averaging tesseral effects: closed form relegation versus … · averaging tesseral effects: closed...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 570127 11 pageshttpdxdoiorg1011552013570127

Research ArticleAveraging Tesseral Effects Closed Form Relegation versusExpansions of Elliptic Motion

Martin Lara1 Juan F San-Juan2 and Luis M Loacutepez-Ochoa2

1 CColumnas de Hercules 1 San Fernando Spain2Universidad de La Rioja Logrono Spain

Correspondence should be addressed to Juan F San-Juan juanfelixsanjuanuniriojaes

Received 7 February 2013 Accepted 26 March 2013

Academic Editor Antonio F Bertachini A Prado

Copyright copy 2013 Martin Lara et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Longitude-dependent terms of the geopotential cause nonnegligible short-period effects in orbit propagation of artificial satellitesHence accurate analytical and semianalytical theories must cope with tesseral harmonics Modern algorithms for dealinganalytically with them allow for closed form relegation Nevertheless current procedures for the relegation of tesseral effects fromsubsynchronous orbits are unavoidably related to orbit eccentricity a key fact that is not enough emphasized and constrainsapplication of this technique to small and moderate eccentricities Comparisons with averaging procedures based on classicalexpansions of elliptic motion are carried out and the pros and cons of each approach are discussed

1 Introduction

Soon after Brouwerrsquos dream of a closed form solution to theartificial satellite theory (AST) to higher orders was achievedfor themain problemofAST [1ndash3] the progress inmathemat-ical methods made the closed form solution also possible forthe general case of a tesseral potential This notable successwas feasible thanks to the invention of two algorithms whichare specifically suited for canonical simplifications of Hamil-tonian problems namely the elimination of the parallax [4]and the relegation algorithm [5] The former operates thetesseral averaging in closed form for low Earth orbits (LEO)[6ndash8] while the latter does the same in all other cases [9 10]However many aerospace engineering applications of Earthartificial satellites are related to subsynchronous orbits a casein which both algorithms limit their practical application toorbits with small or moderate eccentricities Indeed high-eccentricity orbits lead to impact with the surface of theEarth in the case of LEO on the one hand while the numberof iterations required by the relegation algorithm makes itimpracticable for high-eccentricity orbits on the other [9]

As reported in the literature the relegation of tesseraleffects from subsynchronous orbits is based on multiplica-tions at each iteration by the ratio of the Earthrsquos rotation

rate to the mean motion of the orbit and hence convergesvery slowly except for low-altitude orbits Another drawbackof the standard algorithm is that it assumes nonresonanceconditions and finds a gradual deterioration in the solutionwhen approaching synchronous condition [9] On the con-trary usual series expansions of elliptic motion (see [11ndash14]eg) allow for the effective isolation and handling of tesseralresonances [15ndash19]

Both approaches tesseral relegation and series expan-sions have been implemented in analytical and semianalyti-cal theories used in renowned laboratoriesThus relegation isreported to be implemented in the Analytic orbit propagatorprogram (AOPP) of the Naval Research Laboratory [20]whereas the Draper Semianalytical Satellite Theory (DSST)[21] these daysmaintained atMIT Lincoln Laboratory or thenumeric-analytical theory THEONA developed at KeldyshInstitute of Applied Mathematics [22] makes recourse toseries expansions which are computed by means of efficientrecurrences of different special functions (see [23] eg)However to our knowledge the literature lacks comparisonsin the relative efficiency of these two radically differentapproaches when applied to the tesseral geopotential

Therefore the question emerges whether averaging tech-niques based on expansions of ellipticmotion are competitive

2 Mathematical Problems in Engineering

or not with algorithms devised for dealing in closed formwith short-period effects due to longitude-dependent termsof the geopotential In the case of LEO the advantageclearly favors the closed form solution whose based onthe different perturbation orders of the Keplerian and theCoriolis term tesseral effects can be averaged in closed form[24 25] Furthermore in this particular case the eliminationof the parallax works wonders succeeding in the closed-formelimination of all tesseral terms except for those related to119898-dailies [6 26] This is a typical case in which the complexityof relegation makes it less efficient than normalization [5]or canonical simplification in the present context [27] Onthe other hand Keplerian and Coriolis terms can no longerbe assumed to be of different orders when approachingresonance conditions which is the common case of orbits inthe medium Earth orbits (MEOs) region and above

Trying to give an answer to this question in the case ofMEO we carry out several comparisons in which we restrictthe test model to a 2 times 2 truncation of the geopotentialWaiting for more effective implementations of the relegationalgorithm which makes it useful also for the canonicalsimplification of tesseral resonance problems we limit ourcomparisons to the case of nonresonantmotion In particularwe explore the performances of our algorithm in a set ofselected cases borrowed from [9] However we also considersample orbits with the same semimajor axis as Galileo orbitsand different eccentricities a case that better supports ourclaim that the new formulation of the tesseral relegationalgorithm is valid for orbits with any semimajor axis Besidesbecause both approaches series expansion and tesseral rel-egation after averaging may well end up being Brouwerrsquoslong-term Hamiltonian (see [11 page 569]) the comparisoncriteria will be based on the size of the transformationequations from mean to osculating elements required bythe different theories for obtaining analogous precision inthe results together with the relative time spent in theirevaluation

The paper is organized as follows For completeness inSection 2 we recall the basics of the relegation algorithm forthe particular case of the geopotential to whichwe contributea slightly different but definitely enlightening point of viewwhich justifies the application of the relegation techniqueto low-eccentricity orbits independently of the ratio of theorbital period to the Earthrsquos rotation period [28] In this sec-tion a brief mention to the standard procedure for averaginglongitude-dependent terms when using classical expansionof the elliptic motion is also included Next in Section 3we briefly describe the whole procedure of computing thelong-term Hamiltonian and the transformations equationsfrom mean to osculating elements which we implementfor the simplest case of a second-order truncation of thegeopotential While application of the tesseral relegation tohigher degree and order truncations does not involve anymodifications to the basic algorithm it produces a notableincrease in the size of the series usedmdashas is also the case whenusing the expansions of elliptic motion approach Finallycomparisons between both procedures relegation and seriesexpansions are presented in Section 4 for a variety of testcases and the pros and cons of each method are discussed

2 Averaging the Geopotential

The motion of artificial satellites close to the Earth departsslightly fromKeplerianmotion because of the noncentralitiesof the geopotential This disturbing potential is written in theusual form

119881 =

120583

119903

sum

119895ge2

(

120572

119903

)

119895

[1198691198951198751198950 (sin120593)

minus

119895

sum

119896=1

(119862119895119896 cos 119896120582 + 119878119895119896 sin 119896120582)

times119875119895119896 (sin120593) ]

(1)

where 120583 is the Earthrsquos gravitational parameter 120572 is theEarthrsquos equatorial radius 119875119895119896 are Legendre polynomials 119869119895 =minus1198621198950 119862119895119896 and 119878119895119896 are harmonic coefficients and (120593 120582 119903)

are spherical coordinatesmdashlatitude longitude and distancerespectivelymdashin an Earth-centered Earth-fixed (ECEF) sys-tem of coordinates

Satellite motion under the influence of the disturbingpotential in (1) is generally nonintegrable even in the sim-plest case of the 1198692 problem [29 30] with the remarkableexception of the equatorial case [31] However analytical orsemianalytical integration of perturbed Keplerian motion iscustomarily approached by perturbation methods Namelyshort-period effects are analytically averaged to formulate theproblem in mean elements next either the mean elementsequations are numerically integrated with very long stepsizesor a new averaging of long-period effects provides the secularevolution of the problem

21 Lie Transforms Depritrsquos Triangle In particular the Lietransforms technique [32] allows for automatic computationof the perturbation theory by machine to higher ordersand these days is taken as a standard procedure Brieflyin a Hamiltonian framework the initial Hamiltonian inosculating variables is cast in the form

H = sum

119898ge0

120598119898

119898

1198671198980 (2)

where the small parameter 120598 which establishes the order ofeach perturbation term may be either physical or formal

The averaged Hamiltonian in mean elements

H1015840= sum

119898ge0

120598119898

119898

1198670119898 (3)

is to be constructed by the perturbation method The newHamiltonian terms1198670119898 are one of the sides of ldquoDepritrsquos trian-glerdquo a triangular array of terms derived from the recurrence

