End-of-life disposal of libration point orbit spacecraft

Zubin P. Olikara,∗ Gerard Gomez,† Josep J. Masdemont‡

Abstract

In this work we investigate end-of-life trajectories forspacecraft in orbit about the Sun-Earth L1 and L2libration points. A plan for decommission is often re-quired during the mission design process. We studythe spacecraft’s natural dynamics in both a high-fidelity model and the circular restricted three-bodyproblem. In particular, we consider the role of theunstable manifold and forbidden regions in determin-ing disposal outcomes. A simple maneuver schemeto prevent returns to the Earth vicinity is also an-alyzed. We include discussion on potential collisionorbit schemes.

1 Introduction

An end-of-life plan is a fundamental aspect of currentand future space missions. This is a well-recognizedrequirement for Earth-orbiting satellites given theneed to prevent the accumulation of space debris inthese important orbital zones [1]. Libration point or-bits are also important regions of interest for currentand future missions. The most commonly used libra-tion points, the Sun–Earth L1 and L2 points, have anunstable dynamical component, causing spacecraft tonaturally depart this regime. These departure tra-jectories, however, can have many possible outcomesincluding returns to the Earth vicinity. Thus, consid-eration should be given to the selection of the desiredoutcome subject to mission constraints.

The design of libration point trajectories hasevolved over the last several decades. The first space-craft to orbit a libration point was the InternationalSun–Earth Explorer (ISEE-3) launched in 1978. Af-ter its original mission at an L1 halo orbit and ex-tended mission as a cometary explorer were complete,it was decommissioned into a heliocentric orbit and isdue for a passage by the Earth in August 2014 [2]. In1995 the Solar and Heliospheric Observatory (SOHO)spacecraft, developed by ESA and NASA, became the

∗Institut d’Estudis Espacials de Catalunya (IEEC) & Uni-versitat de Barcelona (UB), Spain, zubin@maia.ub.es

†IEEC & UB, Spain, gerard@maia.ub.es‡IEEC & Universitat Politecnica de Catalunya (UPC),

Spain, josep@barquins.upc.edu

first to use a trajectory designed using dynamical sys-tems techniques [3]. These techniques, particularlythe use of stable and unstable manifolds, will be lever-aged in the current study. Example of more recentmissions include ESA’s HERSCHEL and PLANCKspacecraft launched together in 2009. They wereplaced into a halo and Lissajous orbit, respectively,about the Sun–Earth L2 libration point. Their de-commissioning was done earlier this year, and it wasselected to do the following (update) [4]. Several fu-ture libration point missions are planned as well [5].

Constraints present during the decommissioningprocess are distinguishable from other mission phaserequirements. These constraints play a fundamentalrole in the end-of-life analysis and the determinationof feasible trajectories. In the current study, the pri-mary requirements we consider are the following:

• Protection of the Earth and its satellite orbitzones. This is where the Earth-orbiting space de-bris problem connects with libration point mis-sion design. A spacecraft passing through pro-tected regimes such as the low-Earth or geosyn-chronous orbits have strict requirements to miti-gate the possibility of collisions [1]. Furthermore,if a spacecraft reenters the Earth’s atmosphere, adetailed analysis is required to ensure it does soin a safe manner. For a spacecraft using nuclearpower generation, strict avoidance of the Earthis necessary.

• Limited fuel remaining aboard the spacecraft.After a spacecraft completes its primary mission,a majority of its fuel will be spent. It is expected,however, that future missions will have some fuelbudgeted specifically for the end-of-life phase. Inour analysis, we assume that about 100 m/s of∆v to be about the maximum available maneuvercapability with potentially less being possible.

• Restricted time window for final maneuvers. Dueto mission operating costs, it is desirable fordecommissioning to be completed in the leastamount of time possible. Ideally, all maneu-vers would be completed within 3 to 6 months,though this may not always be feasible. We con-sider a decommission period of about 1 year to

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be about the maximum practical time available.

