Dynamics and Stability of Tethered Satellites at Lagrangian Points - PowerPoint PPT Presentation

Deimos and Phobos G D T Mars - Deimos Mars - Phobos Magnitude Value Unit Magnitude Value Unit L T 10 km L T 5 km 22.34 km 16.96 km x E x E 200 m 125 m ∆ x ∆ x 4 3 k ξ k ξ 72 hours 72 hours T T 1000 kg 1000 kg m m 45 deg 45 deg φ φ Λ 0.1 Λ 0.1 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 11/100

A RIADNA P ROGRAM G D T Gentil Introduction ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 12/100

Gentil Introduction G D T G OAL : TO GAIN AN INSIGHT INTO THE DYNAMICS OF FAST ROTATING TS 1. Derive the equations of motion of tethered satellites • CRTBP approximation • inert tethers, constant length • coupled (5-DOF) rotational-traslational motion 2. Fast rotating tethers • Average over the (fast) rotation angle 3. Motion “close” to the smaller primary: Hill’s approach • In some interesting cases, rotational and translational motion decouple! 4. Show relevant characteristics of specific applications Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 13/100

Inertial motion of an inert tether G D T • Only gravitational forces • Dumbbell model ⇒ rigid body dynamics: x ) + 1 � µ T = 1 2 m ( ˙ x · ˙ 2 Ω · I · Ω , V = − � y � d m m • m total mass, I central inertia tensor • x position of c.o.m., y position of mass elements • Ω tether’s rotation vector • µ gravitational parameter of the attracting body � ∂ L � d − ∂ L • Lagrangian approach: L = T − V → ∂q j = 0 , d t ∂ ˙ q j • q j = d q j / d t , q j : generalized coordinates ˙ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 14/100

The dumbbell model G D T • Inertial frame Oxyz • Tether-attached frame: origin G , unit vectors ( u 1 , u 2 , u 3 ) , • u 1 = u , u 2 = k × u / � k × u � , u 3 = u 1 × u 2 , • Ω = ( Ω · u 1 , Ω · u 2 , Ω · u 3 ) T ˙ u = Ω × u ⇒ z m 2 z u y y ϕ x G θ O m 1 x x

Gravitational Potential G D T m 2 L 2 s α dm = ρ L ds L 1 G f dm � µ V = − � y � d m u y x m m 1 r = � x � , cos α = u · x /r u G O 1 • Usual expansion of � y � in Legendre polynomials � L ∞ � n � V = − µ 1 + 2( s/r ) cos α + ( s/r ) 2 = − µm d m � a n P n (cos α ) r � r r m n ≥ 0 • a n = a n ( m 1 , m 2 , m T ) are functions of tether & end masses ( a 0 = 1 , a 1 = 0 ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 16/100

Kinetic analysis G D T T = 1 x ) + 1 • Kinetic energy: 2 m ( ˙ x · ˙ 2 Ω · I · Ω • Rotational motion (tether attached frame): • Inertia tensor (null moment of inertia around u ):   0 0 0 I = mL 2 a 2 = 1 3 sin 2 2 φ − 2Λ 12 mL 2 �   � I = 0 I 0     0 0 I • φ = φ ( m 1 , m 2 , m T ) , Λ = m T /m u � 2 • Ω · ˙ u = 0 ⇒ Ω · I · Ω = I � ˙ • Position of the tether: u = (cos ϕ cos θ, cos ϕ sin θ, sin ϕ ) T = 1 + 1 x 2 + ˙ y 2 + ˙ � ϕ 2 + ˙ θ 2 cos 2 ϕ � • Kinetic energy: z 2 � � 2 m ˙ 2 I ˙ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 17/100

Circular Restricted Three Body Problem G D T ω z m x 1 y x 2 G P � k m P 1 � j O m P 2 � i ℓ ν x ℓ ℓ = 2 ( 1 − ν ) ℓ • + Perturbations acting on the c.o.m (ED forces, size, varying length . . . ) • + Attitude dynamics Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 18/100

Motion relative to the synodic frame G D T 1. Two attractive bodies (primaries). Gravitational forces calculated under two 1) spherical attracting bodies and, 2) small ratios for L assumptions: r i • Gravitational potential V i ( i = 1 , 2 ) of each primary: � � V i = − mµ i 1 + ( L r i ) 2 a 2 P 2 (cos α i ) + O ( L r i ) 3 , cos α i = u · x i /r i r i 2. Rotating frame ⇒ inertia forces • Constant rotation rate ω in the z axis direction • Generalized potential includes • Coriolis term: m ω ( x ˙ y − y ˙ x ) • Rotation term: m ( ω 2 / 2) ( x 2 + y 2 ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 19/100

Lagrangian function and non-dimensional variables G D T 1. Lagrangian function x ) + ω 2 m = 1 L 2 ( x 2 + y 2 ) + µ 1 r 1 + µ 2 x 2 + ˙ y 2 + ˙ z 2 ) + ω ( x ˙ 2( ˙ y − y ˙ r 2 � µ 1 � P 2 (cos α 1 ) + µ 2 P 2 (cos α 2 ) + 1 + L 2 a 2 ϕ 2 + ˙ θ 2 cos 2 ϕ ] + O ( L 3 /r 3 ) 2[ ˙ r 3 r 3 1 2 Here, a 2 = I / ( m L 2 ) , 0 < a 2 < 1 / 4 2. Generalized coordinates: x, y, z, ϕ, θ 3. We use non-dimensional variables with characteristics values: total mass of the system, distance ℓ between primaries, τ = ω t (primaries’ orbit period = 2 π ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 20/100

Governing equations for a constant length inert librating tether G D T � � 1 + x ν (1 − ν ) ν ρ 3 ) = ǫ 2 [3( ρ 1 · u )( � ρ 5 [3( ρ · u )( � − 2 ˙ y − (1 − ν )(1 − ) − x (1 − i · u ) − (1 + x ) S 2 ( α 1 )] + i · u ) − xS 2 ( α 2 )] 0 ρ 3 ρ 5 1 1 � � (1 − ν ) ν (1 − ν ) ν ρ 3 ) = ǫ 2 [3( ρ 1 · u )( � ρ 5 [3( ρ · u )( � y + 2 ˙ ¨ x + y − y (1 − j · u ) − yS 2 ( α 1 )] + j · u ) − yS 2 ( α 2 )] 0 ρ 3 ρ 5 1 1 � � ν (1 − ν ) (1 − ν ) ν = ǫ 2 [3( ρ 1 · u )( � ρ 5 [3( ρ · u )( � z + z ¨ ρ 3 + z k · u ) − zS 2 ( α 1 )] + k · u ) − zS 2 ( α 2 )] 0 ρ 3 ρ 5 1 1 (1 − ν ) ν θ cos ϕ − 2(1 + ˙ ¨ θ ) ˙ ϕ sin ϕ = 3 ( ρ 1 · u )( ρ 1 · u 2 ) + 3 ρ 5 ( ρ · u )( ρ · u 2 ) ρ 5 1 (1 − ν ) ν θ ) 2 sin ϕ cos ϕ = 3 ϕ + (1 + ˙ ¨ ( ρ 1 · u )( ρ 1 · u 3 ) + 3 ρ 5 ( ρ · u )( ρ · u 3 ) ρ 5 1 L 3 (5 cos 2 α i − 1) ε 2 ) 2 a 2 , ν ≡ reduced mass of the small primary , 0 = ( S 2 ( α i ) = ℓ 2 ρ 1 ρ cos α 1 = · u , cos α 2 = · u ρ 1 ρ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 21/100

