Solution to the Main Problem of the Artificial Satellite by Reverse

Nonlinear Dynamics manuscript No. (will be inserted by the editor)

Solution to the main problem of the artificial satellite by reverse normalization

Martin Lara

draft of April 16, 2020

Abstract The non-linearities of the dynamics of Earth arti- tion of the zonal harmonic of the second degree, whose co- ficial satellites are encapsulated by two formal integrals that efficient is usually denoted J2, is traditionally known as the are customarily computed by perturbation methods. Stan- main problem of artificial satellite theory. The J2-truncation dard procedures begin with a Hamiltonian simplification that of the gravitational potential is known to give rise to noninte- removes non-essential short-period terms from the Geopo- grable dynamics [12,26,5,8] that comprise short- and long- tential, and follow with the removal of both short- and long- period effects, as well as secular terms [36,31]. However, period terms by means of two different canonical transfor- due to the smallness of the J2 coefficient of the Earth, the mations that can be carried out in either order. We depart full system can be replaced by a separable approximation, from the tradition and proceed by standard normalization to which is customarily obtained by removing the periodic ef- show that the Hamiltonian simplification part is dispensable. fects by means of perturbation methods [53]. Decoupling first the motion of the orbital plane from the in- When written in the action-angle variables of the Kepler plane motion reveals as a feasible strategy to reach higher problem, also called Delaunay variables, the main problem orders of the perturbation solution, which, besides, permits Hamiltonian immediately shows that the right ascension of an efficient evaluation of the long series that comprise the the ascending node is a cyclic variable. In consequence, its analytical solution. conjugate momentum, the projection of the angular momen- tum vector along the Earth’s rotation axis, is an integral of Keywords main problem · Hamiltonian mechanics · the main problem, which, therefore, is just of two degrees normalization · canonical perturbation theory · floating- of freedom. Then, following Brouwer [6], the main prob- point arithmetic lem Hamiltonian is normalized in two steps. First, the short period effects are removed by means of a canonical transfor- mation that, after truncation to some order of the perturba- 1 Introduction tion approach, turns the conjugate momentum to the mean anomaly (the Delaunay action) into a formal integral. Next, The dynamics of close Earth satellites under gravitational a new canonical transformation converts the total angular effects are mostly driven by perturbations of the Keplerian momentum into another formal integral. The main problem arXiv:2007.00058v1 [math.DS] 30 Jun 2020 motion induced by the Earth oblateness. For this reason, the Hamiltonian is in this way completely reduced to a function approximation obtained when truncating the Legendre poly- of only the momenta in the new variables, whose Hamilton nomials expansion of the Geopotential to the only contribu- equations are trivially integrable. M. Lara In spite of the normalization procedure is standard from Scientific Computing Group–GRUCACI, University of La Rioja, the point of view of perturbation theory, it happens that not Madre de Dios 53, 26006 Logrono,˜ La Rioja, Spain Tel.: +34-941-299440 all the action-angle variables of the Kepler problem appear Fax: +34-941-299460 explicitly in the Geopotential, as is usually advisable in the E-mail: [email protected] construction of perturbation solutions [32,43,44,49]. In par- M. Lara ticular, the mean anomaly remains implicit in the gravita- Space Dynamics Group, Polytechnic University of Madrid–UPM, tional potential through its dependence on the polar angle. Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Unavoidably, Kepler’s equation must be solved to show the 2 Martin Lara mean anomaly explicit, a fact that entails expanding the el- could seem to be unavoidably attached. On the other hand, liptic motion in powers of the eccentricity [23,7]. This stan- when extended to higher orders, it removes more terms than dard procedure is quite successful when dealing with orbits those needed in a partial normalization. Last, the fact that the of low eccentricities, like is typical in a variety of astronomy technique was originally devised in the canonical set of po- problems [14,57]. However, it involves handling very long lar variables, to which the argument of the perigee does not Fourier series in the case of orbits with moderate eccentric- pertain, neither helps in grasping the essence of the trans- ities [19,33], and hence the application of this method to formation. Reimplementation of the procedure in the usual different problems of interest in astrodynamics is de facto set of action-angle variables makes the process of convert- prevented. ing the argument of the perigee into a cyclic variable much On the contrary, the integrals appearing in the solution more evident [45], but it still bears the same differences with of the artificial satellite problem can be solved in closed respect to a classical normalization procedure. form of the eccentricity [6,36]. It only requires the help of We disregard the claimed benefits of Hamiltonian sim- the standard relation between the differentials of the true plification procedures and compute the solution to the main and mean anomalies that is derived from the preservation problem of the artificial satellite theory by standard normal- of the total angular momentum of the Keplerian motion. ization. It is called reverse normalization because we ex- Regrettably, the closed form approach soon finds difficul- change the order in which the formal integrals are tradition- ties in achieving higher orders of the short-period elimi- ally introduced when solving the artificial satellite problem. nation, which stem from the impossibility of obtaining the More precisely, the total angular momentum is transformed antiderivative of the equation of the center (the difference into a formal integral in the first place, in this way decou- between the true and mean anomalies) in closed form of pling the motion of the orbital plane from the satellite’s mo- the eccentricity in the realm of trigonometric functions [28, tion on that plane.1 Then, a second canonical transformation 54]. Nonetheless, the difficulties are overcome by the arti- converts the mean anomaly into a cyclic variable, in this way fact of grouping the equation of the center with other func- achieving the total reduction of the main problem Hamilto- tions appearing in the procedure previously to approaching nian. their integration [37,1,17]. Alternatively, the application of The procedure for making the argument of the perigee a preliminary Hamiltonian simplification, the elimination of cyclic in the first place follows an analogous strategy to the the parallax [16,46], eases the consequent removal of short- one devised in the classical elimination of the perigee trans- period effects to some extent [9,25]. formation [2]. However, in our approach it is applied directly From the point of view of the perturbation approach, re- to the original main problem Hamiltonian, and differs from moving the short-period effects in the first place seems the the original technique, as well as from an analogous proce- more natural in view of the degeneracy of the Kepler prob- dure carried out in [56], in which the parallactic terms (in- lem. Indeed, the Kepler Hamiltonian in action-angle vari- verse powers of the radius with exponents higher than 2) are ables, on which the perturbation approach hinges on, only not removed from the new, partially normalized Hamilto- depends on the Delaunay action [22,41]. However, the order nian. In spite of that, we did not find trouble in dealing with in which the formal integrals are sequentially introduced in the equation of the center in closed form in the subsequent the perturbation solution is not relevant in a total normaliza- Delaunay normalization [17], a convenience that might had tion procedure, whose result is unique [3]. In fact, it hap- been anticipated from the discussions in [50]. pens that relegating the transformation of the Delaunay ac- The Hamiltonian reduction has been approached in De- tion into a formal integral to the last step of the perturbation launay variables. Unfortunately, these variables share the de- approach provides clear simplifications in dealing with the ficiencies of their partner Keplerian elements, which are sin- equation of the center [2]. In this way the task of extending gular for circular orbits and for equatorial orbits. Because the solution of the main problem to higher orders is notably of that, the secular terms are reformulated in a set of non- simplified. singular variables that replaces the mean anomaly, the argu- Converting the total angular momentum into a formal in- ment of the perigee, and the total angular momentum, by tegral of the main problem requires making cyclic the argu- the mean argument of the latitude and the projections of ment of the perigee, up to some truncation order of the per- the eccentricity vector in the nodal frame, which are some- turbation approach, in the transformed Hamiltonian. How- times denoted semi-equinoctial variables [34]. For the pe- ever, as it appeared in the literature, the transformation called riodic corrections, we find convenience in using polar vari- by their authors the elimination of the perigee [2] is not the ables, which in the particular case of the main problem are typical normalization procedure, although it operates anal- 1 ogous results. Indeed, on the one hand, the elimination of The advantages of decoupling the motion of the instantaneous or- bital plane from the in-plane motion are well known, and are com- the perigee is only applied to a Hamiltonian obtained af- monly pursued in the search for efficient numerical integration meth- ter the elimination of the parallax, to which simplification it ods, q.v. [42] and references therein. Solution to the main problem of the artificial satellite by reverse normalization 3 free from small divisors of any kind except for those related 2 Construction of the analytical solution to the critical inclination resonance, and are known to pro- duce compact expressions that yield faster evaluation [27, The solution of Laplace’s equation in spherical coordinates 47]. provides the gravitational potential in the form of an expan- We extended the complete normalization to the order six sion in harmonic functions. In the case of the Earth, the mass of the perturbation approach, which, to our knowledge, is distribution is almost symmetric with respect to the rotation the maximum order that has been reported in the literature axis. For this reason, the Geopotential is simplified in dif- (yet limited to partial normalization cases) [25,56]. The aim ferent applications to just the contribution of the zonal har- of computing such a high order is not to enter a competi- monics (see [13], for instance) tion. On the contrary, we did it simply because the particu- m µ X R⊕ −3 V = J P (sin ϕ), (1) lar value of the Earth’s J2 coefficient is ∼ 10 , and hence r rm m m the computed solution should be exact to the numerical pre- m≥0 cision of standard floating-point arithmetic [30] already at where r is distance from the origin, ϕ is latitude, µ is the the fifth order. We checked that it is exactly the case, and Earth’s gravitational parameter, R⊕ is the Earth’s mean equa- extending the computations to the sixth order only served torial radius, Jm are zonal harmonic coefficients, and Pm us to verify that we don’t find observable improvements in denote Legendre polynomials of degrees m. our tests. In order to compute this high-order solution we re- The flow stemming from the potential (1) admits Hamil- sorted to the practicalities of standard commercial software, −3 tonian formulation. Besides, because J2 ∼ 10 whereas in which Deprit’s perturbation algorithm based on the Lie 2 Jm (m > 2) are O(J2 ) or smaller, the Hamiltonian transforms method [15] is easily implemented. On account 2 of the current computational power, it should be quite fea- µ µ R⊕ 1 2
H = − − J2 1 − 3 sin ϕ , (2) sible to extend the perturbation solution, if desired, to even 2a r r2 2 higher orders, although we didn’t try that. is representative of the main characteristics of the dynam- On the other hand, making cyclic the argument of the ics of close Earth satellites, and is commonly known as the perigee with the standard normalization in action-angle vari- main problem of artificial satellite theory. The symbol a in ables seems to be a particularly efficient procedure from Eq. (2) stands for the orbit semi-major axis, and, from stan- the computational point of view, resulting in a generating dard trigonometric relations, sin ϕ = sin I sin θ, where I function whose size is astonishingly smaller than alternative and θ are used to denote the inclination and the argument of proposals in the literature. This fact, combined with the re- the latitude, respectively. duction in the number of transformations required by the When using the classic set of Keplerian elements, the ar- perturbation solution to just two, as opposite to the three gument of the latitude is θ = f + ω, where f and ω denote transformations needed when the elimination of the paral- the true anomaly and the argument of the periapsis, respec- lax is carried out in the first place, might make this latter tively. The radius is computed from the conic equation simplification dispensable, as well as the discussions in [18] p questionable, at least in what respects to the computation of r = , (3) 1 + e cos f explicit analytical solutions as opposite to partial normaliza- tions to be integrated semianalytically. where p = a(1 − e2) is the parameter of the conic and Tests carried out on typical Earth orbits of different types e is the eccentricity of the orbit. Note, however, that, be- confirm that the computed solution bears exactly the ex- cause we are using Hamiltonian formulation, the symbols pected characteristics of a perturbation solution of perturbed appearing in Eq. (2) need to be expressed as functions of Keplerian motion. In particular, we verified the degree of some set of canonical variables. While the Keplerian vari- convergence of successive approximations. We confirmed ables are not canonical, they are conveniently expressed in too that, as expected, the quality of the analytical solution the set of Delaunay canonical variables (`, g, h, L, G, H), in degrades in the vicinity of the critical inclination resonance which ` is the mean anomaly, g = ω, h is the argument of √ L = µa —a case that, of course, would require a specific treatment the ascending√ node, is known as the Delaunay [10,29,48]. In addition, we checked that the computation action, G = L 1 − e2 is the total angular momentum, and of the initialization constants of the analytical solution from H = G cos I is the projection of the angular momentum corresponding initial conditions gets a clear benefit of us- along the Earth’s rotation axis. That the latter is an integral ing a higher order of the periodic corrections than the order of the main problem becomes evident after checking that h needed for ephemeris evaluation, yet this additional accu- is an ignorable variable in Hamiltonian (2), which, in conse- racy is not needed in all the variables and can be limited to quence, is of just two degrees of freedom. the computation of the formal integral given by the Delau- Besides, the main problem Hamiltonian is conservative, nay action [4,19,24]. and, therefore, Eq. (2) remains constant (the total energy) 4 Martin Lara for a given set of initial conditions. On the other hand, the in which {Wm; K0,0} stands for the computation of the Pois- existence of a third integral has not been proved. Then, exact son bracket of Wm and the zeroth order term of the Hamilto- solutions of the main problem are not known. Alternatively, nian. Terms Ke0,m are known from the original Hamiltonian it is approached with the usual tools of non-linear dynam- as well as from previous computations to the step m, which ics, such as Poincare´ surfaces of section or the computation are carried out using Deprit’s fundamental recursion of families of periodic orbits [12,5,38,39]. Still, in some re- n X  n  gions of phase space, and for particular energy values, the F = F + {F ; W }. (11) n,q n+1,q−1 m n−m,q−1 m+1 third integral can be constructed formally with the help of m=0 perturbation methods, obtaining in this way useful analyti- Equation (11) can be used to formulate any function F of cal approximations to the main problem dynamics in these the canonical set of original variables like a function of the particular regions. new variables, the Hamiltonian (4) being just one among the different possibilities. Finally, the term K0,m is chosen at our will, with the only condition of making the homological 2.1 Perturbation approach equation solvable in Wm. We rewrite the main problem in the form of a perturbation When using Delaunay variables the homological equa- tion of the main problem Hamiltonian is solved by indefinite Hamiltonian 1 2 2 integration. Indeed, Eq. (5) turns into K0,0 = − 2 µ /L , X εm H = H(`, g, −, L, G, H) ≡ K , (4) and hence the Lie derivative operator L0 takes the extremely m! m,0 m≥0 simple form

