spin-orbit interaction and chaos in celestial mechanics
TRANSCRIPT
Spin-Orbit Interaction and Chaos in Celestial Mechanics
Manuel Maria Murteira Barreira da Cruz
Thesis to obtain the Master of Science Degree in
Engineering Physics
Supervisor(s): Prof. Doutor Rui Manuel Agostinho Dilão
Examination Committee
Chairperson: Profa¯ . Doutora Maria Joana Patrício Gonçalves de SáSupervisor: Prof. Doutor Rui Manuel Agostinho DilãoMember of the Committee: Prof. Doutor Paulo Jorge Soares Gil
May 2017
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In memory of my grandfather...
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Acknowledgements
I would like to start by thanking my supervisor, Professor Rui Dilao, for his guidance, valuable advices
and support throughout this work. He has been responsible for introducing me to the world of dynamical
systems and celestial mechanics research in particular, and for broadening my horizons in physics and
mathematics in general, and I am very grateful for the opportunity of working with him.
Finally, I wish to thank my friends who accompanied me during this journey at Instituto Superior
Tecnico, and my family, especially my grandmother, my mother and my two grandaunts, for giving me
the means to complete this course, and for their continued support and encouragement. A special word
to my grandfather, who passed away shortly after I entered the course, and without whom I couldn’t have
reached this far.
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Resumo
No contexto da interaccao spin-orbita em Mecanica Celeste podem ocorrer sincronizacoes ou res-
sonancias (instabilidades) entre dois movimentos periodicos.
Nesta dissertacao encara-se o problema da interaccao spin-orbita considerando um sistema con-
stituıdo por um corpo, que se assume ter a forma de um haltere, e uma massa pontual, que interagem
atraves da forca gravitacional. O modelo do haltere e util, pois este constitui o exemplo mais simples
de um corpo rıgido. A este sistema e dado o nome de Haltere Kepleriano. Sao obtidas as equacoes
do movimento do Haltere Kepleriano e estudadas as suas solucoes estacionarias, bem como a sua
estabilidade. Mostra-se que todas as solucoes estacionarias do Haltere Kepleriano sao instaveis, e que
o sistema e estruturalmente instavel no caso de massas iguais do haltere.
Depois de analisado o Haltere Kepleriano, o estudo da dinamica do haltere e incorporado num
contexto do Problema Restrito dos Tres Corpos. Derivam-se as equacoes do movimento do Problema
Restrito dos Tres Corpos com satelite em forma de haltere e analisam-se dois casos particulares — o
Problema Planar Restrito dos Tres Corpos e o Problema de Sitnikov — sendo determinadas condicoes
necessarias a sua ocorrencia. Com base nesta analise, sao encontradas as solucoes estacionarias do
sistema. Estas sao os analogos aos pontos de Lagrange do Problema Circular Restrito dos Tres Corpos
convencional.
Por ultimo, e realizada uma simulacao numerica do caso em que o haltere esta confinado a mover-
se e a rodar no plano de Lagrange, sendo encontradas evidencias de caos nalgumas das solucoes do
problema.
Palavras-chave: Haltere Kepleriano, Interaccao Spin-Orbita, Sincronizacao, Ressonancia,
Caos, Problema Restrito dos Tres Corpos
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Abstract
A synchronisation or a resonance (instability) between two periodic motions may occur in the context of
spin-orbit interaction in Celestial Mechanics.
We approach the study of the spin-orbit interaction by considering a dumbbell-shaped body, which
has the minimal features of a rigid body, moving in the gravitational field of a point mass, a system that
we call the Keplerian Dumbbell system. We derive the equations of motion of the Keplerian Dumbbell
system and analyse the steady states and their stability. We show that all the steady states of the
Keplerian Dumbbell system are Lyapunov unstable. For the case where the two masses of the dumbbell
are equal, the Keplerian Dumbbell system is structurally unstable.
Following up our study of the Keplerian Dumbbell, we incorporate the analysis of the dumbbell dy-
namics into the framework of the Restricted Three-Body Problem. We derive the equations of motion
of the Restricted Three-Body Problem with dumbbell satellite and study two special cases — the Planar
Circular Restricted Three-Body Problem and the Sitnikov Problem. Necessary conditions for these spe-
cial cases to occur are determined, and, based on that, the steady states of the system are found. These
steady states are the direct analogues of the Lagrangian points in the conventional Circular Restricted
Three-Body Problem.
Finally, we do a numerical analysis of the case in which the dumbbell is constrained to move and
rotate in the Lagrange plane. Evidences of chaotic dynamics are found in several solutions to this
problem.
Keywords: Keplerian Dumbbell, Spin-Orbit Interaction, Synchronisation, Resonance, Chaos,
Restricted Three-Body Problem
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Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
1 Introduction 1
1.1 The Spin-Orbit Effect in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Tidal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 An Averaged Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Capture into Synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Surface of Section and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Stationary Solutions of the Keplerian Dumbbell System 11
2.1 Equations of motion of the Keplerian Dumbbell system . . . . . . . . . . . . . . . . . . . . 12
2.2 Equations of motion in dimensionless form . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Steady states and stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Steady state 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Steady state 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Steady state 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.4 Steady states 4 to 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Numerical analysis of the non-stability of some steady state orbits . . . . . . . . . . . . . 24
3 Restricted Three-Body Problem with Dumbbell Satellite 27
3.1 The Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Equations of motion of the dumbbell satellite . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Equations of motion in the synodic reference frame, in dimensionless form . . . . . . . . 33
3.4 Steady states and other solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 PCR3BP with dumbbell satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.2 Sitnikov Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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3.5 Numerical analysis of the case under category 4 of the PCR3BP with dumbbell satellite . 63
4 Conclusions 69
Bibliography 71
A Fixed Points and Steady States of Differential Equations 75
A.1 Differential equations as dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.2 Stability of the fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B Particular Solutions of the General Three-Body Problem 81
B.1 Eulerian solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.2 Lagrangian solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.3 Other periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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List of Tables
1.1 Solar System data, including the rotation states of some of the major, natural satellites. . 2
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List of Figures
1.1 Inertial (a) and rotating (b) reference frames used in the calculation of the asymptotic spin
rate of a satellite in the presence of tidal friction. . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The coordinates used in the averaging theory of spin-orbit interaction. . . . . . . . . . . . 5
1.3 A Surface of Section (SOS) for α = 0.2 and e = 0.1 and ten different trajectories in(θ, θ
)phase space. Image taken from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 A SOS for α = 0.89 and e = 0.1, values appropriate for Hyperion. Image taken from [16]. . 9
2.1 Reference frames and coordinates used in the study of the Keplerian Dumbbell (Keplerian
Dumbbell (KD)) system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Steady state 1 of the KD system, represented in configuration space (Eulerian solutions,
see Appendix B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Steady states 2 (a) and 3 (b) of the KD system in configuration space. . . . . . . . . . . . 19
2.4 Steady state 6 of the KD system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Eigenvalues λ1 and λ2 of the reduced matrix Mss, (2.37), as a function of the steady state
orbital radius u0 of the dumbbell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Motion of the primariesm1,m2 in the inertial reference frame S centred at their barycentre.
An eccentricity of 0.5, a mass ratio m2/m1 of 0.2, and a longitude of the pericentre of π/6
were used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Reference frames and coordinates used in the study of the Restricted Three-Body Prob-
lem (Restricted Three-Body Problem (R3BP)) with dumbbell satellite. . . . . . . . . . . . . 30
3.3 Inertial S and synodicR reference frames used in the analysis of the R3BP with dumbbell
satellite. The reference frame R corotates with the masses m1 and m2. . . . . . . . . . . 34
3.4 Possible configuration of the dumbbell and the primaries in the Planar Circular Restricted
Three-Body Problem (PCR3BP) with dumbbell satellite. . . . . . . . . . . . . . . . . . . . 44
3.5 Two steady states of the PCR3BP with dumbbell satellite, for η = 0.2 and ε = 0.15,
represented in the inertial S and synodic R frames at θ = 3π/5 ((a), (c)) and at θ = 5π/4
((b), (d)), for which the dumbbell lies in the Lagrange plane (ψ = π/2) and maintains a
right angle to the line that joins the primaries (θ − φ = π/2). . . . . . . . . . . . . . . . . . 47
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3.6 Steady states (3.53) of the PCR3BP with dumbbell satellite represented in the inertial S
((a), (b)) and synodic R ((c), (d)) frames at θ = 5π/6, for ε = 0.15. (a), (c) The dumbbell
lies in the Lagrange plane (ψ = π/2) and maintains a right angle to the line that joins the
primaries (θ − φ = π/2); (b), (d) the dumbbell is lined up with the W axis (ψ = 0). . . . . . 49
3.7 Steady states (3.58) of the PCR3BP with dumbbell satellite, for ε = 0.15, in which the
dumbbell lies in the Lagrange plane (ψ = π/2) and is aligned along the direction of the
segment line that joins the primaries (θ = φ), represented in the inertial S ((a), (b)) and
synodic R ((c), (d)) reference frames. (a), (c) v = 0: the dumbbell and primaries are
depicted at θ = 4π/3; (b), (d) v(ε = 0.15, ψ = π/2) ≈ 0.867: the dumbbell and primaries
are depicted at θ = 7π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.8 Steady state (3.66) of the PCR3BP with dumbbell satellite, for η = 0.2 and ε = 0.15,
represented in the inertial S (a) and synodic R (b) reference frames at θ = 5π/3. The
dumbbell is aligned perpendicularly to the Lagrange plane (ψ = 0). . . . . . . . . . . . . . 54
3.9 Schematic representation of a configuration of the dumbbell and the primaries in the Sit-
nikov Problem (SP), in the inertial frame S. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.10 Trajectory of the centre of mass of the dumbbell in inertial space (a) and time evolution
of the angular difference θ − φ (b) of a solution of (3.70) which starts close to one of the
fixed points (3.51). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.11 Trajectory of the centre of mass of the dumbbell in inertial space (b), its initial phase (a),
ending phase (c), and time evolution of the angular difference θ − φ (d) of a solution of
(3.70) which starts close to another of the fixed points (3.51). . . . . . . . . . . . . . . . . 65
3.12 Ejection of the dumbbell from the system (c) and initial phase of its trajectory (a) for a
solution of (3.70) which starts close to one of the fixed points (3.51). . . . . . . . . . . . . 66
3.13 Trajectory of the centre of mass of the dumbbell in inertial space (b), its initial phase (a),
ending phase (c), and time evolution of the angular difference θ − φ (d) of a solution of
(3.70) which does not start close to any fixed point. . . . . . . . . . . . . . . . . . . . . . . 67
A.1 Lyapunov stable fixed point (a). Asymptotically stable fixed point (b). . . . . . . . . . . . . 77
A.2 Graph of a Lyapunov function (a). Geometrical meaning of the Lyapunov theorem (b). . . 78
A.3 Orbits of a nonlinear dynamical system in phase space (left), and corresponding approxi-
mate linear system (right), in a neighbourhood of an hyperbolic fixed point. . . . . . . . . 79
B.1 Eulerian solutions of the General Three-Body Problem. (a) unequal masses; (b)–(c) all
masses equal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
B.2 Lagrangian solutions of the General Three-Body Problem for the case of equal masses.
(a) the bodies move in identical ellipses; (b) the bodies trace the same circular orbit. . . . 82
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List of Symbols
Greek Symbols
α (Ch. 1) Asphericity parameter.
α,β,γ (Chs. 2, 3) Unit vectors defining the principal
axes of inertia of the dumbbell.
αc Critical value of the asphericity parameter,
above which chaos is expected.
αH Asphericity parameter of Hyperion.
γ (Ch. 1) Angle between the long axis of the
satellite and the satellite-planet centre line at
pericentre.
γ0 Equilibrium value of γ.
δ (Ch. 1) Phase lag of the tidal bulge. (Ch. 2, 3)
Dimensionless parameter measuring the rela-
tive weight of the masses of the dumbbell.
∆E Energy dissipated over one cycle of tidal work-
ing.
ε Lag angle of the tidal bulge.
ε Dimensionless parameter measuring the length
of the dumbbell.
η (Ch. 3) Dimensionless parameter measuring
the relative weight of the masses of the pri-
maries.
η (Ch. 1) Satellite’s spin rate in a frame centred
on and rotating with the satellite’s mean motion.
θ (Ch. 1) Angle between the long axis of the
satellite and the major axis of the satellite’s or-
bit. (Ch. 2) Angular coordinate of the orbit of
the centre of mass of the dumbbell. (Ch. 3) An-
gular coordinate of the Keplerian orbits of the
primaries.
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θ (Ch. 1) Satellite’s spin rate. (Ch. 2) Angular ve-
locity of the centre of mass of the dumbbell.
(Ch. 3) Angular velocity of the primaries about
each other and their barycentre.
θ∗ Angular velocity for the circular orbit of the Ke-
pler problem.
λ (Ch. 2) Positive eigenvalues of the Hessian ma-
trix Ms (steady states 4–6). (Ch. 3) Semi-major
axis of the orbit of one of the primaries about
the other normalised to the radius of the circu-
lar orbit of the Kepler problem.
λ1, λ2 (Ch. 2) Eigenvalues of the reduced Hessian
matrix Mss (steady states 4–6).
µ (Ch. 2) Dimensionless parameter measuring
the relation between the masses of the dumb-
bell and of the primary. (Ch. 3) The product of
the gravitational constant G and the sum of the
masses of the primaries, m1 +m2.
ν (Ch. 3) Dimensionless parameter measuring
the relation between the masses of the dumb-
bell and the masses of the two primaries.
ξ (Ch. 2) Inverse of the steady state orbital radius
of the dumbbell, u0 (steady states 4–6). (Ch. 3)
Longitude of the pericentre.
ρ (Ch. 3) Dimensionless distance between the
primaries.
ρij (Ch. 2, 3) Dimensionless vector pointing from
mass i to mass j.
σ (Ch. 2) Normalisation frequency.
τ Dimensionless time.
ϕ Angle between the radius vector from the pri-
mary to the satellite and the line joining the
satellite to the empty focus of its elliptical orbit.
φ (Chs. 2, 3) Azimuthal angle on the sphere of
rotation of the dumbbell.
χ (Ch. 2) Canonical variable (difference between
the angles θ and φ).
xviii
ψ (Ch. 1) Angle between the long axis of the
satellite and the satellite-planet centre line.
(Chs. 2, 3) Polar angle on the sphere of rota-
tion of the dumbbell.
ω (Ch. 2) Steady state angular velocity of the cen-
tre of mass of the dumbbell.
ω (Chs. 2, 3) Rotational angular velocity of the
dumbbell.
ω0 Libration frequency (pendulum equation).
Latin Symbols
a Semi-major axis of the Keplerian elliptical orbit.
A,B,C (Ch. 1) Satellite’s principal moments of inertia.
c (Ch. 2) Constant (steady state 1).
C Centre of mass of m1,m2,m3 (Ch. 2) or m1,m2
(Ch. 3).
C ′ Centre of mass of the dumbbell.
d day (unit of time).
D Magnitude of the tidal torque.
e Eccentricity.
e1, e2, e3 Unit vectors of the coordinate axes of the refer-
ence frames S and S ′.
E Satellite’s total energy.
E0 Peak energy stored in the body’s tidal distortion
over one cycle of tidal working.
f (Chs. 1, 3) True anomaly.
f, h,m Functions.
g (Ch. 1) Guiding centre. (Ch. 2) Energy scale of
the Keplerian Dumbbell (KD) system.
G Gravitational constant.
H (Ch. 2) Hamiltonian of the KD system. (Ch. 3)
Hamiltonian of the dumbbell.
H∗,Heff Effective Hamiltonians.
H(p, e) Power series in the eccentricity.
Iα, Iβ, Iγ Principal moments of inertia of the dumbbell.
k2 Love number of degree 2.
` Dumbbell’s length.
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`2, `3, `4 (Ch. 2) `2, `3: distances of the masses m2 and
m3 to the centre of mass C ′ of the dumbbell,
respectively. (Ch. 3) `3, `4: distances of the
masses m3 and m4 to the centre of mass C ′
of the dumbbell, respectively.
L, L (Ch. 2) Total angular momentum of the KD sys-
tem.
Lz (Ch. 2) Third component of the total angular
momentum of the KD system. (Ch. 3) Angular
momentum integral of the Kepler problem.
L1+2 (Ch. 3) Total orbital angular momentum of the
primaries.
L,L (Ch. 2) Lagrangian of the KD system. (Ch. 3)
L: Lagrangian of the Four-Body Problem.
L1+2 Lagrangian of the Kepler problem.
Ldumbb.,Ldumbb. (Ch. 3) Lagrangian of the dumbbell.
mp Mass of the planet (primary).
ms Mass of the satellite.
m1,m2,m3,m4 (Ch. 2) m1: mass of the primary; m2,m3:
masses of the dumbbell. (Ch. 3) m1,m2:
masses of the primaries; m3,m4: masses of
the dumbbell.
M Mean anomaly.
Ms Hessian matrix.
Mss Reduced Hessian matrix.
n Average orbital angular velocity (mean motion).
Ns Tidal torque.
p Ratio of the satellite’s spin rate to its mean mo-
tion (rational number).
px Canonical momenta.
Q Specific dissipation function.
Qp Planet’s specific dissipation function.
Qs Specific dissipation function of the satellite.
r Instantaneous radius of the Keplerian orbit.
r0 (Ch. 2) Characteristic length of the KD system
(normalisation distance).
r∗ Radius of the circular orbit of the Kepler prob-
lem.
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r, θ, z (Ch. 2) Cylindrical coordinates of the centre of
mass of the dumbbell, referred to mass m1, in
the inertial reference frame S.
rij (Ch. 2, 3) Vector pointing from mass i to mass
j.
Rs Satellite radius.
R (Ch. 3) Synodic reference frame.
S Inertial reference frame centred at the centre of
mass of m1,m2,m3 (Ch. 2) or m1,m2 (Ch. 3).
S ′ Reference frame centred at the centre of mass
of the dumbbell, with coordinate axes parallel
to those of S.
t Time.
T (Ch. 1) Orbital period. (Ch. 2) Total kinetic en-
ergy of the KD system. (Ch. 3) Kinetic energy
of the dumbbell.
Trot Rotation period of Nereid.
u0 (Ch. 2) Steady state orbital radius of the dumb-
bell.
u, v, w (Ch. 3) Dimensionless, cartesian coordinates
of the centre of mass of the dumbbell in the
synodic reference frame R.
u, θ, v (Ch. 2) Dimensionless, cylindrical coordinates
of the centre of mass of the dumbbell, referred
to mass m1, in the inertial reference frame S.
U (Ch. 2) Total potential energy of the KD system.
(Ch. 3) Potential energy of the dumbbell.
U, V,W (Ch. 3) Dimensionless, cartesian coordinates
of the centre of mass of the dumbbell in the in-
ertial reference frame S.
V (Ch. 3) Effective potential.
x, y, z Cartesian coordinates in the reference frames
S,S ′ (Ch. 2) and in the synodic reference frame
R (Ch. 3).
X,Y, Z Cartesian coordinates in the reference frames
S,S ′ (Ch. 3).
y year (unit of time).
Superscripts
T Transpose.
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xxii
List of Acronyms
CR3BP Circular Restricted Three-Body Problem
KD Keplerian Dumbbell
PCR3BP Planar Circular Restricted Three-Body Problem
R3BP Restricted Three-Body Problem
SOS Surface of Section
SP Sitnikov Problem
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Chapter 1
Introduction
Nix is a small satellite that orbits the Pluto-Charon system. It was discovered in 2005 and confirmed in
2006, by the Hubble Space Telescope, together with the other small moon Hydra [1]. Analysis of the
variations in the light reflected by these moons shows unpredictable changes in their brightness. The
spin axis of both moons tumbles chaotically, likely due to the time-varying, asymmetric gravitational field
created by Pluto and Charon, as these orbit their common barycentre, and this effect is only enhanced
by the non-spherical shape of the moons. Showalter and Hamilton [2] have made dynamical simulations
of the rotational period and orientation of Nix versus time which support this hypothesis. Their results
demonstrate that Nix’s orientation is fundamentally unpredictable and that, most of the time, its instanta-
neous rotational period is completely unrelated to its orbital period, although, for brief periods, the small
moon’s long axis tends to oscillate about the direction towards the barycentre of the Pluto-Charon binary
system [2].
