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

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Grupo de Dinámica de Tethers (GDT)

Dynamics and Stability

• f Tethered Satellites at Lagrangian Points†

by

• J. Peláez,

Technical University of Madrid (UPM), 28040 Madrid, Spain Workshop on Stability and Instability in Mechanical Systems: Applications and Numerical Tools Thursday, 4 December 2008, Barcelona † Supported by ESTEC Contract 21259

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 1/100

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Grupo de Dinámica de Tethers (GDT-UPM) - Cisas (Padova)

Dynamics and Stability of Tethered Satellites at Lagrangian Points† by

Claudio Bombardelli, Dario Izzo, ESTEC-ESA, Noordwijk, 2201 AZ, The Netherlands Enrico C. Lorenzini, Davide Curreli, University of Padova, Padova, 35131, Italy

• M. Sanjurjo-Rivo, Fernando R. Lucas, J. Peláez,

Technical University of Madrid (UPM), 28040 Madrid, Spain Daniel J. Scheeres, The University of Colorado, Boulder, CO 80309-0429, USA

• M. Lara

Real Observatorio de la Armada, 11110 San Fernando, Spain Final Report for the Advanced Concept Team † Supported by ESTEC Contract 21259

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 2/100

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Tethers fundamentals

In a tethered system, two masses orbiting at different heights share a common orbital frequency Ω0 ⇒ The third Kepler law is broken by the tether tension m1 m2 r1 r2 r0 T T G g1 g2

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Tethers fundamentals

Stable equilibrium along the local vertical

θ Local Vertical 3mΩ2

0d

d G x z Any deviation θ from the local vertical gives place to a torque. In effect, the gravity gradi- ent force breaks down in

• one component along the tether, which

is basically balanced by the tether tension

• one component orthogonal to the tether,

which provides the restoring torque. This torque leads the tether again to the local vertical Thus, the local vertical is a stable equilibrium position for the tethered system

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 4/100

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Electrodynamic tethers fundamentals

• B
• v
• E
• E =

v × B q+ q− Conductive rod In a frame attached to the rod, a motional elec- tric field appears

• E =

v × B It is induced by the magnetic field B inside which the rod is moving. For a conductive rod moving in the vacuum, a redistribution of sur- face charge takes place, leading to a vanishing electric field inside the rod. Thus, a steady state is reached with no motion of charged particles in the rod. If the rod is moving inside a plasma environ- ment, this picture changes drastically

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 5/100

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2007 AAS/AIAA Space Flight Mechanics Meeting

Sedona, Arizona, January 28 - February 1, 2007

Electrodynamic tethers in a plasma environment

e− e− e− e− e− e− i+ i+ i+ i+

• B
• v

I q+ q− The ionospheric plasma makes the electrons begin to be attracted by the anodic end of the

• rod. Similarly, the ions will be attracted by the

cathodic end. Some of these charges will be trapped by the rod and they produce a current I which flows inside the conductive mater-

• ial. The amount of current I can be increased

with the help of plasma contactors placed in the tethers ends. However, the interaction be- tween the tether current I and the magnetic field B gives place to forces acting on the

• rod. They will break (or accelerate) its mo-

tion (in the figure they are breaking the mo- tion). Their resultant is:

• F =

I × B L

A permanent tethered observatory at Jupiter. Dynamical analysis – p.5/24

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2007 AAS/AIAA Space Flight Mechanics Meeting

Sedona, Arizona, January 28 - February 1, 2007

Electrodynamic tethers in a plasma environment

e− e− e− e− e− e− i+ i+ i+ i+

• B
• v

I q+ q− The ionospheric plasma makes the electrons begin to be attracted by the anodic end of the

• rod. Similarly, the ions will be attracted by the

cathodic end. Some of these charges will be trapped by the rod and they produce a current I which flows inside the conductive mater-

• ial. The amount of current I can be increased

with the help of plasma contactors placed in the tethers ends. However, the interaction be- tween the tether current I and the magnetic field B gives place to forces acting on the

• rod. They will break (or accelerate) its mo-

tion (in the figure they are breaking the mo- tion). Their resultant is:

• F =

I × B L

A permanent tethered observatory at Jupiter. Dynamical analysis – p.5/24

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ARIADNA PROGRAM The two basic tether configurations

Spherical collector m1 m2

Uniform intensity

Insulated cable Bare tether m1 m2 Exposed cable

NON uniform intensity

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 6/100

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Advanced Topics In Astrodynamics – Tethered Systems

TSS-1R. The Tether is partially deployed

Tether Fundamentals – p.13/48

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Advanced Topics In Astrodynamics – Tethered Systems

STS-46 Tethered Satellite System 1 (TSS-1) satellite deployment from OV-104

The satellite is reeled out via its thin Kevlar tether into the blackness of space during deployment operations from the payload bay of Atlantis. At the bottom of the frame is the satellite upper boom including (bottom to top) the 12-m deployment boom, tip can, the docking ring, and concentric ring damper. The Langmuir probe and the dipole-field antenna are stowed at either side of the TSS-1 satellite.

Tether Fundamentals – p.10/48

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Background: Tethered satellites and Lagrangian Points

• The Earth-Moon external point (L2). Excellent scenario for many different missions
• Tethers are useful in some specific missions

* Stabilization at Lagrange points (Colombo, Farquhar, Misra)

• ED tethers: may produce power and propulsion in appropriate environments

* Jovian world (Sanmartín et all., Peláez and Scheeres, . . . ) * plasma collection is better with straightened tethers

• Tether’s stabilization needs mechanical tension

* Close to a body: gravity gradient (self balanced EDT) * Far from body: fast rotating tethers

• Dynamics may be quite different from mass-point satellites

* coupled roto-traslational motion (long tethers) * varying length

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 7/100

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Outline of Presentation

◮ • Gentil Introduction ◮ • Inert Tethers and Periodic Motions ◮ • Io Exploration with Electrodynamic Tethers ◮ • Stability of tethered satellites at lagrangian collinear points

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 8/100

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Diffusion

• Dynamics of tethered libration-points satellites, by

Manuel Sanjurjo Rivo, Fernando R. Lucas & Jesús Peláez, XI Jornadas de Trabajo en Mecánica Celeste, 25—27 June 2008, Albergue de la Real Fábrica, Ezcaray, La Rioja, Spain

• Dynamic behavior of a fast-rotating tethered satellite, by
• M. Lara & Jesús Peláez,

XI Jornadas de Trabajo en Mecánica Celeste, 25—27 June 2008, Albergue de la Real Fábrica, Ezcaray, La Rioja, Spain

• On the dynamics of a tethered system near the colineal libration points, by
• M. Sanjurjo-Rivo, F. R. Lucas, J. Peláez, C. Bombardelli, E. C. Lorenzini, D. Curreli, D. J. Scheeres & M. Lara,