Rotating Tether Attitude Dynamics G D T The Attitude Dynamics of the tether can H G be analyzed more intuitively using the Newton-Euler formulation. The core of the m 1 u 3 analysis is the angular momentum equa- tion: d H G = M G G dt u 2 Here M G is the resultant of the external u 1 ≡ u torques applied to the center of mass G of the tethered system and H G is the angu- lar momentum of the system, at G , in the m 2 motion relative to the center of mass. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 22/100

Rotating Tether Attitude Dynamics G D T In the Dumbbell Model the angular velocity of the tether and its angular momentum H G are: � Ω = u × ˙ u + u ( u · Ω ) , H G = I ◦ Ω = I ( u × ˙ u ) Attached to the tether we take a reference frame Gu 1 u 2 u 3 where the unit vectors are given by: u ˙ u 1 = u , u 2 = u � , u 3 = u 1 × u 2 � ˙ In this body frame the angular momentum is: H G = I ˜ ˜ Ω ⊥ u 3 , Ω ⊥ = � u × ˙ u � = � ˙ u � where and the angular momentum equation takes the form: d ˜ Ω ⊥ Ω ⊥ d u 3 = 1 dt u 3 + ˜ I M G dt Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 23/100

Rotating Tether Attitude Dynamics G D T This way we obtain the following equa- tions: z 0 φ 1 d u 1 = ˜ Ω ⊥ u 2 φ 2 dt u 3 d u 3 = M 2 u 2 ˜ dt u 2 Ω ⊥ I d ˜ φ 3 Ω ⊥ = M 3 s dt I φ 1 G We introduce the Tait-Bryant angles (or y 0 Cardan angles) as generalized coordinates. Notice that, from a mathematical point x 0 of view, we have a four-orden system of ODE’S. u 1 ≡ u Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 24/100

Rotating Tether Attitude Dynamics G D T In terms of the Bryant angles the unit vectors u 1 and u 3 are given by u 1 = (cos φ 2 cos φ 3 , cos φ 1 sin φ 3 + sin φ 1 sin φ 2 cos φ 3 , sin φ 1 sin φ 3 − cos φ 1 sin φ 2 cos φ 3 ) u 3 = (sin φ 2 , − sin φ 1 cos φ 2 , cos φ 1 cos φ 2 ) The equations governing the time evolution of the Bryant angles are: dφ 1 = − M 2 1 · cos φ 3 ω 2 I · dτ Ω ⊥ cos φ 2 dφ 2 = − M 2 1 ω 2 I · · sin φ 3 dτ Ω ⊥ dφ 3 = Ω ⊥ + M 2 1 ω 2 I · · cos φ 3 tan φ 2 dτ Ω ⊥ d Ω ⊥ = M 3 ω 2 I dτ where Ω ⊥ = ˜ Ω ⊥ /ω is the non-dimensional form of ˜ Ω ⊥ . These equations should be integrated from the initial conditions: at τ = 0 : φ 1 = φ 10 , φ 2 = φ 20 , φ 3 = φ 30 , Ω ⊥ = Ω ⊥ 0 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 25/100

Fast Rotating Tether G D T For a fast rotating tether the value of Ω ⊥ is large, Ω ⊥ ≫ 1 . There are two characteristic times: 1) the period of the orbital dynamics of both primaries, τ = O (1) and 2) the period of the intrinsic rotation of the tether τ 1 = Ω ⊥ τ = O (1) For example, consider the governing equation u 3 ≡ v 3 dφ dτ = f ( φ 1 , φ 2 , φ 3 , Ω ⊥ ) Its averaged equations form is: u 2 G � 2 π φ 3 < dφ dτ > = 1 f ( φ 1 , φ 2 , φ 3 , Ω ⊥ ) dτ 1 2 π 0 v 2 v 1 To integrate the function f ( φ 1 , φ 2 , φ 3 , Ω ⊥ ) the slow φ 3 u 1 ≡ u variables ( φ 1 , φ 2 , Ω ⊥ ) take constant values and the fast variable φ 3 is approximated by φ 3 ≈ τ 1 + φ 30 . Stroboscopic frame Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 26/100

Averaged equations for an inert fast rotating tether G D T  � ǫ 2 � � � 1 + x ν ν − 1 n 1 ˜ ν n ˜ 0   − 2 ˙ y − (1 − ν )(1 − ) − x (1 − ρ 3 ) = 3(sin φ 2 + ˜ n ) sin φ 2 − (1 + x ) S 2( ) 3˜ n sin φ 2 − xS 2( ) − ρ 3 ρ 5 ρ 5 2 ρ 1 ρ   1 1 ǫ 2  (1 − ν ) 1 − ν � ˜ � � ˜ � ν n 1 ν n 0  y + 2 ˙ ¨ x + y − y (1 − ρ 3 ) = 3(sin φ 2 + ˜ n ) cos φ 2 sin φ 1 + yS 2( ) + 3˜ n cos φ 2 sin φ 1 + yS 2( ) ρ 3 ρ 5 ρ 5 2 ρ 1 ρ  1 1  ǫ 2 � � � ν (1 − ν ) ν − 1 n 1 ˜ ν n ˜ 0  z + z ¨ ρ 3 + z = 3(sin φ 2 + ˜ n ) cos φ 2 cos φ 1 − zS 2( ) 3˜ n cos φ 2 cos φ 1 − zS 2( ) − ρ 3 ρ 5 ρ 5 2 ρ 1 ρ  1 1 � sin φ 2 cos φ 1 n )(cos φ 2 + ˜ n ˜ � cos φ 1 cos φ 2 3 (1 − ν ) (sin φ 2 + ˜ b ) 3 ˜ dφ 1 ν b = 1 + + + ρ 5 ρ 5 cos φ 2 dτ 2Ω ⊥ cos φ 2 2 Ω ⊥ 1 cos φ 2 2 Ω ⊥   � � dφ 2 cos φ 1 cos φ 2 3 (1 − ν ) (sin φ 2 + ˜ n ) 3 ν n ˜   = − 1 + sin φ 1 + ( y cos φ 1 + z sin φ 1) + ρ 5 ρ 5 dτ 2Ω ⊥ 2 Ω ⊥ 2 Ω ⊥   1 d Ω ⊥ = sin φ 1 sin φ 2 cos φ 1 dτ n, ˜ where the quantities (˜ b ) and the fast variable φ 3 are given by n = x sin φ 2 − ( y sin φ 1 − z cos φ 1) cos φ 2 ˜ ˜ b = x cos φ 2 + ( y sin φ 1 − z cos φ 1) sin φ 2 � sin2 φ 2 cos φ 1 n )(cos φ 2 + ˜ n ˜ � dφ 3 cos φ 1 cos φ 2 3 (1 − ν ) (sin φ 2 + ˜ b ) 3 ν ˜ b = Ω ⊥ − 1 + − tan φ 2 − ρ 5 tan φ 2 ρ 5 2Ω ⊥ cos φ 2 2 Ω ⊥ 2 Ω ⊥ dτ 1 Remember: for a FRT the non-dimensional variable Ω ⊥ is a large number, that is, Ω ⊥ ≫ 1 . Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 27/100

The Hill approach for fast rotating tethers G D T • Frequently, the parameter ν is small. The Hill approximation give us the right order of magnitude of distance to the small primary. We introduce this approximation by means of the change of variables: x = ℓ ν 1 / 3 ξ, y = ℓ ν 1 / 3 η, z = ℓ ν 1 / 3 ζ, ρ = ℓ ν 1 / 3 ˆ ρ • For a FRT the parameter Ω ⊥ is large ( Ω ⊥ ≫ 1 ). For example, taking Jupiter and Io as primaries, T p ≈ 1 . 769 days. If T F RT ≈ 25 minutes then Ω ⊥ > 100 . It is reasonable to take the limit Ω ⊥ → ∞ in the governing equations. � 2 � • The tether’s characteristic length, λ = a 2 L appears in a natural way in the ν 1 / 3 ℓ 2 problem (here ν = µ/ ( µ 1 + µ ) is the reduced mass of the small primary or central body). Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 28/100