The main goal of this work is to characterise the spin-orbit coupling mechanism responsible for
the observed chaotic behaviour of Nix and to describe the synchronisation and the transition to chaos
of Nix’s rotation, in the framework of the Restricted Three-Body Problem (R3BP), Szebehely [3]. We
shall start, though, by analysing the simpler problem which we call the Keplerian Dumbbell (KD), which
concerns the spin-orbit interaction of a dumbbell satellite moving around a primary planet in an elliptic
Keplerian orbit. In the next section, we recall and review some of the main results and procedures of the
theory of spin-orbit interaction.
1.1 The Spin-Orbit Effect in the Solar System
The so-called spin-orbit interaction, or coupling, in Celestial Mechanics refers to the dependence of the
rotational period of a satellite on its orbital period around a primary planet or star, and it manifests itself
in the form of a synchronisation of two periodic motions, [4, 5]. There are commensurabilities between
various types of frequencies or periods; for instance, the orbit-orbit coupling involves the orbital periods
of two or more bodies. However, our main focus is on the synchronisation and/or resonance between
the rotational and the orbital periods of a single body. The most obvious example of these is the Moon,
1
whose orbital period is exactly equal to its rotational period, which leads to the well-known fact that it
keeps the same face towards the Earth at all times: we say that the Moon is in a 1:1 or synchronous
state. Another example of a synchronous 1:1 rotation of particular interest is that of Charon, the major
moon of the dwarf planet Pluto. In this case, not only are the rotational and orbital periods of Charon
equal, they are also equal to the period of rotation of Pluto, which results in the satellite being seen
always in the same position in the sky from the planet.1 In fact, though, most of the major natural
satellites in the solar system are rotating in a 1:1 synchronous state, as shown in Table 1.1. The notable
exceptions to this spin-orbit state among the satellites are Hyperion and Nereid; the rotation period of
the former is chaotic [6, 7], while the rotation period of the latter seems to be near a 750:1 state [8].
Nereid is a mystery, and its rotation appears to be rather irregular [9]. Mercury, on the other hand, was
found to be in a 3:2 synchronous spin-orbit state as it revolves around the Sun, thereby completing three
rotations on its axis while making two revolutions around the Sun [10].
Planet Satellite e T (d) Rotation StateMercury 0.206 87.97 3:2Earth Moon 0.054900 27.321661 1:1Mars Phobos 0.0151 0.318910 1:1
Deimos 0.00033 1.262441 1:1Jupiter Io 0.0041 1.769138 1:1
Europa 0.0101 3.551810 1:1Ganymede 0.0015 7.154553 1:1Callisto 0.007 16.689018 1:1
Saturn Epimetheus 0.009 0.694590 1:1Enceladus 0.0045 1.370218 1:1Titan 0.0292 15.945421 1:1Hyperion 0.1042 21.276609 chaotic
Uranus Miranda 0.0027 1.413 1:1Ariel 0.0034 2.520 1:1Umbriel 0.0050 4.144 1:1Oberon 0.0008 13.463 1:1
Neptune Proteus 0.000 1.122315 1:1Triton 0.0004 -5.876854 1:1Nereid 0.7512 360.13619 Trot = 0.48 d†
Pluto Charon 0.0076 6.387223 1:1
Table 1.1: Orbital period (T ) and synchronisation state of some of the major, natural satellites in thesolar system. We also show the eccentricity (e) of the orbit of the satellites. Data taken from [4]. † Thevalue of the rotation period of Nereid Trot has been taken from [8].
1.1.1 Tidal Theory
A body raises a tide on another because the gravitational force varies across bodies, and a tidal bulge is
created in a position directly under the perturber since no body is truly rigid. As is known from the theory
of the tides, tidal oscillations are dissipative processes, because there is friction between and inside the
various layers of a planet or satellite, and so, energy is lost and a phase lag δ is generated in the tidal
response of the planet or satellite. When the rotational angular velocity or spin rate of a satellite differs
from its instantaneous orbital angular velocity or the rate of change of its true anomaly2f , a tidal torque
is created which acts to either brake or spin up the satellite, according to whether its tidal bulge is carried1This is called a state of mutual “tidal-locking” [4].2In celestial mechanics, the true anomaly f is an angular coordinate that determines the position of a body moving along a
Keplerian ellipse. It is defined as the angle between the position of the body and the direction to the pericentre, as seen from thefocus of the ellipse which is filled by the primary body [4].
2
ahead of (positive phase shift) or lags behind (negative phase shift) the direction towards the perturber,
respectively [11]. As Murray and Dermott [4] pointed out, in order that a zero mean tidal torque acts
over one orbital period and equilibrium is reached, the satellite has to spin faster than its average orbital
angular velocity or mean motion n, which suggests that the 1:1 spin-orbit state is not stable. This was
first demonstrated by Peale and Gold [12], which showed that certain models of tidal friction could make
an axially symmetric planet in an eccentric orbit approach an asymptotic spin rate which is faster than its
mean motion, the value of which depends on the variation of the planet’s specific dissipation function3
Qp with the frequency and amplitude of the tides. Consequently, other torques must be present if we
are to have synchronous rotation. These torques are related to permanent deviations from sphericity, or
quadrupole moments, which require the bodies to be partially solid in order to be able to sustain such
deformations [13]. Goldreich [14] calculated the asymptotic spin rate for a satellite in a low eccentricity
orbit in the absence of permanent bulges. In terms of a tidal drag model due to MacDonald [11] that
takes a constant (frequency-independent) lag angle ε = δ/2 for the tidal bulge, the tidal torque he used
was ([4])
Ns = −D(ar
)6sgn (η − ϕ) , (1.1)
where
D =3
2
k2Qs
n4
GR5s (1.2)
is a positive constant, a is the semi-major axis of the orbit of the satellite, r is the instantaneous radius
of this orbit, G is the gravitational constant, G = 6.67408 × 10−11 m3 kg−1 s−2 in SI units, Rs, Qs and
k2 are the radius, specific dissipation function and Love number4 of the satellite, respectively, the time
derivative η is the spin rate of the satellite in a frame that is centred on the satellite and rotates with
its mean motion n (Figure 1.1b), ϕ is the rate of change of the angle between the radius vector from
the primary to the satellite and the line joining the satellite to the empty focus of its elliptical orbit (see
Figures 1.1a, 1.1b), and sgn(·) is the sign function.
The result obtained was η = 192 e
2n [14, 4]. For the Moon, this would amount to about 3% faster
rotation than the synchronous rate, and we would see both sides of the natural satellite over a period of
2.6 y [4].
1.1.2 An Averaged Equation of Motion
In the following we will always assume, unless otherwise stated, that the axis of rotation or spin axis of
the satellite, modelled as an ellipsoid, coincides with its axis of largest principal moment of inertia (the
shorter axis of the satellite) and is normal to the orbital plane of its elliptical path.5 We’ll also neglect any
secular perturbations6, and so this ellipse and spin orientation may be regarded as fixed. Under these
3The specific dissipation function Q of a body is defined by Q = 2πE0/∆E, where ∆E is the energy dissipated and E0 is thepeak energy stored in the body’s tidal distortion over one cycle of tidal working [11].
4The Love number of degree 2 k2 is a dimensionless parameter describing the ratio of the gravitational potential of a tidally-deformed body at a point to the perturbing potential at that point. It measures the susceptibility of a body’s shape to change inresponse to a tidal potential and is zero for a rigid body [15].
5Tidal torques tend to drive the obliquity, the angle between the spin axis and the orbit normal, to zero, thus it is natural to studythis problem in which the spin axis of the satellite is normal to the orbital plane [16].
6A secular perturbation is a non periodic perturbation over the time frame of interest.
3
empty focus planet
rϕ
ε
f
satellite
n+ η
(a)
satellite
η
g
r
planet
ϕε
(b)
Figure 1.1: (a) The elliptical path of a satellite, as seen in a reference frame centred on the primary(planet). The dashed line indicates the axis of the tidal elongation raised on the satellite by the planet,which is carried ahead of the line that joins the two bodies by the angle ε, in this position of the satellite’sorbit. The arrow marks the direction from the centre of mass of the satellite to the empty focus of its orbit.In this reference frame, the satellite’s spin rate is denoted by n + η, n being its mean motion. (b) Thepath of the planet in a reference frame centred on and rotating with the satellite’s mean motion. In thisrotating reference frame, and for small values of the eccentricity e, the planet moves about its guidingcentre, g, on an ellipse with semi-major and semi-minor axes in the ratio 2:1 (see Sections 2.6 and 4.10of [4] on the guiding centre approximation). The satellite’s spin rate equals η in this frame.
conditions, the (full) equation for the rotational motion of the satellite, whose permanent bulge is acted
upon by a surrounding planet, is ([4, 5, 13])
Cθ − 3
2(B −A)
Gmp
r3sin 2ψ = 0, (1.3)
where A < B < C are the three principal moments of inertia of the satellite, θ is the angle between the
long axis of the satellite (the axis of the ellipsoid with the smallest principal moment of inertia, A) and
the semi-major axis of the satellite’s orbit, ψ is the angle the long axis makes with the line that joins the
centres of the two bodies (see Figure 1.2), r is the radius of the orbit, and mp is the mass of the planet.
As r and ψ vary with the true anomaly f , which is a nonlinear function of time for eccentric orbits, the
4
elliptical path
pericentreplanet
mp
r
θ
ψ
f
satellite
Figure 1.2: The coordinates used in the averaging theory of spin-orbit interaction. The long axis of thesatellite, which is modelled as an ellipsoid, makes an angle ψ with the line that connects the planet,seen as a point mass, to the satellite’s centre of mass, and an angle θ with an axis that coincides withthe major axis of the elliptical orbit of the satellite, which is assumed to be fixed in inertial space. Thesatellite rotates in the orbital plane.
equation is nonintegrable. However, near those cases of interest to us, namely, when the spin rate θ is a
rational multiple of the mean motion, an approximate equation may be derived and the motion analysed
in terms of a slowly varying variable γ = θ−pM , where M is the mean anomaly and p a rational number.
One starts by expanding the full equation in a Fourier series in terms of e and M and then averages all
the terms over one orbital period holding γ fixed to obtain ([4])
γ = −sgn(H(p, e))1
2ω20 sin 2γ . (1.4)
This is just the well-known pendulum equation, where
ω0 = n
(3
(B −AC
)|H(p, e)|
) 12
is the libration frequency, and the H(p, e) are power series in the eccentricity, which depend on the
synchronisation state p and are tabulated in Table I of [13]. Moreover, the leading term in the series is
of order e2|p−1| [4, 13].
In this analysis it is implicitly assumed that each synchronisation may be investigated independently
from all others, in that these average out to zero. The angle γ corresponds physically to the deviation of
the long axis of the satellite from the line joining the centres of the planet and satellite at pericentre.
We should emphasise that the only terms which can contribute to the averaged equation of motion
(1.4) are those for which p is an integer multiple of 1/2. This can be seen by inspecting the arguments of
the cosines in the full, series-expanded equation [4]. Therefore, in the context of the underlying theory,
spin-orbit coupling can be expected to arise only when θ is an integer multiple of half the mean motion
5
(p = −1,− 1
2 ,+12 ,+1,+ 3
2 ,etc...).
If we assume that the rotational period of the satellite was much shorter in the beginning of time and
that its spin has been tidally braked, then a mean tidal torque term should be added to the right hand
side of the pendulum equation (1.4) in order to bring the spin closer to a synchronous state. Of course,
it should be at the same time sufficiently close to one of these states for the equation to be a good
approximation to the motion. We then have
γ = −sgn(H(p, e))1
2ω20 sin 2γ +
|〈Ns〉|C
. (1.5)
The magnitude of the tidal torque term should be lower than the amplitude of the torque on the perma-
nent bulge of the satellite; in other words, |〈Ns〉|C < 12ω
20 , so that γ changes sign and the satellite will
librate about γ = 0(θ = pn
), remaining trapped in a synchronous state. This is called the strength
criterion [4]. In this case, γ will librate about a value γ0 for which γ = 0, given by
γ0 =1
2arcsin
(2 |〈Ns〉|
sgn(H(p, e))ω20C
). (1.6)
If the tidal torque is much smaller than the torque on the quadrupole moment, then the value of γ0 is
determined by the sign of H(p, e): if H(p, e) > 0, γ0 ≈ 0 or π, and the long axis of the satellite will point
in the direction of the planet-satellite line at pericentre; otherwise, γ0 ≈ π2 or 3π
2 , and it will be the shorter
axis that will point in the direction of the planet-satellite line at pericentre.
A sufficient stability condition for the pth synchronisation may be derived by substituting equations (1.1)
and (1.2) for the tidal torque due to MacDonald [11] into the strength criterion. A straightforward calcu-
lation shows that (B −A)/C must exceed a threshold value given by ([4])
(B −AC
)t
=5
2
k2Qs
(Rsa
)3mp
ms
1
|H(p, e)|, (1.7)
where ms is the mass of the satellite. It has been shown in [4, 13] that all the spin-orbit states with
1.0 ≤ p ≤ 2.5 for the Moon, for which (B−A)/C ≈ 2.28× 10−4 ([17]), as well as those with 1.0 ≤ p ≤ 3.0
for Mercury, where it is assumed that (B − A)/C for Mercury is of the same order, are stable. Thus, we
now understand the stability of the present spin-orbit states for both the Moon and Mercury. In the next
section we make a brief review of what is believed to be the mechanics behind capture into a spin-orbit
synchronisation.
1.1.3 Capture into Synchronisation
The effective mechanism of capture into synchronisation remains relatively unknown. However, it is
widely accepted that one has to invoke dissipative torques in order to explain the state of a satellite or
planet [18]. One way of accomplishing this is by assuming that the initial spin period of the body was
short and introducing tidal torques, which act to brake this spin. The body will then pass through several
spin-orbit states, and may be captured into any one of them if the value of the spin rate is below a critical
6
value when the angle γ in equation (1.5) enters its first libration, in analogy with a pendulum [19]. In
general, though, one cannot know a priori the initial conditions of the body and, consequently, if this
requirement will be satisfied. So, a suitable probability distribution over the initial angular velocity should
be introduced in order to calculate capture probabilities. Goldreich and Peale [13, 19], for instance,
define these capture probabilities as the ratio of the range of the energy integral of (1.5)7,
E =1
2Cγ2 − 3
4(B −A)n2H(p, e) cos 2γ, (1.8)
for which capture results, to the whole range of E for which γ becomes an angle of libration. However,
they also argue that the mean tidal torque, 〈Ns〉, has to somehow vary with γ for capture to occur,
because if it is constant, the body will simply pass through synchronisation and continue to despin
[13, 19].
Estimating capture probabilities is then just a matter of incorporating a model of tidal dissipation that
accounts for the variation of 〈Ns〉 with γ into the theory. This has been done in [13, 19] for two models in
which the tidal potential is expanded in a Fourier time series and each component of the tide is given a
phase lag [20]. In the first model, which assumes that the phase lags depend on the tidal frequencies,
a very low value of about 7% was calculated for the probability of capture of Mercury into the 3:2 state,
for e = 0.206, and this probability was found to vary with the value of (B−A)/C and the eccentricity. On
the other hand, the second model, in which only the signs, but not the magnitudes of the lags, depend
on frequency, gave a much higher value of about 70% for the corresponding probability, and this was
found to be determined by the synchronisation state p and the eccentricity alone. This model has some
shortcomings though, as it leads to unrealistic constant torques and doesn’t account for the damping of
the librations of the body [4].
Goldreich and Peale [13] have also computed capture probabilities for Mercury using the tidal torque
in equations (1.1) and (1.2), but this torque is actually unrealistic and is only studied for its simplicity.
First, in keeping the geometric lag angle ε constant, it makes the time lag of the tide vary with a body’s
position in its orbit for a nonvanishing eccentricity [19], and second, it permits for abrupt switches of the
torque and is inherently contradictive as it hinges on a tacit assumption that the time lag is constant [21].
An alternative mechanism for Mercury’s capture into the 3:2 state has been proposed in [22], where it
is argued that for any eccentricity, the spin rate of a body is naturally driven towards an equilibrium value
which depends on its current eccentricity. Then, since the eccentricity varies due to the chaotic evolution
of Mercury’s orbit, the spin rate can be raised and lowered, thus making it possible for a synchronisation
to be repeatedly crossed, and increasing Mercury’s probability of becoming trapped. Correia and Laskar
[22] have computed a probability of capture into the 3:2 state for Mercury of 55.4%.
1.1.4 Surface of Section and Chaos
The spin-orbit problem, as defined in Section 1.1.2, may be analysed by producing a Surface of Sec-
tion (SOS) of the motion (Poincare [23], Henon and Heiles [24]), which allows us to test the validity of
7The energy integral is just the result of the integration with respect to time of equation (1.5).
7
Figure 1.3: A SOS for α = 0.2 and e = 0.1 and ten different trajectories in(θ, θ
)phase space. Values of
θ and θ are plotted at successive pericentre passages, and units have been chosen such that the meanmotion is one. The image is taken from [6].
the averaging theory. This is obtained by looking at the system stroboscopically every time the satellite
passes through pericentre and plotting the corresponding values of θ against θ that result from a numer-
ical integration of the full equation of motion, equation (1.3). The variable γ (taken modulo π) is simply
θ at pericentre passage, so librations in γ are equivalent to librations in θ on the SOS [6]. This is why
the choice of the pericentre is a convenient one. A SOS for α =√
3(B −A)/C = 0.2 and e = 0.1 is
shown in Figure 1.3, which illustrates the clear separation between chaotic behaviour and quasiperiodic
motion characteristic of the phase space of Hamiltonian systems with two or more degrees of freedom
[16]. While close to the stable equilibrium points p = +1, p = + 12 and p = + 3
2 trajectories follow a closed
path, successive points in non-synchronous rotation trajectories trace a smooth curve which covers all
values of θ, and motion at the separatrices is chaotic. According to a discussion in [4], there is a very
good agreement between the trajectories shown in Figure 1.3 and the analytic solutions that result from
the averaging method. This is not always the case though, as the half-widths of the libration regions,
measured by the libration frequency ω0, grow with the asphericity parameter α, and, at some point,
the synchronous states will begin to overlap. Chirikov’s resonance overlap criterion then states that as
soon as the sum of two unperturbed half-widths matches the separation of the stable equilibrium points,
chaos will develop [25]. An estimate of the critical value of α above which chaos is expected, made with
Chirikov’s criterion applied to the p = +1 and p = + 32 states, is given by
αc =1
2 +√
14e, (1.9)
and this prediction is actually in excellent agreement with the numerical results [6].
In Figure 1.4 a SOS for values of α and e appropriate for Hyperion is shown, and it is readily apparent
that much of the simplicity of the previous picture is gone. A large chaotic sea now surrounds all the
8
stable islands from p = + 12 to p = +2, and the averaging method is no longer applicable, since each
synchronous state can no longer be taken in isolation. This is primarily a consequence of the irregular
shape of Hyperion, which grants the moon of Saturn a value of α much larger than the critical one,
αH = 0.89, but also of its high forced eccentricity, e ≈ 0.1, due to an orbit-orbit synchronisation with Titan
[6]. Furthermore, the p = + 32 island has disappeared altogether, and a new, second-order8, p = + 9
4
island is now visible in the top centre of the chaotic sea. There is also a displacement associated with
a forced libration with the same period as the orbital period and amplitude equal to the displacement of
the island centres from the theoretical p values [6].
Figure 1.4: A SOS for α = 0.89 and e = 0.1, values appropriate for Hyperion. The image is taken from[16].