AIAA paper 2008-7380, The 2008 AAS/AIAA Astrodynamics Specialist Conference and Exhibit, 18—21 Agosto 2008 Hawaii Convention Center and Hilton Hawaiian Village, Honolulu, Hawaii, USA

• Io exploration with electrodynamic tethers, by
• C. Bombardelli, E. C. Lorenzini, D. Curreli, M. Sanjurjo-Rivo, F. R. Lucas, J. Peláez, D. J. Scheeres & M. Lara,

AIAA paper 2008-7384, The 2008 AAS/AIAA Astrodynamics Specialist Conference and Exhibit, 18—21 Agosto 2008 Hawaii Convention Center and Hilton Hawaiian Village, Honolulu, Hawaii, USA Abstracts sent:

• Plasma torus exploration with electrodynamic tethers, by
• E. C. Lorenzini, D. Curreli, C. Bombardelli, M. Sanjurjo-Rivo, F. R. Lucas, J. Peláez, D. J. Scheeres & M. Lara,

19th AAS/AIAA Space Flight Mechanics Meeting, February 8—12, 2009, Savannah, Georgia, USA

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Conclusions

• Gentil Introduction: We have developed the theory that permit the analysis theoretical and

the simulations of tethers, of constant or varying length, inert or electrodynamic, rotating or non-rotating in the environment provided for a binary system. Some results are completely new and there is nothing similar in the literature.

• Inert Tethers and Periodic Motions:The influence of the tether length in the existence and

stability of periodic motions associated to Halo orbits has been studied in detail for inert

• tethers. Some results are completely new and open interesting research lines and possibilities.
• Io Exploration with Electrodynamic Tethers: All the analysis is an important novelty.

At least two different, and important, research lines have been set up.

• Stability of tethered satellites at lagrangian collinear points: We show that the Hill

approach permits to clarify some obscure details underneath the previous analysis. An interesting application for the Mars satellites —Deimos or Phobos— can be deduced from our analysis.

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 10/100

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Deimos and Phobos Mars - Deimos Magnitude Value Unit LT 10 km xE 22.34 km ∆x 200 m kξ 4 T 72 hours m 1000 kg φ 45 deg Λ 0.1 Mars - Phobos Magnitude Value Unit LT 5 km xE 16.96 km ∆x 125 m kξ 3 T 72 hours m 1000 kg φ 45 deg Λ 0.1

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 11/100

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ARIADNA PROGRAM

Gentil Introduction

◮ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 12/100

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Gentil Introduction GOAL: 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

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Inertial motion of an inert tether

• Only gravitational forces
• Dumbbell model

⇒ rigid body dynamics: T = 1

2 m ( ˙

x · ˙ x) + 1 2Ω · I · Ω, V = −

• m

µ y dm

• 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
• Lagrangian approach:

L = T − V → d dt ∂L ∂ ˙ qj

• − ∂L

∂qj = 0,

• ˙

qj = dqj/dt, qj: generalized coordinates

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The dumbbell model

• Inertial frame Oxyz
• Tether-attached frame: origin G, unit vectors (u1, u2, u3),
• u1 = u,

u2 = k × u/k × u, u3 = u1 × u2,

• ˙

u = Ω × u ⇒ Ω = (Ω · u1, Ω · u2, Ω · u3)T O m1 m2 G x x x y y z z θ ϕ u

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Gravitational Potential

V = −

• m

µ y dm dm = ρLds α G m2 uG m1 L

1

L

2

x y u r = x, cos α = u · x/r f dm s O

• Usual expansion of

1 y in Legendre polynomials V = −µ r

• m

dm

• 1 + 2(s/r) cos α + (s/r)2 = −µm

r

∞

• n≥0

L r n an Pn(cos α)

• an = an(m1, m2, mT ) are functions of tether & end masses (a0 = 1, a1 = 0)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 16/100

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Kinetic analysis

• Kinetic energy:

T = 1 2m ( ˙ x · ˙ x) + 1 2Ω · I · Ω

• Rotational motion (tether attached frame):
• Inertia tensor (null moment of inertia around u):

I =     I I     I = mL2a2 = 1 12mL2 3 sin2 2φ − 2Λ

• φ = φ(m1, m2, mT ),

Λ = mT /m

• Ω · ˙

u = 0 ⇒ Ω · I · Ω = I ˙ u2

• Position of the tether: u = (cos ϕ cos θ, cos ϕ sin θ, sin ϕ)
• Kinetic energy:

T = 1 2m

• ˙

x2 + ˙ y2 + ˙ z2 + 1 2I

• ˙

ϕ2 + ˙ θ2 cos2 ϕ

• Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 17/100

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Circular Restricted Three Body Problem

m x1 x2 GP ℓ mP1 mP2 x y z ω

• i
• j
• k

O ℓ

2

= ( 1 − ν ) ℓ ν ℓ

• + Perturbations acting on the c.o.m (ED forces, size, varying length . . .)
• + Attitude dynamics

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Motion relative to the synodic frame

1. Two attractive bodies (primaries). Gravitational forces calculated under two assumptions: 1) spherical attracting bodies and, 2) small ratios for L

ri

• Gravitational potential Vi (i = 1, 2) of each primary:

Vi = −mµi ri

• 1 + ( L

ri )2 a2 P2(cos αi) + O( L ri )3

• ,

cos αi = u·xi/ri 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 (ω2/2) (x2 + y2)

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Lagrangian function and non-dimensional variables

1. Lagrangian function L m = 1 2( ˙ x2 + ˙ y2 + ˙ z2) + ω (x ˙ y − y ˙ x) + ω2 2 (x2 + y2) + µ1 r1 + µ2 r2 +L2 a2 µ1 r3

1

P2(cos α1) + µ2 r3

2

P2(cos α2) + 1 2[ ˙ ϕ2 + ˙ θ2 cos2 ϕ]

• + O(L3/r3)

Here, a2 = I/(m L2), 0 < a2 < 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)

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Governing equations for a constant length inert librating tether

− 2 ˙ y − (1 − ν)(1 − 1 + x ρ3

1

) − x(1 − ν ρ3 ) = ǫ2

• (1 − ν)

ρ5

1

[3(ρ1 · u)( i · u) − (1 + x)S2(α1)] + ν ρ5 [3(ρ · u)( i · u) − xS2(α2)]

• ¨

y + 2 ˙ x + y (1 − ν) ρ3

1

− y(1 − ν ρ3 ) = ǫ2

• (1 − ν)

ρ5

1

[3(ρ1 · u)( j · u) − yS2(α1)] + ν ρ5 [3(ρ · u)( j · u) − yS2(α2)]

• ¨

z + z ν ρ3 + z (1 − ν) ρ3

1

= ǫ2

• (1 − ν)