119867119899119902 = 119867119899+1 119902minus1 + sum

0le119898le119899

(

119899

119898) 119867119899minus119898 119902minus1119882119898+1 (4)

Mathematical Problems in Engineering 3

where minus minus notes the Poisson bracket operator and terms119882119898+1 define the generating function

W = sum

119898ge0

120598119898

119898

119882119898+1 (5)

of the transformation from mean to osculating elementsThe transformed Hamiltonian (3) is stepwise constructed

by solving at each step the homological equation

L0 (119882119898) = 119898 minus 1198670119898 (6)

where terms 119898 are known from previous evaluations ofDepritrsquos triangle terms 1198670119898 are chosen at our convenienceand terms 119882119898 must be solved from a partial differentialequation The operator L0(minus) equiv minus11986700 is known as theLie derivative generated by the Keplerian flow11986700

Finally the transformation from mean to osculatingelements is in turn computed from Depritrsquos triangle by thesimple expedient of replacing terms 119867119894119895 in (4) by 120585119894119895 where12058500 = 120585 notes any of the osculating elements and 1205851198980 equiv 0 for119898 gt 0

22 Removing Tesseral Effects from the Geopotential Usualrelations between spherical coordinates and orbital elements[12] permit to express the Hamiltonian of the geopotentialin terms of orbital elements Thus the zeroth order of (2)corresponds to the Kepler problem in the ECEF frame

11986700 = minus120583

2119886

minus 119899119864radic120583119886120578 cos 119894 (7)

where (119886 119890 119894 120596 Ω119872) are usual Keplerian elements 120578 =

radic1 minus 1198902 is commonly known as the eccentricity function and

119899119864 notes the Earthrsquos rotation rate which is assumed to beconstant The first order term is

11986710 = minus120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (8)

where 119891 is the true anomalyThe term 1198672 = 2(119881 minus 1198671) with 119881 given in (1) carries

effects of the second order of 1198692 which are contributed byother zonal and tesseral harmonics Namely it takes the formof a trigonometric series whose coefficients are functions of119890 and 119894 and whose arguments are combinations of the type119895119891 + 119896] + 119898120596 with 119895 119896 and 119898 integers where ] = Ω minus 120599

is the longitude of the ascending node in the rotating frameand 120599 is Greenwich Sidereal TimeThese series are particularinstances of multivariate Fourier series of the form

sum

11989511198952119895119898ge0

1199011198951

1 1199011198952

2 sdot sdot sdot 119901119895119898

119898

times sum

11989611198962119896119897ge0

(11987611989611198962119896119897

11989511198952119895119898

cos120573 + 1198781198961 1198962 11989611989711989511198952119895119898

sin120573) (9)

where 120573 = 11989611205731 + 11989621205732 + sdot sdot sdot + 119896119897120573119897 119901 are polynomialvariableswith integer coefficients and119876 and 119878 are coefficientsExpressions of the form of (9) are commonly known asPoisson series of the type (119898 119897) [33]

The Hamiltonian (2) must be formulated in canonicalvariables whereas Keplerian elements are not canonical Thenatural alternative is using Delaunay variables the canonicalcounterpart of Keplerian elements although other sets ofcanonical variables as nonsingular Poincare elements arecustomarily used [20] (for a survey on orbital elementssets see [34]) In what follows it may be convenient usingthe orbital elements notation because of the natural insightthat it provides to orbital motion problems When thishappens orbital elements must be understood as functionsof canonical variables instead of variables by themselves

Using Delaunay-type canonical variables

ℓ = 119872 119892 = 120596 ℎ = ]

119871 = radic120583119886 119866 = 119871120578 119867 = 119866 cos 119894(10)

where119866 is themodulus of the angularmomentum vector and119867 its projection on the Earthrsquos rotation axis the zeroth-orderHamiltonian is written

11986700 = minus1205832

21198712minus 119899119864119867

(11)

Therefore the Lie derivative is

L0 = 119899120597

120597ℓ

minus 119899119864

120597

120597ℎ

(12)

where 119899 = 12058321198713 is the orbital mean motion At each step

119898 of the perturbation algorithms we use (12) to compute thegenerating function term119882119898 from (6)

221 Series Expansion In spite of the simplicity of (12) thefact that the dependence of the geopotential on the meananomaly does not happen explicitly but through the trueanomaly which is an implicit function ℓ = ℓ(119891 119890) of themean anomaly involving the solution of Kepler equationcomplicates finding a closed form solution to the homologicalequation Hence expansions of the elliptic motion are cus-tomarily used to make explicit the dependence of the tesseralHamiltonian on the mean anomaly (see [11ndash14 35] eg)Once the mean anomaly shows explicitly in the geopotentialthe solution of the homological equation for the Lie derivative(12) is straightforward not only allowing for the eliminationof short-period terms related to tesseral harmonics in the caseof nonresonant motion [36] but also for resonance problems[19]

222 Relegation Alternatively one may realize that exceptfor synchronous orbits either 120575 equiv 119899119899119864 lt 1 or 120575 equiv 119899119864119899 lt 1Then at each step 119898 of the perturbation algorithm one canfeel satisfied with finding an approximate solution

119882119898 = sum

119895ge0

120575119895119882119898119895 (13)

which is computed as follows We describe the subsyn-chronous orbits case 120575 equiv 119899119864119899 lt 1 but the same philosophycan be applied to the supersynchronous case


Replacing (13) into the homological equation (6) we get

sum

119895ge0

119899120575119895(

120597119882119898119895

120597ℓ

minus 120575

120597119882119898119895

120597ℎ

) = 119865 (119891 119892 ℎ 119871 119866119867) (14)

with 119865 equiv 119898 minus 1198670119898 which we reorganize as

sum

119895ge0

120575119895120597119882119898119895

120597ℓ

= sum

119895ge0

120575119895119865119895 (15)

where

1198650 =1

119899

119865 119865119895 =

120597119882119898119895minus1

120597ℎ

(119895 gt 0) (16)

Then (15) is solved iteratively by equating equal powers of120575 At each iteration the corresponding differential equationis solved by quadrature The corresponding solution maybe reached in closed form with the help of the differentialrelation

1198862120578119889ℓ = 119903

2119889119891 (17)

which is derived from the preservation of the angularmomentum of Keplerian motion

A rule for ceasing iterations may be choosing 119895 suchthat 120575119895+1 = O(120598) which would imply neglecting fromthe generating function terms of higher order than thecorresponding one of the perturbation theory Thus forexample in the usual case 120598 = O(1198692) = O(10minus3) for an orbitwith 119886 = 10500 km we get 120575 asymp 18 and hence 119895 = 2

Typically terms 119882119898119895 grow in size with each iteration 119895

making the relegation process unpractical for a relative lownumber of iterations [9]This fact limits the practical applica-tion of the relegation algorithm to orbits relatively close to theEarthrsquos surface where nonimpact orbits must have small ormoderate eccentricities However it has been recently shownthat a slight modification of the relegation algorithm allowsto extend its application to orbits with any semimajor axisbut constrained to the case of low-eccentricities [28] Forcompleteness we summarize the procedure in what followsFurthermore we state the algorithm directly in Delaunayvariables instead of the polar-nodal variables used in theoriginal approach thus relieving the relegation algorithmfrom the intricacies related to the specific algebra of theparallactic functions

Indeed using (17) the Lie derivative is written as

L0 = 1198991198862120578

1199032

120597

120597119891

minus 119899119864

120597

120597ℎ

(18)

which is rewritten as

L0 =119899

1205783(

120597

120597119891

minus 1205783120575

120597

120597ℎ

+ 119890R + 1198902S) (19)

where

R = 2 cos119891 120597

120597119891

S = cos2119891 120597

120597119891

(20)

Therefore the homological equation (6) may be writtenas

120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

=

1205783

119899

119865 minus 2119890 cos119891120597119882119898

120597119891

minus 1198902cos2119891

120597119882119898

120597119891

(21)

showing that independently of the value of the ratio 120575

(or equivalently for orbits with any semimajor axis) ananalogous approach to the classical relegation algorithm canbe taken also in the case of orbits with small eccentricitiesNow the integration of the homological equation is carriedout in the true anomaly instead of the mean one so there isno problem in solving the main part of the partial differentialequation (21)

120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

asymp

1205783

119899

119865 (119891 119892 ℎ 119871 119866119867) (22)

which is taken as the first step in the relegation iterationsEquation (13) is now written as

119882119898 = sum

119895ge0

119890119895119882119898119895 (23)

