(Preprint) AAS 19-909 FLYBY IN THE SPATIAL THREE-BODY PROBLEM Davide Menzio,∗ and Camilla Colomboy The spatial flyby map originates from the planar one to extend its applica- bility to the 3D dynamics of the circular restricted three-body problem. A novel parametrisation enables to give new insights on the effect of the flyby on inclined orbits. The main contributions of this paper consists first of all in the development of the method itself, secondly, in the identification of two type of trajectories: pro- grade (type I) and retrograde (type II) flyby. Finally, the paper demonstrates that direct gravity assist (type I) are more efficient when compared to the retrograde one. The new approach will enable a large-scale of applications in which inclined orbits are necessary for targeting non-coplanar objects but also to meet some spe- cific mission requirements. INTRODUCTION Since the beginning of space exploration, analysts have widely exploited gravity assist in the tra- jectory design to reduce the fuel consumption and contain the overall mission cost. The idea behind the flyby consists in leveraging the interaction with the gravitational field emitted by a secondary body, different from the primary one which is been orbited, in order to modify the overall trajec- tory of the spacecraft with the respect of the latter in a predictable manner. Historically, we have seen a constant increase in the number of flybys implemented in more and more complex orbits, in which the gravity assist is not only implemented to accelerate/decelerate the spacecraft during its interplanetary journey, such as in any mission from Mariner II, passing through Voyager II and Cassini-Huygens to ultimately Bepicolombo, but tailored to meet specific scientific, mission and safety requirements in planetary moon systems. Conic approximation represents the baseline for multi-gravity assisted trajectory in which the total orbit consists of different branches connecting several minor bodies with keplerian arcs dy- namically bounded to the central one and in flybys resolving the discontinuities at each secondary. The simplicity of Keplerian motion represent a great advantage in a mission analysis perspective but at the same a great limit given the chaotic nature of the interaction between two or more grav- itational fields and the energy restriction imposed by using purely keplerian motion, which lead, over the time, to favour a treatment in the three-body dynamics. Two divergent approaches have tackled the trajectory design task: a more mathematically rigorous method which exploits invariant manifolds and homoclinic and heteroclinc connections1,2 and a more engineering one exploiting maps to represent the effect of the third body perturbation on an osculating orbit. Depending on how it is modelled, semi-analytical and numerical mapping techniques are distinguished. In par- ticular, among the former the Keplerian map3 assesses the orbital variation via integration through ∗PhD, Dipartimento di Scienze e Tecnoligie Aerospaziali, Politecnico di Milano, via La Masa, 34, 20134 Milano, IT. yProf, Dipartimento di Scienze e Tecnoligie Aerospaziali, Politecnico di Milano, via La Masa, 34, 20134 Milano, IT. 1 Picard’s iteration of a kick function approximating the effect of the close encounter. Instead, the Period-periapsis map, the Tisserand graph and the Tisserand-Poincare´ graph constitute the leading exponents and the evolution in time of the numerical maps treatment, offering a method to group family of flybys on their Tisserand Parameter (or Jacobi constant) for ballistic elliptic,4 parabolic and hyperbolic orbits5 and low energy ones.6 In the end, the flyby map7 can be considered as an hybrid approach that originated by the observation that the kick approximation of the dynamics is affected by not negligible deviation for energy that are still outside the limits of fully ballistic trajectories.8 The flyby map restores the accuracy of the Keplerian map to the one of numerical propagation and at the same time reduces the computational demand required by the Tisserand- Poincare´ graph implementing conics formulae far from the secondary. All the aforementioned methods were developed for planar motion, however several conditions requires to pass to a spatial formulation. In particular: • the natural inclination of orbital bodies such as our Moon,?,9 the dwarf planets Vesta and Ceres, some Near Earth Asteroids,10, 11 the Galilean moons, Io, Europa, Ganymede and Cal- listo, and Uranus’ largest moons, Triton and Charon; • interesting sites from a scientific purpose or landing opportunity, such as the observation of Saturn’s aurora,12 located at high latitude; • specific mission requirements, such as occultation,13 coverage14 and containment of the radi- ation dose15, 16 To overcame this need, Gomez et al.,17 