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Paper AAS 07-143

DESIGN USING GAUSS’ PERTURBING EQUATIONS WITH APPLICATIONS TO LUNAR SOUTH POLE COVERAGE K. C. Howell,∗ D. J. Grebow,† Z. P. Olikara†

Perturbations due to multiple bodies and gravity harmonics are modeled numerically with Gauss’ equations. Using the linear variational equa- tions, are calculated with shapes and orientations desirable for lunar south pole coverage. Families of periodic orbits are first computed in the - Restricted Three-Body Problem (R3BP) from linear predic- tions along the tangent subspace. Using multiple shooting, the trajectories for two spacecraft are then transitioned to the full model; the full model incorporates solar perturbations and a 50 × 50 Lunar Prospector gravity model. The results are verified with commercial software and vari- ous aspects of the coverage are also discussed. Finally, a ten-year simulation is 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 pole 1 has prompted studies in frozen architectures for con- stant south pole surveillance (including Elipe and Lara, 2 Ely, 3 Ely and Lieb, 4 as well as Folta and Quinn 5). Many of the methods used in these studies were first pioneered by Lidov 6 in 1962. In gen- eral, the Earth is modeled as a third-body in the two-body, Moon-spacecraft problem. requirements are computed directly from Lagrange’s planetary equations. An approxi- mate disturbing function is averaged over the system mean anomalies. The frozen orbit conditions are calculated analytically by fixing , argument of periapsis, and inclination, i.e., allowing no variations in these quantities. Of course, approximating and averaging the disturbing function introduces some errors into the analysis. Furthermore, in the three-body problem an exactly frozen orbit, or periodic orbit, requires that the semi-major axis (a), eccentricity (e), argument of periapsis (ω), inclination (i), and (θ∗) all return simultaneously to their initial values at a later point along the path. Correspondingly, the motion of the ascending node (Ω) must be commensurate with the motion of the third body. Even with the simplifying assumptions, it is still difficult to generalize the analytical methods to include additional perturbations from other bodies and gravity harmonics. However, some frozen orbit conditions can still be numerically integrated in the full model with only minimal secular drift. For example, Ely 3 identifies a potential stable con- stellation of three spacecraft for continuous south pole coverage over 10 years in the full model. Folta and Quinn 5 consider many different sets of initial conditions for frozen orbits, and note secular drift after 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, Purdue University, Grissom Hall, 315 North Grant St., West Lafayette, Indiana 47907-1282; Fellow AAS; Associate Fellow AIAA. †Student, School of Aeronautics and Astronautics, Purdue University, Grissom Hall, 315 North Grant St., West Lafayette, Indiana 47907-1282; Student Member AIAA.

1 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 accurately modeled as a three-body problem. Since there are no analytical, closed-form solutions available in the three-body problem, motion is simulated numerically. Historically, the traditional formulation of the Restricted Three-Body Problem (R3BP) has proven useful in mission design of lunar orbits, with an array of numerical techniques. For example, Farquhar 7 first examined halo orbits in 1971 for continuous communications between the Earth and the far side of the Moon; station-keeping costs are included as part of the analysis. For lunar south pole coverage, Grebow et al. 8 demonstrate that constant 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 point orbits. Apoapsis altitudes all exceed 10,000 km, however, an undesirable feature for lunar ground- based communications. More recently, in a R3BP formulation, Russell and Lara 9 investigate many different repeat 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 orbit favorable for lunar south pole coverage. However, since all the orbits must be synchronized with the Moon’s rotation, only particular solutions can be isolated without making additional assumptions about 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 south pole coverage, it is desirable to control the shape and orientation. For example, the orbits must be eccentric and oriented such that apoapsis occurs over the lunar south pole. The orbits must also possess feasible periapsis and apoapsis altitudes. These parameters are, perhaps, better controlled when 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 R3BP depend on minimizing specific Cartesian coordinates downstream along the path by updating specific Cartesian components of the initial state vector. (For some examples, see Szebeheley. 10) However, in 1977, Markellos and Halioulias 11 developed a strategy for computing families of periodic orbits in the two-dimensional St¨ormerproblem by generating predictions with the tangent subspace. A more recent application is the computation of asymmetric periodic orbits in the classical formulation of the R3BP. 12 The numerical methods and related techniques are easily generalized to compute periodic orbits with Keplerian elements by integrating Gauss’ equations.

According to Folta and Quinn, 5 altitudes above approximately 750 km are dominated by third-body (Earth) disturbances, whereas altitudes below about 100 km are completely dictated by spherical harmonics. Then, for altitudes ranging between 100 and 750 km, both third-body effects and spherical gravity harmonics are significant. In general, orbits designed for lunar south pole coverage are likely to possess apoapsis altitudes greater than 750 km with periapsis altitudes from 100 to 750 km. Russell and Lara 9 also note significant changes when integrating their initial states (corrected to be periodic orbits in the R3BP and incorporating a 50 × 50 harmonic gravity model) in the full ephemeris model, including Earth and solar perturbations. Therefore, the effects of 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). Since 1998, 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 differential corrections 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 trajectory and then locating a continuous path with velocity discontinuities at the target points; the disconti-

2 nuities are subsequently reduced or eliminated. Alternatively, a nearby continuous solution in the full model can be computed with any set of coordinates using a multiple shooting scheme. (See Stoer and Burlirsch. 15) With applications to , G´omez et al. 16 recently used mul- tiple shooting to compute quasihalo orbits in the full model. These methods are easily generalized to 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 of Gauss’ perturbing equations. The state-transition matrix is available, and is, thus, a mechanism for computation of semi-elliptical orbits (e < 1). All trajectories are first designed in the restricted problem, where the R3BP is identified as a special case of the n-body problem with n = 3. Using Gauss’ equations and the numerical techniques developed by Markellos, 11,12 a family of periodic orbits is computed in the R3BP with shape and orientation advantageous for lunar south pole coverage. The accuracy of the solutions are verified by transforming the Keplerian elements into Cartesian coordinates and examining the Jacobi Constant (C). To maximize lunar south pole coverage, two spacecraft are phase shifted in the same orbit by a half-period as described in Grebow et al. 8 The orbits for both spacecraft are transitioned to the full ephemeris model with a multiple shooting scheme using JPL’s DE405 ephemeris file and a 50 × 50 LP165P gravity model. Both spacecraft maintain their initial phasing for over 1,000 days without lunar impact. The results are verified with AGI’s Satellite Tool Kit R where percent access with the Shackleton Crater facility, the Earth, and the White Sands Test Facility is computed. Finally, a 10-year simulation with an optimizing compiler in Fortran 90 is implemented on a compute cluster for two spacecraft using an Adams-Bashforth-Moulton integrator 17 (with variable step size). The elevation angle from the Shackleton 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, F¯pert, applied to the particle, an auxiliary orbit or is defined such that two-body elliptical (e < 1) motion is recovered if F¯pert = 0. (See Figure 1.) Note that ‘overbars’ indicate vectors while ‘hats’ denote vectors of unit ¯P  r θ h T magnitude. The perturbing force Fpert = F F F is comprised of components that are generally 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 F¯pert