The Hill approach for fast rotating tethers G D T � � ˜ η = (3 − 1 ρ 3 ) ξ − λ N ¨ 3 ˜ ξ − 2 ˙ N sin φ 2 − ξS 2 ( ρ ) ρ 5 ˆ ˆ ˆ � � ˜ ξ = − η ρ 3 + λ N η + 2 ˙ 3 ˜ ¨ N cos φ 2 sin φ 1 + ηS 2 ( ρ ) ρ 5 ˆ ˆ ˆ � � ˜ ζ = − ζ (1 + 1 ρ 3 ) − λ N ¨ 3 ˜ N cos φ 2 cos φ 1 − ζS 2 ( ρ ) ρ 5 ˆ ˆ ˆ dφ 1 = cos φ 1 tan φ 2 dτ dφ 2 = − sin φ 1 dτ ˜ N = ξ sin φ 2 − ( η sin φ 1 − ζ cos φ 1 ) cos φ 2 These equations should be integrated from the initial conditions: ξ = ξ 0 , η = η 0 , ζ = ζ 0 , ˙ ξ = ˙ η 0 , ˙ ζ = ˙ at τ = 0 : ξ 0 , ˙ η = ˙ ζ 0 , φ 1 = φ 10 , φ 2 = φ 20 If the initial conditions are φ 10 = φ 20 = 0 ⇒ φ 1 ( τ ) = φ 2 ( τ ) ≡ 0 . Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 29/100

The Hill approach for fast rotating tethers G D T T ETHER ROTATION PARALLEL TO THE PRIMARIES PLANE : 1 − 5 ζ 2 3 ξ − ξ ρ 3 − 3 2 λ ξ � � ξ ′′ − 2 η ′ = ρ 5 ρ 2 1 − 5 ζ 2 − η ρ 3 − 3 2 λ η � � η ′′ + 2 ξ ′ = ρ 5 ρ 2 3 − 5 ζ 2 − ζ − ζ ρ 3 − 3 2 λ ζ � � ζ ′′ = ρ 5 ρ 2 Formally equal to a Hill- J 2 problem: • J 2 helps to stabilize high inclination orbits • we have control over L and, therefore, over λ ! • λ might take “high” values with feasible tether lengths • λ = 0 . 01 ⇒ Metis: L ≈ 7 . 2 km ( R ⊕ ≈ 50 km) • ...when consistent with our O ( L/r ) 3 approximation Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 30/100

Hill+oblateness long-term dynamics G D T • Studied since the dawn of the space era (Lidov, Kozai) • β : ratio J 2 to third body perturbations ( β 2 ∝ a − 5 ) 140 130 120 I ( deg ) 110 100 90 0.0 0.5 1.0 1.5 2.0 2.5 3.0 β

A RIADNA P ROGRAM G D T The END of this Gentil Introduction ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 32/100

A RIADNA P ROGRAM G D T Inert Tethers and Periodic Motions ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 33/100

Inert tethers and periodic motions G D T In the CRTBP for a mass particle there are three well known families of periodic mo- Vertical oscillations tion originated from the collinear points: ◮ 1) Lyapunov periodic orbits, starting from highly unstable small ellipses with retro- Lyapunov orbits grade motion around the collinear points ◮ 3) Eight-shaped orbits ◮ 2) Halo orbits bifurcated from the Lya- punov family Halo orbits Figure is an sketch of three starting orbits We carried out numerical explorations of the tethered-satellite problem. More precisely, we discuss how the known periodic solutions of the Hill problem are modified in the case of tethered satellites. The most favorable configuration is found for tethers rotating parallel to the plane of the primaries, a case in which the attitude of the tethered-satellite remains constant on average. In this case, the effect produced by a non-negligible tether’s length is equivalent to introducing a J 2 perturbation on the primary at the origin, or intensifying it if the primary at the origin is an oblate body, and it is shown that either lengthening or shortening the tether may lead to orbit stability. Promising results are found for eight-shaped orbits, but regions of stability are also found for Halo orbits. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 34/100

Summary G D T • Lyapunov orbits: there is no beneficial effects on orbit stability by using tethered satellites • Eight-shaped orbits: the presence of the tether in general, reduces instability. Stability regions are found for certain lengths of the tether when the Jacobi constant remains below roughly C = 3 . • Halo orbits: stability regions for the tethered satellites are generally found, even when starting from highly unstable orbits of the Hill problem. The general effect of lengthening the tether is to narrow and twist the Halo, sometimes converting the Halo in a thin, eight-shaped orbit. However, depending on the Jacobi constant value of the starting Halo orbit, the required length of the tether to stabilize the orbit may be small and the stabilized orbit may retain most of the Haloing characteristics. Two different motivations to use a tether: 1) to stabilize unstable Halo orbits and 2) to take advantage of the tether; it enjoys the stability properties of some stable Halo orbits (for values of C close to the lower limit). Therefore, at first sight, the most promising regions are close to the Halo family’s stability region of the Hill problem , where tethers of few tenths of kilometers may stabilize Halo orbits of the simplified model. For specific applications one needs to check that the tether length required for stabilization is feasible, say of few tenths of km, and that the corresponding Halo orbit does not impact the central body. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 35/100

Summary G D T 1622 1622 0.97 Jupiter Maximum distance curve Phobos The Moon 215 215 0.78 Distance � Hill units � Europa Stability indices Stability indices k 1 stability curve Minimum distance curve 28 k 1 stability curve Deimos 28 0.58 Enceladus Io 4 4 0.39 0 k 2 stability curve 0 k 2 stability curve 0.2 0.01 � 4 � 4 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.1 0.2 0.3 0.4 0.5 Jacobi constant Minimum radius � Hill problem units � • Left: Stability diagram of the Halo family of the Hill problem (no tether) showing the maximum and minimum distance to the origin. Right: Stability diagram of the Halo family of the Hill problem (no tether) as a function of the minimum distance to the origin. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 36/100

Summary G D T 4 Eros Europa Mercury Io The Moon Enceladus 2 Stability indices k 1 stability curve 0 � 2 k 2 stability curve � 4 0.00 0.05 0.10 0.15 0.20 Minimum radius � Hill problem units � Right: Stability diagram of the Halo family of the Hill problem (no tether) as a function of the minimum distance to the origin. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 37/100

Summary G D T 0.4 0.2 0.4 0.0 0.2 0.0 � 0.2 � 0.2 � 0.4 � 0.4 0.0 0.0 � 0.5 � 0.5 0.0 0.0 0.1 0.2 0.2 0.3 0.4 0.4 • Left: Halo orbit close to Io, stabilized with a tether length of ∼ 70 km. The minimum distance to Io is about one half of it radius. Right: Halo orbit about Eros, stabilized with a tether length of ∼ 110 km. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 38/100

A RIADNA P ROGRAM G D T The END of Inert Tethers and Periodic Motions ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 39/100

A RIADNA P ROGRAM G D T Io Exploration with Electrodynamic Tethers ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 40/100

P OWER AND M ASSES OF SOME RTG’ S G D T Mission Thermal W Electrical W Mass (kg) Pionner 10 2250 150 54.4 Voyager 1 y 2 7200 470 117 Ulysses 4400 290 55.5 Galileo 8800 570 111 Cassini 13200 800 168 P ROBLEMS OF THE RTG’ S • Very high cost: from $40,000 to $400,000 per W • High potential risk • Limited power (mass grows strongly) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 41/100

