IWSCFF-2013-04-02

EFFICIENT SEMI-ANALYTIC INTEGRATION OF GNSS ORBITSUNDER TESSERAL EFFECTS

Martin Lara,∗ Juan F. San-Juan,† Luis M. Lopez-Ochoa‡

The algebra underlying the elimination of the parallax transformation is known tobe useful in relegating short-period effects due to tesseral harmonics of the Geopo-tential. In the case of low-eccentricity orbits, a judicious selection of the generat-ing function of the relegation algorithm allows for a straightforward simplification.Application to low-eccentricity orbits in the medium Earth orbits region illustratesthe benefits of the relegation approach in both resonant and non-resonant cases.

INTRODUCTION

Deprit’s efforts in solving the main problem of the artificial satellite theory to higher orders inclosed form led him to devise the elimination of the parallax transformation.11 Based on the invari-ance of Poisson brackets, whose algebra can be performed in a more convenient set of variables atany step of a perturbation approach by Lie transforms,3, 9, 19 Deprit realized that a set of ideal ele-ments for perturbed Keplerian systems10 included the three parallactic functions that provide a veryconvenient connection between the polar-nodal and Delaunay charts. Thus, the semi-latus rectumand the projections of the eccentricity vector in the direction of the ascending node and its perpen-dicular in the orbital plane, allow one to cast the zonal perturbation of the gravitational potential insuch a form that the explicit dependence on the radius r vanishes except for a factor involving 1/r2.The explicit appearance of the argument of the latitude is, then, conveniently averaged by means ofa canonical transformation of the Lie type operating in the polar-nodal chart. The resulting Hamil-tonian still depends on remaining short-period terms related to the radius, so rather than normalizedit has been simplified.13, 14 But the fact that the simplified Hamiltonian depends on the radius onlythrough inverse of its square, allows one to take full advantage of the differential relation betweenthe mean and true anomalies to completely remove the mean anomaly in closed form by means of alater Delaunay normalization.12

The elimination of the parallax simplification has proven very efficient in reducing the num-ber and complexity of perturbation terms of zonal problems, and has been customarily used as apreparatory transformation to obtain higher orders in closed form in a later averaging over the meananomaly.6, 8, 18 Remarkably, it helped to demonstrate that the critical inclination is an intrinsic sin-gularity of the main problem.7 In addition, the elimination of the parallax is demanded at the onsetwhen carrying out the elimination of the perigee simplification,2 a canonical transformation thatfurther paves the way for computing the secular terms of a zonal perturbation theory by additionally

∗C/Columnas de Hercules 1, ES-11100 San Fernando, Spain†Dep. Matematicas y Computacion, Universidad de La Rioja, ES-26004 Logrono, Spain‡Dep. Ingenierıa Mecanica, Universidad de La Rioja, ES-26004 Logrono, Spain

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simplifying the Hamiltonian to be fully normalized.25 Another merit of the elimination of the par-allax transformation is that it can be used in the computation of natural intermediaries of perturbedKeplerian motion.11 This use of the elimination of the parallax is not restricted to the case of orbitalmotion and can also be exploited in the modeling of roto-orbital dynamics.17

In general, the long-term behavior of artificial satellites resulting from Geopotential forces is cor-rectly described by zonal models, with the remarkable exception of tesseral resonances. Still, to beeffective and precise, artificial satellite theories should take into account second order perturbationsdue to tesseral harmonics (see Ref. 29 for a review on the topic) and, specifically, the contributionof the so-called m-daily terms.4 The common approach is to expand the tesseral perturbation inpower series of the eccentricity and Fourier series of the mean anomaly (see Ref. 21, and referencestherein). However, the question on how the elimination of tesseral short-period effects could benefitfrom the elimination of the parallax transformation arose as soon as this simplification techniquewas devised.

The original proposal for an “extended parallax transformation” was to split the generating func-tion into a zonal part and a tesseral part, so that both parts can be constructed within the algebra ofthe parallactic functions to any order.5 When applying the (extended) elimination of the parallaxtransformation to a tesseral problem, the simplification of short-periodic terms is not as radical as inthe zonal case, and the Poisson series representing the tesseral perturbation remains with a similarexplicit dependence on the argument of latitude as it originally had. Nevertheless, because it hasbeen released from its dependence on the radius, the tesseral Hamiltonian after the elimination ofthe parallax is more amenable to be normalized in a later Delaunay normalization —which we recallthat requires the usual expansion of the longitude dependent terms in standard series of the eccen-tricity and mean anomaly. However, the simplification of the Hamiltonian provided by the extendedparallax transformation is not for free, and is usually penalized with a non-negligible increase in thesize of the generating function.

The extended parallax transformation operates away from resonances of the orbit mean motionwith the system’s rotation rate, and it may accept straightforward simplifications when these fre-quencies are of different orders in the perturbation scheme. In that case the Coriolis term decouplesfrom the kernel of the Lie operator and, consequently, the explicit dependence of the Hamiltonianon the argument of the latitude can be fully eliminated. Based on the higher order of the system’srotation rate, it follows that the remaining short-period terms can be completely removed fromthe Hamiltonian in closed form of the eccentricity by means of a standard Delaunay normaliza-tion.23, 24, 28

The original approach of Coffey and Alfriend5 to the extended parallax transformation is rarelycited in the literature. However, it enjoys the undeniable merit of disclosing the relegation effectthat it operates on tesseral terms. Thus, after the elimination of the parallax, the tesseral part ofthe Hamiltonian, which remains essentially as intricate as in the original problem, can be rewrittento show explicitly the ratio of the system’s rotation rate to the satellite’s mean motion as a factor.Therefore, in those cases in which this ratio is small, such factorization reveals that the consequenceof the extended parallax transformation is to relegate the effects of the tesseral perturbation. Hence,a different approach may be taken in the simplification of tesseral terms based on the relegationalgorithm.15, 22, 27

As reported in the literature, the relegation of tesseral effects implicitly assumes the non-com-mensurability of the main frequencies of the motion, finding the expected gradual deterioration in