∂Wm in which ε is a formal small parameter (ε = 1) used to man- L0(Wm) = n , ifest the strength of each summand of Eq. (4), and ∂` µ in which n = pµ/a3 = µ2/L3 is the mean motion of the K = − , (5) 0,0 2a unperturbed problem. Then, from Eq. (10), 2 µ R⊕ 1  2 2  Z K1,0 = − J2 2 − 3s + 3s cos(2f + 2ω) , (6) 1 2 Wm = (Ke0,m − K0,m) d` + Cm, (12) r r 4 n Km,0 = 0, m ≥ 2. (7) where the functions C ≡ C (−, g, L, G; H) play the role The symbol s in Eq. (6) stands for the usual abbreviation of m m of integration “constants” that verify dC /d` = 0. Equa- the sine of the inclination. m tion (12) is solved in closed form of the eccentricity with the The aim is to find a transformation of variables help of the differential relation between the true and mean (`, g, h, L, G, H; ε) 7→ (`0, g0, h0,L0,G0,H0), anomalies provided by the preservation of the angular mo- mentum of the Kepler problem. That is, G = r2df/dt, from given also by an expansion in powers of the small parameter, which such that, up to some truncation order m = k, the Hamil- a2η d` = r2df, (13) tonian in the new variables is transformed into a separable √ Hamiltonian. Namely, where η = 1 − e2 = G/L is usually known as the eccen- tricity function. Hence, k m 0 X ε 0 0 k+1 H = K0,m(−, −, −,L ,G ,H) + O(ε ). (8) 1 Z r2 m! Wm = (Ke0,m − K0,m) df + Cm. (14) m=0 n a2η The desired transformation is derived from a scalar gen- The transformation that makes the main problem Hamil- erating function tonian separable, up to the truncation order, is split into two different canonical transformations. With the first one X εm W = Wm+1. (9) m! 0 0 0 0 0 0 m≥0 (`, g, h, L, G, H; ε) 7→ (` , g , h ,L ,G ,H ), which, in our case, is computed using Deprit’s perturbation we make the argument of the perigee cyclic, thus converting algorithm based on Lie transforms [15]. The procedure is the total angular momentum into a formal integral. That is, summarized in finding a particular solution of the homolog- the motion of the satellite in the orbital plane, whose incli- ical equation nation remains constant in the new variables, is decoupled from the motion of the orbital plane. Therefore, the reduced L0(Wm) ≡ {Wm; K0,0} = Ke0,m − K0,m, (10) system representing the motion on the orbital plane could be Solution to the main problem of the artificial satellite by reverse normalization 5 integrated separately. Rather, we carry out a second canoni- f in Eqs. (15), (16), and (17) are functions of the Delaunay cal transformation variables.

