Characterization of Trajectories Near the Smaller Primaryin Restricted Problem for Applications

Diane Craig Davis∗ and Kathleen C. Howell†

Purdue University, West Lafayette, Indiana 47907-2045

DOI: 10.2514/1.53871

In the restricted problemof three bodies, the gravity of the distant larger primary affects orbits in the vicinity of the

smaller primary. When these effects are significant, predicting the behavior of a trajectory given its initial state is

challenging. To effectively select a trajectory or arc as part of a trajectory design process, one that satisfies a given

mission requirement, it is necessary to organize and simplify the design space. Periapsis Poincaré maps yield useful

information about the evolution of various orbits over both short- and long-term propagations. The maps allow the

classification of various categories of escape and captured orbits based on their initial conditions and provide a tool

for the methodical selection of orbits with desired characteristics.

I. Introduction

I N THE restricted problem of three bodies, a particle ofinfinitesimal mass moves under the gravitational influence of two

primary bodies, P1 and P2. Orbits near the smaller primary P2 areaffected to various degrees by the gravity of the larger body. As theorbits become sufficiently large and evolve toward the vicinity of theL1 and L2 Lagrange points, the gravitational force due to the largerprimary may no longer be considered merely a perturbation.However, the trajectories can be characterized in terms of themagnitude and orientation of the radius vector at periapsis relative toP2. Consider a set of large P2-centered trajectories in the circularrestricted three-body problem (CR3BP). With apoapses near thezero-velocity curves and energy levels such that the zero-velocitycurves are open, such trajectories exist in a regime where the gravityof P1 can cause escape from the P2-centered orbit or impact into thesurface of P2, immediately or after many revolutions. However, acollection of orbits remains bounded for extended periods of time.Some of these captured orbits display notable characteristics either inthe rotating or the inertial reference frames that lead to a betterunderstanding of the evolution of trajectories in this regime; thisunderstanding of the dynamical structure is valuable for trajectorydesign. In this investigation, the key to predicting the behavior oftrajectories in the vicinity of the smaller primary is the location ofperiapsis. Periapsis Poincaré maps are applied in the CR3BP tocondense extensive information on trajectory behavior into a formuseful for mission design applications. The development ofmultibody mission design tools is particularly valuable with theincreasing prevalence of missions using trajectories in environmentswhere multiple gravitational fields are simultaneously significant,such asGenesis [1] andArtemis [2]. Classifying various categories ofescaping and captured orbits facilitates the selection of trajectorieswith certain desired characteristics. In addition to spacecrafttrajectory design, however, a better characterization of orbits in thisregime can be applied to the behavior of natural bodies such as

comets and moons with regard to their temporary or long-termcapture.

The impact of the gravity of a distant larger primary on the designof spacecraft trajectories is familiar, of course, either as a perturbationor as a dominant force that shapes the orbit. Inclusion of such agravitational force as a perturbation to a reference trajectory has along history, and the study of its influence is extensive. A variety ofmethods can be employed for the analysis of lower-altitude orbits.TheLagrange planetary equations are commonly used to describe theeffects of gravitational perturbations on the orbital elements [3,4]. Infurther investigations, Villac et al. [5] produce an extension to thisapproach for larger orbits using Picard’s method of successiveapproximations [6] with the Hill three-body potential. Their func-tions estimate the changes over a single revolution in the orbitalelements, particularly semimajor axis, eccentricity, and inclinationdue to planetary perturbations on an orbiter about a moon. Whilehighly accurate in many applications, for orbits sufficiently large, thepredictions on long-term orbital behavior begin to lose accuracy. Tocombat the limitations of perturbation methods, numerous strategiesexploit the dynamics in the CR3BP or the Hill restricted three-bodyproblem (HR3BP) to explore the long-term behavior of trajectoriesnear the smaller primary in the restricted three-body problem. Hénon[7–11], a pioneer in the study of multibody orbits, investigates theexistence and stability of periodic and nonperiodic orbits aboutP2 inthe HR3BP. His work extends from the late 1960s through the mid-2000s. Advances in computing power over time allow morecomprehensive searches for periodic trajectories. Russell [12]employs a grid search to catalog numerous planar and three-dimensional symmetric periodic trajectory families in the vicinity ofEuropa in the Jupiter–EuropaCR3BP. Tsirogiannis et al. [13] expandon Russell’s work, presenting an updated, adaptive grid searchapproach alongwith a classificationmethod for the resulting periodicorbits.

Themanifolds associatedwith theL1 andL2 Lyapunovorbits havealso been increasingly used to predict the long-term behavior oftrajectories that originate near the smaller primary in the circularrestricted three-body problem [14,15]. The weak stability boundaryconcept, developed by Belbruno [16], offers an additional meth-odology for the design of transfer trajectories. Further discussion onthe weak stability boundary concept appears in García and Gómez[17]. In addition, the use of Poincaré maps to identify trajectorieswith various long-term behaviors is effective, as demonstrated byAnderson and Lo [18]. In particular, periapsis Poincaré maps are firstdefined and introduced by Villac and Scheeres [19] to relate atrajectory escaping the vicinity ofP2 back to its previous periapsis inthe planarHill problem. The authors apply theirmaps to the design ofescape trajectories from a circular orbit about Europa and otherplanetary satellites. Paskowitz and Scheeres [20] extend thisanalysis, using periapsis Poincaré maps to define regions

Presented as Paper 2010-184 at the AAS/AIAA Space Flight MechanicsMeeting, San Diego, CA, 14–17 February 2010; received 1 February 2011;revision received 14 June 2011; accepted for publication 25 July 2011.Copyright © 2011 by Diane Craig Davis and Kathleen C. Howell. Publishedby the American Institute of Aeronautics and Astronautics, Inc., withpermission. Copies of this paper may be made for personal or internal use, oncondition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0731-5090/12 and $10.00 in correspondence with the CCC.

∗Ph.D. Candidate, School of Aeronautics and Astronautics, ArmstrongHall of Engineering, 701 West Stadium Avenue. Student Member AIAA.

†Hsu Lo Professor of Aeronautical and Astronautical Engineering, Schoolof Aeronautics and Astronautics, Armstrong Hall of Engineering, 701 WestStadium Avenue. Associate Fellow AIAA.