D EORBITING SATELLITES WITH ELECTRODYNAMIC TETHERS G D T A NY SATELLITE CAN BE DEORBITED WITH AN ELECTRODYNAMIC TETHER WORKING IN THE GENERATOR REGIME The mechanical energy lost in the deorbiting process is � v i ∆ h = a i − a f mµ ∆ h ≈ mµ ∆ h (1 − ∆ h � ∆ h f e ∆ E = ) 2 a 2 2 a f a i a f f and it is invested in several task. Basically: � v f • to attract the electrons from the infinity to the tether � f e • ohmics loss in the tether when the current is flowing • some useful work that we can obtain in the form of electrical energy (charging batteries, for example) Deorbiting a satellite We need: magnetic field and plasma environment. Both are present in Jupiter. T HE ELECTRODYNAMIC TETHER IS ABLE TO RECOVER A SIGNIFICANT FRACTION OF THE MECHANICAL ENERGY LOST DURING THE DEORBITING PROCESS Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 42/100

S CENE OF THE MISSION G D T The plasma environment is co-rotating with Jupiter (as a rigid body); the an- gular velocity is ω J ≈ 1 . 7579956104 × 10 − 4 s − 1 The orbital velocity in a circular or- � µ bit of radius r is v c = r ( µ = 1 . 26686536 × 10 17 m 3 / s 2 ) . It exists an orbital radius for which the orbital period coincides with the sideral pe- riod of Jupiter (about 9.925 hours); it Jupiter inner moonlets is the stationary radius ( r s ) given by r Adrastea = 128971 km , r Metis = 127969 km r s = ( T √ µ 2 3 ≈ 160009 . 4329 km 2 π ) r Amalthea = 181300 km , r Thebe = 221895 km Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 43/100

F UNDAMENTALS OF THE MISSION G D T We tray to deorbit one of the Jupiter moon- � v o lets. We joint the tether and the moonlet � with a cable. F e is the electrodynamic drag acting on the tether. This force is trans- � f e mitted thought the cable to the small Jupiter moon that will be deorbited. − � f e 2∆ Ea 2 � f f e ∆ h = mµ To deorbit Amaltea 1mm ( ∆ h = 1 mm) im- plies a loss of mechanical energy about Amalthea: m = 2 . 09 × 10 18 kg, a f = 181300 km The cable can be removed by using the gravitational ∆ E ≈ 4 × 10 15 J ≈ 1 . 1 × 10 9 Kwh attraction of Amalthea to link the tether system and the small Jupiter moon. The tether system moves From a practical point of view the reserve of in an equilibrium position relative to the primaries: energy is unbounded Jupiter + Amalthea Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 44/100

Equilibrium Positions at the Synodic Frame G D T Thrust Io atraction L 2 L 1 Io J • A permanent tethered observatory at Jupiter. Dynamical analysis , by J. Pel´ aez and D. J. Scheeres , Proceedings of the 17th AAS/AIAA Space Flight Mechanics Meeting Sedona, Arizona, Vol. 127 of Advances in the Astronautical Sciences, 2007, pp. 1307–1330 • On the control of a permanent tethered observatory at Jupiter , by J. Pel´ aez and D. J. Scheeres , Paper AAS07-369 of the 2007 AAS/AIAA Astrodynamics Specialist Conference, Mackinac Island, Michigan, August 19-23, 2007 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 45/100

E QUILIBRIUM P OSITIONS G D T 1.2 R OOT 1 R OOT 2 1 � 3 � 2 χ = 3 4 0.8 ζ e 0.6 χ max χ max 0.4 0.2 χ = 0 χ = 0 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 L 2 L 1 ξ e Equilibrium Positions on the plane ( ξ e , ζ e ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 46/100

E QUILIBRIUM P OSITIONS (A MALTEA ) G D T 1.2 R OOT 1 R OOT 2 In [1] 1 � 3 � 2 χ = 3 4 0.8 ζ e 0.6 χ max χ max 0.4 0.2 χ = 0 χ = 0 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 L 1 L 2 ξ e Comparison with the results summarized in [1] [1] A permanent tethered observatory at Jupiter. Dynamical analysis, by J. P ELÁEZ & D. J. S CHEERES , (Paper AAS07-190) of The 2007 AAS/AIAA Space Flight Mechanics Meeting, Sedona, AZ, January 2007. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 47/100

T ETHER DESIGN IN M ETIS . N UMERICAL A NALYSIS G D T 10 d w = cte L = cte d w = 5 mm W u (kw) L = 50 km 1 40 km 30 km 20 km d w = 50 mm 10 km L = 5 km 0.1 10 100 35 49 76 140 m T (in kg) D IFFERENT OPTIMUM CONFIGURATIONS AT METIS ( h = 0 . 1 MM ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 48/100

T ETHER DESIGN IN M ETIS . T HEORETICAL A NALYSIS G D T 10 d w = cte L = cte d w = 5 mm W u (kw) L = 50 km 1 40 km 30 km 20 km d w = 50 mm 10 km L = 5 km 0.1 10 100 35 49 76 140 m T (in kg) D IFFERENT OPTIMUM CONFIGURATIONS AT METIS ( h = 0 . 1 MM ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 49/100

T ETHER DESIGN IN I O . N UMERICAL A NALYSIS G D T 10 d w = cte L = cte W u (kw) 4 L = 35 km 3 d w = 10 mm 2 L = 25 km d w = 50 mm d w = 5 mm 1 d w = 36 mm L = 15 km L = 10 km L = 8 km 0.1 10 100 170 200 300 m T (kg) D IFFERENT OPTIMIZED CONFIGURATIONS IN I O ( h = 0 . 05 MM ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 50/100

T HREE DIFFERENT CONFIGURATION G D T A B C Three possible designs L (km) 25 35 35 d w (mm) 50 25 36 • Aluminum tape 0.05 mm thick • Self-balanced m T (kg) 169 117 170 • Mass of the S/C 500 kg Z C (Oh) 640 1265 886 • Tether mass ≤ 170 kg • χ ∈ [0 , 0 . 115] STABLE W u (w) 2100 2100 3000 • Power from 2 – 3 kw I sc (A) 10.94 5.53 7.89 • distance d e large I av (A) 1.43 1.04 1.49 • Tether tension small ≈ mN φ ∗ (deg) • Appropriate for the scientific explo- 39.7 40.06 40.06 ration of the Io plasma torus χ · 10 3 5.79 5.91 8.43 d e (km) 200,052 198,119 165,803 T (mN) 5.28 7.4 7.4 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 51/100

R OTATING ELECTRODYNAMIC TETHER ORBITING I O G D T We take an inertial frame Oxyz (origin Io) Orbital velocity of Io: v Io ≈ 17 . 33 km/s Plasma velocity at Io orbit ≈ 74 . 17 km/s Orbital velocity of S/C (around Io) ≈ 1 km/s Magnetic field B ≈ 2 · 10 − 6 T z E ≈ ( v Io − v pl ) × B E = E π (cos ωt, sin ωt, 0) , v Io E π ≈ 0 . 12 V/m y ϕ Toward Jupiter O E π ω t B x Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 52/100

P OWER G ENERATION I O G D T II Power generation: assumptions • small values of n ∞ no Ohmic effects IC • negligible ion losses at the cathodic segment • control impedance Z C at the cathodic end A B C h • OML regime • control parameter ζ = z ∗ L = 1 − Z C I C E m L V Vp Results of the analysis CHI Vt EML � L • I av = 1 0 I ( z ) dz = I 0 (1 − 2 5 ζ ) ζ 3 / 2 L • ˙ A C W = Z C I 2 C = I 0 E m L (1 − ζ ) ζ 3 / 2 B h � 2 eE m L 3 I 0 = 4 d w E E 3 π en ∞ II m e ZT Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 53/100