3 superscript P on the force F¯pert indicates components associated with the frame P. In general, a vector with an unspecified vector basis does not include a bold-faced superscript. Unit vectors that P ˆP define P includer ˆ directed from P1 to P2, h parallel to the orbital angular momentum vector ˆP ˆP P corresponding to the osculating conic orbit of P2 relative to P1, and θ = h × rˆ completing the right-handed triad. It is assumed that kF¯pertk is small when compared to the central body force of P1 on P2. Then, according to Gauss, the osculating elements are subject to the following variations

2e sin θ∗ 2aζ a˙ = F r + F θ, ζη ηr ζ sin θ∗ ζ a2ζ2  e˙ = F r + − r F θ, ηa ηa2e r ζ cos θ∗ ζ sin θ∗  r  r sin θ ω˙ = − F r + 1 + F θ − F h, ηae ηae aζ2 ηa2ζ tan i (1) r cos θ i˙ = F h, ηa2ζ r sin θ Ω˙ = F h, ηa2ζ sin i ηa2ζ θ˙∗ = − ω˙ − Ω˙ cos i, r2 √ where ζ = 1 − e2, r = aζ2/ (1 + e cos θ∗), and θ = θ∗ + ω. (See Brumberg 18 for a derivation of the perturbing equations using elliptic functions.) Notice that, for computational purposes, the dependent variable corresponding to the sixth equation is selected as true anomaly, θ∗, rather than p 3 . The η in eq. (1) is the of P2, i.e., η = µ1/a where µ1 is the non- dimensional mass parameter of P1. The equations are scaled by non-dimensionalizing with respect to certain characteristic quantities. From a number of options, the characteristic quantities available from the three-body problem possess certain advantages. Let the characteristic length be defined by the mean distance between P3 and P1, where P3 is the significant perturbing gravitational influence (e.g., the Earth). Determine the characteristic time from the mean motion of P3 relative to P1.

T Let the six element state vectorq ¯ be defined asq ¯ = a e ω i Ω θ∗ . Then, the vector ¯P Fpert is expressed as a net perturbing force where

n ¯P ¯P X ¯P Fpert = R1 (¯q) + Fm (¯q). (2) m=3

¯P The vector R1 (¯q) in eq. (2) is the resultant aspherical gravitational force associated with the body Pn ¯P P1. The summation m=3 Fm (¯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 be added to eq. (2) without altering the following analysis. The first-order variational equations are derived and result in the vector differential equation δq¯˙ = A(t)δq¯. The time-varying matrix A(t) is evaluated analytically along the reference, i.e.

¯P n ¯P ! ∂q˙i ∂q˙i ∂R X ∂F A = + 1 + m . (3) ij ∂q ∂F¯P ∂q ∂q j pert j m=3 j

The expressions for the partial derivatives ∂q˙i are available in Appendix A. Since eqs. (1) are linear ∂qj ¯P ∂q˙i in Fpert, computation of the row vector ¯P simply requires an inspection of eqs. (1). Evaluation ∂Fpert

4 ∂R¯P ∂F¯P of the remaining terms in eq. (3), i.e., 1 and m , depends on the expressions for R¯P and F¯P. ∂qj ∂qj 1 m Of course, initial design is accomplished within the context of the R3BP. However, the following ¯P ¯P analysis includes mathematical expressions for R1 and Fm in the full model, and then, appropriately, ∂R¯P ∂F¯P identifies the R3BP as a special case. Once 1 and m are available, the state-transition matrix ∂qj ∂qj 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 are written in the following form

 δq¯k  δq¯k+1 = [Φ(tk+1, tk) q¯˙k+1] . (5) δtk+1

Note that eq. (5) allows non-contemporaneous variations inq ¯k+1.

The N -Body Problem The n-body problem is represented in Figure 2. The perturbing forces due to n − 2 additional ¯P point-masses, P3, P4, ..., Pn are available from the law of gravity. Thus, Fm (¯q) in eq. (2) is expressed P P ! ¯P r¯2m r¯1m Fm (¯q) = µm − , (6) P 3 3 r¯2m kr¯1mk 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 intermediate I frame is defined such thatx ˆ is directed from the point-mass P3 to P1 at . In addition, the unit vectorz ˆI is parallel to the associated two-body angular momentum vector for the motion of I I I I I I P1 with respect to P3. Of course,y ˆ =z ˆ × xˆ . Notice thatx ˆ ,y ˆ , andz ˆ are consistent with an epoch-fixed “rotating” frame in the three-body problem and will therefore be consistent with the formulation in the R3BP. (See Figure 2.) For example, in the Earth-Moon system, where the central I I body is the Moon (P1), at epochx ˆ is directed from the Earth (P3) to the Moon andz ˆ is parallel to the angular momentum vector. Then, the transformation from I to P is defined by the Euler 3-1-3 (Ω-i-θ) sequence such that the transformation matrix for orientation of P with respect to I is written  cos Ω cos θ − sin Ω cos i sin θ sin Ω cos θ + cos Ω cos i sin θ sin i sin θ P I T = − cos Ω sin θ − sin Ω cos i cos θ − sin Ω sin θ + cos Ω cos i cos θ sin i cos θ . (7) sin Ω sin i − cos Ω sin i cos i

The vectorsx ˆI,y ˆI, andz ˆI are instantaneously available from the JPL DE405 ephemeris file and are the vector components in the transformation matrix ITJ from the EMEJ2000 frame (J) to the J I J intermediate inertial frame (I). Note that bothr ¯1m and T are determined from ephemerides and, P P P P II J J therefore, are independent of the state vectorq ¯. Sincer ¯12 = rrˆ andr ¯1m = T T r¯1m, then P P P P II J J P J r¯2m =r ¯1m − r¯12 = T T r¯1m − rrˆ . Of course, the vectorr ¯1m must be non-dimensional relative ¯P to the characteristic length. The partial derivatives of Fm (¯q) with respect to the state variables must be evaluated for implementation in eq. (3). Partial differentiation is accomplished via a matrix chain-rule expansion, i.e.