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 35, No. 1, January–February 2012

116

corresponding to the first four periapse passages after capture intoorbit about P2. Villac and Scheeres [21] also employ periapsisPoincaré maps in the design of optimal plane change maneuvers,using third-body forces to change the inclination of a trajectory withminimal �V. The periapsis Poincaré maps represent a significantstep forward in both the characterization of the design space near P2

and in the design of trajectories in its vicinity.Finally, the influence of the sun’s gravity on large orbits has been

investigated in terms of a set of quadrants in the rotating frame [22–24]. Depending on the location of an orbit apoapsis within thequadrants, solar gravity tends to elongate or circularize the orbit overthe course of one revolution. An understanding of the influence ofsolar gravity in terms of quadrants can be combined with periapsisPoincaré maps of trajectories about P2 to yield useful informationabout the characteristics of various orbits over a long-termpropagation.

This investigation considers both short-term and long-termbehavior of trajectories orbitingP2 in the CR3BP. Periapsis Poincarémaps are employed to define sets of initial conditions for whichtrajectories impact or escape the vicinity ofP2 through theL1 and L2

gateways, to identify regions of initial conditions for which thespacecraft will remain captured for extended periods of time, tolocate quasi-periodic trajectories in both the rotating and inertialframes of reference, and to identify trajectories with other notablecharacteristics, for example, quasi-frozen orbits in the inertial frame.The maps are then applied to a simple trajectory design problem todemonstrate their effectiveness as a preliminary trajectory designtool. While the methodology is applicable to any P1-P2 system, themajority of the analysis is performed in the sun–Saturn system,where the small mass parameter allows easy interpretation of thenear-symmetric maps. The Saturn–Titan system shares nearly thesame mass parameter. Other systems are also investigated to confirmthe applicability of the results. The investigation focuses on planarmotion, with some preliminary examination of 3-D trajectories.Periapsis Poincaré maps provide the tools for the prediction of thebehavior of large sets of trajectories based on their initial states. Thespecific purpose of this investigation is the use of periapsis Poincaremaps to characterize the design space for application to trajectorydesign, that is, to produce a framework for the methodical selectionof initial states that lead to trajectories the may satisfy variousmission requirements.

II. Background

A. Dynamical Model

Themotion of a spacecraft under the gravitational influence of twolarger bodies, such as the sun and a planet, can be described in termsof the CR3BP. In this model, the two primaries P1 and P2 areassumed to be orbiting their barycenter on circular paths, while thespacecraft is assumed to possess negligible mass. The equations ofmotion in the CR3BP possess five equilibrium solutions, or librationpoints. These consist of the three collinear points (L1,L2, andL3) andtwo equilateral points (L4 andL5). The libration pointsL1 andL2 aredepicted for the sun–Saturn system in Fig. 1. Note that themagnitudeof the Hill radius [22] is equal to

rH ���

3

�1=3

(1)

and is approximately equal to the distance between P2 and theLagrange points L1 and L2. A single integral of motion exists in theCR3BP. Known as the Jacobi integral, it is denoted

J� x2 � y2 � 2�

r� 2�1 � ��

d� v2 (2)

where v is the magnitude of the spacecraft velocity relative to therotating frame and d and r are the distances between the spacecraftand the first and second primaries, respectively. Note that the massparameter, as well as the position and velocity variables, arenondimensional quantities. The characteristic length l� is defined asthe distance betweenP1 andP2, the characteristicmassm� is the totalmass of the two large primaries, and the characteristic time is defined

as t� � � l�3Gm��1=2, where G is the gravitational constant. The Jacobi

constant restricts the motion of the spacecraft to regions in spacewhere v2 0; these regions are bounded by surfaces of zero velocity.In the planar problem, the surfaces reduce to the zero velocity curves(ZVCs). For values of the Jacobi integral higher than that associatedwith the L1 libration point, the ZVCs form closed regions around thetwo primaries. As the energy of the spacecraft is increased, the Jacobivalue decreases until, at theL1 value of the Jacobi integral, the ZVCsopen at the L1 libration point and the spacecraft is free to movebetween the two primaries. Similarly, when the value of the Jacobiintegral decreases to the value associated with L2, the ZVCs open atL2 (see Fig. 1) and the spacecraft may escape entirely from thevicinity of the primaries. A sample trajectory appears in Fig. 1. Theinitial state, defined at periapsis, lies at a given radius rp and anglefrom the rotating x-axis !r. With J given, the expression for theJacobi constantfixes themagnitude of the velocity. In the planar case,the velocity direction is selected such that the trajectory is prograde; apreliminary exploration of retrograde orbits appears in [24].

B. Tidal Acceleration by Quadrants

The direction of the perturbing acceleration due to the gravity ofP1 on a spacecraft in an orbit about P2 depends on the orientation ofthe spacecraft orbit relative to the two bodies. To facilitate theinvestigation of the solar gravitational influence, four quadrantscentered at P2 are defined in the rotating frame and appear in Fig. 1,following the convention established by Yamakawa et al. [23]. Thequadrants are defined in a counterclockwise fashion, with quadrant Ion the far side of the primary and leading it in its orbit. When thespacecraft orbit is viewed in this rotating frame, its orientation isdefined by the quadrant that contains the orbit apoapsis. The quadrantangle � is the angle between the positive x-axis and apoapsis, asdepicted in Fig. 1.

In the vicinity ofP2, the net tidal acceleration is generally directedoutwards from P2 along the x-axis. The effects of the tidalacceleration on a P2-centered orbit are greatest near apoapsis. Inquadrants I and III, the perturbations generally oppose the directionof motion in a prograde orbit. In quadrants II and IV, on the otherhand, the net perturbing acceleration at apoapsis is in the samedirection as the motion along a prograde orbit. As a result, the tidaleffects on an orbit are similar in diagonal quadrants. By comparingthe osculating orbital elements measured at two subsequentperiapses, these effects are quantized.Consider an eccentric progradeorbit sufficiently large to be impacted significantly by the gravity ofP1 but affected such that the perturbations do not cause the orbit tobecome retrograde or to escape the vicinity of P2. If apoapsis lies inquadrant I or III, tidal acceleration lowers periapse radius, decreasessemimajor axis, and increases eccentricity. Conversely, if apoapsisinstead lies in quadrant II or IV, tidal acceleration raises periapseradius, increases semimajor axis, and decreases eccentricity. Forfurther discussion of changes in orbital elements due to tidalacceleration, see Villac and Scheeres [5], Yamakawa et al. [23], andDavis et al. [24]. The solar gravitational perturbations are greatestwhen the orbit lies in the ecliptic plane. Also, within each quadrant,

Fig. 1 P2-centered trajectory with periapse angle !r and quadrantangle �.