O PTIMUM P OWER G ENERATION G D T 0.2 0.15 ˙ W I 0 E m L 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 ζ √ √ � � ζ opt = 3 = 3 π 2 · 1 − ζ opt 1 m e E m = π 30 1 m e E m Z opt 5 , · · C e 3 L e 3 L ζ 3 / 2 8 d w n ∞ 12 d w n ∞ opt Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 54/100

R OTATING ELECTRODYNAMIC TETHER ORBITING I O G D T � π/ 2 � 2 E 3 π L 5 e 3 W av = 1 cos 3 / 2 ϕ dϕ · 4 d w � (1 − ζ ) ζ 3 / 2 � ˙ 3 π n ∞ π m e − π/ 2 14 12 10 P (kw) 8 6 4 2 0 0 10 20 30 40 50 60 Tether length (km) First order approximation of averaged generated power Power generated with a 25 km long 5 cm wide tape for a 5 cm wide tape tether of different lengths in Io tether along circular retrograde orbits of different radii: orbit. r = 1 . 1 r Io (grey solid line), r = 2 . 0 r Io (dark solid line), and r = 3 . 0 r Io (dark dotted line). Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 55/100

P OWER G ENERATION IN I ONIAN O RBIT G D T 10 d w = cte L = cte W u (kw) 4 L=35 3 d=10 2 L=25 d=50 d=5 1 d=36 L=15 L=10 L=8 0.1 170 200 300 10 100 m T (kg) D IFFERENT OPTIMIZED CONFIGURATIONS IN I O ( h = 0 . 05 MM ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 56/100

O RBIT S TABILITY G D T • Io is the Galilean moon more affected by the gravitation of Jupiter • Lara & Russel a shows that retrograde equatorial orbits around Europe are stable when eccentricity and semimajor axis are smaller than critical values. • Their results have been numerically extended to Io: if the apocenter is less than 3.5 r Io the equatorial retrograde orbit is stable for at least a year � π/ 2 �� π/ 2 � F av = 1 L I av ( u × B ) dϕ = 1 π L I av u dϕ × B π − π/ 2 − π/ 2 When the load impedance is controlled for maximum power generation ζ = cte: � π/ 2 � π/ 2 � π/ 2 1 I av cos ϕ E π dϕ = E π I av u dϕ = · I av cos ϕ dϕ E π E π − π/ 2 − π/ 2 − π/ 2 This way we have � k ′ L 5 e 3 k ′ ≈ 0 . 0667 d w n ∞ (5 − 2 ζ ) ζ 3 / 2 F av = √ E π ( E π × B ) , m e The averaged force has a constant value, in a first approximation, on the synodic frame. a On the design of a science orbit about Europa , Paper AAS 06-168, 16th AAS/AIAA Space Flight Mechanics Conference. Tampa, Florida,USA, January 2006 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 57/100

O RBIT S TABILITY G D T � E π L 5 e 3 · (5 − 2 ζ ) ζ 3 / 2 F ≈ F ( − sin ω t, cos ω t, 0) with F = 0 . 0667 d w n ∞ B m e 3.5 3 2.5 r r Io 2 1.5 1 0 5 10 15 20 25 30 time (days) Evolution of orbital radius for circular retrograde orbits of different radii (1.2, 2.0, 2.5 and 3 Io radii) under the effect of the Lorentz force of a 25 km long 5 cm wide tape tether with power-optimized impedance control. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 58/100

O RBIT C ONTROL G D T The main contribution to the motional electric field comes from Io orbital velocity rather than the spacecraft orbital velocity around Io. The � B Lorentz force is thrusting the S/C in part of the orbit (thrust arc) and braking it (drag arc) in the F ∝ E π × B rest of the Ionian orbit. Drag arc Preliminary test with a simple control strategy: E π the control parameter ζ is switched to 1 (maxi- mum Lorentz force) on thrust arcs and switched to 0 (no force) on drag arcs. A numerical sim- v IO Io ulation has been conducted assuming a 500 kg Thrust arc spacecraft equipped with a 25 km long and 5 cm wide electrodynamic tether starting from Schematic of electrodynamic thrust and drag arc a low altitude retrograde circular orbit ( r = 1 . 2 for a generic Ionian orbit. We neglect the orbital Io radii). velocity around Io in the determination of mo- tional electric field Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 59/100

O RBIT C ONTROL G D T 10 6 9 4 8 7 2 6 r y r Io 5 r Io 0 4 3 −2 2 1 −4 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 −6 −6 −4 −2 0 2 4 6 time (days) x r Io Spiral-out trajectory (left) and orbital radius evolution (right) for a 500 kg spacecraft propelled by a 25 km long 5 cm wide tape tether with simple current control strategy. Escape from Io gravitational field require less than 5 months, in this case. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 60/100

A RIADNA P ROGRAM G D T The END of Io Exploration with Electrodynamic Tethers ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 61/100

A RIADNA P ROGRAM G D T Stability of tethered satellites at collinear lagrangian points ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 62/100

P REVIOUS ANALYSIS G D T • The Stabilization of an Artificial Satellite at the Inferior Conjunction Point of the Earth-Moon System , G. Colombo , Smithsonian Astrophysical Observatory Special Report No. 80, November 1961 • The Control and Use of Libration-Point Satellites , R. W. Farquhar , NASA TR R-346, September 1970. pp. 89-102 • Tether Stabilization at a Collinear Libration Point , R. W. Farquhar , The Journal of the Astronautical Sciences, Vol. 49, No. 1, January-March 2001, pp. 91-106. • Dynamics of a Tethered System near the Earth-Moon Lagrangian Points , A. K. Misra, J. Bellerose, and V. J. Modi , Proceedings of the 2001 AAS/AIAA Astrodynamics Specialist Conference, Quebec City, Canada, Vol. 109 of Advances in the Astronautical Sciences, 2002, pp. 415–435. • Dynamics of a multi-tethered system near the Sun-Earth Lagrangian point , B. Wong and A. K. Misra , 13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 2003, Paper No. AAS-03-218. • Dynamics of a Libration Point Multi-Tethered System , B. Wong and A. K. Misra , Proceedings of 2004 International Astronautical Congress, Paper No. IAC-04-A.5.09. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 63/100

E QUILIBRIUM P OSITIONS AT THE S YNODIC F RAME G D T L 4 L 2 L 3 L 1 E 2 E 1 Equilibrium position with an inert tether L 5 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 64/100

E XTENDED D UMBBELL M ODEL . H ILL APPROACH G D T V ARYING LENGTH INERT TETHER � � ˜ η − (3 − 1 ρ 3 ) ξ = λ N ¨ 3 ˜ ξ − 2 ˙ N cos ϕ cos θ − ξS 2 ( ρ ) ρ 5 � � ˜ ξ + η ρ 3 = λ N η + 2 ˙ 3 ˜ ¨ N cos ϕ sin θ − ηS 2 ( ρ ) ρ 5 � � ˜ ζ + ζ (1 + 1 ρ 3 ) = λ N ¨ 3 ˜ N sin ϕ − ζS 2 ( ρ ) ρ 5 � ˙ � + 3 cos θ sin θ = 3 ˜ I s N ( − ξ sin θ + η cos θ ) θ + (1 + ˙ ¨ θ ) − 2 ˙ ϕ tan ϕ ρ 5 I s cos ϕ ˙ = 3 ˜ I s N � θ ) 2 + 3 cos 2 θ � (1 + ˙ ϕ + ¨ ϕ + sin ϕ cos ϕ ˙ ρ 5 ( − sin ϕ [ ξ cos θ + η sin θ ] + ζ cos ϕ ) I s ξ 2 + η 2 + ζ 2 and the quantity ˜ � where ρ = N and the function S 2 ( x ) are given by S 2 ( x ) = 3 2 (5 x 2 − 1) ˜ N = ξ cos ϕ cos θ + η cos ϕ sin θ + ζ sin ϕ, Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 65/100

