-PERTURBATION SOLUTION TO THE RELATIVE MOTION PROBLEM

Transcription

1 IAA-AAS-DCoSS1-6- AAS J -PERTURBATION SOLUTION TO THE RELATIVE MOTION PROBLEM M. Lara * and P. Gurfil A solution to the satellite relative motion problem is constructed based on a radial intermediar of the J -problem. The new solution accounts for nonlinearities of the model to a high etent, and hence is adequate for long-term prediction of bounded relative orbits with arbitrar inclinations without limiting to the case of tightl-controlled formations. The integrabilit of the radial intermediar is utilied for finding periodic relative orbits in a local-vertical local-horiontal frame and to determine an initialiation scheme that ields long-term boundedness of the relative distance. INTRODUCTION The Hill-Clohess-Wiltshire model in the relative motion problem for artificial satellites is known to provide a useful set of equations for dealing with some problems of rendevous 1,, 3 and formation-fling, most notabl optimal control problems. However, this set of equations is constrained in practice because of the linearied approimation, and b the underling assumptions of a circular reference orbit and a central Newtonian gravit field. Specificall, the distance between the deput and chief satellites in modern concepts such as disaggregated satellites and cluster flight 4, 5 a group of satellites that remains within bounded distances while appling ver few formation-keeping maneuvers is not small, a fact that clearl violates the linear differential gravit field assumption. Recent efforts have been directed to relieve the limitations of the linearied approach. On one hand, relative motion is completel analticall described b the use of orbital elements of each satellite, where second order nonlinearities from the differential gravit 6, 7, 8, 9, 1 field ma be accounted for. On the other hand, current research has recalled that the non-linearied relative motion problem is integrable. Indeed, in view of the fact that a solution to the relative motion problem is just the difference between two solutions of the same flow for different initial conditions, this is true, as epected, for those cases in which the flow is integrable. Specificall, general solutions of the full nonlinear relative motion problem have been obtained for the case of elliptic Keplerian motion 11 and for the equatorial case of the onal problem, the latter limited to the particular case of even onal harmonics. 1 Columnas de Hercules 1, ES-111 San Fernando, Spain. mlara@gmail.com Distributed Space Sstems Lab, Facult of Aerospace Engineering, Technion - Israel Institute of Technolog, Haifa 3, Israel. pgurfil@technion.ac.il 59

2 The need to find orbits that would guarantee bounded relative motion under orbital perturbations becomes crucial in the case of distributed space sstems (DSS), since some of the modules forming the disaggregated satellite ma not be equipped with a propulsion sstem. Moreover, the obvious sie limits of individual satellite modules ma seriousl limit the on-board propellent needed for station-keeping and/or cluster-keeping. Thus, it is evident that an effort should be initiated in order to establish an astrodnamical theor that would enable accurate long-term prediction of relative satellite orbits. The goal of the current paper is to etend previous research 1 in order to find closed-form solutions for the problem of satellite relative motion under the effect of J the most dominant perturbation for satellites fling on low Earth orbits (LEO) without limiting to the case in which both the chief and deput satellites remain in the equatorial plane. Because the J -problem is not integrable, in general, we are satisfied with finding an integrable approimation for the relative motion problem under the J perturbation, to be adequate for long-term prediction of bounded relative orbits with arbitrar inclinations. To that end, a radial intermediar Hamiltonian 13, 14 is utilied. The intermediar Hamiltonian retains the original structure of the full J Hamiltonian, ecluding the latitude dependence. In that sense, it is different from the averaged Hamiltonian, because the radial terms are not averaged out. Thus, Hamilton s equations capture the main characteristics of the motion, including long- and short-periodic terms. This formalism provides integrabilit via separation, a fact that is subsequentl used to find periodic relative orbits in a chief-fied local-vertical local-horiontal frame (LVLH) frame and determine an initialiation scheme that ields long-term boundedness of the relative distance. Numerical eperiments show that the intermediar-based computation of orbits indeed provides long-term bounded orbits in the full J problem for various inclinations. THE RADIAL INTERMEDIARY The following developments will be carried out in the Whittaker polar-nodal chart (r, θ, ν, R, Θ, N), where r is distance to the attraction center, θ is the argument of latitude, ν is the right ascension of the ascending node, R =dr/dt is the radial velocit, Θ is the modulus of the angular momentum vector, and N is the polar component of the angular momentum. The mapping from the polar-nodal variables to the classical orbital elements (a, e, I, Ω, ω,f), namel the semimajor ais, eccentricit, inclination, right ascension of the ascending node, argument of periapsis and true anomal, respectivel, can be obtained through the relations: 53

