IAC-08-C1.3.14

THE SOLAR SAIL LUNAR RELAY STATION:AN APPLICATION OF SOLAR SAILS IN THE EARTH–MOON SYSTEM

Geoffrey G. Wawrzyniak∗

Purdue University, United States of Americagwawrzyn@purdue.edu

Kathleen C. Howell†

Purdue University, United States of Americahowell@purdue.edu

A solar-sail spacecraft is proposed as a communications relay between a lunar outpost and groundstations on Earth. Unlike a traditional spacecraft that orbits the Moon, a spacecraft outfittedwith a nongravitational capability, such as a solar sail or a low-thrust engine, could be maintainedin a polar hover orbit above the lunar surface. Such a spacecraft could remain in full view ofthe lunar outpost and ground-based Earth elements at all times, offering uninterrupted two-waycommunications, navigation data, and telemetry. This analysis addresses the challenges of creatingan orbit offset from the center of the Moon using a solar sail. Instantaneous equilibrium surfacesare used to characterize the structure of the dynamics in an Earth–Moon system (with the Sunconsidered in motion). Finally, trajectories that orbit below the Moon and are based on currentor near-future estimates of solar-sail technology are computed using a numerical technique.

INTRODUCTIONSince December 1972 when the Apollo 17 astro-

nauts returned from the Moon, no human has ven-tured further than low Earth orbit (LEO). With theannouncement in 2004 of a Vision for Space Explo-ration, the American space agency, NASA, has a cleargoal of returning to the Moon and establishing a lu-nar base by 2020.1 Communication with astronautsand cosmonauts in LEO has been enabled by means ofEarth-based assets and spacecraft in geosynchronousorbits. However, now that NASA plans to return tothe Moon, new communications resources and creativedesign strategies must be explored and developed.

Current likely scenarios for a lunar infrastructure in-clude a constellation with at least two to four commu-nications satellites in elliptical orbits about the Moonat high inclinations. This strategy provides coverage ofnearly any location on the Moon, including the lunarsouth pole (LSP). The LSP is a leading candidate as asite for the establishment of an outpost due to geolog-ical interest at the south pole, the presence of waterice, and access to near-continual sunlight.2 If the LSPwere preselected as the location for the lunar outpost,perhaps only two, or possibly three, spacecraft in lowlunar orbit would be required to maintain nearly con-tinual communications coverage of the outpost.3–5

More exotic orbital solutions to the coverage prob-

∗Graduate Student, School of Aeronautics and Astronautics†Hsu Lo Professor, School of Aeronautics and Astronautics

lem at the LSP have been proposed in addition tothese traditional constellation configurations. Grebowet al. explore combinations of L1 and L2 highly in-clined halo orbits that offer greater than 90% accessto the LSP per individual spacecraft, L1 and L2 ver-tical orbits such that approximately 50% coverage atthe LSP is available per spacecraft, and L2 butterflyorbits that yield 90% coverage at the LSP per space-craft. When a second spacecraft is added, at least onespacecraft retains 100% coverage of the LSP and Earthreceiving station at all times.6

Recently, an alternative has been suggested, thatis, using nongravitational forces to offset the orbit ofone spacecraft so that it alone could supply all ofthe space-based communications coverage to a baseat the LSP. Two concepts using solar sails have pre-viously appeared in the literature. A team from theJet Propulsion Laboratory has designed a non-periodictrajectory in the region of space south of the Earth–Moon L2 Lagrange point.7 Ozimek et al. at Purduehave used implicit integration to solve for periodic or-bits, resulting in hover orbits below the LSP or southof L1 or L2.

The search for non-Keplerian orbits that sit abovethe poles of a planetary body is not new. Robert For-ward patented the concept of a “statite” that hoversabove the Earth’s poles in 1989.8 Colin McInnes iden-tified equilibrium surfaces in the Sun–Earth systemthat posed as artificial Lagrange points.9 The statite ispositioned on that equilibrium surface. While the best

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known application of the artificial Lagrange points isthe early space-weather warning system, Geostorm,control of orbits on the Sun–Earth equilibrium surfacescould extend to regions above the Earth’s poles.10–13

Farrés and Jorba have used computational tools basedon dynamical systems theory for station keeping ofthese orbits.14 In a slightly different dynamical envi-ronment, Morrow et al. explored the use of solar sailsfor hovering above a fixed point on an asteroid.15 Fi-nally, while all of the aforementioned examples havemade use of solar sails, hovering above a body can alsobe accomplished via low-thrust engines. Recent stud-ies by Broschart and Scheeres demonstrate the controlof a spacecraft above Asteroid 25143 Itokawa usinglow-thrust propulsion.16 While an asteroid is generallyless massive than a moon, the goal of that mission wasto maintain close proximity to the asteroid. A com-munications relay station can also remain relatively farfrom the lunar surface, where less thrust is required tocounteract the gravitational acceleration.

