Dynamical Systems and Space Mission Design Jerrold Marsden, Wang Koon and Martin Lo Wang Sang Koon Control and Dynamical Systems, Caltech koon@cds.caltech.edu
Halo Orbit and Its Computation From now on, we will focus on 3D CR3BP. We will put more emphasis on numerical computations, especially issues concerning halo orbit missions, such as Genesis Discovery Mission Outline of Lecture 5A and 5B: Importance of halo orbits. Finding periodic solutions of the linearized equations. Highlights on 3rd order approximation of a halo orbit. Using a textbook example to illustrate Lindstedt-Poincarémethod. Use L.P. method to find a 3rd order approximation of a halo orbit. Finding a halo orbit numerically via differential correction. Orbit structure near L 1 and L 2
Importance of Halo Orbits: Genesis Discovery Mission Genesis spacecraft will collect solar wind from a L 1 halo orbit for 2 years, return those samples to Earth in 23 for analysis. Will contribute to understanding of origin of Solar system. 1E+6 5 y(km) L 1 L 2 5-5 z 1E+6-1E+6-1E+6 1E+6 x(km) 1E+6 5 y -5-1E+6 x Y Z X 1E+6 1E+6 5 5 z(km) L 1 L 2 z(km) -5-5 -1E+6-1E+6-1E+6 1E+6 x(km) -1E+6 1E+6 y(km)
Important of Halo Orbits: Genesis Discovery Mission A L 1 halo orbit (1.5 million km from Earth) provides uninterrupted access to solar wind beyond Earth s magnetoshphere. 1E+6 5 y(km) L 1 L 2 5-5 z 1E+6-1E+6-1E+6 1E+6 x(km) 1E+6 5 y -5-1E+6 x Y Z X 1E+6 1E+6 5 5 z(km) L 1 L 2 z(km) -5-5 -1E+6-1E+6-1E+6 1E+6 x(km) -1E+6 1E+6 y(km)
Importance of Halo Orbits: ISEE-3 Mission Since halo orbit is ideal for studying solar effects on Earth, NASA has had and will continue to have great interest in these missions. The first halo orbit mission, ISEE-3, was launched in 1978. ISEE-3 spacecraft monitored solar wind and other solar-induced phenomena, such as solar radio bursts and solar flares, about a hour prior to disturbance of space enviroment near Earth.
Importance of Halo Orbits: Terrestial Planet Finder JPL has begun studies of a TPF misson at L 2 involving 4 free flying optical elements and a combiner spacecraft. Interferometry: achieve high resolution by distributing small optical elements along a lengthy baseline or pattern. Look into using a L 2 halo orbit and its nearby quasi-halo orbits for formation flight.
Importance of Halo Orbits: Terrestial Planet Finder The L 2 option offer several advantages: Additional spacecraft can be launched into formation later. The L 2 offers a larger payload capacity. Communications are more efficient at L 2. Observations and mission operations are simpler at L 2.
Importance of Halo Orbits: 3D Dynamical Channels In 3D dynamical channels theory, invariant manifolds of a solid torus of quasi-halo orbits could play similar role as invariant manifold tubes of a Lyapunov orbit. Halo, quasi-halos and their invariant manifolds could be key in understanding material transport throughout Solar system, constructing 3D orbits with desired characteristics..6 Unstable Manifold Forbidden Region.4 Stable Manifold y (rotating frame).2 -.2 -.4 -.6 Sun L 1 Periodic Orbit Stable Manifold.88.9.92.94.96.98 1 x (rotating frame) Forbidden Region Unstable Manifold
3D Equations of Motion Recall equations of CR3BP: Ẍ 2Ẏ =Ω X Ÿ +2Ẋ =Ω Y Z=Ω Z where Ω = (X 2 + Y 2 )/2+(1 µ)d 1 1 +µd 1 2. z S/C d 2 d 1 E x S y 1 µ µ
3D Equations of Motion Equations for satellite moving in vicinity of L 1 can be obtained by translating the origin to the location of L 1 : x =(X 1+µ+γ)/γ, y = Y/γ, z = Z/γ, where γ = d(m 2,L 1 ) In new coordinate sytem, variables x, y, z are scale so that the distance between L 1 and small primary is 1. New independent variable is introduced such that s = γ 3/2 t. y (nondimensional units, rotating frame) 1.5 -.5-1 L 3 S L 1 J m S = 1 - µ m J = µ L 4 L 5 comet L 2 Jupiter's orbit -1 -.5.5 1 x (nondimensional units, rotating frame)
3D Equations of Motion CR3BP equations can be developed using Legendre polynomial P n ẍ 2ẏ (1 + 2c 2 )x = c n ρ n P n ( x x ρ ) n 3 ÿ +2ẋ+(c 2 1)y = c n ρ n P n ( x y ρ ) z + c 2 z = z n 3 n 3 c n ρ n P n ( x ρ ) where ρ = x 2 +y 2 +z 2,andc n =γ 3 (µ+( 1) n (1 µ)( γ 1 γ )n+1 ). Useful if successive approximation solution procedure is carried to high order via algebraic manipulation software programs.