• Minimal end-of-life complexity. If very carefulplanning and maneuvering is necessary for a dis-posal scheme, the associated costs and risks maymake it less desirable than a simpler, though per-haps nominally less appealing, option. A robustscheme is preferred to minimize the chance ofcomplications and to maximize the chance of suc-cessful disposal.

Individual mission needs will introduce additionalconstraints to the design process. The aim of thispaper, however, is to give an initial analysis of thedynamics and some general disposal options availableincluding associated costs. This work could be viewedas a starting point for a particular mission’s end-of-life study.

1.1 Modeling

Since the libration points are features associated withthe circular restricted three-body problem (CR3BP),this will be the principal model for the current study.The CR3BP considers the motion of two massive pri-mary bodies (i.e., the Sun and the Earth) that areassumed to move in a circular orbit about their cen-ter of mass. Let a third body (i.e., the spacecraft)be positioned at r = (x,y,z) in a barycentric rotatingframe defined such that the primary bodies are fixedalong the x-axis. The third body’s six-dimensionalstate (r,v) is governed by the non-dimensionalizedequation of motion(

rv

)=

(v

Ur + 2v× z

), (1)

which includes a force potential-like function

U(r) :=1−µ

r1+

µ

r2+

12(x2 + y2) .

The parameter µ ∈ [0,0.5] relates the primaries’masses, r1 and r2 are the distances of the third bodyto each primary (the larger and smaller primaries arelocated at x =−µ and x = 1−µ, respectively), and zis a unit vector along the z-axis, the direction of thesystem’s angular momentum. For the Sun–Earth sys-tem, we use mass parameter µ = 3.0404234× 10−6.1

The CR3BP’s five equilibrium points are the L1–L5libration points. The L1 libration point lies betweenthe Sun and the Earth on the x-axis. The L2 librationpoint lies beyond the Earth on the x-axis. Additional

1This parameter was determined using the combined massesof the Earth and the Moon as the smaller primary’s mass.

discussion of the CR3BP can be found in many ref-erences including [6].

An important consequence of this model (1) is thatit admits an integral of motion. We will use one ver-sion of this integral known as the Jacobi constant,2

C(x) := 2U(r)−‖v‖2 , (2)

which is a function of the spacecraft’s state. Whenthe equations of motion are derived from a Hamil-tonian perspective, an equivalent integral H ≡ −C/2is obtained. This integral serves an energy-like con-stant, so an increase in the Jacobi constant corre-sponds to a decrease in the spacecraft’s “energy.”

We will leverage the existence of an integral ofmotion to analyze end-of-life trajectories. Fixinga Jacobi constant level C0 and examining equation(2), we must have 2U(r) ≥ C0. Regions in posi-tion space where this inequality is violated are re-ferred to as forbidden regions. The boundary sat-isfies the relation 2U(r) = C0, which requires thatv = 0. This so-called zero-velocity surface (ZVS) isa two-dimensional manifold in the three-dimensionalposition space. If we consider motion restricted tothe plane of the primaries, the boundary is a one-dimensional zero-velocity curve (ZVC) that forms abarrier in the two-dimensional position space. We il-lustrate the ZVC and forbidden region in Figure 1for three different Jacobi constant levels. In the firstillustration, there is a “bottleneck” region around theEarth that allows a spacecraft to access the interiorregion about the Sun and the exterior region awayfrom the primaries. As the Jacobi constant is in-creased (i.e., the energy is decreased), the opening atthe L2 libration point closes as shown in the secondillustration. As it is further increased, the opening atL1 closes. These boundaries set access routes to theEarth vicinity and will play a role in the end-of-lifedesign presented.

While the CR3BP captures much of the relevantdynamics, for higher-fidelity end-of-life analysis weuse the precise positions of solar system bodies avail-able in NASA Jet Propulsion Laboratory’s DE422ephemeris [7]. In particular, we include the posi-tions of the Sun, Earth, and Moon. We only in-clude their point mass gravitational influence neglect-ing any non-uniformities in their gravitational field.Additional perturbations such as solar radiation pres-sure are not incorporated. We arbitrarily select anepoch of 18:00 UTC on January 1, 2015 for this pre-liminary study. Numerical integration is performedusing a Bulirsch–Stoer algorithm [8] with error toler-ances of 10−12 (or tighter).