ρ5

1

[3(ρ1 · u)( k · u) − zS2(α1)] + ν ρ5 [3(ρ · u)( k · u) − zS2(α2)]

• ¨

θ cos ϕ − 2(1 + ˙ θ) ˙ ϕ sin ϕ = 3 (1 − ν) ρ5

1

(ρ1 · u)(ρ1 · u2) + 3 ν ρ5 (ρ · u)(ρ · u2) ¨ ϕ + (1 + ˙ θ)2 sin ϕ cos ϕ = 3 (1 − ν) ρ5

1

(ρ1 · u)(ρ1 · u3) + 3 ν ρ5 (ρ · u)(ρ · u3) ν ≡ reduced mass of the small primary, ε2

0 = (

L ℓ )2a2, S2(αi) = 3 2 (5 cos2 αi − 1) cos α1 = ρ1 ρ1 · u, cos α2 = ρ ρ · u

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Rotating Tether Attitude Dynamics

The Attitude Dynamics of the tether can be analyzed more intuitively using the Newton-Euler formulation. The core of the analysis is the angular momentum equa- tion: dHG dt = M G Here M G is the resultant of the external 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 motion relative to the center of mass. u1 ≡ u u2 u3 G H G m1 m2

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Rotating Tether Attitude Dynamics

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 Gu1u2u3 where the unit vectors are given by: u1 = u, u2 = ˙ u ˙ u, u3 = u1 × u2 In this body frame the angular momentum is: H G = I ˜ Ω⊥u3, where ˜ Ω⊥ = u × ˙ u = ˙ u and the angular momentum equation takes the form: d˜ Ω⊥ dt u3 + ˜ Ω⊥ du3 dt = 1 I M G

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Rotating Tether Attitude Dynamics

This way we obtain the following equa- tions: du1 dt = ˜ Ω⊥u2 du3 dt = M2 ˜ Ω⊥I u2 d˜ Ω⊥ dt = M3 I We introduce the Tait-Bryant angles (or Cardan angles) as generalized coordinates. Notice that, from a mathematical point

• f view, we have a four-orden system of

ODE’S. u1 ≡ u u2 u3 G x0 y0 z0 φ3 φ2 φ1 φ1 s

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Rotating Tether Attitude Dynamics

In terms of the Bryant angles the unit vectors u1 and u3 are given by u1 = (cos φ2 cos φ3, cos φ1 sin φ3 + sin φ1 sin φ2 cos φ3, sin φ1 sin φ3 − cos φ1 sin φ2 cos φ3) u3 = (sin φ2, − sin φ1 cos φ2, cos φ1 cos φ2) The equations governing the time evolution of the Bryant angles are: dφ1 dτ = − M2 ω2I · 1 Ω⊥ · cos φ3 cos φ2 dφ2 dτ = − M2 ω2I · 1 Ω⊥ · sin φ3 dφ3 dτ = Ω⊥ + M2 ω2I · 1 Ω⊥ · cos φ3 tan φ2 dΩ⊥ dτ = M3 ω2I where Ω⊥ = ˜ Ω⊥/ω is the non-dimensional form of ˜ Ω⊥. These equations should be integrated from the initial conditions: at τ = 0 : φ1 = φ10, φ2 = φ20, φ3 = φ30, Ω⊥ = Ω⊥ 0

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 25/100

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Fast Rotating Tether

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 τ1 = Ω⊥τ = O(1) For example, consider the governing equation dφ dτ = f(φ1, φ2, φ3, Ω⊥) Its averaged equations form is: < dφ dτ >= 1 2π 2π f(φ1, φ2, φ3, Ω⊥)dτ1 To integrate the function f(φ1, φ2, φ3, Ω⊥) the slow variables (φ1, φ2, Ω⊥) take constant values and the fast variable φ3 is approximated by φ3 ≈ τ1 + φ30. u1 ≡ u u2 v1 v2 u3 ≡ v3 G φ3 φ3 Stroboscopic frame

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Averaged equations for an inert fast rotating tether

− 2 ˙ y − (1 − ν)(1 − 1 + x ρ3 1 ) − x(1 − ν ρ3 ) = ǫ2 2    ν − 1 ρ5 1

• 3(sin φ2 + ˜

n) sin φ2 − (1 + x)S2( ˜ n1 ρ1 )

• −

ν ρ5

• 3˜

n sin φ2 − xS2( ˜ n ρ )    ¨ y + 2 ˙ x + y (1 − ν) ρ3 1 − y(1 − ν ρ3 ) = ǫ2 2    1 − ν ρ5 1

• 3(sin φ2 + ˜

n) cos φ2 sin φ1 + yS2( ˜ n1 ρ1 )

• +

ν ρ5

• 3˜

n cos φ2 sin φ1 + yS2( ˜ n ρ )

• ¨

z + z ν ρ3 + z (1 − ν) ρ3 1 = ǫ2 2    ν − 1 ρ5 1

• 3(sin φ2 + ˜

n) cos φ2 cos φ1 − zS2( ˜ n1 ρ1 )

• −

ν ρ5

• 3˜

n cos φ2 cos φ1 − zS2( ˜ n ρ ) dφ1 dτ =

• 1 +

cos φ1 cos φ2 2Ω⊥ sin φ2 cos φ1 cos φ2 + 3 2 (1 − ν) Ω⊥ (sin φ2 + ˜ n)(cos φ2 + ˜ b) ρ5 1 cos φ2 + 3 2 ν Ω⊥ ˜ n ˜ b ρ5 cos φ2 dφ2 dτ = −

• 1 +

cos φ1 cos φ2 2Ω⊥

• sin φ1 + (y cos φ1 + z sin φ1)

   3 2 (1 − ν) Ω⊥ (sin φ2 + ˜ n) ρ5 1 + 3 2 ν Ω⊥ ˜ n ρ5    dΩ⊥ dτ = sin φ1 sin φ2 cos φ1 where the quantities (˜ n, ˜ b) and the fast variable φ3 are given by ˜ n = x sin φ2 − (y sin φ1 − z cos φ1) cos φ2 ˜ b = x cos φ2 + (y sin φ1 − z cos φ1) sin φ2 dφ3 dτ = Ω⊥ −

• 1 +

cos φ1 cos φ2 2Ω⊥ sin2 φ2 cos φ1 cos φ2 − 3 2 (1 − ν) Ω⊥ (sin φ2 + ˜ n)(cos φ2 + ˜ b) ρ5 1 tan φ2 − 3 2 ν Ω⊥ ˜ n ˜ b ρ5 tan φ2 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

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The Hill approach for fast rotating tethers

• 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 ν1/3 ξ, y = ℓ ν1/3 η, z = ℓ ν1/3 ζ, ρ = ℓ ν1/3 ˆ ρ

• For a FRT the parameter Ω⊥ is large (Ω⊥ ≫ 1). For example, taking Jupiter and Io as

primaries, Tp ≈ 1.769 days. If TF RT ≈ 25 minutes then Ω⊥ > 100. It is reasonable to take the limit Ω⊥ → ∞ in the governing equations.

