A Semi-Analytical Orbit Propagator Program for Highly Elliptical Orbits∗

A SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS∗

Martin Lara† & Juan F. San-Juan‡ Denis Hautesserres

University of La Rioja Centre National d’Etudes´ Spatiales GRUCACI – Scientiﬁc Computation Group Centres de Competence Technique 26004 Logrono,˜ Spain 31401 Toulouse CEDEX 4, France

ABSTRACT and the coupling between J2 and moon’s disturbing effects are neglected. The paper outlines the semi-analytical theory. The algorithms used in the construction of a semi-analytical propagator for the long-term propagation of Highly Ellipti- cal Orbits (HEO) are described. The software propagates 2. DYNAMICS OF A SPACECRAFT IN HEO mean elements and include the main gravitational and non- gravitational effects that may affect common HEO orbits, as, Satellites in earth’s orbits are affected by a variety of perturba- for instance, geostationary transfer orbits or Molniya orbits. tions of a diverse nature. A full account of this can be found in textbooks on orbital mechanics like [1]. All known per- Index Terms— HEO, Geopotential, third-body perturba- turbations must be taken into account in orbit determination tion, tesseral resonances, SRP, atmospheric drag, mean ele- problems. But for orbit prediction the accuracy requirements ments, semi-analytic propagation are notably relaxed, and hence some of the disturbing effects may be considered of higher order in the perturbation model. 1. INTRODUCTION Furthermore, for the purpose of long-term predictions it is customary to ignore short-period effects, which occur on A semi-analytical orbit propagator to study the long-term evo- time-scales comparable to the orbital period. Thus, in the case lution of spacecraft in Highly Elliptical Orbits is presented. of the gravitational potential the focus is on the effect of even- The perturbation model taken into account includes the grav- degree zonal harmonics, which are known to cause secular itational effects produced by the first nine zonal harmonics effects. Odd-degree zonal harmonics may also be important and the main tesseral harmonics affecting to the 2:1 reso- because they originate long-period effects, whereas the effects nance, which has an impact on Molniya orbit-types, of Earth’s of tesseral harmonics in general average out to zero. The lat- gravitational potential, the mass-point approximation for third ter, however, can have an important effect in resonant orbits, body perturbations, which only include the Legendre polyno- as in the case of geostationary satellites (1 to 1 resonance) or mial of second order for the sun and the polynomials from GPS and Molniya orbits (2 to 1 resonance). second order to sixth order for the moon, solar radiation pres- The importance of each perturbation acting on an earth’s sure and atmospheric drag. Hamiltonian formalism is used to satellite depends on the orbit’s characteristics, and fundamen- model the forces of gravitational nature so as to avoid time- tally on the altitude of the satellite, but also on its mean mo- dependence issues the problem is formulated in the extended tion. Thus, for instance the atmospheric drag, which can have phase space. The solar radiation pressure and the atmospheric an important impact in the lower orbits, may be taken as a drag are added as generalized forces. The semi-analytical the- higher order effect for altitudes above, say, 800 km over the ory is developed using perturbation techniques based on Lie earth’s surface, and is almost negligible above 2000 km. transforms. Deprit’s perturbation algorithm is applied up to A sketch of the order of different perturbations when com- the second order of the second zonal harmonics, J2, includ- pared to the Keplerian attraction is presented in Fig.1 based ing Kozay-type terms in the mean elements Hamiltonian to on approximate formulas borrowed from [1, p. 114]. As il- get “centered” elements. The transformation is developed in lustrated in the figure, the non-centralities of the Geopotential closed-form of the eccentricity except for tesseral resonances have the most important effect in those parts of the orbit that are below the geosynchronous distance, where the J2 contri- ∗CNES CONTRACT REF. DAJ-AR-EO-2015-8181 bution is a first order effect and other harmonics cause second † Funded by the Ministry of Economic Affairs and Competitiveness of order effects. To the contrary, in those parts of the orbit that Spain: Projects ESP2013-41634-P and ESP2014-57071-R. ‡Funded by the Ministry of Economic Affairs and Competitiveness of are farther than the geosynchronous distance the gravitational Spain: Project ESP2014-57071-R. pull of the moon is the most important perturbation, whereas that of the sun is of second order when compared to the dis- tion, including at least the effects of the first few terms turbing effect of the moon, and perturbations due to J2 and of the Legendre polynomials expansion of the third- solar radiation pressure (SRP) are of third order. body perturbation for the moon, and at least the effect of the polynomial of the second degree for the sun; 0.01 Moon • solar radiation pressure effects; J2 10-4 Sun • and the circularizing effects of atmospheric drag affecting the orbit semi-major axis and eccentricity.

magnitude -6 In what respects to the ephemeris of the sun and moon 10 SRP HAM = 0.01 m2kgL of needed for evaluating the corresponding disturbing potentials, the precision provided by simpliﬁed analytical formulas Order -8 J J 10 3 2,2 should be enough under the accuracy of a perturbation theory. For this reason truncated series from [5] of Chapront’s solar 2:1 1:1 1:2 1:3 and lunar ephemeris are enough at the precision of the mean 20 40 60 80 100 120 140 elements propagation and notably speed the computations. Distance from the Earth's center Hthousands of kmL