2Some authors include a constant term µ(1−µ) in this def-inition.

2

Figure 1: zero-velocity curve (gray region is forbidden) - not to scale for Sun–Earth (*remake with smallermass parameter)

...

Figure 2: reference L2 periodic orbits (Lyapunovs andhalos)

1.2 Reference solutions

The analysis will consider a set of nominal librationpoint orbits in the Sun–Earth CR3BP that serve asthe starting point for the analysis. The L1 and L2libration points have stability of type center × cen-ter × saddle. In the plane of the primary bodies, weconsider the Lyapunov family of periodic orbits aboutL2 shown in Figure 2 with y-amplitude increments of100,000 km. Note that the plot axes use astronomi-cal units (1 AU = 149,600,000 km). When the fam-ily crosses a y-amplitude of approximately 650,000(?–verify) km, it bifurcates into the out-of-plane L2 haloperiodic orbit family. Some members of the “north-ern” family are shown in Figure 2 with z-amplitudeincrements of 100,000 km. The “southern” family hasan identical structure mirrored across the primaries’plane.

Note that these families of periodic solutions rep-resent just a small part of the libration points cen-ter manifold. Nevertheless, they capture much of

...

Figure 3: reference L2 “square” Lissajous orbits

the relevant motion of nearby quasi-periodic solu-tions. Thus, for this initial analysis, we will primar-ily focus on spacecraft originating in a periodic or-bit. Given the interest in Lissajous-type trajectories,however, we will also consider the quasi-periodic so-lutions shown in Figure 3. These “square” Lissajousorbits have approximately equal y- and z-amplitudes,one with amplitudes of 100,000 km and the other withamplitude 600,000 km.

For simplicity and to limit the scope of the cur-rent end-of-life analysis, we are focusing on the L2families of orbits. The L1 families appear to exhibitsimilar trends due to the small mass parameter ofthe Sun–Earth system. We will note the differenceswhere applicable. For example, an exterior trajectoryoriginating at an L2 orbit and an interior trajectoryoriginating at an L1 orbit may revolve around the ro-tating frame barycenter in opposite directions, butqualitatively they exhibit similar behavior.

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1.3 Analysis approach

The paper is organized as follows. We will first con-sider natural dynamics of a spacecraft originating ina libration point orbit. We will verify that the rele-vant dynamics are captured by the CR3BP and thatthe flow away from the initial orbit is dominated bythe unstable manifold. In addition, we will discussthe role the ZVS plays in bounding the spacecraft’smotion and investigate the associated probabilities ofa return to the Earth vicinity. In order to reduce thisoutcome’s probability, we will then consider perform-ing a simple maneuver to modify the ZVS geometry.The cost associated with various maneuver locationswill be presented. Discussion will also be made on ter-minal trajectories that impact a body. Note that dur-ing the course of this analysis, we will consider simpleimpulsive maneuvers. This helps to simplify the anal-ysis, while also reflecting the desire to avoid complexmaneuver schemes during the mission decommission-ing phase. While other means of thrust, particularlylow-thrust schemes, will not be considered, one of theprimary drivers of this analysis is the limited fuel re-maining during the end-of-life phase. Thus, naturaldynamics play a fundamental role and the particularthrust method will be of somewhat lesser importance.

2 Natural dynamics

In this section, we will investigate the natural mo-tions of a spacecraft originating in a libration pointorbit. Given the limited fuel available for the end-of-life phase, the spacecraft’s behavior will be drivenby the gravitational influence of solar system bod-ies. First, we will study whether the CR3BP cap-tures the relevant dynamical behavior observed in anephemeris model. Then within the framework of theCR3BP, we will consider the role of the unstable man-ifold for trajectories departing a libration point orbit.These ideas can be used as tools for computing end-of-life probabilities.

