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On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model

F. Topputo

Received: date / Accepted: date

Abstract In this paper two-impulse Earth–Moon transfers are treated in the restricted four-body problem with the Sun, the Earth, and the Moon as primaries. The problem is formulated with mathematical means and solved through direct transcription and multiple shooting strategy. Thousands of solutions are found, which make it possible to frame known cases as special points of a more general picture. Families of solutions are defined and characterized, and their features are discussed. The methodology described in this paper is useful to perform trade-off analyses, where many solutions have to be produced and assessed.

Keywords Earth–Moon transfer low-energy transfer ballistic capture trajectory optimization restricted three-body problem restricted four-body problem

1 Introduction

The search for trajectories to transfer a spacecraft from the Earth to the Moon has been the subject of countless works. The Hohmann transfer represents the easiest way to perform an Earth–Moon transfer. This requires placing the spacecraft on an ellipse having the perigee on the Earth parking orbit and the apogee on the Moon orbit. By properly switching the gravitational attractions along the orbit, the spacecraft’s motion is governed by only the Earth for most of the transfer, and by only the Moon in the final part. More generally, the patched-conics approximation relies on a Keplerian decomposition of the solar system dynamics (Battin 1987). Although from a practical point of view it is desirable to deal with analytical solutions, the two-body problem being integrable, the patched-conics approximation inherently involves hyperbolic approaches upon arrival. This in turn asks for considerable costs (i.e. ∆v maneuvers) to inject the spacecraft into the final orbit about the Moon. Low energy transfers have been found in the attempt to reduce the total cost for Earth–Moon transfers. They exploit the concept of temporary ballistic capture, or weak capture, which is defined in the framework of n-body problems (Belbruno 2004). It has been demonstrated that low-energy transfers may lower the arrival ∆v, so entailing savings on the transfer cost (Belbruno and Miller 1993; Topputo et al. 2005b). A low energy transfer was used in the rescue of the Hiten mission (Belbruno and Miller 1993; Belbruno 2004), and, more recently, a similar transfer has been executed in the mission GRAIL (Hoffman 2009; Roncoli and Fujii 2010; Chung et al. 2010; Hatch et al. 2010). Low energy transfers are being preferred to standard patched- conics transfers for their appealing features, such as the lower cost, the extended launch windows, and the property of allowing for a fixed arrival epoch for any launch date within the launch window (Biesbroek and Janin 2000; Chung et al. 2010; Parker et al. 2011; Parker and Anderson 2011). The extended transfer

Francesco Topputo Department of Aerospace Science and Technology Politecnico di Milano Via La Masa, 34 – 20156, Milano, Italy Tel.: +39-02-2399-8351 Fax: +39-02-2399-8334 E-mail: [email protected] 2 F. Topputo time also benefits operations, check-outs, and out-gassing1. It has been demonstrated that the cost for low energy transfers can be further reduced by using low-thrust propulsion (Belbruno 1987; Mingotti et al. 2009a, 2012). The mission SMART-1 performed such a combination successfully (Schoenmaekers et al. 2001). The low energy approach has also been extended to the case of interplanetary transfers (Topputo et al. 2005b; Hyeraci and Topputo 2010, 2013).

1.1 Low Energy Exterior and Interior Transfers

Low energy transfers can be classified into exterior and interior, according to the trajectory geometry. In the exterior transfers the spacecraft is injected into an orbit having the apogee at approximately four Earth– Moon distances. In this region, a small maneuver, combined with the perturbation of the Sun, makes it possible to approach the Moon from the exterior, and to perform a lunar ballistic capture (Belbruno and Miller 1993). It has been shown that the performances of the transfer can be improved by avoiding the mid-course maneuver (Belbruno and Carrico 2000) and by introducing a lunar gravity assist at departure (Mingotti et al. 2012). The exterior transfers exploit the gravity gradient of the Sun in a favorable way (Yamakawa et al. 1992, 1993). It has been found that these transfers can take place when the orbit is properly oriented with respect to the Sun, the Earth, and the Moon. More specifically, in a reference frame centered at the Earth, with the x-axis aligned with the Sun–Earth line, and with x growing in the anti-Sun direction, the exterior transfers can take place when the apogee lies either in the second or in the fourth quadrant (Bell´o-Mora et al. 2000; Circi and Teofilatto 2001; Ivashkin 2002; Circi and Teofilatto 2006). Exterior transfers can be also viewed in the perspective of Lagrangian points dynamics (Belbruno 1994; Koon et al. 2000). In the coupled restricted three-body problem approximation, the four-body problem that governs the dynamics of the exterior transfers, is split into two restricted three-body problems having Sun, Earth and Earth, Moon as primaries, respectively. It can be shown that the first portion of the exterior transfer is close to the stable/unstable manifolds of the periodic orbits about either L1 or L2 of the Sun– Earth problem. Analogously, the second part (i.e. the lunar ballistic capture) is an orbit defined inside the stable manifold of the periodic orbits about L2 in the Earth–Moon problem (G´omez et al. 2001; Koon et al. 2001; Parker 2006; Parker and Lo 2006). With this approach, second and fourth quadrant transfers are obtained by exploiting the dynamics of the Sun–Earth L1 and L2, respectively. In the interior transfers most of the trajectory is defined within the Moon orbit. These transfers exploit the dynamics of the cislunar region in the Earth–Moon system (Conley 1968; Belbruno 1987; Miele and Mancuso 2001). As the equilibrium point is approached asymptotically (to keep the energy of the transfer low), interior transfers usually have long durations (Bolt and Meiss 1995; Schroer and Ott 1997; Macau 2000; Ross 2003). Also, multiple maneuvers are required to raise the orbit starting from low-altitude Earth parking orbits (Pernicka et al. 1995; Topputo et al. 2005a). Although for such long transfers perturbations may in principle produce significant effects, the solar perturbation is not crucial here, and interior transfers may be defined in the Earth–Moon restricted three-body problem. By postulating that the spacecraft would transit from the Earth region to the Moon region at the energy associated to that of L1, it is possible to quantify the minimum cost for a transfer between two given orbits (Sweetser 1991). Other kinds of interior transfers exploit the resonances between the transfer orbit and the Moon. These require shorter flight times, though their energy level is greater than that of L1 (Yagasaki 2004a,b; Mengali and Quarta 2005).

1.2 Statement of the Problem

In this paper, two-impulse Earth–Moon transfers are studied. These are defined as follows. The spacecraft is initially in a low-altitude Earth parking orbit. A first impulse, assumed parallel to the velocity of the parking orbit, places the spacecraft on a translunar orbit. At the end of the transfer, a second impulse inserts the spacecraft into a low-altitude Moon orbit. This second impulse is again parallel to the velocity of the Moon parking orbit. Thus, the transfer is accomplished in a totally ballistic fashion; i.e. neither mid-course maneuvers nor other means of propulsion are considered during the cruise. A two-impulse transfer strategy involves maneuvering in the proximity of the Earth and Moon. Maneuvering in the direction of the local velocity maximizes the variations of spacecraft energy (Pernicka et al. 1995). Moreover, a mission profile

1 http://moon.mit.edu/design.html, retrieved on 13 February 2012 On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 3 including a low-Earth orbit is consistent with major architecture requirements (e.g., use of small/medium size launchers, in-orbit assembly of large structures, etc. (Perozzi and Di Salvo 2008)). For the sake of a better comprehension of the results presented in this work, the departure and arrival orbits are chosen consistently with the existing literature. A 167 km circular Earth parking orbit, as well as a 100 km circular orbit about the Moon have been extensively used to compute Earth–Moon transfers (see Section 1.3). Thus, although the results presented in this paper can be easily extended to other orbits, for which no qualitative difference is expected, quantitatively, the two-impulse Earth–Moon transfers treated in this paper refer to an initial circular Earth orbit of altitude h = 167km; • i a final circular Moon orbit of altitude h = 100km. • f

1.3 Summary of Previous Contributions

A number of authors have dealt with Earth–Moon transfers between the two orbits defined above (Sweetser 1991; Yamakawa et al. 1992, 1993; Belbruno and Miller 1993; Pernicka et al. 1995; Yagasaki 2004a,b; Topputo et al. 2005a; Mengali and Quarta 2005; Assadian and Pourtakdoust 2010; Peng et al. 2010; Mingotti et al. 2012; Da Silva Fernandes and Marinho 2011). Table 1 represents an attempt of harmonization of all the solutions known to the author. For each solution, the cost (∆v) and the time-of-flight (∆t) are reported2. The two values are given with accuracy of m/s and day, respectively. As different dynamical models can be used to compute Earth–Moon transfers, the reported solutions have to be framed into the model they are achieved with; this is also reported in Table 1. The solutions are labeled in terms of transfer type (interior or exterior), as discussed in Section 1.1; the presence of lunar gravity assists (LGS) is also indicated. The reader can refer to the works cited in Table 1 for details.

1.4 Motivations

Figure 1 reports the solutions of Table 1 in the (∆t,∆v) plane. Each solution taken from the literature corresponds to a point in this plane. As expected, the cost of transfers tends to reduce for increasing transfer time. Moreover, it can be noted that the solutions of different studies tend to cluster together. It is conjectured that these points represent just traces of a more intricate structure, which characterizes the (∆t,∆v) space of solutions of Earth–Moon transfers. Thus, questions arise. Is it possible to reduce known examples to special points of a more general picture? What does the solution structure of two-impulse Earth–Moon transfers look like in the (∆t,∆v) plane? No matter if the transfers are interior or exterior; being different solutions of the same problem, they have to be thought of as special branches of a common picture, which may even contain unknown results.

1.5 Assumptions and Scopes

The aim of this work is to characterize the two-impulse Earth–Moon transfers between two given orbits within a restricted four-body model. In doing so, the same nonlinear boundary value problem as in Yagasaki (2004b) is numerically solved to obtain the results. A number of transfer families are found, including many optimal ones previously reported. The focus is to reveal the structure underlying the points in Figure 1. Thus, patched-conics, interior, and exterior transfers have to be obtained in a common dynamical framework. This means that the dynamical model should suffice to allow the reproduction of all these solutions. As patched-conics, interior, and exterior transfers are defined in the two-, three-, and four-body problems, respectively, a four-body model is to be used. On the other hand, the model has to be sufficiently simple to allow the derivation of a general solutions set, without introducing specific mission parameters (e.g., launch dates, angular parameters on departure and arrival orbits, etc). With reference to Table 1, the planar bicircular restricted four-body problem is chosen. This model is deemed appropriate to catch the basic dynamics characterizing the Sun–Earth–Moon problem (Sim´o et al. 1995; Yamakawa et al. 1992; Yagasaki 2004b; Mingotti et al. 2012; Mingotti and Topputo 2011; Da Silva Fernandes and Marinho 2011).

