Nonsingular Recursion Formulas for Third-Body Perturbations in Mean Vectorial Elements M

A&A 634, A61 (2020) Astronomy https://doi.org/10.1051/0004-6361/201937106 & c ESO 2020 Astrophysics

Nonsingular recursion formulas for third-body perturbations in mean vectorial elements M. Lara1,3, A. J. Rosengren2, and E. Fantino3

1 Scientiﬁc Computing Group, University of La Rioja, Calle Madre de Dios 53, 26006 Logroño, Spain e-mail: [email protected] 2 Department of Aerospace & Mechanical Engineering, The University of Arizona, 1130 N. Mountain Ave., PO Box 210119, Tucson, AZ, USA e-mail: [email protected] 3 Department of Aerospace Engineering, Khalifa University of Science and Technology, PO Box 127788, Abu Dhabi, UAE e-mail: [email protected]

Received 13 November 2019 / Accepted 3 January 2020

ABSTRACT

The description of the long-term dynamics of highly elliptic orbits under third-body perturbations may require an expansion of the disturbing function in series of the semi-major axes ratio up to higher orders. To avoid dealing with long series in trigonometric functions, we refer the motion to the apsidal frame and efficiently remove the short-period effects of this expansion in vectorial form up to an arbitrary order. We then provide the variation equations of the two fundamental vectors of the Keplerian motion by analogous vectorial recurrences, which are free from singularities and take a compact form useful for the numerical propagation of the flow in mean elements. Key words. celestial mechanics

1. Introduction formulation of the right side of the variation equations that ren- ders faster evaluation (Allan 1962; Musen 1963; Roy & Moran Perturbed Keplerian motion is a multiscale problem, in which 1973)1, and may even disclose integrability (Deprit 1984; see the orbital elements evolve slowly when compared to the change also Mignard & Henon 1984; Richter & Keller 1995). with time of the ephemeris, whose fast evolution is determined The vectorial formulation has been shown to be useful in by the rate of variation of the mean anomaly. Usual integration the case of third-body perturbations (Breiter & Ratajczak 2005; schemes that look for the separation of fast and slow frequencies Correia et al. 2011; Katz et al. 2011). In particular, it may be of the motion may be superior to the simpler integration with the an efficient alternative to classical formulations when the orbits Cowell method, in which the Newtonian acceleration is directly are highly elliptic, which is a common case in extrasolar plan- integrated in rectangular coordinates, and formulations based on etary systems (Lee & Peale 2003; Migaszewski & Go´zdziewski Hansen’s ideal frame concept (Hansen 1857) closely approach 2008), in artificial satellite theory (Lara et al. 2012, 2018), and this decoupling (see Lara 2017, and references therein). More- in hierarchical N-body systems in general (Hamers et al. 2015; over, the use of symplectic integrators is widely adopted in the Will 2017). These types of orbits may need higher degrees in the case of Hamiltonian perturbation problems (Laskar & Robutel expansion of the third-body disturbing function to provide a rea- 2001; Blanes et al. 2013). sonable approximation of the dynamics even in the simplifica- On the other hand, the effective decoupling of short- and tions offered by the secular dynamics (Beaugé & Michtchenko long-period effects is achieved with perturbation methods. After 2003; Libert & Sansottera 2013; Andrade-Ines et al. 2016; removing short-period effects in an averaging process, the Andrade-Ines & Robutel 2018; Sansottera & Libert 2019). This long-term behavior of the (mean) orbital elements is efficiently fact has motivated the recent appearance of general expressions integrated from the equations of variation of parameters. The of the expansion of the disturbing function in powers of the parameters, which are constant in the pure Keplerian motion, can ratio of the disturbing and disturbed semimajor axes, includ- take different representations, such as classical Keplerian ele- ing the secular terms (Laskar & Boué 2010; Mardling 2013; ments or nonsingular variables (see Hintz 2008 for a survey), but Palacián et al. 2017). These expressions rely on classical ele- are always related to the two fundamental vectors of the orbital ments of the Keplerian motion and apply to any eccentricity and motion: the angular momentum vector, which determines the ori- inclination. entation of the orbital plane, and the eccentricity vector, which We constrain ourselves to the restricted approximation, in determines the shape and orientation of the reference ellipse in which the lower mass body does not affect the motion of the that plane. This last is a nondimensional version of the classi- primaries. This is a reasonable approximation in artificial satel- cal Laplace vector, which has dimensions of the gravitational lite theory as well as in some problems of Solar System dynam- parameter, or of the Runge-Lenz vector, which has dimensions ics, but it does not apply to the dynamics of extrasolar planetary of angular momentum. While vectorial formulations normally introduce redundancy by increasing the dimension of the differ- 1 A brief historical review on the topic was given by Deprit(1975), ential system, they commonly admit a compact and symmetric and additional details were presented by Rosengren & Scheeres(2014).