In order to study the end-of-life dynamics in anephemeris model, we must first generate librationpoint orbits in this model. For the moment, let usconsider the L2 family of Lyapunov orbits. These areperiodic orbits in the CR3BP as shown in Figure 2.The ephemeris model, however, is time dependent,and periodic solutions no longer exist. Nevertheless,similar orbits can be generated, which are shown inFigure 4 for one “period” of approximately 6 months,though the orbits do not exactly repeat. We can viewthese ephemeris solutions as the initial orbits fromwhich end-of-life trajectories will originate.

As a first comparison between the CR3BP and

Figure 4: ephemeris Lyapunov orbits

ephemeris dynamics, we perform a Monte Carlo anal-ysis about the reference Lyapunov orbit in eachmodel. For this study, we apply random positionperturbations of magnitude 200 km to states alongthe orbit. As long as the perturbation is sufficientlysmall, the precise choice of perturbation has little in-fluence on the results. A total of 100,000 trajectoriesare generated from the set of initial conditions. Dueto the influence of the unstable manifold, which willbe investigated in this section, half of the trajectoriesdepart towards the Earth, and half depart away fromthe primaries. Due to the end-of-life restrictions inthe Earth vicinity, we will focus on this latter set ofexterior trajectories.

In order to classify the outcome, we can checkwhether the exterior trajectory eventually returns tothe Earth vicinity. For the time being, let us definethis as being within twice the Moon’s radius of theEarth. When the set of exterior Monte Carlo tra-jectories first depart the libration point orbit, noneof the spacecraft trajectories have entered the Earthvicinity. As we consider longer intervals of time,an increasing percentage of the trajectories will havecrossed this region at some point along their path.This is illustrated in Figure 5 over a 100-year timeperiod. The first spacecraft returns appear after ap-proximately 15 years. A second set of returns ap-pear between about 25 and 30 years from their libra-tion point orbit departure. This trend is seen in boththe CR3BP and ephemeris results. An additional ob-servation is that natural trajectories departing fromlarger Lyapunov orbits tend to have a higher prob-ability of returning to the Earth. It should also benoted that the ephemeris trajectories considered tendto have a one to two percent higher chance of returnwithin 100 years. This is likely due to the influenceof the Moon, which is not included in the Sun–EarthCR3BP, and can be investigated further in the future.

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Figure 6: time for revolution about ZVS (halo orbit)

Nevertheless, the overall return behavior seen in theCR3BP and ephemeris results is quite similar.

The results can be understood by considering theforbidden regions and zero-velocity surface presentedin the previous section. At the Jacobi constant levelof the libration point orbits, the forbidden regionmakes a thin circular shape centered about the Sunwith a small opening at the Earth. For an exte-rior trajectory, the semi-major axis is larger thanthe Earth’s and, thus, has a longer two-body periodabout the Sun. Therefore, in the rotating frame, thetrajectory rotates clockwise while “bouncing” off thethe ZVS. Depending on the particular initial condi-tions, after one revolution the trajectory can eitherenter through the L2 gateway or make a subsequentrevolution around the ZVS. This is the basis for thejumps seen in Figure 5. If we define a surface aty = 0 and include condition y > 0, we can investigatethe time between intersections. This is shown for arepresentative orbit in Figure 6. This shows the timerequired for a revolution of the ZVS by non-returntrajectories is between 10 and 20 years. Note thatsimilar behavior is observed for interior trajectoriesdeparting from an L1 libration point orbit except thattheir two-body period is less than the Earth’s, and,thus, they rotate counter-clockwise.

The ZVS influences the Earth return results in anadditional manner. Larger libration point orbits cor-respond to Jacobi constant levels with larger gatewayopenings to the Earth regime. Therefore, a trajectorydeparting a larger orbit will have a greater probabil-ity of returning to the Earth after a revolution aboutthe ZVS. This behavior is observed in Figure 5.

Figure 7: earth vicinity trajectories passing withinGEO radius (this is for CR3BP)

The focus so far has been on trajectories that de-part a libration point orbit in a direction away fromthe Earth. For trajectories directed towards theEarth, as a simple study we consider the same set ofL2 Lyapunov orbits in the CR3BP. We look at theirprobabilities of passing within the geosynchronous ra-dius over a 10-year interval. In Figure 7 we see thatfor the larger orbits considered well over 20 percentpass inside this radius within 4 years. While this anal-ysis does not consider the Moon, it emphasizes thepotential risk posed by end-of-life trajectories nearthe Earth.