¯P " P I  P I I J J # ∂Fm ∂ T I J J ∂r P P ∂∆m ∂ T T r¯1m = µm T r¯1m − rˆ ∆m +r ¯2m − 3 , (8) ∂qj ∂qj ∂qj ∂qj ∂qj kr¯1mk

1 ∂PTI ∂r where ∆m = 3 . The partial derivatives and in eq. (8) are straightforward. The most P ∂qj ∂qj kr¯2mk complicated expression in eq. (8) is clearly ∂∆m and is derived in Appendix B. ∂qj

5 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 defined in Figure 3. A spherical coordinate frame (L) is defined in terms of unit vectorsr ˆL, λˆL, and φˆL. Latitude is measured with respect to the P1 equator, while longitude is a degree measurement from the P1 prime meridian. To compute φ and λ, it is necessary to define the orientation of P1 with B respect to an inertial frame. The frame B, fixed in P1, is defined such thatx ˆ locates the intersection B between the equator and prime meridian of P1. The third vector in the P1-fixed frame,z ˆ , is parallel B B B to the spin axis of P1 andy ˆ =z ˆ × xˆ . (See Figure 3.) For the Moon, a set of three Euler angles is available from the JPL DE405 ephemeris file to define the orientation of the Moon with respect to the EMEJ2000 (J) reference frame. Since the Euler sequence is 3-1-3 (ϕ-ϑ-ψ), a transformation to orient B relative to J can be derived such that  cos ϕ cos ψ − sin ϕ cos ϑ sin ψ sin ϕ cos ψ + cos ϕ cos ϑ sin ψ sin ϑ sin ψ B J T = − cos ϕ sin ψ − sin ϕ cos ϑ cos ψ − sin ϕ sin ψ + cos ϕ cos ϑ cos ψ sin ϑ cos ψ . (9) sin ϕ sin ϑ − cos ϕ sin ϑ cos ϑ

(See Konopliv 13 for a geometric discussion of the angles ϕ, ϑ, and ψ.) Then, the transformation between the P1-fixed frame (B) and the spherical coordinate frame (L) is cos φ cos λ − sin λ − sin φ cos λ B L T = cos φ sin λ cos λ − sin φ sin λ . (10) sin φ 0 cos φ ˆ Let b be a three-dimensional unit vector directed from the center of P1 to P2 (as seen in Figure 3). ˆP P ˆB ˆ Of course, in the perifocal frame, b =r ˆ . So, to compute b , or b in the P1-fixed frame, consider

ˆbB = BTJJTIITPrˆP, (11) where, in general, yTx = [xTy]T . Thus, ˆbB is expressed as a function of the state vectorq ¯. Once ˆbB is computed, expressions for φ and λ as functions of the state vectorq ¯ can also be evaluated. Let

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

T ˆbB be resolved into the associated three-element vector such that ˆbB = bx by bz . Then, from Figure 3, it is apparent that

φ = sin−1 bz, (12) x ! −1 b λ = ± cos p , bx2 + by2 where the sign of λ in eq. (12) is determined from the sign of by. Then, the perturbing force due to an aspherical central body for a predetermined degree nmax is written     − (n + 1) P˜(m) C˜ cos mλ + S˜ sin mλ  n n,m n,m  nmax n  n   X X µ1 RP     R¯L (¯q) = 1 m sec φ P˜(m) −C˜ sin mλ + S˜ cos mλ , (13) 1 r2 r n n,m n,m n=2 m=0    ˜(m)0  ˜ ˜    cos φ Pn Cn,m cos mλ + Sn,m sin mλ 

19 where R¯1 (¯q) is expressed in terms of the spherical coordinate frame L. The normalized coef- ficients C˜n,m and S˜n,m in eq. (13), that is, the tesseral and sectoral harmonic coefficients, are determined experimentally. The normalized zonal coefficients are defined such that J˜n = −C˜n,0 and are, therefore, implicitly included in the above summation. The scalar quantity RP1 in eq. (13) is the non-dimensional, nominal value for the radius of body P1 associated with the coefficients. The (m) normalized Legendre function P˜n is defined as

1 m h i 2 d P˜ (sin φ) P˜(m) (sin φ) = (2 − δ ) (n−m)! cosm φ n , (14) n m0 (n+m)! d (sin φ)m

th where P˜n (sin φ) is the normalized, n degree Legendre polynomial of the first kind in sin φ (sin φ is the argument of P˜n) and the Kronecker delta function δm0 is one for m = 0 and zero otherwise.

7 (m) The prime in the third element of eq. (13) is the first derivative of P˜n with respect to sin φ. The normalized Legendre functions are determined recursively, that is

 1  1  h i 2 1 (m) h i 2 (m) 2n+1 2 ˜ (n+m−1)(n−m−1) ˜  (2n − 1) sin φ Pn−1 − Pn−2 , m < n,  (n+m)(n−m) 2n−3 ˜(m) Pn = 1 (15) 2n+1  2 (m−1)  cos φ P˜ , m = n,  2n n−1 0, m > n, √ √ ˜(0) ˜(0) ˜(1) where the recursive sequence is initiated by P0 = 1, P1 = 3 sin φ, and P1 = 3 cos φ. Similarly  1  0 1 h i 2 P˜(m) = −n sin φ P˜(m) + (2n+1)(n+m)(n−m) P˜(m) . (16) n cos2 φ n 2n−1 n−1

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 normalized harmonic coefficients and Legendre polynomials as defined in eq. (13) are possible for large nmax.

Recall that Gauss’ equations require that R¯1 (¯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 β aboutr ˆ, i.e. 1 0 0  P L T = 0 cos β sin β , (17) 0 − sin β cos β where β is the angle between φˆ and hˆ. Then

¯P P L ¯L R1 (¯q) = T R1 (¯q) . (18) However, it is nontrivial to determine β. Rather than computing β, where β must also be a function of the state vectorq ¯, the elements of PTL are determined directly from the following sequence of transformations PTL = PTIITJJTBBTL. (19) The transformation matrices ITJ and JTB are independent of the state vectorq ¯ while PTI and B L ¯P T are functions ofq ¯. The partial derivatives of R1 (¯q) with respect to the state variables are required for evaluation of eq. (3). Then, via a matrix chain-rule expansion

¯P P I B L ¯L ∂R1 ∂ T I JJ BB L ¯L P II JJ B ∂ T ¯L P L ∂R1 = T T T R1 + T T T R1 + T . (20) ∂qj ∂qj ∂qj ∂qj

P I B L The partial derivative ∂ T is also necessary for evaluation of eq. (8). The partial derivative ∂ T ∂qj ∂qj involves more terms − cos λ sin φ 0 − cos λ cos φ − sin λ cos φ − cos λ sin λ sin φ  ∂BTL ∂φ ∂λ = − sin λ sin φ 0 − sin λ cos φ +  cos λ cos φ − sin λ − cos λ sin φ , (21) ∂qj ∂qj ∂qj cos φ 0 − sin φ 0 0 0 where ∂φ and ∂λ are evaluated from eq. (12). (See Appendix C.) Finally, the vector R¯L (¯q), as ∂qj ∂qj 1 defined by eq. (13), is of the form  r  Rn,m nx n   ¯L X X λ R1 (¯q) = Rn,m . (22) n=2 m=0  φ  Rn,m