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the solution when approaching to the resonance condition. Another drawback of the relegation isthat it may converge slowly for orbits with moderate eccentricity. In these cases, because of thegrowth in the number of terms of the generating function with each iteration, one needs to find abalance between number of iterations and accuracy.27

Here we obtain the solution of the relegation algorithm in a slightly different way from that pro-posed by Segerman and Coffey.27 In our approach, the displacement of tesseral effects to higherorders is not based on having a small ratio between the mean frequencies of the motion, althoughanalysis of sub-synchronous orbits may benefit of it. The relegation is produced as a result from thefactorization of tesseral terms by the logarithmic derivative of the radius and operates in both sub-and super-synchronous orbits with small or moderate eccentricities. In addition, it clearly disclosesthe relevance ofm-daily terms in the case of low-altitude orbits, meaning that the relegation is moreeffective for the higher altitude orbits. The relegation of the tesseral effects is straightforward, andprepares the Hamiltonian for a subsequent closed form normalization of non-resonant cases up tothe truncation order. Furthermore, the new approach also allows for Hamiltonian simplification ofresonant cases without need of expanding the center equation. Thus, Hamiltonian terms relatedto tesseral resonances, which can be traced directly in polar-nodal coordinates, are left untouchedand the relegation is only applied to tesseral terms that are free from resonances. The proper treat-ment of the resonance problem in the simplified Hamiltonian can then be approached via standardprocedures.21

The relegation algorithm finds immediate application to the case of Global Navigation SatelliteSystems (GNSS). Because usual GNSS satellites are placed in low-eccentricity orbits in the MediumEarth Orbit (MEO) region, their logarithmic derivative of the radius is very small. Hence, the effectof tesseral terms is immediately relegated to higher orders. The new approach is fully documentedfor a 2 × 2 tesseral model, which includes the main resonant terms that drive the pendular dynam-ics of GPS orbits,1, 16, 20, 26 but the relegation has been implemented for a 5 × 5 truncation of theGeopotential, which is the model used in the simulations.

ELIMINATING THE PARALLAX FROM THE GRAVITATIONAL POTENTIAL

The elimination of the parallax11 is a canonical transformation of the Lie type designed to removeshort-period terms from the gravity potential up to a factor of 1/r2, where r is the radius from theorigin. It makes use of the “parallactic functions” p, C and S that, in the polar-nodal setting, arestate functions of the canonical variables (r, θ, ν, R,Θ, N) given by

p =Θ2

µ, C =

Θ

p

(pr− 1)

cos θ +R sin θ, S =Θ

p

(pr− 1)

sin θ −R cos θ, (1)

where µ is the gravitational constant, θ is the argument of the latitude, ν the argument of the node,R is radial velocity, Θ is the modulus of the angular momentum vector, N = Θ cos I , and I is theorbit inclination. Then, the conic solution for a Keplerian orbit

r =p

1 + e cos f=

p

1 + e cos θ cosω + e sin θ sinω, (2)

where e is the eccentricity, f is the true anomaly, and ω is the argument of the perigee, can be castin the form of the identity

1

r=

1

p

(1 +

pC

Θcos θ +

pS

Θsin θ

), (3)

3

which can be used to express inverse powers of the radius as Fourier series in the argument oflatitude whose coefficients are functions of p, C, and S.

An important property of the parallactic functions is that if a function F can be expressed asF ≡ F (p, C, S, θ,Θ, N), then, in the Keplerian flow

H0 =1

2

(R2 +

Θ2

r2

)− µ

r− ωEN, (4)

where ωE is the system rotation rate, the Lie derivative of F is simply11

L0(F ) =Θ

r2∂F

∂θ. (5)

The fundamental equation (5) can be used in a perturbation scheme to simplify a class of Hamil-tonians of the form

H = H0,0 +Θ2

r2

∑j≥0

εj

j!Hj,0, (6)

where H0,0 = H0 is given in Eq. (4), ε is a small parameter which manifests the influence ofeach term of the Hamiltonian (6), and Hj,0 are trigonometric polynomials in θ whose coefficientscan be written in terms of (p, C, S,N,Θ). Using Deprit’s inductive algorithm by Lie transforms,9

the generating function of the simplifying transformation W =∑

j≥0(εj/j!)Wj+1, is obtained by

solving the homological equation

L0(Wj) ≡Θ

r2∂Wj

∂θ= H0,j −H0,j , (7)

where H0,j carries terms derived from previous computations and is always factored by Θ2/r2, andH0,j remains unknown. If we select

H0,j =1

2π

∫ 2π

0H0,j dθ,

then factors 1/r2 cancel out in the homological equation (7) and Wj is computed by quadrature upto any order j of the perturbation approach without leaving the algebra of parallactic functions.

As a result of the elimination of the parallax transformation the Hamiltonian is averaged overshort-period terms up to a factor 1/r2. This fact is clearly noted when reformulating the sim-plified Hamiltonian in Delaunay variables, where the state functions C = (G/p) e cosω, andS = (G/p) e sinω, G ≡ Θ, are obviously free from short-periodics. Full details on the elimi-nation of the parallax simplification can be consulted in the original work of Deprit.11

Zonal problem

In the artificial satellites theory, the Hamiltonian is formulated as a perturbation of the Keple-rian motion due to the non-centralities of the Geopotential. The zonal problem only deals withterms of the disturbing function independent of the longitude. When using polar-nodal variables thedisturbing function of the zonal problem can be cast into the form

Θ2

r2

∑j≥2

αj

pjVj (8)

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where α is the Earth’s equatorial radius, and terms Vj are Fourier series in θ whose coefficients arefunction of p, C, S, Θ and N . Therefore, the zonal problem Hamiltonian of disturbing functionEq. (8) pertains to the class of Hamiltonians defined by Eq. (6), and the elimination of the paral-lax can be applied to any desired order, thus reducing dramatically the number and complexity ofperturbation terms.

