diversity and validity of stable-unstable transitions in the algorithmic weak stability boundary

7/31/2019 Diversity and Validity of Stable-Unstable Transitions in the Algorithmic Weak Stability Boundary

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Diversity and Validity of Stable-Unstable

Transitions in the Algorithmic Weak Stability

Boundary *

Priscilla A. Sousa Silva Maisa O. Terra

Departamento de Matematica, Instituto Tecnologico de Aeronautica,

Praca Marechal Eduardo Gomes, 50, 12.228-900, Sao Jose dos Campos, SP, [email protected]

Abstract

This paper is devoted to verify the consistency of the algorithmic Weak

Stability Boundary definition concerning the achievement of capture-escape

detection, through examining the transitions produced by the implemen-

tation of this definition. Our main goal is to show that many types of spu-

rious transitions concerning capture-escape behavior are found besides the

expected transitions due to the separatrix role of the hyperbolic invariant

manifolds of the central manifold of the collinear equilibria of the Planar

Circular Restricted Three-Body Problem. We identify and characterize

authentic and spurious transitions and discuss their spatial distribution

along the boundary for sets of initial conditions with high eccentricity,

showing the frequent occurrence of spurious transitions and of collisional

trajectories. Also, we investigate smooth and fractal-like portions of the

boundary. Finally, we propose an alternative stability boundary definition

based on the effective detection of capture-escape transitions.

Keywords Weak Stability Boundary; Restricted Three-Body Problem;

Earth-to-Moon Transfers; Capture-Escape Transitions

1 Introduction

The Weak Stability Boundary (WSB) concept was heuristically proposed by

E. Belbruno (Belbruno, 1987) as an initiative to establish regions of effectivestability related to temporary capture orbits around the Moon aiming to reducefuel consumption at the arrival portion of Earth-to-Moon transfer trajectories.

At least three distinct definitions share this nomenclature: (i) a qualitativeregion defined in the phase space where gravitational forces cancel each other(Belbruno, 1987; Belbruno and Miller, 1993; Circi and Teofilatto, 2001), (ii) analgorithmic definition based on the dynamical evolution of a specific set of ini-tial conditions which are classified according to a prescribed stability criterion(Belbruno and Miller, 1993; Belbruno, 2004; Garca and Gomez, 2007), and (iii)an analytical approximation defined just as the intersection of three subsets of

* Additional contact information: Priscilla A. Sousa Silva - priandss at maia.ub.es.

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the phase space (Belbruno, 2004; Belbruno et al, 2008). For the time being, no

precise mathematical correspondence has been established among the stabilityboundaries generated by these three definitions. So, although they share thesame name, they do not necessarily share the same properties and characteris-tics.

This work deals with the algorithmic version of the WSB. In its currentconstruction, sets of initial conditions are defined around the Moon, fixing nullradial velocity and the oscullating ellipse eccentricity. Then, these sets of initialconditions are evolved under the dynamics of the Planar Circular RestrictedThree-Body Problem (PCR3BP) and classified according to a certain stabilitydefinition based on the measurement of the Kepler energy, which is inherent ina two-body dynamics, such that the WSB corresponds to the stable-unstabletransitions. Accordingly, the stable subset of initial conditions is expected togenerate captured orbits around the Moon.

Recently, many authors have built the algorithmic WSB, introducing modi-fications in the algorithm and using several dynamical models and different sub-system of the Solar System to exploit specific aspects of interest in the context ofspace mission design (see, for instance Topputo and Belbruno (2009); Romagnoliand Circi (2009); Belbruno et al (2010); Mako et al (2010)). However, an un-restricted and general verification of the correspondence of the stable-unstabletransitions produced by the algorithmic definition with actual capture-escapetransitions of trajectories in the PCR3BP is still lacking.

This issue was first addressed by Garca and Gomez (2007) which presentednumerical results that show the relation of the WSB set with the invariant hy-perbolic manifolds associated to the central manifolds of the collinear librationpoints L1 and L2 of the PCR3BP. However, their numerical investigation was

not general, but was restricted to only one chosen energy value for an alter-native version of the algorithm which used a different set of initial conditions,obtained by fixing the energy constant of the PCR3BP instead of the osculatingellipse eccentricity1. Also they employed an extended stability definition thatconsidered n-turns around the Moon instead of only one turn.

Later, Belbruno et al (2010) gave a geometric argument for the fact that,for some energy range, the points of the WSB are the points with zero radialvelocity that lie on the stable manifolds of Lyapunov orbits, provided that thesemanifolds satisfy some topological conditions. Their argument is supported bynumerical experiments for a restricted subset of WSB points with moderate ec-centricity, namely e = 0.4. Given that the topological conditions considered byBelbruno et al (2010) are sufficient but not necessary, it may occur that the in-variant manifolds are responsible for the transitions even when these conditionsfail. These authors have conjectured that the algorithmic WSB can provide a

1In Garca and Gomez (2007), Section 1 presents a revision of the algorithmic definitionproposed by Belbruno (2004) to obtain regions of weak stability around the Moon. Following,Section 2 presents the stable regions obtained through an implementation of this definition.Section 3 deals with the possibility of obtaining rough estimations of the stable zones usingJacobis first integral of the PCR3BP. Finally, Section 4 gives evidence of the connectionbetween the weak stability regions and the invariant manifolds of the central manifold of thecollinear equilibria of the PCR3BP for just one energy value (C = 3.09998). However, theset of initial conditions used in this last numerical experiments is built fixing the energy level,while the eccentricity is kept free. Moreover, an extended stability criterion for n-turns isapplied. Thus, in fact, the stability boundaries regarded in this verification differ from thoseof the stable sets presented in Section 2.

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good substitute for the hyperbolic invariant manifolds of a considered mathe-

matical model, even when these objects are no longer well defined, such as intime-dependent Hamiltonian systems (Haller, 2001; Gawlick et al, 2009; Shortet al, 2011). However, their approach is not suited to identify or to treat tran-sitions that may appear due to other mechanisms nor when unrestricted rangesof energies are considered and their topological conditions fail.

Our contribution aims a broad investigation of the nature of the stabilityboundary generated by the algorithmic definition, considering no energy restric-tion, in order to verify the consistency of the classification procedure regardingthe detection of capture-escape transitions.

We present examples of different types of stability transitions, identifyingdynamical and constructional agents that may act individually or in associationto determine a stable-unstable pair detection. We show that, besides the hyper-bolic manifolds of Lyapunov orbits, there exist other mechanisms that generate

stability transitions, namely, the proximity of the Moon and constructional el-ements of the algorithmic definition. Also, we show that spurious transitionscaused by the constructional mechanisms are not rare but constitute a very sig-nificant part of the boundary, specially for high eccentricity values. In addition,through a refined detection procedure, we characterize the fractal-like structureof the boundary, showing that it also appears due to the combination of dynami-cal and constructional mechanisms, and consists in both authentic and spurioustransitions.