DAVIS AND HOWELL 117

solar perturbations are at a maximum when apoapsis lies at approx-imately 45 from the sun–Saturn line, although the precise value isorbit-dependent.

C. Periapsis Poincaré Maps

After considering the effects of tidal acceleration on individualtrajectories, it is useful to generalize the results. One tool thatfacilitates exploration of the design space is the periapsis Poincarémap. The Poincarémap is commonly used to interpret the behavior ofgroups of trajectories, relating the states at one point in time to afuture state forward along the path. By fixing the value of the Jacobiintegral and selecting a surface of section, the dimensionality of theproblem is reduced by two; the four-dimensional planar problem isthus reduced to two dimensions. The surface of section can be a planein configuration space, corresponding, for example, to y� 0.However, another useful type ofmap is a periapsis Poincarémap,firstdefined and applied by Villac and Scheeres [19]. In this type of map,the surface of section is the plane of periapsis passage, defined by theconditions _r2 � 0 and �r2 > 0. Villac and Scheeres [19], as well asPaskowitz and Scheeres [20], employ periapsis Poincaré maps toexplore the short-term behavior of escaping trajectories and capturedtrajectories within the context of the Hill three-body problem withapplications to the Jupiter–Europa system. Building on these results,the short-term behavior of trajectories in the CR3BP is explored;subsequently, periapsis Poincaré maps are employed to extractinformation concerning the long-term evolution of largeP2-centeredtrajectories.

The current focus is on trajectories characterized by Jacobi valuesjust below that corresponding to the L2 libration point. For a givenJacobi value, the second condition defining periapsis, �r2 > 0,outlines a region in the vicinity of P2. Within the line defined by�r2 � 0 (see Figs. 2 and 3), a state satisfying thefirst condition, _r2 � 0,is a periapsis; outside the line, it is an apoapsis. For each positionwithin the periapse region, the velocity is defined such that the initialstate is a periapsis along a P2-centered trajectory. Each orbit is thenparameterized by its initial polar coordinates: periapse radius rp0 andthe angle with respect to the rotating x-axis, defined as !r0 andrepresented in Fig. 1. Impact trajectories are defined as thosepossessing a position vector, at any time, that passes at or within theradius of the body P2.

III. Short-Term Behavior Near P2

A. Prograde Planar Behavior

An investigation of short-term orbit behavior is based on theconstruction of periapsis Poincaré maps. For a given Jacobi value, aregion on the surface of section is isolated within the zero-acceleration line that forms the boundary between periapses andapoapses [25]. A set of initial periapse conditions is then definedwithin the region. Each point within the region corresponds to theinitial condition associatedwith a specific prograde, planar trajectoryabout P2; each trajectory is propagated forward in time to itssubsequent periapsis. Four possible outcomes of this propagationexist: the spacecraft impacts P2; the spacecraft escapes out the L1

gateway; the spacecraft escapes through the L2 gateway; or, thespacecraft remains captured near P2, that is, it continues to evolvewithin the ZVCs. As an example, consider a spacecraft in the vicinityof Saturn in the sun–Saturn system. For a Jacobi valueJ� J1 � 3:0173046596239 �J < JL2�, each initial condition ispropagated forward to its next periapsis; the integration is terminatedearly if the state reaches escape or impact. The resulting map, whichdisplays approximately 60,000 periapsis points, appears in Fig. 2(left). Each point is colored consistent with the outcome of thepropagation, as defined in the legend in Fig. 2. Note that well-definedlobes exist that identify the escaping trajectories in Fig. 2 (left). Theselobes are analogous to the lobes defined for the HR3BP in Villac andScheeres [19] and Paskowitz and Scheeres [20]. These lobesrepresent regions inwhich a periapsis occurs just before direct escapefrom the vicinity of Saturn; any trajectory with a periapsis in one ofthese lobes escapes before reaching its next periapsis. Conversely, atrajectory with periapsis lying outside a lobe does not escape beforeits next periapse passage. These lobes can, therefore, be consideredgateways to escape: all escaping trajectories pass through one ofthese regions at the final periapse passage near P2 before escape.(While some trajectories pass through an additional periapse statenear the L1 or L2 gateways, the final periapse passage close to thesmaller primary occurs within one of the lobes near P2. Of course, aLyapunovorbit exists around bothL1 andL2 at this value of J, but thefocus of the investigation is on P2-centered trajectories and transitsthrough the gateways.) Similarly, the black regions represent initialconditions associated with orbits that impact Saturn before the nextperiapse passage. The initial condition maps are, thus, a useful tool

Fig. 2 Initial condition map for one revolution (left) and six revolutions (right), sun–Saturn system, J � J1 < JL2.

Fig. 3 Escapes out L1 (left) and L2 (right) numbered by periapse passages before escape in the sun–Saturn system.

118 DAVIS AND HOWELL

for the design of escape or impact trajectories. By selecting aperiapsis within one of the lobes on the map, a designer guaranteesthe trajectory will escape the vicinity of P2 or impact its surfacebefore reaching its subsequent periapsis.

If the initial conditions are propagated for a longer span of time, asimilar initial condition map is created. Figure 2 (right) represents aninitial condition map that results when each trajectory is propagatedfor up to six revolutions. Again, integration is terminated if thetrajectory escapes or impacts Saturn; if it reaches its sixth periapsepassage without impacting or escaping, it remains classified ascaptured. As before, the colors represent the fate of each trajectory.As expected, during the longer propagations, more trajectoriesescape or impact Saturn. The lobes from Fig. 2 (left) remain; they arejoined by additional regions of escape and impact. Similar lobes aredefined for up to four periapse passages in the HR3BP by Paskowitzand Scheeres [20]. A single trajectory’s periapses cycle through thelobes in a predictable pattern.

