IAC-08-C THE SOLAR SAIL LUNAR RELAY STATION: AN APPLICATION OF SOLAR SAILS IN THE EARTH MOON SYSTEM

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1 IAC-08-C THE SOLAR SAIL LUNAR RELAY STATION: AN APPLICATION OF SOLAR SAILS IN THE EARTH MOON SYSTEM Geoffrey G. Wawrzyniak Purdue University, United States of America Kathleen C. Howell Purdue University, United States of America A solar-sail spacecraft is proposed as a communications relay between a lunar outpost and ground stations on Earth. Unlike a traditional spacecraft that orbits the Moon, a spacecraft outfitted with a nongravitational capability, such as a solar sail or a low-thrust engine, could be maintained in a polar hover orbit above the lunar surface. Such a spacecraft could remain in full view of the lunar outpost and ground-based Earth elements at all times, offering uninterrupted two-way communications, navigation data, and telemetry. This analysis addresses the challenges of creating an orbit offset from the center of the Moon using a solar sail. Instantaneous equilibrium surfaces are used to characterize the structure of the dynamics in an Earth Moon system (with the Sun considered in motion). Finally, trajectories that orbit below the Moon and are based on current or near-future estimates of solar-sail technology are computed using a numerical technique. INTRODUCTION Since December 1972 when the Apollo 17 astronauts returned from the Moon, no human has ventured further than low Earth orbit (LEO). With the announcement in 2004 of a Vision for Space Exploration, the American space agency, NASA, has a clear goal of returning to the Moon and establishing a lunar base by Communication with astronauts and cosmonauts in LEO has been enabled by means of Earth-based assets and spacecraft in geosynchronous orbits. However, now that NASA plans to return to the Moon, new communications resources and creative design strategies must be explored and developed. Current likely scenarios for a lunar infrastructure include a constellation with at least two to four communications satellites in elliptical orbits about the Moon at high inclinations. This strategy provides coverage of nearly any location on the Moon, including the lunar south pole (LSP). The LSP is a leading candidate as a site for the establishment of an outpost due to geological interest at the south pole, the presence of water ice, and access to near-continual sunlight. 2 If the LSP were preselected as the location for the lunar outpost, perhaps only two, or possibly three, spacecraft in low lunar orbit would be required to maintain nearly continual 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 to these traditional constellation configurations. Grebow et al. explore combinations of L 1 and L 2 highly inclined halo orbits that offer greater than 90% access to the LSP per individual spacecraft, L 1 and L 2 vertical orbits such that approximately 50% coverage at the LSP is available per spacecraft, and L 2 butterfly orbits that yield 90% coverage at the LSP per spacecraft. When a second spacecraft is added, at least one spacecraft retains 100% coverage of the LSP and Earth receiving station at all times. 6 Recently, an alternative has been suggested, that is, using nongravitational forces to offset the orbit of one spacecraft so that it alone could supply all of the space-based communications coverage to a base at the LSP. Two concepts using solar sails have previously appeared in the literature. A team from the Jet Propulsion Laboratory has designed a non-periodic trajectory in the region of space south of the Earth Moon L 2 Lagrange point. 7 Ozimek et al. at Purdue have used implicit integration to solve for periodic orbits, resulting in hover orbits below the LSP or south of L 1 or L 2. The search for non-keplerian orbits that sit above the poles of a planetary body is not new. Robert Forward patented the concept of a statite that hovers above the Earth s poles in Colin McInnes identified equilibrium surfaces in the Sun Earth system that posed as artificial Lagrange points. 9 The statite is positioned on that equilibrium surface. While the best 1 of 13

2 known application of the artificial Lagrange points is the early space-weather warning system, Geostorm, control of orbits on the Sun Earth equilibrium surfaces could extend to regions above the Earth s poles Farrés and Jorba have used computational tools based on dynamical systems theory for station keeping of these orbits. 14 In a slightly different dynamical environment, Morrow et al. explored the use of solar sails for hovering above a fixed point on an asteroid. 15 Finally, while all of the aforementioned examples have made use of solar sails, hovering above a body can also be accomplished via low-thrust engines. Recent studies by Broschart and Scheeres demonstrate the control of a spacecraft above Asteroid Itokawa using low-thrust propulsion. 