The simplification of a zonal Hamiltonian provided by the elimination of the parallax greatlyexpedites a later normalization of the problem, and hence is customarily used as a preparatorytransformation when computing higher order normalizations in closed form.6, 8, 18

Tesseral case

In a rotating frame with constant rotation rate ωE, the non-central part of the gravity potential isconveniently written in the tesseral case as

Θ2

r2

∑j≥2

αj

pj

∑k≥0

Vk,j

where the functions Vk,j are now Poisson series in θ and ν, in general, with coefficients that areexpressed in terms of p, C, S, Θ and N . The appearance of longitude dependent terms of the formF ≡ F (p, C, S, θ, ν,Θ, N) in the tesseral problems complicates the Lie derivative

L0(F ) =Θ

r2∂F

∂θ− ωE

∂F

∂ν.

Because the elimination of the parallax transformation requires to remain within the requiredalgebra of the parallactic functions, Coffey and Alfriend develop the extended parallax elimination,5

in which a solution to the homological equation

L0(Wj) ≡Θ

r2∂Wj

∂θ− ωE

∂Wj

∂ν= H0,j −H0,j (9)

is found by separating the contribution of zonal and tesseral terms in the involved functions. Now,

H0,j =Θ2

r2Kj,Z +

Θ2

r2Kj,T , (10)

where the term Kj,Z does not depend on ν, while Kj,T carries the longitude dependent terms. Thelatter itself is split into a part Kj,S that depends on ν but not on θ (related to the so-called m-dailyterms), and another part Kj,P depending on both ν and θ

Kj,T = Kj,P +Kj,S .

This latter splitting is done to prevent the introduction of secular terms in the transformation whenintegrating m-dailies with the proposed method of solution.

The generating function is in turn split into two partsWj = Wj,Z+Wj,T , so that the homologicalequation (9) takes the form

Θ

r2∂Wj,Z

∂θ+

Θ

r2∂Wj,T

∂θ− ωE

∂Wj,T

∂ν=

Θ2

r2Kj,Z +

Θ2

r2Kj,P +

Θ2

r2Kj,S −H0,j . (11)

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A solution to Eq. (11) is then obtained by, first, imposing the constraint

Θ

r2∂Wj,T

∂θ=

Θ2

r2Kj,P , (12)

from which r cancels out and, therefore, Wj,T = Wj,T (p, C, S, θ, ν,Θ, N) is obtained by quadra-ture. The new Hamiltonian term H0,j is then chosen

H0,j = ωE∂Wj,T

∂ν+

Θ2

r2Kj,S +

Θ2

r2〈Kj,Z〉θ. (13)

Finally, the zonal part of the generating function is computed from

Wj,Z = Θ

∫(Kj,Z − 〈Kj,Z〉θ) dθ, (14)

which is also free from r. In this way, Wj has the required form so that subsequent Poisson bracketcomputations in the Lie triangle9 allow the resulting terms to remain within the algebra of parallacticfunctions.

We note that the extended parallax transformation does not prevent the explicit appearance of theargument of the latitude in tesseral terms, except for m-dailies, rather limiting to the elimination ofthe parallactic factors. Although it reaches a clear reduction in the number of terms of the simplifiedHamiltonian, this reduction is balanced with a non-negligible increase in the number of terms of thegenerating function, c.f. Table I of Ref. 5. Besides, since a later normalization of tesseral termsdoes require the usual expansion of the true anomaly in terms of the mean one, the efficiency ofthis simplification as part of a normalization process may be questioned. A relevant exception isthe case in which the system’s rotation rate is of higher order than the Keplerian motion. Then, thekernel of the Lie derivative is the usual Keplerian motion in inertial space while the Coriolis termis taken as a perturbation, and hence the term ωE (∂Wj,T /∂ν) is no longer part of the homologicalequation.23, 28

RELEGATION OF TESSERAL EFFECTS

It is worth noting that Eq. (12) may be written

Wj,T =1

n

∫Θ2

r2Kj,P dθ,

where n = Θ/r2 is the main part of the rate of the argument of latitude. Then, the part of the newHamiltonian (13) that carries the tesseral terms may be rewritten

ωE∂Wj,T

∂ν=ωE

n

(Θ2

r2

∫∂Kj,P

∂νdθ

)(15)

showing that the tesseral terms remain essentially unaltered except for the factor ωE/n betweenthe rotation rate of the system and the main frequency of the orbital motion. When ωE n wemay say that the tesseral effects have been relegated to a higher order, a case that has been namedinverse relegation as opposite to the case of super-synchronous orbits.27 This relegation effect of theextended parallax transformation prompts a different perspective in finding the perturbtion solutionto a tesseral problem.

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The relegation approach

Similarly to the extended elimination of the parallax algorithm, in a relegation approach thegenerating function is also split into different parts. However, because of its general character, therelegation algorithm may be applied to a wide variety of problems, and is not constrained at all to thespecific algebra of the parallactic functions, not even to the use of polar-nodal variables.15 Specifi-cally, in what is relevant to the tesseral problem, the generating function of the relegation may alsodepend on the variable r.27 Therefore, the Lie derivative of a function F ≡ F (r, θ, ν, C, S, p,Θ, N)must be taken in its whole form

L0(F ) = R∂F

∂r+

Θ

r2∂F

∂θ− ωE

∂F

∂ν,

thus producing three different types of Hamiltonian terms in the Lie triangle iterations.

Then, if we separate into zonal and tesseral parts the known Hamiltonian terms as well as thegenerating function, at each order j of the simplification algorithm we obtain the homological equa-tion

L0(Wj,Z) + L0(Wj,T ) =Θ2

r2(Kj,Z +Kj,T )−H0,j . (16)

The ideal situation would be to solve Wj,T from the constraint

L0(Wj,T ) =Θ2

r2Kj,T (17)

which incorporates all tesseral effects to the generating function. Then, H0,j would be chosen as inthe zonal case

H0,j =Θ2

r2〈Kj,Z〉θ.

Finally, Wj,Z would be solved from

L0(Wj,Z) =Θ2

r2Kj,Z −H0,j .