Analysis by Wisdom [6, 16, 26] of the stability of the spin axis of Hyperion at the libration regions
and in the chaotic sea has shown that the p = +1 state as well as the chaotic zone are mainly attitude
unstable, which means that the slightest deviation of the spin axis from the orbit normal while the satellite
is in one of these zones grows exponentially on a short timescale, and the satellite tumbles chaotically
in three-dimensional space. Ground-based observations have actually confirmed this behaviour of Hy-
perion’s axis [7]. In fact, though, the half-width of the chaotic separatrix also grows exponentially with α
(while depending only linearly on the eccentricity), and hence, even the small, irregularly shaped satel-
lites in the solar system rotating in the p = +1 state have significant chaotic zones associated with the
separatrix. It has also been shown by Wisdom [16, 26] that this chaotic zone around the 1:1 state is
attitude unstable, so he has argued that all satellites with irregular shapes presently in this spin-orbit
state may have spent a considerable9 amount of time tumbling chaotically before the spin finally left this
zone.
We end this section by noting that capture into synchronisation will only occur as described in Sec-
tion 1.1.3 if the reduction in energy over one librational cycle due to tidal torques is significantly greater
8In addition to primary synchronisations, those studied in 1.1.2 under the averaging theory, the phase space may containsecondary, or even more complicated synchronisations, which only appear under a second-order treatment of spin-orbit dynamicsor in a SOS.
9of the order of the tidal despinning time.
9
than the width of the chaotic separatrix. Otherwise, the capture process becomes essentially a random
process [6].
Although the tidal dissipation hypothesis may be invoked to justify capture into synchronisation, this
hypothesis seems to be weak. Not only the models of tidal dissipation rely on poorly determined pa-
rameters which are related to the internal structure of the bodies involved in the spin-orbit interaction,
tides on small, solid bodies like some moons or asteroids are much weaker than those that act on larger
bodies in hydrostatic equilibrium [5]. In Chapter 2, we follow a different approach. We examine a simpler
and minimal model for the spin-orbit interaction, which we hope describes the different types of spin-orbit
synchronisation effects and resonances — the Keplerian Dumbbell (KD) system. This model describes
the satellite as two point masses joined by a rigid massless rod, resembling the shape of a dumbbell,
hence the name. We study in detail the dynamics of the KD system, by deriving the exact equations of
motion and then analysing the steady state orbits and their stability.
Chapter 3 deals with the extension of the KD model into the framework of the R3BP, where Nix (the
satellite) is modelled by a dumbbell which is subjected to the gravitational fields of Pluto and Charon
(the primaries). The exact equations of motion for this problem are derived, and two special cases are
studied: the Planar Circular Restricted Three-Body Problem (PCR3BP) with dumbbell satellite, in which
the dumbbell’s centre of mass is constrained to move in the plane of the orbits of the primaries, and
the so-called Sitnikov Problem (SP), where it is constrained to move along an axis that is orthogonal to
this plane and that passes through the barycentre of the primaries [27]. Necessary conditions for these
motions to occur are determined, and the steady states of the system and other types of solutions are
found. We also obtain sufficient conditions for the existence of a Jacobi-like invariant ([28]) in the circular
problem. The chapter concludes with a numerical analysis of a subcase of the PCR3BP with dumbbell
satellite, for which the spin axis of the dumbbell is perpendicular to the orbital plane of the primaries.
In the last chapter, we summarise our results and draw some conclusions.
10
Chapter 2
Stationary Solutions of the Keplerian
Dumbbell System
The Keplerian Dumbbell (KD) system is a three-body problem where two of the bodies are connected
by a rigid massless rod — the dumbbell. The KD system is the simplest possible system exhibiting the
minimal features of a rigid body, namely, it has more than one point mass, it is extended in space, and
the distance between masses is conserved during motion. Besides, such a system may prove useful in
describing the dynamics of elongated asteroids with a prolate spheroidal shape like 4769 Castalia [29],
433 Eros [30] or Nix [1], or small satellites [31].
The simplest system showing spin-orbit coupling and eventually synchronisation and resonance be-
tween the translational and rotational motion is the KD system. Despite the apparent simplicity of the
KD system, an exact analysis of its full dynamics, steady state orbits, eventual synchronisation and res-
onance is lacking. Recently, it has been found numerically that the KD system, restricted to a planar
elliptic orbit, shows chaotic dynamics and is nonintegrable [32, 33]. For the planar cases and under
approximate Hamiltonian equations of motion, the existence of steady state orbits has been analysed
by Celletti and Sidorenko [34]. The stability of the steady state orbits starting from an approximate La-
grangian approach has been partially analysed by Elipe et al. [35]. In all these approaches the equations
of motion are approximate. For an extensive review of previous works on the KD system, we refer to
[34].
In this chapter, we derive the exact equations of motion of the KD system, following a Lagrangian
approach. The equations of motion are not restricted to the plane containing the three masses and are
written in the inertial reference frame of the three-body system. We then analyse the steady or stationary
orbits of the KD system. For the case where the two masses of the dumbbell are equal and the motion
of the centre of mass of the dumbbell is planar, the steady states are Lyapunov unstable (Appendix A).
As these steady states are not preserved if the masses of the dumbbell are different, this shows that
this system is structurally unstable. Some of these structurally unstable systems are Eulerian solutions
of the general three-body problem (see Appendix B). For the case where the dumbbell is aligned with
the direction connecting its centre of mass to the centre of mass of the KD system, there are unstable,
11
planar steady state trajectories. This result is exact, suggesting that synchronised spin-orbit effect in
celestial mechanics is an intrinsic non-linear phenomenon.
This chapter is organised as follows: in the next section we derive the Lagrangian and Hamiltonian
equations of motion of the KD system. In Section 2.2, we rescale these equations to a dimensionless
form. The new, dimensionless form simplifies the numerical analysis of the equations of motion. In
Section 2.3, we analyse all the steady states of the KD system and their stability. Finally, in Section 2.4
we complement the stability analysis with numerical simulations.
2.1 Equations of motion of the Keplerian Dumbbell system
The Keplerian Dumbbell (KD) system consists of a primary point mass m1 and a dumbbell, interacting
through their mutual gravitational force. The dumbbell is formed by two point masses m2 and m3,
connected by a rigid massless rod of length `. To describe the motion of the KD system, we consider
the inertial reference frame S = (Cxyz) centred at the centre of mass of the three masses (Figure 2.1).
Figure 2.1: Reference frames of the KD problem. In the reference frame S of the centre of mass of thethree-body system, the centre of mass C ′ of the dumbbell has cylindrical coordinates (r, θ, z) referred tomass m1. In S ′, the orientation of the dumbbell is specified by the angles ψ (polar) and φ (azimuthal).The orientations of the coordinate axes of S and S ′ are the same.
The dumbbell is allowed to rotate in the three-dimensional ambient space, and the configuration
manifold of each of the dumbbell masses is a sphere S2, centred at the centre of massC ′ of the dumbbell.
To describe the attitude dynamics of the dumbbell relative to the reference frame S ′ = (C ′xyz), we
consider the azimuthal and the polar spherical angles φ and ψ, respectively. The distances of the masses
m2 and m3 to the centre of mass C ′ of the dumbbell are `2 = `m3/(m2 +m3) and `3 = `m2/(m2 +m3).
The unit vectors of the coordinate axes (x, y, z) are {e1, e2, e3}. We denote by γ the unit vector directed
along the dumbbell towards mass m2. In spherical coordinates, γ = cosφ sinψ e1 + sinφ sinψ e2 +
12
cosψ e3. The projection of the rod on the (x, y)-horizontal plane of S ′ is p(φ) = cosφ e1 + sinφ e2,
then we define a new unit vector α = p(φ+ π/2) = − sinφ e1 + cosφ e2 (Figure 2.1). As γ • α = 0, α is
perpendicular to γ. On the other hand, as α•e3 = 0, the two vectors {γ, e3} define a plane perpendicular
to α and, therefore, the angular velocity of the dumbbell around the instantaneous direction of rotation
α is ψ. Let β = γ ∧ α = − cosψ cosφ e1 − cosψ sinφ e2 + sinψ e3 be a third unit vector. Then the unit
vectors {α,β,γ} are mutually perpendicular and define the principal axes of inertia of the dumbbell.
In the reference frame {C ′αβγ}, the inertia tensor of the dumbbell is a diagonal matrix, whose
diagonal elements are
Iα = Iβ = m2`22 +m3`
23 =
m2m3
m2 +m3`2 and Iγ = 0. (2.1)
To calculate the equations of motion of the KD system we follow a Lagrangian perspective. Let r1 and
rC′ be the position vectors of the mass m1 and of the centre of mass of the dumbbell in the reference
frame S (Figure 2.1). As the kinetic energy of the dumbbell is the sum of the kinetic energy of its centre
of mass, assuming that the total mass m2 +m3 is concentrated at C ′, plus the kinetic energy of rotation,
the total kinetic energy of the KD system is
T =1
2m1 ‖r1‖2 +
1
2(m2 +m3) ‖ ˙rC′‖2 +
1
2ωT
↔
Iω, (2.2)
where↔
I is the tensor of inertia of the dumbbell in the reference frame (C ′αβγ), and ωT, the instanta-
neous rotational angular velocity of the dumbbell evaluated in the same frame, is
ωT = ψα+ φ e3 = ψα+ φ sinψ β + φ cosψ γ. (2.3)
By construction, e3 = sinψ β + cosψ γ. Introducing (2.1) and (2.3) into (2.2), we finally conclude that
T =1
2m1 ‖r1‖2 +
1
2(m2 +m3) ‖ ˙rC′‖2 +
1
2
m2m3
m2 +m3`2(ψ2 + φ2 sin2 ψ
). (2.4)
The potential energy of the KD system, in the inertial reference frame S, is
U = −Gm1
(m2
‖r12‖+
m3
‖r13‖
), (2.5)
where ‖r12‖ and ‖r13‖ are the distances between mass m1 and each of the other two masses, m2 and
m3, respectively. Hence, in the inertial reference frame S, by (2.4) and (2.5), the Lagrangian of the KD
system is
L = T − U =1
2m1 ‖r1‖2 +
1
2(m2 +m3) ‖ ˙rC′‖2
+1
2
m2m3
m2 +m3`2(ψ2 + φ2 sin2 ψ
)+Gm1
(m2
‖r12‖+
m3
‖r13‖
).
(2.6)
We can now rewrite the Lagrangian (2.6) as a function of the relative distances between masses.
13
Defining rC′ = r1 + r1C′ and, as in the center of mass reference frame S, m1r1 + (m2 +m3) rC′ = 0,
we have r1 = − m2 +m3
m1 +m2 +m3r1C′
rC′ =m1
m1 +m2 +m3r1C′ .
(2.7)
Therefore, the Lagrangian (2.6) rewrites as
L =1
2
m1 (m2 +m3)
m1 +m2 +m3‖ ˙r1C′‖2 +
1
2
m2m3
m2 +m3`2(ψ2 + φ2 sin2 ψ
)+Gm1
(m2
‖r12‖+
m3
‖r13‖
),
(2.8)
where ‖r1C′‖ is the distance between the mass m1 and the centre of mass of the dumbbell. Introducing
cylindrical coordinates (r, θ, z), r1C′ = r cos θ e1 + r sin θ e2 + z e3, the Lagrangian (2.8) becomes
L =1
2
m1 (m2 +m3)
m1 +m2 +m3
(r2 + r2θ2 + z2
)+
1
2
m2m3
m2 +m3`2(ψ2 + φ2 sin2 ψ
)+Gm1
(m2
‖r12‖+
m3
‖r13‖
),
(2.9)
where the distances between masses are
r12 = r1C′ + rC′2 =
r cos θ + m3
m2+m3` sinψ cosφ
r sin θ + m3
m2+m3` sinψ sinφ
z + m3
m2+m3` cosψ
,
r13 = r1C′ + rC′3 =
r cos θ − m2
m2+m3` sinψ cosφ
r sin θ − m2
m2+m3` sinψ sinφ
z − m2
m2+m3` cosψ
,
rC′2 = `2γ,
rC′3 = −`3γ.
(2.10)
Due to the large number of parameters in the Lagrangian (2.9), we rescale this Lagrangian to dimen-
sionless variables.
2.2 Equations of motion in dimensionless form
We introduce the scaling constants and new variables through the relations
r = r0u, z = r0v, ` = r0ε, t =1
στ, (2.11)
where (u, v) are the new dimensionless variables, r0 is assumed to be a characteristic length of the KD
system, ε is a rescaled small parameter, σ is a normalisation unit of frequency, and τ is a dimensionless
14
parameter measuring time in units of 1/σ. In the rescaled variables, the Lagrangian (2.9) transforms into
L =Gm1 (m2 +m3)
r0
[1
2
σ2r03
G (m1 +m2 +m3)
(u′
2+ v′
2+ u2θ′
2)
+1
2
(1− δ)δGm1
σ2r03ε2(ψ′
2+ φ′
2sin2 ψ
)]+Gm1 (m2 +m3)
r0
1− δ‖ρ12‖
+Gm1 (m2 +m3)
r0
δ
‖ρ13‖,
where‖ρ12‖2 = u2 + v2 + δ2ε2 + 2 δ ε (u sinψ cos(θ − φ) + v cosψ)
‖ρ13‖2 = u2 + v2 + (1− δ)2 ε2 − 2 (1− δ) ε (u sinψ cos(θ − φ) + v cosψ)
(2.12)
and δ = m3/(m2 +m3). The prime symbol (′) denotes derivation with respect to dimensionless time
τ . The constant g = Gm1 (m2 +m3) /r0 has dimensions of an energy and represents the energy scale
of the problem. We take as our new Lagrangian, which we call L, the dimensionless quantity which
multiplies this prefactor. This is just the Lagrangian written in units of g. We further make a choice of the
units of length and time such that σ2r03 = G (m1 +m2 +m3), and we finally get the new dimensionless
Lagrangian
L =1
2
(u′
2+ v′
2+ u2θ′
2)
+1
2
(1− δ)δµ
ε2(ψ′
2+ φ′
2sin2 ψ
)+
1− δ‖ρ12‖
+δ
‖ρ13‖, (2.13)
where µ = m1/(m1 +m2 +m3), and the dimensionless distances between the masses are ‖ρ12‖ and
‖ρ13‖, as defined in (2.12).
Therefore, from the dimensionless Lagrangian (2.13), the equations of motion in the (u, θ, v, φ, ψ)
coordinates are
u′′ − uθ′2 = −(1− δ)u+ δ ε sinψ cos (θ − φ)
‖ρ12‖3+ δ
(1− δ) ε sinψ cos (θ − φ)− u‖ρ13‖3
u2θ′′ + 2uu′θ′ = u(1− δ)δ ε sinψ sin (θ − φ)
(1
‖ρ12‖3− 1
‖ρ13‖3
)v′′ = −(1− δ)v + δ ε cosψ
‖ρ12‖3− δ v − (1− δ) ε cosψ
‖ρ13‖3
ε(φ′′ sin2 ψ + 2φ′ ψ′ sinψ cosψ
)= −µu sinψ sin (θ − φ)
(1
‖ρ12‖3− 1
‖ρ13‖3
)
ε(ψ′′ − φ′2 sinψ cosψ
)= −µ (u cosψ cos (θ − φ)− v sinψ)
(1
‖ρ12‖3− 1
‖ρ13‖3
).
(2.14)
The dimensionless equations of motion (2.14) depend on the three parameters δ, ε and µ. The
parameter δ ∈ (0, 1) measures the relative weight of the masses of the dumbbell. For a symmetric
dumbbell with m2 = m3, δ = 1/2. The parameter ε measures the length of the rod of the dumbbell in
units of r0, the approximate radius of the trajectory of the dumbbell. For a dumbbell satellite or asteroid,
ε is close to zero. If ε = 0, we have a Kepler problem. The mass parameter 0 < µ ≤ 1, measures the
relation between the masses of the primary and of the dumbbell. For a small dumbbell satellite, µ ' 1.
The left hand side of the second equation in (2.14) can be written in the formd(u2θ′
)dτ
, and u2θ′ is
15
proportional to the angular momentum of the centre of mass of the dumbbell with respect to the centre
of mass of the three-body system.
The dimensionless equations (2.14) can be derived from an effective Hamiltonian, with the standard
techniques of Hamiltonian dynamics, [36]. In fact, from the Lagrangian (2.13), the conjugate momenta
are
pu = u′, pθ = u2θ′, pv = v′, pφ =(1− δ)δ
µε2φ′ sin2 ψ, pψ =
(1− δ)δµ
ε2ψ′, (2.15)
and the dimensionless Hamiltonian associated with the Lagrangian (2.13) is
H =1
2
(pu
2 +pθ
2
u2+ pv
2
)+ µ
pφ2
2(1− δ)δε2 sin2 ψ+ µ
pψ2
2(1− δ)δε2
− 1− δ‖ρ12‖
− δ
‖ρ13‖. (2.16)
The five second order equations of motion (2.14) have the Hamiltonian (2.16) and the angular mo-
mentum as conservation laws. The angular momentum of the KD system is
L =m1(m2 +m3)
m1 +m2 +m3r1C′ ∧ ˙r1C′ +
m2m3
m2 +m3`2(ψα+ φ sinψβ
)=
m1(m2 +m3)
m1 +m2 +m3
((rz sin θ − z(r sin θ + rθ cos θ))e1
+((zr − rz) cos θ − rzθ sin θ)e2 + r2θe3
)+
m2m3
m2 +m3`2(ψα+ φ sinψβ
),
(2.17)
where, in the first line, the first term on the right hand side of (2.17) is the angular momentum of the
centre of mass of the dumbbell, relative to the centre of mass of the KD system, and the second term
is the angular momentum of the dumbbell relative to its own centre of mass. Introducing the rescaling
variables (2.11) into (2.17), the angular momentum written in units of g/σ (units of angular momentum)
isL = (uv′ sin θ − v(u′ sin θ + uθ′ cos θ))e1
+((vu′ − uv′) cos θ − uvθ′ sin θ)e2 + u2θ′e3
+δ(1− δ)ε2
µ(ψ′α+ φ′ sinψβ) .
(2.18)
As L is a constant of motion, by (2.18) it follows that the dumbbell has planar motion only if
u2θ′ +δ(1− δ)ε2
µφ′ sin2 ψ = pθ + pφ = Lz = constant. (2.19)
This planar conservation law, derived from (2.18), could also have been deduced from (2.14), by adding
the second and fourth equations. In general, the dumbbell has an intrinsic three-dimensional motion in
configuration space.
16
2.3 Steady states and stability analysis
The fixed points of the system of equations (2.14) are the steady states of the KD system. Here, we call
steady states to periodic trajectories of the dumbbell. To calculate the coordinates of the fixed points of
the system of equations (2.14), we impose that some of the components of the vector field defined by
(2.14) are zero (see Section A.1 of Appendix A).
2.3.1 Steady state 1
With the choice u = 0 in the first equation in (2.14), the centre of mass of the KD system coincides
with the position of the primary mass m1. Therefore, to fulfill a fixed point condition we must also have
v = 0, together with the two speed conditions u′ = 0 and v′ = 0. In this case, by (2.12), ‖ρ12‖ = δε and
‖ρ13‖ = (1 − δ)ε, and the right hand side of the first equation in (2.14) is zero only if δ = 1/2, implying
that m2 = m3 and ‖ρ12‖ = ‖ρ13‖. The second equation in (2.14) is identically zero, the orbital angular
momentum of the dumbbell is also zero, and the dumbbell masses rotate with constant angular speed
θ′ = ω = ±(2√
2/ε)3/2, as we shall prove below in (2.24). Therefore, in the configuration space, the KD
system is constrained to move along a circumference of arbitrary radius, with centre at the primary body
(Figure 2.2). If m2 6= m3, this steady state does not exist.
We now analyse the possible trajectories of the dumbbell around the primary body m1. Under the
conditions just derived, by (2.14), the attitude of the dumbbell around the primary is described by the
equations d
dt
(φ′ sin2 ψ
)= 0
ψ′′ − φ′2 sinψ cosψ = 0.(2.20)
The first equation in (2.20) is a conservation law, implying that φ′ sin2 ψ = c, where c is constant. If
c = 0, then φ(t) is constant, for every t ∈ R, and the second equation in (2.20) reduces to ψ′′ = 0. In
this case, the dumbbell system rotates around the primary, with fixed azimuthal angle φ =constant. In
Figure 2.2a), we depict a solution trajectory of the KD system for φ(t) = 0, for every t ∈ R.