3 μr a = r R +Θ μr e = 1 Θ μa I =cos 1 N Θ Ω=ν cos f = Θ μr, sin f = Θ R μre μe ω = θ f (1a) (1b) (1c) (1d) (1e) (1f) where μ is the gravitational parameter. The Hamiltonian for the J -perturbed dnamics is defined b H (r, θ, R, Θ,N)= 1 ) (R + Θ μ { 1 1 [ r r J α 3sin θ (1 N ) ]} 1, () r Θ where α is the mean equatorial radius. Although the Hamiltonian flow derived from Eq. () is not integrable in general (ecept the equatorial and polar cases), approimate analtical solutions computed b perturbation methods are commonl used in artificial satellite theor. Thus, Brouwer s first-order Hamiltonian 15 has been successfull used in the development of relative motion theories 6, 8, 9 based on mean elements. In this paper, the relative motion theor builds on a radial intermediar, the so-called Cid intermediar 13, 14 H (r, R, Θ,N)= 1 (R + Θ r ) μ [ 1 1 ( r J α 1 r 3 )] N, (3) Θ which, contrar to Brouwer s first-order Hamiltonian, retains short-periodic effects related to the radius evolution. From the common definition of an intermediar, Eq. (3) is just a truncation of Eq. (). In addition, as a natural intermediar, 16 Eq. (3) is obtained as the 17, 18, 19 first-order Hamiltonian of the elimination of the latitude simplification. Since θ and ν are cclic in Eq. (3), both Θ and N are integrals of the motion, and therefore the inclination remains constant. In addition, from Hamilton equations, d(r, θ, ν) H = dt (R, Θ,N), d(r, Θ,N) = H dt (r, θ, ν), so that the intermediar takes into account the precession the node of the orbital plane (as opposed to the central-force approimation of 1 ), dν dt = 3 μ r J α r N Θ. (4) 531

4 In addition, dθ dt = Θ r + 3 μ r J α N r Θ = Θ 3 r dν cos I. (5) dt Equations (4) and (5) include all the secular variations of the node and periapsis of the full J problem up to O(J ), because the intermediar (3) and the full Hamiltonian () average eactl to the same epression. 14 Moreover, Eq. (4) shows that the node remains fied onl in the case of polar orbits, where N =. The analtical solution of the motion determined b the radial intermediar (3) was previousl obtained in for the case of an elliptic-tpe motion utiliing lineariation. The lineariation process has the merit of converting the orbit evolution in the new time scale into a Keplerian solution, but onl for the radius. However, the solution for the angular variables still relies on the use of elliptic integrals and, because the lineariation is based on the solution of a cubic polnomial, recovering the original time scale requires in turn the use of elliptic integrals and functions. Therefore, we find it more convenient to find the solution b direct integration; this approach clearl shows that the two cases mentioned in are related to low- and high-inclination orbits. ANALYTICAL SOLUTION Since, in the radial intermediar (3), Θ and N are constant, the flow is separable and the reduced flow (r, R) uncouples from the complete dnamics. We limit the discussion to the case of elliptic-tpe motion, in which r evolves between two limit values. In this case, since the reduced dnamics are of one degree-of freedom (DOF), the motion in the (r, R) phase-space is made of closed curves, showing that r is a periodic function. First, we can solve the 1-DOF problem dr dt = R, dr dt = μ r + Θ r + 3 μj α 3 r 4 53 ( 1 3 ) N, (6) Θ which, for given initial conditions, is convenientl integrated using the energ integral H = h. Thus, ( ) 1 dr + 1 Θ dt r μ r + μj α 1 (1 3 N ) = h, r 3 4 Θ from which where χ (Θ, N) μj α ( 1 3 dr dt = h + μ r Θ r χ r, (7) 3 ) ( N 1 = μj Θ α 3 ) cos I = const., (8) which ma be positive or negative depending on the inclination. If I =cos 1 ( ±3 1/) then χ =, so the radial motion evolves similarl to a Keplerian orbit, and χ> for I deg.