16 While an asteroid is generally less massive than a moon, the goal of that mission was to maintain close proximity to the asteroid. A communications relay station can also remain relatively far from the lunar surface, where less thrust is required to counteract the gravitational acceleration. The broad objective of this analysis is the development of an approach that exposes useful orbits in the Earth Moon frame that emerge via an additional force. Specifically, the goal of this effort is the design of orbits that allow a spacecraft to remain in view of a lunar pole and the Earth at all times. Ideally, the additional force is provided by a solar sail; this investigation aims to demonstrate whether such capability is possible using current technology. The sail s acceleration does offset some of the Moon s gravitational acceleration and also assists the spacecraft in maintaining line-of-sight to the LSP. However, if a sail currently available is not sufficient, the advances in sail technology that are required to create such an orbit are identified. The design techniques are also useful when considering other sources of an additional force, for example, low-thrust propulsion. Such tools also support the design of possible coverage scenarios that can be accomplished using solely low-thrust propulsion or a hybrid system involving a both a sail and low-thrust engines. In a circular restricted three-body (CR3B) model, five equilibrium points are located by evaluating the equations of motion when the relative acceleration and velocity terms are zero. Missions such as Genesis and ACE have demonstrated that a gravitational equilibrium point, such as the Sun Earth L1 point, can serve as the basis of an orbit for a traditional spacecraft. These equilibrium point solutions can evolve into three-dimensional artificial equilibrium surfaces in a CR3B system using a solar sail. 9 The sail-face normal is aligned in a direction opposite to that of the gravity gradient associated with the primaries. The size of the surface depends on the physical sail properties and extends beyond the poles of the primaries for a 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, the Sun s rays are constantly moving with respect to the Earth Moon gravitational field, generating a surface of instantaneous equilibrium locations that are time variant. McInnes discusses short-term (approximately 3-hour) solutions on an Earth Moon equilibrium surface by assuming they are fixed over such a time span. 9 Forcing a spacecraft to follow this instantaneous equilibrium surface for longer times necessarily violates the concept of equilibrium. However, an understanding of the dynamics of the system and the requirements for a periodic orbit can be achieved by observing the instantaneous equilibrium surface as it changes. By the nature of their dependence on the moving Sun as viewed in a fixed Earth Moon system, the orbits of spacecraft outfitted with solar sails are a special and more complex example of hover orbits. A set of instantaneous equilibrium points from the instantaneous equilibrium surface in the Earth Moon system can add insight to the dynamics of a hover orbit. Since the spacecraft must be in motion, the instantaneous equilibrium surface highlights certain quantifiable components of the acceleration vector that must be exploited or offset. A discussion of general equilibrium in the Earth Moon CR3B system and an examination of the accelerations from a solar sail are the basis for the development of an instantaneous equilibrium surface. Parameters from recently designed sails are used in the generation of these instantaneous equilibrium surfaces. The surfaces alone do not account for all of the accelerations required for a periodic orbit, but aid in isolating the remaining acceleration components. Thus, the surfaces represent a starting point for generating periodic orbits. Recent examples of periodic hover orbits and their acceleration profiles are examined within the context of the instantaneous equilibrium surfaces. APPROACH Equilibrium in the Earth Moon System The first step in developing useful orbits is the isolation of equilibrium conditions in the Earth Moon system. In the Earth Moon CR3B system, the equations of motion including an applied acceleration are b d 2 r dt ( e ω b b dr dt ) + U(r) = a (1) where µ represents the mass fraction of the smaller primary, e ω b is the normalized angular velocity vector relating the orientation of the rotating frame, b, to the inertial frame, e. The position of the third body, of negligible mass, with respect to the system barycenter, r, is differentiated relative to an observer in the rotating frame. The Coriolis and centripetal terms are Vectors are denoted with boldface. 2 of 13

3 subsequently revealed. Following McInnes, 10 the gravitational acceleration is combined with the centripetal acceleration and labeled the gradient of the pseudopotential, U(r), that is, ( U(r) = e ω b (e ω b r ) ) ( (1 µ) + r1 3 r 1 + µ ) r2 3 r 2 (2) r 1 = (µ + x) 2 + y 2 + z 2 (3) r 2 = (µ + x 1) 2 + y 2 + z 2 (4) where r 1 and r 2 are the distances from the larger and smaller bodies (in this case, the Earth and Moon, respectively). The terms on the left in Eqn. (1) are considered the natural acceleration in the CR3B system. The term on the right is the applied acceleration. The source of the applied acceleration is not initially specified. The system is defined consistent with the familiar CR3B problem. Thus, Eqn. (1) is nondimensionalized such that e ω b = 1ê 3 = 1ˆb 3, the period of the system is 2π, the characteristic mass is equal to the sum of the masses of the two primaries, and the distance between the primaries is unity. Thus, in the Earth Moon system, the characteristic values used for nondimensionalizing the system equations are L = km, (Earth Moon distance) M = M e + M m = kg µ = M m /M = t L = 3 G (M e + M m ) 2πt = days The system characteristic acceleration is then a = L t 2 = G (M e + M m ) L 2 = mm/s 2 