8 ∂R¯L Correspondingly, the expression for 1 is written ∂qj ∂Rr   n,m     ∂qj    ¯L nx n  λ  ∂R X X ∂Rn,m  1 = , (23) ∂q ∂q j n=2 m=0  j   φ  ∂Rn,m     ∂qj  where ! ∂Rr ∂Rr ∂r ∂Rr ∂P˜(m) ∂φ ∂Rr ∂λ n,m = n,m + n,m n + n,m , (m) ∂qj ∂r ∂qj ∂P˜n ∂φ ∂qj ∂λ ∂qj ! ∂Rλ ∂Rλ ∂r ∂Rλ ∂P˜(m) ∂Rλ ∂φ ∂Rλ ∂λ n,m = n,m + n,m n + n,m + n,m , (24) (m) ∂qj ∂r ∂qj ∂P˜n ∂φ ∂φ ∂qj ∂λ ∂qj 0 ! ∂Rφ ∂Rφ ∂r ∂Rφ ∂P˜(m) ∂Rφ ∂φ ∂Rφ ∂λ n,m = n,m + n,m n + n,m + n,m . (m)0 ∂qj ∂r ∂qj ∂P˜n ∂φ ∂φ ∂qj ∂λ ∂qj

0 ∂P˜(m) ∂P˜(m) The expressions n and n are also defined recursively. (See Appendix C.) The partials ∂φ ∂φ ∂φ ∂qj and ∂λ appear in eq. (21) as well. The remaining partial derivatives in eqs. (24) can be evaluated ∂qj simply 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 be recovered from eq. (2) by selecting n = 3 in the summation. In the Earth-Moon system, periodic ¯P 9 orbits, or repeat lunar ground-track orbits, exist in the R3BP for non-zero R1 (¯q). Ultimately, the goal is a strategy for the computation of families of quasi-periodic solutions in the full model possessing specific characteristics for shape and orientation. Therefore, for this analysis, the initial ¯P design phase incorporates R1 (¯q) = 0. That is, the only perturbing force is due to the third-body point 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 body P1, as demonstrated by the dashed circle in Figure 4. The rotating frame (S), centered at P1, is consistent with the rotating frame in the R3BP. The frame S is defined such thatx ˆS is directed S from P3 toward P1,z ˆ is parallel to the angular velocity vector associated with the orbit of P3, andy ˆS =z ˆS × xˆS. For convenience, it is assumed that the rotating frame (S) is aligned with the intermediate inertial frame (I) at the initial time (t0 = 0). Then, the position of P3 with respect to P1, that isr ¯13, is not accessed from ephemerides, but is computed from the non-secular expression

I I I r¯13 = − cos t xˆ − sin t yˆ , (25)

I where t is the non-dimensional time as characterized by the mean motion of P3. Then,r ¯13 defines the position of P3 with respect to P1, with components expressed in terms of the intermediate inertial P P I I P P P P I I P I frame (I). (See Figure 4.) Therefore,r ¯13 = T r¯13 andr ¯23 =r ¯13 − r¯12 = T r¯13 − rrˆ . Sincer ¯13 is independent of the state-vectorq ¯, the form of eq. (8) remains unaltered. However, the position vectorr ¯13 in eq. (25) is already expressed in terms of the intermediate inertial frame (I). That is, I r¯13 does not require transformation from the EMEJ2000 frame (J) to the intermediate inertial frame

9 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

¯P  P I  P I  ∂F3 ∂ T I ∂r P P ∂∆3 ∂ T I = µ3 r¯13 − rˆ ∆3 +r ¯23 − r¯13 , (26) ∂qj ∂qj ∂qj ∂qj ∂qj

1 ∂∆3 ∂∆m where ∆3 = 3 . An expression for is derived from the more general expression for in P ∂qj ∂qj kr¯23k Appendix B.

INITIAL DESIGN IN THE RESTRICTED THREE-BODY PROBLEM

All orbits are initially designed in the R3BP without the gravitational harmonics associated with P1 (the Moon). Of course, the variational relationships governing δq¯ in eq. (5) cannot be directly applied 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 by Markellos 11,12 in 1977. Markellos also discusses a methodology for exploiting the tangent subspace to generate families of periodic orbits. The theory and associated numerical techniques are applied to Gauss’ equations to determine families of periodic orbits within the context of the R3BP. For all computations, 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 third body. For the orbit to be periodic, the elements a, e, ω, i, and θ∗ must all simultaneously return to their initial values at a later time t along the path. Correspondingly, the motion of the ascending node 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 to P1, 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, θ0) , ∗ i = Fi (a0, e0, i0, Ω0, θ0) , (28) ∗ Ω = FΩ (a0, e0, i0, Ω0, θ0) , ∗ ∗ θ = Fθ∗ (a0, e0, i0, Ω0, θ0) . ∗ The mappings Fa, Fi, FΩ, and Fθ∗ bring a0, e0, i0,Ω0, and θ0 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 is ∗ the determination of the variations δa0, δi0, δΩ0, and δθ0, such that, at a later time, the value of the element is equal to its original value plus the variation. Thus, the following will be true at time ∗ ∗ ∗ t: a = a0 + δa0; e = e0 + δe0; Ω = Ω0 + δΩ + t + δt − 2kπ; and θ = θ0 + δθ0. In terms of the mappings

∗ ∗ a = Fa (a0 + δa0, e0 + δe0, i0 + δi0, Ω0 + δΩ0, θ0 + δθ0) = a0 + δa0, ∗ ∗ i = Fi (a0 + δa0, e0 + δe0, i0 + δi0, Ω0 + δΩ0, θ0 + δθ0) = i0 + δi0, (29) ∗ ∗ Ω = FΩ (a0 + δa0, e0 + δe0, i0 + δi0, Ω0 + δΩ0, θ0 + δθ0) = Ω0 + δΩ0 + t + δt − 2kπ, ∗ ∗ ∗ ∗ ∗ θ = Fθ∗ (a0 + δa0, e0 + δe0, i0 + δi0, Ω0 + δΩ0, θ0 + δθ0) = θ0 + δθ0.