5 Equation (7) is integrated b quadrature, (t t )mod T = I t, (9) where T is the fundamental period of the solution, and, denoting s =1/r r dr sma I t = r min h +(μ/r) (Θ /r ) (χ/r 3 ) = ds s s χs 3 Θ s +μs+h, (1) with s ma denoting the maimum value of s. A solution for the node can be obtained from Eq. (4), (ν ν )modν = 3 μj α N Θ I ν (11) where ν is the angle traveled b the node in a fundamental period T/ and, taking into account Eq. (7), dt sma I ν = r(t) = s ds 3 s χs3 Θ s +μs+h. (1) Finall, from Eq. (5), (θ θ )modθ = 3 μj α N Θ I 3 ν +ΘI θ (13) where θ is the latitude angle traveled in a fundamental period T/, I ν is given in Eq. (1) and, b a repeated use of Eq. (7), dt sma I θ = r(t) = ds χs3 Θ s +μs+h. (14) s On one hand, the fact that the integration of Cid s intermediar depends on elliptic integrals does not give immediate insight into the phsical solution. On the other hand, the reduced flow in (r, R) is 1-DOF and, for given values of the integrals Θ and N, can be represented b simple contour plots of the Hamiltonian (3). Circular orbits We immediatel note from Eq. (6) that circular orbits ma eist when R =and N = Θ 3α J 3J α +4(Θ /μ) r 4r (15) admits an real solution. B circular we mean closed orbits with constant radius in the (r, θ) plane; in general, these orbits will not be closed in the configuration space, ecept for the equatorial or polar cases, because of the precession of the nodes, cf. Eq. (4). 533

6 If we use non-dimensional units of length and time such that J =1/α and μ =1, then the inclination condition (15) can be written as N = Θ 3 3+4Θ r 4r. (16) In the case of equatorial orbits, this gives the condition Θ = r +3/(r), and for circular polar orbits Θ = r 3/(4r). In the general case, Eq. (16) shows that there is a critical value r = 3/ ( km for the Earth) below which circular orbits can eist for an value of Θ, but above which circular orbits onl eist if Θ r 3/(4r). We note also that while circular orbits ma eist with inclinations closer to the equator than I = arccos ±3 1/ for an r, this is no longer true for high inclination orbits, as illustrated in Fig. 1. The critical value r = 3/ marks also the value of r above which one can find circular orbits at an inclination r Figure 1. Contours of admissible inclination [deg] leading to circular solutions, cf. Eq. (16). Cubic solution As seen in Eqs. (1) and (14), the solution to Cid s intermediar motion depends on the solution of the fundamental cubic (ecluding the critical case cos I =1/3) P s 3 + Θ χ s μ χ s h =, (17) χ which, in general, must have at least one real root, s 3. Moreover, for bounded motion to eist, P must posses at least two positive roots, which in turn implies that all three roots For Earth orbits, this would mean that the unit of length is km, and the unit of time s 534

7 must be real, s 1 s s 3. In the case of bounded absolute motion, i.e., closed orbits, the solution of the cubic P can be written as 1 s i = 1 Θ 3 χ + Θ 4 3 χ + 6μ λ +πi cos, (i =1,, 3) (18) χ 3 where ( ) λ =cos 1 7hχ 9μ Θ χ Θ 6 signχ (Θ 4 +6μχ) 3/ Finall, we note that for χ<, s 1 s s, while for χ>, s s s 3. (19) Auiliar variables and constants In what follows, we define the following auiliar constants and variables k = (s s 1 )/(s 3 s 1 ) < 1, k =1/k, () κ = 1 k < 1, κ =1/κ, (1) n = (s s 1 )/s 1, ñ = n/k, () ν = (s 3 s )/s 3, ν = ν/κ. (3) = (s s 1 )/(s s 1 ) 1, φ = arcsin, = k, (4) = (s s 1 )/(s 3 s 1 ) 1 (5) ζ = (s 3 s)/(s 3 s ) 1, ϕ = arcsin ζ, ζ = κ ζ (6) ζ = 1 1, (7) In what follows we adhere to the convention of using the elliptic parameter (the square of the elliptic modulus) as one of the arguments of the elliptic integrals, contrar to the elliptic modulus used in other conventions. Solution to I ν The case χ<: s 1 s s We write the quadrature I ν in Eq. (1) as I ν = s s s ds χ (s s1 )(s s )(s s 3 ) (8) where s i s i, i =1,, 3, are computed from Eqs. (18)-(19). The change of variables in Eq. (4) leads to I ν = s n χ s3 s d, (9) k 535

