A Mean Elements Orbit Propagator Program for Highly Elliptical Orbits

CEAS Space Journal manuscript No. (will be inserted by the editor)

HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits

Martin Lara · Juan F. San-Juan · Denis Hautesserres

CEAS Space Journal (ISSN: 1868-2502, ESSN: 1868-2510) (2018) doi:10.1007/s12567-017-0152-x (Pre-print version)

Abstract The algorithms used in the construction of a semi- intermediary solutions to the J2 problem [34,11], the contin- analytical propagator for the long-term propagation of High- uous increase in the accuracy of observations demanded the ly Elliptical Orbits (HEO) are described. The software prop- use of more complex dynamical models to achieve a sim- agates mean elements and include the main gravitational and ilar precision in the orbit predictions. In particular higher non-gravitational effects that may affect common HEO or- degrees in the Legendre polynomials expansion of the third- bits, as, for instance, geostationary transfer orbits or Mol- body disturbing function are commonly required (see [17, niya orbits. Comparisons with numerical integration show 24], for instance). Useful analytical theories needed to deal that it provides good results even in extreme orbital config- with a growing number of effects, a fact that made that the urations, as the case of SymbolX. trigonometric series evaluated by the theory comprised tens of thousands of terms [5]. Keywords HEO · Geopotential · third-body perturbation · In an epoch of computational plenty, the vast possibili- tesseral resonances · SRP · atmospheric drag · mean ties offered by special perturbation methods clearly surpass elements · semi-analytic propagation those of general perturbation methods in their traditional application to orbit propagation. Apparently by this reason analytical perturbations have these days been cornered to a 1 Introduction downgraded role of providing some insight into the problem under investigation, a task for which a first order averag- The subject of analytical or semi-analytical propagation is ing is usually considered to be enough, yet the computations very old. Since the first analytical orbit propagators based on of higher orders may provide important details on the dynamics [33,21]. However, analytical theories like the pop- Funded by CNES contract Ref. DAJ-AR-EO-2015-8181. Preliminary ular SGP4 [16] still enjoy a wide number of users mainly results were presented at the 6th ICATT, Darmstadt, Germany, March involved in catalog propagation duties, a role in which other 14 – 17, 2016 tools like the Draper Semi-Analytic Satellite Theory [29,6] M. Lara or the numeric-analytic theory THEONA [12] can compete GRUCACI – University of La Rioja to numerical integration up to a limited accuracy.1 C/ Madre de Dios 53, Edificio CCT, 26006 Logrono,˜ Spain On the other hand, new needs in satellite propagation, arXiv:1808.05462v1 [astro-ph.EP] 18 Jul 2018 Tel.: +34-941-299440 like the challenges derived of compliance with Space Law, Fax: +34-941-299460 motivate the development of software tools based on analyti- E-mail: [email protected] cal or semi-analytical methods, as, for instance, STELA2. In J.F. San-Juan addition, design of end of life disposal strategies may require GRUCACI – University of La Rioja E-mail: [email protected] the long term propagation of thousands of trajectories to find an optimal solution; accurate ephemeris are not needed in D. Hautesserres CCT – Centre National d’Etudes´ Spatiales 18 Av. Edouard Belin, 1 A partial list of orbit propagators can be found in http://facul- 31401 Toulouse CEDEX 4, France ty.nps.edu/bneta/papers/list.pdf, accessed September 29, 2016. E-mail: [email protected] 2 https://logiciels.cnes.fr/content/stela 2 Martin Lara et al. the preliminary design and using semi-analytic propagation be important because they give rise to long-period effects, makes the approach quite feasible [1]. whereas the effects of tesseral harmonics in general aver- Current needs for long-term propagation at the Centre age out to zero. The latter, however, can have an important National d’Etudes´ Spatiales motivate the present research. impact on resonant orbits, as in the case of geostationary HEOSAT, a semi-analytical orbit propagator to study the satellites (1 to 1 resonance) or GPS and Molniya orbits (2 to long-term evolution of spacecraft in Highly Elliptical Orbits 1 resonance). (HEO) is presented. The perturbation model used includes The importance of each perturbation acting on an earth’s the gravitational effects produced by the more relevant zonal satellite depends on the orbit characteristics, and fundamen- harmonics as well as the main tesseral harmonics affecting tally on the altitude of the satellite. Thus, for instance the to the 2:1 resonance of earth’s gravitational potential, which atmospheric drag, which can have an important impact in has an impact on Molniya-type orbits; the third body per- the lower orbits, may be taken as a higher order effect for turbations in the mass-point approximation, which only in- altitudes above, say, 800 km over the earth’s surface, and is clude the Legendre polynomial of second order for the sun almost negligible above 2000 km. and the polynomials from second order to sixth order in the A sketch of the order of different perturbations when case of the moon; solar radiation pressure, in the cannonball compared to the Keplerian attraction is presented in Fig. 1 approximation, and atmospheric drag. based on approximate formulas borrowed from [32, p. 114]. The forces of gravitational origin are modeled taking ad- As illustrated in the figure, the non-centralities of the Geopo- vantage of the Hamiltonian formalism. Besides, the prob- tential have the most important effect in those parts of the or- lem is formulated in the extended phase space in order to bit that are below the geosynchronous distance, where the J2 avoid time-dependence issues. The solar radiation pressure contribution is a first order perturbation and other harmonics and the atmospheric drag are added as generalized forces. cause second order effects. To the contrary, in those parts of The semi-analytical theory is developed using perturbation the orbit that are farther than the geosynchronous distance techniques based on Lie transforms. Deprit’s perturbation the gravitational pull of the moon is the most important per- algorithm [7] is applied up to the second order of the second turbation, whereas that of the sun is of second order when zonal harmonics, J2. In order to avoid as far as possible the compared to the disturbing effect of the moon, and pertur- lost of long-period effects from the mean elements Hamilto- bations due to J2 and solar radiation pressure (SRP) are of nian, the theory is corrected by the inclusion of long-period third order. terms of the Kozai-type [20,10]. The transformation is developed in closed-form of the eccentricity except for tesseral 0.01 Moon resonances, and the coupling between J2 and the moon’s disturbing effects are neglected. J2 10-4 Sun This paper describes the semi-analytical theory and pres- ents relevant examples of the numerical validation. An ex-

magnitude -6 tensive description of the tests performed in the validation 10 SRP HAM = 0.01 m2kgL of of the HEOSAT software is given in [22].

Order -8 10 J3 J2,2

2 Dynamics of a spacecraft in HEO 2:1 1:1 1:2 1:3 20 40 60 80 100 120 140 Satellites in earth’s orbits are affected by a variety of per- Distance from the Earth's center Hthousands of kmL turbations of different nature. A full account of this can be Fig. 1 Perturbation order relative to the Keplerian attraction (after [1]). found in textbooks on orbital mechanics (see, for instance, An area to mass ratio of 0.01m2/kg was taken for estimating the SRP. −6 [32,35]). All known perturbations must be taken into ac- The horizontal, dashed line corresponds to 3.33 × 10 times the Kep- lerian attraction. count in orbit determination problems. However, for orbit prediction the accuracy requirements are notably relaxed, and hence some of the disturbing effects may be considered Finally, we recall that integration of osculating elements of higher order in the perturbation model. is properly done only in the True of Date system [9]. For this Furthermore, for the purpose of long-term predictions it reason Chapront’s solar and lunar ephemeris are used [4,3], is customary to ignore short-period effects, which occur on which are directly referred to the mean of date in this way time-scales comparable to the orbital period. Thus, in the including the effect of equinoctial precession. case of the gravitational potential the focus is on the ef- In the case of a highly elliptical orbit the distance from fect of even-degree zonal harmonics, which are known to the satellite to the earth’s center of mass varies notably along cause secular effects. Odd-degree zonal harmonics may also the orbit, a fact that makes particularly difficult to establish HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 3 the main perturbation over which to hinge the correct pertur- which is written in terms of the x and X, into a new Hamil- bation arrangement. This issue is aggravated by the impor- tonian tance of the gravitational pull of the moon on high altitude εm Hf= ∑ H0,m, (2) orbits, which requires taking higher degrees of the Legendre m≥0 m! polynomial expansion of the third-body disturbing function in the new, prime variables, which is free from short-period into account. Because this expansion converges slowly, the terms. size of the multivariate Fourier series representing the moon The transformation is derived from a generating function perturbation will soon grow enormously. Besides, for oper- εm W = Wm+ , (3) ational reasons, different HEOs may be synchronized with ∑ m! 1 the earth rotation, and hence, being affected by resonance m≥0 each term of which is computed as a solution of the homo- effects. Also, a HEO satellite will spend most of the time in logical equation the apogee region, where, depending on the physical characteristics of the satellite, the solar radiation pressure may have L0(Wm) + Hf0,m = H0,m, (4) an observable effect in the long-term. Finally, the perigee of as follows: usual geostationary transfer orbits (GTO) will enter repeat- – terms Hf0,m are known. They are obtained from previous edly the atmosphere, yet only for short periods. computations based on Deprit’s recurrence Therefore, a long-term orbit propagator for HEO aiming n n Hn,q+ = Hn+ ,q + {Hn−m,q;Wm+ }, (5) at describing at least qualitatively the orbit evolution must 1 1 ∑ m 1 consider the following perturbations: m≥0 where { ; } means the Poisson bracket operator. – the effects of the main zonal harmonics of the Geopoten- – the new term H0,m is chosen as the part of Hf0,m that is tial (J —J in our case), as well of second order effects 2 10 free from short-period terms of J2; – the term Wm is then solved from Eq. (4), where – the effects of the tesseral harmonics of the Geopotential ∂Wm that affect the most important resonances, and, in partic- L (Wm) = {H , ;Wm} + , (6) 0 0 0 ∂t ular, the 2:1 resonance that impacts Molniya orbits is customarily known as the Lie derivative of Wm. – lunisolar perturbations (mass-point approximation). The first few terms of the Legendre polynomials expansion After solving the homological equation up to the desired of the third-body perturbation are needed in the case of order, the new Hamiltonian in Eq. (2) is obtained by chang- the moon, whereas the effect of the polynomial of the ing the old variables in which the terms H0,m are written by second degree is enough for the sun; the new ones. – solar radiation pressure effects; Once the generating function is known, the transforma- – and the circularizing effects of atmospheric drag affect- tion from old to new variables is itself computed by applying ing the orbit semi-major axis and eccentricity. Deprit’s recurrence in Eq. (5) to the vector functions εm In what respects to the ephemeris of the sun and moon, x = ∑ xm,0(x,X), m≥0 m! needed for evaluating the corresponding disturbing poten- m tials, the precision provided by simplified analytical formu- ε X = ∑ Xm,0(x,X), las is enough for the accuracy requirements of a perturbation m≥0 m! theory. For this reason, truncated series from Chapront’s so- where x0,0 = x, xm,0 = 0 (m > 0), and X0,0 = X, Xm,0 = 0 lar and lunar ephemeris are taken from [31]. The precision (m > 0). of these short truncations is enough for mean elements prop- For further details in the procedure the interested reader agation and notably speed the computations. is referred to Deprit’s seminal reference [7] or to modern textbooks in perturbation theory or celestial mechanics.