Recall that tidal acceleration causes the periapse radius of atrajectory following an apoapsis in quadrant I or III to decrease overthe course of one revolution and the eccentricity of the orbit toincrease. Therefore, trajectories oriented such that their apoapses liein quadrants I and III have the potential to either impact uponreaching periapsis (if the decrease in rp results in a periapsis belowthe surface ofP2) or to escape (due to the increase in eccentricity). Ingeneral, a periapsis located in quadrant I is followed (or preceded) byan apoapsis in quadrant III, and vice versa. The locations of theescape and impact trajectories in Fig. 2 can therefore be understoodin terms of quadrants. In summary, for the lobes associated withimmediate escape in Fig. 2 (left), the trajectory is oriented in quadrantI or III such that the increase in eccentricity due to the tidalacceleration results in escape before another periapsis near Saturn isreached. For the immediate impacts in Fig. 2 (left), the tidalacceleration does not result in escape, but the periapsis is loweredsufficiently that the radius falls below the surface of P2.

The regions defining initial conditions that result in escapingtrajectories may be categorized by the number of periapse passagesthe spacecraft completes before escaping the vicinity of P2. Thesemaps appear in Fig. 3, with each lobe numbered according toperiapse passages before escape. The zones labeled “1” correspondto immediate escape; trajectories originating in zones labeled “2”pass through one additional periapsis before escaping the vicinity ofSaturn; and so on. A trajectory that remains in the vicinity ofP2 afterthe sixth periapse passage is classified as captured. Analogousregions are defined for up to three revolutions in the Hill problem byPaskowitz and Scheeres [20]. Figure 3 therefore maps each regionforward: for example, an initial state in region 3 will, when it reachesits first subsequent periapsis, lie in region 2; on its second passage itwill lie in region 1, and it will escape on the following revolution. Asnoted in Paskowitz and Scheeres [20], some trajectories reachperiapse conditions near L1 or L2 just before escape. This phenom-enon is responsible for the overlap in the regions, for example, thelighter points bordering the otherwise darker immediate escape lobe.Regardless, the final periapse passage near P2 for every escapingtrajectory lies in one of the lobes corresponding to immediate escape.If these figures are flipped across the x-axis, they represent regions ofentrance into the system rather than escape. That is, a trajectory will

enter the vicinity of Saturn through one of the two gateways, and passits first periapsis in the 1 lobe, its second periapsis in the 2 lobe, etc.

Recall that the invariant stable manifold tubes associated with theL1 andL2 Lyapunov orbits, at a given value of the Jacobi integral, actas separatrices between escaping and capture trajectories in thevicinity of P2 [14]. In a traditional Poincaré map created with asurface of section that is defined, for example, as the plane y� 0,escaping trajectories correspond to points on the map that lie withinthe curve formed by a stable manifold tube; nonescaping trajectorieslie outside the curves on the map that correspond to the manifoldtubes. Similarly, in the periapsis Poincarémaps, regions of escape aredelineated by the stable manifolds associated with the L1 and L2

Lyapunov orbits. The periapses of the stable manifold trajectoriesassociated with the L1 Lyapunov orbit appear overlaid on the initialcondition map in Fig. 4 (left). Note that the periapses of the manifoldtrajectories neatly outline the regions of escape through L1.Similarly, the periapses of the trajectories lying on the stablemanifold corresponding to the L2 Lyapunov orbit appear in red inFig. 4 (right). Again, these periapses outline the lobes of initialconditions escaping through L2. Similarly, the unstable manifoldtubes corresponding to the L1 and L2 Lyapunov orbits delineateregions in which the trajectories enter the vicinity of P2 through theL1 or L2 gateways. The periapses of the unstable manifoldtrajectories are the mirror image (reflected across the x-axis) of thestable manifold periapses.

As previously analyzed by Gomez et al. [14], a trajectory that liesboth within a stable L1 tube and an unstable L2 tube can represent atransit trajectory; that is, it enters the P2 vicinity through L2 andsubsequently escapes, after an unspecified number of revolutions,through L1. Similarly, a transit trajectory may enter through the L1

gateway and depart through L2. A sample transit trajectory(L2 ! L1) appears in Fig. 5. The first two lobes representingperiapses within theL2 unstablemanifold appear to the left ofP2; thefirst two lobes associated with the L1 stable manifold appear in thefigure to the right of P2. A periapse state is selected that lies withinboth of the tubes; it appears as a black dot. Note that it is located in theL2 unstable lobe that corresponds to the second periapsis afterentering the vicinity of Saturn. The same initial state also lies in anL1

stable lobe [the 2 lobe in Fig. 3 (left)], and in a forward propagation,will pass through one more periapsis [in the 1 direct escape lobe in

Fig. 4 Periapses along L1 (left) and L2 (right) stable manifold trajectories overlaid on the initial condition map, sun–Saturn system.

Fig. 5 Initial condition map representing two periapse passages withinthe L2 unstable and L1 stable manifold tubes; transit trajectory overlaidin black.

DAVIS AND HOWELL 119

Fig. 3 (left)] before escaping through L1. The result is a transittrajectory that enters the vicinity of Saturn throughL2 and completesthree periapse passages before escaping through the L1 gateway,passing through three periapse lobes in sequence that highlight itspassage. Such a transit trajectory is also selected consistent with thenumber of periapse passages, between entering the P2 vicinity andescaping, by employing themaps in Fig. 3 alongwith their reflectionsacross the x-axis. One advantage of the periapsis Poincaré map fortrajectory design applications is that the maps exist in configurationspace, allowing the selection of initial conditions based on thephysical location of periapsis.

Thus far, the maps reflect a single value of the Jacobi integral,slightly less than the value at L2. By changing J, the initial conditionmaps also change. Initial condition maps for three values of theJacobi integral appear in Fig. 6. For a Jacobi integral equal to thevalue at L2 (top left), the gateway has not yet opened and notrajectories escape through L2. As the energy of the spacecraft isincreased and the Jacobi value decreases, the corresponding mapsevolve: the regions of impact and escape expand, and the regionsrepresenting captured trajectories contract. These regions are coloredconsistent with the legend appearing in Fig. 2. At high energies, thetrajectories escape quickly, after fewperiapse passages. For example,at a Jacobi value J� JL2 � 0:0009 (top right), most trajectoriesescape within three revolutions. At a Jacobi value J� JL2 � 0:0018(bottom), the majority of the trajectories escape the vicinity of P2

before two revolutions are completed. Such maps offer informationon the existence of various types of trajectories at various energylevels (i.e., trajectories that escape immediately or remain capturedfor several revolutions) that is applicable to preliminary missiondesign.