8 with k given b Eq. () and n b Eq. (). Then, [ I ν = Iν s3 s3 F (φ k ) χ s 3 s 1 ] s3 s 1 E(φ k ) s 3 (3) where [ ] Iν s3 s3 = K(k s3 s 1 ) E(k ), (31) χ s 3 s 1 s 3 In Eqs. (3) and (31), F (φ k ) is the incomplete elliptic integral of the first kind, k is the elliptic modulus, and φ = arcsin, cf. Eq. (4). In addition, E(k ) is the complete elliptic integral of the second kind, E(φ k ) is the incomplete elliptic integral of the second kind, and K(k ) is the complete elliptic integral of the first kind. We note that Iν is the particular value of Eq. (8) for the lower limit s = s 1. Then, the fundamental angle ν of the solution is ν 1 = 3 μj α N Θ I ν (3) cf. Eq. (11). The case χ>: s s s 3 We write the quadrature I ν in Eq. (1) as I ν = s3 s s ds χ (s s1 )(s s )(s s 3 ) (33) where s i s i, i =1,, 3, are computed from Eqs. (18)-(19). Using again the change of variables (6) ields s 3 1 νζ I ν = χ s3 s dζ (34) 1 ζ 1 ζ 1 κ ζ with κ given b Eq. (1), and ν in Eq. (3). Then, [ s1 s1 I ν = F (ϕ κ ) χ s 3 s 1 where ϕ = arcsin ζ, cf. Eq. (6). The fundamental angle ν of the solution is ] s3 s 1 E(ϕ κ ). (35) s 1 ν = 3 μj α N Θ I ν (36) where, now, [ ] Iν s1 s1 = K(κ s3 s 1 ) E(κ ), (37) χ s 3 s 1 s 1 is the particular value of Eq. (33) for the lower limit s = s. 536

9 Solution to I θ The case χ<: s 1 s s We write the quadrature I θ in Eq. (14) as s ds I θ = (38) s χ (s s1 )(s s )(s s 3 ) where s i s i, i =1,, 3, are computed from Eqs. (18)-(19). The change of variable given b Eq. (4) ields I θ = with k given b Eq. (). Then, 1 d χ s3 s (39) k I θ = I θ χ s3 s 1 F (φ k ), (4) where I θ = χ s3 s 1 K(k ), (41) We note that I θ is the particular value of Eq. (38) for the lower limit s = s 1. Then, the fundamental angle θ of the solution is cf. Eq. (13). The case χ>: s s s 3 s3 θ 1 =ΘI θ ν 1 N Θ, (4) We write the quadrature I θ in Eq. (14) as ds I θ = (43) s χ (s s1 )(s s )(s s 3 ) where s i s i, i =1,, 3, are computed from Eqs. (18)-(19). Now, we make the change of variables given in Eq. (6) to obtain with κ given in Eq. (1). Then, dζ I θ = χ s3 s (44) 1 1 ζ 1 κ ζ I θ = where ϕ = arcsin ζ, cf. Eq. (6). The fundamental angle θ of the solution is where, now, I θ = ζ χ s3 s 1 F (ϕ κ ) (45) θ =ΘI θ ν is the particular value of Eq. (43) for the lower limit s = s. N Θ, (46) χ s3 s 1 K(κ ) (47) 537