3 Deprit’s perturbation approach 4 Gravitational perturbations: Hamiltonian approach The aim is to find a canonical transformation 0 0 The Hamiltonian Eq. (1) is arranged as follows (x,X) → (x ,X ;ε), µ = − , m m H0,0 where x ∈ R are coordinates, X ∈ R their conjugate mo- 2a 2 menta, and ε is a small parameter, that converts the Hamil- µ R⊕ H , = − C , P (sinϕ), tonian 1 0 r r2 2 0 2 m ε H2,0 = 2(Z + T + V(| + V ), H = ∑ Hm,0 (1) m≥0 m! Hm,0 = 0, m ≥ 3, 4 Martin Lara et al. where µ is the earth’s gravitational parameter, (r,ϕ,λ) are body, which will always be the case when dealing with per- the spherical coordinates of the satellite and a is the semi- turbed Keplerian motion, Eq. (9) can be expanded in power major axis of the satellite’s orbit, R⊕ is the earth’s equatorial series of the ratio r/r? radius, Pm are the usual Legendre polynomials, and Cm,0 are n2a3 rm = − ? ? P ( ), zonal harmonic coefficients. V? β ∑ m m cosψ? (10) r? m≥2 r? where m? 4.1 Geopotential second order perturbations β = , (11) m? + m and The second order Hamiltonian term H2,0 comprises the sec- xx? + yy? + zz? ond order perturbations of gravitational origin. Thus, the cosψ? = . (12) rr? higher order terms of the zonal potential are The semi-analytic theory only considers P2 in Eq. (10) m µ R for the sun disturbing potential, whereas P2–P6 are taken for = − ⊕ C P (sinϕ). Z ∑ m m,0 m the moon. Both solar and lunar ephemerides are taken from r m≥3 r the low precision formulas given by [31] (see also [30]). The tesseral disturbing function is The Lie transforms averaging is only developed up to m µ R⊕ the second order in the small parameter, so there is no cou- T = − ∑ m (7) r m≥2 r pling between the different terms of the disturbing function. m Hence, the generating function of the averaged Hamiltonian × ∑ [Cm,n cosnλ + Sm,n sinnλ]Pm,n(sinϕ) can be split into different terms which are simply added at n=1 the end. where Pm,n are associated Legendre polynomials, and Cn,m and Sn,m, m 6= 0, are the non-zonal harmonic coefficients. Besides, for orbital applications, instead of using spherical 4.3 Time dependency coordinates the satellite’s radius vector is expressed in or- Orbits with large semi-major axis will experience high per- bital elements. This can be done by first replacing the spher- turbations due to the moon’s gravitational attraction, and ical coordinates by Cartesian ones may be better described in the restricted three-body prob- z y x sinϕ = , sinλ = , cosλ = . lem model approximation than as a perturbed Keplerian or- r p 2 2 p 2 2 x + y x + y bit. However, since our approach is based on perturbed Ke- Then, the orbital and inertial frame are related by means of plerian motion, we still take the Keplerian term as the zero simple rotations, to give order Hamiltonian.  x   r  Because sun and moon ephemeris are known functions of time, the perturbed problem remains of three degrees of  y  = R3(−Ω)R1(−I)R3(−θ) 0  (8) z 0 freedom, but the Lie derivative in Eq. (6) must take the time dependency into account, viz. where R and R are usual rotation matrices, 1 3 ∂W ∂W ∂W L (W ) ≡ {H , ;W } + = n + , θ = f + ω, 0 0 0 ∂t ∂M ∂t p where n stands for mean motion. r = , 1 + ecos f Dealing explicitly with time can be avoided by moving (a,e,I,Ω,ω,M) are traditional orbital elements and to the extended phase space. Then, assuming that the semi- major axis, eccentricity, and inclination of the third-body or- 2 p = a(1 − e ), bits, remain constant is the semi-latus rectum of the osculating ellipse. ∂W ∂W ∂W dω(| ∂W dΩ(| L0(W ) = n + n(| + + ∂M ∂M(| ∂ω(| dt ∂Ω(| dt ∂W ∂W dω ∂W dΩ 4.2 Third-body perturbations +n + + , ∂M ∂ω dt ∂Ω dt Under the assumption of point masses, the third-body dis- which, in view of the period of the lunar perigee motion of turbing potential is 8.85 years (direct motion) and the period of the lunar node motion of 18.6 years (retrograde motion), and the higher pe- µ? r? r · r? riods in the case of the sun relative orbit, can be safely ap- V? = − − 2 (9) r? ||r − r?|| r? proximated by where V? ≡ V(| for the moon, and V? ≡ V in the case of the ∂W ∂W ∂W L0(W ) ≈ n + n(| + n . (13) sun. If the disturbing body is far away from the perturbed ∂M ∂M(| ∂M HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 5

4.4 Third body direction where we abbreviate s ≡ sinI, and all the symbols, viz. a, r, I, and θ, are assumed to be functions of some set of canoni- For the semi-analytic theory, the time dependency will man- cal variables. In particular, the averaging is carried out based ifest only in the short-period corrections, which are derived on the canonical set of Delaunay variables, which is made from the generating function W . In view of the form of the of the coordinates ` = M, g = ω, h = Ω, and their conjugate Lie derivative in Eq. (13), the directions of both the sun and momenta the moon must be expressed as functions of the sun and √ moon mean anomaly, respectively. In the case of the sun, L = µa, G = Lη, H = GcosI, we use respectively, where     x r p 2  y  = R1(−ε)R3(−λ )R2(β ) 0 , (14) η = 1 − e , z 0 is customarily known as the “eccentricity function”. The De- where ε is the mean obliquity of the ecliptic, r is the ra- launay variables are action-angle variables for the Keplerian dius of the sun, and β and λ are the ecliptic latitude and motion, and fit particularly well in the construction of per- longitude of the sun, respectively. The sun’s latitude can be turbation theories for perturbed Keplerian motion. neglected because it never exceeds 1.2 arc seconds when re- The homological equation of the zonal problem is ob- ferred to the ecliptic of the date, whereas the longitude of tained from Eq. (4) as the sun is expressed in terms of the mean anomaly using standard formulae. ∂Wm −n + Hf0,m = H0,m, (18) For the moon, because the lunar inclination to the equa- ∂` tor is not constant, the orbit is rather referred to the mean In consequence, the terms of the generating function are equator and equinox of the date by means of four rotations computed from quadratures of the form     x(| r(| 1 Z y Wm = (Hf0,m − H0,m)d`. (19)  (|  = R1(−ε)R3(−Ω(|)R1(−J)R3(−θ(|) 0 , n z(| 0 where θ(| is the argument of the latitude of the moon (which is also expressed as a function of the moon’s mean anomaly 5.1 Elimination of the parallax by standard formulas), J ≈ 5.◦15 is the inclination of the moon orbit over the ecliptic, which is affected of periodic It is known that the removal of short-period terms from the oscillations whose period slightly shorter than half a year Hamiltonian is facilitated to a large extent by a preprocess- because the retrograde motion of the moon’s line of nodes, ing of the original Hamiltonian in order to remove paral- and Ω(| is the longitude of the ascending node of the moon lactic terms [8]. That is, by first applying the “parallactic orbit with respect to the ecliptic measured from the mean identity” equinox of date. 1 1 1 1 (1 + ecos f )m−2 Note, however, that in the present stage, the semi-analyt- = = , m > 2, (20) rm r2 rm−2 r2 pm−2 ical theory only deals with mean elements and corresponding homological equations of the type in Eq. (13) do not and next selecting H0,m by removing all the trigonometric need to be solved at the second order. Future evolutions of terms that explicitly contain the true anomaly in the expan- the theory will take the short-period corrections due to third- sion of Hm,0 as a Fourier series [26,25]. body perturbations into account. 5.1.1 First order 5 Averaging zonal terms At the first step of Deprit’s recurrence in Eq. (5), the known The zonal part of the Hamiltonian is first transformed. Be- terms are simply Hf0,1 ≡ H1,0, where cause of the actual values of the zonal coefficients of the n h = C ( − s2)( + e f ) + s2 e ( f + ) earth, the old Hamiltonian is arranged in the form H1,0 2,0 4 6 1 cos 3 cos 2ω µ 2 2 2 ioR⊕ n p H0,0 = − (15) +2cos(2 f + 2ω) + ecos(3 f + 2ω) 2a r2 8η6 2 µ R⊕ H1,0 = − C2,0P2(ssinθ) (16) Then, the new Hamiltonian term H , is selected as, r r2 0 1 µ Rm 2 2 = −2 ⊕ C P (ssin ) (17) µ R⊕ 1 3 2 p H2,0 ∑ m m,0 m θ H0,1 = C2,0 − s (21) r m≥3 r p p2 2 4 r2 6 Martin Lara et al. and the generating function term W1 must be computed from 5.1.2 Second order Eq. (19) with m = 1. That is, Deprit’s recurrence now gives Z 2 2 2 1 R⊕ n p h 2 W1 = C2,0 4 − 6s ecos f (22) n r2 8η6 H0,2 = {H0,1,W1} + H1,1, 3 = { ,W } + { ,W } + . 2 i H1,1 H0,0 2 H1,0 1 H2,0 +s ∑ jE1, j cos( j f + 2ω) d`, j=1 Hence, the computable terms of the homological equation in which (18) are