Unfortunately, finding a solution to Eq. (17) is not possible in general. Hence, algorithms in theliterature make the implicit assumption that the effect of R∂Wj,T /∂r and either (Θ/r2) ∂Wj,T /∂θor ωE ∂Wj,T /∂ν is small when compared to the relevant part of the Lie derivative, which will beeither ωE ∂Wj,T /∂ν or (Θ/r2) ∂Wj,T /∂θ. In consequence, an ad hoc solution to the main part ofEq. (17) is found in agreement with this implicit assumption. In spite of the fact that subsequentdifferentiation of such computed Wj,T with respect to r and either ν or θ, which is required inthe solution of the homological equation (16), produces new tesseral terms, they are shown to befactored by the ratio between the main frequencies of the motion and, hence, have smaller influencethan corresponding terms of the original Hamiltonian. Successive iterations of the procedure, ifrequired, will relegate the effect of the tesseral terms to higher orders.

Basically, the assumptions of the relegation algorithm are that the eccentricity is not large andthat the ratio of the rotation rate of the system to the satellite’s mean motion is small. In the case ofsuper-synchronous orbits, the algorithm may be reformulated under the assumption that the ratio ofthe satellite’s mean motion to the rotation rate of the system is small. As mentioned before, the morebeneficial case occurs when the ratio of the rotation rate of the system to the satellite’s mean motion

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is of the order of the small parameter, a fact that notably simplifies the perturbation approach in thecomputation of the simplification solution.23, 28 On the contrary, the convergence of the algorithmwill be poor when any of these quantities is not small. Full details on the procedure can be consultedin Ref. 27.

The case of close periods

Tests performed by Segerman and Coffey on the inverse relegation algorithm showed that, asexpected, the best results are obtained for the lower eccentricities, whereas a gradual degradationoccurs closer to the tesseral resonance altitudes.27 We propose a slightly different alternative forthe relegation of tesseral effects, which, on one side, simplifies the relegation of non-resonant cases,and, on the other side, clarifies how the relegation should be applied to the cases of resonant motion.

To begin with, the homological equation (16) is written explicitly

Θ

r2∂Wj,Z

∂θ+R

∂Wj,T

∂r+

Θ

r2∂Wj,T

∂θ− ωE

∂Wj,T

∂ν=

Θ2

r2Kj,T +

Θ2

r2Kj,Z −H0,j (18)

where, as before, terms Kj,Z do not depend on ν, and those Kj,T take account of all tesseral terms,including m-dailies. Equation (18) is solved as follows.

First, at difference from Segerman and Coffey, who solved Wj,T by quadrature for all the dif-ferent kind of terms that can appear either in the Geopotential or are derived in the computationprocedure,27 we compute Wj,T as a solution of the linear partial differential equation

Θ

r2∂Wj,T

∂θ− ωE

∂Wj,T

∂ν=

Θ2

r2Kj,T . (19)

Then, after solving Wj,T from Eq. (19), we choose,

Rj = −R∂Wj,T

∂r(20)

and

H0,j =Θ2

r2〈Kj,Z〉θ +Rj . (21)

Finally, Wj,Z is computed as in Eq. (14)

Wj,Z = Θ

∫(Kj,Z − 〈Kj,Z〉θ) dθ. (22)

Because terms ofKj,T are of the general formK = Q cos(k θ+mν), where the non-dimensionalcoefficients Q are of the form Q ≡ Q(p, S, C,Θ, N ;α), and m and k are integers, it is easilychecked that Eq. (19) admits a solution Wj,T with corresponding terms of the form

Y = Θn

k n−mωEQ sin(k θ +mν). (23)

Tesseral effects remain in the new Hamiltonian term H0,j through the summand Rj of Eq. (21),which, c.f. Eqs. (20) and (23) is made of terms of the form

R∂Y

∂r=R

r

2mωE

k n−mωEY

8

where the radial velocity may be expressed as

R = C sin θ − S cos θ =Θ

p

(C sin θ − S cos θ

)where the non-dimensional state functions C = (p/Θ)C, S = (p/Θ)S, when written in orbitalelements are checked to be

C = e cosω, S = e sinω.

Therefore, choosing Wj,T as solution of Eq. (19) introduces a residualRj in the homological equa-tion which is made of trigonometric, sine or cosine terms that are multiplied by

Θ2

r2r

p

2mωE n

(mωE − k n)2Q,

where Q ≡ Q(p, S, C,Θ, N ;α) is, at least, of the order of the eccentricity. That is, we obtainanalogous terms to the original tesseral terms except for the factor

λ = er

p

2mωE n

(mωE − k n)2. (24)

Consequently, away from tesseral resonances k n = mωE , the relegation of tesseral effects isdirectly reached for the lowest eccentricity orbits, where e r/p ≈ e. Note that |e r/p| < 1 ∀frequires e < 1/2, a condition that we assume thereon. Besides, the expansion

q =2mωE n

(mωE − k n)2=

2

k

∑i>0

i(mωE

k n

)±i(25)

where the plus sign applies to the case mωE < k n and the minus sign to k n < mωE, shows thatin the case of sub-synchronous orbits (resp. super-synchronous) a small enough ratio δ = ωE/n(resp. 1/δ) will make the relegation of tesseral terms more effective.

Further relegation of tesseral terms up to the order of λm ∼ em can be achieved as follows. First,we replace

Wj,T =

m∑i=1

W ij,T

in Eq. (18). Then, we iterate m times the solution of Eqs. (19)–(20), which we rewrite

Θ

r2∂W i

j,T

∂θ− ωE

∂W ij,T

∂ν= Ri−1j , Rij = −R

∂W ij,T

∂r, (26)

an iteration that starts withR0j = (Θ2/r2)Kj,T . Once the iteration is completed, we choose

H0,j =Θ2

r2〈Kj,Z〉θ +Rmj . (27)

Finally, Wj,Z is computed from Eq. (22).

