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