A qualitative analysis of bifurcations to halo orbits
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1 1/28 spazio A qualitative analysis of bifurcations to halo orbits Dr. Ceccaroni Marta ceccaron@mat.uniroma2.it University of Roma Tor Vergata Work in collaboration with S. Bucciarelli, A. Celletti, G. Pucacco
2 hird AstroNet-II School, June 2014, Zielona Gora 2/28 spazio Tale of Contents: The Model The way for the central manifold The Poincaré Maps Frequency analysis Fast Lyapunov Indicators Results Conclusions
3 The Model: A CR3BP with a radiating (β) major primary and an oblate (A) second primary 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 rotating system of reference (uniformly ω = 1, centered in the baricenter of the primaries) P 1 set in [µ, 0, 0], P 2 in [ 1 + µ, 0, 0]. Third AstroNet-II School, June 2014, Zielona Gora 3/28 spazio
4 4/28 spazio Figure: The system in the rotating frame
5 5/28 spazio The Oblateness parameter A = J 2 r 2 e, J 2 dynamical oblateness coefficient and r e is the equtorial radius of the planet (scaled with d P1,P 2 ) A measures the ellipticity of the smaller primary (but relative to the A.U. considered) The solar pressure parameter β: dimensionless parameter sail loading parameter or lightness number of the sail L S A β = 2πGM S c m, Where L S is the solar luminosity. β measures the ratio between the acceleration due to Solar pressure and the gravitational one. Hereafter we will consider 0 < µ 1 2, 0 A 10 4, 0 β 0.5
6 6/28 spazio The Model: The system has equations of motion: Ẍ 2nẎ = Ω X Ÿ + 2nẊ = Ω Y Z = Ω Z Ω = n2 2 (X2 + Y 2 ) + q(1 µ) r 1 + µ r 2 [ 1 + A 2r 2 2 with n = A mean motion 2 r 1 = (X µ) 2 + Y 2 + Z 2 r 2 = (X µ + 1) 2 + Y 2 + Z 2 ( 1 3Z2 r 2 2 )] The equilibria of the system (x e, y e, 0), are the solution of Ω X = 0 Ω Y = 0 Ω Z = 0
7 7/28 spazio Figure: The Collinear equilibria of the system
8 8/28 spazio β = 10 2, A = , µ = , β = 10 2, µ [0, 1 2 ] A [0, 10 4 ] Γi Μ Γ1,Γ2, 1 Γ A µ = , A = , space β [0, 1 2 ] Γ1,Γ2, 1 Γ Β Figure: Collinear points location variyng the parameters; dots γ 1, dash γ 2, continuous line γ 3 /1 γ 3.
9 The way for the central manifold: Shift and scale the equilibrium points by x = X xe γ, y = Y ye γ, z = Z γ with γ distance of (x e, y e, 0) from the closer primary. (symplectic of parameter γ 2 ) Expand in Taylor series 1 1 r 1, r 2 and their powers Search for eigenvalues and eigenvectors of the linearised system with ẍ 2nẏ (n 2 2a)x = 0 ÿ + 2nẋ + ( n 2 + b)y = 0 z + cz = 0, a = q(1 µ) α 3 µ 1+α 3 3Aµ 1+α 5 b = q(1 µ) α 3 + µ 1+α 3 + 3Aµ 2 1+α 5 c = q(1 µ) α 3 + µ 1+α 3 + 9Aµ 2 1+α 5 and α = 1 ± γ for L 1, L 2 respectively. Remark: c > 0, the vertical direction decouples. Third AstroNet-II School, June 2014, Zielona Gora 9/28 spazio