Our results imply that the WSB generated by the current algorithmic defini-tion cannot substitute the hyperbolic invariant manifolds unrestrictedly. Through-out our investigation, various vulnerable aspects of the algorithmic definitionregarding the adequate detection of capture-escape transitions are identified,

leading to the proposition of necessary modifications of the stability definitionto avoid these inconsistencies.

The paper is organized as follows. In Section 2, we present the dynamicalmodel (the PCR3BP) and state the revised version of the WSB algorithmicdefinition as in Garca and Gomez (2007). In Section 3, we report two imple-mentations of the WSB algorithmic definition, extracting the stability boundaryfor sets of initial conditions with clockwise and counterclockwise initial veloci-ties, and discussing its relation with trajectories that collide with the surface ofthe Moon. In Section 4 we present and characterize several types of transitions.The broad distribution of these types of transitions is illustrated in Section 5for sets of initial conditions with a high value of eccentricity and with bothclockwise and counterclockwise initial velocities. In addition, a refinement pro-cedure for the detection of the boundary is presented, allowing the investigationof smooth and fractal-like portions of the boundary. In this section, we also pro-pose an alternative stability boundary definition based on the effective detectionof capture-escape transitions. The last section is devoted to the conclusions.

2 Theoretical framework

2.1 Dynamical model

The Planar Circular Restricted Three-Body Problem (PCR3BP) provides thedynamical framework in which the algorithmic WSB has been defined. This

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mathematical model describes the motion of a particle P3 of negligible mass in

the gravitational field of two main bodies, called primaries, P1 and P2, of massesm1 and m2, which move in circular orbits around their mutual center of mass(Szebehely, 1967). The motion of P3 is restricted to the orbital plane of theprimaries. This dynamical model is expressed in non-dimensional variables, insuch a way that the distance between P1 and P2, the sum of their masses andtheir angular velocity around the barycenter are normalized to one. Thus, theonly parameter of the model is the mass ratio = m2/(m1 + m2), m1 > m2.For the Earth-Moon (EM) system, = 0.0121506683.

In the synodic barycentric reference frame, P1 and P2 are located at (, 0)and ( 1, 0), respectively. In that coordinate system, (x,y, x, y) represents thestate of P3 and the particles equations of motion are given by

x 2y = x,

y + 2x = y, (1)

where

(x, y) =1

2(x2 + y2) +

1

r1+

r2+

(1 )

2, (2)

is the effective potential and r1 = [(x)2+y2]1/2 and r2 = [(x+1 )

2+y2]1/2

are the distances from P3 to P1 and P2, respectively.This system has an integral of motion given by

J(x,y, x, y) = 2(x, y) (x2 + y2) = C, (3)

where C is the Jacobi constant. The conservation associated to J defines a

three-dimensional invariant manifold immersed in the four-dimensional phasespace byM(, C) =

(x,y, x, y) R4|J(x,y, x, y) = C

. (4)

The regions obtained by the projection of the M surface onto the positionspace x-y are called the Hill regions, which constitute the accessible areas to thetrajectories for each given C value.

The dynamical model has five equilibrium points, Lk, k = 1,2,3,4,5, alsocalled libration or Lagrangian points, for which the Jacobi constant values aregiven by Ck. For a given , the energy values associated to Ck define fivepossible Hill regions configurations which correspond to particular transportpossibilities through the phase space. For the Earth-Moon system, we haveC1 = 3.20034490, C2 = 3.18416414, C3 = 3.02415026, and C4 = C5 = 3.0.

For future reference, we refer as Moon realm to the accessible region of theposition space contained in the circle centered at P2 with radius equal to thedistance from L1 to P2, where L1 is the Lagrangian point between P1 and P2.Analogously, Earth realm refers to the accessible region of the position spacecontained in the circle centered at P1 with radius equal to the distance from L3to P1, excluding the Moon realm. Finally, the accessible region of the positionspace outside these two delimited areas will be called the exterior realm.

There is a uniparametric family of periodic orbits, called Lyapunov orbits,around each collinear Lagrangian point Lk, k = 1,2,3, (Moser, 1958). The un-stable and stable invariant manifolds associated to a Lyapunov orbit are locallyhomeomorphic to two-dimensional cylinders and act as separatrices of the energy

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shell, defining four categories of trajectories in the neck region near each equi-

librium, namely, transit, nontransit and asymptotic orbits (hyperbolic invariantmanifolds), besides the periodic orbit itself (Conley, 1969). These invariantmanifolds constitute transport channels (Koon et al, 2000) in the solar systemand have been exploited in modern mission design, such as in the Petit GrandTour to the Galilean Jovian moons (Gomez et al, 2001).

2.2 The WSB algorithmic definition

The current numerical algorithmic definition was stated in Belbruno (2004)(Chapter 3.2.1) and the construction procedure was reviewed and extended inGarca and Gomez (2007).

Let l() be the radial segment connecting the positions of P2 and P3, where is the counterclockwise angle measured from the x-axis, for x > 1 + , to

l(). Sets of initial conditions are constructed for which P3 is assumed to startits motion on the periapsis of an osculating ellipse around P2 with the modulusof the sidereal initial velocity, , given by

2 =(1 + e)

r2, (5)

where e is the eccentricity of the osculating ellipse. The initial Kepler energyof P3 with respect to P2 is negative for e [0, 1) since the two-body energycomputed at the periapsis (Bate et al, 1971) is hK = (e 1)/2r2.

For a fixed position on l(), prograde osculating motions about P2 are gen-erated by initial conditions with counterclockwise (positive) velocity given by

x = 1 + + r2 cos , y = r2 sin ,x = r2 sin sin , y = r2 cos + cos ,

(6)

while retrograde osculating motions about P2 are generated by initial conditionswith clockwise (negative) velocity given by

x = 1 + + r2 cos , y = r2 sin ,x = r2 sin + sin , y = r2 cos cos .

(7)

For the trajectories generated by these initial conditions, the following defi-nition of stable behavior was proposed by Belbruno (2004):

Definition 1 The motion of a particle is said to be stableabout P2, under the

PCR3BP dynamics, if after leaving l() it makes a full cycle about P2

withoutgoing around P1 and returns to l() at a point with negative Kepler energy withrespect to P2. Otherwise, the motion will be unstable.

This stable behavior is said to be related to a type of capture, called ballisticcapture2, which was analytically defined by Belbruno (2004) as:

Definition 2 P3 is ballistically captured by P2 at time t = tc if, for a solution(t) = (x(t), y(t), x(t), y(t)) of the R3BP, hK((tc)) < 0, where hK is the two-body energy of P3 with respect to P2.

2We remark that this definition for ballistic capture is not unique. For instance, we referto Koon et al (2001) and Marsden and Ross (2005) for alternative definitions.