3. HAMILTONIAN ARRANGEMENT Fig. 1. Perturbation order relative to the Keplerian attraction. The averaged Hamiltonian is constructed by the Lie trans- Finally, we recall that integration of osculating elements forms technique [6], including the time dependence in the so- is properly done only in the True of Date system [2]. For this lution of the homological equation. reason Chapront’s solar and lunar ephemeris are used [3,4], Thus, the Hamiltonian is arranged as a power series in a which are directly referred to the mean of date thus including small parameter , viz. H = P (m/m!)H , with the effect of equinoctial precession. m≥0 m,0

In the case of a highly elliptic orbit (HEO) the distance H0,0 = −µ/(2a) of the satellite to the earth’s center of mass varies notably 2 H1,0 = −(µ/r)(R⊕/r) C2,0P2(sin ϕ) along the orbit, a fact that makes particularly difficult to es- 1 X tablish the main perturbation over which to set up the correct H = −(µ/r) (R /r)mC P (sin ϕ) + T + V + V 2 2,0 ⊕ m,0 m (| perturbation arrangement. This issue is aggravated by the im- m>2 portance of the gravitational pull of the moon on high altitude where µ is the earth’s gravitational parameter, (r, ϕ, λ) are orbits, which requires taking higher degrees of the Legendre spherical coordinates and a is the semi-major axis of the satel- polynomial expansion of the third-body disturbing function lite’s orbit, R is the earth’s equatorial radius, P are Leg- into account. Since this expansion converges slowly, the size ⊕ m endre polynomials, and C are zonal harmonic coefficients. of the multivariate Fourier series representing the moon per- m,0 The tesseral disturbing function is turbation will soon grow enormously. Besides, for operational reasons, different HEOs may be synchronized with the earth m m µ X R⊕ X rotation, and hence, being notably affected by resonance ef- T =− Pm,n(sin ϕ)[Cm,n cos nλ+Sm,n sin nλ] r rm fects. Also, a HEO satellite will spend most of the time in the m≥2 n=1 apogee region, where the solar radiation pressure may have in which P are associated Legendre polynomials while an observable effect in the long-term, yet depending on the m,n C and S , m 6= 0, are non-zonal harmonic coefficients. physical characteristics of the satellite. Finally, the perigee n,m n,m Under the assumption of point masses, of usual geostationary transfer orbits (GTO) will enter repeat- 2 edly the atmosphere, although only for short periods. V? = −(µ?/r?) r?/||r − r?|| − r · r?/r? (1) Therefore, a long-term orbit propagator for HEO aiming at describing at least qualitatively the orbit evolution must where V? ≡ V(| for the moon, and V? ≡ V in the case of the consider the following perturbations: sun. The disturbing body is far away from the perturbed body when dealing with perturbed Keplerian motion. Then Eq. (1) • the effects of the main zonal harmonics of the Geopo- can be expanded in power series of the ratio r/r? tential as well as second order effects of J2; 2 3 X m V? = −β (n a /r?) (r/r?) Pm(cos ψ?), (2) • tesseral effects for the most common resonances. In ? ? m≥2 particular, the 2:1 affecting Molnya orbits, or super syn- chronous resonances affecting space telescope orbits; where β = m?/(m? + m) and

• lunisolar perturbations in the mass-point approxima- cos ψ? = (xx? + yy? + zz?)/(rr?) (3) The semi-analytic theory only considers P2 in Eq. (2) for In the case of the moon, because the lunar inclination to the sun potential, whereas P2–P6 are taken for the moon. the equator is not constant, the orbit is rather referred to the The Lie transforms averaging is only developed up to the mean equator and equinox of the date by means of second order in the small parameter, so there is no coupling r = R (−ε) R (−Ω ) R (−J) R (−θ )(r , 0, 0)τ between the different terms of the disturbing function. Hence, (| 1 3 (| 1 3 (| (| the generating function of the averaged Hamiltonian can be where θ(| is the argument of the latitude of the moon, J ≈ split into different terms which are simply added at the end. 5◦.15 is the inclination of the moon orbit over the ecliptic, which is affected of periodic oscillations whose period 3.1. Time dependency slightly shorter than half a year because the retrograde motion of the moon’s line of nodes, and Ω(| is the longitude of the Orbits with large semi-major axis will experience high pertur- ascending node of the moon orbit with respect to the ecliptic bations from the moon’s gravity, and may be better described measured from the mean equinox of date. Solar and lunar under the three-body problem model. However, since our ap- ephemerides are taken from the low precision formulas in [5]. proach is based on perturbed Keplerian motion, we still take Besides, for orbital applications, instead of using spheri- as the zero order Hamiltonian the Keplerian term. cal coordinates the satellite’s radius vector is expressed in or- Because the sun and moon ephemeris are known func- bital elements; This can be done by ﬁrst replacing the spheri- tions of time, the perturbed problem remains of three degrees cal coordinates by Cartesian ones: sin ϕ = z/r, sin λ = y/q, of freedom, but the Lie derivative must take the time depen- cos λ = x/ρ, where q = px2 + y2. Then, the orbital and dency into account, viz. L (W) ≡ {H ; W} + ∂W/∂t = 0 0,0 inertial frame are related by means of simple rotations R (α): n ∂W/∂` + ∂W/∂t. i τ τ Dealing explicitly with time can be avoided by moving (x, y, z) = R3(−Ω) R1(−I) R3(−θ)(r, 0, 0) (6) to the extended phase space. Then, assuming that the semi- major axis, eccentricity, and inclination of the third-body or- where θ = f + ω, r = p/(1 + e cos f), and (a, e, I, Ω, ω, M) bits, remain constant are traditional orbital elements.

