Paper AAS 07-143

DESIGN USING GAUSS’ PERTURBING EQUATIONS WITHAPPLICATIONS TO LUNAR SOUTH POLE COVERAGE

K. C. Howell,∗ D. J. Grebow,† Z. P. Olikara†

Perturbations due to multiple bodies and gravity harmonics are modelednumerically with Gauss’ equations. Using the linear variational equa-tions, orbits are calculated with shapes and orientations desirable for lunarsouth pole coverage. Families of periodic orbits are first computed in theEarth-Moon Restricted Three-Body Problem (R3BP) from linear predic-tions along the tangent subspace. Using multiple shooting, the trajectoriesfor two spacecraft are then transitioned to the full ephemeris model; thefull model incorporates solar perturbations and a 50×50 Lunar Prospectorgravity model. The results are verified with commercial software and vari-ous aspects of the coverage are also discussed. Finally, a ten-year simulationis investigated in the full LP165P gravity model for long term communica-tions with a site at the Shackleton Crater.

INTRODUCTION

Interest in the lunar south pole1 has prompted studies in frozen orbit architectures for con-stant south pole surveillance (including Elipe and Lara,2 Ely,3 Ely and Lieb,4 as well as Folta andQuinn5). Many of the methods used in these studies were first pioneered by Lidov6 in 1962. In gen-eral, the Earth is modeled as a third-body perturbation in the two-body, Moon-spacecraft problem.Frozen orbit requirements are computed directly from Lagrange’s planetary equations. An approxi-mate disturbing function is averaged over the system mean anomalies. The frozen orbit conditionsare calculated analytically by fixing orbital eccentricity, argument of periapsis, and inclination, i.e.,allowing no variations in these quantities. Of course, approximating and averaging the disturbingfunction introduces some errors into the analysis. Furthermore, in the three-body problem an exactlyfrozen orbit, or periodic orbit, requires that the semi-major axis (a), eccentricity (e), argument ofperiapsis (ω), inclination (i), and true anomaly (θ∗) all return simultaneously to their initial valuesat a later point along the path. Correspondingly, the motion of the ascending node (Ω) must becommensurate with the motion of the third body. Even with the simplifying assumptions, it is stilldifficult to generalize the analytical methods to include additional perturbations from other bodiesand gravity harmonics. However, some frozen orbit conditions can still be numerically integrated inthe full model with only minimal secular drift. For example, Ely3 identifies a potential stable con-stellation of three spacecraft for continuous south pole coverage over 10 years in the full model. Foltaand Quinn5 consider many different sets of initial conditions for frozen orbits, and note secular driftafter several months; they employ the full LP100K and LP165P models for some combinations of

∗Hsu Lo Professor of Aeronautical and Astronautical Engineering, School of Aeronautics and Astronautics, PurdueUniversity, Grissom Hall, 315 North Grant St., West Lafayette, Indiana 47907-1282; Fellow AAS; Associate FellowAIAA.

†Student, School of Aeronautics and Astronautics, Purdue University, Grissom Hall, 315 North Grant St., WestLafayette, Indiana 47907-1282; Student Member AIAA.

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e, ω, and i. A method of ‘centering’ is discussed to keep the spacecraft from lunar impact for 10 years.

The Moon-spacecraft system that also incorporates Earth gravity is, perhaps, more accuratelymodeled as a three-body problem. Since there are no analytical, closed-form solutions available inthe three-body problem, motion is simulated numerically. Historically, the traditional formulationof the Restricted Three-Body Problem (R3BP) has proven useful in mission design of lunar orbits,with an array of numerical techniques. For example, Farquhar7 first examined halo orbits in 1971 forcontinuous communications between the Earth and the far side of the Moon; station-keeping costsare included as part of the analysis. For lunar south pole coverage, Grebow et al.8 demonstrate thatconstant communications with a ground station at the Shackleton Crater (very near the south pole)can be achieved with two spacecraft in many different combinations of Earth-Moon libration pointorbits. Apoapsis altitudes all exceed 10,000 km, however, an undesirable feature for lunar ground-based communications. More recently, in a R3BP formulation, Russell and Lara9 investigate manydifferent repeat ground track orbits for global lunar coverage, incorporating a 50× 50 LP150Q grav-ity model. Their study also includes a 10-year propagation of a ‘73-cycle’ repeat ground track orbitfavorable for lunar south pole coverage. However, since all the orbits must be synchronized with theMoon’s rotation, only particular solutions can be isolated without making additional assumptionsabout the problem.

The equations describing the motion of a spacecraft within the context of the R3BP are tradi-tionally written in terms of Cartesian coordinates. However, in the design of orbits for lunar southpole coverage, it is desirable to control the shape and orientation. For example, the orbits must beeccentric and oriented such that apoapsis occurs over the lunar south pole. The orbits must alsopossess feasible periapsis and apoapsis altitudes. These parameters are, perhaps, better controlledwhen the problem is reformulated in terms of Keplerian elements by numerically integrating Gauss’equations. Unfortunately, many corrections schemes for computation of periodic orbits in the R3BPdepend on minimizing specific Cartesian coordinates downstream along the path by updating specificCartesian components of the initial state vector. (For some examples, see Szebeheley.10) However,in 1977, Markellos and Halioulias11 developed a strategy for computing families of periodic orbitsin the two-dimensional Stormer problem by generating predictions with the tangent subspace. Amore recent application is the computation of asymmetric periodic orbits in the classical formulationof the R3BP.12 The numerical methods and related techniques are easily generalized to computeperiodic orbits with Keplerian elements by integrating Gauss’ equations.

According to Folta and Quinn,5 lunar orbit altitudes above approximately 750 km are dominatedby third-body (Earth) disturbances, whereas altitudes below about 100 km are completely dictatedby spherical harmonics. Then, for altitudes ranging between 100 and 750 km, both third-bodyeffects and spherical gravity harmonics are significant. In general, orbits designed for lunar southpole coverage are likely to possess apoapsis altitudes greater than 750 km with periapsis altitudesfrom 100 to 750 km. Russell and Lara9 also note significant changes when integrating their initialstates (corrected to be periodic orbits in the R3BP and incorporating a 50 × 50 harmonic gravitymodel) in the full ephemeris model, including Earth and solar perturbations. Therefore, the effectsof spherical gravity harmonics and even fourth-body solar perturbations cannot be neglected. How-ever, in modeling the spherical harmonics, information about the far side of the Moon is limited.The existence of lunar mascons, or mass concentrations, further complicates baseline modeling. Cur-rently, accurate modeling data is available from NASA’s Lunar Prospector Mission (1998). Since1998, Lunar Prospector models have been enhanced from degree 75 up to degree 165 (LP165P).13

When transitioning orbits from the R3BP to the full LP165P model, including solar perturbations,it is also desirable to maintain the nominal shape as initially designed in the R3BP. One differentialcorrections scheme that can accomplish this goal was developed in 1986 by Howell and Pernicka.14

However, the methodology depends on targeting positions at specified intervals along the trajectoryand then locating a continuous path with velocity discontinuities at the target points; the disconti-

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nuities are subsequently reduced or eliminated. Alternatively, a nearby continuous solution in thefull model can be computed with any set of coordinates using a multiple shooting scheme. (SeeStoer and Burlirsch.15) With applications to celestial mechanics, Gomez et al.16 recently used mul-tiple shooting to compute quasihalo orbits in the full model. These methods are easily generalizedto locate natural solutions in the full LP165P model including solar perturbations, using Gauss’perturbing equations.

In this current analysis, the n-body problem with spherical harmonics is formulated in terms ofGauss’ perturbing equations. The state-transition matrix is available, and is, thus, a mechanismfor computation of semi-elliptical orbits (e < 1). All trajectories are first designed in the restrictedproblem, where the R3BP is identified as a special case of the n-body problem with n = 3. UsingGauss’ equations and the numerical techniques developed by Markellos,11,12 a family of periodicorbits is computed in the R3BP with shape and orientation advantageous for lunar south polecoverage. The accuracy of the solutions are verified by transforming the Keplerian elements intoCartesian coordinates and examining the Jacobi Constant (C). To maximize lunar south polecoverage, two spacecraft are phase shifted in the same orbit by a half-period as described in Grebowet al.8 The orbits for both spacecraft are transitioned to the full ephemeris model with a multipleshooting scheme using JPL’s DE405 ephemeris file and a 50 × 50 LP165P gravity model. Bothspacecraft maintain their initial phasing for over 1,000 days without lunar impact. The results areverified with AGI’s Satellite Tool KitR© where percent access with the Shackleton Crater facility,the Earth, and the White Sands Test Facility is computed. Finally, a 10-year simulation with anoptimizing compiler in Fortran 90 is implemented on a compute cluster for two spacecraft usingan Adams-Bashforth-Moulton integrator17 (with variable step size). The elevation angle from theShackleton site for both spacecraft is also computed and discussed in relationship to coverage.

