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Asymptotic Solution for the Orbit Motion Under Constant Tangential Acceleration1

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Asymptotic Solution for the Orbit Motion Under Constant Tangential Acceleration1 ASYMPTOTIC SOLUTION FOR THE ORBIT MOTION UNDER CONSTANT TANGENTIAL ACCELERATION1 Claudio Bombardelli(1), Giulio Baú (2), Jesús Peláez(1) (1) Universidad Politécnica de Madrid, 28040 Madrid, Spain, +34 913 366 306, [email protected], (2) University of Padova, CISAS G. Colombo, via Venezia 15, 35131 Padova, Italy, +39 049 82768485, [email protected] Abstract: We present an analytical method for propagating the orbital motion of a spacecraft perturbed by a constant tangential acceleration. The solution is obtained starting from the orbital element formulation recently pro- posed by Pelaez et al. and applying perturbation theory. Analytical expressions valid for elliptic orbits with generic eccentricity, describe the instantaneous time variation of all orbital elements. A comparison with high-accuracy numerical re- sults is carried out showing that the analytical method can be eectively applied to multiple-revolution low-thrust orbit transfer around planets and in interplan- etary space. The limits of the applicability of the method, which depend on the the ratio between the tangential ecceleration and the local gravity, are discussed. Keywords: Low thrust, Orbital Dynamics, Perturbation Theory 1 Introduction The study of the orbit evolution of a spacecraft under the perturbation of a constant tangential acceleration has important implications in orbital dyan- mics. It is well known, for instance, that a purely tangential thrust provides a maximum instantaneous rate of orbital energy increase and can be useful in the process of nding optimal solutions for the maximization of such energy along a generic time span2. In addition, as high-specic-impulse propulsion systems provide a relatively small thrust primarily due to the cost, in terms of hardware mass, of the required onboard power, the tangential thrust magnitude is typi- cally small compared with local gravity and is applied contnuously over a time period that is at least of the same order of the period of the osculating orbit or even orders of magnitude larger. For these reasons many authors have proposed analytical or semianalytical solutions for the propagation of orbits under constant or variable tangential 1An extended version of this article was submitted to the journal of Celestial Mechanics and Dynamical Astronomy. 2The maximum increase in the orbital energy over a xed period of time or, equivalently, the minimum time for a specic energy increase is obtained when the thrust is aligned with the primer vector minimizing the Hamiltonian associated with the optimum control problem. 1 acceleration ([1],[2], [3], [4],[5]). These methods, which can be eective in re- producing the secular variations of slowly spiralling circular and even elliptical orbits, cannot in general reproduce oscillatory variations of the orbital elements and lose accuracy when the ratio beteen the thrust magnitude and local gravity increases beyond say 1 × 10−3. For these reasons they may show degraded per- formance when applied to the propagation of interplanetary trajectories when realistic level of achievable accelerations can make a trajectory consideably far from keplerian. In order to overcome this limitation we have developed an approximate, yet accurate analytical model[6] to represent the average and oscillatory time evolution of all orbital elements by exploiting the use of perturbation theory and of a non-singular variation of parameter formulation of the orbital dyanmics[7]. Assuming the acceleration magnitude is small (say < 0:1) when compared to the local gravity we formulate the orbit dynamics problem as a general per- turbation problem. After a straightforward series expansion the rst-order time evolution of the three generalized orbital elements associated to the planar or- bital motion are obtained, in analytical close form, by simple quadrature. Secu- lar and oscillatory terms are both computed, which are generally a combination of elliptic integrals of the rst and second kind. Trigonometric series for the os- cillatory terms are given in order to speed up the computation process and allow the computation of second-order terms. The eectiveness of the approximate analytical solution is tested for a transfer from GTO to Earth escape and from Earth to Mercury, in both cases assuming continuous and constant tangential acceleration and no orbit plane change. 