∂W ∂W ∂W ˙ ∂W ∂W 4. AVERAGING ZONAL TERMS L0 = n + n(| +g ˙(| + h(| + n + ... ∂` ∂`(| ∂g(| ∂h(| ∂` The zonal part of the Hamiltonian is first transformed. Be- which, in view of the period of the lunar perigee of 8.85 years cause of the actual values of the zonal coefficients of the earth, (direct motion) and of the lunar node of 18.6 years (retrograde the old Hamiltonian is arranged in the form motion), can be safely approximated by H0,0 = −µ/(2a) (7) L0 ≈ n ∂W/∂` + n(|∂W/∂`(| + n ∂W/∂` (4) 2 H1,0 = −(µ/r)(R⊕/r) C2,0P2(sin I sin θ) (8) X 2 Note, however, that in the present stage, the theory only deals H2,0 = −2(µ/r) (R⊕/r) Cm,0Pm(sin I sin θ) (9) with mean elements and corresponding homological equa- m>2 tions does not need to be solved. Future versions of the theory where all the symbols, viz. a, r, I, and θ, are assumed to may take the short-period corrections due to third-body per- be functions of some set of canonical variables. In particular, turbations into account. the averaging is carried out based on the canonical set of De- launay variables, which is made of the coordinates ` = M, √ 3.2. Third body direction g = ω, h = Ω, and their conjugate momenta L = µa, G = Lη, H = G cos I, respectively. These variables are the For the semi-analytic theory, the time dependency will man- action-angle variables of the Keplerian motion. ifest only in the short-period corrections, which are derived The homological equation is −n(∂W /∂`) + H = from the generating function W. In view of the form of the m e0,m H , and hence terms of the generating function are com- Lie derivative in Eq. (4), the directions of both the sun and 0,m puted from quadratures. the moon must be expressed as functions of the sun and moon mean anomaly, respectively. For the sun, we use 4.1. Elimination of the parallax r = R (−ε) R (−λ ) R (β )(r , 0, 0)τ (5) 1 3 2 It is known that the removal of short-period terms from the where τ means transposition, ε is the mean obliquity of the Hamiltonian is facilitated to a large extent by a preprocess- ing of the original Hamiltonian in order to remove parallactic ecliptic, and r is the radius of the sun, and β and λ are the ecliptic latitude and longitude of the sun, respectively. The terms [7]. That is, by first applying the parallactic identity sun’s latitude can be neglected in view of it never exceeds 1.2 1 1 1 1 (1 + e cos f)m−2 = = , m > 2, (10) arc seconds when referred to the ecliptic of the date. rm r2 rm−2 r2 pm−2 n5 21 21 3 5 and then selecting H0,m by removing all the trigonometric × − s2 + s4 + c2 − s4 e2 terms of the expansion of Hm,0 as a Fourier series that ex- 4 8 16 8 8 2 plicitly contain the true anomaly [8,9]. 3h15 35 2 2 η i 2 2 o − − s − 4 − 5s 2 e s cos 2ω The 1st step of Deprit’s recurrence is He0,1 ≡ H1,0, where 8 2 4 (1 + η)

2 2 2 6 2 H1,0 = C2,0(R⊕/r) n p /(8η ) (4 − 6s )(1 + e cos f) with 2 +3 [e cos(f + 2ω) + 2 cos(2f + 2ω) + e cos(3f + 2ω)] s bi/2c−1 Ri X J ∗ = C ⊕ elQ slB ˙ıım exp(˙ıı lω) (14) i i,0 pi i,l i,l Then, the new Hamiltonian term H0,1 is selected as j=0 2 2 2 6 2 H0,1 = C2,0(R⊕/r) n p (1/η )(1/4) 2 − 3s (11) where l = 2j + m, b c notes an integer division, m ≡ i mod 2, and ˙ıı = (−1)1/2. The eccentricity polynomials Q and W i,j and the generating function term 1 must be solved from the inclination polynomials B are given in Tables1,2, and3, −n(∂W /∂`) + H = H i,j homological equation 1 e0,1 0,1. That is, respectively. Note that Eq. (14) applies also to i = 2; indeed 2 ∗ Z H0,1 = (µ/p)(p/r) J2 . n 2 2 2 6 W1 = (1/n) C2,0(R⊕/r) n p /(8η ) (12) Finally, the simplified Hamiltonian is obtained by replacing the old variables by the new ones in all the H0,m terms. h 2 2 X io × 4 − 6s e cos f + s jE1,j cos(jf + 2ω) d` That is, the new Hamiltonian is obtained by assuming that all j=1,3 symbols that appear in the H0,m terms are functions of the new (prime) Delaunay variables. in which E1,1 = 3e, E1,2 = 3, E1,3 = e. Equation (12) is solved in closed form by recalling the differential relation 4.2. Delaunay normalization dM = (r/p)2η3 df, (13) After the preparatory simplification, it is trivial to remove the based on the preservation of the angular momentum of the remaining short-period terms from the simplified Hamilto- 0 P m 0 0 Keplerian motion. We get nian H = ( /m!)Hm,0, where H0,0 is the same as H0,0, 0 0 H1,0 is the same as H0,1 in Eq. (11), and, H2,0 is the same as 3 H0,2, but all of them expressed in the new, prime variables. 2 C2,0 h 2 2X i W1 = k+nR (4−6s )e sin f+s E1,jsin(jf+2ω) H0 ⊕ 8η3 The new Hamiltonian term 0,1 is chosen by removing j=1 0 the short-period terms from H1,0, namely where k is an arbitrary function independent of `. 1 Z 2π µ R2 1 3 To avoid the appearance of hidden long-period terms in 0 0 ⊕ 3 2 H0,1 = H1,0dM = C2,0 2 η − s 1 R 2π 2π 0 p p 2 4 W1, k is chosen to guarantee that 2π 0 W1 dM = 0. Using, again, Eq. (13) to compute the quadrature, we get which was trivially solved using Eq. (13). The first term of the new generating function is solved by k = (1/8)nR2 C η−3(1 + 2η)(1 − η)(1 + η)−1s2 sin 2ω. ⊕ 2,0 quadrature from the homological equation from which