If c 6= 0, the second equation in (2.20) reduces to
ψ′′ − c2 cosψ
sin3 ψ= 0. (2.21)
In the two-dimensional (ψ,ψ′) phase space, equation (2.21), with ψ ∈ (0, π), has a unique fixed point with
coordinates (ψ = π/2, ψ′ = 0). By (2.16) and (2.15), equation (2.21) is derivable from the Hamiltonian
H∗(ψ,ψ′) = ψ′2/2 + c2/(2 sin2 ψ). For the fixed point (ψ = π/2, ψ′ = 0), H∗(π/2, 0) = c2, φ′ = c, and the
dumbbell has a three-body Eulerian type solution (Figure 2.2b)), [37].
It is easily shown that the fixed point (ψ = π/2, ψ′ = 0) of equation (2.21) is stable of the centre
type. For any initial condition away from the fixed point in the (ψ,ψ′) phase space, the dumbbell librates
around (ψ = π/2, ψ′ = 0). Due to the first equation in (2.20), φ(t) is also periodic, with the same period
as ψ(t), and the trajectory in the configuration space is also periodic (Figure 2.2c)).
The solutions of the KD system depicted in Figure 2.2 are also Eulerian type solutions of the general
17
Figure 2.2: Steady solutions in the three-dimensional configuration space of the dumbbell with equalmasses (m2 = m3). The positions of the dumbbell masses m2 and m3 at time τ = 40 are shown. In a),c = 0 and, in the configuration space, the KD rotates around the primary body. This trajectory has beencalculated for the azimuthal angle φ(t) = 0. In b) and c), c = 1. In b), the dumbbell has been obtainedwith the initial condition (ψ = π/2, ψ′ = 0), and φ′ 6= 0 is a constant. In c), the dumbbell librates aroundthe fixed point (ψ = π/2, ψ′ = 0) in the (ψ,ψ′) phase space, which corresponds to a closed trajectory inthe configuration space. This orbit has been calculated with the initial conditions φ(0) = 0, ψ(0) = π/6and φ′(0) = 0. All the solutions depicted are also Eulerian solutions of the general three-body problem(see Appendix B).
three-body problem, provided m2 = m3 (Appendix B). These solutions can be obtained by numerical
integration of the three-body equations with an appropriate choice of initial conditions, [37, 38].
The periodic orbits of the KD system in Figure 2.2 occur for the following conditions:
u = 0; v = 0; u′ = 0; v′ = 0; m2 = m3. (2.22)
This fixed point, or the Eulerian trajectories in Figure 2.2, are Lyapunov unstable (Section A.2 of Ap-
pendix A). In fact, linearising the first equation in (2.14) around (2.22), with θ′ = ω and ψ = π/2 (Fig-
ure 2.2b), we obtain
u′′ = (ω2 + 16/ε3)u,
showing that the dumbbell Eulerian trajectories are unstable for perturbations in the radial coordinate u.
On the other hand, as these steady trajectories only exist for m2 = m3, any infinitesimal variation on
the value of the masses destroys the periodic orbits. This implies that the dynamical system (2.14), with
m2 = m3, is structurally unstable, [39].
2.3.2 Steady state 2
Assuming that the masses of the dumbbell are equal, m2 = m3, we have δ = 1/2. Then the system of
equations (2.14) has the equilibrium solution
u = u0 > 0; v = 0; ψ = 0, π;
u′ = v′ = ψ′ = 0; m2 = m3,(2.23)
according to which the dumbbell is always oriented vertically. As the centre of mass of the dumbbell
describes a circular orbit of radius u0 with v = 0 in configuration space, by the second equation in (2.14),
18
the angular momentum of the centre of mass of the dumbbell Lz = u2θ′ is conserved, and the distances
‖ρ12‖ and ‖ρ13‖ are equal. Equating u′′ to zero in the first equation in (2.14), and by (2.12), we obtain
θ′2
:= ω2 =1(
u02 + ε2
4
)3/2 . (2.24)
Conditions (2.23) and (2.24) define a continuous family of periodic solutions of the KD system, pa-
rameterised by u0, with angular velocity θ′, which can be positive or negative. This relationship is the
equivalent of the third law of Kepler for KD systems. As this fixed point only exists for m2 = m3, any
infinitesimal perturbation of the values of the masses destroys the steady state 2 and the dynamical
system (2.14) is structurally unstable.
To show that the steady state 2 is unstable, we linearise the first equation in (2.14) around (2.23),
obtaining u′′ = (u − u0)3u20/(u20 + ε2/4)5/2, and u0 is calculated from (2.24). This system is clearly
unstable for small perturbations around u = u0.
In Figure 2.3a), we show the orbit of steady state 2 in configuration space.
Figure 2.3: Circular orbits (steady states 2 and 3) in the three-dimensional configuration space of theKD system, with m2 = m3. These are the dumbbell isosceles steady state configurations. In both cases,these trajectories are unstable. The positions of the dumbbell masses m2 and m3 at time τ = 29 areshown. In a), we show a circular orbit for the steady state 2, where ψ(t) = 0 or ψ(t) = π, for every t ≥ 0.The angular velocity of the dumbbell is given by (2.24). In b), we show the stationary circular orbit forthe steady state 3, where ψ(t) = π/2 and θ(t) − φ(t) = ±π/2, for every t ≥ 0. The rotational angularvelocity of the dumbbell has the same sign as the angular velocity of translation.
2.3.3 Steady state 3
The third steady state is similar to the previous one, but occurs for a different orientation of the dumbbell:
u = u0 > 0; v = 0; θ − φ = ±π/2; ψ = π/2;
u′ = v′ = ψ′ = 0; θ′ = φ′; m2 = m3.(2.25)
19
For this family of solutions of equations (2.14), the centre of mass of the dumbbell describes a circular
orbit of radius u0 at v = 0 in configuration space, and the radius and the angular velocity are also related
through (2.24).
In Figure 2.3b), we show the orbit in configuration space of this circular steady state. As m2 = m3,
the angular momentum is also conserved. From the point of view of an observer at the origin of the
coordinate system S, m2 and m3 maintain the same relative orientation, and the rotation of the dumbbell
in the local reference frame S ′ is locked with the translation. This corresponds to spin-orbit coupling in a
1 : 1 synchronisation. However, as m2 = m3, any infinitesimal perturbation on the masses destroys the
periodic point as in the previous steady states. As before, linearising the first equation in (2.14) around
(2.25), we obtain u′′ = (u− u0)3u20/(u20 + ε2/4)5/2, and, therefore, this steady state is also unstable.
We can summarise the stability properties of the steady states 1, 2 and 3 in the theorem:
Theorem 1. The steady states (2.22), (2.23) and (2.25) of the KD system, described by equations (2.14)
and depicted in Figures 2.2 and 2.3, are unstable (to small variations on the initial conditions). Moreover,
if m2 = m3, the KD equations of motion (2.14) are structurally unstable.
2.3.4 Steady states 4 to 6
The system of equations (2.14) has three more steady states:
u = u0 > 0; v = 0; θ − φ = 0, π; ψ = π/2;
u′ = v′ = ψ′ = 0; θ′ = φ′.(2.26)
These steady states are numbered according to the conditions:
steady state 4: m2 < m3, φ = θ
steady state 4b: m2 > m3, φ = θ + π
steady state 5: m2 > m3, φ = θ
steady state 5b: m2 < m3, φ = θ + π
steady state 6: m2 = m3, φ = θ or φ = θ + π.
(2.27)
Interchanging m2 and m3, the pairs of steady states 4 and 4b, and 5 and 5b, correspond to the same
geometric arrangement of the dumbbell in configuration space.
From the first equation in (2.14), u0 is a solution of the equation
f(u) = uθ′2 − (1− δ) u± δ ε
‖ρ12‖3+ δ±(1− δ) ε− u‖ρ13‖3
= 0, (2.28)
where‖ρ12‖2 = (u± δ ε)2 ,
‖ρ13‖2 = [u∓ (1− δ) ε]2 ,(2.29)
and the upper signs correspond to the case θ − φ = 0, while the lower signs correspond to θ − φ = π.
20
As these two choices can be converted into one another by interchanging m2 and m3, in the following, it
is enough to consider the upper signs in (2.28) and (2.29). So, under these conditions, the u coordinate
of each steady state is a solution of
f(u) = uθ′2 − (1− δ) 1
(u+ δε)2− δ 1
(u− (1− δ)ε)2= 0, (2.30)
provided u0 − (1 − δ)ε > 0, which implies u0 > (1 − δ)ε. For these steady states, the analogous of
Kepler’s third law is given by the dependence of u0 on θ′, u0 ≡ u0(θ′, δ, ε), obtained by solving equation
(2.30). Therefore, for any choice of θ′ = ω, positive or negative, there is a unique circular trajectory of the
centre of mass of the dumbbell with radius u0, relative to the centre of mass of the KD system, obtained
from (2.30).
That there is indeed a unique solution of equation (2.28) is easily shown. In fact, as limu→+∞ f(u) =
+∞ and f(0) = −(1−3δ(1−δ))/(δ2ε2(1−δ)2) < 0, there exists at least one solution of f(u) = 0, provided
θ′ 6= 0. The unicity follows from the monotonicity of f.
As φ(t) = θ(t), the dumbbell rotates in the direction of the translational motion, and the two masses
m2 and m3 are aligned with the direction defined by the origins of the reference frames S and S ′,
Figure 2.1. The families of equilibrium solutions (2.26)–(2.27) correspond to the 1 : 1 synchronisation of
the translational and rotational motions.
In Figure 2.4, we show a sequence of dumbbell positions along a circular orbit in configuration space
S.
Figure 2.4: Circular orbit in the three-dimensional configuration space of the KD system, with m2 = m3,corresponding to steady state 6. The positions of the dumbbell masses are calculated at times τ =0, 1.4, 3.0 and 4.5. The orbit of the dumbbell is circular around the centre of mass of the KD system. Theperiod of rotation of the dumbbell around its centre of mass is the same as the period of translation ofthe dumbbell, corresponding to a 1 : 1 synchronisation.
21
To analyse the stability of the steady states (2.27), we first consider the planar case. With the
conditions ψ = π/2, pψ = 0, v = 0 and pv = 0 in (2.16), the equations of motion in Hamiltonian form are
u′ = pu
p′u =pθu3− (1− δ) u+ δε cos(θ − φ)
(u2 + δ2ε2 + 2δεu cos(θ − φ))3/2
−δ u− (1− δ)ε cos(θ − φ)
(u2 + (1− δ)2ε2 − 2(1− δ)εu cos(θ − φ))3/2
θ′ =pθu2
p′θ =u(1− δ)δε sin(θ − φ)
(u2 + δ2ε2 + 2δεu cos(θ − φ))3/2
− u(1− δ)δε sin(θ − φ)
(u2 + (1− δ)2ε2 − 2(1− δ)εu cos(θ − φ))3/2
φ′ =µ
(1− δ)δε2pφ
p′φ = − u(1− δ)δε sin(θ − φ)
(u2 + δ2ε2 + 2δεu cos(θ − φ))3/2
+u(1− δ)δε sin(θ − φ)
(u2 + (1− δ)2ε2 − 2(1− δ)εu cos(θ − φ))3/2.
(2.31)
As the total angular momentum is conserved, by (2.31) or (2.19), we can eliminate p′φ from the above
equations, obtaining
u′ = pu
p′u =pθu3− (1− δ) u+ δε cos(θ − φ)
(u2 + δ2ε2 + 2δεu cos(θ − φ))3/2
−δ u− (1− δ)ε cos(θ − φ)
(u2 + (1− δ)2ε2 − 2(1− δ)εu cos(θ − φ))3/2
θ′ =pθu2
p′θ =u(1− δ)δε sin(θ − φ)
(u2 + δ2ε2 + 2δεu cos(θ − φ))3/2
− u(1− δ)δε sin(θ − φ)
(u2 + (1− δ)2ε2 − 2(1− δ)εu cos(θ − φ))3/2
φ′ =µ
(1− δ)δε2(Lz − pθ),
(2.32)
where we have introduced the conservation law
pθ + pφ = Lz, (2.33)
and Lz is a constant. Introducing the new variable χ = θ − φ and making the choice pχ = pθ, equations
22
(2.32) simplify to
u′ = pu
p′u =pχu3− (1− δ) u+ δε cosχ
(u2 + δ2ε2 + 2δεu cosχ)3/2
−δ u− (1− δ)ε cosχ
(u2 + (1− δ)2ε2 − 2(1− δ)εu cosχ)3/2
χ′ =pχu2− µ
(1− δ)δε2(Lz − pχ)
p′χ =u(1− δ)δε sinχ
(u2 + δ2ε2 + 2δεu cosχ)3/2
− u(1− δ)δε sinχ
(u2 + (1− δ)2ε2 − 2(1− δ)εu cosχ)3/2.
(2.34)
Equation (2.31) is Hamiltonian, with effective Hamiltonian function
Heff =1
2
(pu
2 +pχ
2
u2− 2
µ
(1− δ)δε2Lzpχ +
µ
(1− δ)δε2pχ
2
)− 1− δ‖ρ12‖
− δ
‖ρ13‖.
(2.35)
To analyse the stability of the family of fixed points 4 to 6, we use the fact that, if the Hamiltonian
has a local maximum or minimum at the fixed point, then that fixed point is Lyapunov stable, [40]. So,
we calculate the Hessian matrix of Hamiltonian (2.35), evaluated at the fixed points under analysis. If
the Hessian matrix of Heff is positive or negative definite, then the fixed points are Lyapunov stable. On
the contrary, if the eigenvalues have different signs and at least one of them is zero, then we are in the
presence of an unstable fixed point.
In the Hamiltonian (2.35), the coordinates of the fixed points (2.29) are u = u0, χ = 0, pu = 0 and
pχ = u20ω. Then, the Hessian matrix (2.35) calculated at these fixed points is
Ms =
1 0 0 0
0 1u20
+ µ(1−6Lzu0ω)(1−δ)δε2 − 2ω
u20
0
0 − 2ωu20
3ω2
u20− 2(A(1− δ) +Bδ) 0
0 0 0 (B −A)u0(1− δ)δε
, (2.36)
where
A =1
(u0 + δε)3, B =
1
|(u0 − (1− δ)ε)|3
and (B − A) > 0. As Ms is a symmetric matrix, all its eigenvalues are real. The matrix Ms has two
positive eigenvalues, λ = 1 and λ = (B −A)u0(1− δ)δε. We now prove the following theorem:
Theorem 2. For a sufficiently large radius u0 of the dumbbell circular trajectory, the steady states (2.26)–
(2.27) of the KD system, described by equations (2.14) and depicted in Figure 2.4, are Lyapunov unsta-
ble. Moreover, if m2 = m3, the KD equations of motion (2.14) are structurally unstable.
Proof. The proof is based on the analysis of the eigenvalues of the Hessian matrix Ms in (2.36) of the
Hamiltonian (2.35), calculated at the steady states. Due to the particular form of the matrixMs, it has two
positive eigenvalues λ = 1 and λ = (B −A)u0(1− δ)δε. The other two eigenvalues are the eigenvalues
23
of the reduced matrix
Mss =
1u20
+ µ(1−6Lzu0ω)(1−δ)δε2 − 2ω
u20
− 2ωu20
3ω2
u20− 2(A(1− δ) +Bδ)
. (2.37)
Introducing the new variable ξ = 1/u0 into (2.37) and developing this matrix in Taylor series around
ξ = 0, the reduced matrix Mss becomes, up to fourth order in ξ,
Mss =
ξ2 + µ(1−δ)δε2 0
0 −2δξ2
+O(ξ5). (2.38)
Therefore, for ξ sufficiently small (or equivalently, u0 sufficiently large), Mss has one positive and one
negative eigenvalue. This property is also valid for the Hessian matrix (2.36). So, this implies that the
Hamiltonian function (2.35) is at least a 1−saddle near the fixed point, [41], and, therefore, the steady
states (2.26)–(2.27) of the KD system are unstable. The structural instability of the equations of motion
(2.14), for equal masses of the dumbbell m2 = m3, has been shown above.
In the next section, the stability properties of the steady states (2.26)–(2.27) of the KD system are
analysed numerically.
2.4 Numerical analysis of the non-stability of some steady state
orbits
In the previous section, we have shown that the steady state orbits of the KD system are Lyapunov
unstable. This has been shown rigorously for steady states (2.22), (2.23) and (2.25), and partially for
steady states (2.26)–(2.27).
In Figure 2.5, we show the numerically calculated values of the eigenvalues of the reduced matrix
Mss, as a function of the steady state orbital radius of the dumbbell, u0, for different choices of the
parameters. These numerical simulations show that the reduced matrix (2.37) can have one positive
and one negative eigenvalue, or two negative eigenvalues. This demonstrates that the Hessian matrix
(2.36) can be locally a 1−saddle or a 2−saddle, which implies globally that the fixed points (2.26)–(2.27)
of the KD system are Lyapunov unstable.
24
1 2 3 4 5u0
-100
100
200
300
400
500
λ1
λ2
a)
1 2 3 4 5u0
-100
100
200
300
400
500
λ1λ2
b)
Figure 2.5: Eigenvalues λ1 and λ2 of the reduced matrix Mss, (2.37), as a function of the steady stateorbital radius u0 of the dumbbell. In a), δ = 0.5, µ = 1.0 and ε = 0.1. In b), δ = 0.8, µ = 1.0 and ε = 0.3.
25
26
Chapter 3
Restricted Three-Body Problem with
Dumbbell Satellite
Nix is a small natural satellite of the Pluto-Charon system, discovered in 2005 along with Hydra, another
moon of the binary system [1]. In recent years, interest has risen in these small moons, not only because
the New Horizons spacecraft flew through the double planet system and captured images of their sur-
faces, but also because it was discovered that they spin and wobble unpredictably [2]. The effect is likely
due to the constantly shifting gravitational field produced by the larger bodies, Pluto and Charon, and is
only enhanced by the prolate spheroidal shape of the moons, which are the subject of asymmetric grav-
itational torques. In this chapter we study the dynamics of the three-body system constituted by Pluto,
Charon and Nix, by using a convenient toy model for Nix — that of a dumbbell satellite. This system
may be strictly regarded as a special case of the four-body problem, for which there is a constraint on
the distance between two of the bodies. This is the most simple and minimal model of a rigid body that
will enable us to analyse the mechanism of resonance and the transition to chaos of the small moon.
Furthermore, since the mass of Nix is much smaller than that of Pluto or Charon ([42]), we treat the case
in study in the framework of the Restricted Three-Body Problem (R3BP), [3].
The problem of analysing the dynamics of a dumbbell satellite in the context of the R3BP is not new.
For instance, Vera [43] considered a rigid dumbbell satellite placed at the equilateral point L4 of the
R3BP (see [44, 3] for more details) and studied the attitude dynamics of the satellite, providing sufficient
conditions for the existence of periodic orbits via Averaging Theory (Chapter 1). However, as far as
the author is aware, the full dynamics of the dumbbell, including spin-orbit interaction, has never been
explored in the literature. This is the problem we shall tackle in this chapter.
This chapter is organised as follows: in the next section we formulate the problem in more precise
terms. In Section 3.2 we derive the exact equations of motion of the system, following a Lagrangian
approach. The equations of motion are written in the inertial reference frame of the centre of mass of
the two more massive bodies. In order to simplify the analysis, we rewrite and rescale the equations
of motion in a rotating reference frame in Section 3.3. In Section 3.4 we restrict our attention to two
special cases of the R3BP with dumbbell satellite, as we call it: the Planar Circular Restricted Three-
27
Body Problem (PCR3BP) with dumbbell satellite, in which the dumbbell’s centre of mass is constrained
to move in the plane of the orbits of the two massive bodies, and the so-called Sitnikov Problem (SP)
(Sitnikov [27]), where it is constrained to move along an axis that is orthogonal to this plane and that
passes through the barycentre of these bodies. Necessary conditions for these motions to occur are
determined, and the steady states of the system and other types of solutions are found. We also obtain
sufficient conditions for the existence of a Jacobi-like invariant ([28]) in the circular problem. Lastly,
Section 3.5 concludes with a numerical analysis of a subcase of the PCR3BP with dumbbell satellite, for
which the axis of rotation of the dumbbell is perpendicular to the orbital plane of the massive bodies.
