonical units)

(a)

0

001

002

003

004

005

0

001

002

003

004

005

0

001

002

003

004

005

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0

001

002

003

004

005


e998400= 02

e998400= 03

e998400= 0

e998400= 01

(b)

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(c)

Figure 2 These figures show the evolution of the eccentricity as a function of time for low-inclination orbits On the top of each figure isshown the corresponding eccentricity of the perturbing body (1198901015840) In the first column the results are for the case of Low-altitude orbits Thesecond column is for the case ofMedium-altitude orbits The third column is for the case of High-altitude orbits The lines are as follows fullelliptic restricted three-body problem (solid line) and single-averaged (dashed line) and double-averaged models (dashed-dotted line) Forthe colors 119894

0

= 30∘ (red line) 20∘ (blue line) 10∘ (pink line) and zero (orange line) The increase of the semimajor axis and the eccentricity

of the primaries reduce the average distance from the spacecraft to the Moon making the perturbations stronger

third-body perturbations are stronger This fact acceleratesthe dynamics of the system and the period of the oscillationof the eccentricity is shorter when compared with the low-altitude orbits It is visible that high-altitude orbits haveoscillations of eccentricity increased in amplitude with the

increase of the eccentricity of the primaries and the averagedmodels are not able to predict this propertyThey have resultsthat decrease in quality with the increase of the eccentricityof the primaries The amplitude of the oscillations increasedabout ten times with respect to the previous lower orbits


066

068

07

072In

clina

tion

(rad

)

064

066

068

07

072

Incli

natio

n (r

ad)


0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

064

066

068

07

072

Incli

natio

n (r

ad)

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

8000

080

000

8000

0

Double

(a)

066

067

068

069

07

071

072

066

067

068

069

07

071

072

066

067

068

069

07

071

072

0

5000

1000

0

1500

0


40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0

2000

020

000

2000

0

Double

(b)

0

5000

1000

0

1500

0

066

068

07

072

066

068

07

072

064

066

068

07

072

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0


2000

020

000

2000

0

Double

(c)

Figure 3 These figures show the evolution of the inclination as a function of time for near critical inclinations On the top of each figure isshown the corresponding eccentricity of the perturbing body (1198901015840) In the first column the results are for the case of Low-altitude orbits Thesecond column is for the case ofMedium-altitude orbits The third column is for the case of High-altitude orbits The lines are as follows fullelliptic restricted three-body problem (solid line) and single-averaged (dashed line) and double-averaged models (dashed-dotted line) Forthe colors 119894

0

= 41∘ (red line) 40∘ (orange line) 39∘ (blue line) and 38∘ (pink line) The increase of the semimajor axis and the eccentricity of

the primaries reduce the average distance from the spacecraft to the Moon making the perturbations stronger

due to the increase of the perturbation effects The periodof the oscillations is also reduced as an effect of the increaseof the perturbations The increase of the eccentricity of theprimaries has the same effect of increasing the amplitudesand reducing the period of oscillations The evolutions of theinclinations are not shown here because they remain constantfor all of the situations considered in Figure 2

32 Near Critical Inclination Orbits Figures 3 and 4 show theresults of the evolutions of the inclination and the eccentricity

respectively as a function of time for initial inclinations 1198940=

41∘ (red line) 40∘ (orange line) 39∘ (blue line) and 38∘ (pink

line) The same representations for the models are used hereThe general behaviors are also according to the expected fromthe same literature cited above and the increase of the initialinclination causes oscillations with larger amplitudes Bothaveraged models have results that have good accuracy forvalues of 1198901015840 up to 02 For eccentricities of 03 and above theaveraged models start to have results that are not so goodand those deviations from the real cases increase with the


0

001

002

003

004

005

006Ec

cent

ricity

0

001

002

003

004

005

006

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

001

002

003

004

005

006

Ecce

ntric

ity

0 20000 40000 60000 80000Time (canonical units)

(a)

00

5000 10000 15000 20000

003

006

009

012

015

00

5000 10000 15000 20000

003

006

009

012

0150

05000 10000 15000 20000

003

006

009

012

015


(b)

00

5000 10000 15000 20000

005

01

015

02

025

03

0 5000 10000 15000 200000

005

01

015

02

025

03

00

5000 10000 15000 20000

005

01

015

02

025

03


(c)

Figure 4 These figures show the evolution of the eccentricity as a function of time for near critical inclination spacecraft orbits The firstsecond and third rows correspond to the results for eccentricity of the perturbing body (1198901015840) 00 01 and 02 In the first column the resultsare for the case of Low-altitude orbits The second column is for the case of Medium-altitude orbits The third column is for the case ofHigh-altitude orbits The lines are as follows full elliptic restricted three-body problem (solid line) and single-averaged (dashed line) anddouble-averaged models (dashed-dotted line) For the colors 119894

0

= 41∘ (red line) 40∘ (orange line) 39∘ (blue line) and 38∘ (pink line) The

increase of the semimajor axis and the eccentricity of the primaries reduce the average distance from the spacecraft to the Moon making theperturbations stronger

eccentricity of the primaries so they are omitted here It isclear that for values of the eccentricity of the primaries up to01 both averaged models are very accurate When 1198901015840 = 02the single-averaged model gives more accurate results for theevolution of the eccentricity of the perturbed body whilethe double-averaged model gives more accurate results forthe evolution of the inclination The increase of 1198901015840 also doesnot change the dynamics of the system in a noticeable formfor low- andmedium-altitude orbits For high-altitude orbitsthere is a noticeable acceleration of the dynamics