Therefore, calling E1,0 = (1 + 2η)(1 − η)/(1 + η) 1 h Z i φ W 0 = − H0 ` + H0 (r/p)2η3 df = H0 , 1 n 0,1 1,0 n 0,1 C2,0 h X i W = nR2 (4−6s2)e sin f +s2 E sin(jf +2ω) 1 ⊕ 8η3 1,j j=0,3 where φ = f − M is the equation of the center. Since φ is made only of short-period terms, there is no need of introduc- At order 2, Deprit’s recurrence gives: H0,2 = {H0,1,W1}+ ing additional integration constants. H1,1, and H1,1 = {H0,0,W2} + {H1,0,W1} + H2,0. There- At the second order, the computable terms of the homo- 0 0 0 0 0 fore, the homological equation is −n(∂W2/∂`) + He0,2 = logical equation are He0,2 = {H0,1,W1} + {H1,0,W1} + 0 0 H0,2, where He0,2 = {H0,1,W1} + {H1,0,W1} + H2,0. H2,0, from which expression the new Hamiltonian term H0,2 After performing the required operations, parallactic is chosen by removing the short-period terms. terms are removed from He0,2 using Eq. (10). Then, He0,2 After performed the required operations and replac- is written in the form of a Poisson series, and H0,2 is chosen ing prime variables by new, double prime variables in all the H0 terms, we get the averaged Hamiltonian H00 = by removing the trigonometric terms of He0,2 that explicitly 0,m 0 0 depend on the true anomaly, to give: H0,1 + (1/2)H0,2, viz. X µ X µ H = 2(µ/p)(p/r)2 J ∗ − (µ/p)(p/r)2C2 (R /p)4 H00 = η3 J ∗ + η3C2 (R /p)4(3/16) (15) 0,2 i 2,0 ⊕ p i p 2,0 ⊕ i=3,10 i=2,10 n 1 17 × (1 − 5c2)c2 − + s2 − s4 e2 − (1 − 3c2)2 avoid the explicit appearance of time in the Hamiltonian, we 3 8 move to a rotating frame with the same frequency as the ro- 1 5 (1 − 5c2)η2 o tation of the earth. The argument of the node in the rotating × η − (1 − 7c2) − e2s2 cos 2ω , 2 4 (1 + η)2 frame is h = Ω − n⊕ t, where n⊕ is the earth’s rotation rate, and, to preserve the Hamiltonian character, we further introduce the Coriolis term −n H. It is then simple to check that 5. THIRD-BODY AVERAGING ⊕ H = Θ cos I still remains as the conjugate momentum to h. Thus, the tesseral Hamiltonian is arranged as Now, the perturbation Hamiltonian is arranged