and is useful as a reference. Static equilibrium exists when the particle is fixed relative to the rotating frame. Thus, the acceleration and velocity terms are zero in Eqn. (1), reducing it to U(r) = a (5) The pseudo-gravity gradient is balanced by the applied acceleration. In a simple case where the magnitude of a is independent of all other terms in the equations of motion (e.g., position) as well as the direction corresponding to any external energy sources (e.g., the Sun), equilibrium can be achieved anywhere in the Earth Moon region as long as a force sufficient to overcome the pseudo-gravity gradient is available. Equilibrium also requires the direction of the applied acceleration, n, to be parallel to the pseudo-gravity gradient, n = U(r) (6) U(r) Contours of the magnitude of the pseudo-potential gradient in the rotating CR3B Earth Moon system appear in Fig. 1. The applied acceleration must equal U(r) for the vehicle to remain in equilibrium in any position. The vacant areas near the primaries represent saturation, i.e., acceleration beyond the system characteristic acceleration, a. The five traditional Lagrange points are apparent where a = 0. The equilibrium condition consistent with Eqn. (5) implies velocity is zero, resulting in no Coriolis acceleration. It is noted and obvious that any change in position will result in a Coriolis term; therefore, dynamical equilibrium (velocity without acceleration) is not possible. Acceleration from a Solar Sail Solar sails, reflecting light from the Sun, are one option to provide the additional acceleration term in Eqn. (1). Although little explored, low-thrust propulsion, laser-driven light sails, or some hybrid system combining aspects of all three are also possible sources of the applied acceleration. The advantage of solar sails resides in the limited thrust that is available without propellant. Of course, the acceleration from a sail depends on its orientation with respect to the Sun. Given the system as defined in Fig. 2, the nondimensional sail acceleration at 1 AU can be derived as a(t) = β(ˆl(t) n) 2 n (7) where n is the sail-face normal, ˆl(t) is the sunlight direction, and β is the sail s characteristic acceleration in nondimensional units. Multiply β by the system characteristic acceleration, a, to recover the sail characteristic acceleration in dimensional units, a 0 (see the Appendix for details). 10 The sunlight direction is expressed relative to the rotating frame and is a function of time, that is, ˆl(t) = cos(ωt)ˆb 1 sin(ωt)ˆb 2 + 0ˆb 3 (8) where Ω is the ratio of the synodic rate of the Sun to the system rate, i.e., Ω = = (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 four body model 17 creates a slight difference in the map of U; the first and second Lagrange points shift, at most, 620 km and 1707 km (when the Sun is not aligned with either axis), respectively. Solar gravitation is not included in this analysis. Recall that the Moon s radius is 1737 km. 3 of 13

4 a) Earth Moon Fig. 2 Vector definitions within the context of a rotating coordinate frame; viewed in the x y plane. The dot product, ˆl(t) n is often expressed as cos α. 10 Unit vectors relating the rotating frame (the x y axes), b i, and the inertial frame, e i, are labeled in the upper right. b) Zoom near Moon Fig. 1 A contour plot of U in the Earth Moon system. The color scale extends from 0 to 2.73 mm/s 2, or one nondimensional acceleration unit. and t = 0 is defined when the Sun line coincides with the x-axis. The system time, t, can be phased so that it coincides with τ when both are zero. Because geometry plays an essential role, expressing time in terms of τ is more illuminating. For an application to a solar sail, the direction of the net force from solar radiation pressure is necessarily away from the Sun, establishing a constraint on the pointing vector, that is, ˆl(t) n 0, or ˆl(t) U(r) 0 (10a) (10b) Hence, a region is identified where no instantaneous equilibrium condition, due to a solar sail, can exist. However, a spacecraft outfitted with a sail is not prohibited from moving in this region. For a given instant, the direction of the sail-face normal that is required for instantaneous equilibrium appears in Fig. 3. Consequently, the regions where instantaneous equilibrium is impossible when t or τ = 0 are also apparent. Because the acceleration from the solar sail depends on a light source moving in the Earth Moon frame, no permanent equilibrium solution is ever established for a spacecraft outfitted with a solar sail. However, solutions corresponding to instantaneous equilibrium conditions can be calculated and are useful for designing solar-sail trajectories in the Earth Moon system using current and future sail technology. Solar Sail State-of-the-Art The largest challenge with solar sails is the advancement of their technological readiness level (TRL) so that sails can be considered as viable options for inspace use. The next step for advancing the TRL of solar sails is a deployment in space. Unfortunately, previous attempts to deploy a sail have ended in launch-vehicle failure; other mission concepts that incorporate sails were not developed beyond the design phase. Most recently, Nanosail-D, a student project sponsored by NASA, was to piggyback on the launch of a SpaceX Falcon The Planetary Society plans to launch Cosmos 2 to test the deployment and control of a solar sail in LEO. 19 Engineers at L Garde designed a sail for NASA s ST-9 competition, whose main purpose was to advance sailing technology to TRL Team Encounter proposed an advanced sail to deliver digitized material and biological signatures out of the Solar System. 21 Although a solar sail has not yet been proven in space, these missions offer estimates for solar sail characteristic accelerations. Table 1 lists the characteristic accelerations in dimensional (a 0 ) and nondimensional (β) units and their associated areal densities (σ). 4 of 13