We have thus far considered the correspondencebetween the natural motions in the CR3BP andthe ephemeris models using a Monte Carlo analysis.However, there is an underlying dynamical structure,namely the unstable manifold, that dominates themotion away from the initial libration point orbit.The benefit of taking advantage of this special struc-ture is that it allows us to easily parameterize statesdeparting the initial orbit using two (for periodic or-bits) or three (for quasi-periodic orbits) variables.

In order to numerically generate the manifold, thetypical approach (see, for example, [9]) is to deter-mine a linear approximation of the unstable directionrelative to the libration point orbit. A small pertur-bation in this direction is made, and these initial con-ditions are propagated forwards in time to globalizethe manifold. For the Sun–Earth system, scaling theperturbation’s position component to about 200 kmis a reasonable choice. One important observation isthat the sign of this perturbation is free. This allowsus to generate two “halves” of the unstable manifold:one departing towards the Earth and one departingaway from the Earth. In Figure 8, we show the half ofan L2 orbit’s unstable manifold departing away fromthe Earth. We also show trajectories starting from

5

...

Figure 5: exterior trajectories returning to Earth region (CR3BP left, ephemeris right)–match axes limits

random perturbations on the orbit state. We can seesee the underlying influence of the unstable manifoldin these results.

An important consequence of the manifold struc-ture is that following solely the natural dynamics,there is a fifty percent chance of heading towards theEarth (which, unless done intentionally, should beavoided), and a fifty percent chance of departing awayfrom the Earth. However, a small maneuver shouldbe possible to switch between the outcomes. Thefundamental idea is that the projection of the post-maneuver state onto the unstable subspace should bedirected away from the Earth.

An additional benefit of considering the unstablemanifold, along with the complementary stable man-ifold, is that it provides an additional option for lo-cating trajectories that return to the Earth vicinity.Specifically, for planar motion in the CR3BP, the sta-ble manifold serves as a boundary between return andnon-return trajectories [10]. Using a suitable surfaceof section, in this case y = 0, y > 0, we plot in Figure9 the the closed curves representing the first intersec-tions of an L2 Lyapunov orbit’s stable and unstablemanifolds. The segments of the unstable manifold ly-ing in the interior of the stable manifold will returnto the Earth after one revolution of the ZVS. Thesecond intersections of the remaining unstable man-ifold segments can be used to find returns after tworevolutions, and so forth. The primary challenge ofthis approach is that the geometry can become quitecomplicated. Even for the first intersections shownin Figure 9, the closed curves are distorted into nar-row spiral-like shapes. The situation becomes evenmore complex in the spatial CR3BP where the stablemanifold of the center manifold should be considered.

While the direct application of the stable manifoldto the computation of returns can be challenging, itstill provides an important geometric understanding

Figure 9: stable (green) and unstable (blue) intersec-tions

of the return behavior. For example, assume that alarge number of trajectories on the unstable manifoldare generated and return trajectories are located bymonitoring the distance from the Earth. The bound-aries of the return interval can be be determined us-ing a bisection approach based on the return or non-return outcome. Refinements to this basic schemecan make it more robust, but it suffices to give a firstlook at intervals on manifold that naturally returnto Earth without recourse to a Monte Carlo-style ap-proach. This is used to generate the first return seg-ments shown in Figure 9. Similar ideas could be ap-plied to the study of out-of-plane orbits, but they arenot investigated now. However, the behavior shouldfollow a similar trend to the planar orbits since theout-of-plane dynamics are uncoupled to first-order(?).

It is apparent overall that there a close correspon-dence between the dynamics in an ephemeris modeland the CR3BP. In particular, the dynamics arestrongly influenced by the unstable manifold, whichprovides a set of natural end-of-life trajectories. Some

6

...

Figure 8: caption

of these trajectories have more desirable outcomesthan others. If the probability of a successful disposalis not deemed sufficient, maneuvers can be incorpo-rated to achieve a desired result.