• The tether’s characteristic length, λ = a2

2

• L

ν1/3 ℓ 2 appears in a natural way in the problem (here ν = µ/(µ1 + µ) is the reduced mass of the small primary or central body).

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The Hill approach for fast rotating tethers

¨ ξ − 2 ˙ η = (3 − 1 ˆ ρ3 )ξ − λ ˆ ρ5

• 3 ˜

N sin φ2 − ξS2( ˜ N ˆ ρ )

• ¨

η + 2 ˙ ξ = − η ˆ ρ3 + λ ˆ ρ5

• 3 ˜

N cos φ2 sin φ1 + ηS2( ˜ N ˆ ρ )

• ¨

ζ = −ζ(1 + 1 ˆ ρ3 ) − λ ˆ ρ5

• 3 ˜

N cos φ2 cos φ1 − ζS2( ˜ N ˆ ρ )

• dφ1

dτ = cos φ1 tan φ2 dφ2 dτ = − sin φ1 ˜ N = ξ sin φ2 − (η sin φ1 − ζ cos φ1) cos φ2 These equations should be integrated from the initial conditions: at τ = 0 : ξ = ξ0, η = η0, ζ = ζ0, ˙ ξ = ˙ ξ0, ˙ η = ˙ η0, ˙ ζ = ˙ ζ0, φ1 = φ10, φ2 = φ20 If the initial conditions are φ10 = φ20 = 0 ⇒ φ1(τ) = φ2(τ) ≡ 0.

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The Hill approach for fast rotating tethers

TETHER ROTATION PARALLEL TO THE PRIMARIES PLANE:

ξ′′ − 2 η′ = 3 ξ − ξ ρ3 − 3 2 λ ξ ρ5

• 1 − 5 ζ2

ρ2

• η′′ + 2 ξ′

= − η ρ3 − 3 2 λ η ρ5

• 1 − 5 ζ2

ρ2

• ζ′′

= −ζ − ζ ρ3 − 3 2 λ ζ ρ5

• 3 − 5 ζ2

ρ2

• Formally equal to a Hill-J2 problem:
• J2 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

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Hill+oblateness long-term dynamics

• Studied since the dawn of the space era (Lidov, Kozai)
• β: ratio J2 to third body perturbations (β2 ∝ a−5)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 90 100 110 120 130 140 β I (deg)

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ARIADNA PROGRAM

The END of this Gentil Introduction

◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 32/100

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ARIADNA PROGRAM

Inert Tethers and Periodic Motions

◮ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 33/100

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Inert tethers and periodic motions

In the CRTBP for a mass particle there are three well known families of periodic mo- tion originated from the collinear points: ◮ 1) Lyapunov periodic orbits, starting from highly unstable small ellipses with retro- grade motion around the collinear points ◮ 3) Eight-shaped orbits ◮ 2) Halo orbits bifurcated from the Lya- punov family Figure is an sketch of three starting orbits Vertical oscillations Halo orbits Lyapunov 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 J2 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

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Summary

• 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

• rbit. 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

• f 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

• f 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

• rbits 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

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Summary

k1 stability curve k2 stability curve Minimum distance curve Maximum distance curve 1.5 2.0 2.5 3.0 3.5 4.0 4 4 28 215 1622 0.01 0.2 0.39 0.58 0.78 0.97 Jacobi constant Stability indices Distance Hill units

k1 stability curve k2 stability curve Jupiter The Moon Europa Io Enceladus Deimos Phobos 0.0 0.1 0.2 0.3 0.4 0.5 4 4 28 215 1622 Minimum radius Hill problem units Stability indices

• 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

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Summary

k1 stability curve k2 stability curve Eros Mercury The Moon Europa Io Enceladus 0.00 0.05 0.10 0.15 0.20 4 2 2 4 Minimum radius Hill problem units Stability indices

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

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Summary

0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.4 0.2 0.0 0.2 0.4 0.5 0.0

• 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

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ARIADNA PROGRAM

The END of Inert Tethers and Periodic Motions

◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 39/100

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ARIADNA PROGRAM

Io Exploration with Electrodynamic Tethers

◮ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 40/100

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POWER AND MASSES OF SOME RTG’S

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

PROBLEMS 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

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DEORBITING SATELLITES WITH ELECTRODYNAMIC TETHERS

ANY SATELLITE CAN BE DEORBITED WITH AN ELECTRODYNAMIC TETHER WORKING IN THE

GENERATOR REGIME

∆h = ai − af ∆h

• vi
• vf
• fe
• fe

Deorbiting a satellite

The mechanical energy lost in the deorbiting process is ∆E = mµ 2afai ∆h ≈ mµ 2a2

f

∆h(1 − ∆h af ) and it is invested in several task. Basically:

• to attract the electrons from the infinity to the

tether

• hmics 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) We need: magnetic field and plasma environment. Both are present in Jupiter. THE 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

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SCENE OF THE MISSION

Jupiter inner moonlets

rAdrastea = 128971 km, rMetis = 127969 km rAmalthea = 181300 km, rThebe = 221895 km The plasma environment is co-rotating with Jupiter (as a rigid body); the an- gular velocity is ωJ ≈ 1.7579956104 × 10−4s−1 The orbital velocity in a circular or- bit of radius r is vc = µ r (µ = 1.26686536 × 1017m3/s2). It exists an orbital radius for which the orbital period coincides with the sideral pe- riod of Jupiter (about 9.925 hours); it is the stationary radius (rs) given by rs = ( T√µ 2π )

2 3 ≈ 160009.4329 km Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 43/100

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FUNDAMENTALS OF THE MISSION

• vo
• f e
• f e

− f e Amalthea: m = 2.09 × 1018 kg, af = 181300 km The cable can be removed by using the gravitational attraction of Amalthea to link the tether system and the small Jupiter moon. The tether system moves in an equilibrium position relative to the primaries: Jupiter + Amalthea We tray to deorbit one of the Jupiter moon- 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- mitted thought the cable to the small Jupiter moon that will be deorbited. ∆h = 2∆Ea2

f

mµ To deorbit Amaltea 1mm (∆h = 1mm) im- plies a loss of mechanical energy about ∆E ≈ 4 × 1015J ≈ 1.1 × 109Kwh From a practical point of view the reserve of energy is unbounded

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 44/100

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Equilibrium Positions at the Synodic Frame

J Io L1 L2 Thrust Io atraction

• 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

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EQUILIBRIUM POSITIONS

0.2 0.4 0.6 0.8 1 1.2

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 χ = 3

4

2

3

χmax χmax ξe ζe ROOT 1 ROOT 2 L1 L2 χ = 0 χ = 0 Equilibrium Positions on the plane (ξe, ζe)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 46/100