2 The patched-conics solutions (Hohmann, bielliptic, and biparabolic) have been calculated by assuming the gravitational parameters of the Earth and Moon equal to 3.986 105 km3s−2 and 4.902 103 km3s−2, respectively. × × 4 F. Topputo

Table 1 Known examples of Earth–Moon transfers between a hi = 167 km circular orbit about the Earth and a hf = 100 km circular orbit about the Moon. References are ordered by year of publication. The cost (∆v) is either the sum of the two maneuvers (for two-impulse transfers) or the sum of all required maneuvers (for multi-impulse transfers); ∆t is the time-of- flight. When more than four solutions are published (this is the case of Yamakawa et al. (1992); Yagasaki (2004a,b); Mengali and Quarta (2005); Mingotti and Topputo (2011); Da Silva Fernandes and Marinho (2011)), the four most representative results in terms of (∆v, ∆t) are reported. The fourth column indicates the model in which these solutions are obtained. The fifth column specifies the type of transfer according to the geometry of the trajectory (I: Interior; E: Exterior; LGA: Lunar Gravity Assist). (†) 6 (∗) (‡) Apogee radius of 1.5 10 km; hi = 200 km; ∆v of 573 m/s and 676 m/s added to the results published in Yamakawa × et al. (1992) and Yamakawa et al. (1993), respectively, to achieve a 100 km circular orbit around the Moon; (◦)speed unit of 1024.5m/s and time unit of 4.341 days used to extract data from scaled values available.

Reference ∆v [m/s] ∆t [days] Model Type Biparabolic 3947 Patched two-body, 3D E Hohmann 3954∞ 5 I Bielliptic(†) 4221 90 E Sweetser (1991) 3726 – Restricted three-body, 2D I

Yamakawa et al. (1992)(∗),(‡) 3806 93 Bicircular restricted four-body, 2D E 3783 160 E+LGA 3749 111 E+LGA 3741 180 E+LGA Belbruno and Miller (1993) 3838 160 Full ephemeris n-body,3D E+MGA

Yamakawa et al. (1993)(∗),(‡) 3761 83 Restricted four-body, 2D E+LGA 3776 95 E+LGA 3797 134 E+LGA 3896 337 E+LGA Pernicka et al. (1995) 3824 292 Restricted three-body, 2D I

Yagasaki (2004a)(◦) 3916 32 Restricted three-body, 2D E 3924 31 I 3947 14 I 3951 4 I

Yagasaki (2004b)(◦) 3855 43 Bicircular restricted four-body, 2D E+LGA 3901 32 E 3911 36 E+LGA 3942 24 I Topputo et al. (2005a) 3894 255 Restricted three-body, 2D I 3899 193 I Mengali and Quarta (2005) 3861 85 Restricted three-body, 2D I 3920 68 I 3950 14 I 4005 3 I Assadian and Pourtakdoust (2010) 3865 111 Restricted four-body, 3D E 3880 96 E 4080 59 E

Peng et al. (2010)(∗) 3908 112 Patched restricted three-body, 2D E 3908 121 E 3915 113 E 3920 108 E Mingotti and Topputo (2011) 3793 88 Bicircular restricted four-body, 2D E+LGA 3828 51 E 3896 31 I 3936 14 I Da Silva Fernandes and Marinho (2011) 3857 87 Bicircular restricted four-body, 2D E 3866 86 E 3901 32 I 3944 14 I On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 5

4100 Hohmann Theoretical Min, Sweetser (1991) 4050 Yamakawa et al (1992) Belbruno and Miller (1993) 4000 Yamakawa et al (1993) Pernicka et al (1995) Yagasaki (2004a) 3950 Yagasaki (2004b) Topputo et al (2005) 3900 Mengali and Quarta (2005) v (m/s) ∆ Assadian and Pourtakdoust (2010) 3850 Peng et al (2010) Mingotti et al (2011) 3800 Da Silva and Marinho (2011)

3750

3700 0 50 100 150 200 250 300 ∆t (days)

Fig. 1 Solutions of Table 1 reported on the (∆t, ∆v) plane. The horizontal line represents the theoretical minimum (Sweetser 1991); the vertical line separates the region investigated in this paper (white) from the rest of the solutions space.

Another parameter to set is the maximum transfer duration. This has to be limited to avoid the indefinite growing of possible solutions. In literature, solutions with a time-of-flight of up to 300 days exist (see Table 1 and Figure 1), and Earth–Moon transfers requiring as much as two years have been found (Bolt and Meiss 1995). However, in this work, a lunar transfer requiring hundreds of days is not deemed a viable option for real missions. For this reason, the characterization of two-impulse Earth–Moon transfers is restricted to those solutions whose duration is less than 100 days (i.e. the search is limited to the white area in Figure 1). Although this could appear too restrictive, it will be shown that the selected search space possesses a sufficiently rich structure, which may contain novel transfer families. In this search space, the front of Pareto-efficient solutions is reconstructed. (These are the solutions showing the best balance between cost and transfer time; see Definition 2.) The aim is to build a catalogue from which preliminary estimates on the transfer cost could be extracted if the transfer time was specified (or vice-versa). Summarizing, the scopes of the paper are: i. to optimize two-impulse Earth–Moon transfers in the planar bicircular restricted four-body problem; ii. to reconstruct the solution space in the (∆t,∆v) plane, ∆t 100days, and to extract the set of Pareto- ≤ efficient solutions; iii. to characterize the solution structure by identifying families of transfer orbits, some of which may be unknown in literature. The remainder of the paper is structured as follows. In Section 2 the background notions are recalled (restricted three-/four-body problem and temporary ballistic capture). In Section 3 the optimization problem is treated in terms of first guess solutions, optimization algorithms, and implementation issues. In Section 4 the optimized solutions are presented and discussed on a global basis. Specific results are characterized and families of solutions are defined in Section 5. Concluding, final considerations are given in Section 6.

2 Background

2.1 Planar Circular Restricted Three-Body Model

In the Planar Circular Restricted Three-Body Problem (PCRTBP) two primary bodies P1, P2 of masses m1 > m2 > 0, respectively, move under mutual gravity on circular orbits about their common center of mass. The third body, P , assumed of infinitesimal mass, moves under the gravity of the primaries, and 6 F. Topputo in the same plane. The motion of the primaries is not affected by P . In the present case, P represents a spacecraft, and P1, P2 represent the Earth and the Moon, respectively. Let µ = m2/(m1 + m2) denote the mass ratio of P2 to the total mass. The motion of P relative to a co-rotating coordinate system (x,y) with the origin at the center of mass of the two bodies, and in normalized distance, mass, and time units, is described by (Szebehely 1967)

∂Ω ∂Ω x¨ 2˙y = 3 , y¨ +2˙x = 3 , (1) − ∂x ∂y where the effective potential, Ω3, is given by

1 2 2 1 µ µ 1 Ω3(x,y) = (x + y ) + − + + µ(1 µ), (2) 2 r1 r2 2 −

2 2 1/2 2 2 1/2 with r1 = [(x+µ) +y ] , r2 = [(x+µ 1) +y ] as P1, P2 are located at ( µ, 0), (1 µ, 0), respectively. − − − The motion described by Eqs. (1) has five equilibrium points Lk, k = 1,..., 5, known as the Euler– Lagrange libration points. Three of these, L1, L2, L3, lie along the x-axis;, the other two points, L4, L5, lie at the vertices of two equilateral triangles with common base extending from P1 to P2. The system of differential equations (1) admits an integral of motion, the Jacobi integral,

2 2 J(x,y, x,˙ y˙)=2Ω3(x,y) (˙x +y ˙ ). (3) −

The values of the Jacobi integral at the points Li, Ci = J(Li), i = 1,..., 5, satisfy C1 > C2 > C3 > C4 = C5 = 3. Table 2 reports the collinear libration points location and their Jacobi energy for the Earth–Moon system. The projection of the energy manifold

(C) = (x,y, x,˙ y˙) R4 J(x,y, x,˙ y˙) = C , (4) J { ∈ | } onto the configuration space (x,y) is called Hill’s region. In the PCRTBP, the motion of P is always confined to the Hill regions of the corresponding Jacobi energy C. The boundary of Hill’s region the a zero-velocity curve. Hill’s regions vary with the Jacobi energy C (see Figure 2). In this model, for P to transfer from P1 to P2 (and vice versa), the condition J

Table 2 Location of the collinear Lagrange points, xLi , and their Jacobi energy, Ci, i = 1, 2, 3, for µ = 0.0121506683.

xLi Ci

L1 0.8369147188 3.2003449098 L2 1.1556824834 3.1841641431 L3 1.0050626802 3.0241502628 −

2.2 Planar Bicircular Restricted Four-Body Model

The Planar Bicircular Restricted Four-Body Problem (PBRFBP) incorporates the perturbation of a third primary, P3, the Sun in our case, into the PCRTBP described above. P3 is assumed to revolve in a circular orbit around the center of mass of P1–P2. This model is not coherent as all the primaries are assumed to move in circular orbits. Nevertheless, it catches basic insights of the real four-body dynamics (Castelli 2011); this holds as the eccentricities of the Earth and Moon orbits are 0.0167 and 0.0549, respectively, and the Moon orbit inclination on the ecliptic is 5deg. On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 7

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P P2 P P2 1 P1 2 P1

L1 L1 L2 L3 L1 L2

(a) C2 J < C1. (b) C3 J < C2. (c) C4,5 J < C3. ≤ ≤ ≤ Fig. 2 Hill’s regions (white) and forbidden regions (grey) for varying energies.

The equations of motion of the PBRFBP are (Sim´oet al. 1995; Yagasaki 2004b)

∂Ω ∂Ω x¨ 2˙y = 4 , y¨ +2˙x = 4 , (5) − ∂x ∂y

where the four-body effective potential, Ω4, reads

ms ms Ω4(x,y,t) = Ω3(x,y) + 2 (x cos(ωst) + y sin(ωst)). (6) r3(t) − ρ

In Eq. (6) t is the time, ms is the mass of the Sun, ρ is the distance between the Sun and the Earth–Moon barycenter, and ωs is the angular velocity of the Sun in the Earth–Moon rotating frame; the phase angle of the Sun is θ(t) = ωst. Thus, the present location of the Sun is (ρ cos(ωst),ρ sin(ωst)), and therefore the Sun-spacecraft distance is

2 2 1/2 r3(t)=[(x ρ cos(ωst)) +(y ρ sin(ωst)) ] . (7) − − It is worth pointing out that the Sun is not at rest in the reference frame adopted in Eqs. (5). This causes the loss of autonomy, and makes the PBRFBP a time-dependent model (see Eqs. (6)–(7)). This is a great departure from the PCRTBP as both the equilibrium points and the Jacobi integral vanish. Table 3 reports the value of the physical constants used in this study and their associated units of distance, time, and speed according to the normalization of the PCRTBP.

Table 3 Physical constants used in this work (taken from Sim´oet al. (1995)). The constants used to describe the Sun perturbation (ms, ρ, ωs) meet the length, time, and mass normalization of the PCRTBP. The units of distance, time, and speed are used to map scaled quantities into physical units.