It can be noticed that near the time 20000 canonicalunits medium-altitude orbits have presented some devia-tions between the results of the full and the averaged modelsin particular for the higher inclinationsThis is a result of theincrease of the semimajor axis from low to medium orbitswhich places the spacecraft in an orbit that is much moreperturbed by the third body Again those models begin togive results that are also not so accurate when the eccentricityof the primaries increases It is visible that the two-second

order averaged models have results that are very closer toeach other with similar deviations from the full model So inthis range of inclinations and altitudes both averagedmodelshave about the same quality of results The increases of thesemimajor axis accelerate the dynamics also in this situationNote that the time scale of the axis is different for low-altitudeearth orbits and the deviations of the averaged models occurvery early

The third columns of Figures 3 and 4 show that at higheraltitude both averaged models do not fit perfectly even forthe circular case for the perturbing body (1198901015840 = 0) It canbe noticed that near the time 8000 canonical units somedeviations between the full and the averaged models occurin particular for the higher inclinationsThis is a result of theincrease of the semimajor axis which places the spacecraftin an orbit that is much more perturbed by the third bodybecause it is closer to theMoonThe period of the oscillationsis also reduced by this stronger perturbation Again thosemodels begin to give results that are also not so accurate when


0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)

Figure 5 These figures show the evolution of the inclination as a function of time for high-inclination spacecraft orbit On the top of eachfigure is shown the corresponding eccentricity of the perturbing body (1198901015840) In the first column the results are for the case of Low-altitudeorbits The second column is for the case ofMedium-altitude orbits The third column is for the case of High-altitude orbits The lines are asfollows full elliptic restricted three-body problem (solid line) single-averaged (dashed line) and double-averaged models (dashed-dottedline) For the colors 119894

0

= 80∘ (red line) 70∘ (blue line) 60∘ (green line) 50∘ (pink line) and 40∘ (orange line) The increase of the semimajor

axis and the eccentricity of the primaries reduce the average distance from the spacecraft to the Moon making the perturbations stronger

the eccentricity of the primaries increases not predictingcorrectly the oscillations performed by the fullmodel At highaltitudes both averaged approximations begin to lose qualityfor 1198901015840 = 02 It is visible that the two second-order averagedmodels have results that are very closer to each other withsimilar deviations from the full model So in this range ofinclinations and altitudes both averaged models have aboutthe same quality of results for values of the eccentricity of theprimaries below 02

33 Above Critical Inclination Orbits Figures 5 and 6 showthe evolutions of the inclinations and the eccentricities asa function of time for initial inclinations above the criti-cal value The general behaviors are the expected ones (see[17 18]) and the inclinations remain constant for a longtime return very fast to the critical value and then increaseagain to its original valuesThe eccentricity shows an oppositebehavior and increases when the inclination decreases andvice versa The increase of the initial inclinations also causesoscillations with larger amplitudes Both averaged models

have results that are excellent for 1198901015840 = 00 and they still havegood accuracy for eccentricities of the primaries up to 02

The single-averaged model for low-altitude orbits in anaverage over the time gives results that are closer to those ofthe full model This fact is very clear for eccentricities of theprimaries 02 It is also noted that for 1198901015840 = 04 the valuesgiven by the full model for the inclination and eccentricitylie in the middle of the curves described by the averagedmodels showing a decrease to the critical value later than thedecrease predicted by the single-averaged model but beforethe predictions made by the double-averaged model Thisfact is due to the truncation of the expansion The resultsshow that the effect of this truncation depends on the initialconditions of the simulations so the effects of the neglectedterms are different for different orbits For this eccentricityof primaries (1198901015840 = 04) the second-order averaged modelsare no longer accurate enough to study this problem Theaccuracy of the averaged models decreases with the increaseof the initial inclination Note that the averaged models arestill good for 1198901015840 = 04 in the case 119894

0= 40∘ but this is not so


0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

Time (canonical units) Full SingleDouble

(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)

Figure 6These figures show the evolution of the eccentricity as a function of time for high-inclination orbitsThe first second and third rowscorrespond to the results for eccentricity of the perturbing body (1198901015840) 00 02 and 04 In the first column the results are for the case of Low-altitude orbits The second column is for the case ofMedium-altitude orbits The third column is for the case ofHigh-altitude orbits The linesare as follows full elliptic restricted three-body problem (solid line) single-averaged (dashed line) and double-averaged models (dashed-dotted line) For the colors 119894

0

= 80∘ (red line) 70∘ (blue line) 60∘ (green line) 50∘ (pink line) and 40∘ (orange line) The increase of the

semimajor axis and the eccentricity of the primaries reduce the average distance from the spacecraft to the Moon making the perturbationsstronger

good for higher values of the initial inclinations with theworst results happening for 119894

0= 80∘ Another result that

comes from those figures is that the increase of 1198901015840 causes anacceleration of the dynamics of the system Note that the timerequired by the inclination and eccentricity to suffer the large-amplitude oscillation becomes shorterwhen the primaries arein a more elliptic orbit