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EQUILIBRIUM POSITIONS (AMALTEA)

0.2 0.4 0.6 0.8 1 1.2

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

χ = 0 χ = 0 χmax χmax χ = 3

4

2

3

ξe ζe ROOT 1 ROOT 2 L2 L1 In [1] Comparison with the results summarized in [1] [1] A permanent tethered observatory at Jupiter. Dynamical analysis, by J. PELÁEZ & D. J. SCHEERES, (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

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TETHER DESIGN IN METIS. NUMERICAL ANALYSIS

0.1 1 10 10 100 mT (in kg) Wu (kw) L = cte dw = cte L = 50 km L = 5 km dw = 50 mm dw = 5 mm 35 49 76 140

10 km 20 km 30 km 40 km

DIFFERENT OPTIMUM CONFIGURATIONS AT METIS (h = 0.1 MM)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 48/100

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TETHER DESIGN IN METIS. THEORETICAL ANALYSIS

0.1 1 10 10 100 mT (in kg) Wu (kw) L = cte dw = cte L = 50 km L = 5 km dw = 50 mm dw = 5 mm 35 49 76 140

10 km 20 km 30 km 40 km

DIFFERENT OPTIMUM CONFIGURATIONS AT METIS (h = 0.1 MM)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 49/100

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TETHER DESIGN IN IO. NUMERICAL ANALYSIS

0.1 1 10 10 100 200 300 2 3 4 170

mT (kg) Wu (kw) L = cte dw = cte L = 8 km L = 10 km L = 15 km L = 25 km L = 35 km dw = 50 mm dw = 5 mm dw = 10 mm dw = 36 mm

DIFFERENT OPTIMIZED CONFIGURATIONS IN IO (h = 0.05 MM)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 50/100

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THREE DIFFERENT CONFIGURATION

A B C L (km) 25 35 35 dw (mm) 50 25 36 mT (kg) 169 117 170 ZC (Oh) 640 1265 886 Wu (w) 2100 2100 3000 Isc (A) 10.94 5.53 7.89 Iav (A) 1.43 1.04 1.49 φ∗ (deg) 39.7 40.06 40.06 χ · 103 5.79 5.91 8.43 de (km) 200,052 198,119 165,803 T (mN) 5.28 7.4 7.4

Three possible designs

• Aluminum tape 0.05 mm thick
• Self-balanced
• Mass of the S/C 500 kg
• Tether mass ≤ 170 kg
• χ ∈ [0, 0.115] STABLE
• Power from 2 – 3 kw
• distance de large
• Tether tension small ≈ mN
• Appropriate for the scientific explo-

ration of the Io plasma torus

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 51/100

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ROTATING ELECTRODYNAMIC TETHER ORBITING IO

Toward Jupiter Eπ ϕ vIo B x y z ω t O

We take an inertial frame Oxyz (origin Io) Orbital velocity of Io: vIo ≈ 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 E ≈ (vIo − vpl) × B E = Eπ(cos ωt, sin ωt, 0), Eπ ≈ 0.12 V/m

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 52/100

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POWER GENERATION IO

A C II h B IC

A C V Vp Vt EML CHI h B

E E ZT II

Power generation: assumptions

• small values of n∞ no Ohmic effects
• negligible ion losses at the cathodic segment
• control impedance ZC at the cathodic end
• OML regime
• control parameter ζ = z∗

L = 1 − ZC IC EmL Results of the analysis

• Iav = 1

L

L

0 I(z)dz = I0 (1 − 2 5ζ)ζ3/2

• ˙

W = ZCI2

C = I0EmL(1 − ζ)ζ3/2

I0 = 4dw 3π en∞

• 2eEmL3

me

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 53/100

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OPTIMUM POWER GENERATION

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1

ζ

˙ W I0EmL

ζopt = 3 5 , Zopt

C

= 3π √ 2 8 · 1 − ζopt ζ3/2

• pt

· 1 dwn∞

• meEm

e3L = π √ 30 12 · 1 dwn∞

• meEm

e3L

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 54/100

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ROTATING ELECTRODYNAMIC TETHER ORBITING IO

˙ Wav = 1 π π/2

−π/2

cos3/2 ϕ dϕ · 4dw 3π n∞

• 2E3

πL5e3

me

• (1 − ζ) ζ3/2

10 20 30 40 50 60 2 4 6 8 10 12 14

P (kw) Tether length (km) First order approximation of averaged generated power for a 5 cm wide tape tether of different lengths in Io

• rbit.

Power generated with a 25 km long 5 cm wide tape tether along circular retrograde orbits of different radii: r = 1.1 rIo (grey solid line), r = 2.0 rIo (dark solid line), and r = 3.0 rIo (dark dotted line).

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 55/100

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POWER GENERATION IN IONIAN ORBIT

0.1 1 10 10 100 200 300 2 3 4 170 L=8 L=10 L=15 d=50 d=5 d=10 L=25 L=35 d=36

mT (kg) Wu (kw) L = cte dw = cte

DIFFERENT OPTIMIZED CONFIGURATIONS IN IO (h = 0.05 MM)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 56/100

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ORBIT STABILITY

• Io is the Galilean moon more affected by the gravitation of Jupiter
• Lara & Russela 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 rIo the equatorial

retrograde orbit is stable for at least a year F av = 1 π π/2

−π/2

L Iav (u × B)dϕ = 1 π L π/2

−π/2

Iav udϕ

• × B

When the load impedance is controlled for maximum power generation ζ =cte: π/2

−π/2

Iav udϕ = 1 Eπ π/2

−π/2

Iav cos ϕEπ dϕ = Eπ Eπ · π/2

−π/2

Iav cos ϕ dϕ This way we have F av = k′ √Eπ (Eπ × B), k′ ≈ 0.0667 dwn∞

• L5e3

me (5 − 2ζ)ζ3/2 The averaged force has a constant value, in a first approximation, on the synodic frame. aOn 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

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ORBIT STABILITY

F ≈ F(− sin ω t, cos ω t, 0) with F = 0.0667 dwn∞B

• EπL5e3

me · (5 − 2ζ) ζ3/2

5 10 15 20 25 30 1 1.5 2 2.5 3 3.5

r rIo

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

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ORBIT CONTROL

Eπ F ∝ Eπ × B Io vIO Drag arc Thrust arc Schematic of electrodynamic thrust and drag arc for a generic Ionian orbit. We neglect the orbital velocity around Io in the determination of mo- tional electric field The main contribution to the motional electric field comes from Io orbital velocity rather than the spacecraft orbital velocity around Io. The Lorentz force is thrusting the S/C in part of the

• rbit (thrust arc) and braking it (drag arc) in the

rest of the Ionian orbit. Preliminary test with a simple control strategy: 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- ulation has been conducted assuming a 500 kg spacecraft equipped with a 25 km long and 5 cm wide electrodynamic tether starting from a low altitude retrograde circular orbit (r = 1.2 Io radii). B

• Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 59/100

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ORBIT CONTROL

−6 −4 −2 2 4 6 −6 −4 −2 2 4 6

x rIo y rIo

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1 2 3 4 5 6 7 8 9 10

r rIo

time (days) 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

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ARIADNA PROGRAM

The END of Io Exploration with Electrodynamic Tethers

◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 61/100

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ARIADNA PROGRAM

Stability of tethered satellites at collinear lagrangian points

◮ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 62/100

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PREVIOUS ANALYSIS

• 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

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EQUILIBRIUM POSITIONS AT THE SYNODIC FRAME

L1 L2 L3 L4 L5 E1 E2

Equilibrium position with an inert tether Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 64/100

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EXTENDED DUMBBELL MODEL. HILL APPROACH

VARYING LENGTH INERT TETHER

¨ ξ − 2 ˙ η − (3 − 1 ρ3 )ξ = λ ρ5

• 3 ˜

N cos ϕ cos θ − ξS2( ˜ N ρ )

• ¨

η + 2 ˙ ξ + η ρ3 = λ ρ5

• 3 ˜

N cos ϕ sin θ − ηS2( ˜ N ρ )

• ¨

ζ + ζ(1 + 1 ρ3 ) = λ ρ5

• 3 ˜

N sin ϕ − ζS2( ˜ N ρ )

• ¨

θ + (1 + ˙ θ) ˙ Is Is − 2 ˙ ϕ tan ϕ

• + 3 cos θ sin θ = 3 ˜

N ρ5 (−ξ sin θ + η cos θ) cos ϕ ¨ ϕ + ˙ Is Is ˙ ϕ + sin ϕ cos ϕ

• (1 + ˙

θ)2 + 3 cos2 θ

• = 3 ˜

N ρ5 (− sin ϕ[ξ cos θ + η sin θ] + ζ cos ϕ) where ρ =

• ξ2 + η2 + ζ2 and the quantity ˜

N and the function S2(x) are given by ˜ N = ξ cos ϕ cos θ + η cos ϕ sin θ + ζ sin ϕ, S2(x) = 3 2 (5x2 − 1)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 65/100

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EXTENDED DUMBBELL MODEL. HILL APPROACH

λ = Ld ℓ 2 · a2 ν2/3 , a2 = Is m L2

d

∈ [ 1 12, 1 4 ] where, ℓ is the distance between primaries, ν is the reduced mass of the small primary (usually ν ≪ 1), m = m1 + m2 + mT is the total mass of the system and Is is the moment of inertia about a line normal to the tether by the center of mass G of the system; a2 is of order unity and takes its maximum value a2 = 1/4 for a massless tether with equal end masses (m1 = m2). For a tether of varying length the parameter λ is a function of time since the deployed tether mass md and the deployed tether length Ld(t) are changing. Moreover, some terms of these governing equations involve the ratio: ˙ Is Is = 2 ˙ Ld Ld Jg, Jg = 1 − Λd (1 + 3 cos 2φ)

• 3 sin2 2φ − 2Λd

, Λd = md m , cos2 φ = m1 m + Λd 2

• This formulation includes the mass of the tether through the parameter Λd and the mass angle φ. In order

to neglect the tether mass, we only have to introduce the condition Λd = 0 in the above expressions. For a tether of constant length the parameter λ is also constant and the quotient ˙ Is/Is vanishes, that is, ˙ Is/Is = 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

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EXTENDED DUMBBELL MODEL. HILL APPROACH

In the search of equilibrium position we will consider the tether tension given by T0 ≈ m1 m2 m1 + m2 ω2Ld   ˙ ϕ2 + cos2 ϕ{(1 + ˙ θ)2 + 3 cos2 θ} − 1 + 1 ρ3 {3 ˜ N ρ 2 − 1} − ¨ Ld Ld   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 T0 ≈ Tc  cos2 ϕ{1 + 3 cos2 θ} − 1 + 1 ρ3 {3 ˜ N ρ 2 − 1}   , Tc = m1 m2 m1 + m2 ω2Ld 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

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VALUES OF λ FOR SUN AND DIFFERENT PLANETS

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 1000 2000 3000 4000 5000 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

Tether length (km) λ

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 68/100

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VALUES OF λ FOR BINARY SYSTEMS IN THE SOLAR SYSTEM

1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 0.01 1 100 10000 200 400 600 800 1000 Earth-Moon Mars-Phobos Mars-Deimos Saturn-Mimas Saturn-Enceladus Saturn-Rhea Saturn-Titan Neptun-Triton

Tether length (km) λ

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 69/100

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VALUES OF λ FOR BINARY SYSTEMS IN THE JOVIAN WORLD

1e-014 1e-012 1e-010 1e-008 1e-006 0.0001 0.01 1 100 20 40 60 80 100

λ

Metis Adrastea Amalthea Thebe Io Europa Ganymede Callisto

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NON ROT. TETHERS. CONST. LENGTH. EQUILIBRIUM POSITIONS

Small Primary L2 E2 Center of mass G Sketch of the equilibrium position E2 in the neighborhood of L2. There is another one, similar to this, on the left of L1.

ξe = ±ρe, ηe = ζe = ϕe = 0, θe = 0, π, λ = ρ2

e

3 3 ρ3

e − 1 ,

ρ3

e > 1/3

This expression can be expanded when λ ≪ 1 is small and provide the asymptotic solution ξe ≈ ( 1 3)

1 3 + 3 1 3 λ − 9 λ2 + O(λ3)

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

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NON ROT. TETHERS. CONST. LENGTH. STABILITY ANALYSIS

A linear stability analysis shows that the equilibrium positions of that family are unstable for any value of λ.

10

−6

10

−4

10

−2

2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85

R(s5) λ Real part of the unstable eigenvalue as a function of λ

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 72/100

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NON ROTATING TETHERS. VARIABLE LENGTH

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 L2. Dim.

• N. Dim.