10 Solution to I t The case χ<: s 1 s s We write the quadrature I t in Eq. (1) as I t = s s ds s χ (s s 1 )(s s )(s s 3 ) (48) where s i s i, i =1,, 3, are computed from Eqs. (18)-(19). Equation (5) leads to I t = 1 k d χs 1 s s 1 (1 + ñ ) 1 k (49) 1 where k =1/k and ñ = n/k as in Eq. () and Eq. (), respectivel. Then, [ k n I t = T 1 1+n 1 χs 1 s s 1 F (φ k )+ n 1 k sin φ k + n 1+nsin sin φ (5) φ ] n k + n E(φ k )+ (3+n) k +(+n) n Π( n; φ k ) k + n where Π( n; φ k ) is the incomplete elliptic integral of the third kind, and T 1 is the fundamental (half) period, given b [ 1 k (3+n) k +(+n) n T 1 = Π( n k )+ χs 1 s s 1 1+n k + n n ] k + n E(k ) K(k ) (51) where Π( n k ) is the complete elliptic integral of the third kind. Note that T 1 is the particular value of Eq. (48) for the lower limit s = s 1. The case χ>: s s s 3 We write the quadrature I t in Eq. (1) as I t = s s 3 ds s χ (s s 1 )(s s )(s s 3 ) (5) where s i s i, i =1,, 3, are computed from Eqs. (18)-(19). The change of variables in Eq. (7) leads to 1 d ζ I t = χs 3 s3 s ζ (1 ν ζ ) 1 ζ 1 κ ζ (53) where κ =1/κ and ν = ν/κ, cf. Eqs. (1) and (3), respectivel. Then, [ 1 κ ν ν 1 κ sin ϕ I t = χs 3 s3 s 1 ν κ ν 1 ν sin sin ϕ (54) ϕ F (ϕ κ ) ν κ ν E(ϕ κ )+ 3κ ν (κ ν) ν Π ( ν; ϕ κ )] κ ν 538

11 The fundamental (half) period of r is now [ 1 κ (3 ν) κ ( ν) ν T = Π(ν κ ) ν ] χs 3 s3 s 1 ν κ ν κ ν E(κ ) K(κ ) (55) which is the particular value of Eq. (5) for the upper limit s = s. The resonance condition The difference between θ, given b Eqs. (4) and (46), and the Keplerian value θ = π varies depending on inclination. Figure shows this evolution for an Earth satellite with initial semimajor ais a() = 7 km and initial eccentricit e() =.8 (cf. Eq. (1)). The figure remains identical for all the tests made for different a() and e() ecept for an epected scaling in the value of the ordinates. Remarkabl, θ = π when I = arccos 1/5, under the numerical precision used, i.e. at the critical inclination. Lower-order resonances are not found for other inclinations, although the ma eist for almost rectilinear orbits. Π Θ rad inclination deg Figure. Evolution of the θ critical angle with inclination. Hence, all the orbits at the critical inclination are π-periodic in the plane perpendicular to the angular momentum vector. This continuum of periodic solutions that happens as a result of the 1:1-resonance provides an eplanation of the bifurcation phenomenon that takes place at the critical inclination of the artificial satellite theor,, 3 although limited to the degenerate case. Since the solution of Cid s intermediar is purel analtic, the removal of this degenerac b second-order terms in the solution of the full J problem, Eq. (), should be attributed to the contribution of the terms depending on the argument of latitude, which are absent in Cid s intermediar. CONDITIONS FOR RELATIVE MOTION BOUNDEDNESS AND PERIODICITY The aforementioned discussion clearl shows that the formalism of the radial intermediar provides a powerful method for obtaining an integrable approimation of J -perturbed relative motion. In this section, it is shown that the radial intermediar can be used for detecting bounded and periodic J -perturbed relative orbits. 539

12 To begin, let the subscripts ( ) C and ( ) D denote quantities related to the chief and deput satellites, respectivel. The common relative motion boundedness conditions require 1, 4 that dν C dt = dν D dt, dθ C dt + dν C dt cos I C = dθ D dt + dν D dt cos I D. (56) From Eqs. (4) and (5), we find rd 3 rc 3 = Θ C N D, Θ D N C rd rc = Θ D Θ C, (57) Both equations require that the ratio between the instantaneous radii remain constant, which in turn implies that the chief s and deput s fundamental periods and angles must be equal. Because of that, both orbits must have the same eccentricit and semimajor ais, and furthermore the chief and deput must have the same true anomal. In addition, the modulus of the angular momentum must be the same in both orbits, i.e. Θ D = Θ C, and hence r D = r C, which in turn implies N D = N C. This fact onl leaves some freedom in choosing the argument of the periapsis, and of course the right ascension of the ascending node, because of the aial smmetr of the onal potential. The conditions for relative motion periodicit in the chief-fied LVLH frame can be obtained as follows. Let r C =(r C,, ) τ and r D =(r D,, ) τ be the respective position vectors of the chief and deput satellites in their own Earth-centered polar rotating frames. The position vector of the deput in the chief-fied LVLH frame, ρ =(,, ) T, can be obtained through the connection 5 ρ = R 3 (θ C ) R 1 (I C ) R 3 (ν C ν D ) R 1 ( I D ) R 3 ( θ D ) r D r C (58) where the standard rotation matrices from a reference frame to a given frame are R 1 (β) = 1 cosβ sin β sin β cos β,r 3 (β) = For T -periodicit in the LVLH frame, it is required that cos β sin β sin β cos β (59) 1 ρ(t) =ρ(t + T ). (6) As mentioned above, the radial intermediar-based conditions for bounded motion entail r D = r C = r. In addition, I D = I C =cos 1 (N C /Θ C )=const. Therefore, ν D ν C =Δis Note, however, that Cid s intermediar is a truncation that is onl accurate to O(J ) in θ, so the difference in the argument of periapsis ma be limited in practice. 54