E XTENDED D UMBBELL M ODEL . H ILL APPROACH G D T � L d � 2 a 2 I s ∈ [ 1 12 , 1 λ = · ν 2 / 3 , a 2 = 4 ] m L 2 ℓ d where, ℓ is the distance between primaries, ν is the reduced mass of the small primary (usually ν ≪ 1 ), m = m 1 + m 2 + m T is the total mass of the system and I s is the moment of inertia about a line normal to the tether by the center of mass G of the system; a 2 is of order unity and takes its maximum value a 2 = 1 / 4 for a massless tether with equal end masses ( m 1 = m 2 ). For a tether of varying length the parameter λ is a function of time since the deployed tether mass m d and the deployed tether length L d ( t ) are changing. Moreover, some terms of these governing equations involve the ratio: � m 1 ˙ ˙ I s L d J g = 1 − Λ d (1 + 3 cos 2 φ ) Λ d = m d m + Λ d � cos 2 φ = � , = 2 J g , m , 3 sin 2 2 φ − 2Λ d � I s L d 2 This formulation includes the mass of the tether through the parameter Λ d and the mass angle φ . In order to neglect the tether mass, we only have to introduce the condition Λ d = 0 in the above expressions. For a tether of constant length the parameter λ is also constant and the quotient ˙ I s /I s vanishes, that is, ˙ I s /I s = 0 . For the sake of simplicity we will assume a massless tether in what follows. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 66/100

E XTENDED D UMBBELL M ODEL . H ILL APPROACH G D T In the search of equilibrium position we will consider the tether tension given by � ˜   � 2 ¨ m 1 m 2 θ ) 2 + 3 cos 2 θ } − 1 + 1 N L d ϕ 2 + cos 2 ϕ { (1 + ˙ ω 2 L d  ˙ T 0 ≈ ρ 3 { 3 − 1 } −  m 1 + m 2 ρ L d This expression has been derived under the following assumptions: 1) inert tether, 2) massless tether, and 3) the Hill approach has been performed. At any equilibrium position, tether tension must be positive because a cable does not support compression stress and the above expression takes the form � ˜   � 2  cos 2 ϕ { 1 + 3 cos 2 θ } − 1 + 1 N m 1 m 2  , ω 2 L d T 0 ≈ T c ρ 3 { 3 − 1 } T c = ρ m 1 + m 2 We will use this expression for checking the tether tension in a given steady solutions of the equations of motion; if the tension is positive the equilibrium position will exist; if negative, the equilibrium position will not exist. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 67/100

V ALUES OF λ FOR S UN AND DIFFERENT PLANETS G D T 0.0001 1e-006 1e-008 λ 1e-010 Mercury Venus Earth 1e-012 Mars Jupiter Saturn Uranus 1e-014 Neptune 1e-016 0 1000 2000 3000 4000 5000 Tether length (km) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 68/100

V ALUES OF λ FOR BINARY SYSTEMS IN THE S OLAR S YSTEM G D T 10000 100 λ 1 0.01 0.0001 Mars-Phobos 1e-006 Mars-Deimos Saturn-Mimas 1e-008 Saturn-Enceladus Saturn-Rhea Saturn-Titan 1e-010 Earth-Moon Neptun-Triton 1e-012 1e-014 0 200 400 600 800 1000 Tether length (km) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 69/100

V ALUES OF λ FOR BINARY SYSTEMS IN THE J OVIAN WORLD G D T 100 λ 1 0.01 0.0001 1e-006 Metis 1e-008 Adrastea Amalthea Thebe 1e-010 Io Europa Ganymede 1e-012 Callisto 1e-014 0 20 40 60 80 100

N ON R OT . T ETHERS . C ONST . L ENGTH . E QUILIBRIUM POSITIONS G D T Center of mass G Small Primary E 2 L 2 Sketch of the equilibrium position E 2 in the neighborhood of L 2 . There is another one, similar to this, on the left of L 1 . λ = ρ 2 e − 1 � , e � 3 ρ 3 ρ 3 ξ e = ± ρ e , η e = ζ e = ϕ e = 0 , θ e = 0 , π, e > 1 / 3 3 This expression can be expanded when λ ≪ 1 is small and provide the asymptotic solution ξ e ≈ ( 1 1 1 3 λ − 9 λ 2 + O ( λ 3 ) 3 + 3 3) The convergence of this serie is poor but, for really small values of λ the two first terms give a useful approximation. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 71/100

N ON R OT . T ETHERS . C ONST . L ENGTH . S TABILITY ANALYSIS G D T A linear stability analysis shows that the equilibrium positions of that family are unstable for any value of λ . 2.85 2.8 2.75 2.7 R ( s 5 ) 2.65 2.6 2.55 2.5 −6 −4 −2 10 10 10 λ Real part of the unstable eigenvalue as a function of λ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 72/100

N ON R OTATING T ETHERS . V ARIABLE L ENGTH G D T The idea original of Colombo and Farquhar is as follows: let us assume that the tether length is L and for that length we have the equilibrium position labeled with (1) in this figure. On the right of such a position, the system center of mass G is acted by a force that impulses it toward the right. By increasing the tether length up to L (1) = L + ∆ (1) L we move the equilibrium position up to the point labeled with (2) in figure; now the force acting on G impulses it toward the left. Then we decrease the tether length, L (2) = L (1) + ∆ (2) L in order to move the equilibrium position on the left side of G . . . Thus, by changing the tether length in an appropriate way the center of mass G can be stabilized and kept in the neighborhood of the collinear lagrangian point L 2 . L 2 ∆ x 1 1 x L 2 = ( 1 3 L c 3 λ L c 3 Dim. 3 ) O (1) 1 (2) 1 N. Dim. ξ L 2 = ( 1 3 ) 3 λ 3 3 G Distances from the small primary and from the collinear point L 2 when λ ≪ 1 ( L c = ℓν 1 / 3 ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 73/100

N ON R OTATING T ETHERS . V ARIABLE L ENGTH G D T We carried out two different analysis: • Linear approximation for small values of λ • Full problem. Proportional control We finish the analysis of non rotating tethers pointing to some drawbacks associated with this kind of control • Control drawbacks Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 74/100

L INEAR APPROXIMATION FOR SMALL VALUES OF λ G D T ξ = ξ L + 3 1 / 3 λ 0 (1 + u ( τ )) , η = 3 1 / 3 λ 0 v ( τ ) , λ = λ 0 (1 + s ( τ )) d 2 u � 3 cos 2 θ cos 2 ϕ − 1 � s ( τ ) + 27 � cos 2 θ cos 2 ϕ − 1 � = 0 d τ 2 − 2 d v d τ − 9 u ( τ ) + 9 2 2 d 2 v d τ 2 + 2 d u d τ + 3 v ( τ ) − 9 cos 2 ϕ cos θ sin θ (1 + s ( τ )) = 0 d 2 w d τ 2 + 4 w ( τ ) − 9 cos ϕ cos θ sin ϕ (1 + s ( τ )) = 0 (1 + s ( τ )) d 2 θ d τ 2 + d s d τ (1 + d θ d τ ) − 2 tan ϕ d ϕ � 1 + d θ � � 1 + d s � + 12 cos θ sin θ (1 + s ( τ )) = 0 d τ d τ d τ (1 + s ( τ )) d 2 ϕ �� � � 2 d τ 2 + d s d ϕ 1 + d θ + 12 cos 2 θ d τ + (1 + s ( τ )) sin ϕ cos ϕ = 0 d τ d τ s ( τ ) = e − β τ ( A cos Ω τ + B sin Ω τ ) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 75/100