3 Closing zero-velocity surface

In this section, we will consider performing a simplemaneuver to forbid a future return to the Earth vicin-ity. If such a maneuver is possible, the protection re-quirements for the Earth and its satellite zones wouldbe satisfied. We will analyze the associated maneu-ver costs and times to determine whether this is afeasible end-of-life scenario. The primary idea will beto take advantage of the dynamical structure, specifi-cally the unstable manifold and the zero-velocity sur-face (ZVS), present in the CR3BP. It could then beshown that a shadow of this structure persists whena more accurate ephemeris model is considered.

For a spacecraft orbit about the L1 or L2 libra-tion points, the corresponding ZVS must be open atthat libration point. Otherwise, the orbit would passthrough the forbidden region and not be a valid solu-tion. As discussed in the previous section, this meanthat trajectories departing the orbit have the possi-bility of eventually returning to Earth. A maneuver,however, allows us to change the Jacobi constant (or,equivalently, the energy) effectively modifying the ge-ometry of the zero-velocity surface. If the level ischanged such that the Earth region and the interioror exterior region are disconnected by a forbidden re-gion, a return to the Earth vicinity for a spacecraftoutside it is impossible at least under the CR3BPdynamics. (could discuss earlier ZVC figures)

Prior to performing this maneuver, the unstable

manifold will drive the dynamics. In order to separatethe spacecraft from the Earth region, we consider thehalf of the manifold departing away from the Earthtowards the interior region for an L1 spacecraft andtowards the exterior region for an L2 spacecraft. Asmall correction burn could be used to accomplishthis, but this is not currently studied.

For this preliminary analysis as well as in the in-terest of mission simplicity, once the spacecraft hasdeparted the initial libration point orbit, we considerthe most basic option: a single, impulsive maneuverto change the Jacobi constant. Recalling the geom-etry of the ZVS shown in Figure 1, the surface willfirst become closed when the Jacobi constant C is in-creased (energy is decreased) to that of the librationpoint CLi (i = 1,2). Returning to equation (2), for aspacecraft with position and velocity (r,v), the ma-neuver ∆v must satisfy

CLi = 2U(r)−‖v + ∆v‖2 . (3)

Let us first consider when a solution to this equationexists. If CLi > 2U(r), there is no possible maneuverto close the gateway to the Earth at this position.Physically, this means that the current position r liesin the forbidden region associated with Jacobi con-stant CLi , and, thus, this is not a appropriate placefor a maneuver. Even if a solution exists, certainpositions r correspond to a state in the Earth’s re-gion after the maneuver and not in the interior orexterior region. This is an undesirable outcome foran end-of-life disposal trajectory, and these solutionsare discarded.

A simple illustration of the geometry is shown inFigure 10. A trajectory departing away from the pri-maries on the unstable manifold of an L2 Lyapunov

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Figure 10: diagram to describe procedure

orbit is shown in blue. The zero-velocity surface atthis Jacobi constant level is shown in black, and ingray is the ZVS at the Jacobi constant level CL2 ofthe libration point. Initially, the position r is in theexterior region relative to this new ZVS, then passesthrough the forbidden region, the Earth region, theforbidden region again, and finally out to the exteriorregion. The end-of-life scheme described can be ap-plied at any point on this trajectory in the exteriorregion. It is not applicable for the other segments.

Once appropriate positions on the unstable mani-fold have been identified, the maneuver cost can becalculated. From an inspection of equation (3), it canbe observed that most efficient maneuver increasingthe Jacobi constant will be in the direction −v. Thisallows us to compute the cost to be

‖∆v‖= ‖v‖−√

2U(r)−CLi , (4)

which is well defined for positions outside of the for-bidden region.

Let us now consider the results for various fami-lies of initial libration point orbits. In Figure 11, weinclude maneuver costs for various size L2 Lyapunovorbits with amplitudes Ay ranging from 100,000 to600,000 kilometers. Note that the unstable manifoldsassociated with the periodic orbits we study are twodimensional. For presenting the results we parame-terize the manifold as follows. On the vertical axis,an angle uniformly parameterizing the initial periodicorbit in time is used. On the horizontal axis, the timetraveling on the unstable manifold is shown. Eachtrajectory on the unstable manifold, thus, traces outa horizontal line. The coloring corresponds to thecost computed using equation (4) to close the ZVS atthat point along the manifold. Uncolored regions cor-respond to states on the manifold that are either inthe Earth or the forbidden region at the new Jacobiconstant level.