Symbol Value Units Meaning µ 1.21506683 10−2 — Earth–Moon mass parameter × 5 ms 3.28900541 10 — Scaled mass of the Sun ρ 3.88811143 × 102 — Scaled Sun–(Earth+Moon) distance × −1 ωs 9.25195985 10 — Scaled angular velocity of the Sun − × 8 lem 3.84405000 10 m Earth–Moon distance × −6 −1 ωem 2.66186135 10 s Earth–Moon angular velocity × Re 6378 km Earth’s radius Rm 1738 km Moon’s radius hi 167 km Altitude of departure orbit hf 100 km Altitude of arrival orbit DU 3.84405000 108 m Distance unit TU 4.34811305× days Time unit VU 1.02323281 103 ms−1 Speed unit × 8 F. Topputo

2.3 Ballistic Capture

For the analysis below it is convenient to define the ballistic capture of P by P2; i.e. the ballistic capture of the spacecraft by the Moon. Let x(t) be a solution of Eqs. (5), x(t)=(x(t),y(t), x˙(t), y˙(t)), and let H2(x) be the Kepler energy of P with respect to P2,

1 2 µ H2 = v2 , (8) 2 − r2

3 where v2 is the speed of P relative to a P2-centered inertial reference frame ,

2 2 1/2 v2 = [(˙x y) +(˙y + x + µ 1) ] . (9) − −

Definition 1 (Ballistic captures) P is ballistically captured by P2 at time t1 if H2(x(t1)) 0. It is temporary ≤ ballistically captured by P2 for finite times t1,t2, t1 0 for tt2. It is permanently captured by P2 if it is ballistically captured for all times t t3, for a finite time t3. ≥ These are the standard definitions of capture (see Belbruno (2004)). In the frame of Earth–Moon trans- fers, P can experience temporary ballistic captures only. For the capture to be permanent, a dissipative force must act upon P (De Melo and Winter 2006). To assess if Earth–Moon transfers exhibit temporary ballistic capture upon Moon arrival it is sufficient to check the sign of the right-hand side of Eq. (8) at the final time. (See Belbruno (2004); Belbruno et al. (2008); Topputo et al. (2008); Topputo and Belbruno (2009); Belbruno et al. (2010); Circi (2012) for more details on ballistic capture). The angular momentum is another two-body indicator that can be used to understand the motion upon Moon arrival. As the motion is planar, the angular momentum h2 of P with respect to P2 is perpendicular to the plane (x,y), and its magnitude is given by

h2 =(x + µ 1)(˙y + x + µ 1) y(˙x y). (10) − − − − At arrival, direct or retrograde orbits with respect to the Moon can be detected by simply checking the sign of the right-hand side of Eq. (10) at the final time.

3 Optimization of Two-Impulse Earth–Moon Transfers

Two impulse transfers are defined as follows. The spacecraft is assumed to be in a circular orbit about the Earth of scaled radius ri =(Re + hi)/DU. At the initial time, ti, an impulse of magnitude ∆vi injects the spacecraft into a translunar trajectory; ∆vi is aligned with the local circular velocity whose magnitude is (1 µ)/r . At the final time, t , an impulse of magnitude ∆v inserts the spacecraft into the final orbit − i f f aroundp the Moon. This orbit has scaled radius rf =(Rm + hf )/DU; ∆vf is aligned with the local circular velocity whose magnitude is µ/rf . The total cost of the transfer is ∆v = ∆vi + ∆vf and the transfer time is ∆t = t t . p f − i

3.1 Definition of the Optimization Problem

Let xi =(xi,yi, x˙ i, y˙i) be the initial transfer state. This is at a physical distance ri from the Earth, and has the velocity aligned with the local circular velocity if

(x + µ)2 + y2 r2 =0, (11) i i − i (x + µ)(˙x y ) + y (˙y + x + µ)=0, (12) i i − i i i i are both verified (Yagasaki 2004b). (The reader can refer to Eq. (34) in Appendix 1 for the transformation from the synodic to the Earth-centered, inertial frame.) For brevity, these two initial conditions are indicated

3 See Appendix 1 for the coordinate transformation behind (9) and (10). On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 9 as ψi(xi) = 0, where ψi is the two-dimensional vector valued function of xi given by the left-hand sides of (11)–(12). Under these initial conditions, the cost for the initial maneuver is

1 µ ∆v = (˙x y )2 +(˙y + x + µ)2 − , (13) i i − i i i − r r p i which can be thought as a scalar, nonlinear function of xi; i.e. ∆vi(xi). Analogously, let xf =(xf ,yf , x˙ f , y˙f ) be the final transfer state. In order to arrive at a circular orbit of radius rf around the Moon, with the velocity vector aligned with the local circular velocity, xf must satisfy

(x + µ 1)2 + y2 r2 =0, (14) f − f − f (x + µ 1)(˙x y ) + y (˙y + x + µ 1)=0. (15) f − f − f f f f −

Equations (14)–(15) are denoted as ψf (xf ) = 0; the two-dimensional vector valued function ψf is defined by the left-hand sides of (14)–(15). Under these final conditions, the arrival maneuver can be quantified by

2 2 µ ∆vf = (˙xf yf ) +(˙yf + xf + µ 1) , (16) q − − − r rf which is a scalar, nonlinear function of the final state; i.e. ∆vf (xf ). We can now proceed to state the optimization problem as follows.

Statement of the problem. Let x(t) = ϕ(x ,t ; t) be a solution of (5), integrated from (x ,t ) to t t . i i i i ≥ i The optimization problem for two-impulse Earth–Moon transfers consists of finding x ,t ,t , t >t , such that { i i f } f i ψi(xi) = 0, ψf (xf ) = 0, where xf = ϕ(xi,ti; tf ), and the function

∆v(xi,ti,tf ) = ∆vi(xi) + ∆vf (ϕ(xi,ti; tf )) (17) is minimized.

3.2 Forming an Initial Guess

From (17) it can be inferred that a two-impulse Earth–Moon transfer is uniquely specified once the six scalars x ,t ,t are given. In principle, one can guess the initial state, the initial time, and the final time, { i i f } and check if the boundary conditions (11)–(12) and (14)–(15) are satisfied, and the objective function (17) is minimized. In the most likely event in which either the boundary conditions are violated or the objective function is not minimized, the guess can be used to define an initial solution, or first guess solution. This can be treated in an optimization framework, where both the boundary conditions and the minimization of the objective function are enforced. Since this work aims at reconstructing the totality of solutions of two-impulse Earth–Moon transfers, thousands of initial guesses have to be processed to find locally optimal solutions. This can be done by performing a systematic search (i.e. a grid sampling) within the space in which the six scalars are defined. However, systematic searches suffer from the curse of dimensionality, and the high nonlinearities character- izing most Earth–Moon transfers would require a fine sampling of the variables. From this perspective, it would be desirable to guess as few numbers as possible. A transformation has been developed to define first guess solutions by specifying four scalars only. With reference to Figure 3, these quantities are: α, an angle defined on the Earth circular parking orbit; β, the initial-to-circular velocity ratio; ti, the initial time; δ, the transfer duration.

Given α,β,t , δ , the construction of initial guess solutions is straightforward. Let r0 = r and v0 = { i } i β (1 µ)/r0, the initial transfer state, x0(α, β)=(x0,y0, x˙ 0, y˙0), reads − p x0 = r0 cos α µ, y0 = r0 sin α, x˙ 0 = (v0 r0)sin α, y˙0 =(v0 r0)cos α. (18) − − − − In Eq. (18), α determines the location of the initial point on the circular orbit, and β specifies the initial velocity, which is already aligned with the circular velocity. It is easy to verify that x0(α, β) automatically satisfies Eqs. (11)–(12); i.e. ψ (x0(α, β)) = 0, α, β . i ∀{ } 10 F. Topputo

v 0 x0(α, β ) x˜f = ϕ(x0, t i; ti + δ)

α

r Earth r0 Moon f

Fig. 3 Direct shooting of initial guess solutions.

The final state is defined by integrating x0 in the time span [ti, ti+δ]; i.e.x ˜f (α,β,ti, δ) = ϕ(x0(α, β),ti; ti+ δ). If ψ (˜x ) = 0 and ∆v(x0(α, β),t , δ) is minimum, then the combination α,β,t , δ produces an optimal f f i { i } solution; otherwise, it defines an initial solution for a subsequent optimization step (see Figure 3). The search of first guess solutions can be further specialized by defining the set of admissible parameters. The selection of lower and upper bounds for each parameter sets the four-dimensional hypercube to sample (this is the reason why δ is the transfer duration and not the final time; (Armellin et al. 2010)). As for the first parameter, it is straightforward that α [0, 2π]. The second parameter is allowed to be β [1, √2], where ∈ ∈ β = 1 sets the circular velocity, while β = √2 defines the Earth-escape velocity. As for ti, setting the initial time specifies the initial angular position of the Sun through θ = ωst (see Eqs. (6)–(7)). Thus, t [0, 2π/ωs] i i i ∈ yields θ [0, 2π]; i.e. any possible relative configuration of Sun, Earth, and Moon is considered. The transfer i ∈ duration δ [0,T ] is bounded by T , which is the maximum prescribed transfer time. First guess transfers ∈ with durations of up to 100 days are obtained by setting T =∼ 23 T U (see Table 3). Let Iα, Jβ, Kti , Lδ be the number of points in which the domains of α, β, ti, δ defined above are discretized, respectively. A direct shooting of first guess solutions is implemented through Algorithm 1. The generic first guess orbit, γ(ijkl), can be used to initiate the subsequent trajectory optimization.

