Bifurcations thresholds of halo orbits

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 1/23 spazio Bifurcations thresholds of halo orbits Dr. Ceccaroni Marta ceccaron@mat.uniroma2.it University of Roma Tor Vergata Work in collaboration with A. Celletti, G. Pucacco

0 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 2/23 spazio Table of Contents: Why the bifurcation energy? Why Halo orbits? The Model and its equilibria Lyapunov and Halo orbits The waltz of coordinates Bifurcation thresholds Results

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 3/23 spazio Why the bifurcation energy? because it is something we have just demonstrated because it is amazing to predict bifucations...not convinced yet, ah?... because this procedure, inverted, provides initial conditions for Halo orbits

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 4/23 spazio Why Halo orbits? 1968, C.C. Conley: unstable collinear points for low-cost interplanetary routes. The method used relies on Mosers version of Lyapunovs theorem and the existence of transit orbits between the primaries. Unfortunately, orbits such as this require a long time to complete a cycle [...] One cannot predict how knowledge will be applied - only that it often is. From 1978 on: ISEE-3, SOHO, WMAP, Genesis, Herschel-Planck, GAIA, Standing on the shoulders of the Giants : Libration point orbits (Lissajous, Lyapunov, Halo) Simò, Gómez, Masdemont, Jorba, Marsden, Lo...

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 5/23 spazio The Model and its equilibria: A CR3BP with a radiating major primary L S A β = 2πGM S c m, L S solar luminosity, A area to mass ratio, c light speed m units of measure scaled: - mass s.t. µ = m P 2 m P1 +m P2 (0, 1 2 ] 1 µ = m P 1 m P1 +m P2 - distance s.t. d P1,P 2 = 1 G(m - time s.t. t scaled = P1 +m P2 ) t ω = 1, G = 1 (grav. const.) d 3 P 1,P 2 synodic frame: rotating system of reference (uniformly ω = 1, centered in the barycenter of the primaries) P 1 set in [µ, 0, 0], P 2 in [ 1 + µ, 0, 0].

Figure: The system in the rotating frame 10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 6/23 spazio

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 7/23 spazio The Hamiltonian of the system: H 0 = 1 2 (P 2 X + P 2 Y + P 2 Z) + Y P X XP Y The system has equations of motion: Ω = (X2 +Y 2 ) 2 + (1 β)(1 µ) r 1 + µ r 2 Ẍ 2Ẏ = Ω X Ÿ + 2Ẋ = Ω Y Z = Ω Z with r 1 = (X µ) 2 + Y 2 + Z 2, and r 2 = (X µ + 1) 2 + Y 2 + Z 2 (1 β)(1 µ) r 1 µ r 2. The equilibria of the system (x e, y e, 0), are the solutions of Ω X = 0 Ω Y = 0 Ω Z = 0

Figure: The Collinear equilibria of the system 10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 8/23 spazio

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 9/23 spazio Shift and scale system of reference: x = X xe γ, y = Y ye γ, z = Z γ with γ distance of equilibria/closest primary Expand in Legendre polynomials H 1 = 1 ( p 2 2 x + p 2 y + p 2 z) + ypx xp y ( ) x c n (µ)ρ n P n ρ ρ = x 2 + y 2 ( + z 2 ) c n (µ) = (±1)n µ + ( 1) n (1 µ)(1 β)γ n+1 γ 3 (1 γ) n+1 n 2 Linearize H 1 around (x e, y e, 0) and diagonalise it: H 2 = λ 1 x p x + i ω 1 2 (ỹ2 + p 2 y) + i ω 2 2 ( z2 + p 2 z) + n 3 λ 1, ω 1, ω 2 R saddle center center. H (2) n

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 10/23 spazio Lyapunov and Halo orbits Lyapunov center theorem. Equilibrium two 1-parameter families of periodic orbits ( planar and vertical Lyapunov ). Energy increases stability change of Lyapunov periodic orbits bifurcation (Halo orbits, ω 1 = ω 2 ). REMARK: Periodic and quasi-periodic orbits in the center manifold inherit hyperbolicity from equilibria have stable/unstable manifolds suitable for transfer trajectories

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 11/23 spazio Lyapunov and Halo orbits Lyapunov center theorem. Equilibrium two 1-parameter families of periodic orbits ( planar and vertical Lyapunov ). Energy increases stability change of Lyapunov periodic orbits bifurcation (Halo orbits, ω 1 = ω 2 ). REMARK: Periodic and quasi-periodic orbits in the center manifold inherit hyperbolicity from equilibria have stable/unstable manifolds suitable for transfer trajectories

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 12/23 spazio The waltz of coordinates Resonant (ω 1 = ω 2 ) Birkhoff Normal Form: Proposition: There exists a canonical transformation (Lie Series) (p, q) (P, Q) mapping H 2 into H 3 = λq 1 P 1 + iω 1 Q 2 P 2 + iω 2 Q 3 P 3 + N n=3 H (3) n where polynomials of degree n R N+1 remainder of degree N + 1 (negligible). H n (3) satisfies some properties ( { REMARK: independent on Q 1, P 1 alone H (3) n (3) (3) H 0, H n } = 0). H (3) n (Q, P ) + R N+1 (Q, P )

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 13/23 spazio Action-angle variables for the quadratic part: Q 1 = I xe iθx P 1 = I xe iθx Q 2 = i I ye iθy P 2 = I ye iθy Q 3 = i I ze iθz P 3 = I ze iθz, REMARK: I y + I z = 0 E I y + I z = const. Central Manifold reduction: - I x = Q 1 P 1 = const 2 d.o.f. - I x = 0 center manifold (up to order N). H 4 = ω 1 I y + ω 2 I z + αi 2 y + βi 2 z + I y I z (σ + 2τ cos(2(θ y θ z ))) +... for suitable coefficients α abcd, and δ = (ω 1 ω 2 )/ω 2 detuning

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 14/23 spazio Bifurcation thresholds Resonant variables: E = I y + I z R = I y ν = θ z ψ = θ y θ z The equations of motion become: E = 0 Ṙ = 2dR(R E) sin(2ψ) +... ν = ω 2 + 2bE + cr dr cos(2ψ) +... ψ = δ + 2aR + ce + d(2r E) cos(2ψ) +... with a = α + β σ, b = β, c = σ 2β, d = 2τ. REMARK: This is a 1-DOF system in which the equilibria correspond to periodic orbits.