Correspondingly the analog of the homological equation (15)is

sum

119895ge0

119890119895(

120597119882119898119895

120597119891

minus 1205783120575

120597119882119898119895

120597ℎ

) = sum

119895ge0

119890119895119865119895 (24)

where

1198650 =1205783

119899

119865 1198651 = minus2 cos1198911205971198821198980

120597119891

(25)

and for 119895 gt 1

119865119895 = minus2 cos119891120597119882119898119895minus1

120597119891

minus cos2119891120597119882119898119895minus2

120597119891

(26)

Note that this new formulation of the relegation algo-rithm shows explicitly that the effectiveness of the tesseralrelegation is constrained to the case of low-eccentricityorbits The limitation of the tesseral relegation to the caseof small or moderate eccentricities seems to be an intrinsiclimitation of the tesseral relegation for subsynchronous orbitsand although not enough emphasized appears also in theclassical formulation of the tesseral relegation because of theoccurrence of Hamiltonian ldquotype (A)rdquo terms in which theargument of the latitude appears explicitly [9] An additionaladvantage of the present formulation of the tesseral relegationis that it does not need any special treatment of Hamiltonianterms related to119898-dailies namely those terms of the Poissonseries that only depend on the longitude of the ascendingnode in the rotating frame ℎ


3 Semianalytical Theory fora 2 times 2 Geopotential

We illustrate the relegation procedure and investigate itsperformances by constructing a semianalytical theory for atruncation of the geopotential to degree and order 2 Welimit to the second-order terms of the perturbation theoryincluding the transformation equations up to the secondorder of the small parameter 120598 which is taken of the sameorder as the second zonal harmonic 1198692 Then from (1) theterm 11986720 = 2(119881 minus 11986710) in the initial Hamiltonian (2) issimply

11986720 = minus21205831205722

1199033

2

sum

119896=1

(1198622119896 cos 119896120582 + 1198782119896 sin 119896120582) 1198752119896 (sin120593)

(27)

where spherical coordinates are assumed to be expressed asfunctions of Delaunay variables

The construction of the theory consists of three differentsteps First tesseral effects are averaged using the relega-tion algorithm described above This averaging reduces theHamiltonian of the geopotential to the zonal part whichbecause of the 2 times 2 truncation is just the main problemwhich only considers the disturbing effect of the 1198692 harmonicNext short-period effects are partially removed via the elim-ination of the parallax [4] Finally remaining short periodicsare averaged via a trivial Delaunay normalization [37] Itdeserves to mention that in view of we limit the perturbationtheory to the second order of 1198692 the elimination of theparallax is not required for the closed form normalization[38] and therefore the parallax elimination and Delaunaytransformation may be reduced to a single transformationHowever carrying out the elimination of the parallax stillresults convenient because the important reduction in thenumber of terms in the total transformation equationscompare [2]

31 Tesseral Relegation Because the perturbation theory islimited to the second-order there is no coupling between 1198692and the second order tesseral harmonics 11986221 11987821 11986222 and11987822 Therefore11986701 is chosen the same as11986710 which makes1198821 = 0 at the first order of Depritrsquos triangle (4)

Next we use the Keplerian relation 119903 = 1198861205782(1 + 119890 cos119891)to express 11986720 as a Poisson series in 119891 119892 and ℎ whichis made of 84 trigonometric terms Since the solution ofeach iteration of the homological equation (24) amounts tochanging sines by cosines and vice versa and dividing foradequate linear combinations of 119899 and 119899119864 the first termof the generating function 11988220 is also made of 84 termsComputing 1198651 in (25) for the first iteration of the relegationinvolves multiplication of the whole series 12059711988220120597119891 by cos119891an operation that enlarges the resulting Poisson series by 16new trigonometric terms while the first approximation of thegenerating function1198822 = sum119895=011198822119895 is made of a total of 108trigonometric terms Similarly the computation of 1198652 from(26) involves multiplication of Poisson series by cos119891 andcos2119891 which again enlarges the size of the Poisson series

leading to a new approximation1198822 = sum119895=021198822119895 consistingof 132 trigonometric terms

Further iterations of the relegation are found to increasethe generating function by the same amount namely by 24new trigonometric terms at each new step Thus the fifthiteration of the relegation results in a total generating function1198822 = sum119895=051198822119895 that is a Poisson seriesmade of 84+5times24 =204 trigonometric terms We stop iterations at this order

It deserves mentioning that in addition to the lineargrowing of the length of the Poisson series the complexity ofthe coefficient of each trigonometric term notably increaseswith iterations of the relegation as a result of the combinationof the different series resulting from the procedureThus if wetake as a rough ldquocomplexity indicatorrdquo the size of the text filerequired for storing each generating function we find out thatthe complexity of each new approximation to the generatingfunction almost doubles with each iteration Specifically weobtain files of 68 139 272 481 832 and 1356 kB forthe 0 5 iteration respectively (throughout this paper werefer to binary prefixes that is 1 kB = 2

10 bytes) Note thatthese sizes are the result of a standard simplification with ageneral purpose algebra system without further investigatingthe structure of the series

32 Elimination of the Parallax After tesseral relegation weget the well-known Hamiltonian of the main problem

H = minus

120583

2119886

minus

120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (28)

Note that since longitude-dependent terms were removedthe problem is now formulated in inertial space

The standard parallax elimination [4] transforms (28) upto the second order of 1198692 into

H = minus

120583

2119886

minus

120583

2119886

1205722

1199032

1198692

1205782(1 minus

3

2

sin2119894)

minus

120583

2119886

1205724

11988621199032

11986922

1205786[

5

4

minus

21

8

sin2119894 + 21

16

sin4119894

+ (

3

8

cos2119894 minus 15

64

sin4119894) 1198902

minus(

21

16

sin2119894 minus 45

32

sin4119894) 1198902 cos 2119892] (29)

Note that after carrying out the relegation transformation(ℓ 119892 ℎ 119871 119866119867)

R997888rarr (ℓ

1015840 1198921015840 ℎ1015840 1198711015840 1198661015840 1198671015840) the variables in

which we express (28) are no longer osculating elements andhence we should have used the prime notation Analogouslythe elimination of the parallax performs a new transforma-tion from ldquoprimerdquo elements to ldquodouble primerdquo elements inwhich should be expressed (29) In both cases we relieved(28) and (29) from the prime notation because there is no riskof confusion Recall also that the elimination of the parallaxis carried out in polar-nodal variables (119903 120579 ] 119877 Θ119873) instead


of Delaunay variables where 119877 is radial velocity Θ = 119866119873 = 119867 = Θ cos 119894 ] = ℎ and 120579 = 119891 + 119892

33 Long-Term Hamiltonian A new transformation fromdouble-prime to triple-prime variables normalizes (29) byremoving the remaining short-period terms related to theradius Then disregarding the prime notation we get thelong-term Hamiltonian in mean (triple-prime) elements

K = minus

120583

2119886

minus

120583

2119886

1205722

1198862

1198692

1205783(1 minus

3

2

1199042)

minus

120583

2119886

1205724

1198864

11986922

1205787[

15

4

minus

15

2

1199042+

105

32

1199044

+

3

8

(2 minus 31199042)

2120578 minus (

3

4

minus

3

4

1199042minus

15

32

1199044) 1205782

minus(

21

8

1199042minus

45

16

1199044) 1198902 cos 2119892]

(30)

which is easily verified to agree with Brouwerrsquos original long-term Hamiltonian compare [11 page 569]

34 Semianalytical Integration The evolution equations aretheHamilton equations of the long-termHamiltonian in (30)namely

119889 (ℓ 119892 ℎ)

119889119905

=

120597K

120597 (119871 119866119867)

119889 (119871 119866119867)

119889119905

= minus

120597K

120597 (ℓ 119892 ℎ)

(31)

Since ℓ and ℎ have been removed by the perturbation theoryboth 119871 and 119867 are integrals of the mean elements systemdefined by (30)

Then the semianalytical integration of the 2times2 truncationof the geopotential performs the following steps

(1) choose osculating elements(2) compute prime elements related to relegation(3) compute second-prime elements related to the elimi-

nation of the parallax(4) compute mean elements related to normalization(5) numerically integrate (31) which allows for very long

stepsizes

At each step of the numerical integration osculatingelements are computed frommean ones by recovering short-period effects This requires undoing sequentially the trans-formations from triple to double primes from double-primesto primes and from primes to osculating