ξL2 = ( 1

3)

1 3

xL2 = ( 1

3)

1 3 Lc

3

1 3 λ

3

1 3 λ Lc

∆x L2 (1) (2) O G Distances from the small primary and from the collinear point L2 when λ ≪ 1 (Lc = ℓν1/3)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 73/100

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NON ROTATING TETHERS. VARIABLE LENGTH

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

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LINEAR APPROXIMATION FOR SMALL VALUES OF λ

ξ = ξL + 31/3 λ0(1 + u(τ)) , η = 31/3 λ0 v(τ) , λ = λ0(1 + s(τ)) d2 u d τ 2 − 2 d v d τ − 9 u(τ) + 9 2 3 cos2 θ cos2 ϕ − 1 s(τ) + 27 2 cos2 θ cos2 ϕ − 1 = 0 d2 v d τ 2 + 2 d u d τ + 3 v(τ) − 9 cos2 ϕ cos θ sin θ (1 + s(τ)) = 0 d2 w d τ 2 + 4 w(τ) − 9 cos ϕ cos θ sin ϕ (1 + s(τ)) = 0 (1 + s(τ)) d2 θ d τ 2 + d s d τ (1 + d θ d τ ) − 2 tan ϕ d ϕ d τ

• 1 + d θ

d τ 1 + d s d τ

• + 12 cos θ sin θ (1 + s(τ)) = 0

(1 + s(τ)) d2 ϕ d τ 2 + d s d τ d ϕ d τ + (1 + s(τ)) sin ϕ cos ϕ

• 1 + d θ

d τ 2 + 12 cos2 θ

• = 0

s(τ) = e−β τ (A cos Ωτ + B sin Ωτ)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 75/100

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ONE DIMENSIONAL MOTION

3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 200 400 600 800 1000

t (hours) x(km)

940 960 980 1000 1020 1040 1060 1080 1100 1120 1140 1160 200 400 600 800 1000

t (hours) x(km)

• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 200 400 600 800 1000

t (hours) ˙ L(m/s)

0.0145 0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 200 400 600 800 1000

t (hours) T(N)

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 u0 = 0.15 and ˙ u0 = 0. The nominal tether length is L = 1000 km

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 76/100

T D G

BI-DIMENSIONAL MOTION

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 100 200 300 400 500 600 700 800 900 1000

t (hours) x(km)

940 960 980 1000 1020 1040 1060 1080 1100 1120 1140 100 200 300 400 500 600 700 800 900 1000

t (hours) L(km)

• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 100 200 300 400 500 600 700 800 900 1000

t (hours) ˙ L(m/s)

0.007 0.0075 0.008 0.0085 0.009 0.0095 0.01 100 200 300 400 500 600 700 800 900 1000

t (hours) T (N)

T D G

FULL PROBLEM. PROPORTIONAL CONTROL

The length variation λ(τ) is governed by a proportional control law: λ = λe +

• Ki (xi − xe,i)

where Ki are gains and xi stands for the variables ξ, η, θ. d y d t = M y where yT =

• δξ, δη, δθ, δ ˙

ξ, δ ˙ η, δ ˙ θ

• .

The detailed stability analysis of this equation is cumbersome. Nevertheless, we firstly consider the simpler case in which only one gain Kξ is different from zero. The characteristic polynomial takes the form: s3 + 3 Kξ ξ4

e

−

• 2 − 4

ξ3

e

• s2 +

6 Jg + 27 ξ4

e

+ 6 ξ7

e

• Kξ −
• 105 − 6

ξ3

e

+ 4 ξ6

e

• s+

+6 9 ξ4

e

− 3 ξ7

e

• Kξ −
• 45 + 9

ξ3

e

− 2 ξ6

e

• = 0

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 78/100

T D G

FULL PROBLEM. PROPORTIONAL CONTROL

The Descartes rule of signs provides a sufficient condition for stability to be fulfilled by Kξ. Such a condition is drawn in figure as a function of the equilibrium value λe for a massless tether (Jg = 1).

10

−6

10

−4

10

−2

2 2.5 3 3.5 4 4.5 5

K∗

ξ

λe Sufficient condition for Kξ to stabilize the system as a function of λe. Values over the curve (Kξ > K∗

ξ )

provide stability.

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 79/100

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CONTROL DRAWBACKS. TETHER LENGTH

∆L ∆xe = 1 2 3

1 3 √a2 λ

10

−6

10

−4

10

−2

10 10 10

1

10

2

10

3

λ

d L d xe

Non linear Linear Ratio between variation of tether length and deviations

• f the center of mass vs. λ.

10

−2

10 10

2

10

4

10

2

10

4

10

6

Sun-Earth Earth-Moon Mars-Phobos Jupiter-Amaltea Jupiter-Io Saturn-Enceladus

L (km)

d L d xe

Ratio between variation of tether length and deviations

• f the center of mass vs. the tether length (km)

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 80/100

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CONTROL DRAWBACKS. TETHER TENSION

10

−5

10

−4

10

−3

−5 5 10 15 20

T Tc

∆ξ Maximum and minimum tether tension vs. the initial perturbation ∆ξ for λe = 10−2.

10

−4

10

−3

10

−6

10

−5

10

−4

λe ∆ξ Zero tether tension ∆ξ vs. λe (blue line). Zero tether length ∆ξ vs. λe (red line). 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

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ROTATING TETHERS. EQUILIBRIUM POSITIONS

ξe = ±ρe, ηe = ζe = 0, λe = 4ρ5

e

3

• 3 − 1

ρ3

e

• ,

ρ3

e > 1/3

For small values of λ the above solution provides the asymptotic solution ξe ≈ ( 1 3)

1 3 + 3 1 3 λ

4 − 9 ( λ 4 )2 + O(λ3) Comparing with the same results for a non-rotating tether ξe = ±ρe, ηe = ζe = 0, λ = ρ2

e

3

• 3 ρ3

e − 1

• ,

ρ3

e > 1/3

ξe ≈ ( 1 3)

1 3 + 3 1 3 λ − 9 λ2 + O(λ3) Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 82/100

T D G

ROTATING TETHERS. PROPORTIONAL CONTROL

λ = λe + Kξδξ + K ˙

ξδ ˙

ξ + Kηδη + K ˙

ηδ ˙

η Through the Routh-Hurwitz theorem, it has been found that asymptotic stability is guaranteed if the gains

• f the control law satisfy the relations

Kξ ≥ 4 3 ρe

• 15ρ3

e − 2

• K ˙

ξ > 0

Kη = 0 K ˙

η = 0

We carried out two simulations of a rotating tether with different values of Kξ and K ˙

ξ in order to see the

qualitative behavior of the system. The characteristics of the simulations are: λe = 0.0401 (ξe = 0.707); ξ0 − ξe = 10−3, ηe = 10−3; Ω⊥ = 50.