5 Table 1 Parameters of recently designed sails Mission a 0 (mm/s 2 ) β σ (g/m 2 ) Nanosail-D Cosmos ST-9, high σ a ST-9, low σ b Team Encounter Critical sail c a This configuration of the ST-9 sail includes the sail, structure, control, and payload. b This configuration of the ST-9 sail includes only the sail and structure. If the sail is scalable, this is the limit of the ST-9 configuration. c This hypothetical sail is included to illustrate the values required for a sail with critical loading, or characteristic acceleration equal to the system characteristic acceleration. a) x y view a) ST-9, high σ b) ST-9, low σ b) x z view Fig. 3 At t = 0: Pseudo-gravity gradient directions and regions where instantaneous equilibrium is impossible (in gray). Moon is three times scale. Instantaneous Equilibrium Surfaces Equations (5) and (7) can be combined and rearranged to solve for the required characteristic acceleration as a function of position and time, that is, β(r, t) = U(r) 3 (ˆl(t) U(r) ) 2 (11) The results from Eqn. (11) are then contoured, consistent with various values of β, to reveal equilibrium solutions for sails with different characteristics. Contouring in three dimensions reveals that the set of instantaneous equilibrium solutions for a given β is best described as a surface. Because the Sun moves in a clockwise fashion as viewed from above, the instantaneous equilibrium surface moves with time. Initially, the Sun direction is aligned with the Earth Moon axis c) Team Encounter d) Critical sail Fig. 4 Earth Moon equilibrium surfaces for four representative sails at τ = 0. The direction of sunlight is indicated by the black arrows. The Earth and Moon are three times scale. (τ = 0). At the initial time, the instantaneous equilibrium surface for four different sails from Table 1 appear in Fig. 4. In the four subfigures, Earth is at the center and the Moon is near the small lobe on the right. The dots representing the Earth and the Moon are enlarged to three times their actual size. The Earth and Moon are fixed in these plots; the direction of sunlight is indicated by the black arrow, that is, along the Earth Moon axis for this instant in time. The plots are nondimensionalized such that the Earth Moon distance is the characteristic length. For a given value of a 0 or β, as the Sun direction 5 of 13

6 changes, the surface evolves. To demonstrate this 2 evolution, β is selected to be 1.70 mm/s, which corresponds to the surface in Fig. 4b. Initially, at τ = 0, an in-plane equilibrium torus wraps around the Earth and may extend in the vertical directions on the Sun side of the Earth Moon barycenter, as is apparent in Fig. 4b. A second, smaller equilibrium surface exists on the far side of the Moon, and appears in Fig. 5 in a zoomed view. As the Sun moves around the Earth Moon system, the torus changes shape, separates, and rejoins. The vertical extensions of the surface are always on the Sun side of the barycenter. The smaller surface near the Moon joins the larger surface as the Sun shifts to the Moon side of the barycenter. A smaller surface always exists on the dark side of the Moon. The surfaces for the ST-9 sail in the vicinity of the Moon appear in Fig. 5, throughout a partial revolution of the Sun. The equilibrium surface in the Earth Moon system is instantaneous. However, part of the surface exists out-of-plane at all times during the synodic cycle. Therefore, if a solar-sail spacecraft follows that equilibrium surface, the spacecraft would always be in view of the Earth and the lunar base. Assume that the instantaneous equilibrium surface is exploited as a starting point for designing an orbit that always remains in view of the LSP. To meet this goal, a point on the southern side of the instantaneous equilibrium surface at each time step is selected, establishing a set of discrete control points that form the basis for the design of an orbit. However, any of the surfaces for a given β is a candidate to supply the point. Depending on the sail characteristics, the surface may be connected or disconnected at various times. From Fig. 5, it is apparent that a surface always exists on the dark side of the Moon (retrograde). It is also clear that there is a lobe that originates on the Sun side of the Moon, but rotates in a direction opposite to that of the Sun as the solar direction moves around the Moon (prograde). This prograde surface extends further south of the orbit plane, but is not a candidate for selection of control points since prograde motion conflicts with the natural tendency of orbits in the CR3B system to be retrograde due to the Coriolis acceleration. Therefore, the retrograde, bubble-like surface on the anti-sun side of the Moon is the better candidate for selecting control points to establish an orbit. Initially, the retrograde bubble is tangent to the L2 point on the anti-sun side of the Moon. As the Sun shifts clockwise around the system, the bubble remains anchored to the L2 point but is pulled toward the extensive toroidal surface around the Earth Moon system and reconnects with the toroidal surface if not already connected. As the Sun proceeds around the Earth Moon system, another bubble emerges that is It is helpful to consider the Earth Moon rotating frame as fixed 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 corresponding to a0 = 1.70 mm/s2 in the vicinity of the Moon. The Moon is three times scale. The direction of sunlight is indicated by the arrow and τ is the 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 Sun has completed one revolution in the Earth Moon system. The larger the characteristic acceleration of the sail, the larger the overall equilibrium surface. At higher values of a0, the disconnected parts of the surface 6 of 13