For medium altitudes when 1198901015840 = 04 the accuracy ofthe averaged models increases with the increase of the initialinclinations Note that the averaged models are still good forthe case of initial inclination 80∘ and the situation with theworst results occurs for 60∘ Inclined orbits are less perturbedby a third body because the mean distance between thespacecraft and the Moon is larger than in planar orbits

Note that the strong changes in inclination and eccen-tricity occur earlier as a result of the stronger perturbationsresulting from the increase of the semimajor axis of the orbit

The increase of the eccentricity of the primaries has the sameeffect showing that an elliptic orbit for the perturbing bodycauses more perturbation on the spacecraft when comparedwith circular orbits with the same semimajor axis becausethe distance between the bodies decreases at the periapsisThe correspondent increase of that distance at the apoapsisdoes not compensate the previous aspect For values of 1198901015840 upto 02 the region where the second-order averaged modelswork better and the differences between the full model andthe averaged models are larger in higher orbits This showsthat the increase of the perturbation effects also increases thedifferences between the models

For high-altitude orbits both averaged models haveresults that are excellent for 1198901015840 = 00 and that still have goodaccuracy for eccentricities of the primaries up to 02 atleast for times up to 17000 canonical units After that theapproximations are no longer good enough to study this


problem under the second-order averaged hypotheses Ingeneral in this situation the single-averaged model givesbetter results For 1198901015840 = 01 and higher the averaged modelsdo not predict very well the time for the second jump of theinclination and eccentricity for all of the inclinations studiedThe eccentricity of the primaries still accelerates the systemreducing the period of the oscillations in the same way madeby the higher altitudes Note that at this high altitude thesystem shows two jumps for the inclination and eccentricity

4 Conclusions

The equations of motion of a spacecraft perturbed by athird body using a single-averaged technique are developedconsidering an elliptic orbit for the disturbing body

Looking at the overall behavior the results are accordingto the literature short oscillations in the inclination andeccentricity for initial inclinations below the critical valuewhich increases fast in amplitude around the critical valuethen the typical behavior of having the inclinations remainingconstant for a long time and then returning very fast to thecritical value to increase fast again to its original values Theeccentricity shows an opposite behavior and increases whenthe inclination decreases and vice versa

The results also showed that circular orbits exist butfrozen orbits do not exist under this model The circularorbits in general do not keep the inclination constant ashappened in the double-averaged model

The increase of the semimajor axis causes an increaseof the third-body perturbation and this fact accelerates thedynamics of the system It was noticed that the period of theoscillations of the inclination and the eccentricity are shorterwhen compared with lower-altitude orbits The increase ofthe eccentricity of the primaries considering the semimajoraxis constant has the same effect of accelerating the dynam-ics So elliptic orbits for the disturbing body have the effect ofincreasing the perturbation if all other elements remain thesame

It was also showed that inclined orbits are less perturbedby a third body since the mean distance between the space-craft and the Moon is larger than in planar orbits

A comparison with the double-averaged technique up tothe second order and with the full restricted elliptic problemwas made and it showed some other facts The second-orderaveraged models are very accurate when the perturbing bodyis in a circular orbitThis accuracy decreases with the increaseof the eccentricity of the primaries The use of second-orderaveragedmodels is not recommended for eccentricities of theprimaries of 03 or larger and a better expansion includingmore terms is required in this situation In general bothsecond-order averaged models have results that are muchcloser with each other than closer to those of the full modelIt means that both averages tend to give similar errors whencompared with the full model

Acknowledgments

The authors wish to express their appreciation for the supportprovided by CNPq under Contracts nos 1501952012-5 and

3047002009-6 by Fapesp (contracts nos 201109310-7 and201108171-3) and by CAPES

References

[1] Y Kozai ldquoOn the effects of the Sun andMoon upon the motionof a close Earth satelliterdquo Smithsonian Institution AstrophysicalObservatory Special Report 22 1959

[2] M M Moe ldquoSolar-lunar perturbation of the orbit of an Earthsatelliterdquo ARS Journal vol 30 no 5 pp 485ndash506 1960

[3] P Musen ldquoOn the long-period lunar and solar effects onthe motion of an artificial satellite IIrdquo Journal of GeophysicalResearch vol 66 pp 2797ndash2805 1961

[4] G E Cook ldquoLuni-solar perturbations of the orbit of an earthsatelliterdquo The Geophysical Journal of the Royal AstronomicalSociety vol 6 no 3 pp 271ndash291 1962

[5] P Musen A Bailie and E Upton ldquoDevelopment of the lunarand solar perturbations in the motion of an artificial satelliterdquoNASA-TN D494 1961

[6] Y Kozai ldquoSecular perturbations of asteroids with high inclina-tion and eccentricityrdquoThe Astrophysical Journal Letters vol 67no 9 p 591 1962

[7] L Blitzer ldquoLunar-solar perturbations of an earth satelliterdquo TheAmerican Journal of Physics vol 27 pp 634ndash645 1959

[8] W M Kaula ldquoDevelopment of the lunar and solar disturbingfunctions for a close satelliterdquoThe Astronomical Journal vol 67no 5 p 300 1962

[9] G E O Giacaglia ldquoLunar perturbations on artificial satellitesof the earthrdquo Smithsonian Astrophysical Observatory SpecialReport 352 1 1973