Article published by EDP Sciences A61, page 1 of9 A&A 634, A61 (2020) systems. For the restricted case, we make use of the apsidal central body, χ? is the mass ratio of the system, cos ψ? = rˆ · rˆ?, frame formulation, which effectively displays the fast and slow with rˆ = r/r and rˆ? = r?/r?, and the Legendre polynomials Pi components of the orbital motion, and we use perturbation the- are given by the usual expansion of Rodrigues’ formula, ory to remove the short-period components of the third-body bi/2c ! ! disturbing function. The third-body direction is assumed to be 1 X i 2i − 2l P (cos ψ ) = (−1)l cosi−2l ψ , (3) a known function of time, given by an ephemeris, but we avoid i ? 2i l i ? time-related issues in the averaging by constraining ourselves l=0 to the usual case in which the position of the third body can with bi/2c denoting the integer part of the division i/2. We recall be taken as fixed during one orbital period. Instead of relying that as usual, the term −µ?/r? is neglected in Eq. (2) because it on the classical Hansen expansions or related eccentricity func- does not contribute to the disturbing acceleration of the massless tions (Hansen 1855; Kaula 1962; Giacaglia 1974; Lane 1989; body. Celletti et al. 2017), we average the Legendre polynomial expan- We define now the apsidal moving frame (O, eˆ, b, n) with the sion of the third-body potential in closed form of the eccentricity same origin as before and the unit vectors n in the direction of after the usual reformulation in terms of the eccentric anomaly the (instantaneous) angular momentum vector (per unit of mass), (Deprit 1983; Kelly 1989). The decoupling of short-period effects is achieved after G = Gn = r × u, (4) the standard Delaunay normalization (Deprit 1982), which we extend only to the first order in the Hamiltonian perturbation where G = kGk and u = dr/dt, with t denoting time. The approach; that is, we do not consider the possible coupling unit vector eˆ has the direction of the (instantaneous) eccentricity between the different terms in which the third-body perturbation vector, is expanded. However, while the normalization is properly car- u × G r ried out in Delaunay canonical variables, which are the action- e = eeˆ = − , (5) angle variables of the Kepler problem, the canonical variables µ r related to the Keplerian elements are collected in vectorial form where e = kek is the (instantaneous) eccentricity of the orbit of throughout the whole normalization procedure. In this way, we the massless body, and µ is the gravitational parameter of the obtain alternative, general expressions for the expansion of the central body. Finally, the unit vector third-body disturbing function in mean vectorial elements up to an arbitrary degree. For completeness, the generating function of b = n × eˆ (6) the infinitesimal contact transformation from mean to osculating elements is also provided in vectorial elements up to an arbitrary defines the binormal direction, in this way completing a direct degree of the expansion. orthonormal frame. The Hamilton equations of the mean elements Hamiltonian For convenience, we define the unit vector ` in the direction are then computed to obtain analogous general expressions for of the ascending node, given by the variation equations of the flow in Delaunay (mean) elements, k × n = ` sin I, (7) which, as expected, are flawed by the appearance of the eccentricity and the sine of the inclination as divisors. Singularities are where avoided by reformulating the flow in different sets of canonical and noncanonical variables. In particular, we provide vectorial, I = arccos(k · n), (8) nonsingular expressions for the variation equations of the eccentricity and angular momentum vectors in mean elements. These is the inclination angle of the (instantaneous) orbital plane with expressions can be used to extend to an arbitrary degree existing respect to the (i, j) plane. In addition, lower-order truncations in the literature (Allan 1962; Katz et al. ` · eˆ = cos ω, ` · b = − sin ω, (9) 2011). where ω is the argument of the pericenter. Then, it is simple to 2. Third-body disturbing potential check that Let (O, i, j, k) be an orthonormal frame with origin O in the eˆ · k = sin I sin ω = −` · b sin I, (10) center of mass of a central attracting body and fixed directions b · k = sin I cos ω = ` · eˆ sin I. (11) defined by the unit vectors i, j, k. Let r and r? be the position vectors of a massless and a massive body, respectively, of modu- Finally, by denoting with h the longitude of the node, the components of the apsidal frame in the inertial frame are given lus r = krk and r? = kr?k. When r r?, the third-body disturbing potential by the rotations ! µ r r · r (eˆ, b, n) = R3(−h) R1(−I) R3(−ω), (12) V − ? ? − ? , ? = 2 (1) r? kr − r?k r? where   where µ? is the third-body gravitational parameter, is customar- 1 0 0   ily replaced by the Legendre polynomial expansion R1(α) =  0 cos α sin α  ,  0 − sin α cos α  2 3 i n?a? X r   V? = − χ? Pi(cos ψ?), (2)  cos α sin α 0  r ri   ? i≥2 ? R3(α) =  − sin α cos α 0   0 0 1  in which µ? has been rewritten in terms of the mean motion n? and the semimajor axis a? of the third-body orbit relative to the are standard rotation matrices.

A61, page 2 of9 M. Lara et al.: Third-body perturbations in mean vectorial elements