O NE DIMENSIONAL MOTION G D T 5.2 1160 1140 5 1120 4.8 1100 1080 4.6 1060 x (km) x (km) 4.4 1040 4.2 1020 1000 4 980 3.8 960 3.6 940 0 200 400 600 800 1000 0 200 400 600 800 1000 t (hours) t (hours) 1.4 0.019 1.2 0.0185 1 0.018 0.8 0.0175 0.6 ˙ 0.017 L (m/s) T (N) 0.4 0.0165 0.2 0.016 0 0.0155 -0.2 0.015 -0.4 -0.6 0.0145 0 200 400 600 800 1000 0 200 400 600 800 1000 t (hours) t (hours) Massless tether with two equal masses (500 kg) at both ends in the Earth-Moon system. The selected parameters are β = 0 . 5 , Ω = 1 . 5 , for the initial conditions u 0 = 0 . 15 and ˙ u 0 = 0 . The nominal tether length is L = 1000 km Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 76/100

B I - DIMENSIONAL MOTION G D T 5.2 1140 5 1120 4.8 1100 4.6 1080 4.4 1060 4.2 x (km) L (km) 1040 4 1020 3.8 1000 3.6 980 3.4 960 3.2 3 940 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 t (hours) t (hours) 1 0.01 0.8 0.0095 0.6 0.009 0.4 ˙ L (m/s) T (N) 0.0085 0.2 0.008 0 0.0075 -0.2 -0.4 0.007 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 t (hours) t (hours)

F ULL P ROBLEM . P ROPORTIONAL C ONTROL G D T The length variation λ ( τ ) is governed by a proportional control law: � λ = λ e + K i ( x i − x e,i ) where K i are gains and x i stands for the variables ξ, η, θ . d y d t = M y � � where y T = δξ, δη, δθ, δ ˙ η, δ ˙ ξ, δ ˙ θ . The detailed stability analysis of this equation is cumbersome. Nevertheless, we firstly consider the simpler case in which only one gain K ξ is different from zero. The characteristic polynomial takes the form: � 3 K ξ �� 6 J g + 27 � 2 − 4 �� + 6 � � 105 − 6 + 4 �� s 3 + s 2 + − K ξ − s + ξ 4 ξ 3 ξ 4 ξ 7 ξ 3 ξ 6 e e e e e e �� 9 − 3 � � 45 + 9 − 2 �� +6 K ξ − = 0 ξ 4 ξ 7 ξ 3 ξ 6 e e e e Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 78/100

F ULL P ROBLEM . P ROPORTIONAL C ONTROL G D T The Descartes rule of signs provides a sufficient condition for stability to be fulfilled by K ξ . Such a condition is drawn in figure as a function of the equilibrium value λ e for a massless tether ( J g = 1 ). 5 4.5 4 K ∗ ξ 3.5 3 2.5 2 −6 −4 −2 10 10 10 λ e Sufficient condition for K ξ to stabilize the system as a function of λ e . Values over the curve ( K ξ > K ∗ ξ ) provide stability. Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 79/100

C ONTROL DRAWBACKS . T ETHER LENGTH G D T ∆ L 1 = 3 √ a 2 λ 1 ∆ x e 2 3 3 6 10 10 Sun-Earth Earth-Moon Non linear Mars-Phobos Linear Jupiter-Amaltea 2 4 10 10 Jupiter-Io d L d L Saturn-Enceladus d x e d x e 1 10 2 10 0 10 −6 −4 −2 0 −2 0 2 4 10 10 10 10 10 10 10 10 λ L (km) Ratio between variation of tether length and deviations Ratio between variation of tether length and deviations of the center of mass vs. λ . of the center of mass vs. the tether length (km) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 80/100

C ONTROL DRAWBACKS . T ETHER TENSION G D T 20 −4 10 15 10 T ∆ ξ T c −5 10 5 0 −6 10 −5 −4 −3 −5 −4 −3 10 10 10 10 10 λ e ∆ ξ Zero tether tension ∆ ξ vs. λ e (blue line). Maximum and minimum tether tension vs. the Zero tether length ∆ ξ vs. λ e (red line). initial perturbation ∆ ξ for λ e = 10 − 2 . To avoid the zero tension problem (slack tether) is the most strong requirement of this control strategy Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 81/100

R OTATING TETHERS . E QUILIBRIUM P OSITIONS G D T λ e = 4 ρ 5 � 3 − 1 � e ρ 3 ξ e = ± ρ e , η e = ζ e = 0 , , e > 1 / 3 ρ 3 3 e For small values of λ the above solution provides the asymptotic solution ξ e ≈ ( 1 3 λ 4 − 9 ( λ 1 1 4 ) 2 + O ( λ 3 ) 3 + 3 3) Comparing with the same results for a non-rotating tether λ = ρ 2 e 3 ρ 3 ρ 3 � � ξ e = ± ρ e , η e = ζ e = 0 , e − 1 , e > 1 / 3 3 ξ e ≈ ( 1 1 1 3 λ − 9 λ 2 + O ( λ 3 ) 3 + 3 3) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 82/100

R OTATING TETHERS . P ROPORTIONAL CONTROL G D T ξ δ ˙ λ = λ e + K ξ δξ + K ˙ ξ + K η δη + K ˙ η δ ˙ η Through the Routh-Hurwitz theorem, it has been found that asymptotic stability is guaranteed if the gains of the control law satisfy the relations K ξ ≥ 4 15 ρ 3 � � 3 ρ e e − 2 K ˙ ξ > 0 K η = 0 K ˙ η = 0 We carried out two simulations of a rotating tether with different values of K ξ and K ˙ ξ in order to see the qualitative behavior of the system. The characteristics of the simulations are: λ e = 0 . 0401 ( ξ e = 0 . 707 ); ξ 0 − ξ e = 10 − 3 , η e = 10 − 3 ; Ω ⊥ = 50 . Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 83/100

S IMULATIONS G D T −3 3 x 10 −4 5 x 10 Kξ = 4 K ˙ ξ = 3 K ˙ ξ = 0 . 5 2 0 1 δ ˙ δ ˙ ξ ξ 0 −5 −1 −10 −2 −3 −15 0.7055 0.706 0.7065 0.707 0.7075 0.708 0.7085 0.709 0.7065 0.707 0.7075 0.708 δξ δξ −3 −3 1 x 10 2 x 10 K ˙ ξ = 3 Kξ = 4 1.5 K ˙ ξ = 0 . 5 0.5 1 0.5 0 δ ˙ δ ˙ η η 0 −0.5 −0.5 −1 −1.5 −1 −2 −1.5 −2.5 −1.5 −1 −0.5 0 0.5 1 1.5 −5 0 5 10 −3 −4 x 10 x 10 δη δη λ e = 0 . 0401 ( ξ e = 0 . 707 ); ξ 0 − ξ e = 10 − 3 , η e = 10 − 3 ; Ω ⊥ = 50 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 84/100