H0,0 = −µ/(2a) − n⊕Θ cos I, H1,0 = 0, H2,0 = 2T , H0,0 = −µ/(2a), H1,0 = 0, H2,0 = 2V(| + 2V where, now, H0,0 is the Keplerian in the rotating frame. where the sun and moon potentials are computed from Eq. (2). Hence, the Lie derivative reads {H ; W } = −n∂W /∂` + m 0,0 Since the maximum power of χ in Pm(χ) is χ , in view n⊕∂W /∂h, and the solution of the homological equation of Eqs. (2) and (3), we check that the satellite’s radius r ap- will introduce denominators of the type (in − jn⊕), with pears now in numerators, contrary to the geopotential case. i and j integer. Therefore, resonances between the rotation Therefore, the closed form theory is approached now using rate of the node in the rotating frame and mean motion of the the eccentric anomaly u instead of the true one f. Hence, satellite i/j = n⊕/n introduce the problem of small divisors. r = a(1−e cos u), and the Cartesian coordinates of the satel- In fact, resonant tesseral terms introduce long-period lite are obtained from Eq. (6) using the known relations on the terms in the semi-major axis that may be not negligible even ellipse: r sin f = aη sin u, r cos f = a(cos u − e). at the limited precision of a long-term propagation. There- Because H1,0 ≡ 0 we choose H0,1 = 0 and trivially ﬁnd fore, these terms must remain in the long-term Hamiltonian. W1 = 0. Then, the second order of the homological equation Furthermore, these terms must be traced directly in the mean, is L0(W2) + He0,2 = H0,2, where, from Deprit’s recurrence, contrary to true, anomaly to avoid leaving short-period terms He0,2 = H2,0. The new Hamiltonian term H0,2 is chosen in the Hamiltonian, which will destroy the performance of the by removing the short-period terms in H2,0. Again, this is semi-analytical integration. Therefore, trigonometric func- 1 R 2π done by closed-form averaging H0,2 = 2π 0 H2,0(r/a) du tions of the true or the eccentric anomaly must be expanded where we used the differential relation as Fourier series in the mean anomaly whose coefﬁcients are power series in the eccentricity. dM = (1 − e cos u) du = (r/a) du (16) After the short-period terms have been removed from the tesseral Hamiltonian, we come back to the inertial frame by which is obtained from Kepler equation. dropping the Coriolis term and replacing h by the RAAN, in The averaging produces the long-term Hamiltonian [10] this way explicitly showing the time into resonant terms of

2 ∗ 3 2 X m−2 the long-term Hamiltonian. H0,2 = 2(na) β (a?/r?) (n?/n) (a/r?) Γm From Kaula expansions [11], we ﬁnd that the main terms m≥2 of the Geopotential that are affected by the 2:1 tesseral reso- 2 with the non-dimensional coefﬁcients nance are R2:1 = −(µ/a)(R⊕/a) (3/4)R2:1, with

bm/2c m R2:1 = F2,2,0G2,0,−1 [C2,2 cos(α + 2ω) + S2,2 sin(α + 2ω)] X X ? ? Γm = Am,j Pm,j,l Sm,l cos α + Tm,l sin α +F2,2,1G2,1,1 (C2,2 cos α + S2,2 sin α) j=0 l=−m +F G [C cos(α + 2ω) + S sin(α − 2ω)] (17) 2,2,2 2,2,3 2,2 2,2 where α = (2j + k)ω + lΩ, and k = m mod 2. where α = 2(Ω − n⊕t) + M is the resonant angle, The eccentricity coefﬁcients Am,j(e), the inclination 2 2 2 ones Pm,j,l(i), and the third-body direction coefﬁcients F2,2,0 = (1 + c) ,F2,2,1 = 2s ,F2,2,2 = (1 − c) (18) T ? (u, v, w) S? (u, v, w) m,l , m,l , are given in Tables4,5–9, and 16 10, respectively. They are valid for both the moon (? ≡ (|) and, up to O(e ), and the sun (? ≡ ) by using, when required, the proper 1 1 5 143 9097e9 ? ? ? 3 5 7 third-body direction vector (u, v, w) ≡ (u , v , w ). G2,0,−1 = − e + e − e − e − 2 16 384 18432 1474560 878959e11 121671181e13 4582504819e15 − − − 6. TESSERAL RESONANCES 176947200 29727129600 1331775406080 3 27 261 14309 423907 G = e + e3 + e5 + e7 + e9 The tesseral potential is no longer symmetric with respect 2,1,1 2 16 128 6144 163840 to the earth’s rotation axis. Therefore, longitude dependent 55489483 30116927341 2398598468863 + e11 + e13 + e15 terms will explicitly depend on time in the inertial frame. To 19660800 9909043200 739875225600 1 3 11 5 313 7 3355 9 Then, calling F = −F /µ, Kozai’s analytical expres- G2,2,3 = e + e + e + e srp 48 768 30720 442368 sions for perturbations due to SRP [13] are easily recov- 1459489e11 187662659e13 33454202329e15 + + + ered from the usual Gauss equations. After averaging over 247726080 39636172800 8561413324800 the mean anomaly, which is done in closed form based on Other 2:1-resonant terms can be found in [12]. For the 1:1 Eq. (16), we get da/dt = 0, and 2 resonance, R1:1 = −(µ/a)(R⊕/a) (3/4)R1:1, with de 2 = (3/4)na F η sin ω (cos ε − 1) cos(λ + Ω) R = [C cos(2α + 2ω) + S sin(2α + 2ω)] F dt 1:1 2,2 2,2 2,2,0 −(cos ε + 1) cos(λ − Ω) + cos ω 2s sin ε ×G2,0,0 + [C2,2 cos 2α + S2,2 sin 2α] F2,2,1G2,1,2 +F G [C sin(α + 2ω) − S cos(α + 2ω)] × sin λ + c(cos ε + 1) sin(λ − Ω) 2,1,0 2,0,−1 2,1 2,1 +c(cos ε − 1) sin(λ + Ω) +F2,1,1G2,1,1 (C2,1 sin α − S2,1 cos α) dI +F2,1,2G2,2,3 [−S2,1 cos(α − 2ω) + C2,1 sin(α − 2ω)] = (3/4)na2(e/η)F cos ωs(cos ε + 1) sin(λ − Ω) dt where now α = Ω − n⊕t + M. −2c sin ε sin λ + s(cos ε − 1) sin(λ + Ω) −10 −9 For the earth, C2,1 = O(10 ) and S2,1 = O(10 ); dΩ 3 2 e 1 hence corresponding terms are commonly neglected from the = na F sin ω s(cos ε + 1) sin(λ − Ω) resonant tesseral potential R1×1. Therefore, the needed in- dt 4 η s clination polynomials are only F2,2,0 and F2,2,1, which were −2c sin ε sin λ + s(cos ε − 1) sin(λ + Ω) already given in Eq. (18), whereas the required eccentricity 16 dω 3 2 F functions, up to O(e ), are = − na sin ω (cos ε + 1) sin(λ − Ω) dt 4 eη 2 5 2 13 4 35 6 5 8 49 10 ×c + 2 s − e /s sin ε sin λ + c(cos ε − 1) G2,0,0 = 1 − e + e − e − e − e 2 16 288 576 3600 × sin(λ + Ω) + η2 cos ω(cos ε + 1) 3725 7767869 5345003 − e12 − e14 − e16 331776 812851200 650280960 × cos(λ − Ω) + (1 − cos ε) cos(λ + Ω) 2 9 2 7 4 141 6 197 8 62401 10 dM 3 e + 1 G2,1,2 = e + e + e + e + e = n + na2 F sin ωc(cos ε + 1) 4 4 64 80 23040 dt 4 e 262841 9010761 8142135359 + e12 + e14 + e16 × sin(λ − Ω) + c(cos ε − 1) sin(λ + Ω) 89600 2867200 2438553600 +2s sin ε sin λ + cos ω (cos ε + 1) 7. GENERALIZED FORCES × cos(λ − Ω) + (1 − cos ε) cos(λ + Ω)