[10] Y Kozai ldquoA new method to compute lunisolar perturbationsin satellite motionsrdquo Smithsonian Astrophysical ObservatorySpecial Report 349 27 1973

[11] M EHough ldquoLunisolar perturbationsrdquoCelestialMechanics andDynamical Astronomy vol 25 p 111 1981

[12] M E Ash ldquoDoubly averaged effect of the Moon and Sun on ahigh altitude Earth satellite orbitrdquo Celestial Mechanics vol 14no 2 pp 209ndash238 1976

[13] S K Collins and P J Cefola ldquoDouble averaged third bodymodel for prediction of super-synchronous orbits over longtime spansrdquo Paper AIAA 64 American Institute of Aeronauticsand Astronautics 1979

[14] M Sidlichovskyy ldquoOn the double averaged three-body prob-lemrdquo Celestial Mechanics vol 29 no 3 pp 295ndash305 1983

[15] J H Kwok ldquoA doubly averaging method for third body per-turbations in planet equator coordinatesrdquo Advances in theAstronautical Sciences vol 76 pp 2223ndash2239 1991

[16] J H Kwok ldquoLong-term orbit prediction for the venus radarmapper mission using an averaging methodrdquo in Proceedingsof the AIAAAAS Astrodynamics Specialist Conference SeatleWash USA August 1985

[17] R A Broucke ldquoLong-term third-body effects via double aver-agingrdquo Journal of Guidance Control and Dynamics vol 26 no1 pp 27ndash32 2003

[18] A F B A Prado ldquoThird-body perturbation in orbits aroundnatural satellitesrdquo Journal of Guidance Control and Dynamicsvol 26 no 1 pp 33ndash40 2003

[19] C R H SolorzanoThird-body perturbation using a single aver-aged model [MS thesis] National Institute for Space Research(INPE) Sao Jose dos Campos Brazil


[20] T A Ely ldquoStable constellations of frozen elliptical inclined lunarorbitsrdquo Journal of the Astronautical Sciences vol 53 no 3 pp301ndash316 2005

[21] T A Ely and E Lieb ldquoConstellations of elliptical inclinedlunar orbits providing polar and global coveragerdquo Journal of theAstronautical Sciences vol 54 no 1 pp 53ndash67 2006

[22] J P S Carvalho A Elipe R V deMoraes and A F B A PradoldquoLow-altitude near-polar and near-circular orbits aroundEuropardquo Advances in Space Research vol 49 no 5 pp 994ndash1006 2012

[23] J P D S Carvalho R V de Moraes and A F B A PradoldquoNonsphericity of the moon and near sun-synchronous polarlunar orbitsrdquo Mathematical Problems in Engineering vol 2009Article ID 740460 24 pages 2009

[24] J P S Carvalho R V de Moraes and A F B A Prado ldquoSomeorbital characteristics of lunar artificial satellitesrdquo CelestialMechanics and Dynamical Astronomy vol 108 no 4 pp 371ndash388 2010

[25] J P D S Carvalho R V deMoraes andA F B A Prado ldquoPlan-etary satellite orbiters applications for themoonrdquoMathematicalProblems in Engineering vol 2011 Article ID 187478 19 pages2011

[26] D Folta and D Quinn ldquoLunar frozen orbitsrdquo in Proceedingsof the AIAAAAS Astrodynamics Specialist Conference AIAA2006-6749 pp 1915ndash1932 August 2006

[27] H Kinoshita and H Nakai ldquoSecular perturbations of fictitioussatellites of uranusrdquo Celestial Mechanics and Dynamical Astron-omy vol 52 no 3 pp 293ndash303 1991

[28] M Lara ldquoDesign of long-lifetime lunar orbits a hybrid ap-proachrdquo Acta Astronautica vol 69 no 3-4 pp 186ndash199 2011

[29] M E Paskowitz and D J Scheeres ldquoDesign of science orbitsabout planetary satellites application to Europardquo Journal ofGuidance Control and Dynamics vol 29 no 5 pp 1147ndash11582006

[30] M E Paskowitz andD J Scheeres ldquoRobust capture and transfertrajectories for planetary satellite orbitersrdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 342ndash353 2006

[31] R P Russell andA T Brinckerhoff ldquoCirculating eccentric orbitsaround planetary moonsrdquo Journal of Guidance Control andDynamics vol 32 no 2 pp 423ndash435 2009

[32] D J Scheeres M D Guman and B F Villac ldquoStability analysisof planetary satellite orbiters application to the Europa orbiterrdquoJournal of Guidance Control and Dynamics vol 24 no 4 pp778ndash787 2001

[33] C W T Roscoe S R Vadali and K T Alfriend ldquoThird-bodyperturbation effects on satellite formationsrdquo in Proceedings ofthe Jer-Nan Juang Astrodynamics Symposium AAS Preprint No12-631 College Station Tex USA June 2012

[34] R C Domingos R V de Moraes and A F B A Prado ldquoThird-body perturbation in the case of elliptic orbits for the disturbingbodyrdquoMathematical Problems in Engineering vol 2008 ArticleID 763654 14 pages 2008

[35] X Liu H Baoyin and X Ma ldquoLong-term perturbations dueto a disturbing body in elliptic inclined orbitrdquo Astrophysics andSpace Science vol 339 no 2 pp 295ndash304 2012