GAUSS’ EQUATIONS AND THE VARIATIONAL RELATIONSHIPS

Assume a particle P2 (e.g., a spacecraft) moves relative to a central body P1 (e.g., the Moon)in a nominally conic orbit. For a perturbing force, Fpert, applied to the particle, an auxiliaryorbit or osculating orbit is defined such that two-body elliptical (e < 1) motion is recovered ifFpert = 0. (See Figure 1.) Note that ‘overbars’ indicate vectors while ‘hats’ denote vectors of unitmagnitude. The perturbing force FP

pert =F r F θ Fh

T is comprised of components that aregenerally expressed in terms of the P2-centered perifocal frame (P) that appears in the figure. A

Figure 1: Motion of P2 Relative to Central Body P1 Subject to the Perturbing Force Fpert

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superscript P on the force Fpert indicates components associated with the frame P. In general, avector with an unspecified vector basis does not include a bold-faced superscript. Unit vectors thatdefine P include rP directed from P1 to P2, hP parallel to the orbital angular momentum vectorcorresponding to the osculating conic orbit of P2 relative to P1, and θP = hP × rP completing theright-handed triad. It is assumed that ‖Fpert‖ is small when compared to the central body force ofP1 on P2. Then, according to Gauss, the osculating elements are subject to the following variations

a =2e sin θ∗

ζηF r +

2aζηr

F θ,

e =ζ sin θ∗

ηaF r +

ζ

ηa2e

(a2ζ2

r− r

)F θ,

ω = −ζ cos θ∗

ηaeF r +

ζ sin θ∗

ηae

(1 +

r

aζ2

)F θ − r sin θ

ηa2ζ tan iFh,

i =r cos θηa2ζ

Fh,

Ω =r sin θ

ηa2ζ sin iFh,

θ∗ =ηa2ζ

r2− ω − Ω cos i,

(1)

where ζ =√

1− e2, r = aζ2/ (1 + e cos θ∗), and θ = θ∗ + ω. (See Brumberg18 for a derivationof the perturbing equations using elliptic functions.) Notice that, for computational purposes, thedependent variable corresponding to the sixth equation is selected as true anomaly, θ∗, rather thanmean anomaly. The η in eq. (1) is the mean motion of P2, i.e., η =

√µ1/a3 where µ1 is the non-

dimensional mass parameter of P1. The equations are scaled by non-dimensionalizing with respectto certain characteristic quantities. From a number of options, the characteristic quantities availablefrom the three-body problem possess certain advantages. Let the characteristic length be definedby the mean distance between P3 and P1, where P3 is the most significant perturbing gravitationalinfluence (e.g., the Earth). Determine the characteristic time from the mean motion of P3 relativeto P1.

Let the six element state vector q be defined as q =a e ω i Ω θ∗

T . Then, the vectorFP

pert is expressed as a net perturbing force where

FPpert = RP

1 (q) +n∑

m=3

FPm(q). (2)

The vector RP1 (q) in eq. (2) is the resultant aspherical gravitational force associated with the body

P1. The summation∑n

m=3 FPm(q) simply represents the forces due to the n − 2 additional point-

masses, P3, P4, . . . , Pn. Additional perturbing forces, such as solar radiation pressure, can also beadded to eq. (2) without altering the following analysis. The first-order variational equations arederived and result in the vector differential equation δ ˙q = A(t)δq. The time-varying matrix A(t) isevaluated analytically along the reference, i.e.

Aij =∂qi∂qj

+∂qi

∂FPpert

(∂RP

1

∂qj+

n∑m=3

∂FPm

∂qj

). (3)

The expressions for the partial derivatives ∂qi

∂qjare available in Appendix A. Since eqs. (1) are linear

in FPpert, computation of the row vector ∂qi

∂FPpert

simply requires an inspection of eqs. (1). Evaluation

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of the remaining terms in eq. (3), i.e., ∂RP1

∂qjand ∂FP

m

∂qj, depends on the expressions for RP

1 and FPm .

Of course, initial design is accomplished within the context of the R3BP. However, the followinganalysis includes mathematical expressions for RP

1 and FPm in the full model, and then, appropriately,

identifies the R3BP as a special case. Once ∂RP1

∂qjand ∂FP

m

∂qjare available, the state-transition matrix

Φ(tk+1, tk) is evaluated from the governing matrix differential equation

Φ(tk+1, tk) = A(tk+1)Φ(tk+1, tk). (4)

Using a first-order Taylor series expansion about the reference, the linear variational equations arewritten in the following form

δqk+1 = [Φ(tk+1, tk) ˙qk+1]δqk

δtk+1

. (5)

Note that eq. (5) allows non-contemporaneous variations in qk+1.

The N -Body Problem

The n-body problem is represented in Figure 2. The perturbing forces due to n − 2 additionalpoint-masses, P3, P4, . . ., Pn are available from the law of gravity. Thus, FP

m (q) in eq. (2) isexpressed

FPm (q) = µm

(rP2m∥∥rP2m

∥∥3 −rP1m

‖r1m‖3

), (6)

where µm is the associated non-dimensional gravitational parameter associated with Pm. For conve-nience, the angles Ω, i, θ are measured relative to an intermediate inertial frame (I). The intermediateframe is defined such that xI is directed from the point-mass P3 to P1 at epoch. In addition, theunit vector zI is parallel to the associated two-body angular momentum vector for the motion ofP1 with respect to P3. Of course, yI = zI × xI. Notice that xI, yI, and zI are consistent with anepoch-fixed “rotating” frame in the three-body problem and will therefore be consistent with theformulation in the R3BP. (See Figure 2.) For example, in the Earth-Moon system, where the centralbody is the Moon (P1), at epoch xI is directed from the Earth (P3) to the Moon and zI is parallelto the angular momentum vector. Then, the transformation from I to P is defined by the Euler3-1-3 (Ω-i-θ) sequence such that the transformation matrix for orientation of P with respect to I iswritten

PTI =

cos Ω cos θ − sinΩ cos i sin θ sinΩ cos θ + cos Ω cos i sin θ sin i sin θ− cos Ω sin θ − sinΩ cos i cos θ − sinΩ sin θ + cos Ω cos i cos θ sin i cos θ

sinΩ sin i − cos Ω sin i cos i

. (7)

The vectors xI, yI, and zI are instantaneously available from the JPL DE405 ephemeris file andare the vector components in the transformation matrix ITJ from the EMEJ2000 frame (J) to theintermediate inertial frame (I). Note that both rJ1m and ITJ are determined from ephemerides and,therefore, are independent of the state vector q. Since rP12 = rrP and rP1m = PTIITJrJ1m, thenrP2m = rP1m − rP12 = PTIITJrJ1m − rrP. Of course, the vector rJ1m must be non-dimensional relativeto the characteristic length. The partial derivatives of FP

m (q) with respect to the state variablesmust be evaluated for implementation in eq. (3). Partial differentiation is accomplished via a matrixchain-rule expansion, i.e.

∂FPm

∂qj= µm

[(∂PTI

∂qjITJrJ1m − ∂r

∂qjrP)

∆m + rP2m

∂∆m

∂qj− ∂PTI

∂qj

ITJ rJ1m

‖r1m‖3

], (8)

where ∆m = 1

‖rP2m‖3 . The partial derivatives ∂PTI

∂qjand ∂r

∂qjin eq. (8) are straightforward. The most

complicated expression in eq. (8) is clearly ∂∆m

∂qjand is derived in Appendix B.