The broad objective of this analysis is the devel-opment of an approach that exposes useful orbits inthe Earth–Moon frame that emerge via an additionalforce. Specifically, the goal of this effort is the de-sign of orbits that allow a spacecraft to remain in viewof a lunar pole and the Earth at all times. Ideally,the additional force is provided by a solar sail; thisinvestigation aims to demonstrate whether such capa-bility is possible using current technology. The sail’sacceleration does offset some of the Moon’s gravita-tional acceleration and also assists the spacecraft inmaintaining line-of-sight to the LSP. However, if a sailcurrently available is not sufficient, the advances in sailtechnology that are required to create such an orbit areidentified. The design techniques are also useful whenconsidering other sources of an additional force, for ex-ample, low-thrust propulsion. Such tools also supportthe design of possible coverage scenarios that can beaccomplished using solely low-thrust propulsion or ahybrid system involving a both a sail and low-thrustengines.

In a circular restricted three-body (CR3B) model,five equilibrium points are located by evaluating theequations of motion when the relative accelerationand velocity terms are zero. Missions such as Gen-esis and ACE have demonstrated that a gravitationalequilibrium point, such as the Sun–Earth L1 point, canserve as the basis of an orbit for a traditional space-craft. These equilibrium point solutions can evolveinto three-dimensional artificial equilibrium surfaces ina CR3B system using a solar sail.9 The sail-face nor-mal is aligned in a direction opposite to that of thegravity gradient associated with the primaries. Thesize of the surface depends on the physical sail proper-ties and extends beyond the poles of the primaries fora sufficiently sized sail.

In the Sun–Earth system, this equilibrium surface

is fixed relative to the frame rotating with the Sun–Earth line. In the Earth–Moon system, however, theSun’s rays are constantly moving with respect to theEarth–Moon gravitational field, generating a surfaceof instantaneous equilibrium locations that are timevariant. McInnes discusses short-term (approximately3-hour) solutions on an Earth–Moon equilibrium sur-face by assuming they are fixed over such a time span.9

Forcing a spacecraft to follow this instantaneous equi-librium surface for longer times necessarily violates theconcept of equilibrium. However, an understanding ofthe dynamics of the system—and the requirements fora periodic orbit—can be achieved by observing the in-stantaneous equilibrium surface as it changes.

By the nature of their dependence on the movingSun as viewed in a fixed Earth–Moon system, the or-bits of spacecraft outfitted with solar sails are a specialand more complex example of hover orbits. A set of in-stantaneous equilibrium points from the instantaneousequilibrium surface in the Earth–Moon system can addinsight to the dynamics of a hover orbit. Since thespacecraft must be in motion, the instantaneous equi-librium surface highlights certain quantifiable compo-nents of the acceleration vector that must be exploitedor offset.

A discussion of general equilibrium in the Earth–Moon CR3B system and an examination of the ac-celerations from a solar sail are the basis for thedevelopment of an instantaneous equilibrium surface.Parameters from recently designed sails are used in thegeneration of these instantaneous equilibrium surfaces.The surfaces alone do not account for all of the acceler-ations required for a periodic orbit, but aid in isolatingthe remaining acceleration components. Thus, the sur-faces represent a starting point for generating periodicorbits. Recent examples of periodic hover orbits andtheir acceleration profiles are examined within the con-text of the instantaneous equilibrium surfaces.

APPROACHEquilibrium in the Earth–Moon System

The first step in developing useful orbits is the iso-lation of equilibrium conditions in the Earth–Moonsystem. In the Earth–Moon CR3B system, the equa-tions of motion including an applied acceleration are

bd2rdt2

+ 2

(eωb ×

bdrdt

)+ ∇U(r) = a (1)

where µ represents the mass fraction of the smallerprimary, eωb is the normalized angular velocity vectorrelating the orientation of the rotating frame, b, to theinertial frame, e.∗ The position of the third body, ofnegligible mass, with respect to the system barycen-ter, r, is differentiated relative to an observer in therotating frame. The Coriolis and centripetal terms are

∗Vectors are denoted with boldface.

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subsequently revealed. Following McInnes,10 the grav-itational acceleration is combined with the centripetalacceleration and labeled the gradient of the pseudo-potential, ∇U(r), that is,

∇U(r) =(

eωb ×(eωb × r

))+(

(1− µ)r31

r1 +µ

r32r2

)(2)

r1 =√

(µ + x)2 + y2 + z2 (3)

r2 =√

(µ + x− 1)2 + y2 + z2 (4)

where r1 and r2 are the distances from the larger andsmaller bodies (in this case, the Earth and Moon, re-spectively). The terms on the left in Eqn. (1) areconsidered the natural acceleration in the CR3B sys-tem. The term on the right is the applied acceleration.The source of the applied acceleration is not initiallyspecified. The system is defined consistent with thefamiliar CR3B problem. Thus, Eqn. (1) is nondimen-sionalized such that eωb = 1ê3 = 1b̂3, the period ofthe system is 2π, the characteristic mass is equal tothe sum of the masses of the two primaries, and thedistance between the primaries is unity. Thus, in theEarth–Moon system, the characteristic values used fornondimensionalizing the system equations are