[36] R C Domingos R V de Moraes and A F B A Prado ldquoThird-body perturbation in the case of elliptic orbits for the disturbingbodyrdquo in Proceedings of the AIAAAAS Astrodynamics SpecialistConference Mackinac Island Mich USA August 2007

[37] R C Domingos R V de Moraes and A F B A Prado ldquoThirdbody perturbation using a single averaged model considering

elliptic orbits for the disturbing bodyrdquo Advances in the Astro-nautical Sciences vol 130 p 1571 2008

[38] C D Murray and S F Dermott Solar System Dynamics Cam-bridge University Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the same technique for the prediction of orbits for a long timespan

In the last years several researches based on both ana-lytical and numerical approaches studied the third-bodyperturbationThemajority of them concentrated on studyingthe effects of a perturbation caused by a third body usinga double-averaged technique like those of Sidlichovskyy [14]Kwok [15 16] Broucke [17] and Prado [18] The disturbingfunction is expanded in Legendrersquos polynomials and theaverage is made over the periods of the perturbing and theperturbed body using an approximated model expanded tosome order Some others studied this problem based on asingle-averaged analytical model where the disturbing func-tion is expanded in Legendrersquos polynomials but the averageis made over the short period of the perturbed body onlysee for example [19] and references therein In all of theresearches cited before the perturbing body was in a circularorbit There are also several papers considering the third-body perturbation in constellations of satellites (see [20 21])and in other planetary systems not involving the Earthlike those of Carvalho (see [22ndash25]) Folta and Quinn [26]Kinochita and Nakai [27] Lara [28] Paskowitz and Scheers(see [29 30]) Russel and Brinckerhoff [31] and Scheers etal [32] In particular Roscoe et al [33] used the analyticalequations derived in Prado [20] to study the dynamics of aconstellation of satellites perturbed by a third body

Using the double-averaged analyticalmodel inDomingoset al [34] the analytical expansion to study the third-bodyperturbation was extended to the case where the perturbingbody is in an elliptical orbit Liu et al [35] included the incli-nation of the perturbing bodyrsquos equatorial plane with respectto its orbital plane

The main idea behind the single-averaged technique isthat it eliminates only the terms due to the short-time peri-odic motion of the perturbed body The results are expectedto show smooth curves for the evolution of the mean orbitalelements for a long-time period when compared with thefull restricted three-body problem In other words a betterunderstanding of the physical phenomenon can be obtainedand it allows the study of long-term stability of the orbits inthe presence of disturbances that cause slow changes in theorbital elements

So an interesting point would be a study to show thedifferences between those two averaged models when com-pared with the full elliptic restricted three-body problemThe idea of the present paper is to study this problem butassuming that the perturbing body is in an elliptical orbitStudies under this assumption are available in Domingos etal (see [34 36 37]) In particular in the present paper a studyof the effects of the eccentricity of the perturbing body inthe orbit of the perturbed body is made for several differentconditions regarding the orbit of the primaries as well as theorbit of the spacecraft

So our goal is to make a more complete study of thisproblem and to perform some tests to verify the differencesin the results obtained by those two approximated techniquesA comparative investigation is made to verify the differencesbetween the single-averaged analytical model with the model

based on the double-averaged technique as well as a compar-ison with the full restricted elliptic three-body problem thatcan provide some insights about their applications in celestialmechanics

The assumptions used to develop the single-averagedanalytical model are the same ones of the restricted ellipticthree-body problem (planet-satellite-spacecraft) Our anal-ysis for the evolution of the mean orbital elements will bebased only on gravitational forces The equations of motionare obtained from Lagrangersquos planetary equations and thenwe numerically integrated those equations Different initialeccentricities for the perturbing body are considered

The set of results obtained in this research performs ananalysis of several well-known characteristics of the third-body perturbation like (i) the stability of near-circular orbitsto investigate under which conditions this orbit remains nearcircular (ii) the behavior of elliptic equatorial orbits and (iii)the existence of frozen orbits which are orbits that keep theeccentricity inclination and argument of periapsis constants

A detailed study considering the ldquocritical anglerdquo of thethird-body perturbation which is an inclination that makesa near circular orbit that has an inclination below this valueto remain near-circular is made This work is structured asfollows In Section 2 we present the equations of motionused for the numerical simulations Section 3 is devoted tothe analysis of the numerical results for near circular orbitsThe theory developed here is used to study the behaviorof a spacecraft around the Earth where the Moon is thedisturbing body The choice of the Moon as the disturbingbody is made based on the fact that its effect on the spacecraftis larger than the effect of the Sun Several plots show the timehistories of the Keplerian elements of the orbits involved Ourconclusions are presented in Section 4

2 Dynamical Model

For the determination of the equations of motion we startedby assuming the existence of a central body with mass 119898

0

that is fixed in the center of the reference system 119909-119910-119911 Theperturbing body with mass 1198981015840 is in an elliptic orbit withsemimajor axis 1198861015840 eccentricity 1198901015840 inclination 119894

1015840 argumentof periapsis 1205961015840 longitude of the ascending node Ω1015840 andmean motion 1198991015840 The spacecraft with negligible mass 119898 is ina generic orbit with orbital elements 119886 (semimajor axis) 119890(eccentricity) 119894 (inclination) 120596 (argument of periapsis) Ω(longitude of the ascending node) and it has mean motion119899 This system is shown in Figure 1