When the direction of the massless body is given by its com- 3. Short-period averaging ponents in the apsidal frame, Because third-body perturbations derive from the potential in rˆ = eˆ cos f + b sin f, (13) Eq. (2), we can take advantage of the Hamiltonian formalism. Thus with f being the true anomaly, we obtain µ cos ψ = (eˆ · rˆ ) cos f + (b · rˆ ) sin f, (14) H = − + V?, (23) ? ? ? 2a which immediately discloses the short-period terms affecting where H must be expressed in some set of canonical variables. Eq. (3). In particular, we use Delaunay variables (`, g, h, L, G, H), where However, the short-period terms of the third-body disturbing the coordinates are the mean anomaly `, the argument of the potential in Eq. (2) are not limited to those contributed by the periapsis g, and the longitude of the node h, and their conjugate Legendre polynomials, and they are better handled when writ- √ momenta are the Delaunay action L = µa, the modulus of ten in terms of the eccentric anomaly u, in contrast to the true the angular momentum vector G, and its projection along the k anomaly. This is done using the geometric relations direction H, respectively. A modern derivation of this useful set a a sin f = η sin u, cos f = (cos u − e), (15) of variables can be found, for instance, in Lara(2016). r r Disregarding short-period effects, the orbit evolution is cus- 0 0 0 0 0 0 where the eccentricity function η = (1 − e2)1/2 is used for conve- tomarily studied in mean elements (` , g , h , L , G , H ), which nience. We recall also that aim to represent the average value of the true osculating variables. The transformation from mean to osculating variables is r = a(1 − e cos u), (16) found using the tools of perturbation theory. To avoid time- and after some rearrangement, we obtain dependency issues introduced by the third-body ephemeris rˆ? ≡ rˆ?(t), we used the extended phase-space formulation. 2 3 i−2 µ n? a? a X 1 a In particular, we relied on the Lie transforms method (Hori V? = − χ? Vi, (17) 2a n2 r3 r 2i−1 ri−2 1966; Deprit 1969), which is considered standard and is thor- ? i≥2 ? oughly described in the literature (see, e.g., Meyer & Hall 1992; in which the nondimensional functions Vi take the form Boccaletti & Pucacco 2002). The details of the transformation bi/2c ! ! will be presented elsewhere, and we focus here on the averaging X i 2i − 2l r2l+1 V = (−1)l [(eˆ · rˆ )(cos u − e) of the disturbing potential. However, because its computation is i l i a2l+1 ? immediate once the disturbing potential has been averaged, we l =0 also provide the generating function from which the short-period i−2l + (b · rˆ?)η sin u . (18) corrections are derived. The reasons for the particular arrangement of Eq. (17), in which The short-period elimination is carried out in closed form we excluded the coefficient a/r from the summation, is that the of the eccentricity. It is effectively achieved with the help of the differential relation between the eccentric and true anoma- removal of short-period terms from V?, which is carried out by Eq. (24), is easily achieved in closed form of the eccentricity by lies d` = (1 − e cos u) du, which is obtained by differentiating taking advantage of the differential relation between the mean Kepler’s equation. Thus, because of Eq. (16), and true anomalies. 1 Z 2π 1 Z 2π r Next, we substitute Eq. (16) into Eq. (18) and use the bino- hV?i` = V?d` = V? du, (24) mial expansion to obtain 2π 0 2π 0 a 2l+1 ! r2l+1 X 2l + 1 from which, after replacing Eq. (17), we obtain = (1 − e cos u)2l+1 = (−1) je j cos j u. (19) a2l+1 j µ n2 a3 X 1 ai−2 j=0 hV i = −χ ? ? hV i , (25) ? ` ? 2a n2 3 2i−1 i−2 i u Analogously, the binomial expansion is applied to the last term r? i≥2 r? in the right side of Eq. (18), to yield where (eˆ · rˆ )(cos u − e) + (b · rˆ )η sin ui−2l ? ? Z 2π i−2l ! 1 X i − 2l hViiu = Vidu. (26) = (eˆ · rˆ )k(cos u − e)k(b · rˆ )i−2l−kηi−2l−k sini−2l−k u, 2π k ? ? 0 k=0 (20) In view of Eq. (22), the only terms that remain below the integral symbol of Eq. (26) are of the form sina u cosb u with a where and b integers. Terms of this type are easily integrated when they k ! X k are expressed as Fourier series in u. Straightforward computa- (cos u − e)k = (−1)mem cosk−m u. (21) tions using standard relations between exponentials and circular m m=0 functions (see, e.g., Kaula 1966), yield Plugging Eqs. (19)–(21) into Eq. (18), we finally obtain i−k−2l i−k−2l j+k−m ! j+k−m i−2l−k (−˙ıı) X X i − k − 2l q bXi/2c ! ! 2Xl+1 ! Xi−2l ! Xk ! cos u sin u = (−1) i 2i − 2l 2l + 1 i − 2l k j+m 2i+ j−m−2l q Vi = e q=0 t=0 l i j k m ! l=0 j=0 k=0 m=0 j + k − m j+l+m i−2l−k k i−2l−k × × (−1) η (eˆ · rˆ?) (b · rˆ?) t × cos j+k−m u sini−2l−k u. (22) × cos 2(˜q − q)u + ˙ıısin 2(˜q − q)u , (27)