4 Numerical Experiments

41 Preliminary Tests In order to check the efficiency of ournew variant of the relegation algorithm we apply it to testsome of the representative cases provided in [9] In particular

Table 1 Initial osculating elements of selected test cases from [9]

Orbit 119886 (km) 119890 119894 (deg) Ω 120596 (deg) 119872

1 12159596 001 5 0 270 0

2 13375556 010 5 0 270 0

3 18520000 035 5 0 270 0

4 18520000 035 100 0 270 0

in view of the center and bottom plots of Figure 2 of [9] dealwith the case of low-altitude orbits which are amenable to amore advantageous treatment of tesseral effects by using thestandard parallax elimination [6 8] we focus on the examplesrelated to the top plot of Figure 2 of [9] namely a set of orbitsin the MEO region whose initial conditions are detailed inTable 1

Orbit 1 of Table 1 is the orbit with the smallest eccentricityand hence the better convergence of the relegation algorithmis expected Indeed while the propagation of the meanelements alone results in rough errors of about 200 meter forthe semimajor axisO(10minus4) for the eccentricity just a few arcseconds (as) for the inclination tens of as for the Right Ascen-sion of the ascending node (RAAN) and of just a few degfor both the argument of the perigee and the mean anomalyFigure 1 shows that these errors are greatly improved whenshort-period zonal and tesseral effects are recovered evenwithout need of iterating the relegation algorithm Thuserrors fall down to just one meter for the semi-major axisto O(10minus6) for the eccentricity to one milliarcsecond (mas)in the case of inclination below one as for the RAAN and totenths of deg both for the argument of the perigee and meananomaly The latter are shown to combine to just a few maswhen using the nonsingular element 119865 = 120596 + 119872 the meandistance to the node (not presented in Figure 1) Both theerrors in 119865 and Ω disclose a secular trend which is of about09 mas times orbital period for Ω (center plot of Figure 1)part of which should be attributed to the truncation of theperturbation theory to the second order In the case of themean distance to the node the secular trend is of about 13mas times orbital period roughly indicating an along-trackerror of about 76 cm times orbital period

Figure 2 shows that a single iteration of (24) clearlyimproves the errors in semimajor axis eccentricity andinclination aswell as the errors in the argument of the perigeeand mean anomaly which generally enhance by one orderof magnitude On the contrary improvements in the short-period terms are outweighed by the dominant secular trendin the case of Ω as displayed in the fourth row of Figure 2Now the secular trend of the errors in the mean distance tothe node reduces to sim27mascycle roughly proportional toan along-track error of sim16 cm times orbital period

Improvements in the propagation of orbit 1 obtained withfurther iterations of the relegation algorithm are very smallexcept for the eccentricity whose errors reachO(10minus8)with asecond iteration of the relegationNew iterations of (24) showno improvement for this example in full agreement with thescaling by the eccentricity in (23) Indeed performing a singleiterationmeans neglecting terms factored by the square of the


10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)

Figure 1 Errors in the orbital elements of orbit 1 of Table 1 when tesseral short-period effects are recovered without iterations Abscissas areorbital periods corresponding to one month propagation

10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)

Figure 2 Errors in the orbital elements of orbit 1 of Table 1 when tesseral short-period effects are recovered with a single iteration of (24)Abscissas are orbital periods corresponding to one month propagation

eccentricity which is 1198902 = 00001 lt 1198692 for orbit 1 Thereforea second iteration of the relegation would introduce third-order effects which are not considered in the perturbationtheory

Other orbits in Table 1 have a notably larger eccentricitythan orbit 1 a fact thatmakesmore iterations of the relegationalgorithm necessary Thus in view of orbit 2 has an eccen-tricity 119890 = 01 third order of 1198692 effects will appear only atthe third iteration of (24) where 1198903 = O(1198692) This resultedexactly to be the case where onemonth propagation of orbit 2with the semianalytical theory produced errors of about 3 cmin the semi-major axis of O(10minus8) for the eccentricity belowthe mas for the inclination and below the as for both theargument of the perigee and the mean anomaly The seculartrend in Ω is now of half mas per orbit and about 2 mas perorbit in the case of the mean distance to the node roughlyequivalent to an along track error of sim14 cm per orbit

The moderate eccentricity of orbits 3 and 4 makes neces-sary the fifth iteration of the procedure which takes accountof terms of the order of 1198905 asymp 0005 Results for one monthpropagation of orbit 4 with the semianalytical theory arepresented in Figure 3 for the first iteration of the algorithmand in Figure 4 for the fifth Note the secular growing ofthe errors in the mean anomaly (bottom plot of Figure 4)roughly amounting to an along-track error of two meters perorbit The rate of convergence of the relegation is poor fororbits 3 and 4 of Table 1 in agreement with the similar rateof reduction of successive powers of the eccentricity for themoderate value of 119890 in this example

42 Comparisons with Expansions of Elliptic Motion Oncethe performances of the new relegation algorithm have beenchecked it rests to test its relative efficiency when comparedwith the usual approach based on expansions of elliptic


0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100

minus02106timesΔ119890 07

(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)

Figure 3 Errors in the orbital elements of orbit 4 of Table 1 after a one iteration of (23) Abscissas are orbital periods corresponding to onemonth propagation

0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07

minus02107timesΔ119890

(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0

minus015Δ120596(as)

(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)

Figure 4 Errors in the orbital elements of orbit 4 of Table 1 after five iterations of the relegation algorithm Abscissas are orbital periodscorresponding to one month propagation

motion Having in mind that comparisons must limit tothe case of small or moderate eccentricities we decided tolimit the expansion of the true anomaly as a function ofthe mean anomaly to the order of 11989012 Hence expansions ofboth the tesseral Hamiltonian and the generating functionof the tesseral averaging are also limited to O(11989012) butthe transformation equations for ℓ and 119892 as well as thenonsingular element 119865 = ℓ + 119892 are downgraded to the orderof 11989011 compare [39] With these limits for expansions in theeccentricity we find out that the transformation equationsof the tesseral averaging take up to 286 kB when stored in atext file This size places the transformation equations of theexpansions of the elliptic motion approach between those ofthe first (204 kB) and the second iteration of the relegationalgorithm (434 kB) Note that zonal terms are subsequentlyaveraged in closed form in both approaches and hence the

efficiency of each approach is just related to the way in whichthe tesseral averaging has been carried out

Thus in view of the respective sizes of the files containingthe transformation equations we consider the tesseral aver-aging based on expansions of the elliptic motion to be moreefficient than the tesseral relegation in those cases in whichit produces errors that are at least comparable to those of asecond iteration of the relegation On the contrary in thosecases in which the errors produced by the expansions of theelliptic motion approach are comparable or worse than thoseobtained with a single iteration of the relegation we say thatthe tesseral relegation is more efficient than the expansions inthe eccentricity procedure

It is not a surprise to check that tesseral averaging basedon expansions of the elliptic motion is superior for orbitsof moderate eccentricity Thus when applied to orbit 4 of


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)

Figure 5 Errors in the orbital elements of an almost-circular Galileo-type orbit when tesseral short-period effects are recovered with oneiteration of the relegation Abscissas are orbital periods corresponding to one month propagation

0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)

Figure 6 Same as Figure 5 when using expansions truncated to O(11989012)

Table 1 (119890 = 035) the eccentricity expansions approachreaches errors comparable to but slightly better than those ofFigure 4 where the five iterations of the relegation used inthe construction of Figure 4 require 2 135 kB of text storage75 times bigger than the eccentricity expansions fileThus inthis case we say that the relegation efficiency is about sevenand a half times lesser than the efficiency of the eccentricityexpansion approach On the contrary when applied to orbit1 of Table 1 (119890 = 001) the eccentricity expansions approachgets the same errors as those obtained with two iterations ofthe relegation which are very similar to those of Figure 2Therefore we say in this case that the eccentricity expansionapproach is of comparable efficiency to the relegation withrelative sizes of text storage 434 kB286 kB asymp 15

Finally we provide a comparison for a Galileo-type orbitwith 119886 = 29600 km 119890 = 0001 and 119894 = 56 deg Notethat the efficiency of the classical relegation approach of [9]will be very poor for orbits of these characteristics because

of the small ratio 119899119864119899 asymp 712 that must be used asthe relegation coefficient Besides Galileo orbits have thesmallest eccentricity of all the cases tested Results for this lastcomparison are presented in Figure 5 for a single iterationof the relegation approach and Figure 6 for the eccentricityexpansions case We note that the errors are of a similarmagnitude in both cases with better results for the meandistance to the node when using relegation and better resultsfor the eccentricity when using eccentricity expansions