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The Kepler energy, hK , is

hK = 12

x2 + y2

Gm

2

r2, (8)

where (x, y, x, y) is the state of P3 in an inertial reference frame with origin

in P2; r2 =

x2 + y2 is the distance between P3 and P2; m2 is the mass ofthe primary; and G stands for the universal gravitational constant (Bate et al,1971).

The stability classification associated to this type of capture generates sta-ble and unstable subsets (see Sousa Silva and Terra (2010, 2011) for extensiveinvestigations of these subsets) and leads to the definition of regions in phasespace known as boundaries of stability (Belbruno, 2004; Garca and Gomez,2007; Topputo and Belbruno, 2009; Belbruno et al, 2010).

Definition 3 The Weak Stability Boundary is given by the set

W = {r| [0, 2), e [0, 1)} , (9)

where r(, e) are the points along the radial line l() for which there is a changeof stability in the sense of Definition 1. The subset obtained by fixing the eccen-

tricity e of the osculating ellipse is

We = {r| [0, 2), e = constant} . (10)

In this algorithmic approach, for space mission design, it is implicit thatthe initial conditions defined by Equations 6 and 7 correspond to states whichare candidate solutions for the final portion of a transfer trajectory. Indeed,

the stable subset should correspond to temporary captured orbits around theMoon.

3 Implementations of the WSB algorithmic def-

inition

We implemented two versions of the WSB algorithmic definition. In implemen-tation A, we considered the usual punctual mass idealization for the primaries,while in implementation B, the Moon was regarded as a finite body with meanradius of approximately 1,738 km.

Both implementations generate stable and unstable sets, S and U, respec-tively, according to Definition 1. The unstable set was subclassified accordingto five instability criteria:

E: instability due to non-negative Keplerian energy, when trajectoriesreturn to l() after one turn around the Moon;

G1: primary interchange through the neck around L1 with C > C3;G2: primary interchange through the neck around L2 with C > C3;G3: generic geometric escape with C < C3;T: instability due to exceeding the maximum integration time, without

returning to l() or going around the Earth.In the second, third and fourth unstable cases, P3 is required to complete a fullrevolution around P1. The threshold value C3 separates the cases for which theexits through L1 or L2 are easily distinguishable or not.

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Figure 1: Boundary of stability (black) obtained through implementation A forinitial conditions with e = 0.9 and (a) positive and (b) negative initial velocities.The safe stable (green), unsafe stable (red), and unsafe unstable (brown) subsetsof initial conditions obtained through implementation Bare also shown. The equilibrium points are represented by blue +. The greycircle delimits the approximated lunar SOI. More details about the stabilityboundaries are found in Figures 10 and 12.

ditions prevails around the Moon for low to medium values of the eccentricity3,in such a way that the first stable-unstable transitions occur far away from the

Moon. This stable core shrinks as e increases, so the stability boundary is atlow altitudes from the surface of the Moon only for high eccentricity values.Particularly, for the cases shown in Figure 1, the first transitions occur below500 km of altitude.

As an initial approach, the boundary was built as the set of stable initialconditions which had at least one neighboring unstable point. In this extractionof the boundary we considered first neighbors both in the radial and the angulardirections. This is an approximation of the actual boundary set and is condi-tioned to the grid employed in the discretization of the -r2 plane. Moreover,isolated stable points, surrounded only by unstable initial conditions, appeardue to the discrete nature of the grid. In such cases, these initial conditionswere also extracted as boundary points. As a second approach of extraction,

we considered a refinement procedure to be presented and further discussed inSection 5.2.In agreement with Garca and Gomez (2007), it is common to have more

than one r where stable-unstable transitions occur along a radial line.

3.2 Collision along the stability boundary

Comparing implementations A and B, we find that many boundary points ex-tracted from the stable set of implementation A collide with the Moon when

3Examples for lower eccentricity values are Figures 1 and 2 of Garca and Gomez (2007),Figures 6 to 9 of Sousa Silva and Terra (2010), and Figure 1 of Sousa Silva and Terra (2011).

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Figure 2: Safe (black) and unsafe (red) stability boundary points for initialconditions with e = 0.9 and (a) positive and (b) negative initial velocities.

implementation B is performed. Particularly, for e = 0.9 and positive initialvelocity, 53.84% of the boundary points obtained through implementation Abelong to the C subset generated by implementation B. In the case of negativeinitial velocity, 40.33% of the boundary points correspond to collisional orbits.In Figure 2, the red points depict the initial conditions of the stability boundarywhich collide with the surface of the Moon when implementation B is performed,while the black points correspond to non-collisional boundary points.

The large presence of collisional trajectories in the boundary set is explainedby the fact that the close approach to the primary causes divergence of neigh-boring trajectories in the phase space, implying distinct nonlinear behaviors anddifferent stability classification.

As seen in Figure 1 (a), for the set with positive initial velocity and e = 0.9,the Moon collisional set accompanies a large part of the stability boundary,implying that these transitions are unsafe, therefore not appropriate for practicalapplications. This is a typical feature for sets with positive initial velocity inthe lunar SOI associated to other eccentricity values (see Sousa Silva and Terra(2010) for the associated sets of initial conditions with lower eccentricity values).

In the case of initial conditions with negative velocity and e = 0.9, theMoon collisional set spreads along thin subsets of stable points and also over

other regions of sparse boundary points, as seen in Figure 1 (b). In this case,we observe regions where the boundary of the stable set does not coexist withthe collisional set.

Collisional transitions will be further explored in Section 4.3.

4 Diversity and validity of transitions

In this Section, we perform a detailed characterization of several pairs of stable-unstable trajectories in order to elucidate how the algorithm criteria gener-ate WSB solutions. We also inspect if these transitions correspond to actualcapture-escape behavior.

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We start presenting spurious transitions regarding capture-escape behavior,

that is, transitions that are solely due to the fulfillment of the algorithm criteria,although no divergence of trajectories is observed. Then, we proceed to caseswhere divergence of trajectories indeed occurs. The dynamical agents account-ing for this effective separation are the primary P2 and the Lyapunov orbitsaround L1 and L2 with their stable and unstable invariant manifolds. In thesecases, both spurious and authentic capture-escape transitions appear.

In the following subsections, we present scenarios at which a single decisivefactor determines a stability transition. However, we remark that there aretransitions along the stability boundary produced by the combination of two ormore factors. In the examples shown, all the initial conditions have e = 0.9.Details about each transition are given in Table 1.

4.1 Measurements of the Kepler energy

Although the Kepler energy hK is not an invariant for the PCR3BP dynamics,it is used as an indicator of ballistic capture state in Definition 2 in analogywith the characteristic bounded and unbounded conic solutions of the two-bodyproblem. As known, in the Kepler problem, hK is constant at all points of theorbit and depends only on the semi-major axis a. For all closed orbits (ellipsesor circles), hK is negative; for the limiting case of parabolic orbits with a = ,hK equals zero; and for the case when the solutions of the Kepler equation areunbounded (hyperbolas), hK is positive (Bate et al, 1971).