A61, page 3 of9 A&A 634, A61 (2020) where ˙ıınotes the imaginary unit, and we abbreviated keeping all the trigonometric terms except for those that make i + j − m − 2l − 2q − 2t = 0. Namely, i + j − m q˜ = − l − t. (28) bi/2c ! ! 2l+1 ! i−2l ! 2 Z X i 2i − 2l X 2l + 1 X i − 2l (V − hV i ) du = i i u l i j k Now, it becomes obvious that the only nonperiodic terms of l=0 j=0 k=0 Eq. (27) k ! X k m+ j i−2l−k k i−2l−k D E × e η (eˆ · rˆ?) (b · rˆ?) Ξi−2l−k = cos j+k−m u sini−2l−k u , (29) m j+k−m u m=0 i−2l−k j+k−m ! are those such that q = q˜, a condition that is only accomplished (−˙ıı)i−2l−k X X i − 2l − k × (−1) j+l+m when i + j − m is even, which in turn implies that i + j + m is also 2i−2l+ j−m q q=0 t=0 even. In addition, because Eq. (27) must be free from imaginary q,q˜ terms, i − k, the exponent of the imaginary unit, must be even in ! j + k − m the case of nonperiodic terms, yielding the numeric coefficient × (−1)q t 2a ! ! 1 X 2b 2a sin 2(˜q − q)u − ˙ııcos 2(˜q − q)u Ξ2b = (−1)a−t , (30) × , (36) 2a 22a+2b b + a − t t 2(˜q − q) t=0 where sin 2(˜q−q)u−˙ııcos 2(˜q−q)u = −˙ıı(cos u+˙ıısin u)2(˜q−q) from 1 − 1 − − where a = 2 ( j + k m) and b = 2 (i k) l are integer numbers. which, using Eq. (15), On the other hand, i − k even allows us to replace ηi−k−2l by 2 1 (i−k−2l) !2(˜q−q) the binomial expansion of (1 − e ) 2 . Thus, replacing r · eˆ r · b sin 2(˜q − q)u −˙ııcos 2(˜q − q)u = −˙ıı e + + ˙ıı . (37) a aη i−k −l 2 i−k ! m+ j 2 i−k −l X − l q m+ j+2q e (1 − e ) 2 = 2 (−1) e , (31) Finally, the Hamiltonian in mean elements, up to higher q q=0 order effects, is obtained by replacing osculating by mean elements into both the Keplerian −µ/(2a) and the third-body aver- into, Eq. (26), we finally obtain aged potential, Eq. (25), to yield µ bi/2c ! ! i−2l ! 2l+1 ! k ! X i 2i − 2l X i − 2l X 2l + 1 X k K = − + hV?i`, (38) hV i = Ξi−2l−k 2a i u l i k j m j+k−m l=0 k=0 j=0 m=0 where now all the symbols appearing in K are functions of the i−k − 2 l ! Delaunay prime elements. X i−k − l × 2 (−1)i−l+qem+ j+2q(eˆ · rˆ )k(b · rˆ )i−2l−k, q ? ? q=0 4. Hamilton equations of the averaged flow (32) The flow in mean (prime) elements is obtained from the Hamil- thus completing the calculation of Eq. (25). ton equations. Because the mean anomaly has been removed by The generating function of the infinitesimal contact trans- the perturbation approach up to the truncation order of the per- formation leading to the averaging is computed from the usual turbation solution, the prime Delaunay action is a constant that relation decouples the reduced flow Z 0 1 dg ∂K ∂hV?i` W = (V? − hV?i`) d`. (33) = = , (39) n dt ∂G0 ∂G0 dG0 ∂K ∂hV i That is, = − = − ? ` , (40) dt ∂g0 ∂g0 n2 a3 X 1 ai−2 Z a dh0 ∂K ∂hV i W = −χ L ? ? V − hV i d` = = ? ` , (41) ? n2 3 2i i−2 r i i u dt ∂H0 ∂H0 r? i≥2 r? dH0 ∂K ∂hV i 2 3 i−2 Z ! − − ? ` n? a? X 1 a = 0 = 0 , (42) = −χ?L −hViiu ` + Vidu , (34) dt ∂h ∂h n2 r3 2i ri−2 ? i≥2 ? from the integration of the variation of the prime mean anomaly, which can be written in the form 0 d` ∂K ∂hV?i` 2 3 X i−2 " Z # = = n + , (43) n? a? 1 a dt ∂L0 ∂L0 W = −χ?L hViiu(u − `) + (Vi − hViiu) du , n2 r3 2i ri−2 ? i≥2 ? which is computed by indefinite integration after solving the (35) reduced system of Eqs. (39)–(42). In fact, in view of Eqs. (25) and (32), computing the right where u − ` = e sin u, from the Kepler equation, and the inte- sides of the variation Eqs. (39)–(42) only requires the corre- grand of the last term in the square brackets is only composed sponding partial derivatives of the function of periodic terms in the eccentric anomaly. Then, after replac- p k σ−k ing Eq. (27) into Eq. (22), we solve the indefinite integral by ασ,k,p = e (eˆ · rˆ?) (b· rˆ?) , σ = i−2l, p = m+ j+2q. (44)

A61, page 4 of9 M. Lara et al.: Third-body perturbations in mean vectorial elements