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Figure 2: Position of Pm and Definition of the Intermediate Inertial Frame (I)

Spherical Harmonics

To incorporate the spherical harmonics associated with P1 into the model, the location of P2

(or the vehicle of interest) must be identified in terms of latitude (φ) and longitude (λ), as definedin Figure 3. A spherical coordinate frame (L) is defined in terms of unit vectors rL, λL, and φL.Latitude is measured with respect to the P1 equator, while longitude is a degree measurement fromthe P1 prime meridian. To compute φ and λ, it is necessary to define the orientation of P1 withrespect to an inertial frame. The frame B, fixed in P1, is defined such that xB locates the intersectionbetween the equator and prime meridian of P1. The third vector in the P1-fixed frame, zB, is parallelto the spin axis of P1 and yB = zB × xB. (See Figure 3.) For the Moon, a set of three Euler anglesis available from the JPL DE405 ephemeris file to define the orientation of the Moon with respectto the EMEJ2000 (J) reference frame. Since the Euler sequence is 3-1-3 (ϕ-ϑ-ψ), a transformationto orient B relative to J can be derived such that

BTJ =

cosϕ cosψ − sinϕ cosϑ sinψ sinϕ cosψ + cosϕ cosϑ sinψ sinϑ sinψ− cosϕ sinψ − sinϕ cosϑ cosψ − sinϕ sinψ + cosϕ cosϑ cosψ sinϑ cosψ

sinϕ sinϑ − cosϕ sinϑ cosϑ

. (9)

(See Konopliv13 for a geometric discussion of the angles ϕ, ϑ, and ψ.) Then, the transformationbetween the P1-fixed frame (B) and the spherical coordinate frame (L) is

BTL =

cosφ cosλ − sinλ − sinφ cosλcosφ sinλ cosλ − sinφ sinλ

sinφ 0 cosφ

. (10)

Let b be a three-dimensional unit vector directed from the center of P1 to P2 (as seen in Figure 3).Of course, in the perifocal frame, bP = rP. So, to compute bB, or b in the P1-fixed frame, consider

bB = BTJJTIITPrP, (11)

where, in general, yTx = [xTy]T . Thus, bB is expressed as a function of the state vector q. OncebB is computed, expressions for φ and λ as functions of the state vector q can also be evaluated. Let

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Figure 3: Direction of P2 in Latitude (φ) and Longitude (λ)

bB be resolved into the associated three-element vector such that bB =bx by bz

T . Then, fromFigure 3, it is apparent that

φ = sin−1 bz, (12)

λ = ± cos−1

(bx√

bx2 + by2

),

where the sign of λ in eq. (12) is determined from the sign of by. Then, the perturbing force due toan aspherical central body for a predetermined degree nmax is written

RL1 (q) =

nmax∑n=2

n∑m=0

µ1

r2

(RP1

r

)n

− (n+ 1) P (m)

n

(Cn,m cosmλ+ Sn,m sinmλ

)m secφ P (m)

n

(−Cn,m sinmλ+ Sn,m cosmλ

)cosφ P (m)′

n

(Cn,m cosmλ+ Sn,m sinmλ

), (13)

where R1 (q) is expressed in terms of the spherical coordinate frame L.19 The normalized coef-ficients Cn,m and Sn,m in eq. (13), that is, the tesseral and sectoral harmonic coefficients, aredetermined experimentally. The normalized zonal coefficients are defined such that Jn = −Cn,0 andare, therefore, implicitly included in the above summation. The scalar quantity RP1 in eq. (13) isthe non-dimensional, nominal value for the radius of body P1 associated with the coefficients. Thenormalized Legendre function P (m)

n is defined as

P (m)n (sinφ) =

[(2− δm0)

(n−m)!(n+m)!

] 12

cosm φdmPn (sinφ)d (sinφ)m , (14)

where Pn (sinφ) is the normalized, nth degree Legendre polynomial of the first kind in sinφ (sinφis the argument of Pn) and the Kronecker delta function δm0 is one for m = 0 and zero otherwise.

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The prime in the third element of eq. (13) is the first derivative of P (m)n with respect to sinφ. The

normalized Legendre functions are determined recursively, that is

P (m)n =

[

2n+1(n+m)(n−m)

] 12

(2n− 1)12 sinφ P (m)

n−1 −[

(n+m−1)(n−m−1)2n−3

] 12P

(m)n−2

, m < n,(

2n+12n

) 12 cosφ P (m−1)

n−1 , m = n,

0, m > n,

(15)

where the recursive sequence is initiated by P(0)0 = 1, P (0)

1 =√

3 sinφ, and P(1)1 =

√3 cosφ.

Similarly

P (m)′

n =1

cos2 φ

−n sinφ P (m)

n +[

(2n+1)(n+m)(n−m)2n−1

] 12P

(m)n−1

. (16)

For large nmax, the unnormalized harmonic coefficients are subject to underflow, while the unnor-malized Legendre polynomials experience overflow. For nmax > 50, computation begins to lose ac-curacy even with double precision storage capability. Alternatively, computations with normalizedharmonic coefficients and Legendre polynomials as defined in eq. (13) are possible for large nmax.

Recall that Gauss’ equations require that R1 (q) be expressed in terms of the perifocal frame(P). (See eq. (2).) The perifocal frame and the spherical frame are related by a single rotation βabout r, i.e.

PTL =

1 0 00 cosβ sinβ0 − sinβ cosβ

, (17)

where β is the angle between φ and h. Then

RP1 (q) = PTLRL

1 (q) . (18)

However, it is nontrivial to determine β. Rather than computing β, where β must also be a functionof the state vector q, the elements of PTL are determined directly from the following sequence oftransformations

PTL = PTIITJJTBBTL. (19)

The transformation matrices ITJ and JTB are independent of the state vector q while PTI andBTL are functions of q. The partial derivatives of RP

1 (q) with respect to the state variables arerequired for evaluation of eq. (3). Then, via a matrix chain-rule expansion

∂RP1

∂qj=∂PTI

∂qjITJJTBBTLRL

1 + PTIITJJTB ∂BTL

∂qjRL

1 + PTL ∂RL1

∂qj. (20)

The partial derivative ∂PTI

∂qjis also necessary for evaluation of eq. (8). The partial derivative ∂BTL

∂qj

involves more terms

∂BTL

∂qj=

∂φ

∂qj

− cosλ sinφ 0 − cosλ cosφ− sinλ sinφ 0 − sinλ cosφ

cosφ 0 − sinφ

+∂λ

∂qj

− sinλ cosφ − cosλ sinλ sinφcosλ cosφ − sinλ − cosλ sinφ

0 0 0

, (21)

where ∂φ∂qj

and ∂λ∂qj

are evaluated from eq. (12). (See Appendix C.) Finally, the vector RL1 (q), as

defined by eq. (13), is of the form

RL1 (q) =

nx∑n=2

n∑m=0

Rr

n,m

Rλn,m

Rφn,m

. (22)

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Correspondingly, the expression for ∂RL1

∂qjis written

∂RL1

∂qj=

nx∑n=2

n∑m=0

∂Rrn,m

∂qj

∂Rλn,m

∂qj

∂Rφn,m

∂qj

, (23)

where

∂Rrn,m

∂qj=

∂Rrn,m

∂r

∂r

∂qj+

(∂Rr

n,m

∂P(m)n

∂P(m)n

∂φ

)∂φ

∂qj+∂Rr

n,m

∂λ

∂λ

∂qj,

∂Rλn,m

∂qj=

∂Rλn,m

∂r

∂r

∂qj+

(∂Rλ

n,m

∂P(m)n

∂P(m)n

∂φ+∂Rλ

n,m

∂φ

)∂φ

∂qj+∂Rλ

n,m

∂λ

∂λ

∂qj, (24)

∂Rφn,m

∂qj=

∂Rφn,m

∂r

∂r

∂qj+

(∂Rφ

n,m

∂P(m)′

n

∂P(m)′

n

∂φ+∂Rφ

n,m

∂φ

)∂φ

∂qj+∂Rφ

n,m

∂λ

∂λ

∂qj.

The expressions ∂P (m)n

∂φ and ∂P (m)′n

∂φ are also defined recursively. (See Appendix C.) The partials ∂φ∂qj

and ∂λ∂qj

appear in eq. (21) as well. The remaining partial derivatives in eqs. (24) can be evaluatedsimply by inspection.

The Restricted Three-Body Problem

The R3BP is a special case of the n-body problem where n = 3. Therefore, the R3BP can berecovered from eq. (2) by selecting n = 3 in the summation. In the Earth-Moon system, periodicorbits, or repeat lunar ground-track orbits, exist in the R3BP for non-zero RP

1 (q).9 Ultimately,the goal is a strategy for the computation of families of quasi-periodic solutions in the full modelpossessing specific characteristics for shape and orientation. Therefore, for this analysis, the initialdesign phase incorporates RP

1 (q) = 0. That is, the only perturbing force is due to the third-bodypoint mass P3.