10 10/28 spazio For the µ, β, A considered the collinear are saddle center center Diagonalise the linear part H(x, y, z, p x, p y, p z ) = λ 1 xp x + ω 1 2 (p2 y + y 2 ) + ω 2 2 (p2 z + z 2 ) + H n n 3 where Complexify 2n 2 2a b+ 16an 2 +4a 2 +8bn 2 4ab+b 2 2 λ 1 = ω 1 = 2n2 2a b ω 2 = c. 16an 2 +4a 2 +8bn 2 4ab+b 2 2 obtaining x = q 1, y = q 2+ip 2 2, z = q 3+ip 2 3 p x = p 1, p y = iq 2+p 2 2, p z = iq 3+p 2 3, H(q, p) = λq 1 p 1 + iω 2 q 2 p 2 + iω 3 q 3 p 3 + n 3 H n (q, p),
11 Reduce to Central Manifold: find suitable change of variables to eliminate the hyperbolic direction. Lie series method: Given a ordered Hamiltonian H = H 2 + ɛh 3 + ɛ 2 H , with H i = k =i h k 1,k 2,k 3,k 4,k 5,k 6 q k 1 1 qk 2 2 qk 3 3 pk 4 1 pk 5 2 pk 6 3, finds a canonic change of variables W = W 1 + ɛw 2 + ɛ 2 W , which maps H into an equivalent H = H 2 + ɛ H 3 + ɛ 2 H plus a reminder R O(ɛ n+1 ) such that up to order n H satisfyes some properties (the rule ). Central manifold: p 1 and q 1 only appear with the same exponent. hyperbolicity pertains in q 1, p 1 set q(0) = p(0) = 0 s.t. t q1(t) = p1(t) = 0 Decomplexify: H(y, z, p y, p z ) = ω 1 2 (p2 y+y 2 )+ ω 2 2 (p2 z+z 2 )+ H 3 (y, z, p y, p z )+ H 4 (y, z, p y, p z )... Third AstroNet-II School, June 2014, Zielona Gora 11/28 spazio
12 The Poincaré Maps: Central manifold + Decomplexy H(y, z, p y, p z ) = ω 1 2 (p 2 y+y 2 )+ ω 2 2 (p 2 z+z 2 )+ H 3 (y, z, p y, p z )+ H 4 (y, z, p y, p z )... Fix the Energy E of the system (i.e the value of the Hamiltonian) p y0 = 0 z 0 = 0 y 0 spans from 0 to the horyzontal Lyapunov p z0 is fixed Integrate the system and keep track whenever the resulting orbit hits the z = 0 surface. Third AstroNet-II School, June 2014, Zielona Gora 12/28 spazio
13 13/28 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
14 13/28 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
15 13/28 spazio E=0.2 E=0.35 Figure: Sun-Vesta-spacecraft, y, p y plane Poincaré map, β = 0, A = 0
16 14/28 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
17 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 = Third AstroNet-II School, June 2014, Zielona Gora 15/28 spazio
18 16/28 spazio Frequency analysis Central manifold+decomplexification: H(y, z, p y, p z ) = ω 1 2 (p2 y +y 2 )+ ω 2 2 (p2 z +z 2 )+ H 3 (y, z, p y, p z )+ H 4 (y, z, p y, p z ) Pass to Action-Angle variables (J = (J y, J z ), θ = (θ y, θ z )): p y = 2J y cos θ y y = 2J y sin θ y p z = 2J z cos θ z z = 2J z sin θ z, Ĥ(J, θ) = h(j) + ɛf(j, θ) = ω J + Ĥ3(J, θ) + Ĥ4(J, θ), with ω = ( h(j) θ y, h(j) θ z )
19 17/28 spazio Frequency analysis Apply perturbation theory (one iteration): Ĥ(J, θ) = h(j) + ɛf(j, θ) = h(j) + ɛ f(j) + ɛ f(j, θ), f(j) = 1 2π 4π 2 0 2π 0 f(j, θ)dθ y dθ z Search a change of variables of generating function Φ(J, θ), ( ) n J = J + ɛ Φ(J,θ), θ = θ + ɛ Φ(J,θ) θ J such that: ( ) ( ) nĥ(j, θ ) = h J + ɛ Φ(J,θ) θ + ɛf J + ɛ Φ(J,θ) θ, θ = h(j ) + ɛω(j ) Φ(J,θ) θ + ɛ f(j ) + ɛ f(j, θ) + O(ɛ 2 ). Adjust the change of variables such that the red terms cancel. h(j ) + ɛ f(j ) is now integrable with some frequencies ω (J ).
