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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
[email protected]
Kathleen C. Howell†
Purdue University, United States of [email protected]
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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In Preparation.
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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