0 0 0 0 0 0 00 00 00 00 00 00 At the second order we compute K0,2 from Eq. (11), to (` , g , h ,L ,G ,H ; ε) 7→ (` , g , h ,L ,G ,H ), obtain that makes ignorable the mean anomaly in the transformed K0,2 = {K0,1,W1} + K1,1, (18) Hamiltonian in double-prime variables. In this way, the De- launay action is converted into a formal integral too, and the in which K1,1 is computed using again Eq. (11). Namely, complete Hamiltonian reduction is achieved up to the trun- K1,1 = {K0,0,W2} + {K1,0,W1} + K2,0. (19) cation order of the perturbation solution, whose Hamilton equations are thus trivially integrable. On account of K2,0 = 0 from Eq. (7), the known terms The computation of the secular terms from the normal- hitherto of the homological equation (10) are ized Hamiltonian is only part of the perturbation solution. To Ke0,2 = {K0,1,W1} + {K1,0,W1}, be complete, it requires the correct initialization of the con- stants of the perturbation solution from corresponding initial whose computation only involves partial differentiation. Be- conditions using the inverse transformation (from original fore approaching the solution of Eq. (14), we first check that to double prime variables), on the one hand, and the recov- the term Ke0,2 is made of trigonometric coefficients Ti whose ery of periodic effects using the direct transformation (from arguments always involve the true anomaly as argument, ex- double prime to original variables) to obtain the ephemeris cept for the terms corresponding to the secular terms propagation. Both the di- 3 µ p T = 2 s2 e2(23s2 − 16
+ 8 4s2 − 3) , rect and inverse transformations are obtained from standard 0 4 r r application of the Lie transforms method. Because there are 3 1 T = − 2G2 (15s2 − 14)s2e2 cos 2ω, no specific artifices related to the computation of the current 1 2 r2 solution in what respects to that part, we do not provide de- 1 ∂C1 T = 6G (5s2 − 4) . tails on their computation and an interested reader is referred 2 r2 ∂g to the literature [15]. Integration of these 3 terms in Eq. (14) would yield corre- sponding terms that grow unbounded with f. The term T0 is free from the argument of the periapsis, and hence it can 2.2 Normalization of the total angular momentum be cancelled by choosing the new Hamiltonian K0,2 having a corresponding term T . On the contrary, the terms T and At the first order, we check from Eq. (11) that the known 0 1 T depend on the argument of the perigee, a fact that pre- terms involved in the homological equation consist only of 2 vents their appearance in the new Hamiltonian under our re- K = K . Therefore, we chose the new Hamiltonian e0,1 1,0 quirement of making g a cyclic variable. Nevertheless, since term we had left C1 arbitrary, both terms will cancel each other 2 µ R⊕ 1 2
if C1 is determined from the partial differential equation K0,1 = − 2 J2 2 − 3s , (15) r r 4 T1 + T2 = 0. We readily check that the particular solution which is the part of Eq. (6) that is free from the argument of 15s2 − 14 C = G s2e2 sin 2ω, (20) the perigee, as desired. Then, Eq. (14) turns into 1 8(5s2 − 4) 3 R2 Z p matches this purpose. Remark that the appearance of the di- W = − GJ ⊕ s2 cos(2f + 2ω) df + C , 1 4 2 p2 r 1 visor 5s2 − 4 prevents application of the solution to the res- onant cases that happen at the so-called critical inclinations which is solved by standard integration after replacing the in which sin2 I = 4/5. That is, the supplementary inclina- radius with the conic equation (3). We obtain tions I = 63.435◦ and I = 116.565◦. 3 After computing the partial derivatives of Eq. (20) with 1 2 X 2− 1 i |i−2| W = −G s 3b 2 ce sin(if + 2ω) + C , (16) 1 2 1 respect to L and G, which also appear among the coeffi- i=1 cients of the terms remaining in Ke0,2, the known terms of where C1 is hold arbitrary by now, the symbol b c stands for the homological equation become fully determined. Then, integer part, and we abbreviated we can safely choose K0,2 so that it comprises those terms of Ke0,2 that are free from the argument of the periapsis. We 1 R2  = J ⊕ , (17) obtain, 4 2 p2 1 2 b1− 2 jc 2 µ p2 3s2 X pj X in which p = G /µ, and, therefore  is a function of the total K = 2 e2kγ , (21) 0,2 r r2 8(5s2 − 4)2 rj 2,j,k angular momentum. Recall that the symbols p, s, e, ω, and j=0 k=0 6 Martin Lara in which the inclination polynomials γ2,j,k ≡ γ2,j,k(s) are which contributes three additional inclination functions that provided in Table1. are also listed in Table2. In this way, the second order term of the generating function given by Eq. (22) becomes fully determined. Table 1 Inclination polynomials γ2,j,k in Eq. (21). The needed partial derivatives of C2 appearing in Ke0,3 6 4 2 0,0 −8 (200s − 455s + 345s − 88) are then computed, and the third order term of the new Hamil- 6 4 2 0,1 375s − 930s + 780s − 224 tonian is chosen, as before, to comprise those terms of Ke0,3 6 4 2 1,0 5 (805s − 1878s + 1464s − 384) that are free from the argument of the periapsis. We obtain 6 4 2 2,0 −825s + 1990s − 1616s + 448 1 4 b2− jc µ p2 3s2 X pj X2 K = 3 e2kγ , (24) 0,3 r r2 32(5s2 − 4)3 rj 3,j,k The second order term of the generating function is then j=0 k=0 readily computed from Eq. (14), yielding where the inclination polynomials γ3,j,k are displayed in Ta- 2 2 2l+2 1  G X X X ∗ ble3. W = Γ e2j+k 2 32(5s2 − 4)2 2,j,k,l l=1 k=l−2 j=0 k6=0 Table 3 Inclination polynomials γ3,j,k in Eq. (24). 2l ×s sin(kf + 2lω) + C2, (22) 2 8 6 4 0,0 −8(5s − 4)(313525s − 899030s + 933656s ∗ 2 with k ≡ k mod 2, and the inclination polynomials Γ2,j,k,l −409296s + 61824) 10 8 6 4 are given in Table2. 0,1 4(1551625s − 5675700s + 8148960s − 5706408s +1930272s2 − 248064) 10 8 6 4 0,2 −2(40500s − 99525s + 64840s + 18788s −33936s2 + 9408) Table 2 Non-zero inclination polynomials Γ in Eqs. (22)–(23). 2 8 6 4 2,j,k,l 1,0 2(5s − 4)(2631475s − 7558270s + 7872692s −3470616s2 + 530304) 2 2 2 1,−1,1 −12(5s − 4)(7s − 6)(15s − 14) −3457125s10 + 12282750s8 − 17085020s6 2 4 2 1,1 0,1,1 −48(5s − 4)(195s − 340s + 148) +11554040s4 − 3756000s2 + 459648 2 2 2 1,1,1 24(5s − 4) (15s − 14) −2(5s2 − 4)(1584375s8 − 4536150s6 + 4716436s4 4 2 2,0 1,1,2 −3(225s − 430s + 208) −2082712s2 + 321408) 2 2 2 0,2,1 −96(5s − 4) (9s − 8) 138375s10 − 128250s8 − 351900s6 + 612440s4 2 4 2 2,1 1,2,1 −24(5s − 4)(65s − 116s + 52) −326368s2 + 56448 4 2 1,2,2 −60(50s − 87s + 38) 8(5s2 − 4)(93300s8 − 259915s6 + 264982s4 − 116928s2 2 2 2 3,0 0,3,1 −64(5s − 4) (8s − 7) +18816) 2 2 2 1,3,1 4(3s − 2)(5s − 4)(15s − 14) −20s2(5s2 − 4)(15s2 − 14)(45s4 + 36s2 − 56) 2 2 4,0 0,3,2 −4(5s − 4)(135s − 122) 4 2 1,3,2 −8(75s − 135s + 61) 2 2 2 1,4,1 −12(5s − 4) (7s − 6) 2 2 0,4,2 24(5s − 4) Then, after solving Eq. (14), we find that the third order 2 2 1,4,2 −12(5s − 4)(25s − 23) 2 2 term of the generating function can be arranged in the form 0,5,2 24(5s − 4) −3(5s2 − 4)(15s2 − 14) 1,5,2 3 3 2l+4 2 2 2 ∗ 1,6,2 6(5s − 4)  G X X X 2j+k 2 2 2 W3 = Γ3,j,k,le 0,0,2 (15s − 14) (15s − 13) 8960(5s2 − 4)4 2 2 4 2 l=1 k=l−4 j=0 0,0,1 8(5s − 4) (1215s − 1997s + 824) k6=0 −2(5s2 − 4)(15s2 − 14)(45s4 + 36s2 − 56) 1,0,1 2l ×s sin(kf + 2lg) + C3 (25)

in which the inclination polynomials Γ3,j,k,l that do not van- C Analogously as we did with 1, the arbitrary function ish are listed in Table4. The constant C3 is determined at the C2 in which Eq. (22) depends upon is determined at the next order of the perturbation approach in the same way as next step of the perturbation approach in such a way that the we did previously. We obtain appearance of unbounded terms in the solution of the ho- 3 3−l mological equation of the third order are avoided. Thus, we 3G X X C = Γ e2(j+l)s2l sin 2lω, compute Ke0,3 by successive evaluations of the fundamen- 3 1536(5s2 − 4)5 3,j,0,l tal recursion (11). After identifying the problematic terms l=1 j=0 of Ke0,3, they are cancelled by computing which contributes six additional inclination polynomials Γ3,j,0,l,

2 2−l that are also listed in Table4. 2G X X C = Γ e2(j+l)s2l sin 2lω, (23) The computation of additional orders finds similar struc- 2 64(5s2 − 4)3 2,j,0,l l=1 j=0 tures. Thus, for instance, the fourth order term of the new Solution to the main problem of the artificial satellite by reverse normalization 7

Table 4 Non-zero inclination polynomials Γ3,j,k,l in Eq. (25).