L∗ = 384400 km, (Earth–Moon distance)

M∗ = Me + Mm = 6.0468× 1024 kgµ = Mm/M∗ = 0.01215

t∗ =

√L∗3

G (Me + Mm)

2πt∗ = 27.321 days

The system characteristic acceleration is then

a∗ =L∗

t∗2=

G (Me + Mm)L∗2

= 2.7307 mm/s2

and is useful as a reference.Static equilibrium exists when the particle is fixed

relative to the rotating frame. Thus, the accelerationand velocity terms are zero in Eqn. (1), reducing it to

∇U(r) = a (5)

The pseudo-gravity gradient is balanced by the appliedacceleration. In a simple case where the magnitudeof a is independent of all other terms in the equa-tions of motion (e.g., position) as well as the directioncorresponding to any external energy sources (e.g.,the Sun), equilibrium can be achieved anywhere inthe Earth–Moon region as long as a force sufficientto overcome the pseudo-gravity gradient is available.Equilibrium also requires the direction of the applied

acceleration, n, to be parallel to the pseudo-gravitygradient,

n =∇U(r)|∇U(r)| (6)

Contours of the magnitude of the pseudo-potentialgradient in the rotating CR3B Earth–Moon systemappear in Fig. 1. The applied acceleration must equal|∇U(r)| for the vehicle to remain in equilibrium in anyposition.∗ The vacant areas near the primaries repre-sent saturation, i.e., acceleration beyond the systemcharacteristic acceleration, a∗. The five traditionalLagrange points are apparent where a = 0. Theequilibrium condition consistent with Eqn. (5) impliesvelocity is zero, resulting in no Coriolis acceleration.It is noted—and obvious—that any change in posi-tion will result in a Coriolis term; therefore, dynamicalequilibrium (velocity without acceleration) is not pos-sible.

Acceleration from a Solar Sail

Solar sails, reflecting light from the Sun, are oneoption to provide the additional acceleration term inEqn. (1). Although little explored, low-thrust propul-sion, laser-driven light sails, or some hybrid systemcombining aspects of all three are also possible sourcesof the applied acceleration. The advantage of solarsails resides in the limited thrust that is available with-out propellant. Of course, the acceleration from a saildepends on its orientation with respect to the Sun.Given the system as defined in Fig. 2, the nondimen-sional sail acceleration at 1 AU can be derived as

a(t) = β(ˆ̀(t) · n)2n (7)

where n is the sail-face normal, ˆ̀(t) is the sunlight di-rection, and β is the sail’s characteristic accelerationin nondimensional units. Multiply β by the systemcharacteristic acceleration, a∗, to recover the sail char-acteristic acceleration in dimensional units, a0 (see theAppendix for details).10 The sunlight direction is ex-pressed relative to the rotating frame and is a functionof time, that is,

ˆ̀(t) = cos(Ωt)b̂1 − sin(Ωt)b̂2 + 0b̂3 (8)

where Ω is the ratio of the synodic rate of the Sun tothe system rate, i.e.,

Ω =365.24− 29.531

365.24= 0.91915 (9)

Since the system equations are nondimensionalized,time is expressed in radians. The Sun angle is τ = Ωt,

∗Adding solar gravitation effects via a very-restricted fourbody model17 creates a slight difference in the map of ∇U ; thefirst and second Lagrange points shift, at most, 620 km and 1707km (when the Sun is not aligned with either axis), respectively.Solar gravitation is not included in this analysis. Recall thatthe Moon’s radius is 1737 km.

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a) Earth–Moon

b) Zoom near Moon

Fig. 1 A contour plot of ∇U in the Earth–Moonsystem. The color scale extends from 0 to 2.73mm/s2, or one nondimensional acceleration unit.

and t = 0 is defined when the Sun line coincides withthe x-axis. The system time, t, can be phased so thatit coincides with τ when both are zero. Because geom-etry plays an essential role, expressing “time” in termsof τ is more illuminating.

For an application to a solar sail, the direction of thenet force from solar radiation pressure is necessarilyaway from the Sun, establishing a constraint on thepointing vector, that is,

ˆ̀(t) · n ≥ 0, or (10a)ˆ̀(t) ·∇U(r) ≥ 0 (10b)

Hence, a region is identified where no instantaneousequilibrium condition, due to a solar sail, can exist.However, a spacecraft outfitted with a sail is not pro-hibited from moving in this region. For a given instant,the direction of the sail-face normal that is required forinstantaneous equilibrium appears in Fig. 3. Conse-quently, the regions where instantaneous equilibriumis impossible when t or τ = 0 are also apparent.

Fig. 2 Vector definitions within the context ofa rotating coordinate frame; viewed in the x − yplane. The dot product, ˆ̀(t) · n is often expressedas cos α.10 Unit vectors relating the rotating frame(the x − y axes), bi, and the inertial frame, ei, arelabeled in the upper right.