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 83/100

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SIMULATIONS

0.7055 0.706 0.7065 0.707 0.7075 0.708 0.7085 0.709 −3 −2 −1 1 2 3 x 10

−3

δξ δ ˙ ξ

Kξ = 4

0.7065 0.707 0.7075 0.708 −15 −10 −5 5 x 10

−4

δξ δ ˙ ξ

K ˙ ξ = 3 K ˙ ξ = 0.5

−1.5 −1 −0.5 0.5 1 1.5 x 10

−3

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10

−3

δη δ ˙ η

Kξ = 4

−5 5 10 x 10

−4

−1.5 −1 −0.5 0.5 1 x 10

−3

δη δ ˙ η

K ˙ ξ = 3 K ˙ ξ = 0.5

λe = 0.0401 (ξe = 0.707); ξ0 − ξe = 10−3, ηe = 10−3; Ω⊥ = 50

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 84/100

T D G

SIMULATIONS

5 10 15 20 25 30 48 50 52 54 56 58 60 62 64 66

τ Ω⊥

Kξ = 4

5 10 15 20 25 30 50 51 52 53 54 55 56 57 58

τ Ω⊥

K ˙ ξ = 3 K ˙ ξ = 0.5

5 10 15 20 25 30 0.185 0.19 0.195 0.2 0.205 0.21 0.215

√ λ τ

Kξ = 4

5 10 15 20 25 30 0.19 0.195 0.2 0.205 0.21 0.215

√ λ τ

K ˙ ξ = 3 K ˙ ξ = 0.5 Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 85/100

T D G

ARIADNA PROGRAM

The END of Stability of tethered satellites at collinear lagrangian points

◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 86/100

T D G

Lyapunov orbits

1.5 1.0 0.5 0.0 0.5 1.0 1.5 3 2 1 1 2 3 x y

Horizontal stability curve Vertical stability curve Orbit period curve 15 10 5 38 9 2 2 9 38 161 682 2889 3.4 4. 4.5 5.1 5.7 6.3 6.9 7.4 8. 8.6 Jacobi constant Stability indices Period

• Left: sample orbits of the family of Lyapunov orbits around L1 (dashed) and L2 (full line) for, from

larger to smaller, C = −1, 0, 1, 2, 3, 4. Right: stability-period diagram of the family of Lyapunov orbits

• f 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

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Tethered family of Lyapunov orbits: unstable

Horizontal stability curve Vertical stability curve Orbit period curve Critical points 1 2 3 4 6 64 663 6870 6.2 6.4 6.5 6.6 6.8 6.9 Tether's characteristic length Stability index Period

0.5 0.0 0.5 1.0 1.5 2 1 1 2 Ξ Η

• 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

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Tethered family of Lyapunov orbits: unstable

Horizontal stability curve Vertical stability curve Orbit period curve Critical orbits 1 2 3 2 1 16 186 2164 25188 3. 3.2 3.3 3.5 3.6 3.7 Tether's characteristic length Stability indices Period

0.66 0.7 0.74 0.78 0.2 0.1 0.0 0.1 0.2 Ξ Η

• Left: stability-period diagram of a family of Lyapunov orbits close to L2 for tether’s length variations;

the horizontal gray lines correspond to the critical values k = ±2 (in the arcsinh scale). Rigth: starting

• rbit (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

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Family of eight-shaped orbits

0.0 0.2 0.4 0.6 Ξ 0.2 0.2 Η 1 1 Ζ

k1 stability curve k2 stability curve Orbit period curve Critical point 8 6 4 2 2 4 2 7 31 130 549 2327 3.1 3.7 4.4 5. 5.6 6.2 Jacobi constant Stability indices Period

• 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

T D G

Tethered family of eight-shaped orbits

k1 stability curve k2 stability curve Orbit period curve 1: start 2 3 4 5 6: end 0.00 0.05 0.10 0.15 0.20 2 3 17 90 475 2501 2.79 2.93 3.07 3.21 3.34 3.48 3.62 Tether's characteristic length Stability indices Period

0.0 0.2 0.4 0.6 Ξ 0.07 0.07 0.5 0.0 0.5 Ζ

• 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

• rbits 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

T D G

Tethered family of eight-shaped orbits

k1 stability curve k2 stability curve Orbit period curve 1: start 2 3 4 5 6: end 0.00 0.05 0.10 0.15 0.20 0.25 0.30 2 1 10 87 788 7122 64375 2.12 2.42 2.72 3.02 3.32 3.61 3.91 Tether's characteristic length Stability indices Period

0.0 0.2 0.4 0.6 Ξ 0.06 0.06 0.5 0.0 0.5 Ζ

• 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

• rbits 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

T D G

Tethered family of eight-shaped orbits

k1 stability curve k2 stability curve Orbit period curve Bifurcation orbits 1 2 3 4 5 2 2 9 38 161 682 2.4 2.6 2.7 2.8 2.9 3. Tether's characteristic length Stability indices Period

• 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

T D G

Family of Halo orbits

k1 stability curve k2 stability curve Orbit period curve Reflection 1.5 2.0 2.5 3.0 3.5 4.0 2. 0. 2. 8. 30. 100. 400. 1600. 1.5 1.8 2. 2.2 2.4 2.6 2.8 3.1 Jacobi constant Stability indices Period

0.0 0.2 0.4 0.6 Ξ 0.4 0.2 0.0 0.2 0.4 Η 0.5 0.0 Ζ

• 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 L2 point, respectively). (C is the Jacobi constant)

◮ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 94/100

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Tethered family of Halo orbits

k1 stability curve k2 stability curve Orbit period curve 0.00001 0.00003 0.00005 0.00007 1 1 2 2.26 2.27 2.28 2.29 2.3 2.31 2.32 Tether's characteristic length Stability indices Period

0.0 0.2 0.4 0.6 Ξ 0.5 0.0 0.5 Η 0.5 0.0 Ζ

• 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

T D G

Tethered family of Halo orbits

k1 stability curve k2 stability curve Orbit period curve 1: start 2: reflection 3 4: end 0.000 0.001 0.002 0.003 0.004 0.005 0.006 3 1 2 5 14 2. 2.1 2.2 2.3 2.4 2.5 Tether's characteristic length Stability indices Period

0.0 0.2 0.4 0.6 Ξ 0.5 0.0 0.5 Η 0.5 0.0 Ζ

• 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

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Tethered family of Halo orbits

k1 stability curve k2 stability curve Orbit period curve 1: start 2: reflection 3 4: end 0.000 0.002 0.004 0.006 0.008 0.010 4 1 2 6 19 1.9 2.1 2.2 2.3 2.5 2.6 Tether's characteristic length Stability indices Period

0.0 0.2 0.4 0.6 Ξ 0.5 0.0 0.5 Η 0.5 0.0 Ζ

• Stability-period diagram of the family of Halo orbit with C = 1.2 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 = 0.009. (C is the Jacobi constant)

◮ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 97/100

T D G

Tethered family of Halo orbits

k1 stability curve k2 stability curve Orbit period curve 1: start 2: reflection 3 4 5 6 7: end 0.00 0.02 0.04 0.06 0.08 51 3 1 23 383 6391 2.1 2.3 2.4 2.6 2.8 2.9 Tether's characteristic length Stability indices Period

• Stability-period diagram of the family of Halo orbit with C = 2 for tether’s length variations. The

horizontal, gray lines mark the critical values k = ±2 in the arcsinh scale. (C is the Jacobi constant)

◭ ◭

Dynamics and Stability of Tethered Satellites at Lagrangian Points – p. 98/100