7 merge on the Sun side, connecting with a bubble on the far side of the Moon. Then, the Sun line always intersects the surface on both the Sun and dark sides of the Moon. However, at lower values, the surface remains disconnected on the Sun side (as apparent in Fig. 5), with some portions (or bubbles) associated only with one Lagrange point. Thus, the Sun line does not intersect the surface on the anti-sun-side during some phases of the synodic cycle. A clearer picture of this phenomenon emerges when the surface is viewed in a two-dimensional plane defined by the sunlight vector, ˆl(t), and the axis of rotation, z. This two-dimensional plane is also valuable to view contours of the instantaneous equilibrium surface projected in that plane. Figures 6 and 7 illustrate the contours corresponding to a sail with a characteristic acceleration of 1.70 mm/s 2 in the plane containing ˆl(t) and z. In Figs. 6 and 7, the sail normal, n, is aligned with U. The thin contours lines represent constant values of the magnitude of U and the gray regions reflect a violation of the pointing constraint for instantaneous equilibrium from Eqn. (10) (similar to Figs. 1 and 3). The thick contour curves are projections of the equilibrium surfaces onto the ˆl(t) z plane. Different colors correspond to the sails in Table 1. The thick contour lines on the right side of each plot correspond to the surface on the anti-sun-side of the Moon. The surface for a 0 = 1.70 mm/s 2 (thick, yellow contour curves) at the respective Sun angle is seen accompanying each contour plot. Control Points Contours from the two-dimensional projections in Figs. 6 and 7 are used to select initial control points to define an orbit. A set of contours evolves as the Sun angle, τ, cycles from 0 to 360. At certain times in the cycle, the ˆl(t) z plane includes no contours for lower values of a 0. Nonetheless, it is useful to examine the contours in the ˆl(t) z plane throughout the synodic cycle. In designing an orbit for a spacecraft that remains in view of the LSP for use as a communications relay, a high elevation angle as viewed from the LSP emerges as a figure of merit. If the contours in Figs. 6 and 7 are a guide to establishing a set of control points, the particular point on the contour for each a 0 that yields the highest elevation angle is determined. Originating at the LSP, a line tangent to the contour surface is computed. An iterative process is employed when computing this point of tangency. A low-resolution contour, corresponding to a 0, is computed and the point on the contour surface that, with the LSP, forms the nearest approximation to tangency is selected as a candidate for tangency. The process is repeated by focusing on the region surrounding the candidate point and selecting a new candidate point for tangency and so on until some tolerance is met. The tangent point Fig. 6 Retrograde contours τ of 13

8 a) x y view b) 3-D view c) x z view d) y z view Fig. 8 Control points selected using the southerntangency strategy. Blue identifies a 0.56 mm/s 2 sail, yellow a 1.70 mm/s 2 sail, and orange a 2.26 mm/s 2 sail. The Moon is three times scale. Fig. 7 Retrograde contours τ 150. is defined to be the control point. At the next instant of time, the Sun shifts, the surfaces change, and a new tangent control point is determined. Figure 8 illustrates three sets of control points selected from the contour tangents for three different sails (0.56 mm/s 2 sail, blue; 1.70 mm/s 2 sail, yellow; and 2.26 mm/s 2 sail, orange) during one synodic cycle. Gaps in the set exist where the ˆl(t) z cutting plane does not intersect a surface on the anti-sun-side of the Moon at a specific value of τ. An alternative to the tangent-selection approach is a strategy to fix the distance below the lunar equator in z and solve for the point on the contour that is located at that distance. The fixed-z distance is selected to correspond to the highest point from the set of tangent control points or to correspond to a height that exists on the instantaneous equilibrium surface most of the time. Sets of control points consistent with the fixed-z strategy for three sails appear in Fig. 9: a 0.56 mm/s 2 sail 5000 km below the x y plane (blue), a 1.70 mm/s 2 sail km below the x y plane (yellow), and a 2.26 mm/s 2 sail km below the x y plane (orange). Identification of these control points serves as a preliminary step in a numerical process to create a periodic orbit. The control points also indicate locations where a suitably sized sail is exactly counteracting the pseudo-gravity gradient at each instant as the Sun direction changes relative to the Earth Moon system. 8 of 13