Using the traditional expansion in Legendrersquos polynomi-als (assuming that 1199031015840 ≫ 119903) the second-order term of thedisturbing potential averaged over the eccentric anomaly ofthe spacecraft is given by the following (see [17 18 34])

⟨1198772⟩ =

1205831015840

1198862

11989910158402

2

(

1198861015840

1199031015840

)

3

(1 +

3

2

1198902

) [

3

2

(1205722

+ 1205732

) minus 1]

+

15

4

(1205722

minus 1205732

) 1198902

(1)


Z

Y

X

m0

m

r

r998400

m998400



1199031015840

=

1198861015840

(1 minus 11989010158402

)

1 + 1198901015840 cos1198911015840

cos1198911015840 = cos1198721015840 + 1198901015840 (cos 21198721015840 minus 1)

+

9

8

11989010158402


sin1198911015840 = sin1198721015840 + 1198901015840 sin 21198721015840

+ 11989010158402

(

9

8

sin 31198721015840 minus 78


(2)




where119863 = Ω minus 1198911015840

minus 1205961015840


1198721015840

= 1198721015840

119900

+ 1198991015840

119905 (4)


119889119890

119889119905

= 119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899


minus cos 119894 cos 2120596 sin 2119863]

119889119894

119889119905

= 119870

1

sin 119894[120583119886 (1 minus 1198902)]123

4

1205831015840

11989910158402

1198862

times 1198902

[ minus 5 cos 2120596 sin 2119863cos2119894

+

3

2

(minus sin 2119863 + sin 2119863cos2119894)

minus

5

2

sin119863 cos 2120596 (1 + cos2119894)]

+ (sin119863 minus sin 2119863cos2119894)

minus5 cos 2119863 sin 2120596 cos 119894

119889Ω

119889119905

= 119870

3

4

1205831015840

11989910158402 119886

2

sin 119894[120583119886 (1 minus 1198902)]12

times (1 +

3

2

1198902

) (minussin2119863 sin 2119894)

+ [cos 2119894 cos 2120596 sin2119863

+ sin 119894 sin 2119863 sin 2120596] (52

1198902

)

119889120596

119889119905

= 119870

3

4

1205831015840

11989910158402

minus

1198862

sin 119894[120583119886 (1 minus 1198902)]12

times [(1 +

3

2

1198902

) (minussin2119863 sin 2119894)

+ (

5

2

1198902

) (cos 2119894sin2119863 cos 2120596)

+

1

2


2

119899


minus 5 cos 119894 sin 2119863 sin 2120596

+

6

2

(cos2119863 + cos2119894 sin2119863)]

(5)

where

119870 =

1 + 31198901015840 cos1198911015840 + 311989010158402cos21198911015840

1 minus 61198901015840

+ 1511989010158402

(6)






119889119890

119889119905

= minus119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899

sin 2 (119863 minus 120596) (7)



10158402

+

(158)11989010158404


3 Numerical Results


0= 001 120596

0= Ω0= 0





0lt 119894119888








1198901015840







0= 30∘ (red





0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

0005

001

0015

002

Ecce

ntric

ity

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(a)

0

001

002

003

004

005

0

001

002

003

004

005

0

001

002

003

004

005

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0

001

002

003

004

005


e998400= 02

e998400= 03

e998400= 0

e998400= 01

(b)

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(c)


0






066

068

07

072In

clina

tion

(rad

)

064

066

068

07

072

Incli

natio

n (r

ad)


0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

064

066

068

07

072

Incli

natio

n (r

ad)

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

8000

080

000

8000

0

Double

(a)

066

067

068

069

07

071

072

066

067

068

069

07

071

072

066

067

068

069

07

071

072

0

5000

1000

0

1500

0


40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0

2000

020

000

2000

0

Double

(b)

0

5000

1000

0

1500

0

066

068

07

072

066

068

07

072

064

066

068

07

072

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0


2000

020

000

2000

0

Double

(c)


0









0

001

002

003

004

005

006Ec

cent

ricity

0

001

002

003

004

005

006

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

001

002

003

004

005

006

Ecce

ntric

ity


(a)

00

5000 10000 15000 20000

003

006

009

012

015

00

5000 10000 15000 20000

003

006

009

012

0150

05000 10000 15000 20000

003

006

009

012

015


(b)

00

5000 10000 15000 20000

005

01

015

02

025

03

0 5000 10000 15000 200000

005

01

015

02

025

03

00

5000 10000 15000 20000

005

01

015

02

025

03


(c)


0








0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Z

Y

X

m0

m

r

r998400

m998400



1199031015840

=

1198861015840

(1 minus 11989010158402

)

1 + 1198901015840 cos1198911015840

cos1198911015840 = cos1198721015840 + 1198901015840 (cos 21198721015840 minus 1)

+

9

8

11989010158402


sin1198911015840 = sin1198721015840 + 1198901015840 sin 21198721015840

+ 11989010158402

(

9

8

sin 31198721015840 minus 78


(2)




where119863 = Ω minus 1198911015840

minus 1205961015840


1198721015840

= 1198721015840

119900

+ 1198991015840

119905 (4)