Because the acceleration from the solar sail dependson a light source moving in the Earth–Moon frame,no permanent equilibrium solution is ever establishedfor a spacecraft outfitted with a solar sail. However,solutions corresponding to instantaneous equilibriumconditions can be calculated and are useful for design-ing solar-sail trajectories in the Earth–Moon systemusing current and future sail technology.

Solar Sail State-of-the-Art

The largest challenge with solar sails is the advance-ment of their technological readiness level (TRL) sothat sails can be considered as viable options for in-space use. The next step for advancing the TRLof solar sails is a deployment in space. Unfortu-nately, previous attempts to deploy a sail have ended inlaunch-vehicle failure; other mission concepts that in-corporate sails were not developed beyond the designphase. Most recently, Nanosail-D, a student projectsponsored by NASA, was to piggyback on the launchof a SpaceX Falcon 1.18 The Planetary Society plansto launch Cosmos 2 to test the deployment and controlof a solar sail in LEO.19 Engineers at L’Garde designeda sail for NASA’s ST-9 competition, whose main pur-pose was to advance sailing technology to TRL 6.20

Team Encounter proposed an advanced sail to deliverdigitized material and biological signatures out of theSolar System.21 Although a solar sail has not yetbeen proven in space, these missions offer estimatesfor solar sail characteristic accelerations. Table 1 liststhe characteristic accelerations in dimensional (a0) andnondimensional (β) units and their associated arealdensities (σ).

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a) x − y view

b) x − z view

Fig. 3 At t = 0: Pseudo-gravity gradient direc-tions and regions where instantaneous equilibriumis impossible (in gray). Moon is three times scale.

Instantaneous Equilibrium Surfaces

Equations (5) and (7) can be combined and rear-ranged to solve for the required characteristic acceler-ation as a function of position and time, that is,

β(r, t) =|∇U(r)|3(

ˆ̀(t) ·∇U(r))2 (11)

The results from Eqn. (11) are then contoured, con-sistent with various values of β, to reveal equilibriumsolutions for sails with different characteristics. Con-touring in three dimensions reveals that the set ofinstantaneous equilibrium solutions for a given β isbest described as a surface. Because the Sun moves ina clockwise fashion as viewed from above, the instan-taneous equilibrium surface moves with time. Initially,the Sun direction is aligned with the Earth–Moon axis

Table 1 Parameters of recently designed sails

Mission a0 (mm/s2) β σ (g/m2)Nanosail-D 0.02 0.01 389.Cosmos 2 0.05 0.02 183.ST-9, high σa 0.58 0.22 14.1ST-9, low σb 1.70 0.63 4.8Team Encounter 2.26 0.84 3.36Critical sailc 2.73 1.00 3.342

aThis configuration of the ST-9 sail includes the sail, struc-ture, control, and payload.

bThis configuration of the ST-9 sail includes only the sailand structure. If the sail is scalable, this is the limit of theST-9 configuration.

cThis hypothetical sail is included to illustrate the valuesrequired for a sail with critical loading, or characteristic accel-eration equal to the system characteristic acceleration.

a) ST-9, high σ b) ST-9, low σ

c) Team Encounter d) Critical sail

Fig. 4 Earth–Moon equilibrium surfaces for fourrepresentative sails at τ = 0. The direction of sun-light is indicated by the black arrows. The Earthand Moon are three times scale.

(τ = 0). At the initial time, the instantaneous equilib-rium surface for four different sails from Table 1 appearin Fig. 4. In the four subfigures, Earth is at the cen-ter and the Moon is near the small lobe on the right.The dots representing the Earth and the Moon are en-larged to three times their actual size. The Earth andMoon are fixed in these plots; the direction of sun-light is indicated by the black arrow, that is, along theEarth–Moon axis for this instant in time. The plotsare nondimensionalized such that the Earth–Moon dis-tance is the characteristic length.

For a given value of a0 or β, as the Sun direction

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changes, the surface evolves.∗ To demonstrate thisevolution, β is selected to be 1.70 mm/s2, which corre-sponds to the surface in Fig. 4b. Initially, at τ = 0, anin-plane equilibrium torus wraps around the Earth andmay extend in the vertical directions on the Sun side ofthe Earth–Moon barycenter, as is apparent in Fig. 4b.A second, smaller equilibrium surface exists on the farside of the Moon, and appears in Fig. 5 in a zoomedview. As the Sun moves around the Earth–Moon sys-tem, the torus changes shape, separates, and rejoins.The vertical extensions of the surface are always on theSun side of the barycenter. The smaller surface nearthe Moon joins the larger surface as the Sun shifts tothe Moon side of the barycenter. A smaller surface al-ways exists on the dark side of the Moon. The surfacesfor the ST-9 sail in the vicinity of the Moon appear inFig. 5, throughout a partial revolution of the Sun.