2 10 8 6 4 2 2,−3,1 35 (15s − 14)(87375s − 335550s + 505080s − 371184s + 132096s − 17920) 2 10 8 6 4 2 2,−2,1 105 (15s − 14)(399375s − 1863400s + 3389440s − 3023632s + 1328128s − 230400) 2 10 8 6 4 2 1,−1,1 −840 (5s − 4)(228125s − 549325s + 255940s + 324664s − 352992s + 93824) 2 10 8 6 4 2 2,−1,1 210 (15s − 14)(100875s − 275600s + 228220s − 6408s − 70272s + 23296) 2 2 6 4 2 2,−1,2 −105 (5s − 4)(15s − 14)(13725s − 37680s + 34228s − 10304) 2 2 8 6 4 2 0,1,1 −1680 (5s − 4) (486525s − 1594290s + 1955772s − 1064576s + 216960) 2 10 8 6 4 2 1,1,1 840 (5s − 4)(1531125s − 6503075s + 10982780s − 9224760s + 3855648s − 641920) 2 10 8 6 4 2 2,1,1 −420 (15s − 14)(61875s − 138825s + 51640s + 92200s − 89088s + 22400) 2 8 6 4 2 1,1,2 1680 (5s − 4)(240750s − 775475s + 932445s − 495822s + 98320) 2 8 6 4 2 2,1,2 105 (5s − 4)(226125s − 787950s + 1015020s − 572056s + 118944) 2 6 4 2 2,1,3 −840 (15s − 14)(1125s − 3300s + 3235s − 1058) 2 3 6 4 2 0,2,1 −1680 (5s − 4) (41615s − 97838s + 76016s − 19488) 2 10 8 6 4 2 1,2,1 −1680 (5s − 4)(666875s − 2586600s + 4014940s − 3117320s + 1210368s − 187904) 12 10 8 6 4 2 2,2,1 210 (5398125s − 27480750s + 57999400s − 64973520s + 40757888s − 13579264s + 1878016) 2 2 6 4 2 1,2,2 −3360 (5s − 4) (42850s − 108830s + 92099s − 25958) 2 8 6 4 2 2,2,2 840 (5s − 4)(123750s − 359475s + 378010s − 167724s + 25624) 8 6 4 2 2,2,3 −105 (639375s − 2259750s + 2991200s − 1757840s + 387104) 2 2 8 6 4 2 0,3,1 −1120 (5s − 4) (270650s − 828285s + 945816s − 477232s + 89664) 2 10 8 6 4 2 1,3,1 280 (5s − 4)(634500s − 2623725s + 4300340s − 3496152s + 1412352s − 227584) 2 10 8 6 4 2 2,3,1 −70 (15s − 14)(76875s − 202950s + 167700s − 16544s − 37376s + 12544) 2 2 6 4 2 0,3,2 −560 (5s − 4) (605775s − 1524950s + 1277728s − 356256) 2 2 6 4 2 1,3,2 560 (5s − 4) (9150s − 6435s − 9741s + 7154) 2 8 6 4 2 2,3,2 35 (5s − 4)(104625s − 68850s − 286620s + 378936s − 127232) 2 6 4 2 1,3,3 −280 (5s − 4)(52875s − 129225s + 102900s − 26456) 2 6 4 2 2,3,3 −140 (15s − 14)(7875s − 21225s + 19120s − 5756) 2 10 8 6 4 2 1,4,1 −840 (5s − 4)(516875s − 1956050s + 2950940s − 2217472s + 829440s − 123392) 2 10 8 6 4 2 2,4,1 420 (5s − 4)(173625s − 668250s + 1023720s − 780120s + 295872s − 44800) 2 3 4 2 0,4,2 −6720 (5s − 4) (730s − 1153s + 444) 2 2 6 4 2 1,4,2 −3360 (5s − 4) (66050s − 166215s + 139230s − 38808) 2 2 6 4 2 2,4,2 420 (5s − 4) (20925s − 38700s + 19984s − 1976) 2 6 4 2 1,4,3 840 (5s − 4)(10125s − 32150s + 33500s − 11456) 2 6 4 2 2,4,3 −2100 (5s − 4)(3525s − 9240s + 7996s − 2280) 2 10 8 6 4 2 1,5,1 −168 (5s − 4)(115000s − 434875s + 663700s − 512080s + 199936s − 31488) 2 2 2 6 4 2 2,5,1 −105s (5s − 4)(15s − 14)(825s − 1990s + 1616s − 448) 2 3 4 2 0,5,2 −336 (5s − 4) (15425s − 24050s + 9112) 2 2 6 4 2 1,5,2 −84 (5s − 4) (551625s − 1390850s + 1167040s − 325728) 2 2 2 4 2 2,5,2 105 (5s − 4) (15s − 14)(225s + 288s − 364) 2 2 4 2 0,5,3 3360 (5s − 4) (1575s − 2795s + 1256) 2 6 4 2 1,5,3 840 (5s − 4)(8250s − 24975s + 25080s − 8336) 2 2 2 2 2,5,3 −420 (5s − 4)(15s − 14) (15s − 13) 2 10 8 6 4 2 2,6,1 35 (5s − 4)(171375s − 616950s + 871680s − 600128s + 199168s − 25088) 2 3 4 2 1,6,2 −280 (5s − 4) (8265s − 12874s + 4872) 2 2 6 4 2 2,6,2 −280 (5s − 4) (11475s − 29280s + 24828s − 6992) 2 3 2 0,6,3 560 (5s − 4) (1335s − 1166) 2 2 4 2 1,6,3 560 (5s − 4) (8325s − 14910s + 6764) 2 6 4 2 2,6,3 70 (5s − 4)(21375s − 61950s + 59960s − 19328) 2 3 4 2 1,7,2 −60 (5s − 4) (8385s − 13226s + 5080) 2 3 2 0,7,3 10080 (5s − 4) (90s − 79) 2 2 4 2 1,7,3 12600 (5s − 4) (105s − 191s + 88) 2 2 3 2 2,8,2 −840 (3s − 2)(5s − 4) (15s − 14) 2 3 2 1,8,3 4200 (5s − 4) (96s − 85) 2 2 4 2 2,8,3 210 (5s − 4) (525s − 990s + 472) 2 3 2 1,9,3 7560 (5s − 4) (10s − 9) 2 3 2 2,10,3 315 (5s − 4) (15s − 14) 2 3 4 2 0,0,3 2 (15s − 14) (825s − 1445s + 634) 2 2 8 6 4 2 0,0,2 −6 (5s − 4) (2171250s − 7719525s + 10225470s − 5983260s + 1305248) 2 2 2 6 4 2 1,0,2 −3 (5s − 4)(15s − 14) (1800s + 2655s − 8208s + 3928) 2 2 10 8 6 4 2 0,0,1 48 (5s − 4) (9060750s − 34431275s + 51858720s − 38675200s + 14258176s − 2072064) 2 12 10 8 6 4 2 1,0,1 −12 (5s − 4)(93223125s − 421210500s + 784654200s − 771469840s + 422629664s − 122600960s + 14780416) 2 12 10 8 6 4 2 2,0,1 6 (15s − 14)(2328750s − 8703375s + 13317150s − 10848180s + 5157560s − 1450624s + 200704) 8 Martin Lara