Hf0,2 = {H0,1,W1} + {H1,0,W1} + H2,0. (25) E1,1 = 3e, E1,2 = 3, E1,3 = e. (23) Equation (22) is solved in closed form by recalling the The computation of the Poisson brackets is straightforward differential relation even for the partial derivatives of the true anomaly, which cannot be explicitly written in terms of the Delaunay vari- dM = (r/p)2η3 d f , (24) ables. For the reader’s convenience, a table of partial derivatives is given in Appendix A, where all of them are ex- based on the preservation of the angular momentum of the pressed as functions of classical orbital elements and related Keplerian motion. We get functions rather than in Delaunay variables; this way of pro- C h ceeding eases computations in the whole process by avoid- W = k + nR2 2,0 (4 − 6s2)esin f 1 ⊕ 8η3 ing the appearance of square roots, on the one hand, and 3 retains the engineering insight provided by the orbital ele- 2 i +s ∑ E1, j sin( j f + 2ω) , ments, on the other. j=1 After carrying out the required operations, parallactic where k is an arbitrary function independent of `. terms are removed from Hf0,2 using Eq. (20). Then, Hf0,2 To avoid the appearance of hidden long-period terms in is written in the form of a Poisson series, from which H0,2 is chosen by removing the trigonometric terms of that W1, we choose k in such a way that Hf0,2 explicitly depend on the true anomaly, to give: 1 Z 2π W1 dM = 0. 2 10 2 4 2π 0 µ p ∗ µ p 2 R⊕ 5 21 2 H0,2 = 2 2 ∑ Ji − 2 C2,0 4 − s (26) p r i=3 p r p 4 8 In this way it is guaranteed that W1 only comprises short- 21 3 5 3 period terms. That is + s4 + c2 − s4 e2 − e2s2 cos2ω 16 8 8 8 1 C Z 2π h 2 2,0 2 2 k = − nR⊕ 3 (4 − 6s )esin f 15 35 2 2 η 2π 8η 0 × − s − 4 − 5s 2 4 (1 + )2 3 η 2 i +s ∑ E1, j sin( j f + 2ω) dM. with c ≡ cosI and j=1 i bi/2c−1 Using, again, Eq. (24) to compute this quadrature, we get ∗ R⊕ l l m Ji = Ci,0 i ∑ e Qi,ls Bi,l ıı˙ exp(ıı˙ lω) (27) p j=0 C 1 − η k = nR2 2,0 (1 + 2η)s2 sin2ω. ⊕ 8η3 1 + η where l = 2 j+m, b c notes an integer division, m ≡ i mod 2, and ıı˙ = (−1)1/2. Note that Eq. (27) applies also to i = 2; Therefore, calling indeed, it is simple to check that − + 1 η 1 2η 2 2 E1,0 = (1 + 2η) = 2 e , µ p ∗ 1 + η (1 + η) H , = J . (28) 0 1 p r2 2 the first term of the generating function of the elimination of The eccentricity polynomials Q in Eq. (27) are given in the parallax simplification is written i, j Table 1 of Appendix B, while the inclination polynomials 2 C2,0 h 2 Bi, j are given in Tables 2–3. W1 = nR⊕ (4 − 6s )esin f 8η3 Finally, the simplified Hamiltonian is obtained by re- 3 placing the old variables by the new ones in all the 2 i H0,m +s ∑ E1, j sin( j f + 2ω) terms. That is, the new Hamiltonian is obtained by assum- j=0 ing that all symbols that appear in Eqs. (21) and (26) are which is now free from hidden long-period terms. functions of the new (prime) Delaunay variables. HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 7

5.2 Delaunay normalization where the sun and moon potentials are computed from the Legendre series expansion in Eq. (10). m After the preparatory simplification, it is trivial to remove Because the maximum power of χ in Pm(χ) is χ , in the remaining short-period terms from the simplified Hamil- view of Eqs. (10) and (12), we check that the satellite’s ra- 0 m 0 0 tonian H = ∑(ε /m!)Hm,0 where H0,0 is the same as dius r appears now in numerators, contrary to the Geopo- 0 0 H0,0, H1,0 is the same as H0,1 in Eq. (21), and, H2,0 is tential case where r always appear in denominators. For that the same as H0,2 in Eq. (26), but all of them expressed in reason, the closed form theory for third-body perturbations the new, prime variables. is approached using the eccentric anomaly u instead of the 0 The new Hamiltonian term H0,1 is chosen by removing true one f . Hence, 0 the short-period terms from H1,0, namely r = a(1 − ecosu), Z 2π 2 0 1 0 µ R⊕ 3 1 3 2 H0,1 = H1,0dM = C2,0 2 η − s (29) 2π 0 p p 2 4 and the Cartesian coordinates of the satellite are expressed in terms of the eccentric anomaly replacing in Eq. (8) the which was trivially solved using Eq. (24). known relations from the ellipse geometry The first term of the new generating function is solved by quadrature from the homological equation, from which r sin f = aη sinu, r cos f = a(cosu − e). (34) Z 2 0 1 0 0 r 3 φ 0 W = −H ` + H η d f = H , (30) In view of H1,0 ≡ 0, we choose H0,1 = 0 and hence 1 n 0,1 1,0 p2 n 0,1 W1 = 0. Then, the second order of the homological equation where (4) is L0(W2)+Hf0,2 = H0,2, where Hf0,2 = H2,0 from De- φ = f − M, prit’s recurrence (5). The new Hamiltonian term H0,2 is chosen by removing the short-period terms from H2,0. Again, is the equation of the center. Since φ is made only of short- this is done in closed-form by computing the average period terms, there is no need of introducing additional integration constants in the solution of W 0. 1 Z 2π r 1 H0,2 = H2,0 du, (35) Analogously to Eq. (25), the computable terms of the 2π 0 a second order homological equation are where we used the differential relation 0 = { 0 ,W 0} + { 0 ,W 0} + 0 , Hf0,2 H0,1 1 H1,0 1 H2,0 dM = (1 − ecosu)du = (r/a)du (36) 0 from which expression the new Hamiltonian term H0,2 is chosen by removing the short-period terms. which is obtained from Kepler equation. After performing the required operations and replacing After computing Eq. (35) we obtain the long-term Ham- prime variables by new, double prime variables in all the iltonian H0,0 = −µ/(2a), H0,1 = 0, and 0 00 0 H0,m terms, we get the averaged Hamiltonian H = H0,0 + a3 n2 am−2 0 0 = 2(na)2β ∗ ? ? Γ , (37) H0,1 + (1/2)H0,2, viz. H0,2 3 2 ∑ m−2 m r? n m≥2 r? ( 10 R4 00 µ µ 3 ∗ µ 3 2 ⊕ 3 2 with the non-dimensional coefficients H = − + η ∑ Ji + η C2,0 4 c (31) 2a p i=2 p p 16 bm/2c m 1 17 η Γ = A P (S? cosα + T ? sinα), (38) ×(1 − 5c2) − + s2 − s4 e2 − (1 − 3c2)2 m ∑ m, j ∑ m, j,l m,l m,l 3 8 2 j=0 l=−m 2 2 5 2 (1 − 5c )η 2 2 where α = (2 j + k)ω + lΩ, and k = mmod2, cf. [27]. − (1 − 7c ) − 2 e s cos2ω . 4 (1 + η) The eccentricity coefficients Am, j ≡ Am, j(e), the inclina- and the orbit evolution is obtained from Hamilton equations tion ones Pm, j,l ≡ Pm, j,l(I), and the third-body direction coefficients T ? ≡ T ? (u?,v?,w?), S? ≡ S? (u?,v?,w?), where d(`00,g00,h00) ∂H 00 m,l m,l m,l m,l = , (32) dt ∂(L00,G00,H00) x? y? z? u? = , v? = , w? = , d(L00,G00,H00) ∂H 00 r? r? r? = − , (33) dt ∂(`00,g00,h00) are given in Tables 4, 5–7, and 8 of Appendix B, respectively. They are valid for both the moon (? ≡ (|) and the ? ≡ 6 Third-body averaging sun ( ) by using the proper third-body direction vector (u?,v?,w?). Now, the perturbation Hamiltonian is arranged as The contribution of lunisolar perturbations to the mean µ elements equations is by adding to Eqs. (32)–(33) the terms H , = − , H , = 0, H , = 2(V +V ) 0 0 2a 1 0 2 0 (| of Eq. (37) derived from corresponding Hamilton equations. 8 Martin Lara et al.