∗ The expressions in eqs. (29) can be expanded about the reference values a0, e0, i0,Ω0, and θ0. The first-order expansions are written

∗ ∂Fa ∂Fa ∂Fa ∂Fa ∂Fa ∗ Fa (a0, e0, i0, Ω0, θ0) + δa0 + δe0 + δi0 + δΩ0 + ∗ δθ0 = a0 + δa0, ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0

∗ ∂Fi ∂Fi ∂Fi ∂Fi ∂Fi ∗ Fi (a0, e0, i0, Ω0, θ0) + δa0 + δe0 + δi0 + δΩ0 + ∗ δθ0 = i0 + δi0, ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0 (30) ∗ ∂FΩ ∂FΩ ∂FΩ ∂FΩ ∂FΩ ∗ FΩ (a0, e0, i0, Ω0, θ0) + δa0+ δe0+ δi0+ δΩ0+ ∗ δθ0 = Ω0 + δΩ0 + t + δt − 2kπ ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0

∗ ∂Fθ∗ ∂Fθ∗ ∂Fθ∗ ∂Fθ∗ ∂Fθ∗ ∗ ∗ ∗ Fθ∗ (a0, e0, i0, Ω0, θ0) + δa0 + δe0 + δi0 + δΩ0 + ∗ δθ0 = θ0 + δθ0, ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0 where, in general, all of the reference Keplerian elements vary with time for the R3BP. Substituting eqs. (28) into eqs. (30) and rearranging results in the targeter matrix relationship     δa0 a0 − a      δi0   i0 − i    = [C] δe0 , (31) t − (Ω − Ω0) − 2kπ     δΩ0  θ∗ − θ∗    0  ∗  δθ0 where  ∂a ∂a ∂a ∂a ∂a  − 1 ∗ ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0     ∂i ∂i ∂i ∂i ∂i   a˙   − 1     ∂a ∂e ∂i ∂Ω ∂θ∗  1  i˙   ∂ω ∂ω ∂ω ∂ω ∂ω   0 0 0 0 0  C = − ∗ . (32)  ∂Ω ∂Ω ∂Ω ∂Ω ∂Ω  ω˙ Ω˙ − 1 ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ  − 1    0  ∗   ∗   ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0  θ˙  ∗ ∗ ∗ ∗ ∗   ∂θ0 ∂θ0 ∂θ0 ∂θ0 ∂θ0  ∗ ∂a0 ∂e0 ∂i0 ∂Ω0 ∂θ0

11 The components of C are evaluated by isolating the necessary sensitivity partials in eq. (5), where ∂qi = Φij (t, 0). Since interest is in semi-elliptical orbits (e < 1), it is desirable to constrain ∂qj,0 variations in eccentricity during the solution process in eq. (32). If e0 is established as a fixed element in eq. (32), that is δe0 = 0, the solution space can be expanded. The targeter relationship then becomes  a − a  δa   0   0       i0 − i   δi0  = [D] , (33) t − (Ω − Ω ) − 2kπ δΩ  0   0  ∗ ∗   ∗  θ0 − θ δθ0 where the matrix D is defined by removing the appropriate column from the matrix C to satisfy the constraint δe0 = 0. Since the matrix D is square, the state variations are obtained by directly inverting eq. (33). However, the matrix D may become ill-conditioned for some orbits (e.g., stable periodic orbits). In such cases, the speed of convergence is increased by incorporating a weighted and/or regularized minimum-norm solution. Equation (33) can then be solved using

δa   a − a   0   0       δi0  −1  i0 − i  = WDT DWDT − γI , (34) δΩ t − (Ω − Ω ) − 2kπ  0  0   ∗   ∗ ∗  δθ0 θ0 − θ for a positive, diagonal weighting matrix W and 4 × 4 identity matrix I. The diagonal elements of W are determined experimentally. A value of the regularizing parameter γ is selected such that h i a 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 solution for the state variations in eq. (33) is obtained and added to the corresponding parameters in the initial state vector. Integrating the new initial conditions yields an orbit that more closely satisfies the conditions for periodicity. An iterative process generally converges to an orbit that meets the requirements.

Example Periodic Solution The targeting scheme successfully computes periodic orbits corresponding to the approximate frozen orbit conditions in the traditional third-body perturbing problem 5 sin 2ω = 0 and e2 + cos2 i = 1. (35) 3 (See Prado 21 for a derivation of eq. (35).) Appropriate initial conditions can be extracted directly from eq. (35). Consider the Earth-Moon system in the R3BP where the central body P1 is the Moon, P2 is the spacecraft, and P3 is the Earth. For orbits possessing geometries desirable for lunar ◦ south pole coverage, the mappings in eq. (28) are defined such that ω = ω0 = 90 (rather than ◦ ◦ ω = ω0 = 270 ). An orbit with e0 = 0.6 requires an inclination equal to i0 = 51.71 as determined ◦ from eq. (35). Also, the value of the ascending node, Ω0 = −90 , positions the initial state in the xˆS−zˆS plane corresponding to the rotating frame (S). Since it is desirable to implement corrections ∗ ◦ near apoapsis, the initial true anomaly is defined to be θ0 = 180 . Consider an orbit with these initial conditions and semi-major axis a0 = 15, 000 km. The , e−ω phase space plot for this trajectory 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 is also 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 the orbit is 25.039 days. The trajectory is commensurate with the orbit of the Earth-Moon system. The spacecraft motion is comprised of 13 conic, semi-elliptical revolutions for each three-body . These two frequencies are apparent when the Keplerian elements are transformed into inertial 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 24,339.4 km. Additionally, all the elements are plotted over one full period in Figure 7. It is clear that the elements are well-behaved. The elements e, ω, and i include both long and short periods, without secular effects. The accuracy of the state equations are verified by examining the well-known integral of the motion, or Jacobi Constant (C), in the R3BP. 10 The Jacobi Constant associated with this orbit is 3.43372 km2/. 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 this neighborhood. Furthermore, this value of C is about 3% greater than the value of C associated 2 2 with the L1 libration point (CL1 = 3.33557 km /s ). Of course, in the R3BP, all trajectories in the lunar 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 15500

15000 (km) a 14500 0 5 10 15 20 25

0.7

e 0.6

0.5 0 5 10 15 20 25

100

(deg) 90 ω 80 0 5 10 15 20 25

54

52 (deg) i

50 0 5 10 15 20 25

−50

−100 Ω(deg) −150 0 5 10 15 20 25

5000 (deg) ∗ θ 0 0 5 10 15 20 25

) 2 2 /s 2 0 C km 0 − 14 C −

10 −2

× 0 5 10 15 20 25 ( time (day)