9 a) x y view b) 3-D view c) x z view d) y z view Fig. 9 Control points selected by fixing the desired z coordinate. The control points are defined for three sails: 5000 km below the x y plane for a 0 = 0.58 mm/s 2 (blue), km for a 0 = 1.70 mm/s 2 (yellow), and km for a 0 = 2.26 mm/s 2 (orange). The Moon is three times scale. Thus, the contours and control points provide insight into the dynamical contribution of the sail as well as the pseudo-gravity gradient. ANALYSIS OF ACCELERATIONS REQUIRED TO LINK CONTROL POINTS The accelerations required to connect some set of control points are calculated and compared to the accelerations present in the system. Any discrepancy between these two accelerations can be attributed to the acceleration required to link the control points and the Coriolis acceleration that results. The equations of motion in the CR3B system can now be rewritten in the form b d 2 r dt ( e ω b b dr dt ) + U = a sail + a thrust (12) where the natural terms are the Coriolis and pseudogravity gradient on the left and the applied accelerations from the sail and any augmenting thrust are on the right in Eqn. (12). The applied accelerations are critical to drive the sail through some desired trajectory defined in terms of the control points. One option is a sail that delivers all of the applied acceleration without any requirement for augmenting the sail force with thrust. Alternatively, a system that explicitly 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 numerically differentiated to determine the contributions of these different components. In the previous analogies, the sail has offset only the pseudo-gravity gradient, U, but this numerical-differentiation process is exploited to determine the level of acceleration that the sail must contribute to supply all applied accelerations. For an arbitrary set of control points, the required sail is infeasible because pointing constraints in Eqn. (10) are violated, variable a 0 is necessary, or both. However, the results offer an improved initial guess for a feasible trajectory and also reveal the nature of the dynamics in the system. Understanding the dynamical structure and the contribution of the various acceleration terms is the main purpose of this analysis. The acceleration components in Eqn. (12) can be rearranged for numerical integration so that the total acceleration of the spacecraft relative to the rotating frame is isolated on the left side and the natural and applied accelerations are summed on the right, i.e. ( ( ) ) b d 2 r b dt 2 = 2 e ω b dr + U + (a sail + a thrust ) dt (13) Control points from the z-fixed strategy, where a 0 = 2.26 mm/s 2 and the z-height is km (i.e., the fully connected orange trajectory in Fig. 9), are used as an example for inspection of the acceleration components. The breakdown of the acceleration components, including the additional thrust required to complete an orbit, appears in Fig. 10. The sail and additional thrust components are defined as the applied components (titled App in Fig. 10) and the natural components are subtracted to render the total acceleration relative to the rotating frame (identified as RotTot in the plot). For a sail-only system, the additional thrust (in the subplot of Fig.10 labeled as Thrust ) component must be zero. The magnitude of the additional thrust is small in this case, except at two control points along the orbit. To further examine the required dynamics to create this orbit, the sail acceleration component can be fictionalized so that it can absorb the additional acceleration. It is termed fictional because the sail can violate pointing constraints from Eqn. (10) and does not require a constant characteristic acceleration. The applied acceleration from the solar sail is a dimensionalized version of Eqn. (7), that is, a sail (t) = a 0 (ˆl(t) n) 2 n (14) The terms in this equation can be calculated for the example orbit and appear in Fig. 11. The first subplot repeats the required applied acceleration from Fig. 10. The second subplot reveals the back-calculation of the required a 0, given the attitude profile in the rotating frame (labeled n, fourth subplot) and sunlight direction (labeled ell, fifth subplot). The plots in Fig of 13

10 Fig. 10 The breakdown of the acceleration components for a 0 = 2.26 mm/s 2 at a fixed-z height of km. Time is measured in terms of the Sun angle, τ (radians). The x component is in blue, y in green, z in red; the magnitude in aqua. Fig. 11 The applied acceleration is compensated via 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 pointing constraint is violated, resulting in a small (ˆl(t) n) term and requiring a large a 0. Otherwise, the variation in a 0 is small. The trajectory is nearly realistic with the sail and could conceivably be adjusted to maintain constant a 0, except for these two points, which occur at a sharp turn in the left side of the orbit pictured in Fig. 9a. Perfectly Circular Orbits Below the Moon If a circular orbit below the Moon is the goal, investigation of fictionalized sail acceleration components reveal that some combination of radius and z-distance produces orbits that require a small variation in a 0 or reasonably achievable a 0 values. The colored maps in Fig. 12 represent the average values of a 0 and the variation in a 0 that are necessary to achieve a circular orbit and satisfy the pointing constraint ((ˆl(t) n) is equivalent to cos α). Note