3.1 The Restricted Three-Body Problem
Nix’s mass is, according to estimations from orbital integration, about five and six orders of magnitude
smaller than those of Charon and Pluto, respectively [42, 45]. Consequently, it is reasonable to assume
that its gravitational influence on the much more massive bodies is quite negligible. This is the premise
of the R3BP: one of the masses of the bodies is taken to be infinitely small, so that it does not perturb
the motion of the other two, also called the primaries [3]. The motion of the primaries becomes then
decoupled of the third body, and one investigates the motion of the latter subject to the a priori known
motions of the former. This model plays an important role in the analysis of the motions of artificial
satellites or small asteroids ([38]), and its study was started by Newton, d’Alembert, Euler, Lagrange
and Poincare more than three centuries ago [46, 44, 28, 23, 47]. For an extensive review of previous
work done on the R3BP, we refer the reader to Musielak and Quarles [38] and references therein. For a
mathematical description of the problem, we refer to Szebehely [3].
In the context of the R3BP, let us then consider two point masses m1 and m2 describing Keplerian
orbits around their common centre of mass in a plane — the Lagrange plane. This plane is conserved
throughout the motion and is normal to the direction of the angular momentum of the two masses.
Without loss of generality, let (X, Y ) be cartesian coordinates on this plane. According to the theory
of the Kepler problem, the equation governing the motion of these masses in the centre of mass or
barycentric reference frame is (see, for instance, Murray and Dermott [4] or Fitzpatrick [5])r =
Lz2
r3− µ
r2
θ(t) = θ(0) +
∫ t
0
Lzr2dt,
(3.1)
where r > 0, the relative distance between the bodies, is given as a function of the angular coordinate θ
by
r(θ − ξ) =Lz
2/µ
1 + e cos(θ − ξ), (3.2)
which is the equation of a conic section in polar coordinates, e being the eccentricity of the conic,
Lz = r2θ the angular momentum integral, associated to the relative motion of the two masses1, and
1Lz is not the actual orbital angular momentum of the Keplerian subsystem, L1+2. Rather, it relates to L1+2 through Lz =∥∥L1+2
∥∥ (m1 +m2) /(m1m2).
28
µ = G(m1 + m2), where G is the gravitational constant. Here we will focus solely on elliptical orbits of
the primaries, for which e ∈ [0, 1), since most bodies in the solar system, with the exception of comets,
have e � 1. In this case, the length Lz2/µ is related to the eccentricity via Lz2/µ = a(1 − e2), where a
is the semi-major axis of the ellipse, and we also have
r(θ − ξ) =a(1− e2)
1 + e cos(θ − ξ). (3.3)
The angle ξ, measured with respect to an arbitrary reference direction, is the angle at which the closest
approximation between the primaries occurs, also called the longitude of the pericentre.
Denoting by r1 and r2 the position vectors of masses m1 and m2, referred to their common centre
of mass, respectively, it is true that
m1r1 +m2r2 = 0. (3.4)
This means that r1 is always in the opposite direction to r2, and thus, the centre of mass is always on
the line joining the masses. It follows from this thatr1 = − m2
m1 +m2(r(θ − ξ) cos θ, r(θ − ξ) sin θ, 0) = (X1, Y1, 0)
r2 =m1
m1 +m2(r(θ − ξ) cos θ, r(θ − ξ) sin θ, 0) = (X2, Y2, 0) .
(3.5)
Therefore, in the inertial reference frame with origin at the centre of mass of the primaries, each mass
also traces a path in space given by the same conic section as the one describing their relative motion,
albeit with a different scale, and the centre of mass sits at one of the foci of both ellipses. The eccen-
tricities are the same, although the semi-major axes are not, therefore all the ellipses are similar; the
periods of their orbits, the mean motions and the instantaneous angular velocities are also equal, but
the pericentres differ by π (Figure 3.1).
Let us consider additionally a dumbbell-shaped satellite formed by two point masses m3 and m4,
connected by a rigid massless rod of length `, such that m3 +m4 � m1, m2. As pointed out, we assume
that these bodies don’t influence the motions of the other two. To study the dynamics of the system of
four masses, we take the inertial reference frame S = (CXY Z) with origin at the centre of mass of the
primaries m1 and m2 (Figure 3.2).
As in Chapter 2, the dumbbell is allowed to rotate in the three-dimensional ambient space, and the
configuration manifold of each of the dumbbell masses is a sphere S2, centred at the centre of mass
C ′ of the dumbbell. To describe the attitude dynamics of the dumbbell relative to the reference frame
S ′ = (C ′XY Z), we consider the azimuthal and the polar spherical angles φ and ψ, respectively. The
distances of the masses m3 and m4 to the centre of mass C ′ of the dumbbell are `3 = `m4/(m3 +m4)
and `4 = `m3/(m3 +m4), respectively. The unit vectors of the coordinate axes (X,Y, Z) are {e1, e2, e3}.
We denote by γ the unit vector directed along the dumbbell towards mass m3. In spherical coordinates,
γ = cosφ sinψ e1 + sinφ sinψ e2 + cosψ e3. The projection of the rod on the (X,Y )-horizontal plane of
S ′ is p(φ) = cosφ e1 + sinφ e2, then we define a new unit vector α = p(φ + π/2) = − sinφ e1 + cosφ e2
29
m2
m1X
Y
C
Figure 3.1: The motion of the masses m1 and m2 in the inertial reference frame S centred at the centreof mass, C. An eccentricity of 0.5, a mass ratio m2/m1 of 0.2, and a longitude of the pericentre of π/6were used.
x y
Z, z
S,R
θm1
m2
X
e1 Ye2
e3
rC′C
φX
e1 Ye2
Z
e3
S ′
m3
m4
α
βγ
C′
ψ
Figure 3.2: Reference frames used in the study of the R3BP with dumbbell satellite. In the referenceframe S of the centre of mass of the primaries, the centre of mass C ′ of the dumbbell has cartesiancoordinates (X,Y, Z). In S ′, the orientation of the dumbbell is specified by the angles ψ (polar) and φ(azimuthal). The orientations of the coordinate axes of S and S ′ are the same. The reference frame R,also called synodic, corotates with the primaries, which are assumed to move in the (X,Y )-plane, andis rotated by an angle θ with respect to S. The centre of mass of the dumbbell has cartesian coordinates(x, y, z) in R.
(Figure 3.2). As γ • α = 0, α is perpendicular to γ. On the other hand, as α • e3 = 0, the two vectors
{γ, e3} define a plane perpendicular to α and, therefore, the angular velocity of the dumbbell around the
instantaneous direction of rotation α is ψ. Let β = γ ∧α = − cosψ cosφ e1 − cosψ sinφ e2 + sinψ e3 be
a third unit vector. Then the unit vectors {α,β,γ} are mutually perpendicular and define the principal
axes of inertia of the dumbbell.
In the reference frame {C ′αβγ}, the inertia tensor of the dumbbell is a diagonal matrix, whose
30
diagonal elements are
Iα = Iβ = m3`23 +m4`
24 =
m3m4
m3 +m4`2 and Iγ = 0. (3.6)
3.2 Equations of motion of the dumbbell satellite
To determine the equations of motion of the dumbbell we once again follow a Lagrangian perspective.
Let r3, r4 and rC′ be the position vectors of masses m3, m4 and of the centre of mass of the dumbbell
in the reference frame S. Moreover, let us define vectors rij = rj − ri, which point from mass mi to
mass mj . Then, in S, the Lagrangian of the system of four masses is
L =1
2m1 ‖r1‖2 +
1
2m2 ‖r2‖2 +
1
2m3 ‖r3‖2 +
1
2m4 ‖r4‖2 +G
m1m2
‖r12‖
+Gm1m3
‖r13‖+G
m2m3
‖r23‖+G
m1m4
‖r14‖+G
m2m4
‖r24‖
+Gm3m4
‖r34‖,
(3.7)
where ‖rij‖ is the distance between masses i and j. As the distance between mass m3 and mass m4,
or the length of the dumbbell, is a constant of the motion, the last term of (3.7) is just a constant and does
not contribute towards the equations of motion. Furthermore, since, by assumption, the motion of the
primaries is independent from the other masses, their Lagrangian, L1+2 = m1 ‖r1‖2 /2 +m2 ‖r2‖2 /2 +
Gm1m2/‖r12‖, becomes decoupled from L. As such, the motion of the dumbbell under the action of
the gravitational field generated by m1 and m2, as these whirl about each other, is governed in S by the
Lagrangian
Ldumbb. =1
2m3 ‖r3‖2 +
1
2m4 ‖r4‖2
+Gm1
(m3
‖r13‖+
m4
‖r14‖
)+Gm2
(m3
‖r23‖+
m4
‖r24‖
).
(3.8)
On the other hand, the motions of m1 and m2 are to be determined from (3.1), which is derived from
L1+2, and whose solution is given parametrically by (3.3) and (3.5).
It is straightforward to show that the kinetic energy of the dumbbell T = m3 ‖r3‖2 /2 + m4 ‖r4‖2 /2
decomposes into a translational part, associated to the motion of C ′, plus a rotational part, that is,
T =1
2(m3 +m4) ‖ ˙rC′‖2 +
1
2ωT
↔
Iω, (3.9)
where↔
I is the inertia tensor of the dumbbell in the reference frame (C ′αβγ), and ωT, the instantaneous
angular velocity of the dumbbell evaluated in the same frame, is
ωT = ψα+ φ e3 = ψα+ φ sinψ β + φ cosψ γ. (3.10)
31
By construction, e3 = sinψ β + cosψ γ. Plugging (3.6) and (3.10) into (3.9), we conclude that
T =1
2(m3 +m4) ‖ ˙rC′‖2 +
1
2
m3m4
m3 +m4`2(ψ2 + φ2 sin2 ψ
), (3.11)
and, accordingly, the Lagrangian (3.8) of the dumbbell satellite in S rewrites as
Ldumbb. =1
2(m3 +m4)
(X2 + Y 2 + Z2
)+
1
2
m3m4
m3 +m4`2(ψ2 + φ2 sin2 ψ
)+Gm1
(m3
‖r13‖+
m4
‖r14‖
)+Gm2
(m3
‖r23‖+
m4
‖r24‖
),
(3.12)
where we have introduced cartesian coordinates (X,Y, Z) for the position of C ′ in S, rC′ = X e1 +
Y e2 + Z e3 (Figure 3.2). The distances between masses are, by (3.5),
r13 = rC′ + rC′3 − r1 =
X + m4
m3+m4` sinψ cosφ−X1
Y + m4
m3+m4` sinψ sinφ− Y1
Z + m4
m3+m4` cosψ
,
r14 = rC′ + rC′4 − r1 =
X − m3
m3+m4` sinψ cosφ−X1
Y − m3
m3+m4` sinψ sinφ− Y1
Z − m3
m3+m4` cosψ
,
r23 = rC′ + rC′3 − r2 =
X + m4
m3+m4` sinψ cosφ−X2
Y + m4
m3+m4` sinψ sinφ− Y2
Z + m4
m3+m4` cosψ
,
r24 = rC′ + rC′4 − r2 =
X − m3
m3+m4` sinψ cosφ−X2
Y − m3
m3+m4` sinψ sinφ− Y2
Z − m3
m3+m4` cosψ
,
rC′3 = `3γ,
rC′4 = −`4γ. (3.13)
Under these circumstances, we note that Lagrangian (3.12) depends explicitly on time through the an-
gular coordinate θ, which, when the orbits of the primaries are elliptical, is a non-linear function of time.2
2It is worth mentioning that the R3BP is a fundamentally different problem than the General Three-Body Problem. In therestricted problem neither total angular momentum nor energy is conserved, whereas they are in the General Three-Body Problemand, specifically, in the KD system, as we saw in the previous chapter.
32
From this Lagrangian, the Euler-Lagrange equations for the motion of the dumbbell are
(m3 +m4) X = −Gm1
((m3
‖r13‖3+
m4
‖r14‖3
)(X −X1) +
m3m4
m3 +m4` sinψ cosφ
(1
‖r13‖3− 1
‖r14‖3
))
−Gm2
((m3
‖r23‖3+
m4
‖r24‖3
)(X −X2) +
m3m4
m3 +m4` sinψ cosφ
(1
‖r23‖3− 1
‖r24‖3
))
(m3 +m4) Y = −Gm1
((m3
‖r13‖3+
m4
‖r14‖3
)(Y − Y1) +
m3m4
m3 +m4` sinψ sinφ
(1
‖r13‖3− 1
‖r14‖3
))
−Gm2
((m3
‖r23‖3+
m4
‖r24‖3
)(Y − Y2) +
m3m4
m3 +m4` sinψ sinφ
(1
‖r23‖3− 1
‖r24‖3
))
(m3 +m4) Z = −Gm1
((m3
‖r13‖3+
m4
‖r14‖3
)Z +
m3m4
m3 +m4` cosψ
(1
‖r13‖3− 1
‖r14‖3
))
−Gm2
((m3
‖r23‖3+
m4
‖r24‖3
)Z +
m3m4
m3 +m4` cosψ
(1
‖r23‖3− 1
‖r24‖3
))
`(φ sin2 ψ + 2φ ψ sinψ cosψ
)= −Gm1 sinψ ((Y − Y1) cosφ− (X −X1) sinφ)
(1
‖r13‖3− 1
‖r14‖3
)
−Gm2 sinψ ((Y − Y2) cosφ− (X −X2) sinφ)
(1
‖r23‖3− 1
‖r24‖3
)
`(ψ − φ2 sinψ cosψ
)= −Gm1 (cosψ [(X −X1) cosφ+ (Y − Y1) sinφ]− Z sinψ)
(1
‖r13‖3− 1
‖r14‖3
)
−Gm2 (cosψ [(X −X2) cosφ+ (Y − Y2) sinφ]− Z sinψ)
(1
‖r23‖3− 1
‖r24‖3
),
(3.14)
where (X,Y, Z) are cartesian coordinates and (φ, ψ) are angular coordinates for the attitude of the
dumbbell.
By virtue of the complicated nature of the equations (3.14), in the next section we simplify the La-
grangian (3.12) and these equations, by making a change to another frame of reference.
3.3 Equations of motion in the synodic reference frame, in dimen-
sionless form
Let us consider a new reference frame R with the same origin as the inertial frame, but which is rotating
in the positive direction, and a new set of cartesian coordinates (x, y, z) associated to this frame. We
choose the direction of the x axis such that the two primary masses always lie along it (Figures 3.2,3.3).
This is commonly called the synodic reference frame [46, 3]. It is assumed that the coordinate axes of
the old and the new reference frames coincide at the origin of time. The coordinates of the centre of
mass of the dumbbell with respect to this frame are now (x, y, z).
By construction, R is rotated with respect to S by the angle θ of (3.1) and (3.5). The new cartesian
33
m2
m1
x
y
θ
Xe1
Y
e2
Z, z
S
R
C
m3
m4
Figure 3.3: Inertial S and synodic R = (Cxyz) reference frames used in the study of the R3BP withdumbbell satellite. The reference frame R corotates with the masses m1 and m2.
coordinates are thus related to the ones in the inertial frame by the simple rotationX
Y
Z
=
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
x
y
z
. (3.15)
Applying transformation (3.15) to (3.12), the Lagrangian of the dumbbell satellite is rewritten in the
synodic reference frame as
Ldumbb. =1
2(m3 +m4)
(x2 + y2 + z2 + θ2
(x2 + y2
)+ 2θ (xy − xy)
)+
1
2
m3m4
m3 +m4`2(ψ2 + φ2 sin2 ψ
)(3.16)
+ Gm1
(m3
‖r13‖+
m4
‖r14‖
)+Gm2
(m3
‖r23‖+
m4
‖r24‖
),
where, in view of (3.5) and (3.13), the squared distances between each of the masses of the dumbbell
m3 and m4 and the primaries m1 and m2 are now given by
‖r13‖2 =
(x+
m2
m1 +m2r(θ − ξ)
)2
+ y2 + z2 +
(m4
m3 +m4`
)2
+ 2m4
m3 +m4`
((x+
m2
m1 +m2
× r(θ − ξ))
sinψ cos (θ − φ)− y sinψ sin (θ − φ) + z cosψ
)
‖r14‖2 =
(x+
m2
m1 +m2r(θ − ξ)
)2
+ y2 + z2 +
(m3
m3 +m4`
)2
− 2m3
m3 +m4`
((x+
m2
m1 +m2
× r(θ − ξ))
sinψ cos (θ − φ)− y sinψ sin (θ − φ) + z cosψ
)
‖r23‖2 =
(x− m1
m1 +m2r(θ − ξ)
)2
+ y2 + z2 +
(m4
m3 +m4`
)2
+ 2m4
m3 +m4`
((x− m1
m1 +m2
34
× r(θ − ξ))
sinψ cos (θ − φ)− y sinψ sin (θ − φ) + z cosψ
)
‖r24‖2 =
(x− m1
m1 +m2r(θ − ξ)
)2
+ y2 + z2 +
(m3
m3 +m4`
)2
− 2m3
m3 +m4`
((x− m1
m1 +m2
× r(θ − ξ))
sinψ cos (θ − φ)− y sinψ sin (θ − φ) + z cosψ
), (3.17)
and r(θ − ξ) is given by (3.3). Here θ is the common angular velocity of the primaries about each other
and their centre of mass, which is related to Lz through
θ =
(1 + e cos(θ − ξ)
1− e2
)2Lza2. (3.18)
Due to the large number of parameters in the Lagrangian (3.16), it is convenient to rescale it to
dimensionless variables. As such, let us introduce new variables through the relations
x = r∗u, y = r∗v, z = r∗w, ` = r∗ε, t =1
θ∗τ, (3.19)
where (u, v, w) are the new, dimensionless variables, ε is the rescaled length of the dumbbell, and
where we adopt r∗ = Lz2/µ, the radius of the circular orbit of the Kepler Problem for the motion of m2
about m1, and θ∗ = µ2/Lz3, the angular velocity corresponding to that orbit, as scaling constants.3 In
the rescaled variables, Lagrangian (3.16) is then transformed into
Ldumbb. = (m3 +m4)µ2
Lz2
[1
2
(u′
2+ v′
2+ w′
2+ θ′
2 (u2 + v2
)+ 2 θ′ (uv′ − u′v)
)+
1
2(1− δ)δ ε2
(ψ′
2+ φ′
2sin2 ψ
)]
+ (m3 +m4)µ2
Lz2 (1− η)
(1− δ‖ρ13‖
+δ
‖ρ14‖
)
+ (m3 +m4)µ2
Lz2 η
(1− δ‖ρ23‖
+δ
‖ρ24‖
), (3.20)
where
‖ρ13‖2 =(u+ η ρ(θ − ξ)
)2+ v2 + w2 + δ2 ε2 + 2 δ ε
((u+ η ρ(θ − ξ)
)sinψ cos (θ − φ)− v sinψ
× sin (θ − φ) + w cosψ)
‖ρ14‖2 =(u+ η ρ(θ − ξ)
)2+ v2 + w2 + (1− δ)2 ε2 − 2 (1− δ) ε
((u+ η ρ(θ − ξ)
)sinψ cos (θ − φ)
− v sinψ sin (θ − φ) + w cosψ)
‖ρ23‖2 =(u− (1− η) ρ(θ − ξ)
)2+ v2 + w2 + δ2 ε2 + 2 δ ε
((u− (1− η) ρ(θ − ξ)
)sinψ cos (θ − φ)
3The orbit r = r∗ is easily seen to be an equilibrium solution of (3.1) with vanishing eccentricity. On the other hand, the valueof θ∗ can be obtained for instance by making the substitutions a = r∗ and e = 0 in (3.18).