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

15 Predictions Using the Tangent Subspace

k+1 k+1 A family of periodic orbits may be computed by continuously updating e0 using e0 = k k e0 + ∆e0, where e0 is associated with the corrected initial state and ∆e0 is a predetermined fixed step size. However, in regions where many different families exist (and all possess similar shape k+1 characteristics), even small values of ∆e0 will not ensure that e0 results in orbits within the same k family as e0 . As an alternative, consider an expansion of the solution space via a linear prediction k k k k ∗k along the tangent subspace Γ . Let ∆a0 , ∆e0 , ∆i0 , ∆Ω0 , and ∆θ0 be the changes to the corrected k k+1 initial stateq ¯0 that predict the next initial stateq ¯0 for a neighboring orbit. That is, the linear k+1 prediction forq ¯0 associated with the neighboring orbit is simply

k+1 k k q¯0 =q ¯0 + d · ∆¯q0 , (36)

k  k k k k ∗k T where ∆¯q0 = ∆a0 ∆e0 0 ∆i0 ∆Ω0 ∆θ0 . The parameter d is a predetermined step size. k+1 The magnitude of d must be sufficiently small to ensure thatq ¯0 yields an orbit within the same k characteristic family as the orbit represented by the initial conditionsq ¯0 . Since Γ is in the nullspace k of C (eq. (32)), the components of ∆¯q0 must satisfy

k  ∆a   ca   0   k     ∆i   ci  0 = ∆ek , (37) k 0 c ∆Ω0   Ω       ∗k c ∗ ∆θ0 θ

k for the free variable ∆e0 . Of course, the values of ca, ci, cΩ, and cθ∗ are derivable from the homogeneous system  k   ∆a0   k   ∆e   0  k ¯ [C] ∆i0 = 0. (38)  k  ∆Ω   0   ∗k ∆θ0 k k The components of ∆¯q0 are determined uniquely by the requirement ∆¯q0 = 1. Then,

 ca     1     0  k 1   ∆¯q0 = p . (39) 1 + c2 + c2 + c2 + c2 ci a i Ω θ∗     cΩ    cθ∗ Given the prediction from eq. (36), the initial condition is propagated, and the iterative process repeats until the convergence to a new solution is achieved. (Of course, all state vectors are bounded in 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, due to communications limitations, it may be necessary to compute orbits with radii of apoapsis less than 10,000 km. Let a0 be the parameter that characterizes the family of orbits. (Recall that e0 characterizes an orbit within the family.) A family satisfying an apoapsis constraint such that ra < 10, 000 km is generated from the initial condition a0 = 5, 000 km and e0 = 0.925, where the 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 the colors of the orbits in the top plot match those of the bottom plot. Each orbit within the family possesses 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 i0 decreases. Also, in stepping through the full range in eccentricities using eq. (36), e0 begins near one and ends near zero. Since e0 approaches zero at the intersection between the southern and northern 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 either ◦ by symmetry or by re-computing the entire family with ω0 = 270 . Of course, these orbits would be useful for north pole coverage. The Jacobi Constant for each orbit in the family is significantly greater 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

◦ ◦ ∗ ◦ 2 2 a0 (km) e0 i0( )Ω0( ) θ0( ) T (day) C (km /s ) 1 4,923.6 0.925 72.551 -89.521 186.353 26.539 4.08618 2 4,972.2 0.8940 69.310 -90.100 178.640 26.571 4.07763 3 5,070.2 0.8501 65.518 -89.957 180.704 26.610 4.06029 4 5,124.6 0.8066 62.344 -89.665 186.848 26.658 4.05165 5 5,131.7 0.7633 59.606 -90.626 150.881 26.714 4.05180 6 5,186.8 0.7203 57.160 -89.933 181.649 26.758 4.04307 7 5,192.2 0.6851 55.358 -89.943 181.541 26.800 4.04317 8 5,199.7 0.6324 52.915 -90.241 172.527 26.858 4.04328 9 5,204.8 0.5936 51.292 -90.322 169.125 26.899 4.04335 10 5,210.5 0.5475 49.536 -90.392 165.611 26.943 4.04342 11 5,217.0 0.4926 47.663 -90.045 178.325 26.992 4.04349 12 5,221.4 0.4493 46.342 -90.266 169.599 27.027 4.04353 13 5,225.9 0.3982 44.955 -90.435 162.619 27.064 4.04357 14 5,229.1 0.3578 43.982 -90.551 158.288 27.090 4.04359 15 5,232.2 0.3142 43.051 -90.673 154.596 27.116 4.04361 16 5,235.1 0.2551 41.989 -91.319 134.590 27.145 4.04362 17 5,237.3 0.2035 41.241 -91.868 123.308 27.165 4.04363 18 5,238.7 0.1646 40.789 -92.424 116.269 27.178 4.04364 19 5,239.6 0.1351 40.510 -93.025 111.920 27.186 4.04364 20 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 transitioned to a full ephemeris model (including solar perturbations). Therefore, let n = 4 in eq. (2), where the location of Earth (P3) and Sun (P4), with respect to the Moon (P1) are determined by the JPL DE405 ephemeris file. Furthermore, since the effects of an aspherical central body are significant, ¯P ˜ ˜ consider the case when R1 (¯q) 6= 0. Then the coefficients Cn,m and Sn,m in eq. (13) must also be determined. Of course, these coefficients are based on knowledge of the Moon’s gravity field that is only available experimentally. Consider a 50 × 50 gravity field (nmax = 50 in eq. (13)) using the LP165P model available at the Geosciences Node of NASA’s Planetary Data System. 22 In terms of this lunar gravity model, the largest unnormalized coefficients in the expansion (though clearly not dominant) 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 from Table 1. As noted, all the orbits possess apoapses less than 10,000 km. Other desirable features include high eccentricities and feasible periapsis altitudes. From Figure 9, it is clear that eccentricity increases as radius of periapsis (rp) decreases. Therefore, consider orbits with rp just above the lunar surface (or horizontal line in Figure 9). The ninth initial condition listed in Table 1 corresponds to an orbit that satisfies these conditions with periapsis altitude near 400 km and e0 approximately 0.6. 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. To transition to the full model, the orbit is discretized into a series of n pointsq ¯1,q ¯2, ...,q ¯n, equally spaced in time. Theq ¯k are strategically separated by ∆t = 7.836, days corresponding to every 20th apoapse point, for a baseline 1,000 day mission. Since the number of close passes (69) is odd, two spacecraft can be placed in the same orbit but phase shifted in time by exactly one-half period to maximize 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 variations δq¯k 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 δq¯k with multiple shooting, consider the fixed-time mapping F¯k of the pointq ¯k over ∆t. Then, to compute a continuous trajectory in the full model, the goal is the determination of δq¯k such that

q¯2 + δq¯2 = F¯1 (¯q1 + δq¯1) ,

q¯3 + δq¯3 = F¯2 (¯q2 + δq¯2) , (40) . .

q¯n + δq¯n = F¯n−1 (¯qn−1 + δq¯n−1) .

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

∂F¯1 q¯2 + δq¯2 = F¯1 (¯q1) + δq¯1, ∂q¯1

∂F¯2 q¯3 + δq¯3 = F¯2 (¯q2) + δq¯2, ∂q¯2 (41) . .