The evolution equations are completed adding the averaged 7.2. Atmospheric drag: Averaged effects effects of the generalized forces to the Hamilton equations. Predicting the atmospheric behavior for the accurate evaluation of drag effects seems naive for the long-term scales of in- 7.1. SRP terest in this study. However, the atmospheric drag may domi- In the cannonball approximation, the perturbing acceleration nate over all other perturbations in the case of orbits with low caused by solar-radiation pressure, is always in the opposite perigee heights, even to the extent of forcing the satellite’s direction of the unit vector of the sun αsrp = −Fsrpi . If, de-orbit. besides, it is assumed that the parallax of the sun is negli- The magnitude of the drag force depends on the local den- gible, the solar flux is constant along the orbit of the satel- sity of the atmosphere ρ and the cross-sectional area A of the lite, and there is no re-radiation from the earth’s surface [13], spacecraft in the direction of motion. The drag force per unit 2 1 then Fsrp = (1 + β)P (a /r ) (A/m) where β is the in- of mass m is αdrag = − 2 ndV , where V is velocity of the dex of reflection (0 < β < 1), A/m is the area-to-mass ra- spacecraft relative to the atmosphere, of modulus V , we ab- tio of the spacecraft, a is the semi-major axis of the sun’s breviated orbit around earth, r is the radius of the sun, and P ≈ nd = ρBV > 0, (19) 4.56 × 10−6 N/m2 is the SRP constant at one AU [1, p. 77]. and B = (A/m)C , is the so-called ballistic coefficient, The components of i in the radial, tangent, and normal drag with the dimensionless drag coefficient C ranging from directions, respectively, are obtained by simple rotations drag 1.5–3.0 for a typical satellite. Note that nd ≡ nd(t). τ A reasonable approximation of the relative velocity is i = R3(θ) R1(I) R3(Ω) R1(−ε) R3(−λ ) (1, 0, 0) obtained with the assumption that the atmosphere co-rotates where λ is the ecliptic longitude of the sun, and ε is the with the earth. Then, from the derivative of a vector in a obliquity of the ecliptic. rotating frame, we get V = (dr/dt) − ω⊕ × r. We further take ω⊕ = n⊕k, and compute its projections in the radial, relative influence may notably vary along the orbit. However, normal, and bi-normal directions as modern tools and methods allow to approach the problem by