13 also constant. Hence, Eq. (58) simplifies to = r { (1 + c) cos(θ +Δ)+s cos θ +(1 c) cos(θ Δ) (61a) 4 } +s [cos(θ + +Δ) cosθ + +cos(θ + Δ)] r = r { (1 + c) sin(θ +Δ)+s sin θ +(1 c) sin(θ Δ) (61b) 4 } s [sin(θ + +Δ) sinθ + + sin(θ + Δ)] = r s [(1 c) sin(θ D Δ) + c sin θ D (1 + c) sin(θ D +Δ)] (61c) where θ + θ D +θ C, θ θ D θ C, and we abbreviated s sin I and c cos I. Therefore, the periodicit condition of Eq. (6) implies r C (t) =r C (t + T ), (6) θ C (t) =θ C (t + T )modπ, (63) θ D (t) =θ D (t + T )modπ, (64) As indicated in Section The Radial Intermediar, due to the integrabilit of the motion determined b Cid s intermediar Hamiltonian, r C is periodic with fundamental period T C. Hence, the period T of the relative motion must be some integer multiple of T C. Then, Eq. (63) is rewritten into where m is the number of fundamental periods. Hence, θ C (t) =θ C (t + mt C )modπ, (65) θ C =(k/m) π, (66) with k integer. Then, Eq. (64) is automaticall accomplished because of the condition r C = r D. Algorithm for generating periodic relative motion Algorithm 1 summaries the procedure for determining the chief s and deput s initial conditions that ield periodic relative motion under the Cid radial intermediar model. Note that if Δθ =, Δν is arbitrar, and all the remaining initial conditions are identical, then the relative orbit will be periodic in the Cid s intermediar model as well as in the full J problem for all values of initial conditions. Sample relative periodic orbits We provide several eamples for a nominal chief LEO with initial orbital elements a C () = 7 km, e C () =.4, ω C () = Ω C () = f C () = and different inclinations. Using Eqs. (1), we obtain the nominal values r C () = 67 km, θ C () =, ν C () =, 541

14 Algorithm 1 Generating T -periodic relative motion 1: Inputs: Initial epoch, t ; initial conditions [r C (t ),θ C (t ),ν C (t ),R C (t ), Θ C,N C ] of a nominal chief orbit; Δν ν D (t ) ν C (t ); Δθ θ D (t ) θ C (t ). : Compute the inclination from cos I C = N C /Θ C. 3: Compute the energ h = H (r C,ν C,R C, Θ C,N C ) from Eq. (3). 4: Compute χ (Θ C,N C ) from Eq. (8). 5: Use h from Step 3 and χ from Step 4 to compute s i using Eqs. (18) and (19). 6: Compute θc from Eq. (4) if χ<or Eq. (46) if χ>. 7: if θc is not commensurate with π then 8: Find a rational number k/m such that k/m θc /π, and set θ C =(k/m) π. 9: Set I C as in Step and h as in Step 3. 1: Solve Eq. (4) if χ<or Eq. (46) if χ>to obtain Θ C and N C =Θ C cos I C. 11: Solve for R C from h = H (r C,ν C,R C, Θ C,N C ) using Eq. (3). 1: Output: Chief initial conditions: [r C (t ),θ C (t ),ν C (t ),R C (t ), Θ C,N C ]=[r C (t ),θ C (t ),ν C (t ),R C(t ), Θ C,N C] 13: end if 14: 15: Outputs: Chief initial conditions: Deput initial conditions [r C (t ),θ C (t ),ν C (t ),R C (t ), Θ C,N C ] [r D (t ),θ D (t ),ν D (t ),R D (t ), Θ D,N D ]= [r C (t ),θ C (t )+Δθ, ν C (t )+Δν, R C (t ), Θ C,N C ] 16: Compute the characteristic period T C from Eq. (51) if χ< or Eq. (55) if χ>. 17: Output: Period of the relative motion, T = mt C. 54