∂F¯n−1 q¯n + δq¯n = F¯n−1 (¯qn−1) + δq¯n−1, ∂q¯n−1

∂F¯k where = Φ (tk + ∆t, tk). Rearranging terms in eq. (41) results in the linear relationship ∂q¯k  ¯  ∂F1  δq¯  −I  1   ¯   ∂q¯1    q¯2 − F1 (¯q1)    δq¯2     ∂F¯     q¯ − F¯ (¯q )   2   δq¯   3 2 2   −I   3  = ∂q¯2 , (42) .   .  .   .   .     ..     ¯      q¯n − Fn−1 (¯qn−1)  ¯  δq¯n−1  ∂Fn−1    −I   δq¯n ∂q¯n−1 | {z } M where I is the 6 × 6 identity matrix. Since there are 6n scalar unknowns, i.e., δq¯1, δq¯2, . . . , δq¯n, 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 the minimum-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., without −1 directly computing MMT  . (See G´omez et al. 16) This is particularly useful when MMT is ill- conditioned. Then,q ¯k is iteratively readjusted until

q¯2 = F¯1 (¯q1) ,

q¯3 = F¯2 (¯q2) , (43) . .

q¯n = F¯n−1 (¯qn−1) .

¯ −9 Continuity is established when the largest discontinuity, i.e., q¯k+1 − Fk (¯qk) max is less than 10 non-dimensional units. The multiple shooting scheme is applied to the discretized pointsq ¯1,q ¯2, ...,q ¯n corresponding to two spacecraft in a single orbit, phase shifted by one-half period. In four iterations, 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 of Figure 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 ground site is directly related to coverage. 8 In general, the elevation angle of each spacecraft from a specific latitude φs and longitude λs on the lunar surface is evaluated as

B B ! π −1 r¯1s · r¯s2 α = − cos B B , (44) 2 r¯1s r¯s2

B wherer ¯1s is the position of the site from the center of the Moon in the body-fixed frame (B), i.e.,

cos φs cos λs B   r¯1s = RP1 cos φs sin λs (45)   sin φs

B B B Of course, the position of the spacecraft relative to the site is then evaluated fromr ¯s2 =r ¯12 − r¯1s = ˆB B rb − r¯1s. The Shackleton Crater has been identified as a site of interest. The elevation angle from ◦ ◦ the 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 one spacecraft is always 17.85◦ above the horizon for the entire simulation.

40

20 (deg) α 0 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 days past epoch

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−ω Phase 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 Kit R ’s (STK) Astrogator from AGI. The orbit is verified by targeting ik,Ωk, and θk with Astrogator Connect using the procedure described in Grebow et al. 8 In general, slight differences in modeling result in very small position errors (less than 1 km) when propagating to eachq ¯k in STK. Of course, these errors are minimized with small corrections. For these orbits, the targeting sequence in Astrogator is extremely sensitive to even small errors. However, in general, the orbit is computed with corrections on the order of a few m/s per year. Once the orbits are targeted in STK, percent access times with a ground station at the Shackleton site are available. Additionally, Earth-based transmitting sites are defined and the access times between the satellites and these transmitting sites are computed. A potential location for a ground station is the White Sands Test Facility (WSTF) located in New Mexico (32.3◦N, 106.8◦W). The times when each satellite, either satellite, or both satellites possess line-of-sight with the stations are computed. (See Table 2.) Due to the initial time phase shift, access times are all computed omitting the first and last half periods. As expected, at least one satellite maintains line-of-sight with 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 desirable feature for a communications relay between the satellites if necessary. Finally, a communications link between the satellites is established for possible communications relay. The line-of-sight access for this link is 81.78%.

Table 2: Percent Access Times Shackleton Earth WSTF Only 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 nearby solutions in the full model for long term simulations. Identifying such orbits is desirable for constant communications with a permanent station at the lunar south pole. Since the full LP165P provides much 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 increase computation time. However, since the propagation of eachq ¯k in a multiple shooting process is inde- pendent, the computation time can be reduced dramatically by integrating the states in parallel, or as deputy processes. In general, a chief process distributes theq ¯k to the available deputy processes for integration. When the integration is complete, the deputy returns F¯k(¯qk) to the chief and receives the next available state. When every F¯k(¯qk) is computed, the chief solves for δq¯k in eq. (42) using a 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 compiler and integrator. So, the simulation is implemented in Fortran 90 using an Adams-Bashforth-Moulton integrator. 17 According to Montenbruck, 23 an Adams-Bashforth-Moulton integrator requires fewer function 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 PGI R ’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 factors in the recurrence relations for the normalized Legendre polynomials depending on n and m are pre-calculated and the polynomials are subsequently computed with array arithmetic. Using the MPI standard to interface between 54 AMD OpteronTM 280 dual-core processors, the simulation converges to a continuous solution in seven iterations. (Recall a continuous solution is computed ¯ −9 when q¯k+1 − Fk (¯qk) max is on the order of 10 non-dimensional units.) For a single spacecraft, the process takes less than four hours total. Consider a comparison of this process in terms of real time to a ten-year integration of an uncorrected state on one processor. Since integration of a single, uncorrected state for the full simulation time does not lend itself well to parallelization, a converged solution using discretization and parallelization can be reached significantly faster. The speed increase is approximately equal to the number of processors available divided by the number of iterations necessary for convergence. For the ten-year simulation described here, a corrected solution using multiple shooting can be delivered in less than one-tenth of the time that is necessary for a single 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 Figure 13. It is clear that the process converges to a nearby solution; both spacecraft closely follow one another 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 × 165 LP165P Gravity Model, e−ω Phase Space

25 of approximately 0.52 with a variation of only ±0.06. The minimum periapsis altitude for both spacecraft is roughly 420 km for the entire simulation. When examining the elevation angle of both spacecraft 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 decrease with time. In fact, for the ten-year simulation, at least one spacecraft is always 15.72◦ above the horizon as seen from the Shackleton Crater.

CONCLUSION

Using Gauss’ equations, perturbations from n − 2 additional bodies and central-body gravity harmonics are modeled and the state-transition matrix is computed. The R3BP is identified as a special case of the n-body problem, where n = 3. The tangent subspace is used to compute families of 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 highly eccentric orbit with apoapsis over the lunar south pole and feasible periapsis altitude is a candidate for lunar south pole coverage. Using JPL’s DE405 ephemeris file and the LP165P gravity model, the candidate orbits are transitioned to the full model with a multiple shooting scheme. The resulting trajectories maintain the shape of the orbit as designed in the R3BP. The trajectories are verified in STK’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 communications with a ground station at the Shackleton Crater over 100% of the simulation time. Furthermore, a long-term simulation demonstrates that both spacecraft maintain their initial phasing without lunar impact for over 10 years, where at least one spacecraft is always 15.72◦ above the horizon. Current research efforts are focused on computing transfers into these orbits. Of course, for long-term communications, station-keeping costs are also significant.