35
− v sinψ sin (θ − φ) + w cosψ)
‖ρ24‖2 =(u− (1− η) ρ(θ − ξ)
)2+ v2 + w2 + (1− δ)2 ε2 − 2 (1− δ) ε
((u− (1− η) ρ(θ − ξ)
)× sinψ cos (θ − φ)− v sinψ sin (θ − φ) + w cosψ
), (3.21)
and η = m2/(m1 +m2), and δ = m4/(m3 +m4). The prime symbol (′) here denotes derivation with
respect to dimensionless time τ , and we have made the definition
ρ(θ − ξ) =λ(1− e2)
1 + e cos(θ − ξ), (3.22)
λ being the semi-major axis of the orbit of one of the primaries about the other normalised to r∗, i.e.
λ = a/r∗. In the synodic reference frame, m1 and m2 always lie along the x axis with coordinates
(−η ρ(θ − ξ), 0, 0) and(
(1− η) ρ(θ − ξ), 0, 0), respectively.
The constant (m3 +m4)µ2/Lz2 has the dimensions of an energy and represents the energy scale of
the problem. If we let the units of mass and length be chosen such that µ = 1 and r∗ = 1, respectively,
and we introduce a new parameter ν defined by ν = (m1 +m2) /(m1 +m2 +m3 +m4), it then follows
that in this system of units the four masses are
Gm1 = 1− η, Gm2 = η, Gm3 =(1− δ) (1− ν)
νand Gm4 =
δ (1− ν)
ν, (3.23)
and the energy scale is given by m3 +m4 = (1− ν) /(Gν). Likewise, the common angular velocity of the
primaries becomes θ = ρ−2(θ − ξ). In an analogous manner to what we did in Section 2.2, we take as
our new Lagrangian the dimensionless quantity that multiplies the prefactor (m3 +m4)µ2/Lz2 in (3.20),
and we call it Ldumbb.,
Ldumbb. =1
2
(u′
2+ v′
2+ w′
2+
u2 + v2
ρ4(θ − ξ)+ 2
uv′ − u′vρ2(θ − ξ)
)
+1
2(1− δ)δ ε2
(ψ′
2+ φ′
2sin2 ψ
)(3.24)
+ (1− η)
(1− δ‖ρ13‖
+δ
‖ρ14‖
)+ η
(1− δ‖ρ23‖
+δ
‖ρ24‖
).
The dimensionless distances between the various masses are now ‖ρ13‖, ‖ρ14‖, ‖ρ23‖ and ‖ρ24‖, as
defined in (3.21).
From the dimensionless Lagrangian (3.24), we may thus derive the equations of motion of the dumb-
36
bell in the synodic reference frame, in the new (u, v, w, φ, ψ) coordinates, which are
u′′ − 2v′
ρ2(θ − ξ)=
1
ρ4(θ − ξ)
(u− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
v
)− (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)(u+ η
× ρ(θ − ξ))
+ (1− δ)δ ε sinψ cos (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)
×(u− (1− η) ρ(θ − ξ)
)+ (1− δ)δ ε sinψ cos (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
))
v′′ + 2u′
ρ2(θ − ξ)=
1
ρ4(θ − ξ)
(v + 2e
sin (θ − ξ)1 + e cos (θ − ξ)
u
)− (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)v
− (1− δ)δ ε sinψ sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)v − (1− δ)δ ε
× sinψ sin (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
))
w′′ = − (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)w + (1− δ)δ ε cosψ
(1
‖ρ13‖3− 1
‖ρ14‖3
))− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)w + (1− δ)δ ε cosψ
(1
‖ρ23‖3− 1
‖ρ24‖3
))
ε(φ′′ sin2 ψ + 2φ′ ψ′ sinψ cosψ
)= − (1− η)
((u+ η ρ(θ − ξ)
)sinψ sin (θ − φ) + v sinψ cos (θ − φ)
)×
(1
‖ρ13‖3− 1
‖ρ14‖3
)− η((u− (1− η) ρ(θ − ξ)
)sinψ sin (θ − φ) + v sinψ cos (θ − φ)
)×
(1
‖ρ23‖3− 1
‖ρ24‖3
)
ε(ψ′′ − φ′2 sinψ cosψ
)= − (1− η)
((u+ η ρ(θ − ξ)
)cosψ cos (θ − φ)− v cosψ sin (θ − φ)− w sinψ
)×
(1
‖ρ13‖3− 1
‖ρ14‖3
)− η((u− (1− η) ρ(θ − ξ)
)cosψ cos (θ − φ)− v cosψ sin (θ − φ)
−w sinψ)( 1
‖ρ23‖3− 1
‖ρ24‖3
),
(3.25)
Several comments are in order concerning the dimensionless equations of motion (3.25). First, the
equations depend on the parameters ε, δ, η, and as well on the eccentricity e, the semi-major axis λ and
the longitude of the pericentre ξ of the orbit of the primary masses about each other. The parameter ε
measures, as stated above, the length of the rod of the dumbbell in units of r∗. For a small satellite like
Nix, ε will be close to zero. As in Section 2.2, the parameter δ ∈ (0, 1) measures the relative weight of
the masses of the dumbbell, while η plays the same role in regard to the primaries as δ does in regard
to m3 and m4. For a symmetric dumbbell with m3 = m4, and for m1 = m2, δ = 1/2 and η = 1/2,
respectively. On the other hand, the mass parameter ν doesn’t appear naturally in the equations. This
37
does not mean however that ν is a redundant parameter, as it is an analogue of µ (Chapter 2), providing
a link between the masses of the primaries and of the dumbbell in the framework of the R3BP. The value
of ν will always be close to unity throughout this paper, in contrast to µ, whose value can range between
zero and unity.
Furthermore, spin-orbit coupling is manifestly present (as it has already been the case in Chapter 2)
in the equations (3.25), since the equations for the motion of the centre of mass of the dumbbell depend
on the attitude angles of the body, and, correspondingly, the equations of motion of the attitude angles
depend on the position of the centre of mass. Likewise, the azimuthal angle φ occurs only in (3.25)
coupled to the orbital angle θ of the primaries through the difference θ − φ.
Lastly, we note that the transformation to the synodic reference frame has introduced terms in the
equations proportional to θ′2u and θ′
2v, the components of the centrifugal acceleration, and θ′v′ and
θ′u′, which are the components of the Coriolis acceleration and depend on the velocity of the centre of
mass of the dumbbell in the rotating frame. There are also terms present which are proportional to θ′′v
and θ′′u. These terms are related to the angular acceleration of the reference frame. When the orbits
of the primaries are circular about their common centre of mass, they vanish, and the equations greatly
simplify. This case — the Circular Restricted Three-Body Problem (CR3BP) — can be obtained from
(3.25) by setting e = 0 and ρ(θ − ξ) = 1.
The dimensionless equations of motion (3.25) are associated to Hamilton’s equations [36]
u′ =∂H∂pu
, v′ =∂H∂pv
, w′ =∂H∂pw
pu′ = −∂H
∂u, pv
′ = −∂H∂v
, pw′ = −∂H
∂w
φ′ =∂H∂pφ
, ψ′ =∂H∂pψ
pφ′ = −∂H
∂φ, pψ
′ = −∂H∂ψ
,
(3.26)
with Hamiltonian
H =1
2
((pu +
v
ρ2(θ − ξ)
)2
+
(pv −
u
ρ2(θ − ξ)
)2
+ pw2 +
1
(1− δ)δ ε2
×(pψ
2 +pφ
2
sin2 ψ
))− 1
2
1
ρ4(θ − ξ)(u2 + v2
)(3.27)
− (1− η)
(1− δ‖ρ13‖
+δ
‖ρ14‖
)− η
(1− δ‖ρ23‖
+δ
‖ρ24‖
),
which is a function of the dimensionless variables (u, v, w, φ, ψ), the canonical momenta associated to
38
these variables,
pu = u′ − v
ρ2(θ − ξ), pv = v′ +
u
ρ2(θ − ξ), pw = w′
pφ = (1− δ)δ ε2φ′ sin2 ψ, pψ = (1− δ)δ ε2ψ′,
(3.28)
and the time, through θ. It is worth pointing out that H is an effective Hamiltonian, as we are in a
non-inertial reference frame.
If we define a scalar function V = V(u, v, w, φ, ψ; θ) by
V = − 1
2
1
ρ4(θ − ξ)(u2 + v2
)− (1− η)
(1− δ‖ρ13‖
+δ
‖ρ14‖
)− η
(1− δ‖ρ23‖
+δ
‖ρ24‖
), (3.29)
the equations of motion of the dumbbell (3.25) can also be written in the following way
u′′ − 2v′
ρ2(θ − ξ)= −∂V
∂u− 2e
ρ4(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)v
v′′ + 2u′
ρ2(θ − ξ)= −∂V
∂v+
2e
ρ4(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)u
w′′ = −∂V∂w
(1− δ) δ ε2(φ′′ sin2 ψ + 2φ′ ψ′ sinψ cosψ
)= −∂V
∂φ
(1− δ) δ ε2(ψ′′ − φ′2 sinψ cosψ
)= −∂V
∂ψ.
(3.30)
The function V may be referred to as an effective potential. The term proportional to u2 + v2 in V is the
centrifugal potential, while the terms proportional to the inverse of the distances between the masses
constitute the gravitational potential. Let us multiply the first equation in (3.30) by u′, the second equation
by v′, the third by w′, the fourth by φ′, the fifth by ψ′, and add all of them. We get
u′u′′ + v′v′′ + w′w′′ + (1− δ) δ ε2(ψ′ψ′′ + φ′φ′′ sin2 ψ + φ′
2ψ′ sinψ cosψ
)=
−(∂V∂u
u′ +∂V∂v
v′ +∂V∂w
w′ +∂V∂φ
φ′ +∂V∂ψ
ψ′)
+2e
ρ4(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)(uv′ − u′v) , (3.31)
or,
d
dτ
[1
2
(u′
2+ v′
2+ w′
2+ (1− δ) δ ε2
(ψ′
2+ φ′
2sin2 ψ
))]+
dVdτ
=∂V∂τ
+2e
ρ4(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)(uv′ − u′v) . (3.32)
The left hand side of (3.32) is just the total time derivative of H, now viewed as a function of the di-
39
mensionless coordinates and their velocities. The first term on the right hand side of (3.32) is equal
to
∂V∂τ
=∂V∂θθ′ =
2e
ρ6(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)(u2 + v2
)+ (1− η)
{1− δ‖ρ13‖3
(e η
ρ(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)
((u+ η ρ(θ − ξ)
)+ δ ε sinψ cos (θ − φ)
)− δ ε sinψ
ρ2(θ − ξ)
((u+ η ρ(θ − ξ)
)sin (θ − φ)
+ v cos (θ − φ)))
+δ
‖ρ14‖3
(e η
ρ(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)
((u
+ η ρ(θ − ξ))− (1− δ) ε sinψ cos (θ − φ)
)+ (1− δ) ε sinψ
ρ2(θ − ξ)
×((u+ η ρ(θ − ξ)
)sin (θ − φ) + v cos (θ − φ)
))}+ η
{1− δ‖ρ23‖3
(− e (1− η)
ρ(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)
((u− (1− η)
× ρ(θ − ξ))
+ δ ε sinψ cos (θ − φ))− δ ε sinψ
ρ2(θ − ξ)
((u− (1− η)
× ρ(θ − ξ))
sin (θ − φ) + v cos (θ − φ)))
+δ
‖ρ24‖3
(− e (1− η)
ρ(θ − ξ)sin (θ − ξ)
1 + e cos (θ − ξ)
((u− (1− η) ρ(θ − ξ)
)− (1− δ) ε sinψ cos (θ − φ)
)+ (1− δ) ε sinψ
ρ2(θ − ξ)
((u− (1− η)
× ρ(θ − ξ))
sin (θ − φ) + v cos (θ − φ)))}
. (3.33)
Hence, this implies that, contrary to the conventional CR3BP where H is a constant of the motion —
the so-called Jacobi integral ([28]) — H is not conserved in the R3BP with dumbbell satellite. Moreover,
even when we consider the CR3BP with dumbbell satellite, for which e = 0 and ρ(θ − ξ) = 1, there
cannot in general be a Jacobi integral, since there are still nonvanishing terms in ∂V/∂τ which aren’t
proportional to e:
∂V∂τ
∣∣∣∣ρ(θ−ξ)=1e=0
= (1− η) (1− δ) δ ε sinψ((u+ η
)sin (θ − φ) + v cos (θ − φ)
)
×
(1
‖ρ14‖3− 1
‖ρ13‖3
)
+ η (1− δ) δ ε sinψ((u− (1− η)
)sin (θ − φ) + v cos (θ − φ)
)×
(1
‖ρ24‖3− 1
‖ρ23‖3
), (3.34)
where ‖ρ13‖2, ‖ρ14‖2, ‖ρ23‖2 and ‖ρ24‖2 are as given in (3.21), with ρ(θ − ξ) = 1. The following are
sufficient conditions for ∂V/∂τ to become equal to zero and, consequently, for H to be a constant of the
40
motion in the CR3BP with dumbbell satellite:
ψ = 0, π (3.35)
‖ρ13‖ = ‖ρ14‖ and ‖ρ23‖ = ‖ρ24‖ (3.36)(u+ η
)sin (θ − φ) + v cos (θ − φ) = 0(
u− (1− η))
sin (θ − φ) + v cos (θ − φ) = 0
(3.37)
‖ρ13‖ = ‖ρ24‖ and ‖ρ14‖ = ‖ρ23‖
η = 1/2 and θ − φ = 0, π
(3.38)
‖ρ13‖ = ‖ρ23‖ and ‖ρ14‖ = ‖ρ24‖
u sin (θ − φ) = −v cos (θ − φ) .
(3.39)
From (3.35) we see that whenever the motion of the dumbbell is such that it is perpendicularly oriented to
the Lagrange plane of the primaries at all times, if such motion exists, H will be a conserved quantity for
the CR3BP. For condition (3.36) to happen, it is mandatory that sinψ cos (θ − φ) = 0 holds, independently
of δ, and, specifically for equal dumbbell masses, w cosψ = v sinψ sin (θ − φ) is also verified. Condition
(3.37) is equivalent to v = 0 and θ − φ = 0, π, and it corresponds to a hypothetical motion of the
dumbbell in which it is constrained to move and rotate in the instantaneous plane which contains both
m1 and m2 and is orthogonal to the Lagrange plane. Condition (3.38) only holds for equal masses of the
primaries and 2u = ± (1− 2δ) ε sinψ, and, finally, condition (3.39) takes place for η = 1/2 and u = 0 and
θ − φ = ±π/2, hence the dumbbell is constrained to move and rotate in the instantaneous mediating
plane of the line segment connecting the primaries. More restrictive conditions include, for instance,
all four distances, ‖ρ13‖, ‖ρ14‖, ‖ρ23‖ and ‖ρ24‖, equal. In conclusion, apart from the distance ‖r34‖,
which coincides with the length of the dumbbell and is obviously a conserved quantity, there are in
general no constants of the motion for the (C)R3BP with dumbbell satellite, because neither energy nor
angular momentum is conserved.
3.4 Steady states and other solutions
The equations of motion of the dumbbell in the synodic reference frame (3.25) are still cumbersome, so
we will focus here solely on the study of two particular cases, the Planar Circular Restricted Three-Body
Problem (PCR3BP) with dumbbell satellite and the Sitnikov Problem (SP), Sitnikov [27]. In the present
section we derive the equations of motion for these two systems and we do a systematic analysis of the
solutions of those equations, including periodic or quasi-periodic motions.
It is convenient to adopt the angular coordinate θ as independent variable instead of τ . This amounts
41
to applying the following transformation properties4,
d
dτ=
1
ρ2d
dθ
d2
dτ2=
1
ρ4
(d2
dθ2− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
d
dθ
), (3.40)
to (3.25), which then transforms into
u− 2
(v + e
sin (θ − ξ)1 + e cos (θ − ξ)
(u− v)
)= u− ρ4
((1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)(u+ η ρ
)+ (1− δ)δ ε
× sinψ cos (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))
+ η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)(u− (1− η) ρ
)+ (1− δ)δ ε sinψ cos (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
)))
v + 2
(u− e sin (θ − ξ)
1 + e cos (θ − ξ)(v + u)
)= v − ρ4
((1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)v − (1− δ)δ ε sinψ
× sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))
+ η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)v − (1− δ)δ ε sinψ sin (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
)))
w − 2esin (θ − ξ)
1 + e cos (θ − ξ)w = −ρ4
((1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)w + (1− δ)δ ε cosψ
(1
‖ρ13‖3
− 1
‖ρ14‖3
))+ η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)w + (1− δ)δ ε cosψ
(1
‖ρ23‖3− 1
‖ρ24‖3
)))
ε
((φ− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
φ
)sin2 ψ + 2 φ ψ sinψ cosψ
)= −ρ4
((1− η)
((u+ η ρ
)sinψ sin (θ − φ)
+ v sinψ cos (θ − φ))( 1
‖ρ13‖3− 1
‖ρ14‖3
)
+ η((u− (1− η) ρ
)sinψ sin (θ − φ) + v sinψ cos (θ − φ)
)( 1
‖ρ23‖3− 1
‖ρ24‖3
))
ε
(ψ − 2e
sin (θ − ξ)1 + e cos (θ − ξ)
ψ − φ2 sinψ cosψ
)= −ρ4
((1− η)
((u+ η ρ
)cosψ cos (θ − φ)− v cosψ
× sin (θ − φ)− w sinψ)( 1
‖ρ13‖3− 1
‖ρ14‖3
)
+ η((u− (1− η) ρ
)cosψ cos (θ − φ)− v cosψ sin (θ − φ)− w sinψ
)( 1
‖ρ23‖3− 1
‖ρ24‖3
)),
(3.41)
Here the variables (u, v, w, φ, ψ) are to be understood as functions of θ instead of τ . The point (·) now
4For notational convenience, from now on we will omit the dependence of ρ on (θ − ξ).
42
symbolises the derivative with respect to θ instead of t.
3.4.1 PCR3BP with dumbbell satellite
As pointed out in Section 3.3, the systems of equations (3.25) or (3.41) have no constants of the motion
or first integrals, hence they are called nonintegrable, in the sense that one cannot obtain a general
solution to these systems of equations in closed form. We thus make a further simplification to (3.41),
by confining the motion of the centre of mass of the dumbbell to the plane of the orbits of the primaries,
while assuming that the motion of m1 and m2 is circular about their common barycentre. The dumbbell
itself is still allowed to rotate around its centre of mass in three-dimensional space. This is the PCR3BP
with dumbbell satellite. The PCR3BP is an adequate starting model for many phenomena in celestial
mechanics, including the modelling of comets interacting with Jupiter and the Sun, whose motion is
very close to Jupiter’s orbital plane, or the design of spacecraft trajectories that take into account the
gravitational field of several bodies. For a comprehensive review of the applications of this model, see
[48].