Alessi et al.,18 Campangnola et al.6 considered the effect of the three dimensional dynamics on invariant manifolds, semi-analytical and numerical maps and its possible application to trajectory design. Nevertheless such methods are still affected by the specific problems of their category: in fact, the former requires an extensive research and refinement of initial conditions, the second is more prone to missmodel the perturbative effect of the secondary and the third doesn’t provide enough insights on the impact of three-body interaction on the initial orbit. Therefore, the present paper aims to extend the limited applicability of the flyby map to the spatial case and to bridge the 3D mapping techniques in a similar way to what has been done for the planar case. The generation process of the map remained unchanged, preserving the numerical integration between Poincare´ sections placed ad hoc to capture the flyby effect. The paper offers an insight on the parametrisation required to reduce the six-dimensional continuous dynamical system into a discrete lower-dimensional phase-space, more tractable in a trajectory design perspective. The main contributions of this work can be summarised in the possibility to identify a set of control variables from a reference osculating orbit, in the identification of two families of flyby (type I and II) and the observations that prograde flybys (type I) are in general more efficient than retrograde (type II) ones. BACKGROUND This section includes the state of the art of some key concepts that will be recalled extensively throughout the paper. The first paragraph summarizes the v-infinity globe method for design of gravity assists, while the second paragraph recap the main ideas behind the flyby map and its logic. 2 V-infinity globe mapping In the linked-conic assumption, the velocity of the particle with the respect to the central body, vsc, can be obtained through the sum of the relative velocity at the secondary, v8, and its own velocity in the primary reference, vga, and its visualisation is usually recalled as the velocity triangle, see Fig.1: Figure 1: Schematic of the velocity triangle and the pump and crank angles, α and κ In such representation, it is possible to define the pump angle, α, between the velocity of the secondary and the infinite velocity, and the crank angle, κ, measured between the line connecting the primaries and the projection of the asymptotic velocity on the plane normal to the ecliptic, see Eq.1 2 ´1 cos α “ 1´v8´a tan κ “ tan γsc (1) 2v8 sin i where the flight path angle, γsc, is the angle measured between the osculating velocity of the particle with respect to the velocity of the secondary and i is the inclination measured from the north axis with the angular momentum. Strange et al.19 following the Ocampo’s et al.9 footsteps, observed that expressing the relative velocity at the secondary as a function of such angles, see Eq.2 T v8 “ v8 sin α cos κ ; cos α ; sin α sin κ (2) “ ‰ allowed to develop a method to design flybys in the linked-conic approximation considering the effect of the gravity assist on modifying the direction of the infinite velocity and consequently on changing period, inclination, nodes and orientation of the close approach in latitude and longitude, see Fig.2. 3 90 3:2 60 1:1 2:1 15 5:6 30 3:1 3:4 12 2:3 4:1 9 6 3 0 180 5:1 180 180 4:1 3:1 3 6 2:3 -30 2:1 9 3:4 3:2 5:6 -60 12 1:1 15 -90 -180 -120 -60 0 60 120 180 Figure 2: The v-infinity globe for a 4 km/s of v8 orbit at Europa: all the possible resonant orbits that can be flown considering all the possible orientation of v8 are represented in the red scale, their inclinations in blue and the orientation of the ascending node in green Obviously, not all the possible solutions, represented on the v-infinity globe are reachable at the same time. In fact, starting from a given entry orientation of the asymptotic velocity, the achievable set of orbits, see Fig.3, are those whose close encounter occurs above the minimum altitude or in other terms, whose turning angle is smaller than the maximum one, which is obtained at the minimum close approach, rp, see Eq.3: 2 ´1 δ rpv sin “ 1 ` 8 (3) 2 µ ˆ ˙ 4 Figure 3: The achievable set for a 4:1 resonant orbit at Europa with 4 km/s of v8 and initial crank angle of -30 deg The planar Flyby map The Flyby map captures the effect of the close approach between two Poincare´ sections in the phase space parametrised by pa; T; λ, fq coordinates: the semi-major axis, a, the Tisserand para- menter, T , the longitude of the periapsis in the rotating frame and the true anomaly, f, of the apsis which defines defined the position of the surface of section along the orbit, in particular: ´π if a ¡ 1 f “ 0 if a ă 1 " Sequentially, three steps are required to compute the flyby map F : pa; T; λqB “ F paA;TA; λAq 1.