15 R C () =, Θ C =58.1km /s, andn C =Θ C cos I. Algorithm 1 ields the modified initial conditions of the chief orbit, which are identical to the initial conditions of the deput orbit ecept for the anglular shifts Δν =1deg, Δθ =.3 deg (67) The obtained initial conditions for the chief and deput are then used for simulating the eact relative orbit in the full J problem. The position components in the LVLH frame, (,, ), are compared in both cases (cf. Eqs. (61)). The polar case For I C =9deg, N C =. We compute θc = radians. We find that 148 θc 1479 π, and solve Eq. (55) with θ C = (1479/148) π to obtain Θ C = km /s, and R C () = km/s from the energ equation. Figure 3 shows the, and projections of the relative position components in the LVLH frame for a one-week propagation. In Cid s intermediar model, the orbit is periodic after 148 characteristic periods of T C = s, or 1 das. Critical inclination orbit In this case I C =cos 1 1/5 (N C = 364 km /s) and the characteristic angle is eactl θc = π. We compute T C = s. Figure 4 shows the relative motion for a one-week propagation; in Cid s intermediar model the orbit is periodic after hours. Low inclination orbit Now we choose I C =3deg, which is reached for N C = km /s. We compute θc = radians. We find that 536 θ C 537 π, and solve Eq. (55) with θc = (537/536) π to obtain Θ C = km /s, and R C () = km/s from the energ equation. Figure 5 shows the relative motion for a one-week propagation; in Cid s intermediar model the orbit is periodic after 536 characteristic periods of T C =58.97 s,or 36.1 das. CONCLUSIONS An integrable approimation for the relative motion under the effect of the onal harmonic J was introduced. This approimation is useful for finding relative orbits that retain bounded inter-satellite distance over relativel long time periods for arbitrar inclinations. As opposed to the averaged Hamiltonian, the proposed formalism captures the main characteristics of the motion, including long- and short-periodic variations, and therefore the actual J -perturbed orbits are still bounded over long time scales. The intermediar Hamiltonian can be used for finding periodic orbits through a correction scheme, which was developed and verified in simulations. This research also provided new insight into the critical inclination problem. Without need of averaging the short-periodic terms, the analtical solution of Cid s intermediar problem showed that the orbits at the critical inclination have fundamental angle θ = π, thus resulting in 1:1-periodic orbits in the (r, θ) plane. This periodicit eases the fulfillment of conditions required for relative periodic motion, and hence orbits at the critical inclination ma provide advantageous nominal conditions in the search for bounded relative motion under realistic astrodnamical models. 543

16 34 Cid's intermediar LVLH frame 34 Full J problem LVLH frame Cid's intermediar LVLH frame Full J problem LVLH frame Cid's intermediar LVLH frame Full J problem LVLH frame Figure 3. Comparison of Cid s intermediar-based solution for the relative motion and a full J model for a polar chief orbit; distances are in units of km. 544

17 Cid's intermediar LVLH frame Full J problem LVLH frame Cid's intermediar LVLH frame 1 1 Full J problem LVLH frame Cid's intermediar LVLH frame Full J problem LVLH frame Figure 4. Comparison of Cid s intermediar-based solution for the relative motion and a full J model for the critical inclination I C =cos 1 1/5; distances are in units of km. 545

18 Cid's intermediar LVLH frame Full J problem LVLH frame Cid's intermediar LVLH frame Full J problem LVLH frame Cid's intermediar LVLH frame Full J problem LVLH frame Figure 5. Comparison of Cid s intermediar-based solution for the relative motion and a full J model for I C =3deg; distances are in units of km. 546

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