A map approximation for the restricted three-body problem Shane Ross Engineering Science and Mechanics, Virginia Tech www.esm.vt.edu/ sdross Collaborators: P. Grover (Virginia Tech) & D. J. Scheeres (U Michigan) SIAM Conference on Applications of Dynamical Systems
Infinity to capture about small companion in binary pair? After consecutive gravity assists, large orbit changes 2
Kicks at periapsis Key idea: model particle motion as kicks at periapsis Semimajor Axis vs. Time Δa+ m 1 m 2 Δa Δa Δa+ In rotating frame where m 1, m 2 are fixed 3
Kicks at periapsis Sensitive dependence on argument of periapse ω Semimajor Axis vs. Time Δa+ m 1 m 2 Δa Δa Δa+ In rotating frame where m 1, m 2 are fixed 4
Kicks at periapsis Construct update map (ω 1, a 1, e 1 ) (ω 2, a 2, e 2 ) using average perturbation per orbit by smaller mass (ω 1,a 1,e 1 ) 5
Kicks at periapsis Construct update map (ω 1, a 1, e 1 ) (ω 2, a 2, e 2 ) using average perturbation per orbit by smaller mass (ω 1,a 1,e 1 ) (ω 2,a 2,e 2 ) 6
Kicks at periapsis Cumulative effect of consecutive gravity assists can be dramatic. 7
Not hyperbolic swing-by Occur outside sphere of influence (Hill radius) not the close, hyperbolic swing-bys of Voyager 8
Capture by secondary Dynamically connected to capture thru tubes 9
Starting model: restricted 3-body problem Particle assumed on near-keplerian orbit around m 1 In the frame co-rotating with m 2 and m 1, H rot (l, ω, L, G) = K(L) + µr(l, ω, L, G) G, in Delaunay variables Evolution is Hamitlon s equations: d (l, ω, L, G) = f(l, ω, L, G) dt Jacobi constant, C J = 2H rot conserved along trajectories 1
Change in orbital elements over one particle orbit Picard s method of approximation Let y(t) = x = unperturbed orbital elements Approximate change in orbital elements over one particle orbit is y = t +T t f(x, τ) dτ, where T = period of unperturbed orbit 11
Change in orbital elements over one particle orbit Assume greatest perturbation occurs at periapsis Limits of integration, apoapsis to apoapsis P m 1 m 2 12
Change in orbital elements over one particle orbit Evolution of G (angular momentum) dg dt = µ R ω, Picard s approximation: T/2 R G = µ T/2 ω dt [( = µ π ( ) 3 r sin(ω + ν t(ν)) dν) sin ω G π r 2 ( π ) ] 2 cos(ν t(ν)) dν K = Keplerian energy change over an orbit K = G µ R 13
Energy kick function Changes have form K = µf(ω), f is the energy kick function with parameters K, C J 2 1 f 1 2 1.8.6.4.2.2.4.6.8 1 ω/π 14
Maximum changes on either side of perturber Semimajor Axis vs. Time Δa+ m 1 ω max m 2 ω max Δa Δa Δa+ 2 1 f 1 2 1.8.6.4.2.2.4.6.8 1 ω/π 15
The periapsis kick map (Keplerian Map) Cumulative effect of consecutive passes by perturber Can construct an update map (ω n+1, K n+1 ) = F (ω n, K n ) on the cylinder Σ = S 1 R, i.e., F : Σ Σ where ( ) ( ) ωn+1 ωn 2π( 2(K = n + µf(ω n ))) 3/2 K n + µf(ω n ) K n+1 Area-preserving (symplectic twist) map Example: particle in Jupiter-Callisto system µ = 5 1 5 16
Verification of Keplerian map: phase portrait Keplerian map 17
Verification of Keplerian map: phase portrait Keplerian map numerical integration of ODEs Keplerian map = fast orbit propagator preserves phase space features but breaks left-right symmetry present in original system can be removed using another method (Hamilton-Jacobi) 18