If we take a second iteration of the relegation thenerrors in the eccentricity of both cases match and in view ofthe respective magnitudes of the errors in 119865 the advantagegoes clearly to the side of the relegation Since errors in119865 are always better when using relegation we say thatthe relegation is more efficient than the expansions in theeccentricity approach for Galileo orbits Besides in orderto provide an efficiency estimator analogous to the previouscases we checked that a ldquomixed relegationrdquomdashwhich includes


the transformation equations of a single iteration for ℓ 119892 ℎand 119867 and the transformation equations after two iterationsfor 119871 and 119866 making a text file of 260 kBmdashprovides the sameerrors as the eccentricity expansions approach maintainingthe clear advantage for 119865 Therefore we might say that forGalileo orbits the efficiency is at least 11 times better thanthe eccentricity expansions for all the elements except for 119865for which the efficiency of the relegation is at least 14 timesbetter than the eccentricity expansions

However in view of tesseral relegation for subsyn-chronous orbits seems to be useful only in the case of thelower eccentricity orbits we further investigate this caseThus it is important to note that because the term 1119903

3

in (27) the initial iteration in the solution of (24) givesa term 11988220 that involves eccentricity polynomials (notexpansions) of degree 3 Since each iteration of the relegationinvolves multiplication by the eccentricity we simplify thecomputation of the generating function by neglecting thosepowers of 119890 that are higher than 3 in successive iterationsof the procedure Proceeding in this way the transformationequations of the tesseral relegation can be shortened toaccommodate in a file of 1437 kB for a single iteration of theprocedure and only 1857 kB for two iterations Therefore weconclude that for the tesseral model investigated relegationmay double the efficiency of tesseral averaging for the lowereccentricity orbits

5 Conclusions

The relegation algorithm has been suitably formulated inDelaunay variables contrary to the standard approach inpolar-nodal variables The use of Delaunay variables relievesthe relegation algorithm from unnecessary subtleties relatedto the particular algebra of the parallactic functions anddiscloses the fundamental differences between the casesof super- and subsynchronous orbit relegation Specificallywhile the relegation factor for super-synchronous orbits isthe ratio of one day to the satellitersquos orbital period inthe subsynchronous case the relegation factor unavoidablydepends on the eccentricity a fact that limits applicationof the subsynchronous relegation to orbits with small andmoderate eccentricities Besides it deserves mentioning thatthe present formulation of the tesseral relegation does notneed any special treatment of the 119898-daily terms albeit ofcourse allowing for separate treatment if preferred

Computations in this paper show that the tesseral relega-tion algorithm for subsynchronous orbitsmay be competitivewith classical expansions of ellipticmotion but only for orbitswith small eccentricities a case in which relegation doublesthe efficiency of the series expansions approach This factconstrains the practical application of the relegation but inno way invalidates it because low-eccentricity orbits are usedin a variety of aerospace engineering applications

Acknowledgment

Part of this research has been supported by the Governmentof Spain (Projects AYA 2009-11896 and AYA 2010-18796 andgrants from Gobierno de La Rioja fomenta 201016)

References

[1] A Deprit ldquoThemain problem in the theory of artificial satellitesto order fourrdquo Journal of Guidance Control and Dynamics vol4 no 2 pp 201ndash206 1981

[2] S Coffey and A Deprit ldquoThird-order solution to the mainproblem in satellite theoryrdquo Journal of Guidance and Controlvol 5 no 4 pp 366ndash371 1982

[3] L M Healy ldquoThe main problem in satellite theory revisitedrdquoCelestial Mechanics amp Dynamical Astronomy An InternationalJournal of Space Dynamics vol 76 no 2 pp 79ndash120 2000

[4] A Deprit ldquoThe elimination of the parallax in satellite theoryrdquoCelestial Mechanics vol 24 no 2 pp 111ndash153 1981

[5] A Deprit J Palacian and E Deprit ldquoThe relegation algorithmrdquoCelestial Mechanics amp Dynamical Astronomy vol 79 no 3 pp157ndash182 2001

[6] J F Palacian ldquoAn analytical solution for artificial satellites at lowaltitudesrdquo inDynamics and Astrometry of Natural and ArtificialCelestial Bodies K Kurzynska F Barlier P K Seidelmann andI Wytrzyszczak Eds pp 365ndash370 1994

[7] J F Palacian J F San-Juan and P Yanguas ldquoAnalytical theoryfor the spot satelliterdquoAdvances in the Astronautical Sciences vol95 pp 375ndash382 1996

[8] M Lara J F San-Juan and L Lopez-Ochoa ldquoPrecise analyticalcomputation of frozen-eccentricity lowEarth orbits in a tesseralpotentialrdquo Mathematical Problems in Engineering vol 2013Article ID 191384 13 pages 2013

[9] A M Segerman and S L Coffey ldquoAn analytical theoryfor tesseral gravitational harmonicsrdquo Celestial Mechanics andDynamical Astronomy vol 76 no 3 pp 139ndash156 2000

[10] J F San A Abad M Lara and D J Scheeres ldquoFirst-orderanalytical solution for spacecraft motion about (433) ErosrdquoJournal of Guidance Control and Dynamics vol 27 no 2 pp290ndash293 2004

[11] D Brouwer and G M ClemenceMethods of Celestial Mechan-ics Academic Press New York NY USA 1961

[12] WMKaulaTheory of Satellite Geodesy Applications of Satellitesto Geodesy Dover Mineola NY USA 1966

[13] N X Vinh ldquoRecurrence formulae for the Hansenrsquos develop-mentsrdquo Celestial Mecshanics vol 2 no 1 pp 64ndash76 1970

[14] E Wnuk ldquoTesseral harmonic perturbations for high order anddegree harmonicsrdquoCelestial Mechanics vol 44 no 1-2 pp 179ndash191 1988

[15] F Delhaise and J Henrard ldquoThe problem of critical inclinationcombined with a resonance inmeanmotion in artificial satellitetheoryrdquo Celestial Mechanics amp Dynamical Astronomy vol 55no 3 pp 261ndash280 1993

[16] L D D Ferreira and R Vilhena de Moraes ldquoGPS satellitesorbits resonancerdquo Mathematical Problems in Engineering vol2009 Article ID 347835 12 pages 2009

[17] J C Sampaio A G S Neto S S Fernandes R Vilhenade Moraes and M O Terra ldquoArtificial satellites orbits in 21resonance GPS constellationrdquo Acta Astronautica vol 81 pp623ndash634 2012

[18] J C Sampaio R Vilhena de Moraes and S D S Fernan-des ldquoThe orbital dynamics of synchronous satellites irregularmotions in the 21 resonancerdquo Mathematical Problems in Engi-neering Article ID 405870 22 pages 2012

[19] M Lara J F San-Juan Z J Folcik and P J Cefola ldquoDeepresonant GPS-dynamics due to the geopotentialrdquo Journal of theAstronautical Sciences vol 58 no 4 pp 661ndash676 2012


[20] S L Coffey H L Neal A M Segerman and J J TravisanoldquoAn analytic orbit propagation program for satellite catalogmaintenancerdquo Advances in the Astronautical Sciences vol 90no 2 pp 1869ndash1892 1995

[21] W D McClain ldquoA recursively formulated first-order semiana-lytic artificial satellite theory based on the generalized methodof averaging volume 1 the generalized method of averagingapplied to the artificial satellite problemrdquo Computer SciencesCorporation CSCTR-776010 1977

[22] A R Golikov ldquoNumeric-analytical theory of the motion ofartificial satellites of celestial bodiesrdquo Preprint No 70 KeldyshInstitute of Applied Mathematics USSR Academy of Sciences1990

[23] R J Proulx and W D McClain ldquoSeries representations andrational approximations for Hansen coefficientsrdquo Journal ofGuidance Control and Dynamics vol 11 no 4 pp 313ndash3191988

[24] B Garfinkel ldquoThe disturbing function for an artificial satelliterdquoThe Astronomical Journal vol 70 no 9 pp 699ndash704 1965

[25] B Garfinkel ldquoTesseral harmonic perturbations of an artificialsatelliterdquoThe Astronomical Journal vol 70 no 10 pp 784ndash7861965