As a result of the relegation we obtain the well-known Hamiltonian of the zonal problem after theoriginal parallax transformation,8 except for the small term Rmj = O(em) that may be neglected.

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Then, a further normalization can be carried out in closed form of the eccentricity to fully eliminatethe short-period terms from the Hamiltonian.

We note that δ is not required to be small, although convergence improves when it happens.The only constraint is to avoid resonant cases between the rate of the argument of latitude andthe system’s rotation rate, because they would introduce small divisors in the solution (23) of thepartial differential equation (19). The relegation algorithm also provides simplification in the caseof resonant motion, but in this case corresponding resonant terms cannot be relegated, and must bekept out of the right member of Eq. (19) to be directly incorporated into the new Hamiltonian H0,j .Resonant terms are easily traced in Kj,T .

RELEGATION OF TESSERAL TERMS FROM A 2× 2 GEOPOTENTIAL

For the sake of illustrating the procedure, we only take into account a 2 × 2 truncation of theEarth’s gravitational potential. Taking higher degree and order truncations does not change any-thing, albeit may require an important computational effort because of the larger size of the seriesinvolved. Besides, we choose a rotating frame with constant rotation rate ωE about the polar axis.Then, because the second order zonal harmonic coefficient C2,0 dominates over all other harmoniccoefficients, the orbital motion is derived from the perturbation Hamiltonian

H =∑

0≤j≤2

1

j!Hj,0

where, using polar nodal variables,

H0,0 =1

2

(R2 +

Θ2

r2

)− µ

r− ωEN

H1,0 =1

2

µ

r

(αr

)2C2,0

(1− 3

2s2 +

3

2s2 cos 2θ

)H2,0 =

3

2

µ

r

(αr

)2 s [(1− c) sin(ν − 2θ) + 2c sin ν − (1 + c) sin(ν + 2θ)]C2,1

−s [(1− c) cos(ν − 2θ) + 2c cos ν − (1 + c) cos(ν + 2θ)]S2,1

−[(1− c)2 cos(2ν − 2θ) + 2s2 cos 2ν + (1 + c)2 cos(2ν + 2θ)

]C2,2

−[(1− c)2 sin(2ν − 2θ) + 2s2 sin 2ν + (1 + c)2 sin(2ν + 2θ)

]S2,2

s ≡ sin I , c ≡ cos I , and Ck,m and Sk,m are harmonic coefficients.

Note that, in order to apply the relegation algorithm, the first and second order Hamiltonian termsmust be written in terms of the parallactic functions. Therefore, the first step is to replace the inverseof the radius in the coefficients µ/r = Θ2/(p r) of H1,0 and H2,0 using the identity in Eq. (3). Ifwe further abbreviate the notation by using the non-dimensional state functions C = (p/Θ)C,

10

S = (p/Θ)S, we get

H1,0 =Θ2

r2α2

p2C2,0

1

2− 3

4s2 +

3

4s2 cos 2θ (28)

+C

[(1

2− 3

8s2)

cos θ +3

8s2 cos 3θ

]+ S

[(1

2− 9

8s2)

sin θ +3

8s2 sin 3θ

]

H2,0 =Θ2

r2α2

p23

4

2∑m=1

3∑k=−3

[κm,k cos(mν + kθ) + σm,k sin(mν + kθ)] (29)

where the coefficients κm,k (respectively σm,k) are obtained from those κm,k (respectively σm,k) inTable 1 by applying the simultaneous rule (Ck,m → Sk,m, Sk,m → −Ck,m).

First order

Because the term H1,0 does not depend on the argument of the node, the relegation algorithmprovides the classical result at the first order.11 Thus,

H0,1 =Θ2

r2α2

p2C2,0

(1

2− 3

4s2),

for the simplified Hamiltonian, and

W1 = Θα2

p2C2,0

3

8s2 sin 2θ

+C

[(1

2− 3

8s2)

sin θ +1

8s2 sin 3θ

]− S

[(1

2− 9

8s2)

cos θ +1

8s2 cos 3θ

],

for the first order term of the generating function. It follows the computation of the Poisson bracketsH1,0;W1 and H0,1;W1, which will be required at the second order.

The simplification is achieved by means of a canonical transformation (r, θ, ν, R,Θ, N) →(r′, θ′, ν ′, R′,Θ′, N ′) and, hence, both H0,1 and W1 should be expressed in the new, prime, vari-ables. Nevertheless, to simplify notation we do without the prime notation wherever there is notrisk of confusion.

Second order simplification of non-resonant orbits

The homological equation is now

L0(W2) = H2,0 −H0,2.

Because we limit to a 2 × 2 truncation of the Geopotential, there are no zonal coefficients Ck,0,k > 2, at the second order. Nevertheless, second order effects due to C2,0 do appear in the Lietriangle computation.

Thus, H2,0 = H1,0;W1 + H0,1;W1 + H2,0, which in accordance to Eq. (10) is split into azonal and a tesseral part

H2,0 =Θ2

r2(K2,Z +K2,T ) ,

11

Table 1. Non-vanishing coefficients of W2,T in Eq. (32)

m k κm,k σm,k

1 ±3 ±(1± c) s(C C2,1 ∓ S S2,1

)(1± c) s

(S C2,1 ± C S2,1

)±2 ±2(1± c) sC2,1 ±2(1± c) s S2,1±1 (1± 3c) s S S2,1 ± (1∓ c) s C C2,1 −(1± 3c) s S C2,1 ± (1∓ c) s C S2,1

0 −4c sC2,1 −4c s S2,1

2 ±3 (1± c)2(C S2,2 ± S C2,2

)−(1± c)2

(C C2,2 ∓ S S2,2

)±2 2(1± c)2 S2,2 −2(1± c)2C2,2

±1 (1± c)[(3∓ c) C S2,2 ± (1∓ 3c) S C2,2

]−(1± c)

[(3∓ c) C C2,2 ∓ (1∓ 3c) S S2,2

]0 4s2 S2,2 −4s2C2,2

where

K2,Z =α4

p4C22,0

15

64s4 C cos 5θ +

9

64s4 C S sin 6θ +

9

128s4(C2 − S2

)cos 6θ (30)