also that the circular orbits with 60,000 km radius require the smallest a 0 and possess a 0 variations of approximately 1 2 mm/s 2. These are candidates for minor control point adjustment and should reveal orbits that are close to circular. The diagonal streak in Fig. 12b indicates that little adjustment would be required for orbits with the given initial r and z values; however, the average a 0 corresponding 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 a fixed-z altitude is easily available as a starting point for a numerical scheme in determining a feasible periodic orbit. All plots of the control points demonstrate that the accelerations must change to accommodate the desired orbits, and consequently the dynamics. NUMERICAL ADJUSTMENT OF CONTROL POINTS Ozimek et al. have developed a method using constrained optimization and direct collocation with nonlinear programming to determine orbits that remain in view of the LSP using only a sail for control. 22 The collocation technique is an implicit integration scheme that relies on an approximated trajectory of control points that can be shifted via numerical corrections processes. The strategy is based on fitting a polynomial between the control points and then comparing the derivative of the polynomial against the equations of motion using an intermediate point for the comparison. The difference between these differentials is labeled the defect and is zero if all control points are solutions to the equations of motion. Constraining certain variables, a Newton-Raphson iteration scheme is employed to update the control points until they converge onto a feasible trajectory. This approach is robust, insensitive to the initial guess, and yields an accurate control history. Using these techniques, Ozimek et al. compute a periodic hover orbit that maintains a view of the LSP, as is apparent in Fig. 13. The breakdown of the acceleration components associated with their hover orbit is plotted in Figs. 14 and 15. Note in Fig. 14 that no additional thrust is required and that the characteristic acceleration remains constant in Fig. 15, aside from errors introduced by numeric differentiation. Ozimek et al. refer to these points as nodes of 13

11 a) x y view b) 3-D view a) Required average a 0 (mm/s 2 ) to maintain circular orbit c) x z view d) y z view b) Required range over a 0 (mm/s 2 ) to maintain circular orbit Fig. 13 Periodic hover orbit using a 0 = 1.70 mm/s 2 from Ozimek et al. 22 The Moon is three times scale. c) Minimum ˆl(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 whereby no defect point is required. 23 The difference between two types of accelerations and velocities is compared: 1) accelerations and velocities required to link the control points and 2) the accelerations and velocities present from the equations of motion at the control points. This difference serves as the basis for the construction of the Jacobian in a Newton-Raphson iteration scheme. A constraint is added to force this difference to zero; periodicity and control vector magnitude are also enforced through constraints. Using the maps in Fig. 12, a combination of desired values for r, z, and a 0 are selected as initial guesses for the new technique. Example trajectories using initial guesses of r = km, z = km, as well as Fig. 14 Breakdown of accelerations in hover orbit from Ozimek et al. 22 Note that no additional thrust is required to maintain this orbit. the characteristic accelerations equal to 0.58 mm/s 2, 1.70 mm/s 2, and 2.26 mm/s 2 are plotted in Fig. 16. For each of these characteristic accelerations, no additional fictional thrust is required to complete the corresponding orbit that appears in Fig. 16; the sail delivers all of the applied thrust to both offset the orbit from the x y plane and to shift the spacecraft in its orbit around the Moon. Note that the sail-direction constraint, (ˆl(t) n), is never violated in any case. The 11 of 13

12 Fig. 15 Total applied acceleration in hover orbit from 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 a 0 = 0.58 mm/s 2 (blue), a 0 = 1.70 mm/s 2 (yellow), and a 0 = 2.26 mm/s 2 (orange). The Moon is three times scale. elevation history for each of the three orbits is represented in Fig. 17. Clearly, there is a strong similarity in the two orbits with the larger characteristic acceleration values suggesting that some family of orbits exists based on similar initial guesses. A number of other cases, with different initial guesses and characteristic accelerations are also examined. The sail orientation, represented in terms of (ˆl(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, indicating that the sail is effectively off for part of the orbit. It should be noted that not all combinations of initial guess and characteristic acceleration in other trials converged or conformed to the pointing constraint in Eqn. (10); however, this method, along with that presented in Ozimek et al. 22 show promise for generating orbits for further study. CONCLUSIONS For the design of a periodic orbit in the vicinity of the LSP in a CR3B model, the instantaneous equilibrium surfaces, as well as the difference between the available acceleration and the required acceleration terms, are very insightful. The comparison of the acceleration components and the views of the changing equilibrium surface offer a greater understanding of the dynamical environment. This knowledge of the dynamical structure