Hamiltonian takes the form perturbation solution [19,9]. Thus, for instance, we find 5, 56, 367, 1152, 2627, and 4897 expanded terms for the 1st, 1 5 b3− jc µ p2 9s2 X pj X2 2nd . . . , 6th order terms of the generating function, respec- K = 4 e2kγ , 0,4 r r2 1280(5s2 − 4)6 rj 4,j,k tively, whereas much longer expressions have been reported j=0 k=0 in the literature. For instance, the sixth order of the gener- (26) ating function reported in [56] entails 39630 terms, which and the fourth order term of the generating function takes is almost one order of magnitude larger than the one ob- the form tained with the current approach. It is worth mentioning that further simplifications could be obtained in particular cases, 4 4 2l+6 6  G X X X ∗ W = Γ e2j+k like when constraining the application of the analytical so- 4 30105600(5s2 − 4)6 4,j,k,l l=1 k=l−6 j=0 lution to the case of low eccentricity orbits, in which case k6=0 many of the terms involved can be neglected [40,21]. 2l ×s sin(kf + 2lω) + C4. (27) The comparisons are, nevertheless, inconclusive due to the diversity of approaches used in the computation of the While the former involves 15 inclination polynomials γ , 4,j,k variety of solutions reported in the literature, which involve the later comprises up to 124 non-vanishing inclination poly- representation in different variables, on the one hand, and nomials Γ , of degree 9 in s2. Therefore, these and other 4,j,k,l yield distinct one-degree-of-freedom Hamiltonians, on the polynomials resulting from following orders are not listed other. The partially normalized Hamiltonian obtained here due to their length. is definitely longer than the one obtained after the classi- The normalization of G has been extended up to the or- cal elimination of the perigee. However, this is not of con- der six without major trouble, except for the increase of the cern in the complete, as opposite to partial, normalization computational burden of successive orders due to the no- of the main problem. Indeed, after the following elimina- table growth of the length of the series to be handled, on tion of short-period terms either procedure should arrive to the one hand, and the increasing size of the rational coef- the same Hamiltonian. We didn’t find reported data for the ficients resulting from the integer arithmetic used, on the second normalization after the elimination of the perigee, other. Thus, the new Hamiltonian terms K and K take 0,5 0,6 but one would expect that the figures should be balanced to analogous forms to the previous orders, with 24 inclination some extent —the generating function of the short-period polynomials γ , and 35 γ . Similarly, the generating 5,j,k 6,j,k elimination probably being heavier in our case than in other function terms W and W have been arranged in the same 5 6 prospective approaches. While the needed data for the thor- form as previous orders of the perturbation approach, with ough comparison is not available, our claims must restrict 254 non-vanishing coefficients Γ and 429 Γ , 15 5,j,k,l 6,j,k,l to the checked fact that our approach makes both transfor- of which correspond to the integration constant C and 21 5 mations of manageable size —as we will show in the next to C . 6 section, where the short-period effects are removed in a De- At the end, since the generating function is known, the launay normalization. This feature makes the evaluation of transformation equations from the original Delaunay vari- higher orders of our perturbation solution certainly practica- ables to the new ones, and vice versa, are readily computed ble. by standard application of the fundamental recursion (11) (see [15] for details). The procedure ends by replacing the original variables by the new ones in the computed terms 2.3 Delaunay normalization K0,m. We remark that the procedure described here is not equiv- The partially normalized Hamiltonian is just of one degree alent to the combination of the elimination of the parallax of freedom, yet it is not separable. To get an explicit ana- and the elimination of the perigee into a single transforma- lytical solution we remove the short-period terms that are tion, in spite of the fact that the total angular momentum is associated to the radius by means of a Delaunay normaliza- converted into a formal integral in both cases. On the con- tion [17]. The partially reduced Hamiltonian from which we trary, the perigee has been removed here keeping as much start takes again the form of Eq. (4), but it is now written short-period terms as possible in the new Hamiltonian. With in prime Delaunay variables (`0, g0, h0,L0,G0,H0). In this this strategy, the size of the generating function of the first Hamiltonian the variables g0 and h0 are cyclic, and hence partial normalization is astonishingly smaller than the one H0 = H and G0 remain constant for given initial conditions. that would be obtained with other alternatives in the liter- The zeroth order term is the same as in Eq. (5), the term K1,0 ature. To check that, we fully expanded both the Hamil- is given by Eq. (15), whereas terms Km,0 with m ≥ 2 are tonian and the generating function and reckoned the num- no longer void, and, on the contrary are given by Eqs. (21), ber of terms. This procedure is the one that has been tradi- (24), (26), for m = 2, 3, 4, respectively, and analogous equa- tionally used in the literature to assess the complexity of a tions for higher orders that, due to its length, are not printed Solution to the main problem of the artificial satellite by reverse normalization 9

in the paper. Moreover, the definition in Eq. (17) turns  into Table 5 Inclination polynomials Λ2,j,k in Eq. (33). a physical constant, rather than a function, when witten in 2 6 4 2 1,0 15(3s − 2)(805s − 2448s + 2400s − 768) the prime variables. In consequence  can replace now the 2 6 4 2 1,1 3(3s − 2)(2225s − 8160s + 8928s − 3072) 8 6 4 2 formal small parameter ε used before in the Lie transforms 1,2 3(825s − 3030s + 4064s − 2368s + 512) 2 6 4 2 procedure, assumed, of course, that the corresponding scal- 1,3 −3s (975s − 2250s + 1728s − 448) 8 6 4 2 ing of the Hamiltonian terms is properly made. 2,0 6(1925s − 6210s + 7452s − 3936s + 768) 8 6 4 2 2,1 6(125s − 930s + 1660s − 1120s + 256) The homological equation to be solved at each step of 8 6 4 2 3,0 2625s − 7270s + 7408s − 3264s + 512 2 6 4 2 the new Lie transformation is the same as before, either in 3,1 s (825s − 1990s + 1616s − 448) the form given by Eq. (12) or Eq. (14), except for choosing a non-zero integration constant does not provide any advan- 2 2 tage in the current case. In fact, since the new Hamiltonian where β = 1/(1 + η), Φ2,0 = 8(s − 1)(5s − 4), Φ2,1 = 2 4 must be free from terms depending on the mean anomaly, 8 − 8s − 5s , and the inclination polynomials Λ2,j,k are which are obtained by averaging, the homological equation provided in Table5. can be further particularized. That is, Integrals cease to be trivial at the third order. Indeed, the formal computation of K using the fundamental recursion Z 2π 2 0,3 1 r (11) yields two different types of terms. K0,m = hKe0,mi` = Ke0,m 2 df, (28) 2π 0 a η The first type consists of terms depending on the equa- and hence tion of the center, which can be reduced to the form Z 2 µ p ` 1 r P (e)Q(s)φ sin mf, Wm = − K0,m + Ke0,m df n n a2η r r φ 1 Z  r2  with P and Q denoting arbitrary eccentricity and inclination = K0,m + Ke0,m − K0,m df, (29) n n a2η polynomials. Definite integration of these kinds of terms is carried out from expressions in [52], whereas the indefinite in which φ = f − ` is the equation of the center, and the integration is achieved by parts, to get terms under the integral sign in the final form of the equation are purely periodic in f. Z sin mf sin mf Z ma2η φ d` = −φ cos mf − cos mfd`, Thus, at the first order of the Lie transforms procedure r2 m we find in which the antiderivatives of cosines of multiples of the µ K = hK i ≡ η3 3s2 − 2
, (30) true anomaly are carried out after expressing them in terms 0,1 1,0 ` p of r and R = dr/dt, rather than in trigonometric functions, from which as discussed in [28]. The second type consists of terms free from the equation 0 2
W1 = G 3s − 2 (e sin f + φ). (31) of the center. In these terms, the trigonometric functions of the true anomaly can be replaced by inverse powers of the Recall that the symbols p, η, e, etc. are now functions of the radius, without involving the radial velocity. We found that Delaunay prime variables. the exponents of the inverse of r range from 0 to 8 miss- At the second order, the known terms are given again by ing the exponent 1. Therefore, both definite and indefinite Eqs. (18) and (19), from which, using Eq. (28), integrals of terms of the second type are trivially solved.

2 Proceeding in this way, we compute the third order Hamil- µ 3 X K = − η3 λ ηj, (32) tonian term 0,2 p 4 2,j j=0 4 µ 9η3 X K = λ ηj, (34) 4 2 2 2 0,3 p 16(5s2 − 4)2 3,j with λ2,0 = 5(7s − 16s + 8), λ2,1 = 4(3s − 2) , and j=0 4 2 λ2,2 = 5s + 8s − 8. The second order term of the gen- erating function is then trivially integrated from Eq. (29), to with the inclination polynomials λ3,j given in Table6. yield The corresponding term of the generating function is

1 1 3 3−b jc 6 7−2b jc G0β X X2 G0β2 X X2 W = − Λ ηkej sin jf W = Λ ηk−1ej sin jf 2 32(5s2 − 4)2 2,j,k 3 128(5s2 − 4)3 3,j,k j=1 k=0 j=1 k=0 1 3 4 3 X 3G0 X X −G0 φ Φ e2j, (33) + φ Φ ηkej cos jf, (35) 4 2,j 16(5s2 − 4)2 3,j,k j=0 j=0 k=0 10 Martin Lara