− 3

64(20− 29s2) s2 C cos 3θ − 3

16(10− 13s2) s2 C S sin 4θ

+3

64s2[4s2 + (20− 27s2) S2 − 5(4− 5s2) C2

]cos 4θ

− 1

32(64− 262s2 + 195s4) C cos θ − 3

64(16− 16s2 − 15s4) C S sin 2θ

+1

128

[320s2 − 336s4 + (48− 39s4) S2 − (48− 96s2 + 9s4) C2

]cos 2θ

− 1

64

[80− 168s2 + 84s4 + 3(8− 36s2 + 25s4) C2 + 3(8 + 20s2 − 35s4) S2

]−15

64(4− 7s2) s2 S sin 3θ − 1

32(64 + 46s2 − 141s4) S sin θ +

15

64s4 S sin 5θ

,

and, from Eq. (29),

K2,T =H2,0

Θ2/r2. (31)

Then, the solution of Eq. (19) gives

W2,T = Θ3

4

α2

p2

2∑m=1

3∑k=−3

1

k −mδ[κm,k cos(mν + kθ) + σm,k sin(mν + kθ)] , (32)

where the coefficients κm,k, σm,k are given in Table 1. Remark that, as expected, the generatingfunction of the extended parallax elimination of Coffey and Alfriend5 is recovered from Eq. (32) bythe simple expedient of neglecting δ, That is, by assuming that the system’s rotation rate is of higherorder than the orbital mean motion.

Then, we are ready to choose H0,2 in accordance with Eq. (21), where, from Eq. (30),

〈K2,Z〉θ = −α4

p43

2C22,0

[5

6− 7

4s2 +

7

8s4 +

(1

4− 9

8s2 +

25

32s4)C2 +

(1

4+

5

8s2 − 35

32s4)S2

],

12

and, recalling that R = (Θ/p) (C sin θ − S cos θ),

R∂W2,T

∂r=

Θ2

r2α2

p2r

p

3

2

2∑m=1

3∑k=−3

mδ

(k −mδ)2(33)

×(C sin θ − S cos θ) [κm,k cos(mν + kθ) + σm,k sin(mν + kθ)] ,

which can be reorganized as a longer Poisson series with the summation index k ranging from −4to +4.

Equation (33) shows that tesseral terms, while remaining of a similar nature to those in the orig-inal Hamiltonian Eq. (29), as a result of the transformation are now factored either by C or S, andconsequently by e, which must be less than 1/2 in our previous assumptions. Therefore, assumingthat the orbit is far away enough from resonances k = j δ, for orbits with small or moderate eccen-tricity the effect of the transformation is the relegation of tesseral disturbances to the order of thethe eccentricity.

The case of the m-daily terms, in which k = 0, deserves some further discussion. In this case,the coefficient q = mδ/(k − mδ)2 in Eq. (33) transforms into q = 1/(mδ) and, therefore, therelegation ofm-daily terms will occur only if e < mδ. The latter inequality is generally satisfied forsuper-synchronous orbits, and also for high-altitude sub-synchronous orbits with small or moderateeccentricities. Quite on the contrary, the smaller values of δ in the case of low-altitude orbits mayprevent the relegation of m-daily terms even for the lower eccentricity orbits; more precisely forthose orbits whose eccentricity is comparable to, or higher than δ. Thus, for instance, while Galileoorbits and the International Space Station (ISS) have the same eccentricity e = 0.001, we obtainδ ≈ 0.6 for a Galileo orbit (a = 29 600 km), whereas it is only δ ≈ 0.065 for the ISS (a = 6 830km), showing that the relegation of m-dailies will be ten times more effective in the case of Galileoorbits than for the ISS. In such cases in which δ is too small, the m-daily terms should not berelegated but being directly incorporated to the new Hamiltonian.

From the preceding discussion we conclude that Eq. (33) clearly discloses the important effect ofm-daily terms in the propagation of low-altitude orbits. Hence the necessity of retaining as muchm-daily terms as possible in accurate semianalytical propagations.4 A corollary derived from thepoor relegation of m-dailies in the case of low-altitude orbits is that the relegation algorithm willattain the best results in the case of low-eccentricity orbits with hig altitude.

For other tesserals than m-dailies the relegation of tesseral effects is always achieved for low-eccentricity orbits. Thus, the approximation q ≈ (m/k2) δ holds for low-altitude orbits, whileq ≈ 1/(mδ) in the case of super-synchronous orbits, c.f. Eq. (25). Therefore, q always reinforces,or at least does not worsen, the relegation.

13

Finally, from Eq. (22) we obtain

W2,Z = Θα4

p4C22,0

3

256s4(C2 − S2

)sin 6θ (34)

+3

64

[s2 −

(5− 25

4s2)C2 +

(5− 27

4s2)S2

]s2 sin 4θ

+3

16

[20

3s2 − 7s4 −

(1− 2s2 +

3

16s4)C2 +

(1− 13

16s4)S2

]sin 2θ

−C[(

2− 131

16s2 +

195

32s4)

sin θ +

(5

16− 29

64s2)s2 sin 3θ − 3

64s4 sin 5θ

]+

3

8C S

[(1− s2 − 15

16s4)

cos 2θ +

(5

4− 13

8s2)s2 cos 4θ − 1

16s4 cos 6θ

]+S

[(2 +

23

16s2 − 141

32s4)

cos θ +5

16

(1− 7

4s2)s2 cos 3θ − 3

64s4 cos 5θ

],

in agreement with the corresponding expression for W2 given by Deprit.11

Assuming that the relegation of tesseral terms with this single iteration is enough for our purposes,and in preparation for a later normalization that will remove the remaining short-period terms relatedto the radius, the simplified Hamiltonian can be expressed in Delaunay variables (`, g, h, L,G,H).Thus, by neglecting the contribution ofR = −R∂W2,T /∂r, we get

H = − µ

2a

1 + 2

ωE

nη c+

α2

r2C2,0

2η2(1− 3c2)− α2

r2α2

a23C2

2,0

16η6(35)

×[

1

3− 7c4 −

(2c2 − 5

4s4)e2 +

(7s2 − 15

2s4)e2 cos 2ω

],

where r = a η2/(1 + e cos f), a = L2/µ, η = G/L, e =√

1− η2, c = H/G, s =√

1− c2,ω = g, the mean motion is n =

õ/a3, and the relation between the true anomaly f and the mean

anomaly M = ` involves the solution of the Kepler equation.