[26] S Coffey and K T Alfriend ldquoShort period elimination for thetesseral harmonicsrdquo Advances in the Astronautical Sciences vol46 no 1 pp 87ndash101 1982

[27] A Deprit and S Ferrer ldquoSimplifications in the theory ofartificial satellitesrdquo American Astronautical Society vol 37 no4 pp 451ndash463 1989

[28] M Lara J F San-Juan and L M Lopez-Ochoa ldquoEfficientsemi-analytic integration of GNSS orbits under tesseral effectsrdquoin Proceedings of the 7th International Workshop on SatelliteConstellations and Formation Flying Paper IWSCFF-2013-04-02 Lisbon Portugal March 2013

[29] M Irigoyen and C Simo ldquoNon integrability of the 1198692 problemrdquoCelestialMechanics andDynamical Astronomy vol 55 no 3 pp281ndash287 1993

[30] A Celletti and P Negrini ldquoNon-integrability of the problemof motion around an oblate planetrdquo Celestial Mechanics ampDynamical Astronomy vol 61 no 3 pp 253ndash260 1995

[31] D J Jezewski ldquoAn analytic solution for the 1198692 perturbedequatorial orbitrdquo Celestial Mechanics vol 30 no 4 pp 363ndash3711983

[32] A Deprit ldquoCanonical transformations depending on a smallparameterrdquo Celestial Mechanics vol 1 pp 12ndash30 1969

[33] A Deprit and A Rom ldquoLindstedtrsquos series on a computerrdquo TheAstronomical Journal vol 73 no 3 pp 210ndash213 1968

[34] G R Hintz ldquoSurvey of orbit element setsrdquo Journal of GuidanceControl and Dynamics vol 31 no 3 pp 785ndash790 2008

[35] E Brumberg and T Fukushima ldquoExpansions of elliptic motionbased on elliptic function theoryrdquo Celestial Mechanics ampDynamical Astronomy vol 60 no 1 pp 69ndash89 1994

[36] G Metris P Exertier Y Boudon and F Barlier ldquoLong periodvariations of the motion of a satellite due to non-resonanttesseral harmonics of a gravity potentialrdquo Celestial Mechanics ampDynamical Astronomy vol 57 no 1-2 pp 175ndash188 1993

[37] A Deprit ldquoDelaunay normalisationsrdquo Celestial Mechanics vol26 no 1 pp 9ndash21 1982

[38] K Aksnes ldquoA note on lsquoThe main problem of satellite theory forsmall eccentricities by A Deprit and A Rom 1970rsquordquo CelestialMechanics vol 4 no 1 pp 119ndash121 1971

[39] A Deprit and A Rom ldquoThe main problem of artificial satellitetheory for small andmoderate eccentricitiesrdquoCelestial Mechan-ics vol 2 no 2 pp 166ndash206 1970

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


or not with algorithms devised for dealing in closed formwith short-period effects due to longitude-dependent termsof the geopotential In the case of LEO the advantageclearly favors the closed form solution whose based onthe different perturbation orders of the Keplerian and theCoriolis term tesseral effects can be averaged in closed form[24 25] Furthermore in this particular case the eliminationof the parallax works wonders succeeding in the closed-formelimination of all tesseral terms except for those related to119898-dailies [6 26] This is a typical case in which the complexityof relegation makes it less efficient than normalization [5]or canonical simplification in the present context [27] Onthe other hand Keplerian and Coriolis terms can no longerbe assumed to be of different orders when approachingresonance conditions which is the common case of orbits inthe medium Earth orbits (MEOs) region and above

Trying to give an answer to this question in the case ofMEO we carry out several comparisons in which we restrictthe test model to a 2 times 2 truncation of the geopotentialWaiting for more effective implementations of the relegationalgorithm which makes it useful also for the canonicalsimplification of tesseral resonance problems we limit ourcomparisons to the case of nonresonantmotion In particularwe explore the performances of our algorithm in a set ofselected cases borrowed from [9] However we also considersample orbits with the same semimajor axis as Galileo orbitsand different eccentricities a case that better supports ourclaim that the new formulation of the tesseral relegationalgorithm is valid for orbits with any semimajor axis Besidesbecause both approaches series expansion and tesseral rel-egation after averaging may well end up being Brouwerrsquoslong-term Hamiltonian (see [11 page 569]) the comparisoncriteria will be based on the size of the transformationequations from mean to osculating elements required bythe different theories for obtaining analogous precision inthe results together with the relative time spent in theirevaluation

The paper is organized as follows For completeness inSection 2 we recall the basics of the relegation algorithm forthe particular case of the geopotential to whichwe contributea slightly different but definitely enlightening point of viewwhich justifies the application of the relegation techniqueto low-eccentricity orbits independently of the ratio of theorbital period to the Earthrsquos rotation period [28] In this sec-tion a brief mention to the standard procedure for averaginglongitude-dependent terms when using classical expansionof the elliptic motion is also included Next in Section 3we briefly describe the whole procedure of computing thelong-term Hamiltonian and the transformations equationsfrom mean to osculating elements which we implementfor the simplest case of a second-order truncation of thegeopotential While application of the tesseral relegation tohigher degree and order truncations does not involve anymodifications to the basic algorithm it produces a notableincrease in the size of the series usedmdashas is also the case whenusing the expansions of elliptic motion approach Finallycomparisons between both procedures relegation and seriesexpansions are presented in Section 4 for a variety of testcases and the pros and cons of each method are discussed

2 Averaging the Geopotential

The motion of artificial satellites close to the Earth departsslightly fromKeplerianmotion because of the noncentralitiesof the geopotential This disturbing potential is written in theusual form

119881 =

120583

119903

sum

119895ge2

(

120572

119903

)

119895

[1198691198951198751198950 (sin120593)

minus

119895

sum

119896=1

(119862119895119896 cos 119896120582 + 119878119895119896 sin 119896120582)

times119875119895119896 (sin120593) ]

(1)

where 120583 is the Earthrsquos gravitational parameter 120572 is theEarthrsquos equatorial radius 119875119895119896 are Legendre polynomials 119869119895 =minus1198621198950 119862119895119896 and 119878119895119896 are harmonic coefficients and (120593 120582 119903)

are spherical coordinatesmdashlatitude longitude and distancerespectivelymdashin an Earth-centered Earth-fixed (ECEF) sys-tem of coordinates

Satellite motion under the influence of the disturbingpotential in (1) is generally nonintegrable even in the sim-plest case of the 1198692 problem [29 30] with the remarkableexception of the equatorial case [31] However analytical orsemianalytical integration of perturbed Keplerian motion iscustomarily approached by perturbation methods Namelyshort-period effects are analytically averaged to formulate theproblem in mean elements next either the mean elementsequations are numerically integrated with very long stepsizesor a new averaging of long-period effects provides the secularevolution of the problem

21 Lie Transforms Depritrsquos Triangle In particular the Lietransforms technique [32] allows for automatic computationof the perturbation theory by machine to higher ordersand these days is taken as a standard procedure Brieflyin a Hamiltonian framework the initial Hamiltonian inosculating variables is cast in the form

H = sum

119898ge0

120598119898

119898

1198671198980 (2)

where the small parameter 120598 which establishes the order ofeach perturbation term may be either physical or formal

The averaged Hamiltonian in mean elements

H1015840= sum

119898ge0

120598119898

119898

1198670119898 (3)

is to be constructed by the perturbation method The newHamiltonian terms1198670119898 are one of the sides of ldquoDepritrsquos trian-glerdquo a triangular array of terms derived from the recurrence

119867119899119902 = 119867119899+1 119902minus1 + sum

0le119898le119899

(

119899

119898) 119867119899minus119898 119902minus1119882119898+1 (4)



W = sum

119898ge0

120598119898

119898

119882119898+1 (5)



L0 (119882119898) = 119898 minus 1198670119898 (6)




11986700 = minus120583

2119886

minus 119899119864radic120583119886120578 cos 119894 (7)




11986710 = minus120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (8)




sum

11989511198952119895119898ge0

1199011198951

1 1199011198952

2 sdot sdot sdot 119901119895119898

119898

times sum

11989611198962119896119897ge0

(11987611989611198962119896119897

11989511198952119895119898

cos120573 + 1198781198961 1198962 11989611989711989511198952119895119898

sin120573) (9)




ℓ = 119872 119892 = 120596 ℎ = ]

