Preprint) AAS 19-909

(Preprint) AAS 19-909

FLYBY IN THE SPATIAL THREE-BODY PROBLEM

Davide Menzio,∗ and Camilla Colombo†

The spatial flyby map originates from the planar one to extend its applicability 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: prograde (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 specific mission requirements.

INTRODUCTION Since the beginning of space exploration, analysts have widely exploited gravity assist in the trajectory 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 trajectory 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 gravitational 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 particular, 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. †Prof, 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 ą 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. the initial conditions in ﬂyby map parameters at the Poincare´ section, ΣA deﬁned as:

´ ´ ΣA ” pa, T, λqA |aA ‰ 1,R2 aA,TA, λA, fA ą 5RHill where fA “ f ´ ε ` ˘ ( are converted in synodical coordinates through the transformation ϕ, see Eq.4:

´1 ´ px, y, x,9 y9qA “ ϕ aA,TA, λA, fA (4)

2. the initial state is propagated numerically until` the crossing with˘ the following surface of section, in the three-body dynamics see Eq.5:

5 p1´µqpx`µq µpx`µ´1q x: ´ 2y9 “ x ´ 2 2 1.5 ´ 2 2 1.5 ppx`µq `y q ppx`µ´1q `y q (5) y: ` 2x9 “ y ´ p1´µqy ´ pµyq ppx`µq2`y2q1.5 ppx`µ´1q2`y2q1.5

3. at the Poincare´ section, ΣB:

` ` ΣB ” pa, T, λqB |aB ‰ 1,R2 aB,TB, λB, fB ą 5RHill where fB “ f ` 2π ` ε located at least at a distance of 5` Hill radii from the˘ secondary,( the ﬁnal state is transformed back in ﬂyby map parameters, see Eq.6:

` a, T, λ, f B “ ϕ pxB, yB, x9 B, y9Bq (6) THE SPATIAL FLYBY MAP ` ˘ The 3D Flyby map originates from the planar one to explore the spatial dynamics. The logic behind is similar however the increased number of degrees of freedom makes the mathematical treatment of the parameters more complicated, nevertheless more interesting. Therefore, the following paragraphs concern variables handling, initial states generation and domain coverage.

Parametrisation The motivation behind using the Poincare´ section in the planar dynamics lies in the fact that it enables to reduce a set of coordinates bounded to a continuous dynamics in a lower dimensional set of parameters. Campagnola et al.7 observed that using the semi-major axis and the Tisserand parameter as slowly varying constants would have led to decrease further the dimensionality, allowing to study the flyby effect as a one-variable dependent problem. In the spatial dynamics, such reasoning cannot be applied anymore: in fact, adding the z-component requires to handle six coordinates to define an orbit unambiguosly, leaving three free variables, in addition to the two slowly varying parameters and a dimension that is removed by the surface of section. A convenient choice of coordinates in our particular case could be the following: pa, T, i, $, ω, fq where $ is the longitude of the periapsis computed as the sum of Ωrot the right ascension of the ascending node in the rotating frame, see Fig. 4, and ω is the argument of the periapsis. Such formulation is particularly interesting for several main reasons: the inclination can be added to the slowly Figure 4: The representation of right ascension of the varying parameters, while the longitude and ascending node, Ωrot, measured from the line con- the argument of the periapsis can be connecting the primaries in the inertial frame sidered as control variables, the transformation, ϕ, from cartesian to flyby map coordinates is invertible and the variation of the control variable in longitude of the periapsis remains to a neighbourhood of 0, as we will see afterwords.

6 Initial condition

Identifying a set of coordinates for the spatial flyby map that lead to close encounter is not trivial and a broad research among all parameters might be computationally intensive. However, if the orbital energies under study are high enough, it is possible to use the formulation offered by Strange et al.19 to identify an initial guess associated to an osculating orbit and to study its evolution under the three-body dynamics, perturbing the control variables around an unperturbed initial guess. In particular, for a given semi-major axis and a given Tisserand Parameter, it is possible to derive an expression of the infinite velocity as function depending only on the crank angles, see Eq.2, from which it is possible to derive the flyby parameters from Eq.7:

pa, T, i, $,¯ ω,¯ fq “ ϕ prr1, 0, 0s, v8pκq ` r0, 1, 0ssq (7) where the bar accent,¯, refers to unperturbed quantities. Such formulation results extremely convenient allowing to treat the inclination as discrete parameter similarly to the semi-major axis and the orbital energy and to bound the inclination to only those values that lead to a close encounter with the secondary. Fig.5 displays the trend of relevant orbital parameters depending on the crank angle and for a given set of small varying parameters. It can be observed that the crank angle determines unambiguously an osculating orbit but it constitutes a quantity that its difﬁcult to be measured far from the secondary, which is always the situation required by the Poincare´ section. The inclination instead represents the perfect candidate to be discretised, taking into consideration that a given level identiﬁes symmetric trajectories with the respect of the line connecting the primaries, see Fig.5.