Algorithm 1 Pseudo code for first guess solution generation. for i = 1 Iα do (i) → α = 2π(i 1)/(Iα 1) for j = 1 −J do − → β β(j) =1+(√2 1)(j 1)/(J 1) − − β − for k = 1 Kti do (k) → t = 2π/ωs(k 1)/(Kt 1) i − i − for l = 1 Lδ do (l) → δ = T (l 1)/(Lδ 1) (ijkl) − (i)− (j) (k) (k) (l) γ = ϕ(x0(α ,β ), ti ; ti + δ), δ δ end for { ≤ } end for end for end for

3.3 Solution by Direct Transcription and Multiple Shooting

Optimal two-impulse Earth–Moon transfers are found with a direct transcription and multiple shooting approach. Direct transcription maps a differential optimal control problem into an algebraic nonlinear programming (NLP) problem; see Enright and Conway (1992); Betts (1998) for details. This method has shown robustness and versatility when used to derive trajectories defined in highly nonlinear vector fields (Mingotti et al. 2009a, 2011a,b, 2012). Moreover, it does not require the explicit derivation of the necessary conditions of optimality, and it is less sensitive to variations of the first guess solution. On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 11

Let the PBRFBP Eqs. (5) be written in the first-order formx ˙ = f(x,t), and let x(t) be a solution. This is discretized over a uniform time grid, which is made up by N points such that t1 = ti, tN = tf , and j 1 t = t1 + − (t t1), j =1,...,N. (19) j N 1 N − − Let xj =(xj ,yj, x˙ j , y˙j ) be the solutions at the j-th mesh point; i.e. xj = x(tj), j =1,...,N. The multiple shooting method requires the dynamics (5) to be integrated within the N 1 intervals [t , t ] by taking − j j+1 xj as initial condition (see Figure 4). The continuity of position and velocity requires the defects ζ = ϕ(x ,t ; t ) x , j =1,...,N 1 (20) j j j j+1 − j+1 − to vanish. Let the NLP variable vector be defined as

y = x1, x2,..., x ,t1,t , (21) { N N } where the initial, final time, t1, tN , are let to vary, to allow the optimizer to converge to an optimal initial configuration and transfer duration. Let ψ1 = ψi(x1) and ψN = ψf (xN ). Then the vector of nonlinear equality constraints is given by c(y) = ψ , ζ , ζ ,..., ζ − , ψ . (22) { 1 1 2 N 1 N } The trajectory optimization has to avoid solutions which impact either the Earth or the Moon. This is imposed at each mesh point through 2 2 2 2 2 2 (Re/DU) (x + µ) + y < 0, (Rm/DU) (x + µ 1) + y < 0, j =1,...,N. (23) − j j − j − j Moreover, the transfer duration has to be strictly positive; i.e. t1 t < 0. Thus, the vector of nonlinear − N inequality constraints is g(y) = η , η ,..., η ,τ , (24) { 1 2 N } where ηj = η(xj ) is the two-dimensional vector valued function defined by the left-hand sides of Eqs. (23) and τ = t1 t . The scalar objective function is − N f(y) = ∆v1 + ∆vN , (25) where ∆v1 = ∆vi(x1) and ∆vN = ∆vf (xN ), defined in Eq. (13) and (16), respectively. It is easy to verify that y is (4N + 2)-dimensional, c is 4N-dimensional, and g is (2N + 1)-dimensional; with inequality constraints satisfied, the problem has 4N + 2 variables and 4N constraints.

NLP problem. Optimal, two-impulse Earth–Moon transfers are found by solving the NLP problem: min f(y) subject to c(y) = 0, y (26) gj (y) < 0, j =1,..., 2N +1.

3.4 Continuation of Optimal Solutions

Once an optimal solution is found, it would be desirable to continue it in order to reconstruct, locally, the whole solution space. This is crucial to derive the totality of solutions and characterize them. To do that, it is convenient to use the transfer duration as continuation parameter. Suppose an optimal solution to problem (26) has been found: y = x1,..., x ,t1,t . This solution has { N N } transfer duration ∆t = tN t1. The aim is to use y to derive a new optimal solution having a fixed duration ′ − ∆t = ∆t+δt, where δt has to be small enough to allow the convergence to a new optimal solution. Thus, the new problem has to enforce the constraint on the transfer duration. This is done through the new condition ′ σ = t t1 ∆t =0, (27) N − − that yields the augmented, (4N + 1)-dimensional vector of equality constraints ′ c (y) = ψ , ζ , ζ ,..., ζ − , ψ ,σ . (28) { 1 1 2 N 1 N }

Continuation problem. Optimal solutions are continued in transfer duration by solving the NLP problem: ′ min f(y) subject to c (y) = 0, y (29) gj (y) < 0, j =1,..., 2N +1. 12 F. Topputo

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x1 . . . xN

. . . x r r 3 i Earth f Moon x4

x x x4 N−2 ζ ζ3 N−1 ζ1 x2 xN−1 x . . . 1 x3 ζ2 xN ζN−2 t1 t2 t3 t4 tN−2 tN−1 tN t

Fig. 4 Direct transcription and multiple shooting optimization scheme.

3.5 Implementation and Verification

The optimization of two-impulse Earth–Moon transfers described above has been implemented within a numerical framework. The initial value problems (e.g., ϕ(xj ,tj ; tj+1)) are solved with a variable-order, multi-step Adams–Bashforth–Moulton scheme (Shampine and Gordon 1975) with absolute and relative − tolerances set to 10 14. This stringent tolerance is imposed to have accurate solutions within the N 1 − intervals in which the solution is split (when the N 1 pieces are put together, the solution has to be still − accurate — provided that the equality constraints are satisfied with equally accurate tolerance). The first guess solutions have been generated with the algorithm sketched in Section 3.2. The four- dimensional search space has been sampled with Iα = 361, Jβ = 101, Kti = 13, and Lδ = 101. Thus, about 4.7 107 first guess orbits γ(ijkl) have been generated. × The NLP problems (26) and (29) have been solved with a sequential quadratic programming algorithm implementing an active-set strategy (Gill et al. 1981) with tolerance on equality and inequality constraints − of 10 10. The gradient of the objective function (25) as well as the Jacobians of the equality and inequality constraints (Eqs. (22), (24), (28)) are calculated using a second-order central difference approximation. Once a solution to problem (26) is obtained, the entire family is computed by continuation (problem (29)) with − values of δt of about 5 10 2 T U. In Figure 5 the convergence history of a sample solution is reported. It can × be seen that, although the final position of the initial guess misses the Moon by more than one Earth–Moon distance, the algorithm is still able to converge to an optimal solution. Optimal solutions are verified a posteriori. If the solution is deemed sufficiently accurate, it is stored, otherwise it is discarded. The check is made by numerically integrating the optimal initial condition within the optimal time interval; i.e.

xf = ϕ(x1,t1; tN ). (30)

The final state xf has to be as close as possible to the optimal final state xf . For a solution to be accepted −8 it is required that max ψ1(x1) , ψN (xf ) 10 ; i.e. the real trajectory violates the boundary conditions − {| | | |} ≤ by less than 10 8 scaled units ( 3m in length and 0.01mm/s in speed). Moreover, it is also checked that ∼ ∼ the real optimal solution impacts neither the Earth nor the Moon. As the impact-free constraint is imposed only on the mesh points through Eqs. (23), the verification is done from end to end, by re-integrating the solution starting from the optimal initial condition; i.e.

η (x(t)) < 0, t [t1, t ], j =1, 2, (31) j ∈ N

with x(t) = ϕ(x1,t1; t), and η(x) as from Section 3.3, Eqs. (23). On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 13

2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 y y y −0.5 −0.5 −0.5 −1 −1 −1 −1.5 −1.5 −1.5 −2 −2 −3 −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 x x x

(a) Iter 0; max cj = 1.866. (b) Iter 3; max cj = 0.475. (c) Iter 6; max cj = 0.203.

1.5 1 1.5 1 1 0.5 0.5 0.5 0 0 0 y y y −0.5 −0.5 −0.5 −1 −1 −1 −1.5 −1.5 −1.5 −2 −2 −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 x x x

(d) Iter 10; max cj = 0.078. (e) Iter 20; max cj = 0.397. (f) Iter 29; max cj = 3.9e-11.

Fig. 5 Iterative solutions of the NLP problem (26). Each figure reports the iteration and the maximum equality constraint violation (max cj ). The first guess (Iter 0) is in Figure 5(a). This is generated with the following set of parameters: α = 1.5π, β = 1.41, ti = 0, δ = 7 (see Section 3.2). This solution has N = 23 mesh points (red bullets in the figures); this yields 94 variables and 92 equality constraints. Convergence has been achieved in 29 iterations. The computational time is 32.4 s on an Intel Core 2 Duo 2 GHz with 4 GB RAM, Mac OS X 10.6.

4 Optimal Solutions

4.1 The Global Picture

The optimal solutions found are reported in Figure 6. Each point of the figure is associated to an optimal solution with duration ∆t and cost ∆v (these two quantities are re-scaled using the time and speed units in Table 3). In total, approximately 5 105 optimal solutions have been found, many of which are not displayed × in Figure 6 (having ∆v > 4150m/s or ∆t > 100days). Figure 6 suggests that the solutions are organized into families, and these families constitute a well-defined structure in the (∆t, ∆v) plane, thus confirming the conjecture in Section 1.4. The families have been achieved by continuation with the procedure shown in Section 3.4. This procedure generates also spurious points not belonging to any known structure. These are the points in which continuation of solutions fails, and are not considered in the analysis below. It is noted that the existence of optimal solutions is not guaranteed for every combination of (∆t, ∆v). This explains the existence of empty areas between families.

A question arises: how do these optimal solutions relate to those found in previous studies? To answer this question, the reference solutions in Figure 1 are superimposed on the points of Figure 6. A good matching is found between the solutions found and those taken from the literature. In particular, the reference solutions appear to be sample points of the reconstructed solutions set, and, most importantly, almost all of them are located in the regions about the local minima of the function ∆v(∆t).

For fixed transfer time, the solutions having a lower cost are of more interest in the investigation below. To ease the analysis, the total set of solutions in Figure 6 is subdivided into two regions that enclose “Short” and “Long”transfers (‘S’ and ‘L’ squares in Figure 6, respectively). Considering one region at a time helps characterizing the families of solutions, and allows us to better focus on the solutions deemed of interest. This is done in Section 5. 14 F. Topputo

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Fig. 6 Optimal two-impulse Earth–Moon transfers, shown in the (∆t, ∆v) plane. The two regions S and L enclose “Short” and “Long” transfer solutions, respectively. The reference solutions in Figure 1 are superimposed (legend in Figure 1).

4.2 Pareto-Efficient Solutions

Although Figure 6 is generated within a simplified, planar gravitational model, the total solutions set may serve to mission designers as a catalogue: first guess values for duration and cost can be extracted for preliminary estimations, and the corresponding solutions may be refined in more complete models. In this perspective, it would be desirable to extract those solutions for which the balance between ∆t and ∆v is optimal. In the framework of multi-objective optimization, this means computing the Pareto front of solutions, made up by the nondominated points.

Definition 2 (Pareto optimality) Let yi and yj be two solutions of Problem (26) (or Problem (29)) defined as in (21); let ∆v(y) = f(y) in (25) and let ∆t(y) = t t1. The solution y strictly dominates y , or y y , N − i j i ≻ j if the following conditions are both satisfied:

i) ∆t(y ) ∆t(y ) and ∆v(y ) ∆v(y ), i ≤ j i ≤ j ii) ∆t(yi) < ∆t(yj ) or ∆v(yi) < ∆v(yj ). The Pareto front is the set of points that are not strictly dominated by another point.