7 Tesseral resonances ×[C2,2 cos(α + 2ω) + S2,2 sin(α + 2ω)]

+F2,2,1G2,1,1(C2,2 cosα + S2,2 sinα) + F2,2,2 The tesseral potential is no longer symmetric with respect o to the earth’s rotation axis. Therefore, longitude dependent ×G2,2,3 [C2,2 cos(α + 2ω) + S2,2 sin(α − 2ω)] , terms will explicitly depend on time when referred to the in which inertial frame. α = 2(Ω − n t) + M, To avoid the explicit appearance of time in the Hamilto- ⊕ nian, we move to a rotating frame with the same frequency is the (slowly evolving) resonant angle, 3 2 as the earth’s rotation rate n⊕. The argument of the node in F2,2,0 = 4 (1 + c) , the rotating frame is 3 2 F2,2,1 = 2 s , (39) h = − n t, 3 2 Ω ⊕ F2,2,2 = 4 (1 − c) and, in order to preserve the symplectic character, we further and, up to O(e16) introduce the Coriolis term −n⊕H into the Hamiltonian. It 1 1 3 5 5 143 7 9097 9 G2,0,−1 = − e + e − e − e − e is then simple to check that H = Θ cosI still remains as the 2 16 384 18432 1474560 − 878959 e11 − 121671181 e13 − 4582504819 e15 conjugate momentum to h. 176947200 29727129600 1331775406080 3 27 3 261 5 14309 7 423907 9 Then, the tesseral Hamiltonian is arranged as a perturba- G2,1,1 = 2 e + 16 e + 128 e + 6144 e + 163840 e tion problem in which + 55489483 e11 + 30116927341 e13 + 2398598468863 e15 µ 19660800 9909043200 739875225600 H0,0 = − − n⊕Θ cosI, G = 1 e3 + 11 e5 + 313 e7 + 3355 e9 2a 2,2,3 48 768 30720 442368 1459489 11 187662659 13 33454202329 15 H1,0 = 0, + 247726080 e + 39636172800 e + 8561413324800 e H2,0 = 2T , Other 2:1-resonant terms can be found in [23]. For the 1:1 tesseral resonance, we ﬁnd where, now, H0,0 is the Keplerian in the rotating frame, and 2 the tesseral potential is given in Eq. (7). Now, the Lie deriva- µ R⊕ n h R1:1 = − F2,2,0G2,0,0 C2,2 cos(2α + 2ω) tive in Eq. (6) reads a a2 i ∂W ∂W +S2,2 sin(2α + 2ω) + F2,2,1G2,1,2 {H0,0;W} = −n + n⊕ , ∂` ∂h o and the solution of the homological equation (4) will intro- ×[C2,2 cos2α + S2,2 sin2α] duce denominators of the type (in − jn⊕), with i and j inte- 2 µ R⊕ n h gers. Therefore, resonances n/n = j/i between the rotation − F2,1,0G2,0,−1 C2,1 sin(α + 2ω) ⊕ a a2 rate of the node in the rotating frame and the mean motion i of the satellite introduce the problem of small divisors. −S2,1 cos(α + 2ω) + F2,1,1G2,1,1

In fact, resonant tesseral terms introduce long-period ef- ×(C2,1 sinα − S2,1 cosα) + F2,1,2G2,2,3 fects in the semi-major axis that may be not negligible even o at the limited precision of a long-term propagation. There- ×[C2,1 sin(α − 2ω) − S2,1 cos(α − 2ω)] , fore, these terms must remain in the long-term Hamiltonian. where, now, Furthermore, these terms must be traced directly in the mean α = Ω − n⊕t + M. anomaly, contrary to true anomaly, to avoid leaving short- −10 In the particular case of the earth, C2,1 = O(10 ) and period terms in the Hamiltonian, which will destroy the per- −9 S , = O(10 ). Due to the smallness of these values, corre- formance of the semi-analytical integration. Hence, trigono- 2 1 sponding terms are commonly neglected from the resonant metric functions of the true anomaly must be expanded as tesseral potential R . Therefore, the only needed inclina- Fourier series in the mean anomaly whose coefﬁcients are 1:1 tion polynomials are F and F , which were already (truncated) power series in the eccentricity. 2,2,0 2,2,1 given in Eq. (39), whereas the required eccentricity func- After the short-period terms have been removed from the tions, up to O(e16), are tesseral Hamiltonian, we return to the inertial frame by drop- 5 2 13 4 35 6 5 8 49 10 ping the Coriolis term and replacing h by the right ascension G2,0,0 = 1 − 2 e + 16 e − 288 e − 576 e − 3600 e 3725 12 7767869 14 5345003 16 of the ascending node (RAAN), in this way the time explic- − 331776 e − 812851200 e − 650280960 e itly appears into resonant terms of the long-period Hamilto- 9 2 7 4 141 6 197 8 62401 10 G2,1,2 = e + e + e + e + e nian. 4 4 64 80 23040 + 262841 e12 + 9010761 e14 + 8142135359 e16 From Kaula expansions [18], we ﬁnd that the main terms 89600 2867200 2438553600 of the Geopotential that are affected by the 2:1 tesseral res- Terms of the Hamilton equations derived from the dis- onance are turbing functions R2:1 or R1:1, will be added to the evolu- 2 tion equations (32)–(33), only for the propagation of those µ R⊕ n R = − F , , G , ,− 2:1 a a2 2 2 0 2 0 1 orbits which experience the corresponding resonance. HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 9