The equilibrium surface in the Earth–Moon systemis instantaneous. However, part of the surface ex-ists out-of-plane at all times during the synodic cycle.Therefore, if a solar-sail spacecraft follows that equi-librium surface, the spacecraft would always be in viewof the Earth and the lunar base.

Assume that the instantaneous equilibrium surfaceis exploited as a starting point for designing an orbitthat always remains in view of the LSP. To meet thisgoal, a point on the southern side of the instantaneousequilibrium surface at each time step is selected, es-tablishing a set of discrete control points that formthe basis for the design of an orbit. However, anyof the surfaces for a given β is a candidate to supplythe point. Depending on the sail characteristics, thesurface may be connected or disconnected at varioustimes. From Fig. 5, it is apparent that a surface alwaysexists on the dark side of the Moon (retrograde). Itis also clear that there is a lobe that originates on theSun side of the Moon, but rotates in a direction op-posite to that of the Sun as the solar direction movesaround the Moon (prograde). This prograde surfaceextends further south of the orbit plane, but is not acandidate for selection of control points since progrademotion conflicts with the natural tendency of orbits inthe CR3B system to be retrograde due to the Corio-lis acceleration. Therefore, the retrograde, bubble-likesurface on the anti-Sun side of the Moon is the bettercandidate for selecting control points to establish anorbit.

Initially, the retrograde bubble is tangent to theL2 point on the anti-Sun side of the Moon. As theSun shifts clockwise around the system, the bubble re-mains anchored to the L2 point but is pulled towardthe extensive toroidal surface around the Earth–Moonsystem and reconnects with the toroidal surface if notalready connected. As the Sun proceeds around theEarth–Moon system, another bubble emerges that is

∗It is helpful to consider the Earth–Moon rotating frame asfixed and the Sun moving with period of one synodic month.

a) τ = 0◦ b) τ = 45◦

c) τ = 90◦ d) τ = 135◦

e) τ = 180◦ f) τ = 225◦

g) τ = 270◦ h) τ = 315◦

Fig. 5 Earth–Moon equilibrium surface corre-

sponding to a0 = 1.70 mm/s2 in the vicinity of the

Moon. The Moon is three times scale. The direc-tion of sunlight is indicated by the arrow and τ isthe Sun angle relative to the x-axis in the Earth–Moon system.

tangent to L1; the Sun and L1 are now in opposition.The evolution of the surfaces progresses until the Sunhas completed one revolution in the Earth–Moon sys-tem.

The larger the characteristic acceleration of the sail,the larger the overall equilibrium surface. At highervalues of a0, the disconnected parts of the surface

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merge on the Sun side, connecting with a bubble onthe far side of the Moon. Then, the Sun line alwaysintersects the surface on both the Sun and dark sidesof the Moon. However, at lower values, the surfaceremains disconnected on the Sun side (as apparentin Fig. 5), with some portions (or bubbles) associ-ated only with one Lagrange point. Thus, the Sunline does not intersect the surface on the anti-Sun-sideduring some phases of the synodic cycle. A clearerpicture of this phenomenon emerges when the surfaceis viewed in a two-dimensional plane defined by thesunlight vector, ˆ̀(t), and the axis of rotation, z. Thistwo-dimensional plane is also valuable to view contoursof the instantaneous equilibrium surface projected inthat plane. Figures 6 and 7 illustrate the contours cor-responding to a sail with a characteristic accelerationof 1.70 mm/s2 in the plane containing ˆ̀(t) and z. InFigs. 6 and 7, the sail normal, n, is aligned with ∇U .The thin contours lines represent constant values ofthe magnitude of ∇U and the gray regions reflect aviolation of the pointing constraint for instantaneousequilibrium from Eqn. (10) (similar to Figs. 1 and 3).The thick contour curves are projections of the equilib-rium surfaces onto the ˆ̀(t)− z plane. Different colorscorrespond to the sails in Table 1. The thick contourlines on the right side of each plot correspond to thesurface on the anti-Sun-side of the Moon. The surfacefor a0 = 1.70 mm/s

2 (thick, yellow contour curves) atthe respective Sun angle is seen accompanying eachcontour plot.

Control Points

Contours from the two-dimensional projections inFigs. 6 and 7 are used to select initial control points todefine an orbit. A set of contours evolves as the Sunangle, τ , cycles from 0 to 360◦. At certain times in thecycle, the ˆ̀(t)− z plane includes no contours for lowervalues of a0. Nonetheless, it is useful to examine thecontours in the ˆ̀(t)− z plane throughout the synodiccycle.