Analytic and Numerical Methods: Overview Lack of general solution motivated researchers to develop semi-analytical method. ISEE-3 halo was designed in this way. See Farquhar and Kamel [1973], and Richardson [198]. Linear analysis suggested existence of periodic (and quasi-periodic) orbits near L 1. 3rd order approximation, using Lindstedt-Poincaré method, provided further insight about these orbits. Differential corrector produced the desired orbit using 3rd order solution as initial guess.
Periodic Solutions of Linearized Equations Periodic nature of solution can be seen in linearized equations: ẍ 2ẏ (1 + 2c 2 )x = ÿ+2ẋ+(c 2 1)y = z+c 2 z= The z-axis solution is simple harmonic, does not depend on x or y. Motion in xy-plane is coupled, has (±α, ±iλ) as eigenvalues. General solutions are unbounded, but there is a periodic solution. y (nondimensional units, rotating frame) 1.5 -.5-1 forbidden region interior region S Zero velocity curve bounding forbidden region L 1 exterior region J L 2 Jupiter region L y (nondimensional units, rotating frame) -1 -.5.5 1 x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)
Periodic Solutions of Linearized Equations Linearized equations has a bounded solution (Lissajous orbit) x = A x cos(λt + φ) y = ka x sin(λt + φ) z = A z sin(νt + ψ) with k =(λ 2 +1+2c 2 )/2λ.(λ=2.86,ν =2.15,k =3.229.) Amplitudes, A x and A z, of in-plane and out-of-plane motion characterize the size of orbit.
Periodic Solutions of Linearized Equations If frequencies are equal (λ = ν), halo orbit is produced. But λ = ν only when amplitudes A x and A z are large enough that nonlinear contributions become significant. For ISEE3 halo, A z = 11, km, A x = 26, km and A y = ka x = 665, km.
Halo Orbits in 3rd Order Approximation Halo orbit is obtained only when amplitudes A x and A z are large enough that nonlinear contributions make λ = ν. Lindstedt-Poincaré procedure has been used to find periodic solution for a 3rd order approximation of PCR3BP system. ẍ 2ẏ (1 + 2c 2 )x = 3 2 c 3(2x 2 y 2 z 2 ) +2c 4 x(2x 2 3y 2 3z 2 )+o(4), ÿ +2ẋ+(c 2 1)y = 3c 3 xy 3 2 c 4y(4x 2 y 2 z 2 )+o(4), z + c 2 z = 3c 3 xz 3 2 c 4z(4x 2 y 2 z 2 )+o(4). Notice that for periodic solution, x, y, z are o(a z )witha z << 1 in normalized unit.
Halo Orbits in 3rd Order Approximation Lindstedt-Poincaré method: It is a successive approximation procedure. Periodic solution of linearized equation (with λ = ν) will form the first approximation. Richardson used this method to find the 3rd order solution. x = a 21 A 2 x + a 22 A 2 z A x cos τ 1 +(a 23 A 2 x a 24 A 2 z)cos2τ 1 +(a 31 A 3 x a 32 A x A 2 z)cos3τ 1, y = ka x sin τ 1 +(b 21 A 2 x b 22 A 2 z)sin2τ 1 +(b 31 A 3 x b 32 A x A 2 z)sin3τ 1, z = δ m A z cos τ 1 +δ m d 21 A x A z (cos 2τ 1 3) + δ m (d 32 A z A 2 x d 31A 3 z )cos3τ 1. where τ 1 = λτ + φ and δ m =2 m, m =1,3. Details will be given later. Here, we will provide some highlights.
Halo Orbit Amplitude Constraint Relationship For halo orbits, we have amplitude constraint relationship l 1 A 2 x + l 2 A 2 z + =. For halo orbits about L 1 in Sun-Earth system, l 1 = 1.5965314, l 2 =1.7498 and =.29221444425. Halo orbit can be characterized completely by A z. ISEE-3 halo orbit had A z = 11, km.
Halo Orbit Amplitude Constraint Relationship For halo orbits, we have amplitude constraint relationship l 1 A 2 x + l 2 A 2 z + =. Minimim value for A x to have a halo orbit (A z > ) is /l1, which is about 2, km (14% of normalized distance).
Halo Orbit Phase-angle Relationship Bifurcation manifests through phase-angle relationship ψ φ = mπ/2, m =1,3. 2solution branches are obtained according to whether m =1orm=3.