τ means of analytical methods. Indeed, using perturbation the- ω⊕ = R3(θ) R1(I) (0, 0, n⊕) ory we succeeded in the implementation of a fast and efficient Then, the velocity components in the radial, normal, and semi-analytical propagator which is able to capture the main bi-normal direction relative to a rotating atmosphere are frequencies of the HEO orbital motion over large spans. In particular, we used the Lie transforms method, which is stan- τ τ V = (R, Θ/r, 0) + rn⊕ (0, − cos I, cos θ sin I) dard these days in the construction of perturbation theories. √ This method is specifically designed for automatic computa- R =r ˙ = (Θ/p)e sin f Θ = r2θ˙ = µp where , and . tion by machine, and can be easily programmed with modern, Models giving the atmospheric density are usually com- commercial, general purpose software. plex. Furthermore, since it depends on the solar flux which is not easily predictable, reliable predictions of the drag effect 9. REFERENCES are not expected for long-term propagation. Hence, the aim is to show the effect that the atmospheric drag might have in the [1] O. Montenbruck and E. Gill, Satellite Orbits. Models, Methods orbit, as opposite from a drag-free mode. Therefore, to speed and Applications, Physics and Astronomy. Springer-Verlag, evaluation of the semi-analytical propagator, we take advan- Berlin, Heidelberg, New York, 2001. tage of the simplicity of the Harris-Priester density model, [2] M. Efroimsky, “Gauge Freedom in Orbital Mechanics,” Annals which is implemented with the modifications of [14]. of the New York Acad. of Sci., vol. 1065, pp. 346–374, 2005. Replacing αdrag into Gauss planetary equations, the long- [3] M. Chapront-Touze and J. Chapront, “ELP 2000-85 - A semi- term effects are computed after averaging the equations over analytical lunar ephemeris adequate for historical times,” As- the mean anomaly, viz. tronomy & Astrophysics, vol. 190, pp. 342–352, 1988. Z 2π [4] J. Chapront and G. Francou, “The lunar theory ELP revisited. da a 1 2 n⊕ 3 = − 2 nd 1 + 2e cos f + e − η c dM Introduction of new planetary perturbations,” Astronomy & dt η 2π 0 n Astrophysics, vol. 404, pp. 735–742, 2003. Z 2π de 1 h e 2 i [5] J. Meeus, Astronomical algorithms, Willmann-Bell, Rich- = nd δc e + cos f − sin f − e − cos f dM dt 2π 0 2 mond, VA, 2nd edition, 1998. Z 2π dI 1 1 2 [6] A. Deprit, “Canonical transformations depending on a small = − s nd δ cos θ dM parameter,” Celestial Mechanics, vol. 1, pp. 12–30, 1969. dt 2 2π 0 dΩ 1 1 Z 2π [7] A. Deprit, “The elimination of the parallax in satellite theory,” = − nd δ sin θ cos θ dM Celestial Mechanics, vol. 24, no. 2, pp. 111–153, 1981. dt 2 2π 0 [8] M. Lara, J. F. San-Juan, and L. M. Lopez-Ochoa,´ “Proper Av- dω dΩ 1 Z 2πn h e i = −c − d sin f 1 − δc 1 + cos f dM eraging Via Parallax Elimination (AAS 13-722),” 2014, vol. dt dt 2π 0 e 2 150 of Advances in the Astronautical Sciences, pp. 315–331. dM 1 Z 2π e r [9] M. Lara, J. F. San-Juan, and L. M. Lopez-Ochoa,´ “Delaunay = n + nd sin f dM variables approach to the elimination of the perigee in Arti- dt 2π 0 η a ficial Satellite Theory,” Celestial Mechanics and Dynamical 1 Z 2π η h e i + n sin f 1 − δc 1 + cos f dM Astronomy, vol. 120, no. 1, pp. 39–56, 2014. 2π d e 2 0 [10] M. Lara, R. Vilhena de Moraes, D. M. Sanchez, and A. F. B. A. where the known relation sin u = (r/p)η sin f in Eq. (20), Prado, “Efficient computation of short-period analytical cor- 2 3 rections due to third-body effects (AAS 15-295),” 2015, vol. and the abbreviation δ = (n⊕/n)(r/p) η have been used. Both the relative velocity with respect to the rotating at- 155 of Advances in the Astronautical Sciences, pp. 437–455. mosphere V , and the atmospheric density ρ are naturally ex- [11] W. M. Kaula, Theory of satellite geodesy. Applications of satel- pressed as a function of the true anomaly [14], then it happens lites to geodesy, Blaisdell, Waltham, Massachusetts, 1966. that nd ≡ nd(f) from the definition of nd in Eq. (19). Hence, [12] M. Lara, J. F. San-Juan, Z. J. Folcik, and P. Cefola, “Deep Res- the quadratures above are conveniently integrated in f rather onant GPS-Dynamics Due to the Geopotential,” The Journal of than in M using the differential relation in Eq. (13). Besides, the Astronautical Sciences, vol. 58, no. 4, pp. 661–676, 2011. due to the complex representation of the atmospheric density, [13] Y. Kozai, “Effects of Solar Radiation Pressure on the Motion of these quadratures are evaluated numerically. an Artificial Satellite,” SAO Special Report, vol. 56, pp. 25–34, 1961. 8. CONCLUSIONS [14] A. C. Long, J. O. Cappellari, C. E. Velez, and A. J. Fluchs, “Mathematical Theory of the Goddard Trajectory Determi- HEO propagation is a challenging problem because the differ- nation System,” Technical Report FDD/552-89/001, NASA, ent causes that have an effect in these kinds of orbits, whose 1989. k m = 2 m = 4 m = 6 m = 8 0 1 2 + 3e2 8 + 40e2 + 15e4 3(16 + 168e2 + 210e4 + 35e6) 2 1 6 + 3e2 48 + 80e2 + 15e4 4 1 10 + 3e2 6 1 k m = 5 m = 7 m = 9 1 4 + 3e2 3(8 + 20e2 + 5e4) 3(64 + 336e2 + 280e4 + 35e6) 3 1 8 + 3e2 5(16 + 20e2 + 3e4) 5 1 12 + 3e2 7 1 k m = 10 0 3(128 + 2304e2 + 6048e4 + 3360e6 + 315e8) 2 15(32 + 112e2 + 70e4 + 7e6) 4 15(8 + 8e2 + e4) 6 14 + 3e2 8 1