When P3 returns to l() at t = tf, the trajectory is classified as stable if

rf((xf + 1 + yf)2 + (xf yf)

2) < 2, (11)

where rf = [(xf + 1 )2

+ y2

f]1/2

and (xf, yf, xf, yf) is the state of P3 at tf.Given that the interaction of P2 and P3 cannot be approximated by a two-

body subsystem outside the lunar SOI, by definition, the validity of the mea-surement ofhK is questionable in this region. Indeed, it is possible to verify thatEq. 11 can be satisfied far away from the lunar SOI, even when P3 is orbitingP1.

Following, we will present two examples showing that the measurement ofthe Kepler energy at l() alone is not always consistent as a capture indicatorfor a solution of the PCR3BP, both if it is measured inside or outside the lunarSOI.

Case 1: In Figure 3, we show two neighbor orbits that return to l() almostsimultaneously outside the lunar SOI. Then, the measurement ofhK determinesa S-E transition according to Definition 1, although the trajectories are indis-tinguishable. After the classification, these trajectories remain together in thephase space for a long time, wandering in the exterior realm. So, this erroneousstable-unstable pair does not correspond to an actual capture-escape transition.

Case 2: Figure 4 shows another pair of trajectories detected as a S-E transi-tion. Both orbits resemble tadpole orbits (Murray and Dermott, 1999), wander-ing through the Earth and the exterior realms for a long time before they returnto the Moon realm and are classified almost simultaneously by the measurementof the Kepler energy inside the lunar SOI. Then, both trajectories escape to-gether from the Moon realm to the Earth realm soon after their classification.So, clearly, no capture-escape behavior occurs, given that the dynamical behav-iors of the stable and unstable orbits do not differ. Moreover, the trajectory

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Figure 3: (a) Projection onto the x-y plane of the stable (black solid) and theunstable (red dashed) orbits of the S-E transition described in Case 1. The insetshows a magnification of the trajectories around the initial conditions (indicatedby ICs). The grey dotted area depicts the lunar SOI and the black + represent

the equilibrium points. (b) hK of P3 w.r.t P2 as a function of the dimensionlesstime for the stable (black solid) and the unstable (red dashed) trajectories. Theblue labeled with S,U indicates where the stable and the unstable trajectoriesare classified.

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Figure 4: (a) Projection onto the x-y plane of the stable (black solid) and theunstable (red dashed) orbits of the S-E transition described in Case 2. Theinitial conditions are indicated by ICs, the grey dotted area depicts the lunarSOI and the black + represent the equilibrium points. (b) hK ofP3 w.r.t P2 and(c) r2 as a function of the dimensionless time for the stable (black solid) andthe unstable (red dashed) trajectories. The blue labeled with S,U indicateswhere the stable and the unstable trajectories are classified.

said to be stable by Definition 1 is not consistent with qualitative aspects of thefull cycle orbits that may be required or appropriate for a temporary capture

orbit in practical applications. Once again, the stability criteria of Definition 1,namely, the association of the geometrical criterion (return to l() after a fullturn around the Moon) with the energy threshold defined by hK = 0, imply thatsimilar dynamical behaviors are artificially separated as stable and unstable.

We remark that when an escaping trajectory returns to l() before complet-ing a full turn around the Earth, and this returning point of the phase space issuch that Eq. 11 is satisfied, the algorithm produces a artificial stable classifi-cation, in the sense that a capture behavior is assigned to a transit trajectorythrough L1 or L2 that wanders through the exterior or the Earth realms beforeits classification.

4.2 Geometrical criteria

Case 3: A stability transition can appear when the return to l() criterionimplies that the classification of similar trajectories in the phase space is per-formed at very different characteristic times. As an example, consider the S-G2transition shown in Figure 5. In this case, the trajectories classified as stableand unstable have very similar long-term behaviors. However, one trajectoryreturns to l() transversally at Ts = 2.237 d.t.u. and is classified as stable,while its first outer radial neighbor is only almost tangent to the radial line, soit continues up until it completes a full turn around the Earth at Tu = 11.160d.t.u. and is classified as unstable due to primary interchange through L2. Boththe stable and the unstable solutions transit through L2 to the exterior realmsoon after the stable orbit is classified and the trajectories remain together in

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Figure 5: (a) Projection onto the x-y plane of the stable (black solid) andthe unstable (red dashed) orbits of the S-G2 transition described in Case 3.The forbidden region for Cs is shown in grey, the approximated lunar SOI isdepicted by the grey dotted area, and the equilibria are marked with +. (b)

Magnification of (a) showing the l() line segment (brown straight line) and thepoint at which the stable trajectory is classified. (c) hK of P3 w.r.t P2 and (d)r2 as a function of the dimensionless time for the stable (black solid) and theunstable (red dashed) trajectories. The blue labeled with S indicates wherethe stable trajectory is classified.

phase space for a long interval of time after that. Also, the time histories ofhKof the two trajectories are almost indistinguishable.As expected, we also find that this mechanism produces transitions of type S-Eand S-G1, and observe pairs for which the stable and the unstable orbits transittogether through L1 into the Earth realm soon after the stable orbit is classified.

Case 4: The requirement of a complete turn around the Earth can alsocause inappropriate transitions. Take, for example, the G2-S transition shownin Figure 6. Both trajectories leave the Moon realm through the neck regionaround L2 and visit the exterior realm before their classifications. A capturebehavior is erroneously assigned to one of the trajectories at Ts = 15.022 d.t.u.,when it returns to l() inside the lunar SOI, just before completing 360 aroundthe Earth. Conversely, its neighbor, although presenting a remarkably similarbehavior, completes a full turn around the Earth just before returning to l(),thus being classified as unstable. This incorrect transition appears because therequirement of completing a full turn around the Earth postpones the classifica-tion of orbits which clearly present escaping behavior. Additionally, the stable

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Figure 6: Projection onto the x-y plane of trajectories of the transition of typeG2-S described in Case 4. The stable trajectory is shown as a black solid curve,while the unstable trajectory is shown as a red dashed curve. The forbiddenregion for Cs is shown in grey, the approximated lunar SOI is depicted by thegrey dotted area, and the points of equilibrium are marked with +.

classification inconveniently neglects the large distance from P3 to P2 as thetime goes by.

4.3 Collision and close approach to P2

The force acting on P3, as well as the particles velocity, increases as P3 ap-proaches the vicinity of one of the primaries, going to infinity as the distancebetween the bodies goes to zero. Considering the punctual mass idealization ofthe PCR3BP, collision with the primaries occurs when r1 or r2 is zero. So, theequations of motion present non-essential singularities that can be eliminatedthrough regularization.

We find that a close approach to P2 can produce a stable-unstable transitiondue to the divergence of two close orbits in the phase space, mainly of theirvelocities, as they approach the singularity. Particularly, there are cases in whichthe orbit at the WSB obtained at the limiting case (r2 0) collides with thecenter of mass of P2, playing the role of an effective dynamical separatrix.