In designing an orbit for a spacecraft that remainsin view of the LSP for use as a communications relay,a high elevation angle as viewed from the LSP emergesas a figure of merit. If the contours in Figs. 6 and 7are a guide to establishing a set of control points, theparticular point on the contour for each a0 that yieldsthe highest elevation angle is determined. Originat-ing at the LSP, a line tangent to the contour surfaceis computed. An iterative process is employed whencomputing this point of tangency. A low-resolutioncontour, corresponding to a0, is computed and thepoint on the contour surface that, with the LSP, formsthe nearest approximation to tangency is selected as acandidate for tangency. The process is repeated by fo-cusing on the region surrounding the candidate pointand selecting a new candidate point for tangency andso on until some tolerance is met. The tangent point Fig. 6 Retrograde contours τ ≤ 120

◦.

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Fig. 7 Retrograde contours τ ≥ 150◦.

a) x − y view b) 3-D view

c) x − z view d) y − z view

Fig. 8 Control points selected using the southern-

tangency strategy. Blue identifies a 0.56 mm/s2

sail, yellow a 1.70 mm/s2 sail, and orange a2.26 mm/s2 sail. The Moon is three times scale.

is defined to be the control point. At the next instantof time, the Sun shifts, the surfaces change, and a newtangent control point is determined. Figure 8 illus-trates three sets of control points selected from thecontour tangents for three different sails (0.56 mm/s2

sail, blue; 1.70 mm/s2 sail, yellow; and 2.26 mm/s2

sail, orange) during one synodic cycle. Gaps in the setexist where the ˆ̀(t) − z cutting plane does not inter-sect a surface on the anti-Sun-side of the Moon at aspecific value of τ .

An alternative to the tangent-selection approach isa strategy to fix the distance below the lunar equa-tor in z and solve for the point on the contour thatis located at that distance. The fixed-z distance is se-lected to correspond to the highest point from the setof tangent control points or to correspond to a heightthat exists on the instantaneous equilibrium surfacemost of the time. Sets of control points consistent withthe fixed-z strategy for three sails appear in Fig. 9: a0.56 mm/s2 sail 5000 km below the x−y plane (blue),a 1.70 mm/s2 sail 12728 km below the x−y plane (yel-low), and a 2.26 mm/s2 sail 13733 km below the x− yplane (orange).

Identification of these control points serves as a pre-liminary step in a numerical process to create a peri-odic orbit. The control points also indicate locationswhere a suitably sized sail is exactly counteracting thepseudo-gravity gradient at each instant as the Sun di-rection changes relative to the Earth–Moon system.

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a) x − y view b) 3-D view

c) x − z view d) y − z view

Fig. 9 Control points selected by fixing the desiredz coordinate. The control points are defined forthree sails: 5000 km below the x− y plane for a0 =0.58 mm/s2 (blue), 12728 km for a0 = 1.70 mm/s

2

(yellow), and 13733 km for a0 = 2.26 mm/s2 (or-

ange). The Moon is three times scale.

Thus, the contours and control points provide insightinto the dynamical contribution of the sail as well asthe pseudo-gravity gradient.

ANALYSIS OF ACCELERATIONSREQUIRED TO LINK CONTROL POINTSThe accelerations required to connect some set of

control points are calculated and compared to the ac-celerations present in the system. Any discrepancybetween these two accelerations can be attributed tothe acceleration required to link the control points andthe Coriolis acceleration that results.

The equations of motion in the CR3B system cannow be rewritten in the form

bd2rdt2

+ 2

(eωb ×

bdrdt

)+ ∇U = asail + athrust (12)

where the natural terms are the Coriolis and pseudo-gravity gradient on the left and the applied acceler-ations from the sail and any augmenting thrust areon the right in Eqn. (12). The applied accelerationsare critical to drive the sail through some desired tra-jectory defined in terms of the control points. Oneoption is a sail that delivers all of the applied acceler-ation without any requirement for augmenting the sailforce with thrust. Alternatively, a system that explic-itly employs a low-thrust engine or a hybrid system,for example, can also deliver the applied-acceleration

term. A set of control points can be numericallydifferentiated to determine the contributions of thesedifferent components. In the previous analogies, thesail has offset only the pseudo-gravity gradient, |∇U |,but this numerical-differentiation process is exploitedto determine the level of acceleration that the sail mustcontribute to supply all applied accelerations. For anarbitrary set of control points, the required sail is in-feasible because pointing constraints in Eqn. (10) areviolated, variable a0 is necessary, or both. However,the results offer an improved initial guess for a feasibletrajectory and also reveal the nature of the dynamicsin the system. Understanding the dynamical structureand the contribution of the various acceleration termsis the main purpose of this analysis.

The acceleration components in Eqn. (12) can berearranged for numerical integration so that the totalacceleration of the spacecraft relative to the rotatingframe is isolated on the left side and the natural andapplied accelerations are summed on the right, i.e.

bd2rdt2

= −

(2

(eωb ×

bdrdt

)+ ∇U

)+ (asail + athrust)

(13)Control points from the z-fixed strategy, where a0 =2.26 mm/s2 and the z-height is 13733 km (i.e., thefully connected orange trajectory in Fig. 9), are used asan example for inspection of the acceleration compo-nents. The breakdown of the acceleration components,including the additional thrust required to completean orbit, appears in Fig. 10. The sail and additionalthrust components are defined as the applied compo-nents (titled “App” in Fig. 10) and the natural com-ponents are subtracted to render the total accelerationrelative to the rotating frame (identified as “RotTot”in the plot). For a sail-only system, the additionalthrust (in the subplot of Fig.10 labeled as “Thrust”)component must be zero. The magnitude of the addi-tional thrust is small in this case, except at two controlpoints along the orbit.