The gravitational force of P3 acting on P2 is mathematically described by eq. (6) where m = 3.It is assumed that the perturbing body P3, moves in a circular path relative to the central bodyP1, as demonstrated by the dashed circle in Figure 4. The rotating frame (S), centered at P1, isconsistent with the rotating frame in the R3BP. The frame S is defined such that xS is directedfrom P3 toward P1, zS is parallel to the angular velocity vector associated with the orbit of P3,and yS = zS × xS. For convenience, it is assumed that the rotating frame (S) is aligned with theintermediate inertial frame (I) at the initial time (t0 = 0). Then, the position of P3 with respect toP1, that is r13, is not accessed from ephemerides, but is computed from the non-secular expression

rI13 = − cos t xI − sin t yI, (25)

where t is the non-dimensional time as characterized by the mean motion of P3. Then, rI13 defines theposition of P3 with respect to P1, with components expressed in terms of the intermediate inertialframe (I). (See Figure 4.) Therefore, rP13 = PTI rI13 and rP23 = rP13 − rP12 = PTIrI13 − rrP. Since rI13is independent of the state-vector q, the form of eq. (8) remains unaltered. However, the positionvector r13 in eq. (25) is already expressed in terms of the intermediate inertial frame (I). That is,rI13 does not require transformation from the EMEJ2000 frame (J) to the intermediate inertial frame

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Figure 4: The Intermediate Inertial (I) and Rotating (S) Frames in the R3BP

(I). As a result, eq. (8) can be written more simply in the form

∂FP3

∂qj= µ3

[(∂PTI

∂qjrI13 −

∂r

∂qjrP)

∆3 + rP23∂∆3

∂qj− ∂PTI

∂qjrI13

], (26)

where ∆3 = 1

‖rP23‖3 . An expression for ∂∆3

∂qjis derived from the more general expression for ∂∆m

∂qjin

Appendix B.

INITIAL DESIGN IN THE RESTRICTED THREE-BODY PROBLEM

All orbits are initially designed in the R3BP without the gravitational harmonics associated withP1 (the Moon). Of course, the variational relationships governing δq in eq. (5) cannot be directlyapplied to compute the initial state variations that satisfy the conditions for periodicity. Therefore,a more sophisticated targeting scheme is necessary. One such alternate strategy was developed byMarkellos11,12 in 1977. Markellos also discusses a methodology for exploiting the tangent subspaceto generate families of periodic orbits. The theory and associated numerical techniques are appliedto Gauss’ equations to determine families of periodic orbits within the context of the R3BP. For allcomputations, the initial time t0 is defined such that t0 = 0.

Targeting Periodic Orbits

Consider an orbit near a dominant gravitational body, but perturbed by the gravity from a thirdbody. For the orbit to be periodic, the elements a, e, ω, i, and θ∗ must all simultaneously return totheir initial values at a later time t along the path. Correspondingly, the motion of the ascendingnode must be commensurate with the motion of the third body, or

t− (Ω− Ω0) = 2kπ, for k integers. (27)

Enforcing equation (27) offsets the rotation of the line of nodes relative to the line connecting P3 toP1, thereby satisfying the periodicity requirement relative to the rotating frame (S). (Recall Figure

10

4.) Consider the following mappings

a = Fa (a0, e0, i0,Ω0, θ∗0) ,

i = Fi (a0, e0, i0,Ω0, θ∗0) ,

Ω = FΩ (a0, e0, i0,Ω0, θ∗0) ,

θ∗ = Fθ∗ (a0, e0, i0,Ω0, θ∗0) .

(28)

The mappings Fa, Fi, FΩ, and Fθ∗ bring a0, e0, i0, Ω0, and θ∗0 from the surface of section at ω = ω0,to the respective state at the kth + 1 crossing of the surface. Then, for periodic motion, the goal isthe determination of the variations δa0, δi0, δΩ0, and δθ∗0 , such that, at a later time, the value ofthe element is equal to its original value plus the variation. Thus, the following will be true at timet: a = a0 + δa0; e = e0 + δe0; Ω = Ω0 + δΩ + t + δt − 2kπ; and θ∗ = θ∗0 + δθ∗0 . In terms of themappings

a = Fa (a0 + δa0, e0 + δe0, i0 + δi0,Ω0 + δΩ0, θ∗0 + δθ∗0) = a0 + δa0,

i = Fi (a0 + δa0, e0 + δe0, i0 + δi0,Ω0 + δΩ0, θ∗0 + δθ∗0) = i0 + δi0,

Ω = FΩ (a0 + δa0, e0 + δe0, i0 + δi0,Ω0 + δΩ0, θ∗0 + δθ∗0) = Ω0 + δΩ0 + t+ δt− 2kπ,

θ∗ = Fθ∗ (a0 + δa0, e0 + δe0, i0 + δi0,Ω0 + δΩ0, θ∗0 + δθ∗0) = θ∗0 + δθ∗0 .

(29)

The expressions in eqs. (29) can be expanded about the reference values a0, e0, i0, Ω0, and θ∗0 . Thefirst-order expansions are written

Fa (a0, e0, i0,Ω0, θ∗0) +

∂Fa

∂a0δa0 +

∂Fa

∂e0δe0 +

∂Fa

∂i0δi0 +

∂Fa

∂Ω0δΩ0 +

∂Fa

∂θ∗0δθ∗0 = a0 + δa0,

Fi (a0, e0, i0,Ω0, θ∗0) +

∂Fi

∂a0δa0 +

∂Fi

∂e0δe0 +

∂Fi

∂i0δi0 +

∂Fi

∂Ω0δΩ0 +

∂Fi

∂θ∗0δθ∗0 = i0 + δi0,

FΩ (a0, e0, i0,Ω0, θ∗0) +

∂FΩ

∂a0δa0+

∂FΩ

∂e0δe0+

∂FΩ

∂i0δi0+

∂FΩ

∂Ω0δΩ0+

∂FΩ

∂θ∗0δθ∗0 = Ω0 + δΩ0 + t+ δt− 2kπ

Fθ∗ (a0, e0, i0,Ω0, θ∗0) +

∂Fθ∗

∂a0δa0 +

∂Fθ∗

∂e0δe0 +

∂Fθ∗

∂i0δi0 +

∂Fθ∗

∂Ω0δΩ0 +

∂Fθ∗

∂θ∗0δθ∗0 = θ∗0 + δθ∗0 ,

(30)

where, in general, all of the reference Keplerian elements vary with time for the R3BP. Substitutingeqs. (28) into eqs. (30) and rearranging results in the targeter matrix relationship

a0 − a

i0 − i

t− (Ω− Ω0)− 2kπθ∗0 − θ∗

= [C]

δa0

δi0

δe0

δΩ0

δθ∗0

, (31)

where

C =

∂a

∂a0− 1

∂a

∂e0

∂a

∂i0

∂a

∂Ω0

∂a

∂θ∗0∂i

∂a0

∂i

∂e0

∂i

∂i0− 1

∂i

∂Ω0

∂i

∂θ∗0∂Ω∂a0

∂Ω∂e0

∂Ω∂i0

∂Ω∂Ω0

− 1∂Ω∂θ∗0

∂θ∗0∂a0

∂θ∗0∂e0

∂θ∗0∂i0

∂θ∗0∂Ω0

∂θ∗0∂θ∗0

− 1ω

a

i

Ω− 1θ∗

∂ω

∂a0

∂ω

∂e0

∂ω

∂i0

∂ω

∂Ω0

∂ω

∂θ∗0

. (32)

11

The components of C are evaluated by isolating the necessary sensitivity partials in eq. (5), where∂qi

∂qj,0= Φij (t, 0). Since interest is in semi-elliptical orbits (e < 1), it is desirable to constrain

variations in eccentricity during the solution process in eq. (32). If e0 is established as a fixedelement in eq. (32), that is δe0 = 0, the solution space can be expanded. The targeter relationshipthen becomes

a0 − a

i0 − i

t− (Ω− Ω0)− 2kπθ∗0 − θ∗

= [D]

δa0

δi0

δΩ0

δθ∗0

, (33)

where the matrix D is defined by removing the appropriate column from the matrix C to satisfythe constraint δe0 = 0. Since the matrix D is square, the state variations are obtained by directlyinverting eq. (33). However, the matrix D may become ill-conditioned for some orbits (e.g., stableperiodic orbits). In such cases, the speed of convergence is increased by incorporating a weightedand/or regularized minimum-norm solution. Equation (33) can then be solved using

δa0

δi0

δΩ0

δθ∗0

= WDT[DWDT − γI

]−1

a0 − a

i0 − i

t− (Ω− Ω0)− 2kπθ∗0 − θ∗

, (34)

for a positive, diagonal weighting matrix W and 4× 4 identity matrix I. The diagonal elements ofW are determined experimentally. A value of the regularizing parameter γ is selected such thata well-conditioned matrix

[DWDT − γI

]is ensured, but it is a value that is sufficiently small so

that the accuracy of the solution is not affected. (See Tikhonov and Arsenin.20) Thus, the solutionfor the state variations in eq. (33) is obtained and added to the corresponding parameters in theinitial state vector. Integrating the new initial conditions yields an orbit that more closely satisfiesthe conditions for periodicity. An iterative process generally converges to an orbit that meets therequirements.