Dynamics of Keplerian map 1.8 1.7 a 1.6. 1.5-1 -.8 -.6 -.4 -.2.2.4.6.8 1 ω/π Resonance zone 1 Structured motion around resonance zones 1 in the terminology of MacKay, Meiss, and Percival [1987] 19
Dynamics of Keplerian map 1.8 1.7 a 1.6. 1.5-1 -.8 -.6 -.4 -.2.2.4.6.8 1 ω/π Resonance zone 2 Structured motion around resonance zones 2 in the terminology of MacKay, Meiss, and Percival [1987] 2
Large orbit changes via multiple resonance zones multiple flybys for orbit reduction or expansion P m 1 m 2 21
Large orbit changes, Γ n = F n (Γ ) ω/π.1 Δa.5 Γ -.5-1 -.8 -.6 -.4 -.2.2.4.6.8 1 22
Large orbit changes, Γ n = F n (Γ ) ω/π.1 Δa.5 Γ -.5 Γ 2-1 -.8 -.6 -.4 -.2.2.4.6.8 1 23
Large orbit changes, Γ n = F n (Γ ) ω/π.1 Δa.5 Γ -.5 Γ 2-1 -.8 -.6 -.4 -.2.2.4.6.8 1.1.5 -.5 -.1 Γ 2 -.15 Γ 1.5.1.15.2.25 24
Large orbit changes, Γ n = F n (Γ ) ω/π.1 Δa.5 Γ -.5 Γ 2-1 -.8 -.6 -.4 -.2.2.4.6.8 1.1.5 -.5 -.1 Γ 2 -.15 Γ 1.5.1.15.2.25 Γ 1 -.5 -.1 -.15 Γ 13.5.6.7.8.9.1 25
Large orbit changes, Γ n = F n (Γ ) ω/π.1 Δa.5 Γ -.5 Γ 2-1 -.8 -.6 -.4 -.2.2.4.6.8 1.1.5 -.5 -.1 Γ 2 -.15 Γ 1.5.1.15.2.25 Γ 1 -.5 -.1 -.15 Γ 13.5.6.7.8.9.1.5 Γ 1 -.5 -.1 -.15 Γ 25 b -.2.8461.8463.8465.8467.8469 26
Large orbit changes, Γ n = F n (Γ ) ω/π.1 Δa.5 Γ -.5 Γ 2-1 -.8 -.6 -.4 -.2.2.4.6.8 1.1.5 -.5 -.1 Γ 2 -.15 Γ 1.5.1.15.2.25 Γ 1 -.5 -.1 -.15 Γ 13.5.6.7.8.9.1.5 Γ 1 -.5 -.1 -.15 -.2 Γ 25 b.8461.8463.8465.8467.8469 example trajectory 27
Reachable orbits and diffusion 2.4 2.2 Reachable Orbits 2 C J = 2.99 a 1.8 1.6 C J = 3. 1.4 1.2 n 5 1 15 2 25 Diffusion in semimajor axis... increases as C J decreases (larger kicks) 28
Reachable orbits: upper boundary for small µ A rotational invariant circle (RIC) RIC found in Keplerian map for µ = 5 1 6 29
Identify Keplerian map as Poincaré return map Σ (ω,a) F(ω,a) F Poincaré map at periapsis in orbital element space F : Σ Σ where Σ = {l = C J = constant} 3
Relationship to capture around perturber Σ e : Poincare Section at Periapsis P Jupiter exit from jovicentric to moon region Jovian Moon Exit: where tube of capture orbits intersects Σ 31
Relationship to capture around perturber Jovian Moon L 2 exit from jovicentric to moon region Exit: where tube of capture orbits intersects Σ Orbits reaching exit are ballistically captured, passing by L 2 32
Relationship to capture from infinity K > Hyperbolic orbits Elliptical orbits K < Captured from infinity after previous periapsis 1 1 ω/π 33
Summary and conclusions Consecutive gravity assists Reduced to simple lower-dimensional map nice analytical form many phase space features preserved Dynamically connected to capture to and escape from perturber capture to and escape from infinity Applicable to some astronomical phenomena Preliminary optimal trajectory design 34
Final word Extensions: out of plane motion (4D map) control in the presence of uncertainty eccentric orbits for the perturbers multiple perturbers transfer from one body to another E G Consider other problems with localized perturbations? chemistry, vortex dynamics,... Reference: Ross & Scheeres, SIAM J. Applied Dynamical Systems, 27. more at: www.shaneross.com 35