Examining Figure 11, we see that for each orbit ini-tially there is only a narrow segment that allows theZVS to be closed and the spacecraft to be trapped inthe exterior region. Furthermore, the maneuver ∆valong this band are on the order of hundreds of me-ters per second for the larger Lyapunov orbits con-sidered. Once we pass approximately 6 months onthe unstable manifold, however, a maneuver closingthe ZVS is possible for all the trajectories. The costreduces to the order of meters per second.

The cost profile for L2 halo periodic orbits closeto their bifurcation with the Lyapunov orbits followsa similar trend as shown in the upper left of Figure12. For halo orbits with larger out-of-plane compo-nents, however, the initial orbit is completely con-tained within the forbidden region at the Jacobi con-stant level CL2 . We need to get a sufficient distancealong the unstable manifold away from the initial haloorbit to close the ZVS. Nevertheless, after about 8months, the scheme can be applied at a cost on theorder of X m/s.

Next we consider a Lissajous orbit of amplitude600,000 km about the L2 libration point. As before,we can depart this orbit along the unstable mani-fold and then perform a maneuver to close the zero-velocity surface at the L2 libration point. However,the manifold is three-dimensional, so we must usethree variables to parameterize it. The sequence ofimages shown in Figure 13 all correspond to a singleinitial orbit. Each plot uses two angles to parame-terize the initial state on the torus from which themanifold trajectory departs (subject to a small per-turbation). The sequence is for different amounts oftime along the manifold. This allows us to explorethe full space of maneuvers for this particular initialLissajous orbit. Initially, there are just small regionsthat correspond to potential maneuvers (analogous tothe situation with the Lyapunov orbits). These ma-neuvers have prohibitively high cost. However, thespace of potential maneuvers increases as the space-craft departs along the manifold. Furthermore, theassociated cost decreases. At eight months, any tra-jectory on the manifold has a valid maneuver on theorder of tens of meters per second. The cost decreasesfrom there to the order of meters per second. (canalso include 100,000 km Liss images)

For illustrative purposes, as a modification of thecurrent scheme, we consider the half of the unstablemanifold heading towards the Earth. In this case, wecan consider trajectories that pass through the bot-tleneck and into the interior region. A maneuver canthen be performed to trap the spacecraft in this re-gion away from the Earth. The results for a 400,000-km Lyapunov orbit are shown in Figure 14. The sit-

8

...

...

...(and more)

Figure 11: Lyapunov costs

9

...

...

...(and more)

Figure 12: L2 halo costs

10

...

...

...(and more)

Figure 13: L2 600,000-km Lissajous

11

Figure 14: passing by Earth, 400,000-km Lyapunov(note: Moon will influence true behavior)

uation is much more complex in this region. Onlycertain segments on the manifold satisfy the neces-sary conditions and the maneuver time is at least oneyear after departing the initial orbit. In red highlight,trajectories passing within the geosynchronous radiusare shown, which could lead to additional end-of-lifeconstraints. Furthermore, the Moon’s influence is notincluded, which could significantly affect these transittrajectories. This example emphasizes the simplic-ity gained by considering exterior trajectories ratherthan interior trajectories subject to chaotic dynamics.

(put in discussion for ephemeris computation)

4 Terminal solutions

In order to permanently dispose of a libration pointspacecraft, it must be brought to a solar system body.In the context of this analysis, the most direct optionsare the Sun, Earth, and Moon. The control requiredto impact any other body such as an asteroid would fitmore as a follow-on mission and is beyond the scopeof the current study. In this section, we discuss thegeneral aspects of the terminal outcomes. More de-tailed study would be possible in later work.