Case 5: Figure 7 displays a S-E transition observed for implementationA, which involves collisional orbits in implementation B. In this example, asshown in frames (b) and (c), we observe that the trajectories diverge in thephase space when they approach the Moon. However, both trajectories leavethe lunar SOI almost simultaneously and through the same exit, soon afterthe classification of the stable orbit. Thus, once again, the stable-unstableclassification is not associated to actual capture-escape behavior. Moreover,given that the initial conditions generate collisional orbits in implementation B,the stability boundary is unsafe. Specifically, one orbit is classified as stable

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because it completes one turn around the Moon at Ts = 5.538 d.t.u., and the

Kepler energy is measured inside the lunar SOI (to be more precise, inside thefinite size of the Moon). Conversely, the other orbit fails to complete a turnin this region and only satisfies this requirement at Tu = 12.108 d.t.u. in theexterior realm of the Earth-Moon system with hk > 0. The graph of hk asa function of t shows that both trajectories would be classified as stable if themeasurement of the Kepler energy was performed before t 5 d.t.u. Otherwise,both would be unstable. The extensive investigation of transitions along the -r2 plane shows that transitions of types S-G1 and S-G2 are also determinedbecause of collision.

4.4 Invariant manifolds

Stable-unstable transitions can be determined by the proximity of trajectories to

the stable manifold Wsk of a Lyapunov orbit k, k = 1, 2, around the collinearequilibria Lk. In the desired capture-escape situation, for the correspondingenergy levels, the unstable orbit is a transit solution inside the j-th Poincarecut ofWsk of k, while the stable orbit is a solution outside of the 1-st, . . ., j-thPoincare cuts of Wsk (non-transit for these specific cuts). Note that the stableorbit can be a transit solution for a m-th Poincare cut of Wsk , with m > j.

Although the Jacobi constant C is not kept fixed along l(), by fixing andrefining the radial step between a given stable initial condition ps, with Jacobiconstant Cs, and its first unstable neighbor pu, with Jacobi constant Cu, wehave that |Cs Cu| 0 (or Cs,u C) as r2 0. So, for the ideal transit-non-transit transition, we can infer that the orbit asymptotic to k(C

) lies atthe stability boundary. We call Ts the interval of time after which the trajectory

generated by ps

returns to l() with hK < 0. As r2 diminishes, the successiveinitial conditions ps generate trajectories that approach the orbit asymptotic tok(C

) and, as a result, Ts increases.Some aspects of this type of transition have already been explored in recent

literature when some topological conditions are satisfied, restricted to the Jacobiconstant interval 3.15 < C < C2 = 3.18416414 (Belbruno et al, 2010). Here, wewill treat and exemplify it unrestrictedly. So, besides the typical well-behavedcapture-escape situation, our investigation contemplates cases with the occur-rence of collisions, homoclinic and heteroclinic connections, and also stabilityclassification inconsistent with actual capture-escape behavior.

Case 6: In Figure 8 (a), we present a S-G1 transition determined by theproximity of the stable manifold of 1(C

), C = 3.17323193, |CsCu| 108.For the sake of visualization, the first Poincare cuts of the neighboring stable

and unstable trajectories are shown in Figure 8 (b) for r02 >> r

2 , such thatCs = 3.17432874 and Cu = 3.17227383. The first Poincare cuts of the stablemanifolds Ws

1(Cs) and Ws

1(Cu) associated to 1(C

s) and 1(Cu), respectively,

are also shown. For r02

, we find that Ts = 2.139 d.t.u., while Ts increases to6.172 d.t.u. for r2 . This kind of transition implies that the stable captureorbits approach the Lyapunov orbit around L1 or L2, which means that P3 istaken to the limits of the lunar SOI before completing a full turn around theMoon.

Case 7: When the invariant manifolds of Lyapunov orbits are no longerhomeomorphic to two-dimensional cylinders due to the occurrence of homoclinicand heteroclinic connections (Llibre and Simo, 1980; Koon et al, 2000; Gidea

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Figure 7: (a) Projection onto the x-y plane of the stable (black solid) and theunstable (red dashed) trajectories of the S-E collisional transition described inCase 5. The forbidden region for Cs is shown in grey and the equilibrium pointsare marked with +. (b) Magnification of (a) in a region near the Moon (greyball). ICs labels the initial conditions and the numbered arrows indicate thesense of the motion in temporal sequence along the stable (black solid) andthe unstable (red dashed) trajectories. (c) Projection onto the x-y plane ofthe stable (black solid) and the unstable (red dashed) trajectories. The arrowsindicate the sense of the motion along the trajectories. (d) hK of P3 w.r.t P2and (e) r2 as a function of the dimensionless time for the stable (black solid)and the unstable (red dashed) trajectories. In each frame, the blue labeledwith S (U) indicates where the stable (unstable) trajectory is classified.

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Figure 8: (a) Projection onto the x-y plane of the stable (black solid) and theunstable (red dashed) trajectories of the S-G1 transition described in Case 6.The stable-unstable separation is determined by the left branch of the stablemanifold (green) of 1(C

) for r2 = 2.60148651 109. The forbidden region

for C is shown in grey and the approximated lunar SOI is depicted by thecurve labeled with rH. (b) Now, the stable-unstable transition is shown forr0

2at the Poincare section x = 1 + , x > 0. The first Poincare cuts of the

stable manifolds of 1(Cs) (black solid curve) and 1(C

u) (red dashed curve)are shown along with the Poincare cut of the non-transit stable orbit (black x)located outside the cut of Ws

1(Cs) (black dotted area) and the Poincare cut of

the transit unstable orbit (red +) located inside the cut of Ws1 (Cu) (red dashed

area).

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and Masdemont, 2007), these separatrices become non-smooth, in such a way

that small variations in the initial conditions can lead to uncertainty in the exitof escaping orbits between the necks around L1 and L2 (Alligood et al, 1996).For some regions in the set of initial conditions, we observe that consecutivetrajectories along a radial segment leave the Moon realm alternately throughL1 or L2 as the grid of initial conditions is refined. In such a scenario, if anorbit is able to fulfill the stability requirements of the algorithm before escap-ing, a temporary capture state is assigned to this orbit, producing a stabilitytransition. An example of such a transition is illustrated by Figure 9 (a), wherea triad of G1-S-G2 orbits is observed. The orbit detected as stable has higherenergy than the stable orbit of Case 6 and is between an escaping orbit throughL2 and an escaping orbit through L1. We observe that the stable trajectory alsotransits through L1 soon after its classification and that its spatial extensionprevious to the classification exceeds the lunar SOI. We will refer again to this

transition in Section 5.2, where we investigate successive refinements of the gridof initial conditions.