20 18/28 spazio Frequency analysis Frequencies are translated back to the variables J y, J z, θ y, θ z ω 2nd order y, ω 2nd order z Define ω r := ω2 nd order y ω 2nd order z Fix the Energy E of the system p y0 = 0 z 0 = 0 y 0 spans from 0 to the horyzontal Lyapunov p z0 is fixed ( Jy 0 = 2y 0 )
21 19/28 spazio E=0.2 Ω_r Jy_ E=0.5 Ω_r Jy_ E=0.35 Ω_r space Jy_0 Figure: J y0 /ω r Sun-Vesta-spacecraft β = 0, A = 0
22 19/28 spazio Recall
23 20/28 spazio E=0.04 Ω_r E=0.1 Ω_r Jy_0 Ω_r Jy_ E=0.05 E=0.4 Ω_r Jy_0 Jy_0 Figure: J y0 /ω r Sun-Vesta-spacecraft, β = 10 2, A =
24 20/28 spazio Recall
25 hird AstroNet-II School, June 2014, Zielona Gora 21/28 spazio Fast Lyapunov Indicators (FLI) Definition (Fast Lyapunov Indicator) Given a Hamiltonian system with H(ξ), ξ = [y, z, p y, p z ] T, s.t. Define H = ξ = A(ξ), ( ) A ξ A(ξ) = [ H p y, H, H p z y, H z ]T Solve the system of variational equations { ξ = A(ξ) η = Hη, η = [η 1, η 2, η 3, η 4 ] T for a fixed set of initial conditions ξ 0 = (y 0, z 0, p y0, p z0 ) and η 0 = (η 10, η 20, η 30, η 40 )
26 hird AstroNet-II School, June 2014, Zielona Gora 21/28 spazio Fast Lyapunov Indicators (FLI) Definition (Fast Lyapunov Indicator) Given a Hamiltonian system with H(ξ), ξ = [y, z, p y, p z ] T, s.t. Define H = ξ = A(ξ), ( ) A ξ A(ξ) = [ H p y, H, H p z y, H z ]T H = y ( H(y,z,p y,p z) p y )... y ( H(y,z,p y,p z) z ). p z ( H(y,z,p y,p z) p y )... p z ( H(y,z,p y,p z) z )
27 hird AstroNet-II School, June 2014, Zielona Gora 21/28 spazio Fast Lyapunov Indicators (FLI) Definition (Fast Lyapunov Indicator) Given a Hamiltonian system with H(ξ), ξ = [y, z, p y, p z ] T, s.t. Define H = ξ = A(ξ), ( ) A ξ A(ξ) = [ H p y, H, H p z y, H z ]T Solve the system of variational equations { ξ = A(ξ) η = Hη, η = [η 1, η 2, η 3, η 4 ] T for a fixed set of initial conditions ξ 0 = (y 0, z 0, p y0, p z0 ) and η 0 = (η 10, η 20, η 30, η 40 )
28 Fast Lyapunov Indicators (FLI) Then the Fast Lyapunov Indicator of the system for ξ 0, η 0 at time T is: { } F LI(ξ 0, η 0, t) := sup log η(t) 0<t T where log η(t) is called Lyapunov Characteristic Exponent. FLIs are chaos indicators: they measure the divergence between orbits starting closeby ( directional!). Fix the Energy E of the system z 0 = 0 y 0 spans from 0 to the horyzontal Lyapunov spans from 0 to the horyzontal Lyapunov p y0 p z0 is fixed η 0 = [1, 0, 0, 0] T Third AstroNet-II School, June 2014, Zielona Gora 22/28 spazio
29 23/28 spazio Comments about FLIs Consider a D-sphere of states transformed by the system dynamics in a D-ellipsoid, Lyapunov Exponents measure the divergence of two orbits starting nearby by considering the expanding/contracting relation between the D-sphere and the D-Ellipsoid Highly diverging orbits means chaos Figure: Cretits: Chaos and order in biomedical rhythms J. Braz. Soc. Mech. Sci. & Eng. vol.27 no.2 Print version ISSN (2005)
30 E=0.2 E=0.35 E=0.5 space Figure: FLI Sun-Vesta-spacecraft β = 0, A = 0 Third AstroNet-II School, June 2014, Zielona Gora 24/28 spazio
31 24/28 spazio p_y y Figure: Bifurcation Value: FLI Sun-Vesta-spacecraft, E = 0.35 β = 0, A = 0