In Eq. (44), both direction vectors eˆ and b depend on the orbital 5. Long-term flow in mean vectorial elements inclination I, the argument of the periapsis g, and the longitude of the node h, as shown in Eq. (12). As expected from the singularities of Delaunay variables, Eqs. (52) and (53) are singular in the case of circular and equa- The required partial derivatives of ασ,k,p with respect to the Delaunay variables are computed using the chain rule: torial orbits. However, these singularities are of virtual nature (Henrard 1974) and may be removed when the equations of the ! ∂ασ,k,p ∂ασ,k,p ∂e ∂ασ,k,p ∂eˆ ∂I flow are represented in other sets of canonical or noncanonical = + · rˆ? variables. In particular, the variations experienced by the angular ∂G ∂e ∂G ∂(eˆ · rˆ?) ∂I ∂G ! momentum vector and the eccentricity vector under third-body ∂ασ,k,p ∂b ∂I perturbations are free from singularities and provide a general, + · rˆ?, (45) ∂(b · rˆ?) ∂I ∂G compact, and elegant way of presenting the variation of the mean ! ∂ασ,k,p ∂ασ,k,p ∂eˆ ∂I elements. = · rˆ? When the mean angular momentum vector is scaled by ∂H ∂(eˆ · rˆ?) ∂I ∂H 0 ! L , the variations of the mean vectorial elements h = ∂ασ,k,p ∂b ∂I G/L0 = ηn and e = eeˆ take the neat symmetric form + · rˆ?, (46) ∂(b · rˆ?) ∂I ∂H (Milankovitch 1941; Allan & Ward 1963; Allan & Cook 1964; Rosengren & Scheeres 2014) ∂ασ,k,p ∂ασ,k,p ∂eˆ ∂ασ,k,p ∂b = · rˆ + · rˆ , (47) ∂g ∂(eˆ · rˆ ) ∂g ? ∂(b · rˆ ) ∂g ? dh ∂R ∂R ? ? = h × + e × , (56) ∂ασ,k,p ∂ασ,k,p ∂eˆ ∂ασ,k,p ∂b dt ∂h ∂e = · rˆ? + · rˆ?. (48) de ∂R ∂R ∂h ∂(eˆ · rˆ?) ∂h ∂(b · rˆ?) ∂h = e × + h × , (57) dt ∂h ∂e Because c ≡ cos I = H/G, s ≡ sin I = (1 − c2)1/2, and e = in which (1 − G2/L2)1/2, we easily obtain 1 2 R − hV i ∂e 1 η ∂I 1 1 ∂I 1 c = 0 ? `, (58) = − , = − , = · (49) L ∂G G e ∂H G s ∂G G s and hV?i` is given in Eq. (25). Furthermore, derivation of In addition, because the effect of a differential rotation about Eqs. (56)–(57) is trivial after minor rearrangement of Eq. (32) the axis n (resp. `, k) is an infinitesimal increase of the angle to replace the directions eˆ and n by the nondimensional magni- g (resp. I, h), we find tudes e and h, respectively. Thus, instead of replacing Eq. (31) in Eq. (32), the latter is ∂eˆ ∂eˆ ∂eˆ 1 = n × eˆ = b, = k × eˆ, = ` × eˆ = (eˆ · k)n, (50) written in the more convenient form ∂g ∂h ∂I s bi/2c ! ! 2l+1 ! i−2l ! k ! X i 2i − 2l X 2l + 1 X i − 2l X k and hViiu = l i j k m ∂b ∂b ∂b 1 l=0 j=0 k=0 m=0 × − × × · i−k −l = n b = eˆ, = k b, = ` b = (b k)n. (51) i−2l−k j+l+m j+m−k k h 2 2i 2 ∂g ∂h ∂I s × Ξ j+k−m(−1) e (eeˆ · rˆ?) η (b · rˆ?) . Thus, straightforward computations yield (59)

2 When now we make use of the identity ∂ασ,k,p 1 η h p−1i k σ−k = − pe (eˆ · rˆ?) (b · rˆ?) 2 2 2 ∂G G e (eˆ · rˆ?) + (b · rˆ?) + (n · rˆ?) = 1, (60) 1 c p h k−1i σ−k eˆ · k + e k(eˆ · rˆ?) (b · rˆ?) (n · rˆ?) which gives the square of the modulus of the third-body direction G s s krˆ?k = 1 when it is computed in the apsidal frame, the depen- 1 c h i b · k + ep(eˆ · rˆ )k (σ − k)(b · rˆ )σ−k−1 (n · rˆ ) , dence of the averaged potential on the unit vector b in the binor- G s ? ? ? s mal direction is replaced by a corresponding dependence on the (52) normal direction n. That is

∂ασ,k,p 1 1 p h k−1i σ−k eˆ · k bi/2c ! ! 2l+1 ! i−2l ! k ! = − e k(eˆ · rˆ?) (b · rˆ?) (n · rˆ?) X i 2i − 2l X 2l + 1 X i − 2l X k ∂H G s s hViiu = l i j k m 1 1 p k h σ−k−1i b · k l=0 j=0 k=0 m=0 − e (eˆ · rˆ?) (σ − k)(b · rˆ?) (n · rˆ?) , − j+m−k i k −l G s s i−2l−k j+l+m k h 2i 2 × − · 2 · − · (53) Ξ j+k−m( 1) (e e) (e rˆ?) X (eˆ rˆ?) , (61) ∂ασ,k,p nh i = ep k(eˆ · rˆ )k−1 (b · rˆ )σ−k+1 ∂g ? ? with the scalar function k+1 h σ−k−1io 2 2 2 − (eˆ · rˆ?) (σ − k)(b · rˆ?) , (54) X = X(e, h, rˆ?) ≡ 1 − e + ξ − ζ , (62)

∂ασ,k,p nh i where we introduced the abbreviations ξ = e · rˆ , ζ = h · rˆ . = ep k(eˆ · rˆ )k−1 (b · rˆ )σ−k(eˆ × rˆ ) ? ? ∂h ? ? ? Now, applying the binomial expansion again, we ﬁnd k h σ−k−1i o i−k − + (eˆ · rˆ?) (σ − k)(b · rˆ?) (b × rˆ?) · k, (55) bi/2c ! ! 2l+1 ! i−2l ! k ! 2 l ! X i 2i − 2l X 2l + 1 X i − 2l X k X i−k − l hV i = 2 i u l i j k m s where the dispensable square brackets in these equations are l=0 j=0 k=0 m=0 s=0 − used to emphasize that corresponding enclosed terms never i−2l−k j+l+m+s j+m k −s 2s+k i−k −l−s × − · 2 · 2 introduce denominators. Ξ j+k−m( 1) (e e) (e rˆ?) X , (63)