Thus far, each initial condition map has applied to the sun–Saturnsystem. However, the maps are equally applicable to orbit design inother P1-P2 systems. Initial condition maps for eight systems appearin Fig. 7. To establish an equivalent standard for comparison, in eachcase the value of the Jacobi integral is equal to the value associatedwithL2. For� approximately equal to 0, the ZVCs open atL1 andL2

simultaneously, so at an energy level such that J� JL1 � JL2, notrajectories escape from the vicinity of P2. As � increases, thepercentage of initial conditions corresponding to trajectories thatescape through L1 also increases. For example, in the sun–Earthsystem, the L1 gateway is only narrowly open when J� JL2, so

relatively few trajectories escape theP2 vicinity after six revolutions.For P1-P2 systems with larger values of �, on the other hand, the L1

gateway is open wider when J� JL2. For example, the larger valueof� associated with the sun–Jupiter system leads tomoreL1 escapesat this energy level, a fact that is clearly visible in themaps.Of course,the physical radius of P2 significantly affects the number of initialpoints that result in impact trajectories. This effect leads to manymore impact trajectories (colored black) in the planet–moon systemsthan in the sun–planet systems appearing in Fig. 7. In particular, fewplanar trajectories in the Jupiter–Europa system avoid impact withEuropa. So, for a spacecraft arriving at Europa with an energy suchthat J is just below JL2, nearly every trajectory will impact beforecompleting six revolutions. Similarly, in the Earth–moon, system,the majority of planar trajectories at this energy level impact thesurface of the moon or escape from its vicinity. Such initial conditionmaps are useful for the selection of trajectories that escape from asmaller primary, such as a moon in a planet–moon system, or byexploiting symmetries in the system, for selecting trajectoriescaptured by the smaller primary [19]. At a given energy level, aninitial condition map for a particular P1-P2 system provides a visualtool that allows the selection of an initial periapse state thatautomatically leads to escape through the desired gateway or impactinto the surface of P2. Also of interest are capture regions thatcontinue to exist after multiple apoapse passages. These regionsrepresent sets of initial conditions that neither impact nor escapefrom P2, initial conditions that represent potential long-term orbitsabout P2.

B. Out-of-Plane Behavior

The initial condition maps that appear in Figs. 2–7 representprograde planar trajectories; these maps are now generalized toconsider some out-of-plane trajectories. In 3-D space, the ZVCs andthe curves that separate periapses from apoapses become surfaces.However, adding a z-component to the trajectory position andvelocity complicates the visualization of the solution space. In theplanar case, for each �x; y� position coordinate, the velocitymagnitude is specified by the Jacobi constant, and the velocitydirection is specified by the apse condition, which requires that thevelocity be perpendicular to the position vector. Hence the positioncoordinate defines a single prograde trajectory. In 3-D space,however, the velocity direction is no longer completely constrained

Fig. 6 Initial conditions maps for decreasing values of J in the sun–Saturn system.

120 DAVIS AND HOWELL

by the �x; y; z� position coordinate; the apse condition only restrictsthe velocity to a plane normal to the position vector. To manage thiscomplication, a plane is defined at rp0 such that the position vector isnormal to the plane. Then, a velocity angle is introduced in this planeto isolate the velocity direction. Consider a trajectory characterizedby a periapsis located at a particular position in space denoted by�x; y; z�. The velocity angle � is defined as the angle between the x-yplane and the velocity vector, as depicted in Fig. 8.

In this analysis, the space inside the 3-D periapsis region ispopulated with initial conditions possessing a given Jacobi constantand velocity angle. Each initial state is propagated forward in timeuntil arrival at a subsequent periapsis occurs or until the trajectoryimpacts P2 or escapes through the L1 or L2 gateways. Three-dimensional lobes, analogous to the planar lobes in Fig. 2, areproduced. Consider a velocity angle�� 0, which corresponds to aninitial velocity vector parallel to the x-y plane for each periapselocation in �x; y; z� space. The corresponding initial condition mapappears in Fig. 9 in x-y and x-y projections. As before, the regionscorresponding to impact trajectories and escaping orbits are coloredaccording to the legend in Fig. 2. As the velocity angle � increases,the corresponding lobes representing escape and impact shrink. Thisdecrease in lobe size is consistent with the fact, noted above, that theinfluence of the tidal acceleration is at a maximum for planartrajectories. As in the planar case, when each initial periapsis is

propagated for up to six revolutions, lobes are produced thatrepresent the placement of periapses before escape. Such a mapappears in Fig. 10. This map corresponds to escape trajectories forinitial periapses with four velocity angles: �� 0, 11.5, 17.2, and22.9. At higher values of velocity angle, no trajectories escape fromthe vicinity of P2 within six revolutions at this value of J. Escapesthrough L1 are plotted according to the legend in Fig. 2, with darkercolors signifying fewer periapse passages remaining until escape.

Fig. 7 Initial condition maps for J � JL2 for increasing values of �.

Fig. 8 Velocity angle � between the x-y plane and the velocity vector.

DAVIS AND HOWELL 121

These 3-D initial condition maps retain the structure appearing inthe planar case. The planar lobes evolve into surfaces in the samewaythe ZVCs are merely planar projections of the 3-D zero-velocitysurfaces. As the velocity angle increases, the lobes shrink, and fewertrajectories escape or impact. This is consistent with the analysissummarized in Sec. II: the effects of tidal acceleration decrease as theinclination of a trajectory increases. As in the planar case, the 3-Dinitial condition maps provide a tool for preliminary trajectorydesign. Once a map is available, in a visual environment, a designercan identify and compute a trajectory with a particular result in realtime.

IV. Long-Term Behavior Near P2

The initial condition maps relate initial periapses to the fate of thetrajectory after a givenpropagation time, revealing information aboutthe short-term behavior of the full set of trajectories. Another type ofperiapsis Poincarémap is useful for the investigation of the long-termevolution of trajectories in the vicinity of P2 in the CR3BP. Long-term periapsis Poincaré maps are created with a surface of section atperiapsis; the state corresponding to each periapsis as the trajectoryevolves is recorded over a long-term propagation. Particular attentionis focused on the gray regions of the initial condition map appearingin Fig. 2 (right), that is, trajectories that remain captured about thesmaller primary for multiple revolutions.