To further examine the required dynamics to cre-ate this orbit, the sail acceleration component can befictionalized so that it can absorb the additional accel-eration. It is termed “fictional” because the sail canviolate pointing constraints from Eqn. (10) and doesnot require a constant characteristic acceleration. Theapplied acceleration from the solar sail is a dimension-alized version of Eqn. (7), that is,

asail(t) = a0(ˆ̀(t) · n)2n (14)

The terms in this equation can be calculated for theexample orbit and appear in Fig. 11. The first subplotrepeats the required applied acceleration from Fig. 10.The second subplot reveals the back-calculation of therequired a0, given the attitude profile in the rotatingframe (labeled “n”, fourth subplot) and sunlight direc-tion (labeled “ell”, fifth subplot). The plots in Fig. 11

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Fig. 10 The breakdown of the acceleration com-

ponents for a0 = 2.26 mm/s2 at a fixed-z height of

−13733 km. Time is measured in terms of the Sunangle, τ (radians). The x component is in blue, yin green, z in red; the magnitude in aqua.

Fig. 11 The applied acceleration is compensatedvia a fictional sail. Time is measured in Sun angle,τ (radians). The x component is in blue, y in green,z in red; the magnitude in aqua.

indicate that there are two times when the pointingconstraint is violated, resulting in a small (ˆ̀(t) · n)term and requiring a large a0. Otherwise, the variationin a0 is small. The trajectory is nearly realistic withthe sail and could conceivably be adjusted to maintainconstant a0, except for these two points, which occurat a sharp turn in the left side of the orbit pictured inFig. 9a.

Perfectly Circular Orbits Below the Moon

If a circular orbit below the Moon is the goal, inves-tigation of fictionalized sail acceleration componentsreveal that some combination of radius and z-distanceproduces orbits that require a small variation in a0or reasonably achievable a0 values. The colored mapsin Fig. 12 represent the average values of a0 and thevariation in a0 that are necessary to achieve a circularorbit and satisfy the pointing constraint ((ˆ̀(t) · n) isequivalent to cos α). Note also that the circular orbitswith 60,000 km radius require the smallest a0 and pos-sess a0 variations of approximately 1−2 mm/s2. Theseare candidates for minor control point adjustment andshould reveal orbits that are close to circular. Thediagonal streak in Fig. 12b indicates that little adjust-ment would be required for orbits with the given initialr and z values; however, the average a0 correspond-ing to these initial guess values is unrealistically high.Like the control points that appear in Figs. 8 and 9,a set of control points that yields a circular orbit at afixed-z altitude is easily available as a starting pointfor a numerical scheme in determining a feasible peri-odic orbit. All plots of the control points demonstratethat the accelerations must change to accommodatethe desired orbits, and consequently the dynamics.

NUMERICAL ADJUSTMENT OFCONTROL POINTS

Ozimek et al. have developed a method using con-strained optimization and direct collocation with non-linear programming to determine orbits that remainin view of the LSP using only a sail for control.22 Thecollocation technique is an implicit integration schemethat relies on an approximated trajectory of controlpoints∗ that can be shifted via numerical correctionsprocesses. The strategy is based on fitting a polyno-mial between the control points and then comparingthe derivative of the polynomial against the equationsof motion using an intermediate point for the com-parison. The difference between these differentials islabeled the defect and is zero if all control points aresolutions to the equations of motion. Constrainingcertain variables, a Newton-Raphson iteration schemeis employed to update the control points until theyconverge onto a feasible trajectory. This approach isrobust, insensitive to the initial guess, and yields anaccurate control history.

Using these techniques, Ozimek et al. compute a pe-riodic hover orbit that maintains a view of the LSP,as is apparent in Fig. 13. The breakdown of the accel-eration components associated with their hover orbitis plotted in Figs. 14 and 15. Note in Fig. 14 that noadditional thrust is required and that the characteris-tic acceleration remains constant in Fig. 15, aside fromerrors introduced by numeric differentiation.

∗Ozimek et al. refer to these points as nodes.22

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a) Required average a0 (mm/s2) to maintain circular orbit

b) Required range over a0 (mm/s2) to maintain circular orbit

c) Minimum ˆ̀(t) · n to maintain circular orbit

Fig. 12 Feasibility of circular orbits below the LSP.The deep red in a) and b) and the deep blue in c)are saturated.