A61, page 5 of9 A&A 634, A61 (2020) which only depends on e and h, as desired, as well as on the orbit propagators based on semianalytical integration (Lara et al. direction rˆ? of the disturbing body. 2012, 2016, 2018). The constant term in hV2iu is commonly Explicit expressions for the ﬁrst terms of the Legendre poly- neglected because it has no eﬀect on the long-term motion. nomial expansion yield the compact expressions Finally, on account of hV i = 1 − 3 2e2 − 5ξ2 + ζ2 , (64) ∂(e · e) 2 u = 0, (71) ∂h 5 h 2 2 2i hV3iu = − ξ 3 1 − 8 − 15ζ + 35ξ , (65) ∂(e · e) 2 = 2e, (72) 3 n ∂e hV i = 3 − 20e2 + 80e4 4 u 4 ∂(e · rˆ?) h i = 0, (73) + 10 7 1 − 10e2 ξ2 − 3 − 10e2 ζ2 ∂h ∂(e · rˆ?) 4 2 2 4o = rˆ , (74) + 35 21ξ − 14ζ ξ + ζ , (66) ∂e ? 3 n ∂X hV i = − ξ 35 1 − 8e2 + 40e4 = −2(h · rˆ?)rˆ?, (75) 5 u 4 ∂h h 2 2 2 2i ∂X +490 1 − 12e ξ − 1 − 4e ζ = −2e + 2(e · rˆ?)rˆ?, (76) o ∂e +147 33ξ4 − 30ζ2ξ2 + 5ζ4 , (67) derivation of Eqs. (56)–(57) is straightforward from Eq. (63). 1 n hV i = 5 5 − 42e2 + 168e4 − 560e6 Therefore, using Eq. (25) and taking into account that 6 u 4 (µ/2a)/L = 1 n, Eqs. (56)–(57) are written in the form h 2 + 105 3 3 − 28e2 + 168e4 ξ2 i−2 i dh X 1 a 2 4 2 = n γi(h × rˆ?) + ρi(e × rˆ?) , (77) − 5 − 28e + 56e ζ dt 2i ri−2 h i≥2 ? − e2 ζ4 − − e2 ζ2ξ2 i−2 + 315 5 14 18 3 14 de X 1 a i = n γi(e × rˆ?) + ρi(h × rˆ?) 2 4 dt 2i i−2 +33 1 − 14e ξ i≥2 r? o +231 429ξ6 − 495ζ2ξ4 + 135ζ4ξ2 − 5ζ6 , (68) +4ρi−1(h × e) , (78)

9 n 2 4 6 where we abbreviate = χ (n /n)2(a /r )3. The polynomials hV7iu = − ξ 35 5 − 48e + 224e − 896e ? ? ? ? 16 ρi are computed from the recursion h 2 4 2 + 105 11 3 − 32e + 224e ξ i−k − bi/2c ! ! 2l+1 ! i−2l ! k ! 2 l ! X i 2i − 2l X 2l + 1 X i − 2l X k X i−k − l 2 4 2i ρ = 2 −3 15 − 96e + 224e ζ i l i j k m s j m s h l=0 =0 k=0 =0 =0 2 4 2 2 2 − − − − − i−k − − − + 231 143 1 − 16e ξ − 110 3 − 16e ζ ξ i 2l k j+l+m+s j+m k 2s 2s+k 1 2 l s 1 × Ξ j+k−m(−1) e ξ X i h i +15 5 − 16e2 ζ4 × (k + 2s)X + (i − k − 2l − 2s)ξ2 , (79) o ξ6 − ζ2ξ4 ζ4ξ2 − ζ6 , +429 715 1001 + 385 35 (69) and the polynomials γi from n 5 2 4 6 8 i−k hV i = 7 35 − 360e + 1728e − 5376e + 16128e bi/2c 2l+1 i−2l k −l 8 u X ! − ! X ! X − ! X ! X2 i−k − ! 64 i 2i 2l 2l + 1 i 2l k 2 l γi = h 2 4 6 2 l i j k m s +252 11 5 − 54e + 288e − 1344e ξ l=0 j=0 k=0 m=0 s=0 i i−k 2 4 6 2 i−2l−k j+l+m+s+1 j+m−k−2s 2s+k 2 −l−s−1 − 35 − 270e + 864e − 1344e ζ × Ξ j+k−m(−1) (i − k − l − s)e ξ ζX . (80) h + 1386 143 1 − 12e2 + 96e4 ξ4 The complexity of the polynomials given by Eqs. (79) and (80) 2 4 2 2 is apparent, and corresponding expressions take a compact form, −66 5 − 36e + 96e ζ ξ which is illustrated in Table1 for the lower degrees. i + 35 − 180e2 + 288e4 ζ4 h + 12012 143 1 − 18e2 ξ6 6. Performance evaluation: The case of high Earth orbits +33 5 − 18e2 ξ2ζ4 − 429 1 − 6e2 ξ4ζ2 i The simplicity of the vectorial approach in approximating the − 7 − 18e2 ζ6 long-term dynamics of a system under third-body perturbations 8 2 6 when compared with classical expansions based on trigonomet- +6435 2431ξ − 4004ζ ξ ric terms is evident from the visual comparison of Table1 with o +2002ζ4ξ4 − 308ζ6ξ2 + 7ζ8 . (70) equivalent results based on classical trigonometric expansions. For instance, while the entire Tables 4−8 of Lara et al.(2018) The terms hV2iu and hV3iu have been repeatedly reported are required to evaluate the third-body disturbing acceleration up in the literature (Musen 1961; Rosengren & Scheeres 2013) to degree 6 of the Legendre polynomial expansion, which calls whereas the remaining terms were veriﬁed with corresponding for the additional evaluation of trigonometric functions, only the expressions in Delaunay elements that are customarily used in upper half of Table1 is required when the vectorial approach is