Equation (35) offers a notable simplification over the original problem, revealing that the Hamil-tonianH ≡ H(`, g,−, L,G,H) has been released from its dependence on the argument of the nodein the rotating frame h. Consequently, H is an integral of the simplified problem that decouplesthe motion of the node in the rotating frame from the reduced, two degrees of freedom probleminvolving the mean anomaly and argument of the perigee.

The standard Delaunay normalization12 of Hamiltonian (35) leads to the Hamiltonian in meanelements

H = − µ

2a

1 + 2

ωE

nη c+

α2

a2C2,0

2η3(1− 3c2)− α4

a43C2

2,0

64η7(36)

×[5(8− 16s2 + 7s4) + (4− 6s2)2η − (8− 8s2 − 5s4) η2 − (28s2 − 30s4) e2 cos 2ω

],

14

and generating function

W = −nα2 C2,0

4η3(f − `) (1− 3c2)− nα2 α

2

a2C22,0

128η7

2e

1 + η(2− 3s2)2 (4 sin f + e sin 2f)

+3(f − `)[5(8− 16s2 + 7s4)− (8− 8s2 − 5s4) η2 − (28s2 − 30s4) e2 cos 2ω

]. (37)

The case of the 1:2 tesseral resonance

If we intend to apply the above simplification to orbits close to the 1:2 tesseral resonance, corre-sponding to orbits with a semimajor axis of about 26 560 km which is specifically the case of GPSorbits, we must take into account that there is a variety of terms that may introduce small denomi-nators in the standard generating function Eq. (32). Therefore, these terms should be left out of therelegation procedure.

The resonance phenomenon is motivated by commensurabilities of the system’s rotation ratewith the mean motion, which is the rate of variation of the mean anomaly M . Although M doesnot appear explicitly in the polar nodal variables, the only terms that can introduce 1:2 tesseralresonances are related to those trigonometric functions of ν and θ in which ν is factored by evencoefficients. Therefore, we must leave these terms untouched to be directly incorporated to the newHamiltonian, whereas other tesseral terms can be simplified.

In our 2 × 2 model, this means that all the terms factored by C2,2 and S2,2 cannot be relegatedand must take part in the new Hamiltonian. Hence, we now make

K2,T =3

4

α2

p2

3∑k=−3

[κ1,k cos(ν + kθ) + σ1,k sin(ν + kθ)] . (38)

In consequence, the new tesseral part of the generating function is

W2,T = Θ3

4

α2

p2

3∑k=−3

1

k − δ[κ1,k cos(ν + kθ) + σ1,k sin(ν + kθ)] , (39)

which is obtained from Eq. (32) by eliminating terms factored by the harmonic coefficients C2,2 andS2,2, the lower part of Table 1. The zonal part Eq. (34) is not affected by the resonance.

Now, the new Hamiltonian is not as simple as Eq. (35), to which we must add the tesseral terms

H2,T =µ

2a

3

4η2α2

r2

3∑k=−3

[κ2,k cos(2h+ kf + kω) + σ2,k sin(2h+ kf + kω)] , (40)

affected by the resonance. But it still provides a significant reduction in the number of terms andcomplexity of the original Hamiltonian.

APPLICATION TO ORBITS IN THE MEO REGION

The iteration of the relegation procedure m times will relegate the influence of tesseral terms onlow-eccentricity orbits to the degree of em. Therefore, the relegation appears as a suitable simpli-fication for specific orbits as those of traditional constellations of GNSS, in which the influence of

15

second order Geopotential effects is small because of the altitude (except for tesseral resonances)and besides have low eccentricities. That is the case of Galileo orbits, with eccentricities in theorder of one thousandth, and also of GPS orbits, where the eccentricities are below one hundredth.The latter, however, are in deep 1:2 tesseral resonance, which prevents a complete relegation oftesseral effects, and hence corresponding terms must remain in the new Hamiltonian to avoid theintroduction of small divisors in the generating function. Sample propagations illustrating the er-rors introduced by the relegation simplification in in the case of Galileo orbits are provided in whatfollows.

Recall that third-body perturbations have an important effect on this kind of orbits. However,the third-body disturbing function cannot be simplified by the relegation procedure discussed here.Therefore, the whole third-body effect will be incorporated directly to the new Hamiltonian, thusnot having any contribution to the errors introduced by the relegation.

Galileo-type orbits

The Galileo constellation is designed in such a way that the satellites move in almost-circularMEO orbits at an altitude of about 23 222 km over the surface of the Earth and 56 deg of nominalinclination.∗ This orbital configuration corresponds to a 17 to 10 repeat groundtrack condition thatmight show small tesseral resonance effects starting for higher order harmonics of the Geopoten-tial. However, in the computations of this paper we limit ourselves to a 5 × 5 truncation of theGeopotential, and hence we take Galileo orbits as a sample of the non-resonant case.

The simple, instant propagation of mean elements from Hamilton equations derived of an analo-gous Hamiltonian to Eq. (36) for the 5×5 truncation case, introduces periodic errors when comparedto the direct integration of the original Hamilton equations. Errors of the mean elements propagationare of the order of ±1.5 km for the semi major axis, below 10−4 for the eccentricity and just a fewarc seconds (as) for the inclination and argument of the node. In the case of the mean anomaly andargument of the perigee, the errors grow to about ±2 degrees, but this is because of the almost cir-cular character of Galileo orbits, with the consequent poor definition of the argument of the perigee.When using nonsingular elements, such as F = M + ω, the amplitude of the errors is reduced toabout ±6 as.