Figure 9: (a) Projection onto the x-y plane of the triad of G1-S-G2 orbits ofCase 7. The stable trajectory is shown as a solid black curve and the pointof classification is marked by a blue . The dashed red and the dot-dashedgreen curves correspond to the G1 and the G2 trajectories, respectively. (b)Projection onto the x-y plane of the S-G1 pair of Case 8. The stable trajectoryis shown as a black solid curve until its classification and as a dot-dashed blackcurve after the point of classification. The unstable trajectory is shown as a

dashed red curve. In both frames, the forbidden region at the energy level ofthe stable trajectory is shown in grey and the lunar SOI is depicted as a greydotted area. The points of equilibrium are indicated by black +. In frame (b),the arrows along the trajectories indicate the sense of the motion in temporalsequence.

Case 8: The stable-unstable transitions due to the separatrix characteristicof the invariant manifolds of Lyapunov orbits often involve collisional trajecto-ries. Thus, these unsafe transitions are not applicable, even when, contrastingwith the Case 5, the transitions now correspond to actual capture-escape be-havior. Figure 9 (b) displays the S-G1 transition observed for implementation

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A, which involves collisional orbits in implementation B. Both trajectories enter

the finite size of the Moon before their classification (in this case, a collisionalorbit with the center of mass of the Moon exists between this stable-unstablepair). Then, one orbit transits to the Earth realm and is classified as unstable,while the other completes a full turn around the Moon and is classified as stable.The stable trajectory circles the Moon a second time before escaping throughL1.

Case 9: Even if two neighbor orbits belong to the j-th Poincare cut ofWsk (both transit in the j-th Poincare iteration), in the corresponding energylevels, the proximity to k can cause divergence of these trajectories in thephase space. This fact combined with other effects (possibly some constructionalmechanisms) can also lead to the classification of one orbit as stable and theother as unstable, although both orbits may have the same long term behaviorand neither correspond to a captured state, thus rendering a artificial transition.

Table 1: Specific information about the stability transitions described in Section4. The second column indicates if the initial conditions have clockwise () orcounterclockwise (+) velocities. The third column gives the transition positionin the -r2 plane of initial conditions. The fourth column indicates the radialstep used to visualize the transition. The fifth and the last columns show,respectively, the classification time and the Jacobi constant value of the stabletrajectory. Both r2 and T

s are given in dimensionless units of the EM-system.

Set (, r2

) r2 Ts Cs

Case 1 (0.640, 0.02966207) r02/1000 3.105 2.95179476S-E

Case 2 (1.941, 0.02853277) r02/1000 25.270 2.95637094S-E

Case 3+ (0.931, 0.02814255) r0

2/10 2.237 3.05912120

S-G2

Case 4+ (0.196, 0.07879292) r02/10 15.022 3.07051220G2-S

Case 5 (1.753, 0.01425078) r02/10 5.538 3.01263962S-E

Case 6+ (1.229, 0.00656586) 2.60148651 109 6.172 3.17323193

S-G1

Case 7+ (0.909, 0.04042663) r02/100 8.308 3.05744295G1-S-G2

Case 8 (0.760, 0.00738293) r02/10 6.172 3.10204647S-G1

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5 The global structure of the algorithmic WSB

and the detection of capture-escape behavior

In this section we show how abundant are the transitions presented in Sec-tion 4. Then, we investigate smooth and fractal-like portions of the stabilityboundary. These analyses reveal that large portions of the algorithmic WSBhave no meaning concerning capture-escape behavior and identifies the portionswhich correspond to authentic transitions.

Additionally, we indicate necessary modifications to the stability definitionand outline a procedure to provide the doubtless detection of pairs of trajectoriesthat correspond to capture-escape transitions.

5.1 Spatial distribution of transitions along the boundary

Following, we discuss the spatial distribution of the predominant factors causingthe stability classification for the sets of initial conditions with e = 0.9 and bothpositive and negative velocities. We have identified portions of each boundarywhere the stability classification is mainly due to common factors and labeledthese regions with italic roman numbers in Figures 9 and 11.

Boundary set with negative initial velocity: The stability boundary forthe set of initial conditions with negative velocity and e = 0.9 is represented inFigure 10 in the -r2 plane, which allows a better identification of the variousportions where different mechanisms occur.

In Figure 10, the three portions labeled with I correspond mostly to thetransitions described in Cases 6, 7, and 8. Approximately half of the stable-

unstable pairs in region I correspond to unsafe transitions. Most transitions inthis region are of type S-E with the Kepler energy of the unstable trajectoriesusually measured outside the lunar SOI (sometimes even in the exterior realm).As discussed in Section 4.1, by construction, this measurement is artificial be-cause the two-body approximation is no longer valid and can lead to incorrectdetection of transitions.

In region I, besides S-E transitions, there are also transitions of type S-G1(for some values of in [0, /4] and [7/4, 2)) and S-G2 (for some values of in [3/4, 3/2]).

For e = 0.9 and negative initial velocity, we observe that the WSB subset forwhich the stable-unstable transitions are authentic capture-escape transitions ismostly in region I. In this case, the transitions present two characteristics:(1) are the first stability transitions along each radial line, and (2) occur atlow altitudes (typically 300 to 600 km)4. These authentic transitions usuallycorrespond to the trajectories with the lowest energy values which allow inner

4Our observations that first transitions correspond to capture-escape behavior due to man-ifold separatrix behavior are qualitatively corroborated by Romagnoli and Circi (2009). Thefirst part of their work refers to a version of the WSB algorithmic definition for the PCR3BPimplemented for the set of initial conditions with e = 0.0 and positive initial velocity usinga shorter integration time of 35 days (reminding that one month corresponds to approxi-mately 2 d.t.u.) in which they consider only first transitions. We note that orbits whichcould be classified as stable by our implementation are classified as unstable due to exceed-ing the maximum integration time in this particular implementation of Romagnoli and Circi(2009), consequently the shortened integration time causes the stable set to shrink, shiftingand diminishing the stability boundary.

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Figure 10: The stability boundary for e = 0.9 and negative initial velocityrepresented in the -r2 plane of initial conditions. The colors identify portionsof the boundary where the classification is mainly due to a common mechanism.The grey points indicate the collisional set of implementation B.

(through L1) or outer (through L2) transfers. We remind that, for lower eccen-tricity values, they occur at higher altitudes, while for higher values of e theyoccur closer to the Moon.

Now, we look at the elongated pieces of the boundary labeled with II, whichcorrespond to S-E transitions. However, unlike S-E transitions in region I, nowthe stable-unstable pairs correspond to the spurious transitions described inCases 1 and 2. As seen in Figure 10, region II corresponds to large portionsof the stability boundary. So, the current stability definition imposes a cutoffcondition which generates an incorrect large subset of the boundary.