A. Periapse Profiles

An initial step towards understanding the long-term behavior of acollection of trajectories involves the examination of a singletrajectory. When a single initial condition, selected from a capturedregion in the short-term initial condition map, is propagated for anextended span of time, a pattern emerges in the subsequent periapselocations relative to P2. The integration time required for patterns toclearly appear depends on theP1-P2 system aswell as on the value ofJ; in the sun–Saturn examples presented here, an integration time of1000 years, or approximately 33 revolutions of the primaries, ischosen. For example, consider a set of trajectories in the sun–Saturnsystem characterized by a Jacobi value J� J1 such that the ZVCs areopen at both L1 and L2. Trajectories at this energy level that remaincaptured for extended intervals of time possess periapses that formneat patterns in the rotating frame. These periapse profiles fall into

several types that are illustrated in Fig. 11 in the rotating frame; thetrajectories also appear in Fig. 12 in an inertial reference frame. Eachtrajectory is propagated for �33 periods of the primaries, or1000 years in the sun–Saturn system, and is colored with its periapselocations plotted in black. Note the differing shapes of each set ofblack periapse points; this shape defines the periapse profile. Thefirstexample, appearing in Fig. 11a, is a “figure eight” shape, generated inthis case with an initial periapse defined by rp0 � 0:125rH and!r0 � 91:67. Trajectories generating figure eight-shaped periapseprofiles tend to fill out the ZVCs near x� 0 but remain distant fromthe libration points. In addition, the argument of periapsis in theinertial frame for figure eight-type trajectories can move quiteslowly; the figure eight-type trajectory represented in Figs. 11a and12a is quasi-frozen in the inertial frame. The second periapse profileis an “hourglass” shape. An example of this profile is represented bythe black periapse points in Fig. 11b; this example results from aninitial periapsis with rp0 � 0:125rH and !r0 � 48:70. Suchtrajectories tend to remain clear of the ZVCs near x� 0 and extendfurther towards the libration points than do figure eight-typetrajectories. The arguments of periapsis in the inertial frame forhourglass-type of trajectories tend to rotate more quickly than forfigure eight-type orbits. Another group of hourglass-type trajec-tories, characterized by slightly larger values of rp0, share quasi-periodic behavior. An example of a quasi-periodic hourglass-typetrajectory appears in Fig. 11c. This trajectory results from initialconditions rp0 � 0:213rH and !r0 � 126:16. A fourth type ofperiapse profile is a hybrid of the hourglass- and figure eight-typeprofiles. The periapses along these trajectories form lobes. Twoexamples of lobe-type profiles appear in Figs. 11d and 11e, identifiedby rp0 � 0:125rH and initial angles!r0 � 127:8 and!r0 � 118:6,respectively. Note in Fig. 11d that the trajectory includes revolutionsthat reach the ZVCs near x� 0, as is typical of figure eight-typeprofiles; the trajectory also possesses revolutions that remain far fromthe ZVCs near x� 0, as do the hourglass-type profiles. Thequasi-periodic lobe-type trajectory in Fig. 11e shares the samecharacteristics. Certainly, the characteristics in the inertial views ofthe figure eight- and hourglass-type trajectories in Figs. 12a and 12bare both represented in the lobe-type trajectories in Figs. 12d and 12e.A fifth type of profile exists at this Jacobi value, comprised oftrajectories with periapses always near!r � 180. These trajectoriesare quasi-periodic with apoapses near L2; an example appears in

Fig. 9 Three-dimensional escape lobes; initial condition map for one revolution; x-y view (left) and x-z view (right). Sun–Saturn system, velocity angle�� 0�, J � J1.

Fig. 10 Three-dimensional escape lobes; initial conditionmap for up to six revolutions; velocity angles�� 0, 11.5, 17.2, and 22.9�; x-y view (top) and x-zview (bottom), sun–Saturn system, J � J1.

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Fig. 11f. The periapses of this trajectory are nearly identicallylocated; in an expanded view, the periapses form an “arrowhead“shape. The orbit is also clearly quasi-periodic in the inertial frame, asapparent in Fig. 12f. The “arrowhead” type trajectories are associatedwith a well-known family of stable periodic trajectories, the“g-family” of periodic orbits from Hénon [7].

The trajectories appearing in Fig. 11 share a common energy level,J� J1. However, as J changes, other patterns emerge in the shape oftrajectory periapses. For example, consider a value of Jacobi constantsuch that the ZVCs are closed at the libration points, J� JL1. At thisenergy level, all trajectories remain captured for the full propagationtime. Figure 8- and arrowhead-type trajectories are present, whilehourglass and lobe-type trajectories are not seen. However, severalother profiles appear, some of which are pictured in Fig. 13. Note, inparticular, the presence of profiles without a defined pattern. Theseapparently chaotic trajectories seem to exist only at lower energies;when the ZVCs are open, trajectories that do not follow a definedpattern tend to escape from the vicinity of Saturn over long propa-gation times. The appearance of chaotic trajectories for J < JL1 hasalso been observed in previous investigations [26].

B. Long-Term Periapsis Poincaré Maps

Many researchers have investigated the behavior of trajectories inthe vicinity ofP2 [7–13]. By characterizing each trajectory accordingto its periapse profile, however, information regarding its long-termbehavior is immediately available. In fact, the location of a singleperiapsis point at a given value of Jacobi integral reveals information

about the long-term behavior of the trajectory. This fact is made clearwhen the periapse locations of many trajectories are plotted togetherin a long-term periapsis Poincaré map.

The trajectories and periapse profiles for six pairs of initialconditions appear in Fig. 11. Consider a larger set of trajectories inthe planar sun–Saturn system characterized by a Jacobi value J� J1such that the ZVCs are open at both L1 and L2. For initial periapseradius values selected such that 0:07rH < rp0 < 0:25rH , values of!r0 ranging from 0 to � are selected. Each initial state is propagatedfor approximately 33 revolutions of the primaries: equivalent to1000 years in the sun–Saturn system. Some of the trajectories escapefrom the vicinity of Saturn during the course of the propagation;others remain captured over the full time span. As each trajectoryevolves, the spacecraft state is recorded at each periapse passage. Thecoordinates yp versus xp, relative to the rotating frame, are plotted on

a map as the trajectories evolve over �33 periods of the primaries.For J� J1, the map appears in Fig. 14 (left). The points corre-sponding to each trajectory are colored consistent with their initialcondition !r0; the color scheme appears in Fig. 14 (right). TheSaturn-centered view focuses on the trajectories that remain capturedfor the duration of the propagation. The map consists of two mainregions. First, an hourglass-shaped region is composed of pointscorresponding to several types of captured trajectories. Trajectoriesassociated with four of the periapse profile types combine to createthis region: figure eight-type trajectories, lobe-type trajectories,hourglass-type trajectories, and quasi-periodic hourglass-typetrajectories. (Note that the two white lobes at the center of the

Fig. 11 Periapsis profiles viewed in the rotating frame for six trajectories with J � J1 � 3:0173046596239.