119871 = radic120583119886 119866 = 119871120578 119867 = 119866 cos 119894(10)


11986700 = minus1205832

21198712minus 119899119864119867

(11)


L0 = 119899120597

120597ℓ

minus 119899119864

120597

120597ℎ

(12)





119882119898 = sum

119895ge0

120575119895119882119898119895 (13)




sum

119895ge0

119899120575119895(

120597119882119898119895

120597ℓ

minus 120575

120597119882119898119895

120597ℎ

) = 119865 (119891 119892 ℎ 119871 119866119867) (14)


sum

119895ge0

120575119895120597119882119898119895

120597ℓ

= sum

119895ge0

120575119895119865119895 (15)

where

1198650 =1

119899

119865 119865119895 =

120597119882119898119895minus1

120597ℎ

(119895 gt 0) (16)


1198862120578119889ℓ = 119903

2119889119891 (17)






L0 = 1198991198862120578

1199032

120597

120597119891

minus 119899119864

120597

120597ℎ

(18)


L0 =119899

1205783(

120597

120597119891

minus 1205783120575

120597

120597ℎ

+ 119890R + 1198902S) (19)

where

R = 2 cos119891 120597

120597119891

S = cos2119891 120597

120597119891

(20)


120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

=

1205783

119899

119865 minus 2119890 cos119891120597119882119898

120597119891

minus 1198902cos2119891

120597119882119898

120597119891

(21)



120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

asymp

1205783

119899

119865 (119891 119892 ℎ 119871 119866119867) (22)


119882119898 = sum

119895ge0

119890119895119882119898119895 (23)


sum

119895ge0

119890119895(

120597119882119898119895

120597119891

minus 1205783120575

120597119882119898119895

120597ℎ

) = sum

119895ge0

119890119895119865119895 (24)

where

1198650 =1205783

119899

119865 1198651 = minus2 cos1198911205971198821198980

120597119891

(25)

and for 119895 gt 1

119865119895 = minus2 cos119891120597119882119898119895minus1

120597119891

minus cos2119891120597119882119898119895minus2

120597119891

(26)





11986720 = minus21205831205722

1199033

2

sum

119896=1

(1198622119896 cos 119896120582 + 1198782119896 sin 119896120582) 1198752119896 (sin120593)

(27)










H = minus

120583

2119886

minus

120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (28)



H = minus

120583

2119886

minus

120583

2119886

1205722

1199032

1198692

1205782(1 minus

3

2

sin2119894)

minus

120583

2119886

1205724

11988621199032

11986922

1205786[

5

4

minus

21

8

sin2119894 + 21

16

sin4119894

+ (

3

8

cos2119894 minus 15

64

sin4119894) 1198902

minus(

21

16

sin2119894 minus 45

32

sin4119894) 1198902 cos 2119892] (29)


R997888rarr (ℓ






K = minus

120583

2119886

minus

120583

2119886

1205722

1198862

1198692

1205783(1 minus

3

2

1199042)

minus

120583

2119886

1205724

1198864

11986922

1205787[

15

4

minus

15

2

1199042+

105

32

1199044

+

3

8

(2 minus 31199042)

2120578 minus (

3

4

minus

3

4

1199042minus

15

32

1199044) 1205782

minus(

21

8

1199042minus

45

16

1199044) 1198902 cos 2119892]

(30)



119889 (ℓ 119892 ℎ)

119889119905

=

120597K

120597 (119871 119866119867)

119889 (119871 119866119867)

119889119905

= minus

120597K

120597 (ℓ 119892 ℎ)

(31)





stepsizes





Orbit 119886 (km) 119890 119894 (deg) Ω 120596 (deg) 119872

1 12159596 001 5 0 270 0

2 13375556 010 5 0 270 0

3 18520000 035 5 0 270 0

4 18520000 035 100 0 270 0






10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)


10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)







0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











W = sum

119898ge0

120598119898

119898

119882119898+1 (5)



L0 (119882119898) = 119898 minus 1198670119898 (6)




11986700 = minus120583

2119886

minus 119899119864radic120583119886120578 cos 119894 (7)




11986710 = minus120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (8)




sum

11989511198952119895119898ge0

1199011198951

1 1199011198952

2 sdot sdot sdot 119901119895119898

119898

times sum

11989611198962119896119897ge0

(11987611989611198962119896119897

11989511198952119895119898

cos120573 + 1198781198961 1198962 11989611989711989511198952119895119898

sin120573) (9)




ℓ = 119872 119892 = 120596 ℎ = ]

119871 = radic120583119886 119866 = 119871120578 119867 = 119866 cos 119894(10)


11986700 = minus1205832

21198712minus 119899119864119867

(11)


L0 = 119899120597

120597ℓ

minus 119899119864

120597

120597ℎ

(12)





119882119898 = sum

119895ge0

120575119895119882119898119895 (13)




sum

119895ge0

119899120575119895(

120597119882119898119895

120597ℓ

minus 120575

120597119882119898119895

120597ℎ

) = 119865 (119891 119892 ℎ 119871 119866119867) (14)


sum

119895ge0

120575119895120597119882119898119895

120597ℓ

= sum

119895ge0

120575119895119865119895 (15)

where

1198650 =1

119899

119865 119865119895 =

120597119882119898119895minus1

120597ℎ

(119895 gt 0) (16)


1198862120578119889ℓ = 119903

2119889119891 (17)






L0 = 1198991198862120578

1199032

120597

120597119891

minus 119899119864

120597

120597ℎ

(18)


L0 =119899

1205783(

120597

120597119891

minus 1205783120575

120597

120597ℎ

+ 119890R + 1198902S) (19)

where

R = 2 cos119891 120597

120597119891

S = cos2119891 120597

120597119891

(20)


120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

=

1205783

119899

119865 minus 2119890 cos119891120597119882119898

120597119891

minus 1198902cos2119891

120597119882119898

120597119891

(21)



120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

asymp

1205783

119899

119865 (119891 119892 ℎ 119871 119866119867) (22)


119882119898 = sum

119895ge0

119890119895119882119898119895 (23)


sum

119895ge0

119890119895(

120597119882119898119895

120597119891

minus 1205783120575

120597119882119898119895

120597ℎ

) = sum

119895ge0

119890119895119865119895 (24)

where

1198650 =1205783

119899

119865 1198651 = minus2 cos1198911205971198821198980

120597119891

(25)

and for 119895 gt 1

119865119895 = minus2 cos119891120597119882119898119895minus1

120597119891

minus cos2119891120597119882119898119895minus2

120597119891

(26)





11986720 = minus21205831205722

1199033

2

sum

119896=1

(1198622119896 cos 119896120582 + 1198782119896 sin 119896120582) 1198752119896 (sin120593)

(27)










H = minus

120583

2119886

minus

120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (28)



H = minus

120583

2119886

minus

120583

2119886

1205722

1199032

1198692

1205782(1 minus

3

2

sin2119894)

minus

120583

2119886

1205724

11988621199032

11986922

1205786[

5

4

minus

21

8

sin2119894 + 21

16

sin4119894

+ (

3

8

cos2119894 minus 15

64

sin4119894) 1198902

minus(

21

16

sin2119894 minus 45

32

sin4119894) 1198902 cos 2119892] (29)


R997888rarr (ℓ






K = minus

120583

2119886

minus

120583

2119886

1205722

1198862

1198692

1205783(1 minus

3

2

1199042)

minus

120583

2119886

1205724

1198864

11986922

1205787[

15

4

minus

15

2

1199042+

105

32

1199044

+

3

8

(2 minus 31199042)

2120578 minus (

3

4

minus

3

4

1199042minus

15

32

1199044) 1205782

minus(

21

8

1199042minus

45

16

1199044) 1198902 cos 2119892]

(30)



119889 (ℓ 119892 ℎ)

119889119905

=

120597K

120597 (119871 119866119867)

119889 (119871 119866119867)

119889119905

= minus

120597K

120597 (ℓ 119892 ℎ)

(31)





stepsizes





Orbit 119886 (km) 119890 119894 (deg) Ω 120596 (deg) 119872

1 12159596 001 5 0 270 0

2 13375556 010 5 0 270 0

3 18520000 035 5 0 270 0

4 18520000 035 100 0 270 0






10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)


10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)