In Figure 7, the Pareto front for the set of solutions in Figure 6 is reported. In practice, the points belonging to the Pareto front correspond to the solutions having the best balance between cost and time. This is worthwhile for real Earth–Moon transfers, provided that the mission constraints do not prevent the choice of a solution belonging to the Pareto front. As a matter of example, the coordinates of the ten sample Pareto-efficient solutions shown in Figure 7 are reported in Table 4. Moving from left to right, the solutions range from the Hohmann-like transfer (i) to a particular exterior low energy transfer (x), which is the cheapest solution found in the entire work (∆v = 3769.3m/s). The transfers corresponding to the points in Figure 7, except for solution (vii) whose shape is similar to that of (viii), are reported in Figure 8 in the Earth-centered, inertial frame. The conversion to the Earth-centered inertial frame as well as the initial condition, xi =(xi,yi, x˙ i, y˙i), and the initial, final time, ti, tf , for each of these ten solutions, are reported in Appendix 1. These numerical values are given with high accuracy (15 decimals) to allow an independent reproduction of the results. On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 15

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Fig. 7 Pareto front (black) associated to the global set of optimal solutions (grey). The coordinates of the ten sample Pareto-efficient points (red dots) are reported in Table 4.

Table 4 Example of Pareto-efficient solutions in Figure 7; ∆t in days, ∆v in m/s. The initial condition and initial, final time for each of these solutions are reported in Appendix 1.

Point (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) ∆t 4.6 14.4 30.6 31.9 43.6 51.3 61.9 64.6 66.7 82.6 ∆v 3944.8 3936.6 3892.7 3887.4 3840.0 3828.6 3828.9 3805.9 3801.4 3769.3

4.3 Angular Momentum at Final Time

The problem statement and, more specifically, the final boundary conditions (14)–(15), enforce the final transfer state, xf =(xf ,yf , x˙ f , y˙f ), to be at a physical distance rf from the Moon and to have the velocity parallel to the local circular velocity. Yet, these conditions do not specify the direction of the final velocity, so allowing the possibility of having both direct and retrograde final orbits around the Moon. These are discriminated by computing the angular momentum with respect to the Moon associated to xf , h2,f = h2(xf ), through Eq. (10). Direct orbits are those orbits for which h2,f > 0; retrograde orbits have instead h2,f < 0. The sign of h2,f for the total solution set in Figure 6 is highlighted in Figure 9. It can be noticed that, for fixed transfer time, solutions belonging to the same family have different costs according to the sign of h2,f . Moreover, there is no apparent way to establish a priori whether the direct or retrograde final orbits are more convenient without having fixed the transfer time. For instance, by considering the local minima in Figure 9(a), we can infer that acquiring a direct final orbit around the Moon is more convenient for ∆t . 25days; retrograde final orbits become more convenient in the range 25 . ∆t . 72days (see also Figure 9(b)). For 72 . ∆t . 90days the ∆v of direct final orbits is lower than that of retrograde orbits, and this conditions is again reversed for 90 . ∆t . 100days. In summary, the analysis of the final angular momentum at the Moon shows that the total set of solutions is organized into a nontrivial structure, which can be understood by a thorough characterization of the families.

4.4 Kepler energy at Final Time

The spacecraft may (or may not) experience ballistic capture upon Moon arrival. To detect those solutions that exhibit ballistic capture it is sufficient to compute the Kepler energy with respect to the Moon at the final time, H2,f = H2(xf ), through Eqs. (8)–(9). If H2,f < 0, then P is ballistically captured by the Moon 16 F. Topputo

(a) Sol (i). (b) Sol (ii) (c) Sol (iii)

(d) Sol (iv) (e) Sol (v) (f) Sol (vi)

(g) Sol (viii) (h) Sol (ix) (i) Sol (x)

Fig. 8 The Pareto-efficient Earth–Moon transfers indicated in Figure 7 shown in the Earth-centered inertial frame (the Moon orbit is dashed). Solution (vii) is very similar to solution (viii) and therefore it is not shown. according to Definition 1 (it is straightforward that the Kepler energy with respect to the Moon is positive at departure). If H2,f > 0, there is no ballistic capture at arrival. In Figure 10(a) the sign of H2,f is highlighted for the total set of solutions in Figure 6 (ballistic captured solutions are reported in red). It can be noticed that only a subset of the total solution set performs lunar ballistic capture at arrival. These are those cases located in the bottom-right part of the (∆t, ∆v) plane; i.e. the solutions with long transfer time and low cost. With reference to Figure 8, only solutions (8), (9), and (10) are ballistically captured at the Moon. While solution (10) belongs to the well-known class of exterior low energy transfers (Belbruno and Miller 1993; Belbruno 2004; Parker et al. 2011; Parker and Anderson 2011; Yamakawa et al. 1992, 1993), the other two (solutions (8) and (9)) deserve further analysis.

4.5 Jacobi energy at Final Time

In the PBRFBP, in which the optimal two-impulse Earth–Moon transfers are achieved, the Jacobi energy is no longer an integral of motion. However, computing the Jacobi energy at the final time, Jf = J(xf ) via Eq. (3), helps understanding the geometry of Hill’s regions upon Moon arrival, where the perturbation of the Sun can be neglected. Moreover, Jf is an indicator of the energy level of interior transfers, where the Sun perturbation has little influence (Yagasaki 2004a,b; Mengali and Quarta 2005). In addition, the exterior low energy transfers can be formulated with the patched restricted three-body problem approximation, which fixes the Jacobi energy in the Earth–Moon system (Koon et al. 2001; Parker 2006; Parker and Lo 2006; Mingotti and Topputo 2011; Mingotti et al. 2011b, 2012). On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 17

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(a) S region. (b) L region.

Fig. 9 Angular momentum with respect to the Moon at the final time, h2,f , in the (∆t, ∆v) plane. Direct (blue) and retrograde (red) final orbits around the Moon.

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(a) Kepler energy at final time (H2,f < 0 red, H2,f > 0 blue). (b) Jacobi energy at final time (case b) red, c) green, d) blue).

Fig. 10 Kepler and Jacobi energies at the final time.

The value of Jf for the total solution set in Figure 6 has been calculated through Eq. (3), and the four energy ranges mentioned in Section 2.1 (Figure 2) have been highlighted in Figure 10(b). First of all, it can be noticed that no solutions belonging to case a) are found. This is due to the fact that, in the PCRTBP, orbits defined at energies C2

5 Characterization of Transfers

In this section the optimal two-impulse Earth–Moon solutions are characterized. Given the global set in Figure 6, only those solutions showing the best balance in terms of cost and transfer time are considered for brevity’s sake. In Figure 6 many of the computed optimal two-impulse transfers are observed to construct some sets of similar curves. Using this structure, we classify the optimal transfers into some families below. The solutions are parameterized by using the four scalars α,β,t , δ . In Section 3.2 these numbers are { i } used to define a first guess solution. In this section they are extrapolated a posteriori from the optimal solution at hand. This is done to present the optimal solutions in a concise form, using parameters having geometrical or physical meaning. (Alternatively, x1, t1, tN could have been given, but this involves reporting six numbers – instead of four – with no direct meaning). These four numbers are given with high accuracy for some sample solutions in each family. This allows an independent reproduction of the optimal solutions through the approach sketched in Section 3.2. Let x1 = (x1,y1, x˙ 1, y˙1), t1, tN be the optimal initial condition and the initial, final time, respectively. The angle that locates the initial point on the Earth parking orbit is y α = arctan 1 , (32)  x1 + µ  whereas the initial-to-circular velocity ratio is

r0 2 2 β = r0 + x˙ +y ˙ . (33) r1 µ  1 1 − q For the analysis below it is convenient to keep track of the initial phase of the Sun (in place of the initial time t ). This is given by θ = ωst1, modulo 2π. The last parameter is simply δ = t t1. i i N − A thorough investigation has been conducted to analyze the optimal solutions. In particular, for each solution the following quantities have been tracked (beside α,β,θ , δ ): angular momentum and Kepler { i } energy with respect to the Moon at final time (h2,f and H2,f ), Jacobi energy at final time (Jf ), magnitude of the second impulse (∆vf ), and total cost (∆v). These quantities are reported in the tables below in dimensionless units except for ∆v and ∆vf which are given in m/s. Dimensional values can be achieved through the units in Table 3. Solutions giving rise to similar transfers that differ only by the sign of h2,f are deemed to belong to the same family. Direct and retrograde final orbits are discriminated by using the superscript ‘+’ and ‘ ’, and called ‘positive’ and ‘negative’, respectively (see Section 4.3). For each family − both positive and negative solutions are shown, provided that both of them exist. Some figures are reported in Appendix 2 for brevity. It is worth pointing out that only those families deemed of interest have been characterized. Many other solutions exist, which are not shown below.

5.1 The S Region

The S region contains solutions whose transfer time is less than 50 days (see Figure 6). The dynamics of the solutions lying in this region are mostly governed by the Earth and Moon gravitation attractions. For this reason, each of the sample solutions reported is shown in both the Earth–Moon rotating frame and the Earth-centered, inertial frame. In the latter reference system, it is useful to define nr as the number of revolutions of P (the spacecraft) around P1 (the Earth).

5.1.1 Family a

Family a is made up of transfers with nr = 0; i.e. Earth–Moon transfers that do not perform a complete revolution around the Earth. In the (∆t,∆v) plane, family a has two local minima, located at ∆t 5 days, ≃ ∆t 44 days, respectively (Figure 11(a)). Ten sample solutions are reported in Table 5 (five positive, five ≃ + negative). The solution a1 corresponds to a Hohmann-like transfer with prograde final orbit around the Moon (∆t = 4.59 days, ∆v = 3944.8 m/s). When moving along the family for increasing ∆t, the transfers approach the Moon in no favorable conditions, and this explains the local maximum at about ∆t 27 ≃ days. Then, the cost of the solutions decreases due to the presence of lunar close encounters (see Figures 12 + − and 24(a)). This gives rise to the minimum represented by solutions a5 and a5 . These solutions cost up to 105 m/s less than the Hohmann transfer with a transfer time of approximately 44 days. Some examples in Yagasaki (2004a,b); Mengali and Quarta (2005) fall into family a (see Figure 6). On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 19

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(a) Families a, b. (b) Families c, d.

Fig. 11 Families a, b, c, d in the ∆t, ∆v plane.

Table 5 Data of transfer samples belonging to family a (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + a1 4.25229089 1.40215710 1.66738357 1.05792443 0.0114 0.3056 2.3752 810.4 3944.8 + a2 5.63387979 1.40494588 6.14451517 2.61596283 0.0118 0.5158 1.9557 898.9 4055.0 + a3 0.84297937 1.40676446 1.35668877 4.21874246 0.0120 0.6268 1.7340 944.4 4114.7 + a4 3.34893923 1.40851449 3.84224935 6.90369808 0.0121 0.6608 1.6662 958.2 4142.2 + a5 4.20833847 1.40204421 2.57748965 10.00079555 0.0110 0.0928 2.8000 717.3 3850.9 − a1 4.06517864 1.40344247 1.35319824 0.80536682 -0.0116 0.3894 2.1616 846.0 3990.5 − a2 5.36112426 1.40453947 2.70928637 2.32554805 -0.0117 0.4637 2.0126 877.3 4030.2 − a3 1.19258419 1.40711159 4.83929965 4.60012948 -0.0120 0.6182 1.7030 940.9 4114.0 − a4 3.64846498 1.40818454 4.18482537 7.36164303 -0.0120 0.5875 1.7645 928.4 4109.8 − a5 4.20411724 1.40204448 2.57735436 10.04653597 -0.0109 0.0682 2.8053 706.4 3839.9

5.1.2 Family b

Family b is formed by transfers with nr = 1; i.e. orbits that perform one complete revolution around the Earth. For each of the two branches (positive and negative) family b shows three local minima and two local maxima in the (∆t,∆v) plane (see Figure 11(a)). The three minima are represented by the solutions ± ± ± ± b1 , b3 , b5 , whose parameters are reported in Table 6. The first minimum (b1 ) corresponds to solutions having the same cost of the Hohmann transfer but ∆t three times higher. Again, when moving from this minimum toward increasing ∆t, the cost increases accordingly until lunar close encounters appear and the two local minima arise. The geometry of these solutions can be seen in Figures 13 and 24(b). Reference solutions belonging to family b can be found in Yagasaki (2004a,b); Mengali and Quarta (2005); Mingotti and Topputo (2011); see Figure 6.