Halo Orbit Phase-angle Relationship Bifurcation manifests through phase-angle relationship: For m =1,A z >. Northern halo. For m =3,A z <. Southern halo. Northern & southern halos are mirror images across xy-plane.
Lindstedt-Poincaré Method: Duffing Equation To illustrate L.P. method, let us study Duffing equation: First non-linear approximation of pendulum equation (with λ =1) q+q+ɛq 3 =. For ɛ =, it has a periodic solution q = a cos t if we assume the initial condition q() = a, q() =. For ɛ, suppose we would like to look for a periodic solution of the form q = ɛ n q n (t) =q (t)+ɛq 1 (t)+ɛ 2 q 2 (t)+ n=
Lindstedt-Poincaré Method: Duffing Equation Finding a periodic solution for Duffing equation: By substituting and equating terms having same power of ɛ, we have a system of successive differential equations: q + q =, q 1 +q 1 = q 3, q 2 +q 2 = 3q 2 q 1, and etc. Then q = acos t, forq () = a, q() =. Since q 1 + q 1 = q 3 = a3 cos 3 t = 1 4 a3 (cos 3t +3cos t), the solution has a secular term q 1 = 3 8 a3 t sin t + 1 32 a3 (cos 3t cos t).
Lindstedt-Poincaré Method: Duffing Equation Due to presence of secure terms, naivemethod suchas expansions of solution in a power series of ɛ would not work. To avoidsecure terms, Lindstedt-Poincaré method Notices that non-linearity alters frequency λ (corr. to linearized system) to λω(ɛ) (λ= 1 in our case). Introduce a new independent variable τ = ω(ɛ)t: t = τω 1 = τ(1 + ɛω 1 + ɛ 2 ω 2 + ). Expand the periodic solution in a power series of ɛ: q = ɛ n q n (τ) =q (τ)+ɛq 1 (τ)+ɛ 2 q 2 (τ)+ n= Rewrites Duffing equation using τ as independent variable: q +(1+ɛω 1 +ɛ 2 ω 2 + ) 2 (q+ɛq 3 )=.
Lindstedt-Poincaré Method: Duffing Equation By substituing q (in power series expansion) into Duffing equation (in new independent variable τ) and equating terms with same power of ɛ, we obtain equations for successive approximations: q + q =, q 1 + q 1 = q 3 2ω 1q, q 2 + q 2 = 3q 2 q 1 2ω 1 (q 1 + q 3 )+(ω2 1 +2ω 2)q, and etc. Potential secular terms can be gotten rid of by imposing suitable values on ω n. The general solution of 1st equation can be written as q = acos(τ + τ ). where a and τ are integration constants.
Lindstedt-Poincaré Method: Duffing Equation Potential secular terms can be gotten rid of by imposing suitable values on ω n. By substituting q = acos(τ + τ ) into 2nd equation, we get q 1 + q 1 = a 3 cos 3 (τ + τ ) 2ω 1 a cos(τ + τ ) = 1 4 a3 cos 3(τ + τ ) ( 3 4 a2 +2ω 1 )acos(τ + τ ). In previous naive method, we had ω 1 and a secular term caused by cos t term. Now if we set ω 1 = 3a 2 /8, we can get rid of cos(τ + τ ) term and the ensuing secular term. Then q 1 = 1 32 a3 cos 3(τ + τ ).
Lindstedt-Poincaré Method: Duffing Equation Potential secular terms can be gotten rid of by imposing suitable values on ω n. Similarly, by substituting q 1 = 32 1 a3 cos 3(τ + τ ) into 3rd equation, we get q 2 + q 2 =( 51 128 a4 2ω 2 )acos(τ + τ ) + (terms not giving secular terms). Setting ω 2 =51a 4 /256, we obtain q 2 free of secular terms, and so on.
Lindstedt-Poincaré Method: Duffing Equation Therefore, to 1st order of ɛ, wehaveperiodic solution q = acos(τ + τ ) + 1 32 ɛ cos 3(τ + τ )+o(ɛ 2 ) = acos(ωt + τ ) + 1 32 ɛ cos 3(ωt + τ )+o(ɛ 2 ). Notice that ω =(1+ɛω 1 +ɛ 2 ω 2 + ) 1 = {1 ɛω 1 1 2 ɛ2 (2ω 2 ω1 2 )+ } =(1 3 8 ɛa2 15 256 ɛ2 a 4 + o(ɛ 3 ). Lindstedt method consists in successive adjustments of frequencies.
Halo Orbit and Its Computation We have covered Importance of halo orbits. Finding periodic solutions of the linearized equations. Highlights on 3rd order approximation of a halo orbit. Using a textbook example to illustrate Lindstedt-Poincarémethod. In Lecture5B, we will cover Use L.P. method to find a 3rd order approximation of a halo orbit. Finding a halo orbit numerically via differential correction. Orbit structure near L 1 and L 2