Table 1. Eccentricity polynomials Qm,k in Eq. (14).

k m = 1 m = 2 m = 3 1 2 3 4 2 5 6 4 2 0 4 3c − 1 − 128 35c − 30c + 3 2048 231c − 315c + 105c − 5 15 2 175 4 2 1 − 64 7c − 1 2048 33c − 18c + 1 315 2 2 4096 11c − 1 m = 4 35 8 6 4 2 0 − 786432 6435c − 12012c + 6930c − 1260c + 35 2205 6 4 2 1 − 131072 143c − 143c + 33c − 1 4851 4 2 2 − 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 2. Even inclination polynomials B2m,2k in Eq. (14). k m = 1 m = 2 m = 3 3 2 15 4 2 35 6 4 2 0 − 8 5c − 1 128 21c − 14c + 1 − 8192 429c − 495c + 135c − 5 35 2 315 4 2 1 256 9c − 1 − 16384 143c − 66c + 3 693 2 2 − 16384 13c − 1 m = 4 105 8 6 4 2 0 262144 2431c − 4004c + 2002c − 308c + 7 1617 6 4 2 1 131072 221c − 195c + 39c − 1 3003 4 2 2 131072 85c − 30c + 1 6435 2 3 524288 17c − 1

Table 3. Odd inclination polynomials B2m+1,2k+1 in Eq. (14). 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 4. Eccentricity polynomials Am,j in Eq. (17).

P2,j,l P3,j,l l j = 0 j = 1 j = 0 j = 1 1 2 1 2 15 2 175 3 0 48 (3c − 1) − 16 s − 128 (5c − 1)s − 128 s 1 1 15 2 525 2 ±1 8 cs 8 χs − 512 χ(15c ∓ 10c − 1) − 512 χs 1 2 1 2 75 525 2 ±2 − 32 s 32 χ 256 χ(3c ∓ 1)s − 256 χ s 75 2 175 3 ±3 512 χs 512 χ

Table 5. Inclination polynomials Pm,j,l in Eq. (17)(χ = c ± 1).

l j = 0 j = 1 j = 2 3 4 2 105 2 2 2205 4 0 − 4096 (35c − 30c + 3) − 1024 (7c − 1)s − 4096 s 15 2 105 2 2205 3 ±1 1024 c(3 − 7c )s 512 χ(14c ∓ 7c − 1)s 1024 χs 15 2 2 105 2 2 2205 2 2 ±2 1024 (7c − 1)s 256 χ (7c ∓ 7c + 1) 1024 χ s 105 3 735 2 2205 3 ±3 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 P4,j,l in Eq. (17)(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

Table 7. Inclination polynomials P5,j,l in Eq. (17)(χ = c ± 1).

l j = 0 j = 2 5 6 4 2 2079 2 4 0 − 65536 (231c − 315c + 105c − 5) − 65536 (11c − 1)s 105 4 2 2079 2 3 ±1 − 32768 c(33c − 30c + 5)s 32768 χ(33c ∓ 11c − 2)s 525 4 2 2 10395 2 2 2 ±2 − 262144 (33c − 18c + 1)s − 262144 χ (33c ∓ 22c + 1)s 525 2 3 10395 3 2 ±3 − 65536 c(11c − 3)s 65536 χ (11c ∓ 11c + 2)s 315 2 4 2079 4 2 ±4 32768 (1 − 11c )s − 32768 χ (33c ∓ 44c + 13) 3465 5 22869 4 ±5 − 65536 cs − 65536 χ (3c ∓ 2)s 1155 6 22869 4 2 ±6 262144 s 262144 χ s

Table 8. Inclination polynomials P6,j,l in Eq. (17)(χ = c ± 1). l j = 3 j = 1 99099 6 315 4 2 2 0 − 131072 s − 131072 (33c − 18c + 1)s 297297 5 315 4 3 2 ±1 65536 χs 65536 χ(99c ∓ 66c − 36c ± 18c + 1)s 1486485 2 4 315 2 4 3 2 ±2 − 524288 χ s − 524288 χ (495c ∓ 660c + 90c ± 108c − 17) 495495 3 3 945 2 3 2 ±3 131072 χ s − 131072 χ (55c ∓ 55c + 5c ± 3)s 297297 4 2 945 2 2 2 ±4 − 65536 χ s − 65536 χ (33c ∓ 22c + 1)s 297297 5 10395 2 3 ±5 131072 χ s 131072 (1 ∓ 3c)χ s 99099 6 10395 2 4 ±6 524288 χ 524288 χ s

Table 9. Inclination polynomials P6,j,l in Eq. (17)(χ = c ± 1).

m l Sm,l Tm,l 2 0 −1 + 3w2 0 ±1 −v w ±uw ±2 u2 − v2 ±2u v 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) ±u w (−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

Table 10. Third-body direction polynomials in Eq. (17); u = x?/r?, v = y?/r?, w = z?/r?.