119889119890

119889119905

= 119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899


minus cos 119894 cos 2120596 sin 2119863]

119889119894

119889119905

= 119870

1

sin 119894[120583119886 (1 minus 1198902)]123

4

1205831015840

11989910158402

1198862

times 1198902

[ minus 5 cos 2120596 sin 2119863cos2119894

+

3

2

(minus sin 2119863 + sin 2119863cos2119894)

minus

5

2

sin119863 cos 2120596 (1 + cos2119894)]

+ (sin119863 minus sin 2119863cos2119894)

minus5 cos 2119863 sin 2120596 cos 119894

119889Ω

119889119905

= 119870

3

4

1205831015840

11989910158402 119886

2

sin 119894[120583119886 (1 minus 1198902)]12

times (1 +

3

2

1198902

) (minussin2119863 sin 2119894)

+ [cos 2119894 cos 2120596 sin2119863

+ sin 119894 sin 2119863 sin 2120596] (52

1198902

)

119889120596

119889119905

= 119870

3

4

1205831015840

11989910158402

minus

1198862

sin 119894[120583119886 (1 minus 1198902)]12

times [(1 +

3

2

1198902

) (minussin2119863 sin 2119894)

+ (

5

2

1198902

) (cos 2119894sin2119863 cos 2120596)

+

1

2


2

119899


minus 5 cos 119894 sin 2119863 sin 2120596

+

6

2

(cos2119863 + cos2119894 sin2119863)]

(5)

where

119870 =

1 + 31198901015840 cos1198911015840 + 311989010158402cos21198911015840

1 minus 61198901015840

+ 1511989010158402

(6)






119889119890

119889119905

= minus119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899

sin 2 (119863 minus 120596) (7)



10158402

+

(158)11989010158404


3 Numerical Results


0= 001 120596

0= Ω0= 0





0lt 119894119888








1198901015840







0= 30∘ (red





0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

0005

001

0015

002

Ecce

ntric

ity

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(a)

0

001

002

003

004

005

0

001

002

003

004

005

0

001

002

003

004

005

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0

001

002

003

004

005


e998400= 02

e998400= 03

e998400= 0

e998400= 01

(b)

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(c)


0






066

068

07

072In

clina

tion

(rad

)

064

066

068

07

072

Incli

natio

n (r

ad)


0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

064

066

068

07

072

Incli

natio

n (r

ad)

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

8000

080

000

8000

0

Double

(a)

066

067

068

069

07

071

072

066

067

068

069

07

071

072

066

067

068

069

07

071

072

0

5000

1000

0

1500

0


40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0

2000

020

000

2000

0

Double

(b)

0

5000

1000

0

1500

0

066

068

07

072

066

068

07

072

064

066

068

07

072

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0


2000

020

000

2000

0

Double

(c)


0









0

001

002

003

004

005

006Ec

cent

ricity

0

001

002

003

004

005

006

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

001

002

003

004

005

006

Ecce

ntric

ity


(a)

00

5000 10000 15000 20000

003

006

009

012

015

00

5000 10000 15000 20000

003

006

009

012

0150

05000 10000 15000 20000

003

006

009

012

015


(b)

00

5000 10000 15000 20000

005

01

015

02

025

03

0 5000 10000 15000 200000

005

01

015

02

025

03

00

5000 10000 15000 20000

005

01

015

02

025

03


(c)


0








0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119889119890

119889119905

= minus119870

15

4

1205831015840

11989910158402 119890radic1 minus 119890

2

119899

sin 2 (119863 minus 120596) (7)



10158402

+

(158)11989010158404


3 Numerical Results


0= 001 120596

0= Ω0= 0





0lt 119894119888








1198901015840







0= 30∘ (red





0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

0005

001

0015

002

Ecce

ntric

ity

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(a)

0

001

002

003

004

005

0

001

002

003

004

005

0

001

002

003

004

005

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0

001

002

003

004

005


e998400= 02

e998400= 03

e998400= 0

e998400= 01

(b)

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(c)


0






066

068

07

072In

clina

tion

(rad

)

064

066

068

07

072

Incli

natio

n (r

ad)


0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

064

066

068

07

072

Incli

natio

n (r

ad)

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

8000

080

000

8000

0

Double

(a)

066

067

068

069

07

071

072

066

067

068

069

07

071

072

066

067

068

069

07

071

072

0

5000

1000

0

1500

0


40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0

2000

020

000

2000

0

Double

(b)

0

5000

1000

0

1500

0

066

068

07

072

066

068

07

072

064

066

068

07

072

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0


2000

020

000

2000

0

Double

(c)


0









0

001

002

003

004

005

006Ec

cent

ricity

0

001

002

003

004

005

006

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

001

002

003

004

005

006

Ecce

ntric

ity


(a)

00

5000 10000 15000 20000

003

006

009

012

015

00

5000 10000 15000 20000

003

006

009

012

0150

05000 10000 15000 20000

003

006

009

012

015


(b)

00

5000 10000 15000 20000

005

01

015

02

025

03

0 5000 10000 15000 200000

005

01

015

02

025

03

00

5000 10000 15000 20000

005

01

015

02

025

03


(c)


0








0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0

0005

001

0015

002

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

0005

001

0015

002

Ecce

ntric

ity

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(a)

