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Design Using Gauss' Perturbing Equations

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Design Using Gauss' Perturbing Equations 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, orbits are calculated with shapes and orientations desirable for lunar south pole coverage. Families of periodic orbits are first computed in the Earth-Moon 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 ephemeris 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 orbit 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 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 conditions are 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 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 true anomaly (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 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 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 lunar orbit 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 celestial mechanics, 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
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