The errors are notably reduced, as expected, when recovering the periodic terms removed bythe relegation procedure. This is illustrated in Figure 1 for a 30 days propagation of a Galileotype orbit with initial osculating elements a = 29 600 km, e = 0.001, I = 56 deg, ω = 270deg, M = 89.8854 deg, Ω = 0, and a period of approximately 14.08 hours. The subindex p inlabels mean osculating values obtained when integrating semi-analytically the theory based on therelegation. Thus, after recovering short-period effects using the transformation equations of therelegation without iterations in the computation of the generating function of the relegation, namelyneglecting R1

2, we find periodic errors of a fraction of a meter for the semi-major axis, of the orderof 10−6 for the eccentricity, about a milliarc second (mas) for the inclination and argument of thenode, while the errors rise to tenths of degree for the mean anomaly and argument of the perigee.Again, when using non-singular elements the latter reduce to a small secular trend of about threemas times orbital period —roughly, an along track error of half a meter times orbital period.

If required, the errors introduced by the relegation algorithm are amenable to reduction by carry-ing out additional iterations in the computation of the tesseral part of the second order generating

∗http://www.esa.int/esaNA/galileo.html

16

0 10 20 30 40 50 60-0.3-0.2-0.1

0.00.10.20.3

a-

a PHm

L

0 10 20 30 40 50 60-0.1

0.00.10.20.3

105

He-

e PL

0 10 20 30 40 50 60-1.0-0.5

0.00.51.01.5

I-

I PHm

asL

0 10 20 30 40 50 60

-1.0-0.5

0.00.5

W-

WP

Hmas

L

0 10 20 30 40 50 60-0.1

0.00.10.20.3

Ω-

ΩP

Hdeg

L

0 10 20 30 40 50 60-0.3-0.2-0.1

0.00.1

M-

MP

Hdeg

L

0 10 20 30 40 50 60

0.000.050.10

F-

FP

HasL

Figure 1. Errors in the propagation of a Galileo-type orbit without any iteration ofthe relegation algorithm. Abscissas are orbital periods.

function. In this way we find that with a single iteration of Eq. (26), neglectingR22, the accuracy of

the semianalytic propagation is further improved by one order of magnitude. Thus, in reference toFigure 2, the errors in the semimajor axis are now in the cm level, they are below 10−7 in the caseof the eccentricity, go down to several micro arc seconds (µas) for the inclination and argument ofthe node, and are of the order of several tens of arc seconds in the case of the mean anomaly andargument of the perigee. Again, the combination of this two last variables discloses a small seculartrend in the error of the nonsingular variable F , which in this case is only of∼ 0.2 mas times orbitalperiod —roughly, an along track error of three cm times orbital period.

A new iteration of the relegation improves the accuracy, but only very slightly —in agreementwith a previous observation by Segerman and Coffey.27 Thus, neglecting the contribution of R3

2,the accuracy of the semianalitical propagation remains basically unaltered in the case of the semi-major axis, inclination and argument of the node, whereas the errors in the eccentricity now fallbelow 10−8, and to just a few arc second for M and ω, albeit the improvement in the accuracyof the nonsingular variable F is just of about 0.1 mas times orbital period, c.f. Figure 3, roughlycorresponding to an along-track error of one and a half cm times orbital period.

17

0 10 20 30 40 50 60-2-1

012

a-

a PHc

mL

0 10 20 30 40 50 60-0.6-0.4-0.2

0.00.20.40.60.8

107

He-

e PL

0 10 20 30 40 50 60

-0.050.000.05

I-

I PHm

asL

0 10 20 30 40 50 60-0.10-0.05

0.000.050.10

W-

WP

Hmas

L

0 10 20 30 40 50 60-30-20-10

0102030

Ω-

ΩP

HasL

0 10 20 30 40 50 60-30-20-10

0102030

M-

MP

HasL

0 10 20 30 40 50 60-15-10

-505

F-

FP

Hmas

L

Figure 2. Errors in the propagation of a Galileo-type orbit for a single iteration ofthe relegation algorithm. Abscissas are orbital periods.

CONCLUSIONS

Orbits useful for GNSS constellations may suffer from either deep or shallow tesseral resonances,a fact that prevents the use of simplification algorithms as they originally appeared in the literature.However, orbits of this kind are designed to have low eccentricities and large semi-major axes,two characteristics that make the logarithmic derivative of the radius to be a small quantity whencompared to the mean motion of GNSS satellites. This fact suggests a variation in the classicalsolution of the inverse relegation algorithm, in which the relegation effects are only dependent onthis relegating factor. The new approach allows for dealing with resonant and non-resonant orbits,and, even in the trivial case of a single iteration of the relegation algorithm, it may produce verysmall errors which are comparable to those caused by the truncation order of the theory.

ACKNOWLEDGEMENTS

Part of this research has been supported by the Government of Spain (Projects AYA 2009-11896and AYA 2010-18796 and a grant “Gobierno de la Rioja Fomenta”). This research has made use ofNASA’s Astrophysics Data System Bibliographic Services.

18

0 10 20 30 40 50 60-1.5-1.0-0.5

0.00.51.01.5

a-

a PHc

mL

0 10 20 30 40 50 60-1.0-0.5

0.00.51.0

108

He-

e PL

0 10 20 30 40 50 60

-0.010.000.010.020.030.040.05

I-

I PHm

asL

0 10 20 30 40 50 60-0.10-0.05

0.000.050.10

W-

WP

Hmas

L

0 10 20 30 40 50 60-6-4-2

0246

Ω-

ΩP

HasL

0 10 20 30 40 50 60-6-4-2

0246

M-

MP

HasL

0 10 20 30 40 50 60-6-4-2

0

F-

FP

Hmas

L

Figure 3. Errors in the propagation of a Galileo-type orbit for two iterations of therelegation algorithm. Abscissas are orbital periods.

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