Figure 5: On the left, the domain of attainable inclinations (in blue), right ascensions of the ascending node (in green) and argument of the periapsis (in magenta) for 4:1 resonant orbit at Europa with 4 km/s of inﬁnite velocity. On the right, the orbits with 3 degrees of inclination.

Feasible domain

Integrating an osculating orbit in a more complex dynamics results generally into collision or large deviations. Such situation occurs due to the fact that the dynamics switches from a discrete system with point-less attractors and dimension-less sphere of influence, (either infinite or nill), to a continuous one in which considering multiple bodies with their finite dimension at the same time causes from one side the emergence of collision corridors and chaos due to the combined continuous

7 interaction between multiple gravitational ﬁelds that extend way beyond the limit of the sphere of inﬂuence.

Figure 6: The distribution of initial conditions in term of longitude and argument of the periapsis for a 4:1 resonant orbit at 3 degrees of inclination, 4 km/s infinite velocity and -30 degrees of crank angle. In black, all the perturbed initial conditions, in red, the solutions piercing the sphere of influence within the first period, in blue the unperturbed guess.

Being left with two control variables result particularly convenient reducing the chaoticity of the system and improving the understanding of three-body interaction. The idea behind the flyby map is to perturb the osculating orbit in longitude and argument of the periapsis and observe the effect of the flyby on the new orbits. A wise choice of the mesh-grid size and span, see Fig.6, prevents from one side an undesired growth in dimensionalty, and at the same time enable to capture all the different al- teration an initial orbit can undergo considering only those initial conditions that result into a close approach within the first period, see Fig.7, which results in the feasible domain of initial guesses represented in red in Fig.6. It is possible to observe that perturbing an initial condition, in that case the -30 degrees of crank Figure 7: The distribution of close approaches oc- angled, located in the origin, and represented curring from perturbing the initial longitude and with a blue star marker, identifies among all argument of the periapsis of a 4:1 resonant orbit the other initial guesses the osculating values at 3 degrees of inclination and with 4 km/s infinite associated to the orbit having the line of nodes velocity from the same side, obtained for -150 degrees. It is interesting to notice that the symmetry of the unperturbed orbits for the same set of small varying parameters, see Fig.5, is maintained in the three-body problem, with solutions at ˘30 degrees in crank angles presenting the same distribution but mirrored with the respect to the origin when compared to ˘150 degrees ones. Finally, from Fig.7 it can be observed that the distribution of feasible initial guesses creates an

8 order surface of close approaches. Differently from what common sense would have suggested, the distribution is less chaotic than expected although it folds on itself evolving towards the sphere of inﬂuence from the same side.

RESULTS

In the previous section, it has been shown how it is possible to switch from a planar to a spatial formulation, to reduce the coordinates into slowly varying parameter and looping control variables, to perform a grid-search perturbing the free coordinates and to extract from this set a domain of feasible orbits. In this section, the main results are summarised. In particular, the first paragraph shows the selection of output variables, the extraction process of the information from the feasible domain of the spatial flyby map and finally, the identification of families of trajectories and their properties.

Output variables

Having a bi-dimensional variable space makes the representation of the results less intuitive compared to the planar case. In fact, any solution derived from the ﬂyby map, depending at the same times to both free-variables, requires a three-dimensional visualisation that is affected by the intrin- sic shape of feasible domain. In the end, moving from a planar to spatial dynamics causes not only an increase in number of variables, which has already been addressed, but also of the interesting parameters to be displayed. Therefore, an analysis of the output variables and a proper selection of the visualisation tool is required to augment the understanding of the effect of the ﬂyby on a 3D trajectory. First of all, it can be noticed that three-body dynamics conserves the energy of the system, therefore the variation of the Tisserand parameter doesn’t represent an interesting information to be displayed. Different is the story for the semi-major axis and the inclination. In the former case, a common practice favours the resonance ratio, m : n, which is connected to the non-dimensional semi-major axis though Eq.8:

? m “ mod a3, n (8) ´ ¯ and relates the orbital period of the mass-less particle, m to the one of the secondary, n, where m and n are integer numbers. Orbital eccentricity is generally disregarded as it is evolution is mapped by the Tisserand parameter together with semi-major axis and inclination. In the end, regarding the longitude and argument of the periapsis, their evolution might be relevant to build tour sequences of ﬂyby when compared to initial conditions of different sets of small- varying parameters but they are not the focus of this paper and they will be disregarded. From the point of view of the visualisation, the paper favours a representation through contour plots that represent a three-dimensional solution on a bi-dimensional space, avoiding to deal with the perspective that most of the time makes information less accessible and immediate and complicates the reasoning and deductive process.

9 Attainable set

With that in mind, the information held by each output variable can be evaluated in the light of ﬂyby features and cross-compared to understand better how the close approach modify the orbit. Fig.8 collects the most interesting output parameters captured at the post-encounter Poincar e´ section, ΣB.