2 2 6 Table 6 Inclination polynomials λ3,j in Eq. (34). in s although they are divided by (5s − 4) , as well as Λ 10 8 6 4 134 nonvanishing trigonometric polynomials of the type , 0 5(28700s − 107205s + 158960s − 118492s 2 +45152s2 − 7168) which are of degree 12 in s yet they must be divided by 2 2 2 4 2 2 7 1 60(3s − 2)(5s − 4) (7s − 16s + 8) (5s − 4) . Finally, the figures of the 6th order are 11 and 10 8 6 4 2 2 −2(28675s − 98005s + 130852s − 87164s 70 of degree 13 in s for λ and Φ, respectively, and 218 for +30176s2 − 4608) Λ 2 2 2 4 2 of degree 16. 3 20(3s − 2)(5s − 4) (5s + 8s − 8) 2 2 6 4 2 Once reached the desired order of the second normal- 4 −s (15s − 14)(450s − 925s + 590s − 112) ization, the procedure ends by changing prime variables by double-prime variables in the new Hamiltonian terms K0,m. Table 7 Non-zero coefficients Φ in Eq. (35). Φ = 15 Φ , 3,j,k 3,1,0 2 3,0,3 In order to provide comparative figures of the compu- Φ = − 3 Φ , Φ = 3Φ , and Φ = 1 Φ . 3,1,2 2 3,0,3 3,2,0 3,0,3 3,3,0 2 3,0,3 tational burden of this normalization with other approaches 10 8 6 4 0,0 5(89100s − 323615s + 466320s − 337684s that might be carried out, we expanded the series that com- +125216s2 − 19456) prise the solution and reckon the number of separate terms, 10 8 6 4 0,2 −2(112125s − 374775s + 488460s − 314932s as we already did in the normalization of the total angular +103840s2 − 14848) 2 2 2 4 2 momentum. We found that the 1st, 2nd, . . . 6th-order terms 0,3 8(3s − 2)(5s − 4) (5s + 8s − 8) 2 2 6 4 2 0,4 −3s (15s − 14)(450s − 925s + 590s − 112) of the generating function of the Delaunay normalization comprise 4, 48, 257, 931, 2266, and 4826 terms respectively. Note that different arrangements from the one chosen by in which the inclination polynomials Φ3,j,k are given in Ta- us —Eqs. (35), (37), etc.— may provide different figures ble7, and those Λ3,j,k in Table8. than those reported, yet should be of analogous magnitude. In the process of carrying out the normalization to higher Following the same procedure with the completely reduced orders we need to deal with trigonometric series of notably Hamiltonian, we reckon up to 2, 9, 29, 55, 106, and 152 co- increasing length. However, we only found terms of the same efficients, for the 1st, 2nd, . . . 6th-order term, respectively. two types as before in the solution of the homological equa- tion, which, in consequence, is analogously integrated. In this way, the second normalization has been extended up to 2.4 Secular terms the order 6 in the small parameter, in agreement with the After neglecting higher order effects of J , the normalized order to which the perigee was previously eliminated. Cor- 2 Hamiltonian is responding Hamiltonian and generating function terms are m˜ analogously arranged in the form of Eqs. (34) and (35), re- X m K00 = K00(−, −, −,L00,G00,H00) ≡ K , (38) spectively, yet the inclination polynomials comprise much m! m,0 more longer listings, as expected. Thus, for instance, at the m≥0 fourth order the normalized Hamiltonian term in which m˜ ≤ 6 for the different approximations provided 6 by the computed perturbation solution, and terms Km,0 are µ 9η3 X K = λ ηj, (36) obtained by replacing prime by double-prime variables in 0,4 p 64(5s2 − 4)3 4,j j=0 corresponding terms given by Eqs. (30), (32), (34), as well as higher order Hamiltonian terms K0,m that have not been contributes 7 new inclination polynomials, which are poly- displayed. That is, the symbols p, η, and s in these equations nomials of degree 7 in the square of the sine of the inclina- are assumed to be functions of the double-prime Delaunay tion, whereas the generating function term momenta. Recall that  is obtained from Eq. (17) by making 1 002 6 7−2b jc p = G /µ. G0β3 X X2 W = Λ ηk−3ej sin jf The corresponding solution to the flow stemming from 4 20480(5s2 − 4)6 4,j,k j=1 k=0 Eq. (38) is 0 5 6 00 00 3G X X ` = ` + n`t, + φ Φ ηkej cos jf, (37) 0 256(5s2 − 4)3 4,j,k 00 00 j=0 k=0 g = g0 + nωt 00 00 h = h + nht, is made of 20 non-zero coefficients Φ4,j,k, of degree 7 in 0 2 s , and 72 non-zero coefficients Λ4,j,k, which are of degree in which, from Hamilton equations, 10 in s2. Note that, because of the denominators factoring ∂K00 ∂K00 ∂K00 the summations, the maximum degree amounts to 4 in any n` = 00 , ng = 00 , nh = 00 , case, in agreement with the order of the perturbation. At the ∂L ∂G ∂H 00 00 00 0 00 5th order we find 9 coefficients of the type λ and 41 non- L = L0 , G = G0, and H = H0, are the initializa- 00 00 00 00 vanishing coefficients of the type Φ, which are of degree 11 tion constants of the analytical solution, and `0 , g0 , h0 , L0 , Solution to the main problem of the artificial satellite by reverse normalization 11

Table 8 Non-zero inclination polynomials Λ3,j,k in Eq. (35).

2 3 2 3 1,0 −864(3s − 2) (5s − 4) 12 10 8 6 4 2 1,1 3(22218875s − 104346550s + 202703740s − 209869352s + 123038240s − 39033472s + 5275648) 12 10 8 6 4 2 1,2 12(10925500s − 50711075s + 97386820s − 99715748s + 57863024s − 18199872s + 2445312) 12 10 8 6 4 2 1,3 3(27560125s − 119080550s + 212650740s − 202245448s + 109190304s − 32208768s + 4128768) 12 10 8 6 4 2 1,4 12(3155125s − 10820800s + 13899620s − 7620256s + 944256s + 613248s − 167936) 12 10 8 6 4 2 1,5 3(5410625s − 17331450s + 19448180s − 6842968s − 2742560s + 2556544s − 491520) 12 10 8 6 4 2 1,6 −12(59625s − 415275s + 942920s − 994980s + 529776s − 134592s + 12288) 2 10 8 6 4 2 1,7 −3s (77625s − 568950s + 1256420s − 1222216s + 550816s − 94080) 2 3 2 3 2,0 −1044(3s − 2) (5s − 4) 12 10 8 6 4 2 2,1 24(131000s − 1121875s + 3061340s − 3989664s + 2758768s − 983648s + 143360) 12 10 8 6 4 2 2,2 96(51625s − 437800s + 1183290s − 1528682s + 1049344s − 372240s + 54144) 2 10 8 6 4 2 2,3 −12(5s − 4)(16375s + 64070s − 257508s + 297320s − 145792s + 26624) 12 10 8 6 4 2 2,4 −12(263625s − 1000750s + 1526820s − 1206712s + 539616s − 142592s + 19968) 2 10 8 6 4 2 2,5 −12s (162375s − 576100s + 787020s − 506248s + 145984s − 12992) 2 3 2 3 3,0 −656(3s − 2) (5s − 4) 12 10 8 6 4 2 3,1 −934875s + 605000s + 4973120s − 10412952s + 8554272s − 3281280s + 491520 12 10 8 6 4 2 3,2 −2080125s + 3562000s + 2881300s − 11103360s + 10155456s − 4038912s + 614400 2 10 8 6 4 2 3,3 −(5s − 4)(254925s − 526480s + 318084s − 10672s − 40064s + 8192) 12 10 8 6 4 2 3,4 3(28875s − 147800s + 254260s − 160992s − 11968s + 54016s − 16384) 2 10 8 6 4 2 3,5 −12s (4500s − 4125s − 16075s + 31670s − 20664s + 4704) 2 3 2 3 4,0 −240(3s − 2) (5s − 4) 2 10 8 6 4 2 4,1 6(5s − 4)(50325s − 157660s + 180520s − 90312s + 18112s − 1024) 2 10 8 6 4 2 4,2 12(5s − 4)(47475s − 147000s + 164852s − 79056s + 14208s − 512) 2 2 8 6 4 2 4,3 6s (5s − 4)(44625s − 136340s + 149184s − 67800s + 10304) 2 3 2 3 5,0 −48(3s − 2) (5s − 4) 2 2 2 6 4 2 5,1 6s (5s − 4) (180s − 609s + 530s − 112) 2 2 8 6 4 2 5,2 3s (5s − 4)(1125s − 7440s + 11516s − 6144s + 896) 4 2 2 4 2 5,3 −3s (5s − 4)(15s − 14)(45s + 36s − 56) 2 3 2 3 6,0 −4(3s − 2) (5s − 4)

0 and G0, are computed applying consecutively the inverse [19] transformation of the normalization of the angular momen- F = F + n t, (39) tum and the Delaunay normalization to corresponding initial 0 F conditions of the original problem. C = e cos(g0 + ngt) = C0 cos ngt − S0 sin ngt, (40) On the other hand, Delaunay variables are singular for S = e sin(g0 + ngt) = S0 cos ngt + C0 sin ngt, (41) equatorial orbits, in which the argument of the node is not L = L0, (42) defined, as well as for circular orbits, in which the argument h = h0 + nht, (43) of the periapsis is not defined. In particular, the singularity H = H , (44) for circular orbits reveals immediately by the appearance of 0 the eccentricity in denominators of the transformation equa- in which nF = n` + ng, C0 = e cos g0, S0 = e sin g0, and 2 2 1/2 tions of both the mean anomaly and the argument of the e = (1 − G0/L0) . The double-prime notation has been perigee (see Eqs. (20) and (21) of [6], for instance). This omitted for simplicity. fact not only makes singular the transformation of these el- After standard partial differentiation, we obtain ements for exactly circular orbits, but prevents convergence m˜ 2m−1 of the respective perturbation series for small values of the X m X n = n + n Ψ (s)ηi, (45) eccentricity. F (5s2 − 4)m m,i m=1 i=0 Different sets of non-singular variables can be used to m˜ 2m−2 X m X avoid these issues [51]. In particular, troubles related with n = n ω (s)ηi, (46) g (5s2 − 4)m m,i low eccentricities are commonly avoided by replacing the m=1 i=0 m˜ 2m−2 mean anomaly, the argument of the periapsis, and the total X m X n = nc Ω (s)ηi, (47) angular momentum with the non-canonical variables given h (5s2 − 4)m m,i by the mean distance to the node F = `+g, also called mean m=1 i=0 argument of the latitude, and the components of the eccen- where the inclination polynomials Ψm,i, ωm,i, and Ωm,i are tricity vector in the nodal frame C = e cos ω, S = e sin ω, provided in Tables9, 10, and 11, respectively, up to the third also called semi-equinoctial elements [34,11]. In these vari- order of the perturbation approach. In these tables we can ables, the secular terms of the main problem are given by check that, in spite of the general arrangement of the secular 12 Martin Lara frequencies in Eqs. (45)–(47), the critical inclination divi- the true anomaly. Therefore, we rather formulate them in sors 5s2 −4 start to appear only at the third order truncation. polar variables. In this way we avoid the trouble of small denominators in the J2-problem, yet, of course, the nodes of exactly equatorial orbits remain undefined. For simplicity Table 9 Inclination polynomials Ψi,j in Eq. (45). we do not deal with this latter case, which, if desired, could 2 2 1,0 −3(5s − 4) be approached using different sets of non-singular variables. 2 2 1,1 −3(3s − 2)(5s − 4) 15 2 2 4 2 2,0 8 (5s − 4) (77s − 172s + 88) 9 2 2 4 2 2,1 8 (5s − 4) (155s − 256s + 104) 3 2 2 4 2 3 Performance of the solution 2,2 8 (5s − 4) (189s − 156s + 8) 15 2 2 4 2 2,3 8 (5s − 4) (5s + 8s − 8) 15 12 10 8 The performance of the analytical solution is illustrated in 3,0 − 32 (2439500s − 11312175s + 21772080s −22346500s6 + 12956400s4 − 4043136s2 + 533248) three different cases. The first one is a low-altitude, almost- 45 2 10 8 6 3,1 − 32 (5s − 4)(62300s − 260365s + 431504s circular orbit with the orbital parameters of the PRISMA −356508s4 + 147552s2 − 24576) 3 12 10 8 mission [55]. The second is a high-eccentricity orbit with 3,2 16 (1835625s − 7723875s + 13291500s −12015300s6 + 6064176s4 − 1644928s2 + 192256) the typical characteristics of a geostationary transfer orbit 15 2 10 8 6 4 (GTO), which we borrowed from [35]. The third test has 3,3 16 (5s − 4)(18175s − 85105s + 153172s − 136540s +61408s2 − 11264) been specifically carried out to check the behavior of the 3 (213750s12 − 1441125s10 + 3537000s8 − 4313100s6 3,4 32 analytical solution when approaching the critical inclination +2835280s4 − 967808s2 + 135424) 21 2 2 2 6 4 2 resonance. For this last case we selected an orbit with orbital 3,5 s (5s − 4)(15s − 14)(450s − 925s + 590s − 112) 32 parameters similar to the TOPEX orbit, which departs only ∼ 3◦ from the inclination resonance condition [20]. Orbital elements corresponding to these three cases are presented in Table 12. The reference, true orbits have been propagated Table 10 Inclination polynomials ωi,j in Eq. (46). numerically in extended precision to assure that all the com-