S IMULATIONS G D T 66 58 Kξ = 4 K ˙ ξ = 3 64 57 K ˙ ξ = 0 . 5 62 56 60 55 Ω ⊥ Ω ⊥ 58 54 56 53 54 52 52 51 50 48 50 0 5 10 15 20 25 30 0 5 10 15 20 25 30 τ τ 0.215 0.215 Kξ = 4 K ˙ ξ = 3 K ˙ ξ = 0 . 5 0.21 0.21 0.205 √ √ 0.205 λ λ 0.2 0.2 0.195 0.195 0.19 0.185 0.19 0 5 10 15 20 25 30 0 5 10 15 20 25 30 τ τ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 85/100

A RIADNA P ROGRAM G D T The END of Stability of tethered satellites at collinear lagrangian points ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 86/100

Lyapunov orbits G D T 3 2889 8.6 682 Horizontal stability curve 8. 2 161 7.4 38 6.9 Stability indices Orbit period curve 1 9 6.3 Period 2 5.7 0 y 0 5.1 � 2 4.5 � 1 � 9 4. Vertical stability curve � 38 3.4 � 2 � 15 � 10 � 5 0 Jacobi constant � 3 � 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 x • Left: sample orbits of the family of Lyapunov orbits around L 1 (dashed) and L 2 (full line) for, from larger to smaller, C = − 1 , 0 , 1 , 2 , 3 , 4 . Right: stability-period diagram of the family of Lyapunov orbits of the Hill problem. Note the arcsinh scale used for the stability curves. ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 87/100

Tethered family of Lyapunov orbits: unstable G D T 6.9 6870 2 Horizontal stability curve 663 6.8 1 Stability index 64 6.6 Orbit period curve Period 0 Η 6.5 6 Vertical stability curve � 1 6.4 0 Critical points � 2 � 4 6.2 0 1 2 3 Tether 's characteristic length � 0.5 0.0 0.5 1.0 1.5 Ξ • Left: stability-period diagram of the family of Lyapunov orbits with Jacobi constant C = − 0 . 408295 for tether’s length variations; the horizontal gray lines correspond to the critical values k = ± 2 (in the arcsinh scale). Rigth: Hill’s problem Lyapunov orbit ( λ = 0 , full line) and an orbit with a tehter’s characteristic lenght λ = 1 (dotted). ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 88/100

Tethered family of Lyapunov orbits: unstable G D T 0.2 25188 3.7 Horizontal stability curve 2164 3.6 0.1 Stability indices 3.5 186 Period Orbit period curve 0.0 Η 16 3.3 Critical orbits 3.2 � 0.1 1 Vertical stability curve � 2 3. 0 1 2 3 � 0.2 Tether's characteristic length 0.66 0.7 0.74 0.78 Ξ • Left: stability-period diagram of a family of Lyapunov orbits close to L 2 for tether ’s length variations; the horizontal gray lines correspond to the critical values k = ± 2 (in the arcsinh scale). Rigth: starting orbit ( λ = 0 , full line) and an orbit with λ = 0 . 1 (dotted). ( C is the Jacobi constant) ◭ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 89/100

Family of eight-shaped orbits G D T Ξ 0.0 0.2 0.2 0.4 Η 0 0.6 � 0.2 2327 6.2 Orbit period curve 1 549 5.6 k 1 stability curve Stability indices 130 5. Period 31 Ζ 4.4 0 k 2 stability curve Critical point 7 3.7 2 3.1 � 1 � 8 � 6 � 4 � 2 0 2 4 Jacobi constant • Left: sample eight-shaped orbits for C = 2 (red), 1 (magenta), and 0 (blue). Right: stability-period diagram of the family of eight-shaped orbits of the Hill problem. ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 90/100

Tethered family of eight-shaped orbits G D T 0.07 � 0.07 2501 3.62 475 Orbit period curve 3.48 k 1 stability curve 0.5 Stability indices 90 3.34 Period 17 3.21 Ζ 0.0 2 k 2 stability curve 6: end 3.07 3 3 1: start 5 � 0.5 2.93 0 4 � 2 2.79 0.0 0.00 0.05 0.10 0.15 0.20 0.2 0.4 Tether's characteristic length Ξ 0.6 • Left: stability-period diagram of a family of eight-shaped, periodic orbits with constant C = 2 for tether ’s length variations; the horizontal gray lines correspond to the critical values k = ± 2 . Rigth: stable orbits for λ = 0 . 2 (left) λ = 0 . 215 (center), and unstable orbit of the Hill problem ( λ = 0 , right). ( C is the Jacobi constant) ◮ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 91/100 ◭

Tethered family of eight-shaped orbits G D T 0.06 � 0.06 64375 3.91 Orbit period curve 0.5 6: end 7122 3.61 Stability indices 788 3.32 k 1 stability curve Period 87 3.02 Ζ 0.0 2 10 2.72 k 2 stability curve 1 2.42 3 1: start 5 � 0.5 4 � 2 2.12 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 Tether's characteristic length 0.4 Ξ 0.6 • Left: stability-period diagram of a family of eight-shaped, periodic orbits with constant C = 2 . 7 for tether ’s length variations; the horizontal gray lines correspond to the critical values k = ± 2 . Right: stable orbits for λ = 0 . 17 (left, black) λ = 0 . 184 (center, blue), and unstable orbit of the Hill problem ( λ = 0 , right). ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 92/100

Tethered family of eight-shaped orbits G D T 682 3. 161 2.9 k 1 stability curve Stability indices 38 Period 2.8 9 Bifurcation orbits 2.7 2 2.6 0 Orbit period curve k 2 stability curve � 2 2.4 0 1 2 3 4 5 Tether's characteristic length • Stability-period diagram of a family of eight-shaped, periodic orbits with constant C = 3 for tether ’s length variations; the horizontal gray line corresponds to the critical values k = +2 . ( C is the Jacobi constant) ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 93/100 ◭

Family of Halo orbits G D T 1600. 3.1 Orbit period curve 400. 2.8 0.4 k 1 stability curve Η 0.2 0.0 100. 2.6 Stability indices � 0.2 30. 2.4 � 0.4 Period Reflection 8. 2.2 0.0 2. 2. Ζ 0. k 2 stability curve 1.8 � 0.5 � 2. 1.5 1.5 2.0 2.5 3.0 3.5 4.0 0.0 Jacobi constant 0.2 0.4 Ξ 0.6 • Left: stability-period diagram of the family of Halo orbits of the Hill problem. Right: sample stable orbit for C = 1 . 08 (the blue and red dots linked by a gray line are the origin and L 2 point, respectively). ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 94/100

Tethered family of Halo orbits G D T 0.5 Η 2.32 0.0 2 2.31 � 0.5 k 1 stability curve Stability indices 2.3 Period 1 0.0 2.29 Orbit period curve 2.28 Ζ 0 2.27 � 0.5 k 2 stability curve � 1 2.26 0 0.00001 0.00003 0.00005 0.00007 0.0 0.2 Tether's characteristic length 0.4 Ξ 0.6 • Left: Stability-period diagram of the family of Halo orbits with C = 1 . 07 for tether ’s length variations. Right: stable (full line) and unstable (dashed) Halo orbits of the Hill problem. ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 95/100

Tethered family of Halo orbits G D T 0.5 14 2.5 Η k 1 stability curve 1: start 0.0 5 2.4 � 0.5 2: reflection Stability indices 2 2.3 Period 4: end Orbit period curve 0.0 2.2 0 Ζ 2.1 � 1 k 2 stability curve � 0.5 3 � 3 2. 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.0 0.2 Tether's characteristic length 0.4 Ξ 0.6 • Stability-period diagram of the family of Halo orbit with C = 1 . 15 for tether ’s length variations. Right: unstable Halo orbits of the Hill problem (red and magenta) and stable (blue) Halo orbits with a tethers characteristic length λ = 0 . 0052 . ( C is the Jacobi constant) ◮ ◭ Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 96/100