8 Generalized forces ×sinλ + c(cosε + 1)sin(λ − Ω) io +c(cosε − 1)sin(λ + Ω) ηF The evolution equations must be completed by adding to the right hand side of Hamilton equations, Eqs. (32)–(33), the dI 3 e h = na2 F cosω s(cosε + 1)sin(λ − Ω) (42) averaged effects of the generalized forces. Because these ef- dt 4 η fects are derived from Gauss equations, we recall that i −2csinε sinλ + s(cosε − 1)sin(λ + Ω) dL 1 da = na dt 2 dt dΩ 3 2 e 1 h = na F sinω s(cosε + 1)sin(λ − Ω) (43) dG dL e de dt 4 η s = η − na2 dt dt η dt i −2csinε sinλ + s(cosε − 1)sin(λ + Ω) dH dG dI = c − na2ηs ( dt dt dt dω 3 F h = − na2 sinω (cosε + 1)sin(λ − Ω) (44) dt 4 eη 8.1 SRP e2 ×c − 2 − s sinε sinλ + c(cosε − 1) s Under the simplifying assumption that the solar panels re- i 2 h main oriented to the sun, or that the satellite is a sphere ×sin(λ + Ω) + η cosω (cosε + 1) ) (or “cannonball”), the perturbing acceleration caused by the i solar-radiation pressure ×cos(λ − Ω) + (1 − cosε)cos(λ + Ω) α = −F i , srp srp dM 3 e2 + 1 n h = n + na2 F sinω c(cosε + 1) (45) is always in the opposite direction of the unit vector of the dt 4 e sun i . If, besides, it is assumed that [19], ×sin(λ − Ω) + c(cosε − 1)sin(λ + Ω) i h – the parallax of the sun is negligible +2ssinε sinλ + cosω (cosε + 1) – the solar flux is constant along the satellite’s orbit io – there is no re-radiation from the earth’s surface ×cos(λ − Ω) + (1 − cosε)cos(λ + Ω) the magnitude of the SRP acceleration is a2 A 8.2 Atmospheric drag: Averaged effects Fsrp = (1 + β)P 2 , r m where β is the index of reflection (0 < β < 1), A/m is the Predicting the atmospheric behavior for the accurate eval- area-to-mass ratio of the spacecraft, a is the semi-major uation of drag effects seems naive for the long-term scales axis of the sun’s orbit around earth, r is the radius of the of interest in this study. However, the atmospheric drag may sun’s orbit around earth, and the solar radiation pressure dominate over all other perturbations in the case of orbits −6 2 constant at one AU is P ≈ 4.56 × 10 N/m [32, p. 77], with low perigee heights, even to the extent of forcing the The components of i in the radial, tangent, and normal satellite’s deorbit. directions, respectively, are obtained by simple rotations The magnitude of the drag force depends on the local density of the atmosphere ρ and the cross-sectional area A  1  of the spacecraft in the direction of motion. The drag force i = R (θ)R (I)R (Ω)R (−ε)R (−λ ) 0 . 3 1 3 1 3   per unit of mass m is 0 = −(1/2)n V, Then, calling F = −Fsrp/µ Kozai’s analytical expres- αdrag d sions for perturbations due to SRP [19] are easily recovered where V is velocity of the spacecraft relative to the atmo- from the usual Gauss equations. After averaging over the sphere, of modulus V, we abbreviated mean anomaly, which is done in closed form based on the differential relation (36), we get nd = ρBV > 0, (46) da = 0 (40) and B = (A/m)C , is the so-called ballistic coefficient, in dt drag which the dimensionless drag coefficient Cdrag ranges from de 3 2n h = na sinω (cosε − 1)cos(λ + Ω) (41) 1.5–3.0 for a typical satellite. Note that nd ≡ nd(t). dt 4 i h A reasonable approximation of the relative velocity is −(cosε + 1)cos(λ − Ω) + cosω 2ssinε obtained with the assumption that the atmosphere co-rotates 10 Martin Lara et al. with the earth. Then, from the derivative of a vector in a dM 1 Z 2π e r = n + nd sin f dM (52) rotating frame, dt 2π 0 η a dr 1 Z 2π η h e i V = − ω⊕ × r. + nd sin f 1 − δc 1 + cos f dM dt 2π 0 e 2 2 3 We further take ω⊕ = n⊕k, in the direction of the earth’s ro- in which δ = (n⊕/n)(r/p) η . tation axis, and compute its projections in the radial, normal, Both the relative velocity with respect to the rotating at- and bi-normal directions as mosphere V, and the atmospheric density ρ are naturally ex-  0  pressed as a function of the true anomaly [28], then it hap- pens that n ≡ n ( f ) from the definition of n in Eq. (46). ω⊕ = R3(θ)R1(I) 0 . d d d n⊕ Hence, the quadratures above are conveniently integrated in f rather than in M using the differential relation in Eq. (24). Then, the velocity components in the radial, normal, and Besides, due to the complex representation of the atmos- bi-normal direction relative to a rotating atmosphere are pheric density, these quadratures are evaluated numerically.  R  V =  (Θ/r) − rn⊕ cosI , rn⊕ cosθ sinI 9 Sample tests where To illustrate the performance of the mean elements theory dr Θ R = = esin f , we describe two test cases: one for a Molniya-type orbit, dt p and the other for a SymbolX-type orbit, in which the propa- dθ √ gations are extended to 100 years. Θ = r2 = µ p. dt A full account of the different tests that have been car- Models giving the atmospheric density are usually com- ried out in the development of the HEOSAT software can be plex. Furthermore, since the atmospheric density depends on consulted in [22]. The test cases include GTO, super GTO, the solar flux which is not easily predictable, reliable pre- and SSTO orbits, as well as Tundra orbits, and orbits of the dictions of the disturbing effects caused by the atmospheric telescope satellites’ missions Integral and XMM-Newton. drag are not expected for long-term propagation. Hence, the The semi-analytical theory generally runs one or two orders aim is rather to show the effect that the atmospheric drag of magnitude faster than the Cowell integration, although might have in the orbit, as opposite from a drag-free model. these ratios notably reduce when the atmospheric drag has a Therefore, to speed evaluation of the semi-analytical prop- non-negligible effect. In spite of that, in all the tested cases agator, we take advantage of the simplicity of the Harris- HEOSAT runs more than 5 times faster than the numerical Priester atmospheric density model [13], which is imple- integration. mented with the modifications of [28]. After replacing αdrag into Gauss planetary equations, the long-term effects are computed by averaging the equations 9.1 Molniya orbit over the mean anomaly, viz. The first test presented is for a Molniya type orbit. The ini- da a 1 Z 2π tial conditions used in the test correspond to the osculating = − 2 nd (47) dt η 2π 0 elements n × 1 + 2ecos f + e2 − ⊕ η3c dM n a = 26554.0km de 1 Z 2π e = 0.72 = − nd (48) I = 63.4deg dt 2π 0 (53) h e i Ω = 0.1deg × e + cos f − δc e + cos f − sin2 f dM 2 ω = 280deg Z 2π M = 0 dI 1 1 2 = − s nd δ cos θ dM (49) dt 2 2π 0 The numerical reference has been computed with a Cowell dΩ 1 1 Z 2π method of order 8, and the integration step size was 30 sec- = − nd δ sinθ cosθ dM (50) dt 2 2π 0 onds. In this example, the satellite reaches about the 10% of dω dΩ the earth-moon distance at apogee, thus suffering moderate = −c (51) dt dt third-body perturbations. 2π The time history of the orbital elements corresponding 1 Z nd h e i − sin f 1 − δc 1 + cos f dM to these initial conditions is presented in Fig. 2. As shown in 2π 0 e 2 HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 11

Fig. 2, the numerical reference and the mean elements prop- L km agation ﬁt quite well, with a slight shift to the right of the H 26 650 mean elements orbit. This shift is due to the difference be- axis 26 600

tween the osculating and mean elements used in both kind of major propagations. Indeed, as far as the conversion from osculat- -

Semi 26 550 ing to mean elements is not implemented in the current version HEOSAT, the initial mean elements used in launching 0 20 40 60 80 100 Time HyearsL the semi-analytical propagation do not correspond exactly to the initial conditions used in the propagation of the os- 0.72 culating reference orbit, even though the semi-major axis of 0.70 the mean elements propagation has been manually adjusted 0.68 to the mean value of 26653.5 km, to which the osculating Eccentricity semi-major axis approximately averages. 0.66 Better agreement in the comparisons between the mean 0 20 40 60 80 100 elements propagation and the numeric reference is expected Time HyearsL when the HEOSAT software be completed with the ana- 64.0 lytic transformation from osculating to mean elements. Sec- L 63.5 deg ond order short-period corrections of the initial semi-major H axis value related to this transformation are known to have 63.0 non-negligible effects in the computation of the mean semi- Inclination major axis, and are of similar importance to the ﬁrst order 62.5 periodic corrections of the osculating to mean conversion of 0 20 40 60 80 100 the other elements [15,2]. Time HyearsL The differences between the mean elements provided 350 by HEOSAT software and the osculating elements provided 300

L 250

by the numerical reference are better appreciated in Fig. 3, deg H 200 where it is shown that periodic errors in the semi-major axes 150 are of the order of 100 km. On the other hand, the more rel- RAAN 100 evant discrepancies between the HEOSAT propagation and 50 0 the numerical reference happen to the RAAN, in which case 0 20 40 60 80 100 long-period errors of growing amplitude superimpose to a Time HyearsL L ∼ . 280

liner trend of 0 3 deg/year. deg H

275 perigee 270 9.2 SimbolX orbit the of 265

The second test presented is for a SymbolX type orbit, with 260 Argument initial conditions corresponding to the osculating elements 0 20 40 60 80 100 a = 106247.136454km Time HyearsL e = 0.75173 Fig. 2 Time history of the orbit elements of the Molniya orbit. Dots: I = 5.2789deg mean elements propagation; gray line: numerical reference (54) Ω = 49.351deg ω = −179.992deg about 1000 km. These irregular variations are caused by the M = 0 moon third-body perturbation, that make the SymbolX orbit and the integration step size of the numerical reference is to experience different resonances, which include a 1:7 reso- now 60 seconds. In this case the orbit apogee can reach half nance of the Laplace type, due to the orbital period of 4 days, the earth-moon distance, and, therefore, the SymbolX or- as well as secular resonances of the Kozai type, cf. [14]. bit undergoes important third-body perturbations due to the Because of the irregularities in the time history of the moon’s gravitational pull. osculating semi-major axis, we did not perform any adjust- The time history of both the HEOSAT propagation and ment in the computation of the HEOSAT mean semi-major the numerically integrated reference are depicted in Fig. 4. axis, and the osculating initial elements are directly used As shown in the ﬁgure, the osculating semi-major axis ex- as mean initial elements for launching the semi-analytical periences important variations whose amplitude can reach propagation. In spite of that, the time history of the mean or- 12 Martin Lara et al.

