Orbits That Include Lunar Encounters
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Transfers to L1 and L2 Orbits that Include Lunar Encounters K. C. Howell School of Aeronautics and Astronautics Purdue University School of Aeronautics and Astronautics History: Some Previous Concepts or Missions Involving Lunar Encounters • ISEE-3/ICE • MAP • Double Lunar • Lunar-A Swingbys • Nozomi • Geotail • Hiten • Relict-2 • Genesis • Wind School of Aeronautics and Astronautics Double Lunar Swingbys ISEE-3 Geotail Phase School of Aeronautics and Astronautics WIND Nozomi School of Aeronautics and Astronautics Relict-2 MAP School of Aeronautics and Astronautics Genesis Trajectory Mission •⊕ to L1 / Return •L1 Orbit: Az= 320,000 km Ay=745,000 km •∆V = 8.5 m/s School of Aeronautics and Astronautics Trajectory Design Including Lunar Encounters Apply recent capabilities •Insight? Fast? Efficient? •Target orbit constrained? Timing? •Multiple spacecraft? •New Design Strategies? ⇓ Approaches: Semi-Analytical Numerical School of Aeronautics and Astronautics Direct Transfer: No Encounter Transfer/LOI Small Liss ∆V:LOI 165 -190m/s School of Aeronautics and Astronautics Alternate Solution: No Encounter Single LOI Maneuver ∆V = 95 m/s School of Aeronautics and Astronautics Transfer to L1: Lunar Encounter ∆ VTTI = 3129.43 m/s ∆ VLOI = 2.32 m/s Direct Transfer ∆ VTTI = 3196.69 m/s ∆ VLOI = 185.70 m/s School of Aeronautics and Astronautics Lunar Encounter Phasing Loops ∆V:LOI 21→ 0 m/s School of Aeronautics and Astronautics Trajectory Design Including Lunar Encounters Apply recent capabilities •Insight? Fast? Efficient? •Target orbit constrained? Timing? •Multiple spacecraft? •New Design Strategies? ⇓ Approaches: Semi-Analytical Numerical School of Aeronautics and Astronautics Restricted 4-Body Problem •Natural Dynamics in the Restricted Four-Body Problem •Focus: vicinity of the Moon perturbed Sun-Earth libration points •Computation of nominal orbits •Computation of transfer arcs to the nominal orbits •Shift information to the ephemeris model School of Aeronautics and Astronautics Restricted 4-Body Problem P1 - Sun P2 - Earth P4 - Moon School of Aeronautics and Astronautics Generalized Semi-Analytical Development •Relative EOMs ( wrt P2 ) (M +M ) 4 rrij 2j r +ri2=∑ M- 2i 332i j 3 rrj=1 r 2i ij 2j ji≠ ,2 •Form of the approximation oo oo r=rf ()α r+ˆ f ()βθˆˆ+f (hγ) ,l= 2,j 2i 2l 1 2l 2 2l 3 2l School of Aeronautics and Astronautics Generalized Semi-Analytical Development •Perturbation Approach f(1 α)t +α f(2 β)s +β f(3 γ)z +γ •Trigonometric Series K-1 α =cαα(c) os(ω t)+ (s)sin(ω t) ∑ k=0 kkkk K-1 β =c ββ(c) os(ω t)+ (s)sin(ω t) ∑ k=0 kkkk K-1 γ =cγγ(c) os(ω t)+ (s)sin(ω t) ∑ k=0 kkkk School of Aeronautics and Astronautics Generalized Semi-Analytical Development •Substitute approximation into relative EOMs f1 − 2n2l f2 -3f = U 1 f1 f 2 + 2n2l f1 -3f2 = U f2 f 3 -3f3 = U f3 •Partials are nonlinear functions of f,12f,f3 U=U(f,f,f) ff11123 U=U(f,f,f) ff22123 U=U(f,f,f) ff33123 School of Aeronautics and Astronautics Generalized Semi-Analytical Development • Planar Motion, i.e., γ 2i = 0 and undisturbed: circular motion ⇒ r=roo(1+α )rˆ + βθˆo 2i 2i 2i 2i 2i 2i School of Aeronautics and Astronautics Generalized Semi-Analytical Development •Consider perturbation of libration points r=roo(t+α )rˆ +(s+ βθ) ˆˆo+(z+γ )ho 23 21 23 21 23 21 23 21 •Initial Guess: α23 = β23 = γ 23 =0 •L2 Perturbed School of Aeronautics and Astronautics Generalized Semi-Analytical Development •Consider orbit: same form r=roo(t+α )rˆ +(s+ βθ) ˆˆo+(z+γ )ho 23 21 23 21 23 21 23 21 •Initial Guess: CR3BP Approximations •L2 Orbit Perturbations School of Aeronautics and Astronautics Quasi-Periodic Motion School of Aeronautics and Astronautics Assess Lunar Gravity Impact ∆vfb δ = 2sin−1 2v∞ 2v∞ ∆vfb = 2 1/+ (vQ∞ p µ ) School of Aeronautics and Astronautics Stable Manifold School of Aeronautics and Astronautics Stable Manifold at Lunar Orbit Crossings School of Aeronautics and Astronautics Elongation