32 24/28 spazio Recall
33 E=0.04 E=0.05 E=0.1 E=0.4 Figure: FLI Sun-Vesta-spacecraft, β = 10 2, A = Third AstroNet-II School, June 2014, Zielona Gora 25/28 spazio
34 25/28 spazio p_y y Figure: Bifurcation Value: Sun-Vesta-spacecraft, E = 0.05 β = 10 2, A =
35 25/28 spazio Recall
36 Results Analytical Result β A L 1 L 2 L 1 L 2 E-M E-M S-E S-E S-V S-V Table: Energy tresholds for halo bifurcation. The analytical result have been obtained implementing a Resonant Normal Form (after the Central Manifold Reduction) see [1] [1] Lissajous and halo orbits in the restricted three body problem, Celletti A., Pucacco, G., Stella D. Preprint (2014). 26/28 spazio Third AstroNet-II School, June 2014, Zielona Gora
37 27/28 spazio Coclusions Numerical techniques such as Poincar e Maps, Frequancy Analysis and FLI, provide a fast and accurate qualitative description of the bifurcations to halo orbits. For the circular approximation the oblateness parameter (in the range of concievable values for the solar system) essentially does not affect the bifurcations thresholds for halo orbits. The inclusion in the system of SRP deeply affects for relatively small values of the mass ratio µ = m 2 m 1 +m 2 the halo bifurcation values, actually enabling (i.e. making it appear at reasonable values of the energy) second and third bifurcations.
38 28/28 spazio
39 28/28 spazio Comments about bifurcations When entering the synchronous resonance, normal modes (i.e. the horyzontal and vertical Lyapunov) lose their stability and Halo (or other) families of orbits arise this bifurcation. Normal modes can get stable again through a further bifurcation. The first bifurcation is the one of the halo family from the planar Lyapunov orbit, which becomes unstable. At the second bifurcation the Lyapunov orbit regains its stability, another family of orbits is born, which, for the case examined, is unstable. The halo keep on existing. The third bifurcation is the one the vertical Lyapunov, when the unstable families (located on the vertical axis of the plots above) collapse on the vertical Lyapunov, which loses its stability.
40 28/28 spazio Comments about FLIs FLI are directional white line in the middle Usually FLIs are seen as a B&W map The external Lyapunov is dark also when it is unstable just for a graphical problem.
41 28/28 spazio The mean motion Denote by M as the mass and A oblateness coefficient of the primary, r distance of P from the primary and H is the constant angular momentum. The effective potential is V eff (r) = H2 M 2r 2 r M A, The solutions of 2r 3 V eff H2 (r) = + M + 3M A r 3 r 2 2r 4 = 0 are r = H2 ± H 4 6M A 2M. Being A small, r H2 M (1 3 M A 2 ). H 2 Assuming the orbit of P circular, say r = a, we have that H 2 = n 2 a 4, thus: n 2 = A. Note that the definition of the angular momentum changes to h = q(1 e 2 ), while Kepler s third law becomes n 2 p a 3 p = q, with n p, a p the mean motion and the semimajor axis of the particle.
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