0 L 106 600 L km H km H -50 106 400 a axis D -100 106 200 major

0 20 40 60 80 100 - 106 000

Time HyearsL Semi 0.015 105 800 0.010 0 20 40 60 80 100 0.005 Time HyearsL e 0.000 D - 0.005 0.9 -0.010 -0.015 0.8 0 20 40 60 80 100 0.7 Time HyearsL 0.6 0.3 0.5 0.2 Eccentricity L 0.1 0.4 deg H 0.0 0.3 I

D -0.1 -0.2 0 20 40 60 80 100 Time HyearsL 0 20 40 60 80 100 100 Time HyearsL 35 80

30 L

L 25 deg 20 H 60 deg H 15 W

D 10 40 5 0 Inclination 20 0 20 40 60 80 100 Time HyearsL 0 0 20 40 60 80 100 2 Time HyearsL L 1

deg 350 H 0

Ω 300 D -1

-2 L 250 deg 0 20 40 60 80 100 H 200 Time HyearsL 150

RAAN 100 Fig. 3 Errors between the numerical reference and the mean elements 50 propagation: Molniya case. 0 0 20 40 60 80 100 Time HyearsL bital elements shows very good agreement with the osculat- L 350 deg ing elements provided by the numerical reference. The detail H 300 250 on Fig. 5 shows that this agreement extends for more than perigee 200 70 years, and the discrepancies become important passed 95 the 150 of years. Again, notable improvements are expected when the 100 mean elements theory is completed with the analytic trans- 50 Argument 0 formation from osculating to mean elements. 0 20 40 60 80 100 Time HyearsL

Fig. 4 Time history of the orbit elements of the SymbolX. Dots: ana- 10 Conclusions lytical propagation; gray line: numerical reference

HEO propagation is a challenging problem due to the different perturbations that have an effect in highly elliptical or- these days in the construction of perturbation theories. This bits, the relative influence of which may notably vary along method is specifically designed for automatic computation the orbit. However, modern tools and methods allow to ap- by machine, and is easily implemented with modern, com- proach the problem by means of analytical methods. Indeed, mercial, general purpose software. using perturbation theory we succeeded in the implementa- Future evolutions of the semi-analytical theory should tion of a fast and efficient semi-analytical propagator which incorporate the transformation from osculating to mean ele- is able to capture the main frequencies of the HEO motion ments, in this way enhancing the precision of the mean el- over long time spans, even in extreme cases, as corroborated ements predictions based on it. Also, in spite of common with the tests performed on the SymbolX orbit. In particu- HEO orbits are not affected by singularities, a reformula- lar, we used the Lie transforms method, which is standard tion in non-singular variables will make the orbit propagator HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 13

400 p – the mean motion n = µ/a3 L 200 – the parameter (or semilatus rectum) p = a(1 − e2) km

H 0 √ 2 a – the eccentricity function η = 1 − e D -200 -400 – the cosine of the inclination c =√ cosI 2 0 20 40 60 80 – the sine of the inclination s = 1 − c Time HyearsL – the eccentric anomaly u, given by M = u − esinu 1 1 0.015 – the true anomaly f , given by (1 − e)tan 2 f = η tan 2 u 0.010 – the radial distance r = a(1 − ecosu) = p/(1 + ecos f ) 0.005

e – the radial velocity R = an(a/r)esinu = an(e/η)sin f

D 0.000 -0.005 – the argument of the latitude θ = f + ω -0.010 -0.015 – the projections of the eccentricity vector in the orbital frame: k = 0 20 40 60 80 ecos f = −1 + p/r, and q = esin f = Rη/(na) Time HyearsL are provided both in the Delaunay chart (`,g,h,L,G,H) and in the 1.5 1.0 polar-nodal chart (r,θ,ν,R,Θ,N). L 0.5 We found convenient to express all the partial derivatives by means deg

H 0.0

I -0.5 of the Keplerian functions: D -1.0 -1.5 p,n,e,η,k,q,s,c 0 20 40 60 80 Time HyearsL from whose deﬁnition it is obtained 6 2 4 4 L = n p /η L 2 2 3

deg G = Lη = n p /η = Θ H 0

W -2 2 3 D H = Gc = n p c/η = N -4 -6 R = n pq/η3 0 20 40 60 80 Time HyearsL r = p/(1 + k) 6 4 Recall also that, from the ellipse geometry, the following relations ap- L 2 ply, cf. Eq. (34), deg

H 0

Ω -2 D sin f = (a/r)η sinu, cos f = (a/r)(cosu − e). -4 -6 Finally, it worths to mention that the use of logarithmic deriva- 0 20 40 60 80 tives is helpful in ﬁnding the differentials that eased the computation Time HyearsL of the partial derivatives. Note that only non-vanishing derivatives are Fig. 5 Errors between the numerical reference and the mean elements presented. propagation: SymbolX case.

A.1 With respect to Delaunay variables software more versatile, widening its scope to the propagation of the majority of objects in a catalogue of earth satellite A.1.1 Orbital elements and related functions and debris orbits. – Semi-major axis a: Acknowledgements Partial support is acknowledged from the Min- da L η2 = 2 = 2 (55) istry of Economic Affairs and Competitiveness of Spain, via Projects dL µ n p ESP2013-41634-P (M.L.) and ESP2014-57071-R (M.L. and J.F.S.). – Mean motion n: dn η4 = −3 , (56) A Some useful partial derivatives in elliptic motion dL p2 – Parameter p: When dealing with automatic manipulation of literal expressions, it dp G η3 results practical to limit the symbolic algebra to the basic arithmetic = 2 = 2 . (57) dG µ n p operations, to wit, addition, subtraction, multiplication and division —integer powers being a particular case of multiplication. However, – Eccentricity function η: square roots and trigonometric functions appear naturally in the for- dη 1 η5 = η − = − (58) mulation of the perturbation in canonical variables. To avoid dealing dL L n p2 explicitly with square roots, it is wise to be equipped with a battery of dη 1 η4 partial derivatives that ease handling and simplifying symbolic expres- = η = (59) sions. dG G n p2 Since our perturbation approach relies on the use of Delaunay and – Eccentricity e: polar-nodal canonical variables, the partial derivatives of the classical de η2 1 η6 Keplerian variables (a,e,I,Ω,ω,M), standing for semi-major axis, ec- = = (60) centricity, inclination, right ascension of the ascending node, argument dL e L en p2 of the periapsis and mean anomaly, respectively, as well as the usual de η2 1 η5 = − = − (61) functions of the Keplerian variables dG e G en p2 14 Martin Lara et al.

– Cosine of inclination c: – Polar component of the angular momentum N: dc c η3 dN = − = −c (62) = 1 (84) dG G n p2 dH dc c 1 η3 – Eccentricity vector k: = = = 2 (63) dH H G n p dk pR q = − = − (1 + k)2 (85) – Sine of inclination s: d` r2 n η3 2 3 ds c η dk η4 k 2 = 2 (64) = − (86) dG s n p dL nr2 e2 k + 1 3 ds c η 3 = − dk η k 2 2 (65) = − − (87) dH s n p dG nr2 e2 k + 1 – Eccentric anomaly u: – Eccentricity vector q: du p 1 + k = = (66) dq 1 p2 p k p2 k d` r η2 η2 = − 1 = = (1 + k)2 (88) d` η3 r2 r η3 r2 η3 du Rη8 qη5 = = (67) dq qη4 1 r2 qη4 (1 + k)2 dL e2 n2 p3 e2 n p2 = − = − 1 (89) dL nr2 e2 p2 n p2 e2 du Rη7 qη4 3 2 3 2 = − 2 2 3 = − 2 2 (68) dq qη 1 r qη (1 + k) dG e n p e n p = − − = − − 1 (90) dG nr2 e2 p2 n p2 e2 A.1.2 Polar-nodal variables and related functions

– Radial distance r: A.2 With respect to polar-nodal variables dr R pq = = (69) d` n η3 A.2.1 Delaunay variables dr η4 2e2r 1 1 η4 2 k = + − = − (70) dL e2 n p2 p r n p 1 + k e2 – Delaunay action L: 3 3 dr η 1 1 η k dL Θ k 2 k 2 = − = (71) = − (1 + k) = −np (1 + k) (91) dG e2 n r p e2 n p dr p η3 η6 – Radial velocity R: dL Θ q q = = p 3 (92) 2 dR n p η dR pn p p pn 2 = − 1 = k(1 + k) (72) dL p2 (1 + k)2 d` η6 r2 r η6 = = (93) dΘ η3 r2 η3 dR R 1 r2 q (1 + k)2 = η4 − = η − 1 (73) dL nr2 e2 p2 p e2 – Modulus of the angular momentum Θ: dR η3 R (1 + k)2 q dG = − = − (74) = 1 (94) dG e2 nr2 e2 p dΘ – True anomaly f : – Polar component of the angular momentum N: d f η r dR p2 (1 + k)2 dH = = = (75) = 1 (95) d` an(p − r) d` r2 η3 η3 dN – Mean anomaly `: d f Rη7 1 1 qη4 = + = (2 + k) (76) dL e2 n2 p2 p r e2 n p2 d` q 1 + k 1 = η 2 − (96) d f Rη6 1 1 qη3 dr r e 1 + k = − + = − (2 + k) (77) dG e2 n2 p2 p r e2 n p2 d` q k 2 = η 2 − (97) – Argument of the latitude θ: dR R e 1 + k d` q 2 + k dθ d f = −η (98) = (78) 2 d` d` dΘ Θ e dθ – Argument of the perigee g: = 1 (79) dg dg q 1 + k = − (99) dθ d f dr r e2 = (80) dL dL dg = 1 (100) dθ d f dθ = (81) dG dG dg q k = − (101) – Modulus of the angular momentum Θ: dR R e2 dΘ dg q 2 + k = 1 (82) = 2 (102) dG dΘ Θ e – Argument of the node ν: – Right ascension of the ascending node h: dν dh = 1 (83) = 1 (103) dh dν HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 15