A61, page 6 of9 M. Lara et al.: Third-body perturbations in mean vectorial elements

Table 1. Some polynomials ρi, γi, given by Eqs. (79)–(80).

ρ1 = −3, ρ2 = 30ξ, γ2 = −6ζ, 15 h 2 2 2i ρ3 = − 2 1 − 8e + 5 7ξ − ζ , γ3 = 75ξζ, h 2 2 2i ρ4 = 105 1 − 10e + 7 3ξ − ζ ξ, h 2 2 2i γ4 = −15 3 − 10e + 7 7ξ − ζ ζ, 105 n 2 4 h 2 2 2 2i ρ5 = − 4 1 − 8e + 40e + 14 3 1 − 12e ξ − 1 − 4e ζ o +21 33ξ4 − 18ξ2ζ2 + ζ4 , 2 2 2 γ5 = 735 1 − 4e + 9ξ − 3ζ ξζ, 63 n 2 4 h 2 2 2 2i ρ6 = 2 5 3 − 28e + 168e + 30 11 1 − 14e ξ − 3 3 − 14e ζ o +33 143ξ4 − 110ξ2ζ2 + 15ζ4 ξ, 105 n 2 4 h 2 2 2 2i γ6 = − 2 5 − 28e + 56e + 6 9 3 − 14e ξ − 5 − 14e ζ o +33 33ξ4 − 18ξ2ζ2 + ζ4 ζ, 315 n 2 4 6 ρ7 = − 16 5 − 48e + 224e − 896e h i +9 11 3 − 32e2 + 224e4 ξ2 − 15 − 96e2 + 224e4 ζ2 h i +33 143 1 − 16e2 ξ4 − 66 3 − 16e2 ξ2ζ2 + 3 5 − 16e2 ζ4 o +429 143ξ6 − 143ξ4ζ2 + 33ξ2ζ4 − ζ6 , 189 n 2 4 γ7 = 8 15 15 − 96e + 224e h i +110 11 3 − 16e2 ξ2 − 3 5 − 16e2 ζ2 o +143 143ξ4 − 110ξ2ζ2 + 15ζ4 ξζ, 495 n 2 4 6 ρ8 = 8 7 5 − 54e + 288e − 1344e h i +77 13 1 − 12e2 + 96e4 ξ2 − 3 5 − 36e2 + 96e4 ζ2 h i +1001 13 1 − 18e2 ξ4 − 26 1 − 6e2 ξ2ζ2 + 5 − 18e2 ζ4 o +715 221ξ6 − 273ξ4ζ2 + 91ξ2ζ4 − 7ζ6 ξ, 315 n 2 4 6 γ8 = − 8 35 − 270e + 864e − 1344e h i +11 33 5 − 36e2 + 96e4 ξ2 − 35 − 180e2 + 288e4 ζ2 h i +143 143 1 − 6e2 ξ4 − 22 5 − 18e2 ξ2ζ2 + 7 − 18e2 ζ4 o +715 143ξ6 − 143ξ4ζ2 + 33ξ2ζ4 − ζ6 ζ.

used. On the other hand, the numerical integration of the mean For our efficiency proofs, we assume the initial conditions elements is expected to progress with similar step sizes when either vectorial or other classical mean elements are used. The a = 106 247.136 km e = 0.75173 former requires integrating a redundant differential system of ◦ higher dimension, however, which might counterbalance the pre- I = 5.2789 Ω = 49.351◦ sumed advantages derived from the simplicity of the formulation ◦ and the use of arithmetic operations only, as opposed to evaluat- ω = 180.008 ing trigonometric functions. M = 0 The improvements obtained with the vectorial formulation where a, e, I, Ω, ω, and M stand for classical Keplerian ele- when higher degrees are required in expanding the third-body ments, and remove all perturbations except for the lunisolar ones. disturbing function are illustrated for high Earth orbits. In partic- The initial epoch needed for the computation of the lunisolar ular, we present an example of a SIMBOL-X-type orbit, whose ephemeris is fixed to 2014 July 1 at 20.7208333 h UTC. Then, semianalytical propagation requires the use of at least a P2–P6 we propagate these initial conditions for intervals of 100 years truncation of the lunar disturbing function in order to capture with both the vectorial approach of this paper and with the clas- the main frequencies of the long-term motion (Lara et al. 2018; sical approach used by Lara et al.(2018). The solar e ffect is trun- Amato et al. 2019). cated to the P2 Legendre polynomial whereas lunar perturbations

A61, page 7 of9 A&A 634, A61 (2020)