ACKNOWLEDGEMENT

A significant portion of this work was completed at the NASA Goddard Spaceflight Center under the supervision of Mr. David Folta. The authors also thank Professor Stephen Heister for use of computing resources. Portions of this work were supported by Purdue University and NASA under contract 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, and Dynamics, Vol. 26, No. 2, March-April 2003, pp. 238-243. 3. T. Ely, “Stable Constellations of Frozen Elliptical Inclined Orbits.” Journal of the Astronautical Sciences, Vol. 53, No. 3, July-September 2005, pp. 301-316. 4. T. Ely and E. Lieb, “Constellations of Elliptical Inclined Lunar Orbits Providing Polar and Global Coverage.” Paper No. AAS 05-158, AAS/AIAA Spaceflight Mechanics Meeting, South Lake 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 as Affected by Gravitational Perturbation from External Bodies,” AIAA Journal (Russian Supplement), Vol.1, No. 8, August 1963, 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 Lunar South 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-Body Problem.” 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, New York, 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 Restricted Three-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. G´omez, J. Masdemont, and C. Sim´o,“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, Conf´erences Sur La Relativit´e en M´ecanique C´eleste et en As- trom´etrie. Institut de M´ecanique C´eleste et de Calcul des Eph´em´erides,´ 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. Accessed 1/17/07.] 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 equations with respect to each state variable. Let p = aζ2. Then, the results appear in the following form.

∂a˙ 3µ = 1 F rer sin θ∗ + F θp , (A.1) ∂a rζη3a4 ∂a˙ 2   e2   pe  = F r 1 + sin θ∗ + F θ + cos θ∗ , (A.2) ∂e ζη ζ2 rζ2 ∂a˙ ∂a˙ ∂a˙ = = = 0, (A.3) ∂ω ∂i ∂Ω ∂a˙ 2e = F r cos θ∗ − F θ sin θ∗ , (A.4) ∂θ∗ ζη ∂e˙ ζ = F raer sin θ∗ + F θ ap − r2 , (A.5) ∂a 2ηa3er ( " #) ∂e˙ 1 ae sin θ∗ ap − r2 ζ ap − r2 ζ  2aer + r2 cos θ∗  = − F r +F θ + − a cos θ∗+ , (A.6) ∂e ηa2 ζ rζ re2 e p ∂e˙ ∂e˙ ∂e˙ = = = 0, (A.7) ∂ω ∂i ∂Ω ∂e˙ ζ  ap + r2  = F r cos θ∗ − F θ sin θ∗ , (A.8) ∂θ∗ ηa pa ∂ω˙ ζ   r  er sin θ  = −F r cos θ∗ + F θ 1 + sin θ∗ − F h , (A.9) ∂a 2ηa2e p p tan i " # ∂ω˙ 1 cos θ∗ p (p+r) e2+ζ2 +ζ2er2 cos θ∗ r ep+ζ2r cos θ∗ = F r −F θ sin θ∗+F h sin θ , (A.10) ∂e ζηa e2 e2p2 p2 tan i ∂ω˙ ζr cos θ = −F h , (A.11) ∂ω apη tan i ∂ω˙ ζr sin θ = F h , (A.12) ∂i apη sin2 i ∂ω˙ = 0, (A.13) ∂Ω ∂ω˙ ζ  sin θ∗ p (p + r) cos θ∗+r2e sin2 θ∗  r (er sin θ sin θ∗ + p cos θ) = F r + F θ −F h , (A.14) ∂θ∗ ηa e p2e p2 tan i ∂i˙ ζr cos θ = F h , (A.15) ∂a 2a2pη ∂i˙ r ep + ζ2r cos θ∗ cos θ = −F h , (A.16) ∂e ζap2η ∂i˙ ζr sin θ = −F h , (A.17) ∂ω apη ∂i˙ ∂i˙ = = 0, (A.18) ∂i ∂Ω ∂i˙ ζr (er cos θ sin θ∗ − p sin θ) = F h , (A.19) ∂θ∗ ap2η ∂Ω˙ ζr sin θ = F h , (A.20) ∂a 2a2pη sin i

28 ∂Ω˙ r pe + ζ2r cos θ∗ sin θ = −F h , (A.21) ∂e ζap2η sin i ∂Ω˙ ζr cos θ = F h , (A.22) ∂ω apη sin i ∂Ω˙ ζr sin θ cos i = −F h , (A.23) ∂i apη sin2 i ∂Ω˙ = 0, (A.24) ∂Ω ∂Ω˙ ζr (er sin θ sin θ∗ + p cos θ) = F h , (A.25) ∂θ∗ ap2η sin i ∂θ˙∗ 3p2η ∂ω˙ ∂Ω˙ = − − − cos i , (A.26) ∂a 2aζ3r2 ∂a ∂a ∂θ˙∗ ηp 3pe + 2ζ2r cos θ∗ ∂ω˙ ∂Ω˙ = − − cos i , (A.27) ∂e ζ5r2 ∂e ∂e ∂θ˙∗ ∂θ˙∗ ∂θ˙∗ = = = 0, (A.28) ∂ω ∂i ∂Ω ∂θ˙∗ 2ηpe sin θ∗ ∂ω˙ ∂Ω˙ = − − − cos i . (A.29) ∂θ∗ ζ3r ∂θ∗ ∂θ∗

APPENDIX B

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

− 3 1 P P  2 ∆m = = r¯2m · r¯2m , (B.1) P 3 r¯2m wherer ¯P = PTIITJr¯J − rrˆP. Then, ∂∆m is computed using a vector dot-product, chain-rule 2m 1m ∂qj expansion, that is

P I − 5    ∂∆m P P  2 P ∂ T I J J ∂r P = −3 r¯2m · r¯2m r¯2m · T r¯1m − rˆ . (B.2) ∂qj ∂qj ∂qj

APPENDIX C

The partial derivatives ∂φ and ∂λ are computed by first noting that ∂qj ∂qj

∂ˆbB ∂ITP = BTJJTI rˆP, (C.1) ∂qj ∂qj

B n x y z oT where ˆbB is defined in eq. (11). Let ∂ˆb be resolved into the three element vector ∂b ∂b ∂b . ∂qj ∂qj ∂qj ∂qj Then, from eq. (12) the partials are written

∂φ 1 ∂bz = p , (C.2) ∂qj 1 − bz2 ∂qj

 y x  ∂λ 1 x ∂b y ∂b = 2 2 b − b . (C.3) ∂qj bx + by ∂qj ∂qj

29 ˜(m) ˜(m)0 ∂Pn ∂Pn The partial derivatives ∂φ and ∂φ can be evaluated using the relationships

(m) ∂P˜ 0 n = cos φ P˜(m) , (C.4) ∂φ n and

0 (m)  1  ∂P˜ 1 0 h i 2 0 n = − n P˜(m) + (n − 2) sin φ P˜(m) − (2n+1)(n+m)(n−m) P˜(m) . (C.5) ∂φ cos φ n n 2n−1 n−1

30