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 15/23 spazio Equilibria of the system: Ṙ = 0 for ψ = 0, π and for ψ = ± π 2, ψ = 0 for R 0/π = δ+(c d)e 2(a+d) R ± π = δ+(c+d)e 2 2(a d) Halo orbits: bifurcations of normal modes entering the 1 : 1 resonance. Recall I y, I z 0 and E = I y + I z 0 I y, I z E, yielding Inclined orbits (θ y θ z = 0, π) E E iy Loop orbits (θ y θ z = ±π/2) E E ly δ δ σ 2(α τ) E E iz 2(β τ) σ δ δ σ 2(α+τ) E E lz 2(β+τ) σ

Second order theory Reduce the perturbing function to sixth order. H 4 = ω 1 I y + ω 2 I z + αi 2 y + βi 2 z + I y I z (σ + 2τ cos(2(θ y θ z ))) +α 3300 I 3 y + α 0033 I 3 z + α 1122 I y I 2 z + α 2211 I 2 y I z +2I y I z [α 2013 I z + α 3102 I y ] cos(2(θ y θ z )) +... Following [ 1 ] yields: E (2) ly = E ly + δ 2 γ α 2 γ c 2 (µ 21 3µ 30 ν 21 ), (γ 2( γ+α)) 2 (γ 2( γ+α)) 3 E (2) iy = E iy + δ 2 γ α+2 γ c 2 (µ 21 3µ 30 +ν 21 ), (γ 2( γ+α)) 2 (γ 2( γ+α)) 3 E (2) lz = E lz + δ 2 β c 2 (µ 12 3µ 03 ν 12 ), ( γ 2( γ β)) 2 ( γ 2( γ β)) 3 E (2) iz = E iz + δ 2 β c 2 + (µ 12 3µ 03 +ν 12 ), ( γ 2( γ β)) 2 ( γ 2( γ β)) 3 1 J. Henrard, Periodic orbits emanating from a resonant equilibrium, Cel. Mech. 1, 437-466 (1970) 0 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 16/23 spazio

Figure: Bifurcation Energies function of the mass ratio: L 1 (blue), L 2 (red), L 3 (orange). The dot-dashed part referring to L 3 corresponds to results obtained when the normal form has already reached the optimal order. 10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 17/23 spazio Results Bifurcation Energy 0.40 0.35 0.30 0.25 0.20 0.15 8 6 4 2 0 log Μ

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 18/23 spazio Comparison between results obtained and numerical results (literature) L 1 Hill s case barycenter Sun Earth Moon equal masses µ 0 µ = 3.04 10 6 µ = 0.012 µ = 1/2 I -1.5000005284-1.5004153349-1.5871935227-1.961675 II -1.5000005288-1.5004156609-1.5871946392-1.96153475 III -1.5000005287-1.5004156182-1.5871903100-1.96153625 IV -1.5000005287-1.5004156162-1.5871904695-1.9615365 V -1.5000005287-1.5004156165-1.5871905379-1.9615365 VI -1.5000005287-1.5004156165-1.5871905378-1.9615365 Numerical -1.5000005287-1.5004156135-1.5871945025-1.961535 Table: Results for the analytical bifurcation estimates for L 1 up to a normal form of order 6 and the numerical values (literature), physical energy.

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 19/23 spazio L 2 Hill s case barycenter Sun Earth Moon equal masses µ 0 µ = 3.04 10 6 µ = 0.012 µ = 1/2 I -1.5000005283-1.5004124990-1.5758498606-1.5245097744 II -1.5000005287-1.5004128783-1.5760986316-1.5481915511 III -1.5000005286-1.5004128287-1.5760666830-1.5438635578 IV -1.5000005286-1.5004128337-1.5760712189-1.5448347101 V -1.5000005286-1.5004128331-1.5760710217-1.5446849644 VI -1.5000005286-1.5004128331-1.5760710217-1.5448200770 Numerical -1.5000005286-1.5004128399-1.5760715007-1.5447556912 Table: Results for the analytical bifurcation estimates for L 2 up to a normal form of order 6 and the numerical values (literature), physical energy.

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 20/23 spazio

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 21/23 spazio E=0.2 E=0.35 E=0.5 space Figure: Sun-Vesta-spacecraft, y, p y plane Poincaré map, β = 0, A = 0

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 21/23 spazio E=0.2 E=0.35 E=0.2 E=0.35 Figure: Sun-Vesta-spacecraft, y, p y plane Poincaré map, β = 0, A = 0

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 21/23 spazio E=0.2 E=0.35 Figure: Sun-Vesta-spacecraft, y, p y plane Poincaré map, β = 0, A = 0

10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 22/23 spazio E=0.2 E=0.35 E=0.5 space Figure: Sun-Vesta-spacecraft, y, p y plane Poincaré map, β = 0, A = 0

E=0.04 E=0.05 E=0.1 E=0.4 Figure: Sun-Vesta-spacecraft, y, p y plane Poincaré map, β = 10 2, A = 4.5776 10 14 10 th AstroNet-II Final Meeting, 15 th -19 th June 2015, Tossa Del Mar 23/23 spazio