80 15 60 10 40

5 20 rgruntime vectorial / trig % rgruntime vectorial / trig %

0 0 2 3 4 5 6 7 8 2 3 4 5 6 7 8 Lunar terms Lunar terms Fig. 1. Runtime of the vectorial approach relative to the classical Fig. 2. Same as Fig.1 when the time spent to evaluate the lunisolar trigonometric implementation in percentage for an increasing complex- ephemerides is included. ity of the lunar perturbation.

disturber may be taken as purely Keplerian (Lara et al. 2012), only include from P2 to P2–P8 in successive increments of one that is not at all the case for Earth-orbiting satellites, in which term. The numerical integration of the averaged flow is car- lunar and solar ephemerides must either be read from a data file ried out with the reputed DOPRI 853 integrator, which imple- or be evaluated from analytical expressions. In this last case, the ments an explicit Runge-Kutta method of order 8(5,3) due to added computational effort is not at all negligible, and the ques- Dormand & Prince(1980), with step size control and dense out- tion therefore naturally emerges whether the efforts in improving put (Hairer et al. 2008). the evaluation of the differential equations of the flow might be In order to estimate the gains of the vectorial formulation vacuous. To verify that, we repeat the computations in the actual with respect to the trigonometric expansions, we first make the case in which lunisolar ephemerides are computed at each inte- propagations with constant step size and without updating the gration step of the numerical integration procedure. Correspond- lunisolar ephemerides. The results are depicted in Fig.1 in terms ing results are shown in Fig.2. Runtime improvements are quite of the runtime percentage of the vectorial approach relative to small for the quadrupolar case and only moderate for the octupo- the classical propagation. Because the figures we present are rel- lar case. The vectorial approach still halves the runtime when ative quantities, analogous results are expected using different moderate degrees are required in the expansion of the lunar dis- computational environments and therefore, no additional infor- turbing function, and only about 20% of the runtime of the clas- mation on the hardware and software used in the tests is needed. sical approach is required when the truncation is extended to the However, we did not optimize the evaluation of the disturbing P2–P8 case. acceleration, and except for trivial arrangements, we completely Finally, we recall that an added bonus on the side of the vec- left the optimization task to the compiler in both cases: in the torial formulation is that its redundancy allows us to examine the vectorial formulation of this paper, and in the classical approach accuracy of the numerical integration by evaluating the geomet- of Lara et al.(2018). rical relations h · e = 0 and e · e + h · h = 1, cf. Herrick(1948). Figure1 shows that when the lunar perturbation is limited Using these tests, we found that the errors of the 100-year numer- to the quadrupolar term, the vectorial formulation only takes ical propagation of the flow in mean vectorial elements is con- about one fifth of the time required by the classical formula- strained to the order of 10−14 when the tolerance of the numerical tion in trigonometric functions to complete the 100-year propa- integrator is set to 10−14. gation. The gains of the vectorial formulation notably increase with the fidelity in modeling lunar perturbations. The vectorial approach performs in less than one tenth of the classical 7. Conclusions approach of Lara et al.(2018) when the lunar octupolar term is also taken into account, and takes an impressive 1.8% of the time We have provided a detailed description of the Hamiltonian required by the classical formulation when the terms P2–P8 are reduction process that removes short-period effects from the taken into account. We recall that for this particular orbit, the third-body disturbing function, keeping the slowly evolving long-term dynamics is correctly modeled with a P2–P6 trunca- parameters in vectorial form. Our approach takes advantage of tion of the lunar disturbing function, cf. Lara et al.(2018), a case the apsidal frame formulation to provide general expressions in which the runtime obtained with the vectorial formulation of the mean element potential in vectorial form up to an arbi- is shorter than 2.5% of the time required by the trigonometric trary degree. The variation of parameter equations of the flow in expansions. mean elements admits a general, nonsingular, compact formula- On the other hand, the evaluation of the third-body accel- tion when the elements are the angular momentum vector and the eration, as given by Eqs. (77)–(78) or equivalent expressions eccentricity vector. We verified that the explicit vectorial expres- in Lara et al.(2018), is only a part of the computational bur- sions for the lower degree truncations of our general approach den of the propagation. Evaluating the disturbing acceleration agree with alternative derivations in the literature, and they can requires the previous knowledge of the third-body ephemeris at now be extended to an arbitrary degree. In addition, the com- each step of the numerical integration of the flow in mean ele- pact form of the vectorial expressions permits avoiding the long ments. While this task may be computationally undemanding for listings required when angular variables are used, and printed simple restricted three-body models in which the orbit of the expressions up to moderate orders of the variation equations

A61, page 8 of9 M. Lara et al.: Third-body perturbations in mean vectorial elements can be arranged in only one side of a letter. Runs for particular Hansen, P. A. 1855, Abhandlungen der Koniglich Sachsischen Gesellschaft der examples confirmed the expected notable improvements of the Wissenschaften, 2, 183, English translation by J. C. Van der Ha, ESA/ESOC, performance in the propagation of the mean elements’ flow when Darmstadt, Germany, 1977 Hansen, P. A. 1857, Abhandlungen der Koniglich Sachsischen Gesellschaft der the vectorial approach was used. Wissenschaften, 5, 41 Henrard, J. 1974, Celest. Mech., 10, 437 Herrick, S. 1948, PASP, 60, 321 Acknowledgements. The work of ML and EF has been funded by Khalifa Uni- Hintz, G. 2008, J. Guid. Control Dyn., 31, 785 versity of Science and Technology’s internal grants FSU-2018-07 and CIRA- Hori, G.-I. 1966, PASJ, 18, 287 2018-85. 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