This procedure effectively reduces the dimension of the configuration space of the dumbbell by one,
becoming R2 × S2, and, accordingly, the dimension of the phase space by two. A configuration of the
dumbbell is now fully specified by u, v and the attitude angles on the sphere, φ and ψ. The configuration
space of both primaries is S1. Imposing w = w = w ≡ 0 and setting e = 0 and ρ = 1 in (3.21) and
(3.41), we arrive at the equations for the PCR3BP with dumbbell satellite in the synodic reference frame,
in dimensionless form,
u− 2v = u− (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)(u+ η
)+ (1− δ)δε sinψ cos (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))
− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)(u− (1− η)
)+ (1− δ)δε sinψ cos (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
))
v + 2u = v − (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)v − (1− δ)δε sinψ sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))
− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)v − (1− δ)δε sinψ sin (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
))
ε(φ sin2 ψ + 2 φ ψ sinψ cosψ
)= − (1− η)
((u+ η
)sinψ sin (θ − φ) + v sinψ cos (θ − φ)
)( 1
‖ρ13‖3
− 1
‖ρ14‖3
)− η((u− (1− η)
)sinψ sin (θ − φ) + v sinψ cos (θ − φ)
)( 1
‖ρ23‖3− 1
‖ρ24‖3
)
ε(ψ − φ2 sinψ cosψ
)= − (1− η)
((u+ η
)cosψ cos (θ − φ)− v cosψ sin (θ − φ)
)( 1
‖ρ13‖3
− 1
‖ρ14‖3
)− η((u− (1− η)
)cosψ cos (θ − φ)− v cosψ sin (θ − φ)
)( 1
‖ρ23‖3− 1
‖ρ24‖3
),
(3.42)
43
where the squared distances between the primaries and the masses of the dumbbell are given by
‖ρ13‖2 =(u+ η
)2+ v2 + δ2 ε2 + 2 δ ε sinψ
((u+ η
)cos (θ − φ)− v sin (θ − φ)
)‖ρ14‖2 =
(u+ η
)2+ v2 + (1− δ)2 ε2 − 2 (1− δ) ε sinψ
((u+ η
)cos (θ − φ)− v sin (θ − φ)
)‖ρ23‖2 =
(u− (1− η)
)2+ v2 + δ2 ε2 + 2 δ ε sinψ
((u− (1− η)
)cos (θ − φ)− v sin (θ − φ)
)‖ρ24‖2 =
(u− (1− η)
)2+ v2 + (1− δ)2 ε2 − 2 (1− δ) ε sinψ
((u− (1− η)
)cos (θ − φ)− v sin (θ − φ)
).
(3.43)
Finally, we point out that in this model the masses m1 and m2 are at rest in the synodic reference
frame, with positions given by (−η, 0, 0) and(
(1− η) , 0, 0), respectively. Therefore, they are always at
unit distance from one another in this system. In Figure 3.4, we depict a possible configuration of the
dumbbell and the primaries in the PCR3BP with dumbbell satellite.
Figure 3.4: Schematic representation of a configuration of the dumbbell and the primaries in thePCR3BP with dumbbell satellite, in the inertial frame S, at some arbitrary instant of time. Here wehave used η = δ = 1/2, so the dumbbell has equal masses, and the primaries have equal masses too.The variables (U, V,W ) are dimensionless, cartesian coordinates in the inertial frame. The dumbbellis aligned perpendicularly to the Lagrange plane of the primaries, while the latter share the same orbitaround their common barycentre.
We study now the equations (3.42). Specifically, we want to apply the theory described in this section
to the Pluto-Charon-Nix system. Since it has been shown by numerical integration that the motion of Nix
is nearly in the plane of the orbits of Pluto and Charon about their common barycentre ([45]), and since
the eccentricities of these orbits are almost negligible ([42]), if we assume that the shape of Nix, like so
many asteroids in the solar system, is that of a dumbbell ([49]), then the PCR3BP with dumbbell satellite
defined by equations (3.42) and (3.43) is a suitable candidate for studying such a system. The system
of equations (3.42) is subjected to a restriction which follows from the vanishing of the left-hand side of
the equation in w in (3.41),
(1− η)ε cosψ
(1
‖ρ13‖3− 1
‖ρ14‖3
)= −η ε cosψ
(1
‖ρ23‖3− 1
‖ρ24‖3
). (3.44)
44
The condition (3.44) is a necessary condition for the motion of the centre of mass of the dumbbell to
be confined to the Lagrange plane of the primaries. It is thus mandatory that all solutions to (3.42)
and (3.43) also verify (3.44), otherwise the corresponding motion cannot occur. In the following we do a
systematic analysis of (3.44) in conjunction with (3.42) and (3.43), in order to derive the possible motions
for the dumbbell in the context of the PCR3BP.
All the conditions that verify equation (3.44) fall into one of the following four main categories:
category 1: ‖ρ13‖ = ‖ρ14‖ and ‖ρ23‖ = ‖ρ24‖ ( 6= ‖ρ13‖ , ‖ρ14‖) ;
category 2: ‖ρ13‖ = ‖ρ24‖ and ‖ρ14‖ = ‖ρ23‖ ( 6= ‖ρ13‖ , ‖ρ24‖)
and η =1
2; (3.45)
category 3: ‖ρ13‖ = ‖ρ14‖ = ‖ρ23‖ = ‖ρ24‖ (all 4 distances equal) ;
category 4: ‖ρ13‖ 6= ‖ρ14‖ 6= ‖ρ23‖ 6= ‖ρ24‖ (all 4 distances different) .
Category 1
The case ε = 0, for which the dumbbell reduces to a single point mass, obviously falls under this category
too in general. We don’t pursue this, since it has already been extensively researched in the literature
(see, for instance, Szebehely [3]), and we are solely interested in the study of the dumbbell dynamics.
There are two cases that fall under this category for ε > 0. By inspection of (3.43), we see that we
can make ‖ρ13‖ = ‖ρ14‖ and simultaneously ‖ρ23‖ = ‖ρ24‖ by setting δ = 1/2 (m3 = m4) and either
ψ = 0, π or
(u+ η) cos (θ − φ) = v sin (θ − φ) and(u− (1− η)
)cos (θ − φ) = v sin (θ − φ) , (3.46)
which implies (u+η) cos (θ − φ) =(u−(1− η)
)cos (θ − φ). This equation has the solutions θ−φ = ±π/2,
which, on substitution in (3.46), give v ≡ 0. For the case
δ =1
2and ψ = 0, π ,
the equations governing the motion of the dumbbell become, from (3.42),
u− 2v = u− (1− η)u+ η
‖ρ13‖3− η u− (1− η)
‖ρ23‖3
v + 2u =
(1− 1− η‖ρ13‖3
− η
‖ρ23‖3
)v
ψ = 0 ,
(3.47)
where ‖ρ13‖ = ‖ρ14‖ =
√(u+ η
)2+ v2 + ε2/4 and ‖ρ23‖ = ‖ρ24‖ =
√(u− (1− η)
)2+ v2 + ε2/4 .
Therefore, this type of motion has the dumbbell always pointing in the orthogonal direction to the La-
45
grange plane, with ψ = const., and only exists for equal masses of the dumbbell. The case
δ =1
2and v ≡ 0 and θ − φ = ±π/2
reduces the equations of motion (3.42) to
u− (1− η)u+ η
‖ρ13‖3− η u− (1− η)
‖ρ23‖3= 0
u = 0
ψ sinψ cosψ = 0
ψ − sinψ cosψ = 0 ,
(3.48)
with ‖ρ13‖ = ‖ρ14‖ =
√(u+ η
)2+ ε2/4 and ‖ρ23‖ = ‖ρ24‖ =√(
u− (1− η))2
+ ε2/4 , since, from θ − φ = ±π/2, φ ≡ 1 and φ ≡ 0. The first two equations of
(3.48) tell us that u = u(θ0) is a constant (θ0 is the initial value of the independent variable θ) and that it
is a root of the function h : R→ R defined by
h(u; η, ε) = u− (1− η)u+ η
‖ρ13‖3− η u− (1− η)
‖ρ23‖3, (3.49)
where the distances are as defined immediately above. That h has a root is easily seen. In fact, the
value of h at u = 0, given by
h(u = 0; η, ε) = 8 (1− η) η
1(ε2 + 4 (1− η)
2) 3
2
− 1(ε2 + 4η2
) 32
, (3.50)
is negative for 0 < η < 1/2, zero for η = 1/2 and positive for 1/2 < η < 1, whatever the value of ε > 0 (in
our applications ε will always be much smaller than unity). This, combined with the continuity of h for all
u ∈ R and the fact that limu→−∞ h(u; η, ε) = −∞ and limu→+∞ h(u; η, ε) = +∞ , ensures that there is
always at least one positive root of h for the interval 0 < η < 1/2 or one negative root for 1/2 < η < 1.
On the other hand, when η = 1/2, u = 0 is a solution. Indeed, one may verify via graphical analysis
of the function h that the equation h(u; η, ε) = 0 can have 1, 2, 3, 4, or even 5 solutions, according
to the values of the parameters η and ε. Additionally, from the last two equations in (3.48), it follows
immediately that ψ = const. for all θ ∈ R and that it can only take the values 0, π or π/2. As such, the
present case corresponds to equilibrium solutions of the system of equations (3.42). These solutions
are defined by the following conditions:
u = u(η, ε); v = 0; θ − φ = ±π/2; ψ = 0, π, π/2;
u = v = ψ = 0; φ = 1; δ = 1/2.(3.51)
46
They are the direct analogues of the Lagrangian points L1, L2 and L3 in the (conventional) CR3BP
[44, 3]. For every pair of values (η, ε), we may have several possible motions, depending on the values
taken by u, ψ or θ−φ, but only a subset of them is truly distinguishable, since the masses of the dumbbell
are equal, m3 = m4. All of the solutions (3.51) are characterized by a circular motion of radius u of the
centre of mass of the dumbbell about the common barycentre of the primaries in inertial space. These
are thus steady states of (3.42). We note that u is a function of the parameters η and ε, given implicitly
by h(u; η, ε) = 0. It is easy to verify via graphical analysis of h that u can take on values smaller or
greater than the radii of the orbits of m1 and m2, hence the dumbbell can be found either in-between the
primaries or outside that region. Since θ − φ = ±π/2, the dumbbell rotates in the same direction of the
translational motion of the primaries, always at right angles to the line segment connecting m1 to m2.
There is thus a 1 : 1 synchronisation of the translational motion of the dumbbell with its rotational motion
and the translational motion of the primaries. When ψ = π/2, the dumbbell is constrained to rotate in
the Lagrange plane, and when ψ = 0, π, it is aligned orthogonally to this plane (this is a steady solution
of (3.47)). Figure 3.5 illustrates two of the steady states (3.51) in the inertial S (Figures 3.5a, 3.5b) and
synodic R (Figures 3.5c, 3.5d) reference frames, for ψ = π/2 and θ − φ = π/2.
(a) (b)
(c) (d)
Figure 3.5: Two steady states of the PCR3BP with dumbbell satellite represented in the inertial S andsynodic R frames at θ = 3π/5 ((a), (c)) and at θ = 5π/4 ((b), (d)), for which the dumbbell lies in theLagrange plane (ψ = π/2) and maintains a right angle to the line that joins the primaries (θ − φ = π/2).The length of the dumbbell is ε = 0.15, and we have used η = 0.2 and δ = 0.5 across all figures.The dashed lines show the paths of the primaries in inertial space, whereas the solid line illustratesthe trajectory of the dumbbell’s centre of mass. (a) The centre of mass of the dumbbell describes acircular trajectory in the region between the primaries (u ≈ 0.44); (b) the centre of mass of the dumbbelldescribes a circular trajectory in the region outside the primaries (u ≈ −1.08); (c) same as (a), depictedin the synodic reference frame R; (d) same as (b), depicted in R.
47
Category 2
We set u = 0 and δ = 1/2 in addition to η = 1/2, in order to satisfy (3.44) with ‖ρ13‖ = ‖ρ24‖ and
‖ρ14‖ = ‖ρ23‖. Plugging these values into (3.43) we retrieve the condition ε v sinψ sin (θ − φ) = 0. This
equation is satisfied for ψ = 0, π or v = 0 or θ − φ = 0, π, provided ε is nonvanishing, which is the case
under study. The case ψ = 0, π and u = 0 and η = δ = 1/2 is seen, by inspection of (3.43), to fall under
category 3 of all four distances equal. It is a hypothetical motion associated to (3.47), and we will come
to its analysis ahead. For now, let us focus on the other two cases, which both fall under the category in
study.
The first case is
η = δ =1
2and u = v = 0 .
This solution of (3.42) has the centre of mass of the dumbbell placed at the centre of mass of the
primaries m1 and m2, and it exists only for equal masses of the primaries and equal masses of the
dumbbell. The rotation of the dumbbell in configuration space is governed by the equations
ε(φ sin2 ψ + 2 φ ψ sinψ cosψ
)= −1
2sinψ sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
)
ε(ψ − φ2 sinψ cosψ
)= −1
2cosψ cos (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
),
(3.52)
where
‖ρ13‖2 = ‖ρ24‖2 = 1/4(1 + ε2 + 2 ε sinψ cos (θ − φ)
)‖ρ14‖2 = ‖ρ23‖2 = 1/4
(1 + ε2 − 2 ε sinψ cos (θ − φ)
).
The equations (3.52) and (3.42) admit the following equilibrium solutions or steady states
u = v = 0; θ − φ = ±π/2; ψ = π/2;
u = v = ψ = 0; φ = 1; η = δ = 1/2;
u = v = 0; ψ = 0, π;
u = v = ψ = 0; η = δ = 1/2 ,
(3.53)
for which all four distances are equal to√
1 + ε2/2, and, thus, fall under the third category of solutions.
In the steady state for which ψ = π/2, the dumbbell is allowed to rotate only in the Lagrange plane and,
in view of the constant difference θ − φ = ±π/2, it rotates in inertial space in the same direction of the
translational motion of the primaries, maintaining at all times a right angle to the line that connects them.
This means that there is a 1 : 1 synchronisation between the rotation of the dumbbell and the translation
of the primaries. The dumbbell always shows the same face to each one of the primaries. For the
other steady state, corresponding to ψ = 0, π, the dumbbell maintains an orthogonal configuration with
respect to the Lagrange plane at the barycentre of m1 and m2. Nothing can be said about the rotation
48
in φ. In Figure 3.6 we illustrate the two kinds of steady states.
(a) (b)
(c) (d)
Figure 3.6: Steady states (3.53) of the PCR3BP with dumbbell satellite represented in the inertial S ((a),(b)) and synodic R ((c), (d)) frames at θ = 5π/6, for ε = 0.15. (a), (c) The dumbbell lies in the Lagrangeplane (ψ = π/2) and maintains a right angle to the line that joins the primaries (θ − φ = π/2) (alsorepresented in (a) is the trajectory in inertial space of the masses of the dumbbell); (b), (d) the dumbbellis lined up with the W axis (ψ = 0).
The second case is
η = δ =1
2and u = 0 and θ − φ = 0, π .
Substituting these conditions into (3.42) and (3.43), it leads to the equations
v = 0
v
(1− 1
2
(1
‖ρ13‖3+
1
‖ρ14‖3
))= 0
ψ sinψ cosψ = 0
ε(ψ − sinψ cosψ
)= ∓1
2cosψ
(1
‖ρ13‖3− 1
‖ρ14‖3
),
(3.54)
where
‖ρ13‖2 = ‖ρ24‖2 = 1/4(1 + 4v2 + ε2 ± 2 ε sinψ
)49
‖ρ14‖2 = ‖ρ23‖2 = 1/4(1 + 4v2 + ε2 ∓ 2 ε sinψ
).
Here the upper sign conforms to θ − φ = 0, while the lower sign corresponds to θ − φ = π. The first and
third equations in (3.54) imply that v and ψ are constant for all θ ∈ R, respectively. This in turn leads
to the conclusion that this case corresponds to steady state solutions of (3.42) for η = δ = 1/2. The
second equation in (3.54) has the trivial solution v = 0. Furthermore, it shows that, in general, v is a root
of the function m : R→ R defined by
m(v; ε, ψ) =1
(1 + 4v2 + ε2 + 2 ε sinψ)32
+1
(1 + 4v2 + ε2 − 2 ε sinψ)32
− 1
4, (3.55)
where ψ may also be seen as a parameter, since we have already determined that it is constant. The
function m is the same for both cases, θ − φ = 0 and θ − φ = π, and it implicitly defines the solution
v = v(ε, ψ), if it exists, as a function of ε and the polar angle ψ. We show that m has exactly two
symmetric roots. To do that, let us first collect some of its properties:
i m(−v; ε, ψ) = m(v; ε, ψ) i.e. m is an even function of v;
ii limv→±∞m(v; ε, ψ) = −1/4;
iii m reaches its maximum value for v = 0. In case ε = 1 and ψ = π/2, v = 0 is a singular point of m.
The first of these properties demonstrates that we only have to analyse the behaviour of m in the interval
[0,+∞). If the absolute maximum of m at v = 0 is positive, then, since m is a continuous, monotonically
decreasing function in that interval, there will be a unique root in [0,+∞). Considering that m is even in
v, there will be another, symmetric root in (−∞, 0]. All that is left doing then is to show that, in fact, the
maximum at v = 0 is positive i.e.
m(v = 0; ε, ψ) =1
(1 + ε2 + 2 ε sinψ)32
+1
(1 + ε2 − 2 ε sinψ)32
− 1
4> 0 . (3.56)
The value of the maximum at v = 0, m(v = 0; ε, ψ), is a function of ψ and ε. The partial derivative of this
function with respect to ψ is
∂
∂ψm(v = 0; ε, ψ) = 3ε cosψ
(1
(1 + ε2 − 2 ε sinψ)52
− 1
(1 + ε2 + 2 ε sinψ)52
), (3.57)
whose roots are ψ = 0, π and ψ = π/2. These are the critical points of m(v = 0; ε, ψ). By continuing this
process further and calculating the second derivative of this function with respect to ψ, one may easily
verify that m(v = 0; ε, ψ) has minimums for ψ = 0, π and a maximum for ψ = π/2. This means that the
maximum of m at v = 0 takes its lowest value when either ψ = 0 or ψ = π. In other words, ψ = 0, π is
a worst-case scenario. We thus calculate m(v = 0; ε; ψ = 0, π) and derive the condition that ensures
the maximum is still positive in the worst-case scenario. It turns out that m(v = 0; ε; ψ = 0, π) =
2/(1 + ε2
)3/2 − 1/4, and this is positive if and only if |ε| <√
3. Since in our applications ε is always
smaller than unity, we have demonstrated the following result: there are always two more equilibrium
50
solutions for the present case with symmetric v coordinates, besides the one for which v = 0. This is
independent of the value of ψ at the fixed point, and is valid in both cases θ − φ = 0 and θ − φ = π.
One can readily verify that ψ = 0, π and ψ = π/2 satisfy the last equation in (3.54). Therefore, (3.54)
and (3.42) admit the following equilibrium solutions or steady states
u = 0; v = 0, ± v(ε, ψ = π/2); θ − φ = 0, π; ψ = π/2;
u = v = ψ = 0; φ = 1; η = δ = 1/2;
u = 0; v = 0, ± v(ε, ψ = 0, π); θ − φ = 0, π; ψ = 0, π;
u = v = ψ = 0; φ = 1; η = δ = 1/2 .
(3.58)
Here ± v(ε, ψ) denotes the two symmetric roots of m. These are the direct analogues of the Lagrangian
equilateral points L4 and L5 in the conventional CR3BP [44, 3]. For the last of these two steady states,5
since ψ = 0, π, the dumbbell is aligned orthogonally to the Lagrange plane, and all the four distances
are in fact equal to√
1 + 4v2 + ε2/2, so it will fall under category 3. The motion of the centre of mass of
the dumbbell in the inertial reference frame is different according to whether v is zero or not. If v = 0,
the dumbbell is at rest at the centre of mass of the primaries, in correspondence with the last of the
steady states (3.53). Otherwise, since v = const. in the synodic reference frame, the centre of mass of
the dumbbell describes a circular trajectory of radius v(ε) ≡ v(ε, ψ = 0, π) in inertial space around this
point, much like one of the cases in (3.51). If that is the case, there will be a 1 : 1 synchronisation be-
tween the translational motions of the dumbbell and the primaries. In passing we note that the equation
m(v; ε, ψ) = 0 is actually solvable when ψ = 0, π, because all the distances are equal then. The result
is v(ε, ψ = 0, π) =√
3− ε2/2, and this is the radius of the orbit. From the topological point of view, both
solutions with v = + v(ε, ψ) and v = − v(ε, ψ) are equivalent in the inertial space.