A.2.2 Orbital elements and related functions – Equation of the center φ: dφ q 1 + k η = + (127) – Parameter p: dr r 1 + η 1 + k 3 dp p η dφ q k 2η = 2 = 2 (104) = + (128) dΘ Θ n p dR R 1 + η 1 + k – Eccentricity vector k: dφ q 2 + k 2 = − (129) dk p (1 + k) dΘ Θ 1 + η = − 2 = − (105) dr r p – Cosine of inclination c: dk 2p 2q 2(1 + k)η3 dc c cq cη3 = = = (106) = − = − = − (130) dΘ rΘ r R n p2 dΘ Θ R p n p2 – Eccentricity vector q: dc 1 q η3 3 = = = (131) dq q p η 2 = = = (107) dN Θ R p n p dR R Θ n p – Sine of inclination s: dq q q2 qη3 ds c2 η3 = = = (108) = (132) dΘ Θ R p n p2 dΘ sn p2 – Eccentricity e: ds cη3 = − (133) de k p k (1 + k)2 dN sn p2 = − = − (109) dr e r2 e p de q2 qη3 = = (110) dR eR en p B Tables of coefficients de 2k(1 + k) + q2 3 The coefficients of the trigonometric series used by HEOSAT are pro- = η 2 (111) dΘ en p vided in following tables – Eccentricity function η: dη k p k (1 + k)2 = = (112) dr η r2 η p References dη q2 qη2 = − = − (113) dR η R n p 1. Armellin, R., San-Juan, J.F., Lara, M.: End-of-life disposal of high elliptical orbit missions: The case of INTEGRAL. Advances in dη 2k(1 + k) + q2 2 56 = = −η 2 (114) Space Research (3), 479–493 (2015). DOI 10.1016/j.asr.2015. dΘ n p 03.020. Advances in Asteroid and Space Debris Science and Tech- – Semi-major axis a: nology - Part 1 da k = −2 (1 + k)2 (115) 2. Breakwell, J.V., Vagners, J.: On Error Bounds and Initialization in dr η4 Satellite Orbit Theories. Celestial Mechanics 2, 253–264 (1970). da pq2 q DOI 10.1007/BF01229499 = 2 = 2 (116) 3. Chapront, J., Francou, G.: The lunar theory ELP revisited. Intro- dR Rη4 nη duction of new planetary perturbations. Astronomy and Astro- da p3 (1 + k)2 = 2 = 2 (117) physics 404, 735–742 (2003). DOI 10.1051/0004-6361:20030529 dΘ η4 Θ r2 n pη 4. Chapront-Touze, M., Chapront, J.: ELP 2000-85 - A semi- – Mean motion n: analytical lunar ephemeris adequate for historical times. Astron- dn 3nk (1 + k)2 omy and Astrophysics 190, 342–352 (1988) = (118) dr η2 p 5. Coffey, S.L., Neal, H.L., Segerman, A.M., Travisano, J.J.: An analytic orbit propagation program for satellite catalog mainte- dn 3nq2 qη = − = −3 (119) nance. In: K.T. Alfriend, I.M. Ross, A.K. Misra, C.F. Peters (eds.) dR Rη2 p AAS/AIAA Astrodynamics Conference 1995, Advances in the As- dn 3n p2 3η tronautical Sciences, vol. 90, pp. 1869–1892. American Astronau- = − = − (120) dΘ η2 Θ r2 r2 tical Society, Univelt, Inc., USA (1996) – True anomaly f : 6. Danielson, D.A., Neta, B., Early, L.W.: Semianalytic Satel- lite Theory (SST): Mathematical algorithms. Technical Report d f q 1 + k NPS-MA-94-001, Naval Postgraduate School, Naval Postgradu- = 2 (121) dr r e ate School, Monterey, CA. Dept. of Mathematics. (1994) d f q k = (122) 7. Deprit, A.: Canonical transformations depending on a small pa- dR R e2 rameter. Celestial Mechanics 1(1), 12–30 (1969). DOI 10.1007/ d f q 2 + k = − (123) BF01230629 dΘ Θ e2 8. Deprit, A.: The elimination of the parallax in satellite theory. – Eccentric anomaly u: Celestial Mechanics 24(2), 111–153 (1981). DOI 10.1007/ du η e2 − k k BF01229192 = 1 + (1 + k) (1 + k) (124) dr pq e2 η2 9. Efroimsky, M.: Gauge Freedom in Orbital Mechanics. Annals of the New York Academy of Sciences 1065, 346–374 (2005). DOI du an r − a de da = − − (125) 10.1196/annals.1370.016 dR R r edR adR 10. Exertier, A.: Orbitographie des satellites artificiels sur de grandes du an r − a de da periodes de temps. Possibilites d’applications. PhD. Thesis. Ob- = − − (126) dΘ R r edΘ adΘ servatoire de Paris, Paris (1988) 16 Martin Lara et al.

2 2 4 Table 1 Eccentricity polynomials Qm,k+m mod 2 in Eq. (27); Qm,m−2 = 1, Qm,m−4 = 2m − 6 + 3e and Q6,0 = 8 + 40e + 15e . k m = 7 m = 8 m = 9 m = 10 0 3(8 + 20e2 + 5e4) 3(16 + 168e2 + 210e4 + 35e6) 3(64 + 336e2 + 280e4 + 35e6) 3(128 + 2304e2 + 6048e4 + 3360e6 + 315e8) 2 5(16 + 20e2 + 3e4) 48 + 80e2 + 15e4 15(32 + 112e2 + 70e4 + 7e6) 4 15(8 + 8e2 + e4)

Table 2 Even inclination polynomials B2m,2k in Eq. (27). k m = 1 m = 2 m = 3 m = 4 1 2 3 4 2 5 6 4 2 35 8 6 4 2 0 4 3c − 1 − 128 35c − 30c + 3 2048 231c − 315c + 105c − 5 − 786432 6435c − 12012c + 6930c − 1260c + 35 15 2 175 4 2 2205 6 4 2 1 − 64 7c − 1 2048 33c − 18c + 1 − 131072 143c − 143c + 33c − 1 315 2 4851 4 2 2 4096 11c − 1 − 131072 65c − 26c + 1 3003 2 3 − 131072 15c − 1 m = 5 21 10 8 6 4 2 0 8388608 46189c − 109395c + 90090c − 30030c + 3465c − 63 693 8 6 4 2 1 2097152 4199c − 6188c + 2730c − 364c + 7 9009 6 4 2 2 1048576 323c − 255c + 45c − 1 19305 4 2 3 4194304 323c − 102c + 3 109395 2 4 16777216 19c − 1

Table 3 Odd inclination polynomials B2m+1,2k+1 in Eq. (27). k m = 1 m = 2 m = 3 m = 4 3 2 15 4 2 35 6 4 2 105 8 6 4 2 0 − 8 5c − 1 128 21c − 14c + 1 − 8192 429c − 495c + 135c − 5 262144 2431c − 4004c + 2002c − 308c + 7 35 2 315 4 2 1617 6 4 2 1 256 9c − 1 − 16384 143c − 66c + 3 131072 221c − 195c + 39c − 1 693 2 3003 4 2 2 − 16384 13c − 1 131072 85c − 30c + 1 6435 2 3 524288 17c − 1