Example Periodic Solution

The targeting scheme successfully computes periodic orbits corresponding to the approximatefrozen orbit conditions in the traditional third-body perturbing problem

sin 2ω = 0 and e2 +53

cos2 i = 1. (35)

(See Prado21 for a derivation of eq. (35).) Appropriate initial conditions can be extracted directlyfrom eq. (35). Consider the Earth-Moon system in the R3BP where the central body P1 is theMoon, P2 is the spacecraft, and P3 is the Earth. For orbits possessing geometries desirable for lunarsouth pole coverage, the mappings in eq. (28) are defined such that ω = ω0 = 90 (rather thanω = ω0 = 270). An orbit with e0 = 0.6 requires an inclination equal to i0 = 51.71 as determinedfrom eq. (35). Also, the value of the ascending node, Ω0 = −90, positions the initial state in thexS−zS plane corresponding to the rotating frame (S). Since it is desirable to implement correctionsnear apoapsis, the initial true anomaly is defined to be θ∗0 = 180. Consider an orbit with theseinitial conditions and semi-major axis a0 = 15, 000 km. The polar, e−ω phase space plot for thistrajectory appears in Figure 5. Note that the initial and final points are denoted by the symbols‘’ and ‘×’, respectively. Clearly the orbit is not yet periodic since ‘×’ and ‘’ do not meet. It isalso apparent that since the desired orbit is doubly periodic in the e−ω phase space, k = 1 in eq. (27).

Four iterations are required to generate a perfectly periodic orbit (within a tolerance of 10−11

non-dimensional units). The periodic trajectory appears in the e−ω phase space in the top plot in

12

Figure 5: Path of P3 in Polar, e−ω Phase Space Before Applying Corrections

Figure 6. Note that the initial and final points coincide (the ‘×’ overlaps the ‘’). The period of theorbit is 25.039 days. The trajectory is commensurate with the orbit of the Earth-Moon system. Thespacecraft motion is comprised of 13 conic, semi-elliptical revolutions for each three-body orbitalperiod. These two frequencies are apparent when the Keplerian elements are transformed intoinertial Cartesian coordinates defined in terms of the intermediate frame (I). (See Figure 6, middle.)Of course, the periodicity conditions ensure that the line of nodes is rotating to offset the Earth-Moon line. Therefore, the orbit is also periodic in the rotating frame (S), as is apparent in Figure 6(bottom). The closest approach radius (rp) is 5,839.9 km and the radius at apoapsis (ra) is 24,339.4km. Additionally, all the elements are plotted over one full period in Figure 7. It is clear that theelements are well-behaved. The elements e, ω, and i include both long and short periods, withoutsecular effects. The accuracy of the state equations are verified by examining the well-known integralof the motion, or Jacobi Constant (C), in the R3BP.10 The Jacobi Constant associated with thisorbit is 3.43372 km2/s2. The value of C for the integration remains constant up to 10−14 km2/s2

as evidenced by Figure 7 (bottom). Thus, Gauss’ equations accurately predict trajectories in thisneighborhood. Furthermore, this value of C is about 3% greater than the value of C associatedwith the L1 libration point (CL1 = 3.33557 km2/s2). Of course, in the R3BP, all trajectories in thelunar vicinity with a Jacobi constant greater than CL1 will remain in the lunar vicinity indefinitely,a desirable feature for long-term lunar south pole coverage.

13

Figure 6: Corrected Path of P3 in Polar, e−ω Phase Space (Top), Intermediate Inertial Frame (I)(Middle) and Rotating Frame (S) (Bottom)

14

0 5 10 15 20 2514500

15000

15500

a(k

m)

0 5 10 15 20 250.5

0.6

0.7

e

0 5 10 15 20 2580

90

100

ω(d

eg)

0 5 10 15 20 2550

52

54

i(d

eg)

0 5 10 15 20 25−150

−100

−50

Ω(d

eg)

0 5 10 15 20 250

5000

θ∗

(deg

)

0 5 10 15 20 25−2

0

2

C−

C0

(×10

−14

km2/s

2)

time (day)

Figure 7: The Keplerian Elements and Change in Jacobi Constant (Bottom)

15

Predictions Using the Tangent Subspace

A family of periodic orbits may be computed by continuously updating ek+10 using ek+1

0 =ek0 + ∆e0, where ek

0 is associated with the corrected initial state and ∆e0 is a predetermined fixedstep size. However, in regions where many different families exist (and all possess similar shapecharacteristics), even small values of ∆e0 will not ensure that ek+1

0 results in orbits within the samefamily as ek

0 . As an alternative, consider an expansion of the solution space via a linear predictionalong the tangent subspace Γ . Let ∆ak

0 , ∆ek0 , ∆ik0 , ∆Ωk

0 , and ∆θ∗0k be the changes to the corrected

initial state qk0 that predict the next initial state qk+1

0 for a neighboring orbit. That is, the linearprediction for qk+1

0 associated with the neighboring orbit is simply

qk+10 = qk

0 + d ·∆qk0 , (36)

where ∆qk0 =

∆ak

0 ∆ek0 0 ∆ik0 ∆Ωk

0 ∆θ∗0kT

. The parameter d is a predetermined step size.The magnitude of d must be sufficiently small to ensure that qk+1

0 yields an orbit within the samecharacteristic family as the orbit represented by the initial conditions qk

0 . Since Γ is in the nullspaceof C (eq. (32)), the components of ∆qk

0 must satisfy∆ak

0

∆ik0∆Ωk

0

∆θ∗0k

= ∆ek0

ca

ci

cΩ

cθ∗

, (37)

for the free variable ∆ek0 . Of course, the values of ca, ci, cΩ, and cθ∗ are derivable from the

homogeneous system

[C]

∆ak0

∆ek0

∆ik0∆Ωk

0

∆θ∗0k

= 0. (38)

The components of ∆qk0 are determined uniquely by the requirement

∥∥∆qk0

∥∥ = 1. Then,

∆qk0 =

1√1 + c2a + c2i + c2Ω + c2θ∗

ca

10ci

cΩ

cθ∗

. (39)

Given the prediction from eq. (36), the initial condition is propagated, and the iterative processrepeats until the convergence to a new solution is achieved. (Of course, all state vectors are boundedin eccentricity such that 0 < e < 1.)

Families of Periodic Orbits

Recall that the apoapsis radius ra for the orbit in Figure 6 is 24,339.4 km. However, dueto communications limitations, it may be necessary to compute orbits with radii of apoapsis lessthan 10,000 km. Let a0 be the parameter that characterizes the family of orbits. (Recall thate0 characterizes an orbit within the family.) A family satisfying an apoapsis constraint such thatra < 10, 000 km is generated from the initial condition a0 = 5, 000 km and e0 = 0.925, wherethe other initial values satisfy eq. (35) or are otherwise the same as those defined in the previous

16

example. The orbit family appears in both the inertial and rotating frames in Figure 8, where thecolors of the orbits in the top plot match those of the bottom plot. Each orbit within the familypossesses 69 close passes of the Moon. The initial condition, period (T ), and Jacobi Constant (C)for each orbit in the family are available in Table 1. Over the entire family, a0 increases while i0decreases. Also, in stepping through the full range in eccentricities using eq. (36), e0 begins nearone and ends near zero. Since e0 approaches zero at the intersection between the southern andnorthern orbit families, it is difficult to transition between the two families using Gauss’ equations.Therefore, only the southern orbits appear in Figure 8. The northern orbits can be computed eitherby symmetry or by re-computing the entire family with ω0 = 270. Of course, these orbits wouldbe useful for north pole coverage. The Jacobi Constant for each orbit in the family is significantlygreater than CL1 . Finally, a plot of rp versus ra confirms the desired effect, i.e., all feasible orbits(rp > RP1) in the family have ra less than 10,000 km. (See Figure 9.)