2 2 puted points are accurate with 15 significant digits in the 1,0 −3(5s − 4) 15 2 2 4 2 decimal representation. We requested that precision because 2,0 8 (5s − 4) (77s − 172s + 88) 2 2 3 2,1 9(3s − 2)(5s − 4) this is the maximum number of digits with which exact dec- 3 2 2 4 2 2,2 8 (5s − 4) (45s + 36s − 56) imal operations are guaranteed in standard double-precision − 15 (2439500s12 − 11312175s10 + 21772080s8 3,0 32 [30]. Because the analytical solution is evaluated in double- −22346500s6 + 12956400s4 − 4043136s2 + 533248) 45 2 3 6 4 2 precision the reference numeric orbit is considered exact. 3,1 − 4 (5s − 4) (168s − 497s + 460s − 136) 3 12 10 8 3,2 16 (2150625s − 9409875s + 16968300s −16218180s6 + 8729136s4 − 2535808s2 + 315136) 15 2 3 6 4 2 Table 12 Initial conditions of the test cases. Angles are in degrees. 3,3 − 4 (5s − 4) (105s + 39s − 228s + 104) 3 12 10 8 6 3,4 32 (438750s − 1771125s + 2865000s − 2345100s type a (km) e I Ω ω ` +999760s4 − 199808s2 + 12544) PRISMA 6878.137 0.001 97.42 168.162 20 30 TOPEX 7707.270 0.0001 66.04 180.001 270 180 GTO 24460.00 0.73 30. 170.1 280 0

Table 11 Inclination polynomials Ω in Eq. (47). i,j Two different kind of errors are associated to the trunca- 2 1,0 −6(5s − 4) tion order of the analytical solution. On the one hand, the 15 2 2 2 2,0 2 (5s − 4) (7s − 8) truncation of the secular terms introduces an error in the 2 2 2 2,1 18(3s − 2)(5s − 4) 3 2 2 2 computation of the frequencies of the analytical solution, 2,2 2 (5s − 4) (5s + 4) 15 10 8 6 4 Eqs. (45)–(47), which will make the perturbation solution 3,0 − 8 (215250s − 823025s + 1255040s − 953760s +361088s2 − 54464) to degrade with time. On the other hand, the truncation af- 45 2 3 4 2 3,1 − 4 (5s − 4) (63s − 124s + 56) fects also the generating function from which the periodic 3 (430125s10 − 1553550s8 + 2222340s6 − 1570224s4 3,2 8 corrections are derived. The latter fundamentally affects the +546432s2 − 74624) 15 2 3 4 2 amplitude of the periodic errors, and, therefore, the qual- 3,3 − 4 (5s − 4) (45s + 28s − 40) 3 10 8 6 4 ity of the ephemeris provided by the analytical solution is 3,4 8 (50625s − 168375s + 215900s − 130800s +35840s2 − 3136) not expected to deteriorate significantly with time by effect of this error. However, this is only true for the direct trans- formation, a case in which we will see that it is acceptable Computing the periodic corrections of the semi-equi- to use a lower order truncation than the truncation used for noctial variables would require to carry out expansions of the secular terms. On the contrary, the truncation order of Solution to the main problem of the artificial satellite by reverse normalization 13 the inverse corrections has a direct effect in the precision 0.4

δ a 0.2

with which the initialization constants are computed from 6 0 a given set of initial conditions. Therefore, not computing 10 -0.2 the inverse transformation with the same accuracy as that 0 1 2 3 4 5 6 7 of the secular terms will also contribute to the deterioration 0.3 of the latter. In practice, the highest accuracy of the inverse δ a

9 0

transformation can be limited to the computation of the peri- 10 odic corrections of the initial semimajor axis (or its partner -0.3 0 1 2 3 4 5 6 7 canonical variable, the Delaunay action) because it affects 0.6 directly the secular mean motion n , while the other ele- 0.3 ` δ a 0 ments only affect the frequencies of the secular node and 12 -0.3 perigee, which are O(J2) when compared to n`. 10 -0.6 0 1 2 3 4 5 6 7 0.6

δ a 0.4 3.1 Accuracy of the periodic corrections 0.2

14 0.0

10 -0.2 First of all, we check the accuracy of the periodic corrections -0.4 for increasing orders of the perturbation solution. If the pe- 0 1 2 3 4 5 6 7 riodic corrections were exact, transforming different states Fig. 1 Relative errors of the semimajor axis of the PRISMA-type orbit of the same orbit provided by the reference solution will re- for, from top to bottom, the 1st, 2nd, 3rd, and 4th order truncations of sult in the same, constant secular values of the momenta, the perturbation solution. The latter shows the relative errors of the 5th order truncation superimposed. Abscissas are hours. and in an exactly linear growing of the secular angles. On the contrary, the secular values obtained from the different states of the true orbit oscillate periodically due to the trun- rather than the total angular momentum G0, for the 1st, 2nd, cation order of the solution. This is illustrated in Fig.1 for . . . , 4th order truncations of the periodic corrections, respec- the semimajor axis of the PRISMA test orbit, where the rela- tively. The relative errors of the 4th order truncation in the tive errors with respect to a constant reference value, which bottom plot of Fig.2 (black dots) are superimposed to the has been obtained as the arithmetic mean of the computed third order ones (gray line) for reference. Corresponding ab- secular values, are shown for different truncations of the per- solute errors are of the order of tens of milliarc seconds for turbation solution. We recall that the expected errors of a the 1st order, hundredths of mas for the second, hundredths perturbation solution are of the order of the neglected terms, of microarc seconds for the third, and below the level of as follows from Eq. (8). Therefore, for a truncation to the 1 thousandth of µas for the fourth order. Due to the larger order m, we would expect errors of the order εm+1. semimajor axis, the GTO orbit is less perturbed in general, We found that the relative errors of the secular semima- and we found that the numerical precision is achieved at the jor axis of the first order truncation (top plot of Fig.1) are of fourth order of the perturbation approach. 2 ◦ the order of J2 , as it should be the case, which amounts to 3 In the case of the TOPEX orbit the inclination of ∼ 66 meters in absolute value. The relative errors of the second or- makes the coefficient 5s2 − 4 to become less than 2 tenths. 3 der truncation (second to top plot) are of the order of J2 , or However, this small divisor is not harmful at all in a first or- less than 3 millimeter in absolute value. The third and fourth der truncation because, as follows from Eq. (20), it is multi- order truncations of the solution yield relative errors of the plied by the square of the eccentricity, which is really small 4 5 order of J2 and J2 , respectively (second from bottom and for the TOPEX orbit (about one thousandth, on average). bottom plots, respectively), or absolute errors of microme- Hence, the relative errors in the semimajor axis introduced ters and hundredths of micrometers, respectively. The lat- by the periodic corrections are slightly better than those pre- ter are very close to the 15 exact digits reachable working viously found for PRISMA, as shown in the top plot of in double precision. Still, additional improvements are ob- Fig.3. The improvements should be a consequence of the tained when the truncation of the periodic corrections is ex- larger semimajor axis of the TOPEX orbit when compared tended to the fifth order of J2, now effectively reaching the with that of PRISMA. numerical precision, as shown in the bottom plot of Fig.1, in The effects of the offending divisors are more apparent which the relative errors of the fifth order truncation (black when using higher order truncations of the periodic correc- dots) are superimposed to the previous case. tions. Indeed, the coefficients of sin(f + 2ω) and sin(3f + The behavior is analogous in the case of the GTO orbit, 4ω) in Eq. (22), as well as the coefficient of sin f in Eq. (33), yet now the errors notable peak at perigee passages. This is are multiplied by the factor e/(5s2 −4)2, which, while being illustrated in Fig.2, where, from top to bottom, we present bigger than before, it is neither of worry for TOPEX yet, as now the relative errors of the inclination I = arccos(H/G0), demonstrated in the second to the top plot in Fig.3, where 14 Martin Lara