2 Equations of Motion Let us consider a particle orbiting around a primary at initial radial position r0 measured from the center of the primary and angular position θ0 measured from the initial eccentricity vector. Let us employ, from now on, r0 as unit of distance and 1=Ω0 as unit of time where Ω0 is the angular rate of a circular orbit with radius equal to the initial radius r0: r µ Ω0 = 3 ; r0 with µ indicating the gravitational parameter of the primary. Using the formulation described by Pelaez et al. ([7]), and under the hypoth- esis that all acting perturbation forces have zero component along the normal to the orbital plane, the orbit geometry can be fully described by the three generalized orbital parameters: e q = cos ∆γ; (1) 1 h e q = sin ∆γ; (2) 2 h 2 1 q = ; (3) 3 h where h is the dimensionless angular momentum of the osculating orbit, e its eccentricity and ∆γ is, for the constant orbit plane case, the rotation of the eccentricity vector with respect to the initial orbit. From the above expressions the orbit eccentricity, the eccentricity vector rotation and the non-dimensional angular momentum can be written, for later use as: pq2 + q2 e = 1 2 ; (4) q3 q ∆γ = tan−1 2 ; (5) q1 1 h = : (6) q3 The expression of the non-dimensional semimajor axis is then: 2 h 1 (7) a = 2 = 2 2 2 : 1 − e q3 − q1 − q2 The independent variable used in the Pelaez method is, again for the planar case ([7]): θ = ν + ∆γ; (8) where ν is the true anomaly of the osculating orbit. Note that θ corresponds to the inertial angular position of the particle measured from the initial eccentricity vector. A Sundmann transformation, corresponding the the angular momentum variation equation, relates θ to the dimensionless time t as: dθ h = = q s2; (9) dt r2 3 where s is the dimensionless transverse velocity of the particle and obeys ([7]): s = q1 cos θ + q2 sin θ + q3: (10) The orbit radius as a function of the generalized orbital parameter can be obtained from the two equations above as: −1 2 −1 (11) r(θ) = (q3s) = q3 + q1q3 cos θ + q2q3 sin θ : The evolution of the three generalized orbital parameters obeys (see[7]): 3 0 1 0 1 q1 s sin θ (s + q3) cos θ d 1 ar (12) @ q2 A = 3 @ −s cos θ (s + q3) sin θ A ; dθ q3s aθ q3 0 −q3 where ar and aθ are, respectively, the component of the dimensionless pertur- bative acceleration along the instantaneous radial and transversal direction. If the acceleration is always directed along the instantaneous velocity vector Eq.s (12) become: 0 q 1 0 s sin θ (s + q ) cos θ 1 d 1 3 e sin ν q2 = p −s cos θ (s + q3) sin θ ; @ A 3 2 @ A dθ q3s e + 2e cos ν + 1 1 + e cos ν q3 0 −q3 (13) where p 2 2 is the dimensionless tangential acceleration magnitude. = ar + aθ Note that, by use of Eq.s(4,5,8,10), the above equations can be put in the form: dq = F (q; , θ); (14) dθ T where q = (q1; q2; q3) and F is a nonlinear vectorial function. Equations (14) must be integrated with the appropriate initial conditions namely: q1(θ0) = e0=h0; q2(θ0) = 0; q3(θ0) = 1=h0: 3 Asymptotic Solution When considering high specic impulse electric propulsion systems, currently the most common low-thrust solution employed in space technology, typical val- ues for the achievable acceleration with resonable payload masses range around 100 mN=tonne [8]. In most circumstances (depending on the local gravity value for the particular orbit considered) the resulting dimensionless acceleration will also be a small quantity which will be used to perform an asymptotic ex- pansion of Eqs. (13) and (9), which characterize, respectively, the trajectory geometry and its evolution in time. 3.1 Trajectory In the hypothesis that is a small quantity we write the three generalized orbital elements as power series: 4 2 2 qi(θ; ) = qi0(θ) + qi1(θ) + qi2(θ) + o( ) i = 1::3: (15) Substituting into Eq.s (14) , expanding in Taylor series and solving for like powers of epsilon we obtain, for the zeroth order: dq i0 = 0; dθ showing that the zeroth order terms are just the (constant) generalized orbital elements of the unperturbed trajectory: q10 = e0=h0; (16) q20 = 0; (17) q30 = 1=h0; (18) where e0 and h0 are, respectively, the eccentricity and dimensionless angular momentum of the initial trajectory. The dierential equations for the rst order terms results: 0 1 0 1 q11 3 e0 + 2 cos θ d h0 q21 = 2 sin θ : (19) @ A 2p 2 @ A dθ (1 + e0 cos θ) e + 2e0 cos θ + 1 q31 0 −1 Eq.s (??) can be best integrated by introducing the new variable E~ which obeys: E~ r1 − e θ tan = 0 tan : (20) 2 1 + e0 2 Note that E~, although similar, does not correspond to the eccentric anomaly 3 of the osculating orbit, except when θ = θ0 at the beginning of the integration . The following relations can be derived from Eq. (20): p 1 − e2 sin Ee sin θ = 0 ; 1 − e0 cos Ee cos Ee − e0 cos θ = ; 1 − e0 cos Ee dEe 1 − e0 cos Ee = ; dθ p 2 1 − e0 3 In such case θ and e0 concide with the true anomaly and eccentricity of the osculating orbit, respectively.
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