11. Garfinkel, B.: On the motion of a satellite of an oblate planet. 20. Kozai, Y.: Second-Order Solution of Artificial Satellite Theory The Astronomical Journal 63(1257), 88–96 (1958). DOI 10.1086/ without Air Drag. The Astronomical Journal 67(7), 446–461 107697 (1962) 12. Golikov, A.R.: THEONA—a numerical-analytical theory of mo- 21. Lara, M.: Simplified Equations for Computing Science Orbits tion of artificial satellites of celestial bodies. Cosmic Research Around Planetary Satellites. Journal of Guidance Control Dynam- 50(6), 449–458 (2012). DOI 10.1134/S0010952512060020 ics 31(1), 172–181 (2008). DOI 10.2514/1.31107 13. Harris, I., Priester, W.: Time-Dependent Structure of the Upper At- 22. Lara, M., San-Juan, J., Hautesserres, D.: Semi-analytical propaga- mosphere. Journal of Atmospheric Sciences 19, 286–301 (1962). tor of high eccentricity orbits. Technical Report R-S15/BS-0005- DOI 10.1175/1520-0469(1962)019h0286:TDSOTUi2.0.CO;2 024, Centre National d’Etudes´ Spatiales, 18, avenue Edouard Be- 14. Hautesserres, D.: Extrapolation long terme de l’orbite du satellite lin - 31401 Toulouse Cedex 9, France (2016) SimbolX par la methode de Gragg-Bulirsch-Stoer (GBS). Tech- 23. Lara, M., San-Juan, J.F., Folcik, Z.J., Cefola, P.: Deep Resonant nical Report DCT/SB/OR/2009-2474, Centre National d’Etudes´ GPS-Dynamics Due to the Geopotential. The Journal of the Spatiales, 18, avenue Edouard Belin - 31401 Toulouse Cedex 9, Astronautical Sciences 58(4), 661–676 (2011). DOI 10.1007/ France (2009) BF03321536 15. Hautesserres, D., Lara, M.: Intermediary LEO propagation in- 24. Lara, M., San-Juan, J.F., Lopez,´ L.M., Cefola, P.J.: On the third- cluding higher order zonal harmonics. Celestial Mechan- body perturbations of high-altitude orbits. Celestial Mechanics ics and Dynamical Astronomy 0, in press (2016). DOI 10. and Dynamical Astronomy 113, 435–452 (2012). DOI 10.1007/ 1007/s10569-016-9736-6. URL http://arxiv.org/pdf/1605. s10569-012-9433-z 00525.pdf 25. Lara, M., San-Juan, J.F., Lopez-Ochoa,´ L.M.: Delaunay variables 16. Hoots, F.R., Roehrich, R.L.: Models for Propagation of the NO- approach to the elimination of the perigee in Artificial Satellite RAD Element Sets. Project SPACETRACK, Rept. 3, U.S. Air Theory. Celestial Mechanics and Dynamical Astronomy 120(1), Force Aerospace Defense Command, Colorado Springs, CO 39–56 (2014). DOI 10.1007/s10569-014-9559-2 (1980) 26. Lara, M., San-Juan, J.F., Lopez-Ochoa,´ L.M.: Proper Averaging 17. Kaufman, B.: First order semianalytic satellite theory with recov- Via Parallax Elimination (AAS 13-722). In: Astrodynamics 2013, ery of the short period terms due to third body and zonal per- Advances in the Astronautical Sciences, vol. 150, pp. 315–331. turbations. Acta Astronautica 8(5–6), 611 – 623 (1981). DOI American Astronautical Society, Univelt, Inc., USA (2014) 10.1016/0094-5765(81)90108-9 27. Lara, M., Vilhena de Moraes, R., Sanchez, D.M., Prado, A.F.B.A.: 18. Kaula, W.M.: Theory of satellite geodesy. Applications of satel- Efficient computation of short-period analytical corrections due lites to geodesy. Blaisdell, Waltham, Massachusetts (1966) to third-body effects (AAS 15-295). In: Proceedings of the 25th 19. Kozai, Y.: Effects of Solar Radiation Pressure on the Motion of an AAS/AIAA Space Flight Mechanics Meeting, Williamsburg, VA, Artificial Satellite. SAO Special Report 56, 25–34 (1961) January 11 – 15, 2015, Advances in the Astronautical Sciences, HEOSAT: A mean elements orbit propagator program for Highly Elliptical Orbits 17

Table 4 Eccentricity polynomials Am, j in Eq. (38). m j = 0 1 2 3 2 3(2 + 3e2) −15e2 3 e(4 + 3e2) e3 4 (8 + 40e2 + 15e4) e2(2 + e2) e4 5 e(8 + 20e2 + 5e4) e3(8 + 3e2) e5 6 16 + 168e2 + 210e4 + 35e6 48e2 + 80e4 + 15e6 10e4 + 3e6 e6

Table 5 Inclination polynomials Pm, j,l in Eq. (38) (χ = c ± 1).

P2, j,l P3, j,l P4, j,l l j = 0 j = 1 j = 0 j = 1 j = 0 j = 1 j = 2 1 2 1 2 15 2 175 3 3 4 2 105 2 2 2205 4 0 48 (3c − 1) − 16 s − 128 (5c − 1)s − 128 s − 4096 (35c − 30c + 3) − 1024 (7c − 1)s − 4096 s 1 1 15 2 525 2 15 2 105 2 2205 3 ±1 8 cs 8 χs − 512 χ(15c ∓ 10c − 1) − 512 χs 1024 c(3 − 7c )s 512 χ(14c ∓ 7c − 1)s 1024 χs 1 2 1 2 75 525 2 15 2 2 105 2 2 2205 2 2 ±2 − 32 s 32 χ 256 χ(3c ∓ 1)s − 256 χ s 1024 (7c − 1)s 256 χ (7c ∓ 7c + 1) 1024 χ s 75 2 175 3 105 3 735 2 2205 3 ±3 512 χs 512 χ 1024 cs 512 χ (2c ∓ 1)s − 1024 χ s 105 4 735 2 2 2205 4 ±4 − 2048 s − 512 χ s − 2048 χ

Table 6 Inclination polynomials P5, j,l in Eq. (38) (χ = c ± 1). l j = 0 j = 1 j = 2 105 4 2 735 2 3 14553 5 0 8192 21c − 14c + 1 s 16384 (9c − 1)s 16384 s 105 4 3 2 2205 2 2 72765 4 ±1 16384 χ(105c ∓ 84c − 42c ± 28c + 1) 32768 χ(15c ∓ 6c − 1)s 32768 χs 735 3 2 2205 2 2 72765 2 3 ±2 − 4096 χ(15c ∓ 9c − 3c ± 1)s 8192 χ (15c ∓ 12c + 1)s 8192 χ s 735 2 2 735 3 72765 3 2 ±3 32768 χ(15c ∓ 6c − 1)s 65536 χ (3c ∓ 1)(15c ∓ 13) 65536 χ s 2205 3 6615 3 72765 4 ±4 − 16384 χ(5c ∓ 1)s − 32768 χ (5c ∓ 3)s 32768 χ s 2205 4 6615 3 2 14553 5 ±5 32768 χs 65536 χ s 65536 χ

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Table 7 Inclination polynomials P6, j,l in Eq. (38) (χ = c ± 1). l j = 0 j = 1 j = 2 j = 3 5 6 4 2 315 4 2 2 2079 2 4 99099 6 0 − 65536 (231c − 315c + 105c − 5) − 131072 (33c − 18c + 1)s − 65536 (11c − 1)s − 131072 s 105 4 2 315 4 3 2 2079 2 3 297297 5 ±1 − 32768 c(33c − 30c + 5)s 65536 χ(99c ∓ 66c − 36c ± 18c + 1)s 32768 χ(33c ∓ 11c − 2)s 65536 χs 525 4 2 2 315 2 4 3 2 10395 2 2 2 1486485 2 4 ±2 − 262144 (33c − 18c + 1)s − 524288 χ (495c ∓ 660c + 90c ± 108c − 17) − 262144 χ (33c ∓ 22c + 1)s − 524288 χ s 525 2 3 945 2 3 2 10395 3 2 495495 3 3 ±3 − 65536 c(11c − 3)s − 131072 χ (55c ∓ 55c + 5c ± 3)s 65536 χ (11c ∓ 11c + 2)s 131072 χ s 315 2 4 945 2 2 2 2079 4 2 297297 4 2 ±4 32768 (1 − 11c )s − 65536 χ (33c ∓ 22c + 1)s − 32768 χ (33c ∓ 44c + 13) − 65536 χ s 3465 5 10395 2 3 22869 4 297297 5 ±5 − 65536 cs 131072 (1 ∓ 3c)χ s − 65536 χ (3c ∓ 2)s 131072 χ s 1155 6 10395 2 4 22869 4 2 99099 6 ±6 262144 s 524288 χ s 262144 χ s 524288 χ

Table 8 Third-body direction polynomials in Eq. (38); u ≡ u?, v ≡ v?, w ≡ w?. m l Sm,l Tm,l 2 0 −1 + 3w2 0 ±1 −vw ±uw ±2 u2 − v2 ±2uv 3 0 0 w(5w2 − 3) ±1 ±u(5w2 − 1) v(5w2 − 1) ±2 ±2uvw w(v2 − u2) ±3 ±u(u2 − 3v2) −v(v2 − 3u2) 4 0 3 − 30w2 + 35w4 0 ±1 vw(3 − 7w2) ±uw(−3 + 7w2) 1 2 2 2 2 ±2 2 (u − v )(−1 + 7w ) ±uv(−1 + 7w ) ±3 v(−3u2 + v2)w ±u(u2 − 3v2)w 1 4 2 2 4 2 2 ±4 4 (u − 6u v + v ) ±uv(u − v ) 5 0 0 w(15 − 70w2 + 63w4) ±1 ±u(1 − 14w2 + 21w4) v(1 − 14w2 + 21w4) ±2 ±2uvw(−1 + 3w2)(u2 − v2)w(1 − 3w2) ±3 ±u(u2 − 3v2)(1 − 9w2) v(−3u2 + v2)(−1 + 9w2) ±4 ±4uv(−u2 + v2)w (u4 − 6u2v2 + v4)w ±5 ±u(u4 − 10u2v2 + 5v4) v(5u4 − 10u2v2 + v4) 6 0 5 − 105w2 + 315w4 − 231w6 0 ±1 v(5 − 30w2 + 33w4)w ∓u(5 − 30w2 + 33w4)w ±2 (u2 − v2)(1 − 18w2 + 33w4) ±2uv(1 − 18w2 + 33w4) ±3 v(3u2 − v2)(3 − 11w2)w ±u(u2 − 3v2)(−3 + 11w2)w 1 2 4 2 2 4 2 2 2 ±4 4 (1 − 11w )(u − 6u v + v ) ±(1 − 11w )uv(u − v ) ±5 v(5u4 − 10u2v2 + v4)w ∓u(u4 − 10u2v2 + 5v4)w ±6 (v2 − u2)(u4 − 14u2v2 + v4) ∓2(3u5 − 10u3v2 + 3uv4)v