5.1.3 Families c, d

In families c, d the particle P performs two revolutions around the Earth, and thus nr = 2. These two families are defined in the range ∆t 24–34 days (see Figure 11(b)). No positive family d is found. Family c ≃ has two local minima (at ∆t 24 and 32 days) and two local maxima (both at ∆t 28 days). Family d has ≃ ≃ a local minimum at ∆t 31. The parameters of families c, d are reported in Table 7 and the corresponding ≃ trajectories are shown in Figures 14 and 25. The trajectories in family d are very similar to those of family c, and thus they are not reported. The examples in Yagasaki (2004a,b); Mingotti and Topputo (2011) can be deemed belonging to family c, d; see Figure 6. All of the solutions shown so far may be classified within the class of interior transfers. Solutions similar to those of families a+, b+, and c+ have been found in Yagasaki − − − − (2004a,b). The families a , b , c , and d presented in this paper further enrich the set of two-impulse Earth–Moon interior transfers. 20 F. Topputo

x 105 2 4 1.5 a+ a+ 2 2 2 a+ 1 4 a+ 0.5 + 1 0 a3 a+ 0 1 −2 y (km) y (adim.) −0.5 + a3 a+ −4 −1 5 + a5 −1.5 + −6 a4 −2 −3 −2 −1 0 1 2 −8 −6 −4 −2 0 2 4 6 x (km) 5 x (adim.) x 10 (a) Rotating frame. (b) Earth-centered inertial frame.

Fig. 12 Positive solution samples belonging to family a. Different scales of grey are used from now on to ease the under- standing of the figures.

Table 6 Data of transfer samples belonging to family b (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + b1 0.21510709 1.40255827 3.84979016 3.25979107 0.0115 0.3273 2.3318 819.7 3957.2 + b2 3.27491402 1.40545258 3.75654396 6.60786839 0.0118 0.5180 1.9512 899.8 4059.9 + b3 4.16580666 1.40184677 1.88698381 8.27446872 0.0112 0.2230 2.5400 774.7 3906.7 + b4 3.25940457 1.40740655 3.75017462 6.13644217 0.0121 0.6506 1.6865 954.1 4129.4 + b5 4.15756374 1.40196534 5.35862484 7.15636510 0.0113 0.2271 2.5319 776.5 3909.4 − b1 0.36971517 1.40244726 4.11174688 3.44750881 -0.0114 0.2929 2.3551 804.9 3941.6 − b2 2.44816652 1.40488455 2.92880814 5.70971870 -0.0117 0.4635 2.0130 877.2 4032.8 − b3 4.17458021 1.40184986 5.03455392 8.29639011 -0.0112 0.2009 2.5393 765.1 3897.1 − b4 3.04668668 1.40729105 0.39404154 5.91740429 -0.0120 0.6239 1.6918 943.2 4117.6 − b5 4.00950307 1.40253826 5.33535717 6.93924121 -0.0113 0.2510 2.4390 786.8 3924.2

5.2 The L Region

The L region encloses solutions with transfer time longer than 50 days (and shorter than 100 days; see Figure 6). The dynamics that governs this subset of solutions is characterized by the Sun perturbation, as the corresponding trajectories spend most of the transfer time well outside the Moon orbit (Figure 8). For this reason, the sample solutions analyzed below are shown in both the Earth-centered and the Sun–Earth rotating frame. The latter is a co-rotating system of reference centered at the Sun–Earth barycenter where the Sun and the Earth are at rest. In this frame the Sun–Earth equilibrium points, L1,SE and L2,SE, are shown, too. In normalized Sun–Earth units, these two points are located at xL1,SE = 0.9900266828 and xL2,SE =1.0100340264, and their energy is C1,SE =3.0003004051 and C2,SE =3.0003000446, respectively. − These values are obtained with the Sun-Earth mass parameter µ =3.0034 10 6 (Szebehely 1967). The SE × trajectory visualization in the Sun–Earth rotating frame is done to both catch the dynamics governing the L-region transfers and check which quadrant the solutions belong to (see Section 1.1). Beside the orbit of the Moon, the plots in the Earth-centered inertial frame report the counterclockwise apparent orbit of the Sun (dashed, not to scale; the initial and final positions of the Sun are labelled with bullets). In the Earth-centered inertial frame (Figures 16(b), 17(b), 19(b), 20(b), 22(b), 23(b)), two dash-dot lines are also drawn: these connect the Earth with both the farthest point of the trajectory (Earth–apogee line) and the Sun location when P is at apogee (Earth–Sun line). The role played by the angle formed by these two lines in the exterior transfers dynamics is discussed in Circi and Teofilatto (2001). As many On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 21

x 105

1 3 + b1 2 0.5 + b2 1 b+ 0 0 1 y (km)

y (adim.) + b2 −1 −0.5 −2

−1 −3

−1.5 −1 −0.5 0 0.5 1 1.5 −4 −2 0 2 4 x (km) 5 x (adim.) x 10 (a) Rotating frame. (b) Earth-centered inertial frame.

x 105 1 4 + + + b b4 b3 5 0.5 2

+ b4 0 0 y (km)

y (adim.) −0.5 −2 −1 b+ −4 5 b+ −1.5 3 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −4 −2 0 2 4 6 x (km) 5 x (adim.) x 10 (c) Rotating frame. (d) Earth-centered inertial frame.

Fig. 13 Positive solution samples belonging to family b. families characterize the L region, three solutions are reported for each family, and the trajectory of one sample solution per family is illustrated for brevity’s sake.

5.2.1 Families e, f

With reference to Figure 15(a), families e, f are defined in the range ∆t 50 to 55 days. This is one of ≃ the few cases in which there is a remarkable difference in cost between positive and negative transfers, the latter being about 10 m/s (minimum-to-minimum) more convenient than the former. From a geometrical point of view, families e, f give rise to short-lasting exterior transfers performing lunar gravity assist. Family e is made up of second quadrant solutions (Figure 16), while family f contains fourth quadrant transfers (Figure 26(a)). Unlike the families belonging to the S region, these solutions can be achieved only within narrow ranges of the initial angular position of the Sun (see Table 8), indicating the importance of the Sun perturbation. The peculiarity of these two families is that the Moon is approached from the interior although most of the trajectory is defined outside the Moon orbit. No ballistic capture is performed by these solutions (H2,f > 0 in Table 8), and the final Jacobi energy (Jf in Table 8) indicates that there are − no forbidden regions at arrival. In the literature, a solution close to e2 has been found in Mingotti and Topputo (2011); see Figure 6. 22 F. Topputo

Table 7 Data of transfer samples belonging to families c, d (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + c1 4.06378349 1.40152201 4.72945419 7.03389884 0.0112 0.2210 2.5441 773.8 3903.3 + c2 3.42728510 1.40356082 3.96094555 6.43288971 0.0116 0.3848 2.2171 844.1 3989.4 + c3 2.48070770 1.40221206 3.05881981 5.56800695 0.0114 0.2872 2.4118 802.5 3937.3 + c4 3.61113135 1.40253003 0.97340271 6.83505041 0.0114 0.3034 2.3797 809.4 3946.7 + c5 4.05803911 1.40135493 4.62727797 7.35158462 0.0112 0.2084 2.5693 768.3 3896.5 − c1 2.75420166 1.40254918 0.14046990 5.84532752 -0.0114 0.2892 2.3623 803.3 3940.8 − c2 3.34572601 1.40276780 3.85832529 6.51705828 -0.0114 0.3017 2.3374 808.7 3947.8 − c3 3.64427971 1.40253854 4.21471731 6.86433002 -0.0114 0.2805 2.3799 799.6 3936.9 − c4 4.05933148 1.40137237 1.48998604 7.33236955 -0.0112 0.1872 2.5668 759.0 3887.3 − c5 4.31706412 1.40179564 1.62661238 7.60396758 -0.0112 0.2195 2.5021 773.1 3904.7 − d1 3.63656515 1.40329433 4.23829875 6.62738371 -0.0115 0.3384 2.2639 824.4 3967.7 − d2 3.83012066 1.40247571 4.59939861 6.80677187 -0.0113 0.2740 2.3928 796.8 3933.7 − d3 3.91244561 1.40197220 4.71924905 6.88266686 -0.0113 0.2352 2.4706 780.0 3913.0 − d4 4.08126636 1.40149990 4.71587999 7.05285562 -0.0112 0.1971 2.5470 763.4 3892.7 − d5 4.31951500 1.40185671 4.81442874 7.30813878 -0.0112 0.2246 2.4918 775.4 3907.5

Table 8 Data of transfer samples belonging to families e, f (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + e1 4.14772378 1.40209214 2.60580334 11.92791632 0.0109 0.0654 2.8546 705.1 3839.0 + e2 4.19538197 1.40205106 2.61012003 11.99341945 0.0109 0.0622 2.8609 703.7 3837.3 + e3 4.26019251 1.40207753 2.58325006 12.18915926 0.0109 0.0665 2.8524 705.6 3839.4 − e1 4.15715993 1.40207818 2.60680155 11.79207437 -0.0109 0.0444 2.8531 695.6 3829.4 − e2 4.19883247 1.40205011 2.61552663 11.84504286 -0.0109 0.0422 2.8575 694.7 3828.2 − e3 4.25352559 1.40207113 2.58069532 12.02828384 -0.0109 0.0457 2.8505 696.2 3830.0 + f1 4.15189184 1.40208600 5.75484592 11.94756971 0.0109 0.0665 2.8525 705.6 3839.4 + f2 4.19879952 1.40205042 5.74949182 12.02236141 0.0109 0.0637 2.8580 704.3 3837.9 + f3 4.26135704 1.40207872 5.72513518 12.21462196 0.0109 0.0683 2.8488 706.4 3840.2 − f1 4.16754826 1.40206691 5.76042579 11.81931468 -0.0109 0.0451 2.8517 696.0 3829.7 − f2 4.20419774 1.40204963 5.75349438 11.88174281 -0.0109 0.0436 2.8547 695.3 3828.9 − f3 4.25369743 1.40207123 5.71573294 12.06657536 -0.0109 0.0478 2.8462 697.2 3830.9

5.2.2 Families g, h, i

These three families exist in the range ∆t 60 to 72 days, where they form two minima (at ∆t 64 and ≃ ≃ 67 days, respectively; see Figure 15(b)). Only positive g and negative h, i solutions are found (Table 9). These solutions show unique features: the transfer orbit foresees a lunar gravity assist that places P in a close Earth encounter orbit, followed by an exterior-like transfer (see Figures 17 and 26(b)). Families g, i generate second quadrant transfers, whereas family h contains fourth quadrant solutions. The function H2,f changes sign along these families, and therefore ballistic capture appears according to Definition 1. This occurs even though Jf < 3, which means that there are no forbidden regions for P upon Moon arrival. With reference to Figure 6, no reference solutions having these features are found in the literature (to the best of the author’s knowledge).