0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

119895ge0

119899120575119895(

120597119882119898119895

120597ℓ

minus 120575

120597119882119898119895

120597ℎ

) = 119865 (119891 119892 ℎ 119871 119866119867) (14)


sum

119895ge0

120575119895120597119882119898119895

120597ℓ

= sum

119895ge0

120575119895119865119895 (15)

where

1198650 =1

119899

119865 119865119895 =

120597119882119898119895minus1

120597ℎ

(119895 gt 0) (16)


1198862120578119889ℓ = 119903

2119889119891 (17)






L0 = 1198991198862120578

1199032

120597

120597119891

minus 119899119864

120597

120597ℎ

(18)


L0 =119899

1205783(

120597

120597119891

minus 1205783120575

120597

120597ℎ

+ 119890R + 1198902S) (19)

where

R = 2 cos119891 120597

120597119891

S = cos2119891 120597

120597119891

(20)


120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

=

1205783

119899

119865 minus 2119890 cos119891120597119882119898

120597119891

minus 1198902cos2119891

120597119882119898

120597119891

(21)



120597119882119898

120597119891

minus 1205783120575

120597119882119898

120597ℎ

asymp

1205783

119899

119865 (119891 119892 ℎ 119871 119866119867) (22)


119882119898 = sum

119895ge0

119890119895119882119898119895 (23)


sum

119895ge0

119890119895(

120597119882119898119895

120597119891

minus 1205783120575

120597119882119898119895

120597ℎ

) = sum

119895ge0

119890119895119865119895 (24)

where

1198650 =1205783

119899

119865 1198651 = minus2 cos1198911205971198821198980

120597119891

(25)

and for 119895 gt 1

119865119895 = minus2 cos119891120597119882119898119895minus1

120597119891

minus cos2119891120597119882119898119895minus2

120597119891

(26)





11986720 = minus21205831205722

1199033

2

sum

119896=1

(1198622119896 cos 119896120582 + 1198782119896 sin 119896120582) 1198752119896 (sin120593)

(27)










H = minus

120583

2119886

minus

120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (28)



H = minus

120583

2119886

minus

120583

2119886

1205722

1199032

1198692

1205782(1 minus

3

2

sin2119894)

minus

120583

2119886

1205724

11988621199032

11986922

1205786[

5

4

minus

21

8

sin2119894 + 21

16

sin4119894

+ (

3

8

cos2119894 minus 15

64

sin4119894) 1198902

minus(

21

16

sin2119894 minus 45

32

sin4119894) 1198902 cos 2119892] (29)


R997888rarr (ℓ






K = minus

120583

2119886

minus

120583

2119886

1205722

1198862

1198692

1205783(1 minus

3

2

1199042)

minus

120583

2119886

1205724

1198864

11986922

1205787[

15

4

minus

15

2

1199042+

105

32

1199044

+

3

8

(2 minus 31199042)

2120578 minus (

3

4

minus

3

4

1199042minus

15

32

1199044) 1205782

minus(

21

8

1199042minus

45

16

1199044) 1198902 cos 2119892]

(30)



119889 (ℓ 119892 ℎ)

119889119905

=

120597K

120597 (119871 119866119867)

119889 (119871 119866119867)

119889119905

= minus

120597K

120597 (ℓ 119892 ℎ)

(31)





stepsizes





Orbit 119886 (km) 119890 119894 (deg) Ω 120596 (deg) 119872

1 12159596 001 5 0 270 0

2 13375556 010 5 0 270 0

3 18520000 035 5 0 270 0

4 18520000 035 100 0 270 0






10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)


10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)







0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11986720 = minus21205831205722

1199033

2

sum

119896=1

(1198622119896 cos 119896120582 + 1198782119896 sin 119896120582) 1198752119896 (sin120593)

(27)










H = minus

120583

2119886

minus

120583

2119903

1205722

11990321198692 [1 minus

3

2

sin2119894 + 3

2

sin2119894 cos (2119891 + 2119892)] (28)



H = minus

120583

2119886

minus

120583

2119886

1205722

1199032

1198692

1205782(1 minus

3

2

sin2119894)

minus

120583

2119886

1205724

11988621199032

11986922

1205786[

5

4

minus

21

8

sin2119894 + 21

16

sin4119894

+ (

3

8

cos2119894 minus 15

64

sin4119894) 1198902

minus(

21

16

sin2119894 minus 45

32

sin4119894) 1198902 cos 2119892] (29)


R997888rarr (ℓ






K = minus

120583

2119886

minus

120583

2119886

1205722

1198862

1198692

1205783(1 minus

3

2

1199042)

minus

120583

2119886

1205724

1198864

11986922

1205787[

15

4

minus

15

2

1199042+

105

32

1199044

+

3

8

(2 minus 31199042)

2120578 minus (

3

4

minus

3

4

1199042minus

15

32

1199044) 1205782

minus(

21

8

1199042minus

45

16

1199044) 1198902 cos 2119892]

(30)



119889 (ℓ 119892 ℎ)

119889119905

=

120597K

120597 (119871 119866119867)

119889 (119871 119866119867)

119889119905

= minus

120597K

120597 (ℓ 119892 ℎ)

(31)





stepsizes





Orbit 119886 (km) 119890 119894 (deg) Ω 120596 (deg) 119872

1 12159596 001 5 0 270 0

2 13375556 010 5 0 270 0

3 18520000 035 5 0 270 0

4 18520000 035 100 0 270 0






10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)


10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)







0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












K = minus

120583

2119886

minus

120583

2119886

1205722

1198862

1198692

1205783(1 minus

3

2

1199042)

minus

120583

2119886

1205724

1198864

11986922

1205787[

15

4

minus

15

2

1199042+

105

32

1199044

+

3

8

(2 minus 31199042)

2120578 minus (

3

4

minus

3

4

1199042minus

15

32

1199044) 1205782

minus(

21

8

1199042minus

45

16

1199044) 1198902 cos 2119892]

(30)



119889 (ℓ 119892 ℎ)

119889119905

=

120597K

120597 (119871 119866119867)

119889 (119871 119866119867)

119889119905

= minus

120597K

120597 (ℓ 119892 ℎ)

(31)





stepsizes





Orbit 119886 (km) 119890 119894 (deg) Ω 120596 (deg) 119872

1 12159596 001 5 0 270 0

2 13375556 010 5 0 270 0

3 18520000 035 5 0 270 0

4 18520000 035 100 0 270 0






10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)


10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)







0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10minus1

0 50 100 150

Δ119886(m

)

(a)

20minus2

minus4

106timesΔ119890

0 50 100 150(b)

1050

minus05

minus1

0 50 100 150

Δ119894(m

as)

(c)

0minus005

minus01

minus015

0 50 100 150

ΔΩ(as)

(d)

001

minus003

0 50 100 150

Δ120596(deg)

(e)

003

minus001

0 50 100 150

Δ119872

(deg)

(f)


10minus1

0 50 100 150

Δ119886(cm)

(a)0 50 100 150

2

0107timesΔ119890

(b)

0 50 100 150

04020Δ

119894(m

as)

(c)0 50 100 150

0

minus015ΔΩ(as)

(d)

0 50 100 150

40minus4Δ

120596(as)

(e)0 50 100 150

40minus4Δ

119872(as)

(f)







0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100

5minus4

minus13

Δ119886(m

)

(a)0 20 40 60 80 100


(b)

0 20 40 60 80 100

012

0Δ119894(as)

(c)0 20 40 60 80 100

008

minus002ΔΩ(as)

(d)

0 20 40 60 80 100

02

minus13Δ120596(as)

(e)0 20 40 60 80 100

30

0Δ119872

(as)

(f)


0 20 40 60 80 100

03

minus08Δ119886(m

)

(a)0 20 40 60 80 100

07


(b)

0 20 40 60 80 100

9

0

Δ119894(m

as)

(c)0 20 40 60 80 100

7

minus2ΔΩ(m

as)

(d)

0 20 40 60 80 100

0


(e)0 20 40 60 80 100

2

0Δ119872

(as)

(f)







0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

01

minus06

108timesΔ119890

(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

0

minus18Δ119865(m

as)

(e)


0 10 20 30 40 50

15

minus05Δ119886(cm)

(a)0 10 20 30 40 50

11


(b)

0 10 20 30 40 50

003

0Δ119894(m

as)

(c)0 10 20 30 40 50

003

minus007

ΔΩ(m

as)

(d)

0 10 20 30 40 50

03

minus01Δ119865(as)

(e)









3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3


5 Conclusions



Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









averaging tesseral effects: closed form relegation versus … · averaging tesseral effects: closed...

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