DAVIS AND HOWELL 123

hourglass-shaped region correspond to trajectories that pass too closeto the origin for accurate integration; these trajectories have been leftout of the map.) A second region exists to the left of the hourglass,centered about yp � 0; it is composed of periapse points belonging toarrowhead-type trajectories. Surrounding these two regions is a largezone corresponding to trajectories that eventually escape. Islandsassociated with periodic trajectories exist within the periapsisPoincaré map. Three islands are marked by arrows in Fig. 14, andtheir associated periodic trajectories appear in Fig. 15. On the left,Figs. 15a, 15c, and 15e, the trajectories are plotted relative to therotating frame; the corresponding inertial views appear in Figs. 15b,15d, and 15f. Note that these three periodic orbits are computed bysimply propagating the initial conditions located from the map; nocorrections procedure was employed. These scenarios have beensuccessfully transitioned to the ephemeris model [27]. A designercan take advantage of the patterns present in the CR3BP and yettransition the resulting trajectories into the full model.

Recall that by employing the short-term initial condition mapsdiscussed in Sec. III, the location of an initial periapsis reveals the fateof the trajectory over one to six revolutions. In the same way, thelocation of periapsis in x-y space defines the periapse profile andtherefore the behavior of a planar trajectory over a long-termpropagation. A schematic appears in Fig. 16. Each colored zonerepresents a periapse profile type. If a trajectory possesses a periapsislocated in one of the colored zones, every periapsis over a long-termpropagation will also lie within the same colored zone. For example,if a trajectory with J� J1 in the sun–Saturn system possesses aninitial periapsis statewithin region a, the trajectory is characterized asa figure eight-type trajectory, and each periapsis over a long-termpropagation will remain within region a. Similarly, any trajectoryoriginating from a periapse location within region b will eventuallyimpact Saturn. A periapsis lying within the region c corresponds to alobe-type trajectory, and within region d, to an hourglass-typetrajectory. Periapses located in one of the e regions also belong tohourglass-type trajectories; however, the periapses along these pathsremain within discrete regions, yielding quasi-periodic trajectories.Zone f is composed of periapses from arrowhead-type trajectories,and any trajectory with an initial periapsis within the white regionwill escape from the vicinity of Saturn over a long-term propagation.This schematicmay be used in a design process; simply by placing an

Fig. 12 The six trajectories from Fig. 14 viewed in an inertial frame ofreference.

Fig. 13 Three samples of periapse profiles at a lower energy: J � JL1.

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initial periapsis within one of the colored regions at the specifiedvalue of Jacobi integral, a designer is certain to achieve a trajectorywith particular characteristics. Note, however, that the boundariesbetween the regions are imprecise. Close to the edges of the regions,

trajectories may exhibit characteristics of both types of trajectories,or theymay flip from one type of trajectory to another. The schematicprovides a quick, methodical way to identify trajectories with certaintypes of behaviors.

The map in Fig. 14 and the schematic in Fig. 16 are specific to themass parameter and Jacobi constant associated with this set oftrajectories; changing either of these parameters will change thecharacteristics in the figures. In Fig. 17, periapsis Poincaré mapsappear for the sun–Saturn system (left) and the Earth-moon system(right) at a lower energy, J� JL1. Again, each point is coloredcorresponding to its initial condition !r0, according to the coloringscheme seen in the legend in Fig. 14 (right). Because the ZVCs areclosed, every trajectory remains captured for the full propagationtime, approximately 33 periods of the primaries. In the sun–Saturnmap, note the presence of quasi-periodic islands along the positiveand negative x-axis. In addition, a region of figure eight-typetrajectories is surrounded by a number of chaotic orbits. Thesechaotic trajectories are also evident in the Earth-moon map,surrounding in this case a region of hourglass-, lobe-, and figureeight-type trajectories. Quasi-periodic trajectories again appearalong the positive x-axis. The lack of symmetry across x� 0, aresult of the large value of � in the Earth-moon system, is clearlyapparent.

Fig. 14 Periapsis Poincaré map displaying yp vs xp over 1000 years fortrajectories in the sun–Saturn systemwith J � J1, 0:07rH < rp0 < 0:25rH ,0<!r0 < � (left). Three islands associated with periodic trajectories aremarked with arrows. Each point is colored according to the initialcondition !r0, following the scheme shown in the legend (right).

Fig. 15 Periodic trajectories corresponding to the pointsmarked by arrows in Fig. 14. Rotating views appear on the left, inertial views on the right. lobe-type trajectory (top), hourglass-type trajectory (center), and arrowhead-type trajectory (bottom).

DAVIS AND HOWELL 125

C. Out-of-plane Long-Term Trajectory Evolution

By characterizing the velocity direction in terms of the angle � asdescribed inFig.8, theplanar long-termanalysis isextendedinto threedimensions. PeriapsisPoincarémapsareproduced for the sun–Saturnsystem that illuminate the design space and facilitate the selection of

particular trajectories, again colored consistent with initial condition!r0. For an initial periapse radius rp0 � 0:12rH and values of !r0ranging from 0 to �, maps of yp versus xp appear in Fig. 18 for threeselectedvelocityangles:�� 17:2,�� 68:8, and�� 137:5. Ineachcase, the initial z-height is set to zero, and the Jacobi constant J� J1.As in the planar case, the points are colored according to initialperiapse angle!r0, which ranges from0 to 180. The hourglass shapeof the captured periapses, familiar from the planar examples, persistsfrom �� 0 to approximately �� 20. When the velocity vector isoriented such that � is a larger angle, the figure eight shape isprominent in the shape of the periapses. For angles larger than�� 40, all of the initial conditions result in trajectories that remaincaptured for the duration of the 1000-year propagation: no trajectoryescapes through either the L1 or L2 gateways, even though the zero-velocity surfaces remain open at this value of J. As discussed inHamilton and Krivov [28], retrograde trajectories about P2 are morestable than similar prograde trajectories, due in part to the direction ofthe Coriolis acceleration, which is directed inward toward P2 in theretrograde case but outward, away fromP2, in the prograde case. Theperiapsis Poincaré maps support this conjecture, becoming tightlyordered as thevelocity angle increases toward�� 180, representingplanar retrograde trajectories.