5.2.3 Families j, k

Families j, k are defined within ∆t 72 and 82 days (see Figure 18(a)). These present two minima at ≃ ∆t 75 and 80, respectively. Positive j and negative k solutions are found. Together with families g, h, i, ≃ On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 23

x 105 4 0.8 + c1 0.6 3 + c5 0.4 2 0.2 + 1 c3 0 0 + y (km) c1 y (adim.) −0.2 c+ −1 −0.4 5 −2 −0.6 c+ −0.8 3 −3

−1 −0.5 0 0.5 1 −4 −2 0 2 4 x (km) 5 x (adim.) x 10 (a) Rotating frame. (b) Earth-centered inertial frame.

x 105

+ 3 c2 0.5 2 + c2 1

0 0 c+ y (km) 4 y (adim.) + −1 c4 −0.5 −2

−3 −1 −1 −0.5 0 0.5 1 −4 −2 0 2 4 x (km) 5 x (adim.) x 10 (c) Rotating frame. (d) Earth-centered inertial frame.

Fig. 14 Positive solution samples belonging to family c. these solutions show a transition between ballistic capture and escape upon Moon arrival (see the sign of H2,f in Table 10). However, these solutions have Jf =∼ 3, meaning that the energy level at Moon arrival is close to C4,5. Families j, k are made up by second, fourth quadrant transfers, respectively. The geometry of the trajectories recalls that of an exterior transfer with lunar gravity assist, although from the Sun–Earth rotating frames it can be inferred that the dynamics of L1,SE and L2,SE is not exploited (i.e. the trajectories are neither close to possible L1/2,SE periodic orbit nor their manifolds; see Figure 19). No reference solutions are found in the region families j, k belong to (Figure 6).

5.2.4 Families l, m, n

These three families are defined within ∆t 78 and 90 days. Each of the three families presents a local ≃ minimum, with the one of family l being the global minimum of the whole ∆v(∆t) function (see Figure 18(b)). This is located at ∆t = 82.6 days, ∆v = 3769.3 m/s. From the solution parameters summarized in Table 11 it can be inferred that: 1) only positive transfers are found; 2) all of the solutions perform ballistic capture at Moon arrival (H < 0); 3) the final value of the Jacobi energy at arrival satisfies C3 J

3855 3835 3850 3830 − − g1 h 3845 3825 1 + + f3 3820 f1 + 3840 f2 +

v (m/s) + v (m/s) − − e3 3815 ∆ e ∆ g h 1 + 2 2 − 3835 e2 h− i − 3810 3 1 − f3 f − − 1 f 3805 − − 3830 2 − g3 i2 i − e3 3 e1 − e2 3800 52 53 54 55 62 63 64 65 66 67 68 69 ∆t (days) ∆t (days) (a) Families e, f. (b) Families g, h, i.

Fig. 15 Families e, f, g, h, i in the ∆t, ∆v plane.

x 10−3 x 105

8 6

6 4 4 2 2 0 L1 L

2 y (km)

y (SErf) 0 −2 −2 −4 −4 −6 −6 0.995 1 1.005 1.01 −5 0 5 10 x (km) 5 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

− Fig. 16 Solution e2 belonging to family e.

Reference solutions matching any of these three families are not found, though examples in Yamakawa et al. (1992, 1993); Mingotti and Topputo (2011) are defined in the same region of families l, m, n (Figure 6).

5.2.5 Families o, p

Families o, p span a wide range of transfer times, from ∆t 70 to 100 days, with a transfer cost that drops ≃ 50 m/s along the families. Both positive and negative solutions are found for the two families, the former being cheaper than the latter for ∆t > 75 (see Figure 21(a)). The dynamics of these solutions recall that of a classic exterior transfer (see Figure 22, solution o+, second quadrant transfer, and Figure 27(b), solution − 5 p5 , fourth quadrant transfer). No lunar gravity assist exists, and this explains the variability of α in Table 12 (departure may occur in any direction, if the Moon has not to be encountered, provided that a suitable value of θi is considered). From Table 12, a direct relation between transfer cost and final Kepler energy at Moon arrival can be observed. When H2,f decreases, ∆v decreases accordingly. These families are close to the reference solutions in Mengali and Quarta (2005); Da Silva Fernandes and Marinho (2011); Assadian and Pourtakdoust (2010) and adhere to the trajectories used in GRAIL mission (Hoffman 2009; Roncoli and Fujii 2010; Chung et al. 2010; Hatch et al. 2010); see Figure 6. On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 25

Table 9 Data of transfer samples belonging to families g, h, i (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + g1 4.18237304 1.40268471 0.36355032 14.26905351 -0.0108 0.0282 2.8854 688.4 3826.9 + g2 4.19369433 1.40265901 0.34820064 14.48063955 -0.0108 0.0029 2.9362 676.9 3815.2 + g3 4.21070714 1.40262644 0.34162206 14.77502012 -0.0107 -0.0156 2.9733 668.5 3806.6 − h1 4.14682062 1.40281171 3.48471172 14.29339564 -0.0108 0.0237 2.8945 686.3 3825.9 − h2 4.15885546 1.40276662 3.46143639 14.55097864 -0.0108 -0.0044 2.9508 673.6 3812.8 − h3 4.15635259 1.40277811 3.44611191 14.79936225 -0.0107 -0.0160 2.9741 668.3 3807.6 − i1 4.14224224 1.40286493 0.14736246 14.99949586 -0.0107 -0.0139 2.9699 669.3 3809.2 − i2 4.27648072 1.40258858 0.26443401 15.32147461 -0.0107 -0.0262 2.9946 663.7 3801.4 − i3 4.51532179 1.40284772 0.58605743 15.73544729 -0.0107 -0.0231 2.9884 665.1 3804.9

5 x 10−3 x 10 8 10 6 8 4 6 2 4 L L

1 2 y (km)

y (SErf) 0 2 −2 0 −4 −2 −6 0.995 1 1.005 1.01 −5 0 5 10 x (km) 5 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

− Fig. 17 Solution i2 belonging to family i.

Table 10 Data of transfer samples belonging to families j, k (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + j1 3.96454437 1.40367734 2.56638459 16.72624653 0.0108 -0.0123 3.0096 670.0 3816.3 + j2 4.22463033 1.40207788 3.12430176 17.34374511 0.0107 -0.0155 3.0161 668.5 3802.3 + j3 4.37950445 1.40226193 3.26548328 17.77648273 0.0108 0.0160 2.9531 682.9 3818.1 − k1 3.99864546 1.40294205 5.08214538 17.75063250 -0.0107 -0.0233 2.9886 665.0 3805.5 − k2 4.18510811 1.40206304 5.42476134 18.34078867 -0.0107 -0.0248 2.9916 664.3 3798.0 − k3 4.41582183 1.40234774 5.60169530 18.69107060 -0.0107 -0.0153 2.9726 668.6 3804.5

5.2.6 Families q, r, s, t

The branches defining these families are reported in Figure 21(b) and their parameters are summarized in Table 13. Solutions s, t perform a lunar gravity assist similar to families g, h, i, although no ballistic − capture is accomplished (see Figure 23, solution s3 , second quadrant transfer). Families q, r exploit instead the dynamics of the Sun–Earth Lagrange points and their transfer structure recalls that of families o, p (see + Figure 28, solution q2 , second quadrant transfer). With reference to Figure 6, no reference solutions close to these families are found. 26 F. Topputo

!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~

(a) Families j, k. (b) Families l, m, n.

Fig. 18 Families j, k, l, m, n in the ∆t, ∆v plane.

x 10−3 x 106 1 10

8 0.5 6

4 0 y (km)

y (SErf) 2 L L −0.5 0 1 2

−2 −1 −4 0.995 1 1.005 1.01 −1 −0.5 0 0.5 1 1.5 x (km) 6 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

+ Fig. 19 Solution j2 belonging to family j.

6 Conclusions

Since the discovery of low-energy transfers, the problem of finding alternative, efficient Earth–Moon transfers has been approached with a focus on a small class of solutions, if not on just one kind of solutions. Typically, when a novel solution is found, one compares it to the Hohmann transfer, leaving unanswered the question on the existence of other solutions, possibly having intermediate features between the solution at hand and the Hohmann transfer. This modus operandi prevents the formulation of a complete trade-off between cost and transfer time. Motivated by this need, the present paper reconstructs the global set of solutions for the two-impulse Earth–Moon transfers. The idea is to view the Hohmann, the interior, and the exterior transfers as special solutions of the same problem. This means that known cases may be thought of as belonging to a more general picture. The framework used in this paper is the (∆t, ∆v) plane, which allows direct assessment of solutions for practical purposes. The global set of optimal two-impulse Earth–Moon transfers presented in this paper demonstrates that the intuition on the existence of intermediate solutions was valid. Known solutions reduce to points belonging to different families of transfers. The families deemed of more interest have been characterized and studied in detail, and their main features have been discussed. The parameters defining all the solutions presented are given to allow the reproduction of these transfers for possible further independent investigations. On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 27

Table 11 Data of transfer samples belonging to families l, m, n (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + l1 3.93000540 1.40447433 2.32536708 18.01942133 0.0106 -0.0811 3.1470 638.5 3791.0 + l2 4.22063005 1.40207720 3.07943365 18.99518723 0.0106 -0.0875 3.1597 635.5 3769.3 + l3 4.35687235 1.40222223 3.19400904 19.60848008 0.0106 -0.0830 3.1508 637.6 3772.5 + m1 4.10457544 1.40221448 2.32660368 19.30732266 0.0106 -0.0731 3.1309 642.2 3777.0 + m2 4.22350820 1.40206865 2.48849928 19.98666422 0.0106 -0.0799 3.1445 639.1 3798.0 + m3 4.25793279 1.40208973 2.51902225 20.50160843 0.0106 -0.0786 3.1420 639.6 3773.5 + n1 4.17979291 1.40207328 2.41578647 19.27387149 0.0106 -0.0676 3.1200 644.7 3778.5 + n2 4.22841073 1.40206951 2.46855661 19.54086245 0.0106 -0.0685 3.1219 644.3 3778.0 + n3 4.29017247 1.40212432 2.51636438 19.81143467 0.0106 -0.0648 3.1146 646.0 3780.1

6 x 10−3 x 10 10 1

0.5 5 0 y (km) y (SErf) L1 L2 0 −0.5

−1 −5 0.995 1 1.005 1.01 −1 −0.5 0 0.5 1 1.5 x (km) 6 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

+ Fig. 20 Solution l2 belonging to family l.