Table 1: Initial Conditions, Period (T ), and Jacobi Constant (C) for Orbit Family

a0 (km) e0 i0() Ω0() θ∗0() T (day) C (km2/s2)1 4,923.6 0.925 72.551 -89.521 186.353 26.539 4.086182 4,972.2 0.8940 69.310 -90.100 178.640 26.571 4.077633 5,070.2 0.8501 65.518 -89.957 180.704 26.610 4.060294 5,124.6 0.8066 62.344 -89.665 186.848 26.658 4.051655 5,131.7 0.7633 59.606 -90.626 150.881 26.714 4.051806 5,186.8 0.7203 57.160 -89.933 181.649 26.758 4.043077 5,192.2 0.6851 55.358 -89.943 181.541 26.800 4.043178 5,199.7 0.6324 52.915 -90.241 172.527 26.858 4.043289 5,204.8 0.5936 51.292 -90.322 169.125 26.899 4.0433510 5,210.5 0.5475 49.536 -90.392 165.611 26.943 4.0434211 5,217.0 0.4926 47.663 -90.045 178.325 26.992 4.0434912 5,221.4 0.4493 46.342 -90.266 169.599 27.027 4.0435313 5,225.9 0.3982 44.955 -90.435 162.619 27.064 4.0435714 5,229.1 0.3578 43.982 -90.551 158.288 27.090 4.0435915 5,232.2 0.3142 43.051 -90.673 154.596 27.116 4.0436116 5,235.1 0.2551 41.989 -91.319 134.590 27.145 4.0436217 5,237.3 0.2035 41.241 -91.868 123.308 27.165 4.0436318 5,238.7 0.1646 40.789 -92.424 116.269 27.178 4.0436419 5,239.6 0.1351 40.510 -93.025 111.920 27.186 4.0436420 5,292.1 0.0631 40.073 -96.565 120.912 27.195 4.03476

17

Figure 8: Family of Solutions in the Inertial (I) (Top) and the Rotating (S) (Bottom) Frames

18

Figure 9: Radius of Periapsis (rp) Versus Radius of Apoapsis (ra) for Orbit Family

LUNAR SOUTH POLE COVERAGE AND TRANSITION TO THE FULL MODEL

For mission design, the potential baseline trajectory that appears in Figure 8 must be transitionedto a full ephemeris model (including solar perturbations). Therefore, let n = 4 in eq. (2), wherethe location of Earth (P3) and Sun (P4), with respect to the Moon (P1) are determined by the JPLDE405 ephemeris file. Furthermore, since the effects of an aspherical central body are significant,consider the case when RP

1 (q) 6= 0. Then the coefficients Cn,m and Sn,m in eq. (13) must also bedetermined. Of course, these coefficients are based on knowledge of the Moon’s gravity field thatis only available experimentally. Consider a 50× 50 gravity field (nmax = 50 in eq. (13)) using theLP165P model available at the Geosciences Node of NASA’s Planetary Data System.22 In terms ofthis lunar gravity model, the largest unnormalized coefficients in the expansion (though clearly notdominant) are C2,0 = −J2 = −0.0002032366 and C3,1 = 0.00002843689. Finally, for all simulations,the epoch is set to 7 Jan 2009 12:00:00.00 UTCG.

Orbit Selection

Initial conditions that define potential orbits for lunar south pole coverage are selected fromTable 1. As noted, all the orbits possess apoapses less than 10,000 km. Other desirable featuresinclude high eccentricities and feasible periapsis altitudes. From Figure 9, it is clear that eccentricityincreases as radius of periapsis (rp) decreases. Therefore, consider orbits with rp just above the lunarsurface (or horizontal line in Figure 9). The ninth initial condition listed in Table 1 corresponds toan orbit that satisfies these conditions with periapsis altitude near 400 km and e0 approximately0.6. Therefore, let the orbit defined by the ninth initial condition be the baseline orbit for the full

19

model. (The orbit is isolated in Figure 11.) The period of the orbit in the R3BP is 26.9 days. Totransition to the full model, the orbit is discretized into a series of n points q1, q2, . . ., qn, equallyspaced in time. The qk are strategically separated by ∆t = 7.836, days corresponding to every 20thapoapse point, for a baseline 1,000 day mission. Since the number of close passes (69) is odd, twospacecraft can be placed in the same orbit but phase shifted in time by exactly one-half period tomaximize coverage of the lunar south pole. (See Grebow et al.8)

Transition to the Full Model with Multiple Shooting and Elevation Angle

Multiple Shooting. Discretizing the orbit also serves as a basis for rectification of the osculating el-lipse, thereby minimizing errors associated with integrating the Keplerian elements. The variationsδqk are continuously updated using a multiple shooting scheme where, over the course of the inte-gration, the new orbit maintains the nominal shape as designed in the R3BP. To solve for δqk withmultiple shooting, consider the fixed-time mapping Fk of the point qk over ∆t. Then, to compute acontinuous trajectory in the full model, the goal is the determination of δqk such that

q2 + δq2 = F1 (q1 + δq1) ,

q3 + δq3 = F2 (q2 + δq2) ,

...

qn + δqn = Fn−1 (qn−1 + δqn−1) .

(40)

Of course, a first-order Taylor series expansion of the right-hand side of eq. (40) results in expressionsof the form

q2 + δq2 = F1 (q1) +∂F1

∂q1δq1,

q3 + δq3 = F2 (q2) +∂F2

∂q2δq2,

...

qn + δqn = Fn−1 (qn−1) +∂Fn−1

∂qn−1δqn−1,

(41)

where ∂Fk

∂qk= Φ (tk + ∆t, tk). Rearranging terms in eq. (41) results in the linear relationship

q2 − F1 (q1)q3 − F2 (q2)

...qn − Fn−1 (qn−1)

=

∂F1

∂q1−I

∂F2

∂q2−I

. . .∂Fn−1

∂qn−1−I

︸ ︷︷ ︸

M

δq1

δq2

δq3...

δqn−1

δqn

, (42)

where I is the 6 × 6 identity matrix. Since there are 6n scalar unknowns, i.e., δq1, δq2, . . . , δqn,and only 6n − 6 equations, the system defined in eq. (42) generally has infinitely many solutions.However, a minimum-norm solution yields an option close to the nominal path. When computing theminimum-norm solution, observe that MMT is a banded symmetric matrix. Therefore, Cholesky

20

factorization of MMT allows for the minimum-norm solution to be obtained recursively, i.e., withoutdirectly computing

[MMT

]−1. (See Gomez et al.16) This is particularly useful when MMT is ill-conditioned. Then, qk is iteratively readjusted until

q2 = F1 (q1) ,

q3 = F2 (q2) ,

...

qn = Fn−1 (qn−1) .

(43)

Continuity is established when the largest discontinuity, i.e.,∥∥qk+1 − Fk (qk)

∥∥max

is less than 10−9

non-dimensional units. The multiple shooting scheme is applied to the discretized points q1, q2,. . ., qn corresponding to two spacecraft in a single orbit, phase shifted by one-half period. In fouriterations, a nearby continuous solution is computed for both spacecraft in the full model result-ing in a new trajectory for each spacecraft. (Compare Figure 13 to Figure 11.) In the top plot ofFigure 13, both trajectories maintain an eccentricity of 0.53 with a maximum variation of only ±0.05.

Elevation Angle. The elevation angle α associated with each spacecraft relative to a lunar groundsite is directly related to coverage.8 In general, the elevation angle of each spacecraft from a specificlatitude φs and longitude λs on the lunar surface is evaluated as

α =π

2− cos−1

(rB1s · rBs2∥∥rB1s

∥∥∥∥rBs2∥∥), (44)

where rB1s is the position of the site from the center of the Moon in the body-fixed frame (B), i.e.,

rB1s = RP1

cosφs cosλs

cosφs sinλs

sinφs

(45)

Of course, the position of the spacecraft relative to the site is then evaluated from rBs2 = rB12 − rB1s =rbB − rB1s. The Shackleton Crater has been identified as a site of interest. The elevation angle fromthe Shackleton Crater for both spacecraft is computed from eq. (44) where (λs, φs) = (0,−89.9).Figure 10 is a sample plot of the elevation angle for the orbits in Figure 13 over a 10-day interval.It is clear that both spacecraft maintain the prescribed phasing even after 1,000 days. At least onespacecraft is always 17.85 above the horizon for the entire simulation.