An alternative technique is in development wherebyno defect point is required.23 The difference betweentwo types of accelerations and velocities is compared:1) accelerations and velocities required to link thecontrol points and 2) the accelerations and velocitiespresent from the equations of motion at the controlpoints. This difference serves as the basis for theconstruction of the Jacobian in a Newton-Raphson it-eration scheme. A constraint is added to force thisdifference to zero; periodicity and control vector mag-nitude are also enforced through constraints.

Using the maps in Fig. 12, a combination of desiredvalues for r, z, and a0 are selected as initial guesses forthe new technique. Example trajectories using initialguesses of r = 50000 km, z = −36000 km, as well as

a) x − y view b) 3-D view

c) x − z view d) y − z view

Fig. 13 Periodic hover orbit using a0 = 1.70 mm/s2

from Ozimek et al.22 The Moon is three timesscale.

Fig. 14 Breakdown of accelerations in hover or-bit from Ozimek et al.22 Note that no additionalthrust is required to maintain this orbit.

the characteristic accelerations equal to 0.58 mm/s2,1.70 mm/s2, and 2.26 mm/s2 are plotted in Fig. 16.For each of these characteristic accelerations, no ad-ditional “fictional” thrust is required to complete thecorresponding orbit that appears in Fig. 16; the saildelivers all of the applied thrust to both offset the or-bit from the x− y plane and to shift the spacecraft inits orbit around the Moon. Note that the sail-directionconstraint, (ˆ̀(t) ·n), is never violated in any case. The

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Fig. 15 Total applied acceleration in hover orbitfrom Ozimek et al.22

a) x − y view b) 3-D view

c) x − z view d) y − z view

Fig. 16 Periodic hover orbit constructed using

a0 = 0.58 mm/s2 (blue), a0 = 1.70 mm/s

2 (yellow),and a0 = 2.26 mm/s

2 (orange). The Moon is threetimes scale.

elevation history for each of the three orbits is repre-sented in Fig. 17. Clearly, there is a strong similarityin the two orbits with the larger characteristic accel-eration values suggesting that some family of orbitsexists based on similar initial guesses.

A number of other cases, with different initialguesses and characteristic accelerations are also ex-amined. The sail orientation, represented in terms of(ˆ̀(t) ·n), is nearly zero for a significant length of time

Fig. 17 Elevation plots for the orbits in Fig. 16.

along the orbit for these particular cases, indicatingthat the sail is effectively “off” for part of the orbit.It should be noted that not all combinations of ini-tial guess and characteristic acceleration in other trialsconverged or conformed to the pointing constraint inEqn. (10); however, this method, along with that pre-sented in Ozimek et al.22 show promise for generatingorbits for further study.

CONCLUSIONSFor the design of a periodic orbit in the vicinity of

the LSP in a CR3B model, the instantaneous equilib-rium surfaces, as well as the difference between theavailable acceleration and the required accelerationterms, are very insightful. The comparison of the ac-celeration components and the views of the changingequilibrium surface offer a greater understanding ofthe dynamical environment. This knowledge of thedynamical structure is a key element in creating use-ful trajectories, but also critical to developing a modelfor the dynamical foundation of the problem when con-structing these non-Keplerian orbits. The problem iscomplex, but results indicate that a solution is feasible,an essential step in the design of such nontraditionalorbits.

ACKNOWLEDGEMENTSThe authors wish to thank Daniel Grebow for his

insightful suggestion on a new collocation techniqueto create periodic orbits. Martin Ozimek and DanielGrebow also generously shared numerical data fromtheir analysis and contributed in valuable technicaldiscussions. The first author gratefully acknowledgesthe National Aeronautics and Space Administration’s(NASA) Office of Education for their sponsorship ofhis attendance at the 59th International AstronauticalCongress in Glasgow, Scotland. Portions of this workwere supported by Purdue University.

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APPENDIXThe nondimensional sail characteristic acceleration,

β, can be derived in two ways. The first is to expressβ as a ratio of the sail characteristic acceleration indimensional units, a0, and the system characteristicacceleration, a∗. The second arises from the ratio ofthe critical areal density and the sail areal density.The lower design limit for sail areal density, sometimesdenoted the solar sail loading parameter, is currentlyaround 4 g/m2.20

Solar radiation pressure, measured in micro-Pascals,at 1 AU is

P1 =1368 W/m2

c= 4.563 µPa

The sail characteristic acceleration, a0, is the accelera-tion from solar radiation pressure on the sail at 1 AU,that is,

a0 =2ηP1

σ

Using the lower design limit value for the areal density(σ = 4 g/m2), a0 is approximately 2 mm/s2 when η =90%. Critical areal density “balances” the gravity ofthe system at the Earth–Moon distance, L∗,

σ∗ = 2ηP1

(GM∗

L∗2

)−1= 3.342 g/m2

Thus, the nondimensional sail characteristic accelera-tion in Earth–Moon CR3B system is expressed as

β =a0a∗

=σ∗

σ

clearly based either on characteristic and critical ac-celerations or on critical and characteristic areal den-sities.

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