V. Example: Mission Design for Titan Capture

The short-term and long-term maps developed in Secs. III and IVare now combined to demonstrate an application. In this example,

Fig. 17 Periapsis Poincarémapsmapdisplaying yp vs xp over 33periods of theprimaries for trajectories in the sun–Saturn system (left) andEarth-moonsystem (right) with J � JL1.

Fig. 18 Periapsis Poincaré maps representing trajectories with velocity angles �� 17:2, �� 68:8, and �� 137:5�.

Fig. 16 Periapse profile schematic. Region a corresponds to figureeight-type trajectories, region b to impact trajectories, region c to lobe-type orbits, region d to hourglass-type trajectories, region e to quasi-periodic hourglass-type trajectories, and region f to arrowhead-typetrajectories. The white region corresponds to escaping trajectories.

126 DAVIS AND HOWELL

periapsis Poincaré maps are used to design a capture trajectory into along-term quasi-periodic orbit about Titan. Consider a spacecraft inthe Saturn–Titan system with an energy level such that the ZVCs areopen at L1 and L2 (J� 3:015311< JL2). An initial condition mapcorresponding to regions of entrance into the vicinity of Titanthrough the L1 and L2 gateways at this value of J appears in Fig. 19(left). In this map, the regions corresponding to periapses oftrajectories entering the vicinity of Titan through L1 and L2 arecolored according to the legend in Fig. 19. At this energy level, fewtrajectories remain in orbit around Titan for six revolutions.Therefore, a maneuver is inserted at a selected periapsis of aparticular entering trajectory, to change the energy level and to placethe spacecraft onto a quasi-periodic orbit about Titan. To achieve thisobjective, however, the particular trajectories must be carefullyselected.

To ensure that the orbit remains centered at Titan, the post-maneuver energy level is chosen to just close off theL1 gateway, that

is, J� JL1. A long-term periapsis Poincaré map is created bypropagating 2844 trajectories for 33 periods of the primaries. In theSaturn–Titan system, this corresponds to approximately 1.44 years.Initial conditions are selected such that 0:049rH < rp0 < 0:4rH and0< !0 < �. The map of yp versus xp over time appears in Fig. 19(right). The periapses of each trajectory are colored accordingto the orbit’s initial periapse angle !0. From this map, it is clear thatregions of quasi-periodic trajectories exist centered at !0 � 0 and!0 � �.

By overlaying the post-�V long-term map on the pre-�V initialcondition map, it is simple to determine which long-term orbits withJ� JL1 can be achieved by applying a �V at periapsis soon afterentering the vicinity of Titan. A zoomed view of the overlaid mapsappears in Fig. 20. Consider the periodic orbit with J� JL1 and withperiapsis located at rp0 � 0:233rH. and !0 � 174:9. This coor-dinate ismarked in Fig. 20. The corresponding trajectory is located inthe lobe associatedwith the second periapsis afterL2 capture. That is,the trajectory enters through the L2 gateway and reaches the desiredlocation at its second periapsis passage in the vicinity of Titan, as inFig. 21 (left).When the trajectory passes through this periapsis, a�Vof 7:2 m=s is applied to lower the energy so that J� JL1. Theresulting quasi-periodic trajectory appears in Fig. 21 (right). It ispropagated for 1.44 years, equivalent to 33 periods of the primaries.By combining initial condition maps corresponding to entrance intothe vicinity of Titan with long-term periapsis Poincaré maps, acapture into a quasi-periodic orbit about Titan is produced.

VI. Conclusions

In this study, periapsis Poincaré maps are employed to revealcharacteristics of groups of trajectories over both short-term andlong-term propagations. Initial condition maps are generated tohighlight the behavior of trajectories over the short-term. Byparameterizing orbits by polar coordinates at periapsis, the mapsallow a mission designer to quickly and methodically select initialconditions that result in a trajectory with particular desired behavior.Trajectories escaping (or entering) through theL1 andL2 gateways as

Fig. 20 Long-term map at J � JL1 overlaid on initial condition map atJ � 3:015311< JL2; quasi-periodic trajectorymarked in green lies in thelobe corresponding to the secondperiapsis after entrance into the vicinityof Titan through L2.

Fig. 19 Saturn–Titan initial condition map corresponding to regions of entrance into the vicinity of Titan for J � 3:015311< JL2 (left). Long-termpostperiapsis Poincaré map for J � JL1 (right).

Fig. 21 Pre-�V trajectory overlaid on Saturn-Titan initial condition map (left);�V applied at second periapsis results in the periodic Titan-centeredtrajectory (right).

DAVIS AND HOWELL 127

well as orbits that impact P2 after one or several revolutions areclearly identifiable based on the location of a single periapse passage.

Long-term trajectory evolution is also investigated, beginningwith a categorization in terms of periapse profiles of different types ofP2-centered trajectories that exist at a particular value of Jacobiintegral. For given sets of initial conditions, long-term periapsisPoincaré maps are generated; these maps reveal informationregarding the corresponding trajectories. Based on the location of asingle periapsis, the maps and associated schematics indicate char-acteristics of the long-term behavior of the trajectory. The maps andschematics hence provide a valuable design tool for the selection oforbits in the vicinity of the smaller primary. Without such tools,selecting an initial state that leads to a trajectory with desired char-acteristics is challenging when the gravitational effects of the distantlarger primary are significant. The process is simplified significantlyby the use of periapsis Poincaré maps.

Acknowledgments

The authors thank the editor and the reviewers for theircontributions to improving this paper. The authors also appreciate theassistance of Amanda Haapala with some of the computations. Thisresearch was supported through a Graduate Assistance in Areas ofNational Need Fellowship, Zonta International Amelia EarhartFellowship, and a Purdue Forever Fellowship.

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