The third observed region, labeled with III, corresponds to the two portionsof the boundary that are at the base of the elongated pieces labeled with II.Again, most transitions in this region are of type S-E. Many transitions are dueto the geometrical criteria (Cases 3 and 4), and also due to close approach tothe Moon (Case 5). In this last case, typically, the stable trajectories stay insidethe lunar SOI until the instant of classification, often being classified within thefinite size of the Moon. Some transitions like the ones described in Cases 7 and8 can be found in this region as well.

Now we move to the thin strips labeled with IV. These subsets of theboundary are accompanied by the collisional subset and extend from low tohigh altitudes. Most of the transitions are of type S-G3. Both the stable andthe unstable trajectories correspond to high energy orbits that wander far awayfrom the lunar SOI. Usually it takes a long time span before each trajectory isclassified. Even though the stable and the unstable orbits differ considerably in

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the phase space, both of them present escape behaviors. Moreover, the stability

classification is highly conditioned to the fulfillment of the geometrical criteriaof Definition 1.Finally, we look at the portions labeled with V. In this case, both stable and

unstable trajectories have long classification time and present highly nonlinearbehavior. In this region, we find several cases of trajectories that wander inthe exterior realm, before returning to l() and being erroneously classified asstable due to the measurement of the Kepler energy (combination of Cases 1 to4). Again no capture behavior is observed for the stable orbits.

In Figure 11 (a) and (b), respectively, we see the dependence of the Jacobiconstant with and with r2. From frame (a), it is possible to associate lowervalues of C with regions II, IV, and V of Figure 10. These initial conditionscorrespond to high energy trajectories for which a broad extension of the phasespace is accessible. In Figure 11 (b) we clearly see that the energy increases with

r2. Also, only the low altitude portions of the boundary (regions I and III)correspond to high values of C, that is, low energy levels for which the necksaround L1 and L2 are well defined. In Figure 11 (c), we plot the classificationtime of the stable-unstable pair versus . Regions III, IV, and V correspondto pairs with long classification time, while region I corresponds to stable tra-jectories that are classified after a short time with unstable neighbors that havelonger classification time. Also, it is possible to identify region II, for which theclassification time of the stable and of the unstable trajectories match.

Boundary set with positive initial velocity: Now, we discuss the stabilityboundary for the set of initial conditions with positive initial velocity and e = 0.9shown in Figure 12.

We start considering the portions of the boundary labeled with I and II.While most of the transitions in region I are of type S-G2, most of the transitionsin region II are of type S-G1. In these portions, the transitions are due to theseparatrix behavior of the invariant manifolds of the Lyapunov orbits around L1or L2, however, a large amount of stable-unstable pairs corresponds to unsafetransitions.

For e = 0.9 and positive initial velocity, the WSB subset for which thestable-unstable transitions are authentically related to capture-escape behaviorcorresponds mostly to regions I and II, i.e., the first stability transitions alongeach radial line and that occur at low altitudes.

The elongated arms, labeled with III, present transitions of type S-E, S-G1and S-G2. Again, many stable-unstable pairs correspond to unsafe transitions.For transitions of type S-E that occur at the bases of the arms, the classification

occurs as described in Case 3. Other transitions of type S-E along the arms aresimilar to Cases 5. In these cases and also for transitions of type S-G1 andS-G2, although the trajectories may differ considerably in the phase space, theclassification does not often refers to actual capture-escape behavior. However,some authentic transitions like the ones described in Cases 7 and 8 can also befound in region III.

Finally, the two portions labeled with IV correspond to transitions of typeS-G1 and S-G2 for which both stable and unstable trajectories have long clas-sification time and are spatially extended, presenting highly nonlinear behavior(like the trajectories shown in Figures 4 and 6).

In Figure 13 (a) and (b), respectively, we present the distribution of the

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Figure 11: The distribution of the Jacobi constant at the stability boundary as afunction of (a) , and (b) r2; for initial conditions with negative initial velocitiesand e = 0.9. The colors refer to the portions of the boundary identified inFigure 10. In frame (b), the lines labeled with C3 and C4,5 indicate the valueof C at L3 and L4,5, respectively. (c) Classification time in dimensionless unitsof stable (black) and unstable (cyan) trajectories of boundary transitions as afunction of .

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Figure 12: The stability boundary for e = 0.9 and positive initial velocity repre-sented in the -r2 plane of initial conditions. The colors identify portions of theboundary where common predominant factors cause the stability classification.The grey points indicate the collisional set of implementation B.

Jacobi constant with and with r2. Comparing it with Figure 11 (a) and (b),we see that, initial conditions with positive velocity along the stability boundarybelong to lower energy levels than initial conditions with negative velocity. Thelow altitude portions of the boundary, regions I and II in Figure 12, correspondto the lowest energy levels. In Figure 13 (c) we plot the classification timeof all boundary trajectories versus along with the classification time of itsouter radial neighbor. We see that first transitions at low altitudes correspondto stable trajectories that are classified after a very short time with unstableneighbors that have a slightly longer classification time, while the distributionof the classification time along the arms is scattered. Finally, it is p ossibleto identify region IV, for which the classification time of the stable and theunstable trajectory are comparable.

5.2 Refined extraction

Some of the transitions shown in the examples of Section 4 and many of thetransitions inspected along the boundary to identify the regions presented theSubsection 5.1 were characterized using a grid of initial conditions finer thanthe original one built with r0

2. This was needed to get a better identification

of the elements which are determinant in each transition.In order to extract a finer boundary set, a refinement procedure was per-

formed in the radial segment between the stable and unstable points of a de-tected transition, considering a smaller radial step r2/n and classifying thenew subset of n 1 initial conditions. Such a procedure was found to be more

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Figure 13: The distribution of the Jacobi constant at the stability boundary as afunction of (a) , and (b) r2; for initial conditions with positive initial velocitiesand e = 0.9. The colors refer to the portions of the boundary identified inFigure 12. In frame (b), the lines labeled with C1 and C2 indicate the valueof C at L1 and L2, respectively. (c) Classification time in dimensionless unitsof stable (black) and unstable (cyan) trajectories of boundary transitions as afunction of .

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effective than a simple bisection process, which allows the tracking of only one

transition.By implementing the refinement procedure, we distinguish between smoothand fractal-like grid-dependent portions of the weak stability boundary.

For smooth portions of the boundary, the refinement procedure between thestable-unstable pair leads just to a finer solution of the same type. Take, forexample, = rad, e = 0.9 and positive initial velocity. For a radial incrementr02, the first stable-unstable transition along l() occurs at r

= 5.4318 103

and is of type S-G1. In this case, only new points belonging to S or G1 appearwith the refinement. Then, up to a resolution of r2 = r

02 10

4, we findr = 5.9261 103, which corresponds to an altitude of approximately 540 km.

Typically, smooth portions of the boundary are associated with large stableregions and related both to collisional and non-collisional orbits. They arevery common in the boundary set with negative initial velocity, especially along

region II and pieces of region I of Figure 10. For the boundary set with positiveinitial velocity, smooth portions are common in regions I, II and pieces whereS-E transitions occur within region III of Figure 12.