is a key element in creating useful trajectories, but also critical to developing a model for the dynamical foundation of the problem when constructing these non-keplerian orbits. The problem is complex, but results indicate that a solution is feasible, an essential step in the design of such nontraditional orbits. ACKNOWLEDGEMENTS The authors wish to thank Daniel Grebow for his insightful suggestion on a new collocation technique to create periodic orbits. Martin Ozimek and Daniel Grebow also generously shared numerical data from their analysis and contributed in valuable technical discussions. The first author gratefully acknowledges the National Aeronautics and Space Administration s (NASA) Office of Education for their sponsorship of his attendance at the 59 th International Astronautical Congress in Glasgow, Scotland. Portions of this work were supported by Purdue University. 12 of 13

13 REFERENCES 1 The Vision for Space Exploration. Technical Report NP HQ, National Aeronautics and Space Administration, February Space Communication Architecture Working Group (SCAWG). NASA Space Communication and Navigation Architecture Recommendations for Technical report, National Aeronautics and Space Administration, Washington, DC, May Todd A. Ely. Stable constellations of frozen elliptical inclined lunar orbits. Journal of the Astronautical Sciences, 53(3): , July September Todd A. Ely and Erica Lieb. Constellations of elliptical inclined lunar orbits providing polar and global coverage. In AAS/AIAA Astrodynamics Specialist Conference, Lake Tahoe, CA, August Kathleen C. Howell, Daniel J. Grebow, and Zubin P. Olikara. Design using Gauss perturbing equations with applications to lunar south pole coverage. In Advances in the Astronautical Sciences, February Daniel J. Grebow, Martin T. Ozimek, Kathleen C. Howell, and David C. Folta. Multibody orbit architectures for lunar south pole coverage. Journal of Spacecraft and Rockets, 45(2): , March April John L. West. The lunar polesitter. In AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, August Robert L. Forward. Statite: A spacecraft that does not orbit. Journal of Spacecraft and Rockets, 28(5): , September October Colin R. McInnes, Alastair J. C. McDonald, John F. L. Simmons, and Ewan W. MacDonald. Solar sail parking in restricted three-body systems. Journal of Guidance, Control, and Dynamics, 17(2): , March April Colin R. McInnes. Solar Sailing: Technology, Dynamics and Mission Applications. Space Science and Technology. Springer-Praxis, New York, Chen-wan L. Yen. Solar sail geostorm warning mission design. In AAS/AIAA Space Flight Mechanics Meeting, February Carl G. Sauer. The L1 diamond affair. In AAS/AIAA Space Flight Mechanics Conference, February Hexi Baoyin and Colin R. McInnes. Solar sail halo orbits at the Sun-Earth artificial L1 point. Celestial Mechanics and Dynamical Astronomy, 94(2): , February Ariadna Farrés and Àngel Jorba. Solar sail surfing along families of equilibrium points. In 58th IAC, Hyderabad, India, October Esther Morrow, Dan Lubin, and Daniel Scheeres. Solar sail orbit operations at asteroids. Journal of Spacecraft and Rockets, 38(2): , March April Stephen Broschart and Daniel Scheeres. Control of hovering spacecraft near small bodies: Application to asteroid Itokawa. Journal of Guidance, Control, and Dynamics, 28(2): , March April Kathleen Howell and Diane Davis. Spacecraft trajectory design strategies based on close encounters with a smaller primary in a 4-body model. In AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, August Charles Adams. Technical communications, 6 August Lou Freidman. Technical communications, 18 May David Lichodziejewski, Billy Derbes, David Sleight, and Troy Mann. Vaccum deployment and testing of a 20m solar sail system. In 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, May Billy Derbes, Gordon Veal, Jim Rogan, and Charlie Chafer. Team Encounter solar sails. In 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April Martin Ozimek, Daniel Grebow, and Kathleen Howell. Solar sails and lunar south pole coverage. In AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, August Geoffrey Wawrzyniak, Daniel Grebow, and Kathleen Howell. A direct collocation technique using control points and accelerations. In Preparation. APPENDIX The nondimensional sail characteristic acceleration, β, can be derived in two ways. The first is to express β as a ratio of the sail characteristic acceleration in dimensional units, a 0, and the system characteristic acceleration, a. The second arises from the ratio of the critical areal density and the sail areal density. The lower design limit for sail areal density, sometimes denoted the solar sail loading parameter, is currently around 4 g/m Solar radiation pressure, measured in micro-pascals, at 1 AU is P 1 = 1368 W/m2 c = µpa The sail characteristic acceleration, a 0, is the acceleration from solar radiation pressure on the sail at 1 AU, that is, a 0 = 2ηP 1 σ Using the lower design limit value for the areal density (σ = 4 g/m 2 ), a 0 is approximately 2 mm/s 2 when η = 90%. Critical areal density balances the gravity of the system at the Earth Moon distance, L, σ = 2ηP 1 ( GM L 2 = g/m 2 ) 1 Thus, the nondimensional sail characteristic acceleration in Earth Moon CR3B system is expressed as β = a 0 a = σ σ clearly based either on characteristic and critical accelerations or on critical and characteristic areal densities. 13 of 13