1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 10100

20

40

days past epoch

α(d

eg)

Figure 10: Sample Plot of Elevation Angle (α) for Both Spacecraft after 1,000 Days

21

Figure 11: Selected R3BP Orbit for Lunar South Pole Coverage in Polar, e−ω Phase Space (Top),Intermediate Inertial Frame (I) (Middle) and Rotating Frame (S) (Bottom)

22

Figure 12: Corrected 1,000-Day Orbit for Two Spacecraft (Red and Blue) in the Full Model in e−ωPhase Space (Top), Intermediate Inertial Frame (I) (Middle) and Rotating Frame (S) (Bottom)

23

Percent Access and Verification in Satellite Tool Kit’s Astrogator

The LP165P model is also available in Satellite Tool KitR©’s (STK) Astrogator from AGI. Theorbit is verified by targeting ik, Ωk, and θk with Astrogator Connect using the procedure describedin Grebow et al.8 In general, slight differences in modeling result in very small position errors (lessthan 1 km) when propagating to each qk in STK. Of course, these errors are minimized with smallcorrections. For these orbits, the targeting sequence in Astrogator is extremely sensitive to evensmall errors. However, in general, the orbit is computed with corrections on the order of a few m/sper year. Once the orbits are targeted in STK, percent access times with a ground station at theShackleton site are available. Additionally, Earth-based transmitting sites are defined and the accesstimes between the satellites and these transmitting sites are computed. A potential location for aground station is the White Sands Test Facility (WSTF) located in New Mexico (32.3N, 106.8W).The times when each satellite, either satellite, or both satellites possess line-of-sight with the stationsare computed. (See Table 2.) Due to the initial time phase shift, access times are all computedomitting the first and last half periods. As expected, at least one satellite maintains line-of-sightwith the Shackleton station over 100.00% of the simulation time. In fact, both satellites are in line-of-sight 39.29% of the time. At least one satellite is viewable from the Earth at all times, a desirablefeature for a communications relay between the satellites if necessary. Finally, a communicationslink between the satellites is established for possible communications relay. The line-of-sight accessfor this link is 81.78%.

Table 2: Percent Access Times

Shackleton Earth WSTFOnly Satellite 1 69.60% 96.72% 47.38%Only Satellite 2 69.68% 96.72% 49.22%Both Satellites 39.29% 93.44% 45.56%Either Satellite 100.00% 100.00% 49.22%

Simulation Time 1,062.50 days

A Long-Term Simulation in the Full LP165P Gravity Model

A ten-year simulation demonstrates that a multiple shooting scheme is capable of locating nearbysolutions in the full model for long term simulations. Identifying such orbits is desirable for constantcommunications with a permanent station at the lunar south pole. Since the full LP165P providesmuch better accuracy than previous LP100J and LP100K models for extended mission design,13

consider the full 165 × 165 expansion. Of course, higher-order expansions significantly increasecomputation time. However, since the propagation of each qk in a multiple shooting process is inde-pendent, the computation time can be reduced dramatically by integrating the states in parallel, oras deputy processes. In general, a chief process distributes the qk to the available deputy processesfor integration. When the integration is complete, the deputy returns Fk(qk) to the chief and receivesthe next available state. When every Fk(qk) is computed, the chief solves for δqk in eq. (42) usinga minimum-norm solution and Cholesky factorization. The process repeats until a ten-year, contin-uous solution is converged. Of course, the computational time is also dependent on the compilerand integrator. So, the simulation is implemented in Fortran 90 using an Adams-Bashforth-Moultonintegrator.17 According to Montenbruck,23 an Adams-Bashforth-Moulton integrator requires fewerfunction evaluations for the same degree of accuracy as a Runge-Kutta solver with first-order differ-ential equations. Since derivative evaluations are computationally intensive when integrating Gauss’equations in the full 165× 165 model and the governing matrix differential equations (42 total first-order differential equations), minimizing the number of function evaluations is significant.

24

For a ten-year simulation, the orbit that appears in Figure 11 is discretized into 477 points,where the time between each point is 7.836 days. The simulation is compiled with PGIR©’s PGF95TM

compiler. Since the PGF95 compiler performs various optimizations, including for array opera-tions, computations are written in terms of arrays whenever possible. For example, all the factorsin the recurrence relations for the normalized Legendre polynomials depending on n and m arepre-calculated and the polynomials are subsequently computed with array arithmetic. Using theMPI standard to interface between 54 AMD OpteronTM 280 dual-core processors, the simulationconverges to a continuous solution in seven iterations. (Recall a continuous solution is computedwhen

∥∥qk+1 − Fk (qk)∥∥

maxis on the order of 10−9 non-dimensional units.) For a single spacecraft,

the process takes less than four hours total. Consider a comparison of this process in terms ofreal time to a ten-year integration of an uncorrected state on one processor. Since integration ofa single, uncorrected state for the full simulation time does not lend itself well to parallelization, aconverged solution using discretization and parallelization can be reached significantly faster. Thespeed increase is approximately equal to the number of processors available divided by the number ofiterations necessary for convergence. For the ten-year simulation described here, a corrected solutionusing multiple shooting can be delivered in less than one-tenth of the time that is necessary for asingle integration. Furthermore, there is an even greater time advantage considering that the single,uncorrected initial state would require to be run through a corrections process.

The multiple shooting corrections scheme is applied to both spacecraft on the compute cluster.The results of the simulation for both spacecraft are plotted in the polar e−ω phase space in Figure13. It is clear that the process converges to a nearby solution; both spacecraft closely follow oneanother in the e−ω phase space for the entire ten-year simulation. The orbit maintains an eccentricity

Figure 13: Corrected Ten-Year Orbit for Two Spacecraft (Red and Blue) in the Full 165× 165LP165P Gravity Model, e−ω Phase Space

25

of approximately 0.52 with a variation of only ±0.06. The minimum periapsis altitude for bothspacecraft is roughly 420 km for the entire simulation. When examining the elevation angle of bothspacecraft as measured from the Shackleton site, the spacecraft remain in phase even after ten years.Furthermore, the minimum elevation for both spacecraft does not appear to monotonically decreasewith time. In fact, for the ten-year simulation, at least one spacecraft is always 15.72 above thehorizon as seen from the Shackleton Crater.

CONCLUSION

Using Gauss’ equations, perturbations from n − 2 additional bodies and central-body gravityharmonics are modeled and the state-transition matrix is computed. The R3BP is identified as aspecial case of the n-body problem, where n = 3. The tangent subspace is used to compute familiesof periodic orbits in the R3BP with feasible apoapsis altitudes. In general, modeling with Gauss’equations provides direct control of orbit shape and orientation not otherwise available. A highlyeccentric orbit with apoapsis over the lunar south pole and feasible periapsis altitude is a candidatefor lunar south pole coverage. Using JPL’s DE405 ephemeris file and the LP165P gravity model, thecandidate orbits are transitioned to the full model with a multiple shooting scheme. The resultingtrajectories maintain the shape of the orbit as designed in the R3BP. The trajectories are verified inSTK’s Astrogator, where line-of-sight with the Shackleton Crater, Earth, and WSTF is computed.Two spacecraft, phase shifted in time by one-half period in the same orbit, maintain communicationswith a ground station at the Shackleton Crater over 100% of the simulation time. Furthermore, along-term simulation demonstrates that both spacecraft maintain their initial phasing without lunarimpact for over 10 years, where at least one spacecraft is always 15.72 above the horizon. Currentresearch efforts are focused on computing transfers into these orbits. Of course, for long-termcommunications, station-keeping costs are also significant.

ACKNOWLEDGEMENT

A significant portion of this work was completed at the NASA Goddard Spaceflight Center underthe supervision of Mr. David Folta. The authors also thank Professor Stephen Heister for use ofcomputing resources. Portions of this work were supported by Purdue University and NASA undercontract number NNXO6AC22B.

REFERENCES

1. “The Vision for Space Exploration,” National Aeronautics and Space Administration Publica-tion, NP-2004-01-334-HQ, February 2004.

2. A. Elipe and M. Lara, “Frozen Orbits about the Moon.” Journal of Guidance, Control, andDynamics, Vol. 26, No. 2, March-April 2003, pp. 238-243.

3. T. Ely, “Stable Constellations of Frozen Elliptical Inclined Orbits.” Journal of the AstronauticalSciences, Vol. 53, No. 3, July-September 2005, pp. 301-316.

4. T. Ely and E. Lieb, “Constellations of Elliptical Inclined Lunar Orbits Providing Polar andGlobal Coverage.” Paper No. AAS 05-158, AAS/AIAA Spaceflight Mechanics Meeting, SouthLake Tahoe, California, August 7-11, 2005.

5. D. Folta and D. Quinn, “Lunar Frozen Orbits.” Paper No. AIAA 06-6749, AAS/AIAA Astro-dynamics Specialist Conference, Keystone, Colorado, August 21-24, 2006.

6. M. Lidov, “Evolution of the Orbits of Artificial Satellites of Planets as Affected by GravitationalPerturbation from External Bodies,” AIAA Journal (Russian Supplement), Vol.1, No. 8, August1963, pp. 719-759.

26

7. R. Farquhar, “The Utilization of Halo Orbits in Advanced Lunar Operations.” NASA TND-365,Goddard Spaceflight Center, Greenbelt, Maryland, 1971.