Figure 8: The characteristic information resulting from perturbing longitude and argument of the periapsis of a 4:1 resonant orbit at 3 degrees of inclination, 4 km/s inﬁnite velocity and -30 degrees of crank angle. From the top to the bottom, the period and the inclination of the post-encounter osculating orbit.

The period and inclination trends can be evaluated considering the distance and orientation dis- tributions for the close approaches, see Fig. 10. An interesting node before proceeding regards the fact that for tidally locked moon, the synodical moon-centered frame is equivalent to body ﬁxed one and therefore rotating coordinates, upon translation, can be directly used to compute latitude and longitude of the mass-less particle with the respect of the secondary.

1 3 11:3 4:1 7:2 0.5 10:3 3:1 0 4 5 6 1 2 5:1 -0.5 4:1 14:3 9:2

-1 13:3 3 -7.5 -5 -2.5 0 2.5 5 7.5 -7.5 -5 -2.5 0 2.5 5 7.5

Figure 9: A zoom-in on the areas with largest variations in period/semi-major axis, left, and inclination, right, displayed with the white box in Fig.8

10 Fig.9 catches the areas which experience the largest variations in semi-major axis and inclination, respectively. As expected, there is a perfect perfect overlap between the two, see Fig. 10, showing what was already know, i.e that the largest deviations occurs for the lowest altitude ﬂybys.

Figure 10: The distribution of the ﬂyby properties resulting from perturbing longitude and argument of the periapsis of a 4:1 resonant orbit at 3 degrees of inclination, 4 km/s inﬁnite velocity and -30 degrees of crank angle. From the top to the bottom, the close approach distance, longitude and latitude.

Two interesting insights can be deduce comparing the distribution of output parameters of in- terest, see Fig.8, with the direction of the close approaches, see Fig. 10. In particular, it can be seen that the flyby doesn’t affect the period when the close approach lies on the line connecting the primaries, associated to a longitude of 0 or 180 degrees. At the same time, no variations on the inclination is experienced when the minimum altitude passage lies on the ecliptic, condition obtained for 0 degrees latitude. Moreover an positive increment in inclination is obtained for close approach occurring in the southern hemisphere, reduction in the northern. Finally from a qualitative estimate, it can be seen that the flyby map variations in period and inclination, once reduced of the colliding solutions, represent a way smaller subset compared to attainable solutions identified by the v-infinity globe mapping, displayed in Fig.3. Situation that was expected considering the prior statement on switching from a linked-conic approximation to three-body dynamics.

Families identiﬁcation The comparison of the period/semi-major axis distribution with the longitude of the minimum altitude passage hinders a last intuition. It is possible to represent the longitude of the minimum

11 altitude passage color-coded based on the orientation of the secondary with the respect of the primary, see Fig. 11 , and distinguish the flybys depending whether they are direct (type I) and pass behind the orbit of Europa, having their close approach in the anti-Jovian hemisphere, in magenta, or retrograde (type II) and cross the orbit of the secondary, passing in front of the moon and having their minimum altitude passage sub-Jovian hemispheres, in green. In the end, one can observe that the contours lines associated to the resonances appears way more condensed for retrograde flyby and more outspread for prograde one. Such information suggests that there exist two families of flyby, see Fig. 11, type I and type II , or prograde and retrograde, or anti-Jovian and sub-Jovian, and that type II is less efficient compared to type I.

Figure 11: The identiﬁcation of two families of ﬂyby, type I and type II represented in magenta and green, respectively.

A quick test to verify the hypothesis, is to represent the variation of one of the output parameters where the other one is preserved. Fig. 12 displays the trend of the post-encounter inclination for a resonant ﬂyby, which preserves the semi-major axis. In particular, it can be observed that at the passage, prograde ﬂybys, in green, induces a large deviation in eccentricity compared to retrograde ones, in magenta.

Figure 12: The identiﬁcation of two families of ﬂyby, type I and type II represented in magenta and green, respectively.

12 CONCLUSION In this paper, the spatial flyby map was derived from the planar one and constitutes a fundamental tool to provide new insights on the third-body interaction in spatial dynamics. In particular, the flyby map has shown the existence of two families of flyby, type I and type II, and that retrograde flyby, type II, are less efficient compared to prograde one, type I. Future works foresee the application of the spatial flyby map to low-energy trajectories and different inclinations.

ACKNOWLEDGEMENT I would like to thank Dr. Stefano Campagnola for the help provided throughout my visiting period at Jet Propulsion Laboratory, for the long discussion on a such interesting topic and for sharing all his knowledge, his tricks and ideas in the most spontaneous way. I would like to thank the COMPASS project and Ermenegildo Zegna Founders Scholarship for sponsoring me during my visiting period at the Jet Propulsion Laboratory. The work performed with this paper has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 679086 - COMPASS).

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