Since a spacecraft in an L1 or L2 libration point or-bit is approximately in a circular orbit relative to theSun, a disposal into the Sun would require the semi-major axis to be roughly halved. This would requirea significant change in energy, and this will likely ex-ceed the amount of fuel available to the spacecraft.Using a sequence of flybys, etc. could reduce this cost,but again is beyond the current scope and would re-quire end-of-life planning effort well beyond a typicalmission.

As a second possibility, there is the option of an

Figure 15: geometry of Moon collision

Earth collision orbit. However, there are logistical is-sues associated with this. First, precise control wouldbe necessary to target the Earth. Second, it wouldnecessarily pass through protected orbital regions,which would require careful planning dealing with un-certainties. Third, a spacecraft reentering the atmo-sphere will break-up and needs to have safe disposalsite such as into ocean. This option is not consid-ered in general for the constraints listed in the firstsection, but specific analysis for a particular missioncould be performed to assess feasibility.

The final disposal option to consider is a collisioninto the Moon. As opposed to an Earth collisionorbit, a Moon collision orbit would be subject tofewer restrictions. Furthermore, the unstable man-ifold from libration point orbits often cross the orbitof the Moon. In Figure 15, a trajectory departingtowards the Earth on the unstable manifold of anL2 Lyapunov orbit is shown. It has many crossingswith the Moon’s orbit shown in green. A collisionwith the Moon, however, would require the Moon tohave an appropriate phase in its orbit. This wouldrequire precise planning and targeting, which mayintroduce additional complexity into the end-of-lifeplanning scheme, but the potential for a completelyfinal outcome warrants future consideration. Prelim-inary analysis shows a single maneuver of cost lessthan 100 m/s is possible for several possible phasings.

5 Concluding remarks

This study provides a first look at the dynamics ap-plicable to the end-of-life disposal for libration pointspacecraft. The primary constraints relevant to these

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spacecraft are the avoidance of space debris (plane-tary protection) and limiting the fuel, time, and com-plexity of this mission phase. This leads to the con-sideration of natural motions in both an ephemerisand CR3BP model and associated outcome probabil-ities. It is apparent that the unstable manifold is theprimary driver of motion away from the orbit. Us-ing a small maneuver, it should be possible to placea spacecraft on an unstable manifold external trajec-tory away from the L2 libration point (or an internaltrajectory away from the L1 libration point) avoidingthe Earth bottleneck regime. By including an addi-tional maneuver on the unstable manifold, the bottle-neck can be closed. This was seen to have a relativelylow cost though the maneuver must be performed af-ter about 6 months. Assuming that this time periodis acceptable, this seems like a simple option to dis-pose of a libration point spacecraft. Comments werealso made about the possibility of a terminal disposaloption primarily into the Moon.

There are many directions for future investigation.More complicated maneuver schemes could reducethe time required to close the zero-velocity curve,which may be a worthwhile tradeoff for additionalfuel use and complexity. In a context of a particularmission, more detailed analysis could be conductedand aspects such as the ephemeris epoch could be setaccordingly. In addition, each mission has its ownunique set of constraints that must be balanced. Incertain cases, the terminal options may warrant fur-ther consideration. Further analysis of internal tra-jectories inside the Earth’s orbit from the L1 librationpoint would be insightful. Analysis of the influenceof inaccuracies in the thrust magnitude and directioncould also be studied statistically.

Acknowledgements

The authors would like to acknowledge support fromMarie Curie Research Training Network AstroNet-II.In addition, the first author is grateful for supportfrom the International Astronautical Federation toattend the conference through the Emerging SpaceLeaders Grant Programme.

References

[1] iso-24113 space debris mitigation requirements

[2] ISEE-3 mission and Earth passage

[3] SOHO trajectory design

[4] HERSCHEL and PLANCK end-of-life

[5] future libration point missions

[6] Roy, Orbital Motion

[7] NASA JPL DE422

[8] Bulirsch–Stoer algorithm, ODEX

[9] generating manifold - dynamics and mission de-sign near libration points?

[10] Gomez et al., Connecting orbits and invariantmanifolds in the spatial restricted three-bodyproblem (verify, or maybe use a periapsis mappaper(?))

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