7 Acknowledgments

The author is grateful to Pierluigi Di Lizia, Alexander Wittig, and Koen Geurts for having proof-read the paper, to Mauro Massari and Franco Bernelli-Zazzera for having shared their workstations, and to Roberto Armellin for the computation of the Pareto-efficient solutions.

References

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!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~

(a) Families o, p. (b) Families q, r, s, t.

Fig. 21 Families o, p, q, r, s, t in the ∆t, ∆v plane.

Table 12 Data of transfer samples belonging to families o, p (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v − o1 4.87924505 1.41088384 2.16174675 16.36363453 -0.0109 0.0445 2.8529 695.7 3898.1 − o2 5.18402876 1.41078261 2.45241547 17.07355978 -0.0108 -0.0019 2.9459 674.7 3876.3 − o3 6.01081082 1.41029724 3.09897520 18.13605289 -0.0108 -0.0111 2.9643 670.5 3868.4 − o4 0.93009504 1.40999519 4.16463357 19.50110233 -0.0107 -0.0142 2.9705 669.1 3864.6 − o5 2.63901403 1.41009626 5.70421254 21.24636234 -0.0107 -0.0149 2.9719 668.8 3865.1 + o1 5.15585335 1.41084129 2.45717541 16.73769746 0.0109 0.0337 2.9179 690.8 3892.9 + o2 5.58251216 1.41078410 2.90767751 17.64177466 0.0107 -0.0269 3.0389 663.3 3865.0 + o3 0.20095333 1.41024314 3.61092707 18.94414311 0.0107 -0.0430 3.0710 656.0 3853.4 + o4 1.58324803 1.41004949 4.84316834 20.56435823 0.0107 -0.0484 3.0818 653.5 3849.4 + o5 2.99264083 1.41013880 6.12422036 22.10419030 0.0107 -0.0502 3.0853 652.7 3849.3 − p1 5.02222917 1.41085460 5.45004921 16.55725651 -0.0109 0.0373 2.8672 692.5 3894.7 − p2 5.49763936 1.41058926 5.83714409 17.51562844 -0.0108 -0.0068 2.9556 672.5 3872.6 − p3 0.02048615 1.41022252 0.22874828 18.43533710 -0.0108 -0.0110 2.9640 670.6 3867.9 − p4 1.51967236 1.41000824 1.56238694 20.04968749 -0.0108 -0.0134 2.9688 669.5 3865.1 − p5 3.05842328 1.41011989 2.95811499 21.63452349 -0.0107 -0.0139 2.9699 669.3 3865.7 + p1 5.29149080 1.41082818 5.74447801 16.91153673 0.0108 0.0280 2.9292 688.3 3890.3 + p2 5.70978560 1.41072464 6.16102181 17.85988596 0.0107 -0.0310 3.0469 661.5 3862.7 + p3 0.44578644 1.41019712 0.69440616 19.24256202 0.0107 -0.0427 3.0703 656.1 3853.2 + p4 2.49730625 1.41015273 2.52992889 21.52688793 0.0107 -0.0477 3.0802 653.9 3850.6 + p5 3.34228464 1.41005213 3.32154711 22.47702442 0.0107 -0.0491 3.0830 653.2 3849.2

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x 10−3 x 105 8 10 6 8 4 6 2 4 L1 L2 0 y (km) y (SErf) 2 −2 0 −4 −2 −6 0.995 1 1.005 1.01 −5 0 5 10 x (km) 5 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

+ Fig. 22 Solution o5 belonging to family o.

Table 13 Data of transfer samples belonging to families q, r, s, t (∆v are in m/s).

Sol α β θi δ h2,f H2,f Jf ∆vf ∆v + q1 0.03381194 1.41065912 3.67413052 19.74181230 0.0106 -0.0797 3.1442 639.1 3839.8 + q2 0.10879184 1.41056240 3.71022277 19.96824167 0.0106 -0.0842 3.1532 637.0 3837.0 + q3 0.49122400 1.41031915 3.99219506 20.65689374 0.0106 -0.0844 3.1535 637.0 3835.0 + r1 0.13483581 1.41065431 0.63885104 19.84872767 0.0106 -0.0799 3.1446 639.1 3839.7 + r2 0.22346700 1.41053662 0.67779827 20.06106177 0.0106 -0.0837 3.1522 637.3 3837.0 + r3 0.43650553 1.41039878 0.83354004 20.58467910 0.0106 -0.0849 3.1546 636.7 3835.4 − s1 4.17015923 1.40272187 6.23928017 20.12726592 -0.0109 0.0606 2.8205 703.0 3841.8 − s2 4.19420474 1.40265697 6.24424980 20.38791538 -0.0108 0.0258 2.8903 687.3 3825.6 − s3 4.25274883 1.40257540 0.00339720 20.54123860 -0.0108 0.0207 2.9006 685.0 3822.6 − t1 4.39366912 1.40263808 3.34123544 20.34934312 -0.0109 0.0671 2.8076 705.8 3844.0 − t2 4.40612382 1.40265462 3.35659232 20.54099714 -0.0109 0.0375 2.8668 692.6 3830.8 − t3 4.41997347 1.40267407 3.37438159 20.73265115 -0.0108 0.0223 2.8972 685.7 3824.2

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8 0.5 6

4 0 y (km) y (SErf) 2

L1 L2 0 −0.5

−2

−4 −1 0.995 1 1.005 1.01 −5 0 5 10 15 x (km) 5 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

− Fig. 23 Solution s3 belonging to family s.

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Appendix 1

An orbit in the rotating frame, x(t) = x(t),y(t), x˙(t), y˙(t) , is converted to an orbit expressed in the P1- { } centered (or Earth-centered), inertial frame, X1(t) = X1(t),Y1(t), X˙ 1(t), Y˙1(t) , through { }

X1(t)=(x(t) + µ)cos t y(t)sin t − Y (t)=(x(t) + µ)sin t + y(t)cos t 1 (34) X˙ 1(t)=(˙x(t) y(t))cos t (˙y(t) + x(t) + µ)sin t − − Y˙1(t)=(˙x(t) y(t))sin t +(˙y(t) + x(t) + µ)cos t − where t is the present, scaled time. The transformation into the P2-centered (i.e. Moon-centered), inertial frame is obtained from (34) by replacing ‘µ’ with ‘µ 1’. The initial conditions of the points highlighted − in Figure 7 are reported in Table 14 with high accuracy. This allows to reproduce the optimal solutions by simply integrating the initial conditions under the PBRFBP dynamics; i.e. the i-th transfer orbit is γ(i) = ϕ(x ,t ; t), t t where ϕ is the flow of (5). { i i ≤ f } Appendix 2

The figures corresponding to the solutions characterized in Section 5 are reported below. 32 F. Topputo

Table 14 Initial condition xi = (xi,yi, x˙ i, y˙i) and initial, final time, ti, tf of solutions (i)–(x) in Figure 7, 8 and Table 4.

(i) (ii) (iii) (iv) (v)

xi -0.019710793706630 0.004430665316507 -0.022223338375692 -0.022469248813271 -0.020509279478864 yi -0.015255813698632 0.003867133727872 -0.013727223510871 -0.013543346852322 -0.014833373417121 x˙ i 9.554404402969761 -2.421772563013686 8.593048071999794 8.477163558651014 9.289091958818513 y˙i -4.734752068884238 10.383974704885071 -6.305349228506921 -6.458691168284453 -5.234406611705439 ti -1.803851247295145 -4.242568609277442 28.854324263774792 5.183780363655846 -2.783372600934104 tf -0.745915731505332 -0.928062659449309 35.904880040219062 12.518449758919608 7.256777745724602 (vi) (vii) (viii) (ix) (x)

xi -0.020796119659883 -0.021300676720177 -0.019048265996746 -0.019022730972677 -0.020288471040515 yi -0.014668044128481 -0.014358714903565 -0.015566582598150 -0.015577872203673 -0.014955651584857 x˙ i 9.185639864842763 8.996820692867965 9.751858086858906 9.759101553325923 9.365880586172640 y˙i -5.414082611512081 -5.733172198727265 -4.321076476149917 -4.305155198773667 -5.096246610790654 ti -2.825197110547974 3.020458855979923 2.932062269968396 6.480457482011631 0.068219944002269 tf 8.989837035669957 17.277056927521521 17.819572739337175 21.838729667546875 19.078513034970545

5 x 105 x 10 6 4 − − a4 b5 2 − 4 a1 − a2 0 2 − b3 − b2 −2 0 y (km) − y (km) a3 b− −4 −2 1 − a5 −6 −4 − b4

−8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 x (km) 5 x (km) 5 x 10 x 10 (a) Family a (negative). (b) Family b (negative).

Fig. 24 Negative solution samples belonging to family a and b (Earth-centered inertial frame).

x 105 x 105

3 3

2 2 c− − 2 c4 1 1 − c3 0 0 y (km) y (km) −1 −1 − c5 − −2 c1 −2 −3 −3

−4 −2 0 2 4 −4 −2 0 2 4 x (km) 5 x (km) 5 x 10 x 10 (a) Family c (negative). (b) Family c (negative).

Fig. 25 Negative solution samples belonging to family c (Earth-centered inertial frame). On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model 33

x 10−3 x 10−3 6 4 4 2 2 L 1 L2 L1 L2 0 0 −2 −2 y (SErf) y (SErf) −4 −4 −6 −6 −8 −8

0.995 1 1.005 1.01 0.995 1 1.005 1.01 x (SErf) x (SErf) + − (a) Solution f1 , SE rotating frame. (b) Solution h3 , SE rotating frame.

+ − Fig. 26 Solutions f1 and h3 .

x 105 x 10−3 10 6 8 4

6 2

4 L1 L 0 2 2 y (km)

y (SErf) −2 0 −4 −2 −6 −4 −6 −8 −5 0 5 10 0.995 1 1.005 1.01 x (km) 5 x 10 x (SErf) + − (a) Solution m2 , Earth-centered frame. (b) Solution p5 , SE rotating frame.

+ − Fig. 27 Solution m2 and p5 .

x 10−3 x 105

8 10

6 8

4 6

2 4 y (km) y (SErf) L L 0 1 2 2

−2 0

−4 −2

0.995 1 1.005 1.01 −5 0 5 10 x (km) 5 x (SErf) x 10 (a) Sun-Earth rotating frame. (b) Earth-centered inertial frame.

+ Fig. 28 Solution q2 belonging to family q.