8. D. Grebow, M. Ozimek, K. Howell, and D. Folta, “Multi-Body Orbit Architectures for LunarSouth Pole Coverage.” Paper No. AIAA 06-179, AAS/AIAA Astrodynamics Specialist Confer-ence, Tampa, Florida, January 22-26, 2006.

9. R. Russell and M. Lara, “Repeat Ground Track Orbits in the Full-Potential Plus Third-BodyProblem.” Paper No. AIAA 06-6750, AAS/AIAA Astrodynamics Specialist Conference, Key-stone, Colorado, August 21-24, 2006.

10. V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, NewYork, 1971.

11. V. Markellos and A. Halioulias, “Numerical Determination of Asymmetric Periodic Solutions.”Astrophysics and Space Science, Vol. 46, 1977, pp. 183-193.

12. V. Markellos, “Asymmetric Periodic Orbits in Three Dimensions.” Royal Astronomical Society,Monthly Notices, Vol. 184, 1978, pp. 273-281.

13. A. Konopliv, “Recent Gravity Models as a Result of the Lunar Prospector Mission.” Icarus, Vol.150, No. 1, 2001, pp. 1-18.

14. K. Howell and H. Pernicka, “Numerical Determination of Lissajous Trajectories in the RestrictedThree-Body Problem.” Celestial Mechanics, Vol. 41, 1988, pp. 107-124.

15. J. Stoer and R. Burlirsch, Introduction to Numerical Analysis. Springer-Verlag, New York, 1983.

16. G. Gomez, J. Masdemont, and C. Simo, “Quasihalo Orbits Associated with Libration Points.”Journal of the Astronautical Sciences, Vol. 46, No. 2, 1998, pp. 135-176.

17. L. Shampine and H. Watts, DEPAC–Design of a User Oriented Package of ODE Solvers.SAND79-2374, Sandia National Laboratories, Albuquerque, New Mexico, 1980.

18. V. Brumberg, Conferences Sur La Relativite en Mecanique Celeste et en As-trometrie. Institut de Mecanique Celeste et de Calcul des Ephemerides, Paris, France.[http://www.imcce.fr/fr/formation/cours/Cours Brumberg. Accessed 1/17/07.]

19. G. Spier, Design and Implementation of Models for the Double Precision Trajectory Program(DPTRAJ). Technical Memorandum 33-451, Jet Propulsion Laboratory, Pasadena, California,1971.

20. A. Tikhonov and V. Arsenin, Solutions of Ill-posed Problems. Wiley, New York, 1977.

21. A. Prado, “Third-Body Perturbation in Orbits Around Natural Satellites.” Journal of Guidance,Control, and Dynamics, Vol. 26, No. 1, January-February 2003, pp. 33-40.

22. National Aeronautics and Space Administration Planetary Data System Geosciences Node.Washington University, St. Louis, Missouri. [http://pds-geosciences.wustl.edu. Accessed1/17/07.]

23. O. Montenbruck, “Numerical Integration Methods for Orbital Motion.” Celestial Mechanics,Vol. 53, No. 1, 1992, pp. 59-69.

27

APPENDIX A

For evaluation of eq. (3), it is necessary to compute the partial derivatives of the state equationswith respect to each state variable. Let p = aζ2. Then, the results appear in the following form.

∂a

∂a=

3µ1

rζη3a4

(F rer sin θ∗ + F θp

), (A.1)

∂a

∂e=

2ζη

[F r

(1 +

e2

ζ2

)sin θ∗ + F θ

(pe

rζ2+ cos θ∗

)], (A.2)

∂a

∂ω=∂a

∂i=∂a

∂Ω= 0, (A.3)

∂a

∂θ∗=

2eζη

(F r cos θ∗ − F θ sin θ∗

), (A.4)

∂e

∂a=

ζ

2ηa3er

[F raer sin θ∗ + F θ

(ap− r2

)], (A.5)

∂e

∂e= − 1

ηa2

F r ae sin θ∗

ζ+F θ

[ap− r2

rζ+ζ(ap− r2

)re2

−ζe

(a cos θ∗+

2aer + r2 cos θ∗

p

)], (A.6)

∂e

∂ω=∂e

∂i=

∂e

∂Ω= 0, (A.7)

∂e

∂θ∗=

ζ

ηa

(F r cos θ∗ − F θ ap+ r2

pasin θ∗

), (A.8)

∂ω

∂a=

ζ

2ηa2e

[−F r cos θ∗ + F θ

(1 +

r

p

)sin θ∗ − Fh er sin θ

p tan i

], (A.9)

∂ω

∂e=

1ζηa

[F r cos θ∗

e2−F θ p (p+r)

(e2+ζ2

)+ζ2er2 cos θ∗

e2p2sin θ∗+Fh r

(ep+ζ2r cos θ∗

)p2 tan i

sin θ

], (A.10)

∂ω

∂ω= −Fh ζr cos θ

apη tan i, (A.11)

∂ω

∂i= Fh ζr sin θ

apη sin2 i, (A.12)

∂ω

∂Ω= 0, (A.13)

∂ω

∂θ∗=

ζ

ηa

F r sin θ∗

e+ F θ

[p (p+ r) cos θ∗+r2e sin2 θ∗

p2e

]−Fh r (er sin θ sin θ∗ + p cos θ)

p2 tan i

, (A.14)

∂i

∂a= Fh ζr cos θ

2a2pη, (A.15)

∂i

∂e= −Fh r

(ep+ ζ2r cos θ∗

)cos θ

ζap2η, (A.16)

∂i

∂ω= −Fh ζr sin θ

apη, (A.17)

∂i

∂i=

∂i

∂Ω= 0, (A.18)

∂i

∂θ∗= Fh ζr (er cos θ sin θ∗ − p sin θ)

ap2η, (A.19)

∂Ω∂a

= Fh ζr sin θ2a2pη sin i

, (A.20)

28

∂Ω∂e

= −Fh r(pe+ ζ2r cos θ∗

)sin θ

ζap2η sin i, (A.21)

∂Ω∂ω

= Fh ζr cos θapη sin i

, (A.22)

∂Ω∂i

= −Fh ζr sin θ cos iapη sin2 i

, (A.23)

∂Ω∂Ω

= 0, (A.24)

∂Ω∂θ∗

= Fh ζr (er sin θ sin θ∗ + p cos θ)ap2η sin i

, (A.25)

∂θ∗

∂a= − 3p2η

2aζ3r2− ∂ω

∂a− cos i

∂Ω∂a

, (A.26)

∂θ∗

∂e=ηp(3pe+ 2ζ2r cos θ∗

)ζ5r2

− ∂ω

∂e− cos i

∂Ω∂e, (A.27)

∂θ∗

∂ω=∂θ∗

∂i=∂θ∗

∂Ω= 0, (A.28)

∂θ∗

∂θ∗= −2ηpe sin θ∗

ζ3r− ∂ω

∂θ∗− cos i

∂Ω∂θ∗

. (A.29)

APPENDIX B

To evaluate the partial derivative of ∆m with respect to qj , note that

∆m =1∥∥rP2m

∥∥3 =(rP2m · rP2m

)− 32 , (B.1)

where rP2m = PTIITJrJ1m − rrP. Then, ∂∆m

∂qjis computed using a vector dot-product, chain-rule

expansion, that is

∂∆m

∂qj= −3

(rP2m · rP2m

)− 52

[rP2m ·

(∂PTI

∂qjITJrJ1m − ∂r

∂qjrP)]

. (B.2)

APPENDIX C

The partial derivatives ∂φ∂qj

and ∂λ∂qj

are computed by first noting that

∂bB

∂qj= BTJJTI ∂

ITP

∂qjrP, (C.1)

where bB is defined in eq. (11). Let ∂bB

∂qjbe resolved into the three element vector

∂bx

∂qj

∂by

∂qj

∂bz

∂qj

T

.Then, from eq. (12) the partials are written

∂φ

∂qj=

1√1− bz2

∂bz

∂qj, (C.2)

∂λ

∂qj=

1bx2 + by2

(bx∂by

∂qj− by

∂bx

∂qj

). (C.3)

29

The partial derivatives ∂P (m)n

∂φ and ∂P (m)′n

∂φ can be evaluated using the relationships

∂P(m)n

∂φ= cosφ P (m)′

n , (C.4)

and

∂P(m)′

n

∂φ= − 1

cosφ

n P (m)

n + (n− 2) sinφ P (m)′

n −[

(2n+1)(n+m)(n−m)2n−1

] 12P

(m)′

n−1

. (C.5)

30