0.1 0.0 ing coefficients are found in higher orders of the periodic δ I

6 -0.1 corrections, which make that the relative errors of the fourth

10 -0.2 -0.3 order truncation are clearly larger than corresponding ones 0 5 10 15 20 25 30 of PRISMA. These relative errors are depicted in the bottom 0.3 plot of Fig.3 (gray line) superimposed with the errors of the 0.2 fifth order truncation (black dots), showing that, at the end, δ I

9 0.1 the numerical precision is reachable also in this case with 10 0.0 -0.1 the fifth order truncation of the solution. The absolute error 0 5 10 15 20 25 30 of the different truncations are now of the order of one me- 0.0 ter, a few tenths of cm, sevral hundredths of mm, tenths of δ I -0.1

12 -0.2 µm, and thousandths of µm, respectively.

10 -0.3 -0.4 0 5 10 15 20 25 30 3.2 Accuracy of the analytical solution 2

δ I 1

14 0 The reference orbits of the three test cases are now com- -1 10 -2 pared with the ones provided by the analytical solution. That 0 5 10 15 20 25 30 is, starting from initial conditions in Table 12, we first apply Fig. 2 Relative errors of the inclination of the GTO orbit for, from top the inverse periodic corrections to initialize the constants of to bottom, the 1st, 2nd, 3rd, and 4th order truncations of the perturba- the secular solution. Then, the secular terms are evaluated in tion solution. The latter is superimposed to the relative errors of the 3rd Eqs. (39)–(44) at the same times ti as those in which the true order truncation. Abscissas are hours. solution has been obtained from the numerical integration. It follows the application of the direct periodic corrections 1.5 1.0 to each secular term to get the ephemeris predicted by the δ a 0.5

7 t 0.0 analytical solution at the times i. The accuracy of the dif- 10 -0.5 ferent truncations of the analytical solution is assessed by 0 1 2 3 4 5 6 7 computing the root-sum-square (RSS) of the difference be- 0.4 tween the position and velocity predicted by the analytical

δ a 0.2 solution and those of the reference orbit and the times ti. 9 0 The results of the PRISMA test case are shown in Fig.4, 10 -0.2 in which ordinates are displayed in a logarithmic scale to 0 1 2 3 4 5 6 7 ease comparisons. The letter S in the labels (S:P ) stand for 0.1 0.0 the truncation order of both the inverse corrections and the δ a -0.1

11 -0.2 secular terms, whereas P indicates the truncation order of -0.3 10 -0.4 the direct periodic corrections. As we already pointed out, 0 1 2 3 4 5 6 7 the latter do not need to be computed to the same accuracy 30 as the inverse transformation. We observe in Fig.4 that a

δ a 20 10 simple first order truncation of the analytical solutions — 15 0 10 -10 curve labeled (1:1)— starts with a RSS error of approxi- 0 1 2 3 4 5 6 7 mately 1 meter in position, but, due to the errors introduced by the first order truncation of the secular terms, the RSS Fig. 3 Relative errors of the semimajor axis of the TOPEX-type or- errors reach more than 10 km after one month. The solution bit for different truncations of the perturbation solution. Abscissas are hours. is notably improved when taking the second order terms of the inverse corrections and the secular terms into account. This is shown with the curve labeled (2:1), that only reaches it can be checked that the relative errors in the semimajor about 30 m at the end of the one-month propagation. The 3 axis remain of O(J2 ). On the contrary, at the third order we improvements are obtained with only a slight increase of the find the small divisor alone in coefficients of sin 2m(f +ω), computational burden, because the inverse corrections and m = 1, 2, 3, in Eq. (25), in this way lessening to some ex- the secular frequencies are evaluated only once. The (3:2) 3 tent the corrector effect of the terms factored by J2 . This propagation starts from a much smaller RSS error, of less slight deterioration of convergence is noted in the second- than 1 cm, that only grows to about 10 cm by the end of from-bottom plot of Fig.3, where we see that the relative day 30. Errors fall clearly below the mm level in the case of errors in the semimajor axis are now about 10 times big- the (4:3) truncation, and to just a few µm in the (5:4) case. ger than corresponding errors of PRISMA. Bigger offend- No further improvement of the RSS errors is observed if the Solution to the main problem of the artificial satellite by reverse normalization 15 direct periodic corrections are taken also up to the fifth or- Analogous tests carried out for TOPEX are presented der, yet a slight improvement is achieved in that case in the in Fig.6. The results are similar to those previously pre- preservation of the energy integral, that reaches in this last sented in Fig.4 for PRISMA, and the apparent discrepancies case the numerical precision. when using the (2:1) solution are only due to the logarithmic scale, which encompasses a different range in each figure. In fact, the periodic oscillations of the RSS errors are of the same amplitude in both cases (∼ 2 m), yet the secular trend

1000 (1:1) grows at a low rate of less than 2 cm per day in the case of (2:1) TOPEX while it does almost two orders of magnitude faster ∼ 1 1 for PRISMA ( m/day). This fact should be attributed (3:2) to the less perturbed orbit of TOPEX due to its higher al- titude. The (3:2) solution provides quite similar RSS errors 0.001 (4:3) in both cases, TOPEX and PRISMA, whereas the (4:3) so- lution performs worse for TOPEX, whose RSS errors grow m ) ( error position RSS -6 10 (5:4) now at a secular rate about 5 times faster than in the case of PRISMA. These behavior is in agreement with the slight de- 0 5 10 15 20 25 30 terioration of the solution previously observed when testing Fig. 4 RSS errors of different (S:P) truncations of the secular (S) and the periodic corrections due to the proximity to the critical periodic terms (P) of the analytical solution in the PRISMA test case. inclination. Things are balanced with the (5:4) order solution Abscissas are days. —also in agreement with the previously observed behavior due to the increased precision of the analytical solution— for which the RSS errors reach only the micrometer level at The behavior of the analytical solution is analogous in the end of the 30 day propagation. the case of the high-eccentricity GTO orbit in what respects to the secular terms, yet the errors of the periodic correc- 1000 (1:1) tions are now of larger amplitude, as clearly observed in Fig.5. Now, the periodic oscillations of the (1:1) trunca- 10 (2:1) tion may allow the RSS errors to grow to many tens of km. 0.100 A secular trend in the RSS errors of about half meter per (3:2) day of the (2:1) solution remains almost hidden along the 0.001 30 days propagation under periodic oscillations of about 30 (4:3) 10-5 m. A similar behavior is observed in the (3:2) propagation, ( m ) error position RSS 5:4 where the amplitude of the oscillations is now reduced to the ( ) 10-7 cm order. This amplitude is further reduced to hundredths of mm with the (4:3) truncation. Finally the (5:4) solution im- 0 5 10 15 20 25 30 proves slightly the propagation, and the RSS errors remain Fig. 6 RSS errors of different (S:P) truncations of the analytical solu- of just a few µm along the whole propagation interval, thus tion in the TOPEX test case. Abscissas are days. showing the same quality as in the previous test case. As it is confirmed by the examples carried out, at the end of a long-term propagation the secular trend of the er- rors prevails over their periodic oscillations. Therefore, in practice, perturbation solutions in which the periodic terms 1000 (1:1) (2:1) are recovered to a lower order than the order at which the secular frequencies are truncated make full sense. The com- 1 (3:2) putational burden of these kinds of solutions is notably al- leviated, and hence are definitely much more practicable 0.001 (4:3) for ephemeris computation. This is further illustrated for TOPEX in Fig.7, where it is shown that the (5:3) truncation m ) ( error position RSS 10-6 of the analytical solution might replace the more accurate (5:4) (5:4) truncation in a long-term propagation with substantial 0 5 10 15 20 25 30 reductions in computing time and minimum degradation of Fig. 5 RSS errors of different (S:P) truncations of the analytical solu- accuracy, which, besides, remains much more uniform along tion in the GTO test case. Abscissas are days. the whole propagation interval. Note that the scale of the or- 16 Martin Lara dinates axis has been changed from meters in Fig.6 to mm Conflict of Interest: The author declares that he has no conflict of in Fig.7. interest.

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