Angle School of Aeronautics and Astronautics Ephemeris Model: Stable Manifold Transfer Paths Beyond Lunar Orbit School of Aeronautics and Astronautics School of Aeronautics and Astronautics School of Aeronautics and Astronautics School of Aeronautics and Astronautics Lunar Orbit Parameters School of Aeronautics and Astronautics Preliminary Design Procedure • Manifolds: If properly timed, periselene event will occur •Check ΨΨ e − < 5º (Modify tJD (∆t M ), timing condition) •Ephemerides Model: Check season ⇒ Ze •Select Z-e ZS/C <20,000 km •Adjust displacement value “d” to allow finer control (including altitude) •Phasing Loops? •Differentially correct to meet constraints School of Aeronautics and Astronautics Moon Timing Issues -1 β24 (θ) -1 β21(θ) Ψe(θ)=θ +tan -tan (1+ α24 (θ))(1+ α21(θ)) where θ = θo +Nt • Given Time ⇒→θΨclosed form solution →e • Given Elong →→ Ψθe solve transcendental equation School of Aeronautics and Astronautics Manifolds: Asymptotically Approaching Orbit F FWS XX0 =+d Y School of Aeronautics and Astronautics Algorithm: Earth-to-Lissajous Natural Solutions Lissajous Trajectory Case d Value Periselene Altitude Inclination # (km) Elongation (deg) (km) (deg) Orbit S, Class I 47 201.4764 -62.04 186.90 33.47 Orbit S, Class I 141 201.0680 -36.00 196.20 21.12 Orbit S, Class II 134 201.2000 -47.40 699.87 44.35 Orbit T, Class I 52 199.1130 -64.04 183.62 54.91 Orbit T, Class I 138 201.8585 -47.72 182.71 41.19 Orbit T, Class II 64 198.7386 -72.12 184.17 10.07 Orbit T, Class II 134 201.0986 -50.27 183.53 18.35 Corrected Transfer Trajectories Case Name TTI TTI Lunar Lunar LOI Total Date Cost Encounter Altitude Cost Cost (m/s) Date (km) (m/s) (m/s) Orbit S, I, 47 5/20/01 3139.90 6/26/01 17084.52 0.78 3140.68 Orbit S, I, 141 5/22/01 3140.39 7/02/01 11999.28 5.04 3145.43 Orbit S, II, 134 5/22/01 3128.06 7/01/01 36781.55 8.26 3136.32 Orbit L, I, 52 5/21/01 3133.10 6/29/01 15658.84 6.28 3139.38 Orbit L, I, 138 5/21/01 3133.57 7/01/01 14114.37 8.08 3141.65 Orbit L, II, 64 5/20/01 3127.87 6/29/01 17546.67 4.88 3132.75 Orbit L, II, 134 5/22/01 3133.42 6/30/01 15277.15 1.49 3134.91 School of Aeronautics and Astronautics Transfer 1: Lunar Encounter School of Aeronautics and Astronautics Transfer 2 School of Aeronautics and Astronautics Temporary Capture School of Aeronautics and Astronautics Summary •Semi-Analytical: coherent-R4BP •Perturbation approach •Trigonometric series for location of Sun/Moon •Associated invariant manifolds •Identify manifolds bent by the Moon •Approx lunar timing conditions •Numerical Approach: Full Model •Compute nominal orbit (Liss or halo) •Calculate manifolds •Differentially Correct to meet constraints School of Aeronautics and Astronautics R4BP EOMs • Newton’s Equations of Motion X-θ11Y-2θ Y=UX Y+θ11X+2θ X=UY Z=UZ • Pseudo-Potential 1MM M U(X,Y, Z) = θ X+22Y ++1 23+ 1 2r13 r23 r43 • Contrast with CR3BP 1.No integral of the motion 2.No constant equilibrium solutions 3.Symmetries a.Reflection about the X-Y plane b.Reflection about the X-Z plane (with time reversal) IIF primary motion with same symmetry School of Aeronautics and Astronautics Assess Lunar Gravity ∆vfb δ = 2sin−1 2v∞ 2v∞ ∆vfb = 2 1/+ (vQ∞ p µ ) School of Aeronautics and Astronautics Results: Physical Scenario School of Aeronautics and Astronautics Lissajous Orbit: Ay>Az School of Aeronautics and Astronautics Construction of Transfers Using Manifolds School of Aeronautics and Astronautics Transfer 3: Lunar Encounter School of Aeronautics and Astronautics Transfer 4: Lunar Encounter School of Aeronautics and Astronautics Transfer 5: Lunar Encounter School of Aeronautics and Astronautics Transfer 6: Lunar Encounter School of Aeronautics and Astronautics Transfer 7: Lunar Encounter School of Aeronautics and Astronautics Preliminary Design Procedure • Manifolds: Select Z-e ZS/C <20,000 km • “Place” Moon at appropriateΨ e ⇒ non-physical scenarios •Ephemerides Model: Check season ⇒ Ze •Shift to Ephemeris Model School of Aeronautics and Astronautics.