The second situation refers to grid-dependent portions of the boundary. Inthis case, new stable-unstable pairs appear between the original transition whena refinement procedure is performed, so these fractal-like portions cannot beresolved by a simple bisection process and are conditioned to the grid resolution.Usually, both dynamical and constructional effects are combined to generatethe fractal-like structure of the boundary. These fractal-like portions are notnecessarily related to the sparse regions of boundary points obtained with anunique grid of initial conditions.

Take, for example, the case of initial conditions with positive initial velocity,

e = 0.9 and = 0.909 rad. For r0

2, we detect a S-E transition at r2 =3.8991 102. The next unstable neighbor of the isolated E point is an initialcondition that belongs to the unstable subset G2. Figure 14 displays the stabilityconfiguration on the radial segment between the original S and the G2 points forthe three radial steps considered besides r02. Figure 14 (a) presents the originalS-E-G2 sequence. Here, both S and E are collisional orbits of implementationB. With the first refinement (Figure 14 (b)), one new stable state appears alongwith new G1 unstable points, defining a new transition sequence of type G1-S-G2 with just one new boundary point. This transition occurs as described inCase 7. In the second refinement (Figure 14 (c)), other two isolated stable statesappear, along with several other unstable points, leading to four new stabilitytransitions. Particularly, the new transition sequence G1-S-G2 corresponds tothe triad of solutions shown in Figure 9 (a). At this level of refinement, wealso find transitions which do not correspond to capture-escape behavior. Forexample, the first transition from left to right, which of type S-G1, is similarto Case 5. In addition the E-S pair corresponds to two similar escaping orbitswhich are classified outside the lunar SOI (Case 1). In the last refinement(Figure 14 (d)) no new stability transitions appear.

As the energy of the initial conditions grows, the zero velocity curves departfrom the lunar region, allowing a larger spatial extension of the trajectories.When submitted to the stability classification, the spatially extended nonlinearbehavior of trajectories produces a large diversity of classifications as finnergrids are considered in a small region of initial conditions, accounting for thefractal-like characteristic of the algorithmic WSB.

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Figure 14: Successive refinement of a radial segment containing a stable-unstabletransition in the set of initial conditions with positive initial velocity and e = 0.9for (a) r0

2, (b) r0

2/10, (c) r0

2/100, and (d) r0

2/1000. The circles represent

stable initial conditions, while the polygons depict different types of unstableinitial conditions as indicated by the label. The numbers inside each figureindicate the amount of consecutive initial states with the same classification.

Grid-depended transitions are common along regions III and V, and along

the highest altitudes of region I of Figure 10. In Figure 12, these transitionsappear at highest altitudes of regions I and II, and along region III.

With these observations, we conclude that the stability definition must bereviewed in order to avoid undesirable spurious transitions, allowing a bettercomprehension and possible applications of grid-dependent transitions whichare authentic in the sense of capture-escape.

5.3 Proposal of alternative stability criteria for the detec-

tion of capture-escape transitions

The previous results show that the stability Definition 1 is not suited to solelydetect capture-escape behavior around the Moon.

Specifically, Cases 2 and 4 illustrate that, even inside the lunar SOI, themeasurement of hK does not guarantee that the behavior of the stable trajec-tory corresponds to a temporary capture if the time history between the initialcondition and the final state is neglected. Also, the threshold provided by nullhK renders artificial stable-unstable transitions that do not correspond to actualcapture-escape behavior.

We propose an alternative stability boundary definition based on the effec-tive detection of capture-escape transitions, as stated below: Sets of initial conditions with null radial velocity and fixed Jacobi constantmust be considered. We verified that authentic capture-escape transitions cor-respond to the lowest altitudes above the energy threshold for escape through

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L1 and L2. This indicates that there is an ideal range of the Jacobi constant

values that allow well defined capture-escape behavior. The measurement of hK must be replaced by the distance of the trajectoryto the Moon after a complete revolution around that primary. The tracking of r2(t), the distance from P3 to P2 as a function of the time,must be used as an instability criterion instead of the requirement of a completeturn around the Earth. Namely, a threshold value rlim2 can be established, suchthat if r2(t) exceeds r

lim2 , the trajectory must be classified as unstable. This

new condition fixes, for example, the inadequate classification of the trajectorysaid to be stable in Case 4 and also shortens the time needed to classify theunstable one. The maximum integration time of the stability classification must be dimin-ished. If the maximum integration time is diminished or if the distance of thetrajectory to the Moon is monitored, most of the large region of spurious tran-

sitions labeled as V in Figure 10 will disappear. Then, once the stability classification of single orbits is performed, pairs oftrajectories that are candidate to capture-escape transitions must be certifiedby tracking the distance between these trajectories from the instant at whichthe first of them is classified up to the classification of the second one. If theseorbits remain together in the phase-space they do not correspond to a capture-escape transition. Otherwise, an authentic transition is obtained. This avoidsthe inconsistent detection of transitions like the ones shown in Cases 1, 3 and5, for example. Also, we remark that the length of time between the instantsat which the captured and the escaping trajectories return to l(), although notconclusive, provides a good indication of capture-escape behavior. Finally, in order to select just safe solutions, transitions involving a stable

orbit that enters the finite size of the Moon before its classification must beneglected too.

The investigation of this new definition is the goal of a future contribution.

6 Conclusions

In this paper, a detailed characterization of stable-unstable transitions producedby the Weak Stability Boundary algorithmic definition was carried out withno energy restrictions to verify the effectiveness of the stability classificationregarding capture-escape behavior detection.

We described several types of transitions which occur due to other mecha-nisms besides the separatrix role of the invariant manifolds of periodic orbits

around L1 and L2. Specifically, we showed how spurious transitions (not relatedto capture-escape) are produced when the nonlinear behavior of the dynamicalsystem is submitted to the current stability classification procedure. Then, weshowed that these spurious transitions are not rare, but constitute a very signif-icant part of the boundary. In order to illustrate this statement, we examinedhow these diverse transitions are predominantly distributed along the stabilityboundary for a high eccentricity value, namely, e = 0.9. In addition, a re-finement procedure was implemented, showing that fractal-like structures canappear due to the combination of several possible mechanisms, and not onlydue to the dynamical separatrix effect.

Thus, we conclude that the current algorithmic WSB does not provide an

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unrestricted substitute for the hyperbolic invariant manifold structure, and that

the constructive procedure should be revised in order to solely extract adequatecapture-escape separatrices.Finally, in order to avoid the extraction of spurious boundary transitions

from the initial condition sets, we proposed an alternative stability boundarydefinition and discussed the accurate detection of capture-escape behavior. Theinvestigation of the effectiveness of the new stability criteria is the goal of afuture work.

Acknowledgements This work was partially supported by CAPES. Theauthors are grateful to Prof. Carles Simo for valuable discussions and for hissuggestions for the presentation of this paper.

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