0

001

002

003

004

005

0

001

002

003

004

005

0

001

002

003

004

005

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0 5000 10000 15000 20000

0

001

002

003

004

005


e998400= 02

e998400= 03

e998400= 0

e998400= 01

(b)

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

0 5000 10000 15000 200000

002

004

006

008

01

e998400= 02

e998400= 03

e998400= 0

e998400= 01


(c)


0






066

068

07

072In

clina

tion

(rad

)

064

066

068

07

072

Incli

natio

n (r

ad)


0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

064

066

068

07

072

Incli

natio

n (r

ad)

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

8000

080

000

8000

0

Double

(a)

066

067

068

069

07

071

072

066

067

068

069

07

071

072

066

067

068

069

07

071

072

0

5000

1000

0

1500

0


40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0

2000

020

000

2000

0

Double

(b)

0

5000

1000

0

1500

0

066

068

07

072

066

068

07

072

064

066

068

07

072

40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0


2000

020

000

2000

0

Double

(c)


0









0

001

002

003

004

005

006Ec

cent

ricity

0

001

002

003

004

005

006

Ecce

ntric

ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

001

002

003

004

005

006

Ecce

ntric

ity


(a)

00

5000 10000 15000 20000

003

006

009

012

015

00

5000 10000 15000 20000

003

006

009

012

0150

05000 10000 15000 20000

003

006

009

012

015


(b)

00

5000 10000 15000 20000

005

01

015

02

025

03

0 5000 10000 15000 200000

005

01

015

02

025

03

00

5000 10000 15000 20000

005

01

015

02

025

03


(c)


0








0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










066

068

07

072In

clina

tion

(rad

)

064

066

068

07

072

Incli

natio

n (r

ad)


0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

0

2000

0

4000

0

6000

0

064

066

068

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072

Incli

natio

n (r

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40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

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e998400= 02

Full Single

8000

080

000

8000

0

Double

(a)

066

067

068

069

07

071

072

066

067

068

069

07

071

072

066

067

068

069

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0

5000

1000

0

1500

0


40∘

41∘

38∘

39∘

e998400= 0

40∘

41∘

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39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0

2000

020

000

2000

0

Double

(b)

0

5000

1000

0

1500

0

066

068

07

072

066

068

07

072

064

066

068

07

072

40∘

41∘

38∘

39∘

e998400= 0

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41∘

38∘

39∘

e998400= 01

40∘

41∘

38∘

39∘

e998400= 02

Full Single

0

5000

1000

0

1500

0

0

5000

1000

0

1500

0


2000

020

000

2000

0

Double

(c)


0









0

001

002

003

004

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0

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002

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ity

0 20000 40000 60000 80000

0 20000 40000 60000 80000

0

001

002

003

004

005

006

Ecce

ntric

ity


(a)

00

5000 10000 15000 20000

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5000 10000 15000 20000

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0150

05000 10000 15000 20000

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(b)

00

5000 10000 15000 20000

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0 5000 10000 15000 200000

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01

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02

025

03

00

5000 10000 15000 20000

005

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015

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025

03


(c)


0








0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

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ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

001

002

003

004

005

006Ec

cent

ricity

0

001

002

003

004

005

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ntric

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0 20000 40000 60000 80000

0 20000 40000 60000 80000

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(a)

00

5000 10000 15000 20000

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5000 10000 15000 20000

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05000 10000 15000 20000

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(b)

00

5000 10000 15000 20000

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0 5000 10000 15000 200000

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00

5000 10000 15000 20000

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(c)


0








0 20000 40000 60000 8000006

08

1

12

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clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

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Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

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0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

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cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

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ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

04

06

08

1

Full SingleDouble

(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20000 40000 60000 8000006

08

1

12

14In

clina

tion

(rad

)

40∘

50∘

60∘

70∘

80∘

e998400= 0

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 02

0 20000 40000 60000 8000006

08

1

12

14

Incli

natio

n (r

ad)

40∘

50∘

60∘

70∘

80∘

e998400= 04


(a)

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

2000006

08

1

12

14

40∘

50∘

60∘

70∘

80∘

e998400= 04

0 5000 10000 15000

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 0


(b)

40∘

50∘

60∘

70∘

80∘

06

08

1

12

14

0 5000 10000 15000 20000

06

08

1

12

14

0 5000 10000 15000 20000

40∘

50∘

60∘

70∘

80∘

e998400= 02

06

08

1

12

14

0 5000 10000 15000 20000

e998400= 04

e998400= 0


(c)


0









0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

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1

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ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

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1

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(b)

0 5000 10000 15000 20000

Full Single

0 5000 10000 15000 20000


0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20000 40000 60000 800000

02

04

06

08

1Ec

cent

ricity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity

0 20000 40000 60000 800000

02

04

06

08

1

Ecce

ntric

ity


(a)


0

02

04

06

08

1

00

5000 10000 15000 20000

02

04

06

08

10

05000 10000 15000 20000

02

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06

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1

Full SingleDouble

(b)

0 5000 10000 15000 20000

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0 5000 10000 15000 20000


0

02

04

06

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1

0

02

04

06

08

1

0

02

04

06

08

1

Double

(c)


0












4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











4 Conclusions







Acknowledgments



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a study of single- and double-averaged ... · mathematicalproblems in engineering...

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