For the steady states with ψ = π/2, the dumbbell is confined to rotate in the Lagrange plane of the
primaries, and, in view of θ − φ being equal to 0 or π, it points at all times in the direction of the line
segment connecting the primaries. The motion of the centre of mass of the dumbbell in the inertial
reference frame is exactly as stated above for the case ψ = 0, π. There are 1 : 1 synchronisations for
these solutions too, in accordance with the fact that the masses are all at rest in the synodic reference
frame. If v = 0, the synchronisation happens between the rotational motion of the dumbbell and the
translational motions of m1 and m2. If v = ± v(ε, ψ = π/2), the translational motion of the dumbbell
additionally synchronises with the other two periodic motions. Figure 3.7 depicts some of the steady
states (3.58).
5There are actually 18 steady states in (3.58), although not all of them are distinguishable, because η = δ = 1/2. “Two steadystates” refers only to the manner in which they are grouped, according to the category to which they belong. In the case understudy, some belong to category 2, others to 3.
51
(a) (b)
(c) (d)
Figure 3.7: Steady states (3.58) of the PCR3BP with dumbbell satellite, in which the dumbbell lies in theLagrange plane (ψ = π/2) and is aligned along the direction of the segment line that joins the primaries(θ = φ), represented in the inertial S ((a), (b)) and synodic R ((c), (d)) reference frames. We have usedε = 0.15. (a), (c) v = 0: the dumbbell and primaries are depicted at θ = 4π/3 (also represented in (a) isthe trajectory of the masses of the dumbbell in the inertial space); (b), (d) v(ε = 0.15, ψ = π/2) ≈ 0.867:the dumbbell and primaries are depicted at θ = 7π/4. This steady state is an extension of the equilateralLagrangian points of the CR3BP ([44, 3]) to the case where the satellite has a dumbbell shape.
Category 3
We require that all four distances, ‖ρ13‖, ‖ρ14‖, ‖ρ23‖ and ‖ρ24‖, are equal. Equating ‖ρ13‖2 to ‖ρ23‖2,
and, separately, ‖ρ14‖2 to ‖ρ24‖2 in (3.43), one gets
(u+ η
)2=(u− (1− η)
)2 − 2δ ε sinψ cos (θ − φ) (3.59)(u+ η
)2=(u− (1− η)
)2+ 2 (1− δ) ε sinψ cos (θ − φ) . (3.60)
Subtracting (3.60) from (3.59), it follows that
−2δ ε sinψ cos (θ − φ) = 2 (1− δ) ε sinψ cos (θ − φ) ,
or sinψ cos (θ − φ) = 0 for positive ε. Substituting back into equations (3.59)– (3.60), the next condition
holds (u+ η
)2=(u− (1− η)
)2, (3.61)
with solution u+η = 1/2. This demonstrates that for all four distances to be equal, it is necessary that the
centre of mass of the dumbbell moves along the perpendicular bisector of the line segment connecting
the primaries. The coordinate u of C ′ in the synodic reference frame is then fixed by the value of η.
52
On the other hand, equating ‖ρ13‖2 to ‖ρ14‖2 and ‖ρ24‖2 in (3.43), one obtains the condition
δ2ε2 − 2δ ε v sinψ sin (θ − φ) = (1− δ)2 ε2 + 2 (1− δ) ε v sinψ sin (θ − φ) ,
or, equivalently,
ε2 (2δ − 1) = 2ε v sinψ sin (θ − φ) , (3.62)
where we have already substituted the values for both u + η and sinψ
× cos (θ − φ). The equation (3.62) shows that the quantity v sinψ sin (θ − φ) is a constant of the mo-
tion when all four distances are equal, whose value is ε (δ − 1/2).
The cases under this category thus satisfy the following properties:
sinψ cos (θ − φ) = 0 and u =
1
2− η
ε2 (2δ − 1) = 2ε v sinψ sin (θ − φ) .
We have two possibilities: either ψ = 0, π or θ− φ = ±π/2. For ψ = 0, π, v sinψ sin (θ − φ) is identically
zero and therefore δ = 1/2, corresponding to the case
ψ = 0, π and u =1
2− η and δ =
1
2. (3.63)
The equations of motion for (3.63) are, by (3.42),u− 2v = u
(1− 1
‖ρ13‖3
)
v = v
(1− 1
‖ρ13‖3
),
(3.64)
where ‖ρ13‖ = ‖ρ14‖ = ‖ρ23‖ = ‖ρ24‖ =√
1 + 4v2 + ε2/2 from (3.43), and, of course, ψ = 0. They
govern the motion of the centre of mass of the dumbbell along the perpendicular bisector of the line that
connects m1 to m2. The equations (3.64) and (3.42) take the following equilibrium solutions or steady
statesu = v = 0; ψ = 0, π;
u = v = ψ = 0; η = δ = 1/2 ,(3.65)
for equal masses of the primaries, and
u =1
2− η; v = ± v(ε, ψ = 0, π); ψ = 0, π;
u = v = ψ = 0; δ = 1/2 ,(3.66)
for any value of η (and ε <√
3). We have already encountered the steady states (3.65) in (3.53) and
(3.58). While we knew from equation (3.58) that the fixed points (3.66) exist for η = 1/2 and u = 0,
equation (3.66) tells us that they also exist more generally for different masses of the primaries and
u = 1/2− η. Here, as before, v(ε, ψ = 0, π) =√
3− ε2/2. Figure 3.8 depicts one of these steady states,
53
for η = 0.2 and ε = 0.15. To conclude, we note that H is a constant of the motion for (3.64), because
ψ = 0, π for the case (3.63), in accordance with (3.34)–(3.35).
(a)
(b)
Figure 3.8: Steady state (3.66) of the PCR3BP with dumbbell satellite, for η = 0.2 and ε = 0.15, repre-sented in the inertial S (a) and synodic R (b) reference frames at θ = 5π/3. The dumbbell is alignedperpendicularly to the Lagrange plane (ψ = 0). This is another steady state of the “equilateral type”, thistime for different masses of the primaries. This steady state only occurs for orthogonal configurations ofthe dumbbell satellite with respect to the Lagrange plane.
For θ − φ = ±π/2, we have the second case
θ − φ = ±π/2 and u =1
2− η and ε2 (2δ − 1) = ± 2ε v sinψ , (3.67)
where, as usual, the upper sign denotes θ−φ = π/2, while the lower sign corresponds to θ−φ = −π/2.
The equations of motion (3.42) transform, for (3.67), into
u− 2v = u
(1− 1
‖ρ13‖3
)
v = v
(1− 1
‖ρ13‖3
)ψ sinψ cosψ = 0
ψ − sinψ cosψ = 0 ,
(3.68)
54
with ‖ρ13‖ = ‖ρ14‖ = ‖ρ23‖ = ‖ρ24‖ =
√1 + 4v2 + ε2 − (2v sinψ)
2/2 =
√1 + ε2 + (2v cosψ)
2/2. On
the other hand, the condition ε2 (2δ − 1) = ± 2ε v sinψ implies that the product v sinψ is a constant of
the motion for (3.68). We can get the corresponding value of δ from that condition, which is
δ =1
2± v sinψ
ε. (3.69)
This shows that the value of the constant is fixed by δ and ε. Moreover, since δ ∈ (0, 1), the value of
v sinψ has to lie within the interval from −ε/2 to ε/2. Thus, the equation in (3.68) governing the motion
of the centre of mass of the dumbbell in v becomes completely decoupled from the equations in ψ.
Additionally, we see from the last two equations in (3.68) that ψ is necessarily a constant, so that, by
the constancy of v sinψ, v is also constant. From this we conclude that the solution of (3.67)–(3.68)
is a steady state of (3.42). We can have either ψ = 0, π, which matches the previous case (3.63), or
ψ = π/2. Setting u = v = v ≡ 0 and ψ = π/2 in (3.68), one gets that u = v = 0. By (3.67) and (3.69),
this implies that η = δ = 1/2, hence the steady state associated to (3.68) is
u = v = 0; θ − φ = ±π/2; ψ = π/2;
u = v = ψ = 0; φ = 1; η = δ = 1/2 ,
which was encountered in (3.53) under category 2.
Category 4
The equation (3.44) restricting the motion of the dumbbell in the PCR3BP may also be satisfied by simply
imposing ψ = π/2 in (3.42). This leads to the following system of equations:
u− 2v = u− (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)(u+ η
)+ (1− δ)δ ε cos (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))
− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)(u− (1− η)
)+ (1− δ)δ ε cos (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
))
v + 2u = v − (1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)v − (1− δ)δ ε sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
))
− η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)v − (1− δ)δ ε sin (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
))
ε φ = − (1− η)((u+ η
)sin (θ − φ) + v cos (θ − φ)
)( 1
‖ρ13‖3− 1
‖ρ14‖3
)
− η((u− (1− η)
)sin (θ − φ) + v cos (θ − φ)
)( 1
‖ρ23‖3− 1
‖ρ24‖3
),
(3.70)
55
and ψ = 0, where
‖ρ13‖2 =(u+ η
)2+ v2 + δ2 ε2 + 2 δ ε
((u+ η
)cos (θ − φ)− v sin (θ − φ)
)‖ρ14‖2 =
(u+ η
)2+ v2 + (1− δ)2 ε2 − 2 (1− δ) ε
((u+ η
)cos (θ − φ)− v sin (θ − φ)
)‖ρ23‖2 =
(u− (1− η)
)2+ v2 + δ2 ε2 + 2 δ ε
((u− (1− η)
)cos (θ − φ)− v sin (θ − φ)
)‖ρ24‖2 =
(u− (1− η)
)2+ v2 + (1− δ)2 ε2 − 2 (1− δ) ε
((u− (1− η)
)cos (θ − φ)− v sin (θ − φ)
).
(3.71)
The equations (3.70) study the planar oscillations of the dumbbell satellite in the Lagrange plane of
the primaries, in a context of spin-orbit interaction. We have already encountered some solutions with
ψ = π/2 in this section, corresponding to synchronisations 1 : 1. Those are the steady states associated
to (3.70), and, as we have seen, they are characterized by having two or more of the four distances
equal. The general solutions of (3.70) need not have any two distances equal. Moreover, all the equi-
librium solutions derived so far exist only for η and/or δ equal to 1/2. This shows that those motions are
structurally unstable (Appendix A), for any small perturbation in the values of the masses destroys the
corresponding trajectories in the phase space of the system. On the other hand, the general motions
governed by equations (3.70) do not have any restrictions on the values which η and δ can take between
0 and 1. These solutions fall under category 4. We present several numerically obtained solutions to the
system of equations (3.70) in Section 3.5.
3.4.2 Sitnikov Problem
A very interesting dynamical system is known as the Sitnikov Problem (SP). The SP is a special case
of the R3BP and it has a very simple formulation: a massless body (known as the satellite) is confined
to move along a straight line that is perpendicular to the Lagrange plane of the Keplerian orbits of two
primaries with equal masses and passes through their barycentre. Despite its simple formulation, it has
however a very rich phase space structure when the primaries move in eccentric orbits, where all kinds of
motions, namely periodic orbits, quasi-periodic orbits and chaotic motion, may be found. It is thus often
cited as a model case for the appearance of chaos. The SP is named after Russian mathematician
Kirill Sitnikov, who first showed the existence of oscillatory motions in this case of the R3BP that are
unbounded (Sitnikov [27]).
Several extensions of the SP have been studied over the years (see, for instance, Dvorak and Sui
Sun [50]). For a review of previous works on the SP, we refer to [51]. Here we will extend the problem
by assuming that the satellite has the shape of a dumbbell. The centre of mass of the dumbbell will
thus be restricted to move in a straight line perpendicular to the plane formed by the Keplerian orbits
of the primaries and passing through their common centre of mass, while the dumbbell is still allowed
to rotate in three-dimensional space. This effectively reduces the number of degrees of freedom in the
configuration space of the system of equations (3.41) by two, which becomes R× S2. A configuration of
56
the dumbbell is now fully specified by w and the attitude angles on the sphere, φ and ψ. The equations
for the SP become
w − 2esin (θ − ξ)
1 + e cos (θ − ξ)w = −ρ4
((1− η)
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)w + (1− δ)δ ε cosψ
(1
‖ρ13‖3
− 1
‖ρ14‖3
))+ η
((1− δ‖ρ23‖3
+δ
‖ρ24‖3
)w + (1− δ)δ ε cosψ
(1
‖ρ23‖3− 1
‖ρ24‖3
)))
ε
((φ− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
φ
)sin2 ψ + 2 φ ψ sinψ cosψ
)= −ρ5 η (1− η) sinψ sin (θ − φ)
×
(1
‖ρ13‖3− 1
‖ρ23‖3+
1
‖ρ24‖3− 1
‖ρ14‖3
)
ε
(ψ − 2e
sin (θ − ξ)1 + e cos (θ − ξ)
ψ − φ2 sinψ cosψ
)= −ρ4
((1− η)
(η ρ cosψ cos (θ − φ)− w sinψ
)×
(1
‖ρ13‖3− 1
‖ρ14‖3
)− η(
(1− η) ρ cosψ cos (θ − φ) + w sinψ)( 1
‖ρ23‖3− 1
‖ρ24‖3
)),
(3.72)
and the squared distances between the primaries and the masses of the dumbbell are given by
‖ρ13‖2 = η2ρ2 + w2 + δ2 ε2 + 2 δ ε(η ρ sinψ cos (θ − φ) + w cosψ
)‖ρ14‖2 = η2ρ2 + w2 + (1− δ)2 ε2 − 2 (1− δ) ε
(η ρ sinψ cos (θ − φ) + w cosψ
)‖ρ23‖2 = (1− η)
2ρ2 + w2 + δ2 ε2 + 2 δ ε
(− (1− η) ρ sinψ cos (θ − φ) + w cosψ
)‖ρ24‖2 = (1− η)
2ρ2 + w2 + (1− δ)2 ε2 − 2 (1− δ) ε
(− (1− η) ρ sinψ cos (θ − φ) + w cosψ
). (3.73)
Figure 3.9 depicts a theoretical configuration of the dumbbell and the primaries in the SP, in the reference
frame S.
Similarly to the case of equations (3.42) for the PCR3BP with dumbbell satellite, the system of equa-
tions (3.72), together with (3.73), is restricted by two additional conditions which follow from the vanishing
of the left hand sides of the equations in u and v in (3.41),
(1− η) η ρ
((1− δ)
(1
‖ρ13‖3− 1
‖ρ23‖3
)+ δ
(1
‖ρ14‖3− 1
‖ρ24‖3
))= − (1− δ) δ ε sinψ cos (θ − φ)
×
(η
(1
‖ρ23‖3− 1
‖ρ24‖3
)+ (1− η)
(1
‖ρ13‖3− 1
‖ρ14‖3
)),
(1− η) ε sinψ sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
)= −η ε sinψ sin (θ − φ)
(1
‖ρ23‖3− 1
‖ρ24‖3
).
(3.74)
Equations (3.74) are necessary conditions for the motion of the centre of mass of the dumbbell to
be confined to the coordinate axis that is orthogonal to the Lagrange plane of the primaries. As in
57
Figure 3.9: Schematic representation of a configuration of the dumbbell and the primaries in the SP, inthe inertial frame S, at θ = (9/10)π. The parameters of the orbits of the primaries around their commoncentre of mass are as follows: λ = 3.5, ξ = 0 and e = 0.5. The length of the dumbbell is ε = 0.4. We haveused η = δ = 1/2, so the dumbbell has equal masses, and the primaries have equal masses too. Thedumbbell is aligned parallel to the Lagrange plane (ψ = π/2) and orthogonal to the line segment thatjoins the primaries (θ − φ = −π/2), while its centre of mass moves along the dashed line that coincideswith the W axis.
Section 3.4.1, we look for solutions to the SP, equations (3.72)–(3.73), consistent with these conditions,
specifically equilibrium solutions or steady states. The Hamiltonian H is not in general conserved in the
SP, because the orbits of the primaries have a nonzero eccentricity.
All the conditions that satisfy (3.74) fall into one of the following three main categories:
category 1: ‖ρ13‖ = ‖ρ24‖ and ‖ρ14‖ = ‖ρ23‖ (6= ‖ρ13‖ , ‖ρ24‖)
and η = δ =1
2;
category 2: ‖ρ13‖ = ‖ρ14‖ = ‖ρ23‖ = ‖ρ24‖ (all 4 distances equal) ;
category 3: ‖ρ13‖ = ‖ρ23‖ and ‖ρ14‖ = ‖ρ24‖ (6= ‖ρ13‖ , ‖ρ23‖) . (3.75)
Category 1
We equate ‖ρ13‖2 to ‖ρ24‖2 or ‖ρ14‖2 to ‖ρ23‖2 in (3.73), with η = δ = 1/2, to obtain
ε
(1
2ρ sinψ cos (θ − φ) + w cosψ
)= −ε
(−1
2ρ sinψ cos (θ − φ) + w cosψ
),
which has the solution w cosψ = 0. Three cases fall under this category: the first of these is
η = δ =1
2and w = 0 , (3.76)
58
for which the equations of motion (3.72) becomeε
((φ− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
φ
)sin2 ψ + 2φ ψ sinψ cosψ
)= −1
2ρ5 sinψ sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
)
ε
(ψ − 2e
sin (θ − ξ)1 + e cos (θ − ξ)
ψ − φ2 sinψ cosψ
)= −1
2ρ5 cosψ cos (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
),
(3.77)
with ‖ρ13‖ = ‖ρ24‖ =√ρ2 + ε2 + 2 ε ρ sinψ cos (θ − φ)/2 and ‖ρ14‖ =
‖ρ23‖ =√ρ2 + ε2 − 2 ε ρ sinψ cos (θ − φ)/2. The two equations (3.77) describe the rotation of a dumb-
bell, whose centre of mass lies fixed at the barycentre of the primaries.
The second case is
η = δ =1
2and ψ = π/2 , (3.78)
and it reduces (3.72) to
w − 2e
sin (θ − ξ)1 + e cos (θ − ξ)
w = −1
2ρ4
(1
‖ρ13‖3+
1
‖ρ14‖3
)w
ε
(φ− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
φ
)= −1
2ρ5 sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
),
(3.79)
wherein ‖ρ13‖ = ‖ρ24‖ =√ρ2 + 4w2 + ε2 + 2 ε ρ cos (θ − φ)/2 and ‖ρ14‖ = ‖ρ23‖ =√
ρ2 + 4w2 + ε2 − 2 ε ρ cos (θ − φ)/2. These equations govern the motion and rotation of a dumbbell
which remains parallel to the Lagrange plane, and whose centre of mass moves along the coordinate
axis orthogonal to this plane.
Finally, it may happen that both w = 0 and ψ = π/2. This coincides with the case
η = δ =1
2and w = 0 and ψ = π/2 , (3.80)
whose equation of motion is, by (3.72),
ε
(φ− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
φ
)= −1
2ρ5 sin (θ − φ)
(1
‖ρ13‖3− 1
‖ρ14‖3
), (3.81)
with ‖ρ13‖ = ‖ρ24‖ =√ρ2 + ε2 + 2 ε ρ cos (θ − φ)/2 and ‖ρ14‖ = ‖ρ23‖ =
√ρ2 + ε2 − 2 ε ρ cos (θ − φ)/2.
Equation (3.81) characterises the planar oscillations of a dumbbell that is restricted to rotate in the La-
grange plane, and whose centre of mass lies at the barycentre of the primaries.
The equations (3.77), (3.79) and (3.81), under cases (3.76), (3.78) and (3.80), respectively, admit
the following four steady states
w = 0; θ − φ = 0, π, ±π/2; ψ = π/2;
w = ψ = 0; φ = 1; η = δ = 1/2 .(3.82)
59
In all four steady states, the dumbbell lies in the Lagrange plane, with its centre of mass located at the
barycentre of the primaries. For θ − φ = 0, π, it rotates in inertial space always coinciding with the line
joining the primaries, while for θ − φ = ±π/2, it maintains at all times a right angle to this line. As the
fact that all the masses are at rest relative to one another in the synodic reference frame suggests, all
steady states are characterised by a 1 : 1 synchronisation between the period of rotation of the dumbbell
and the translational period of the primaries. The φ equation in (3.77), (3.79) and (3.81) transforms for
steady states (3.82) into
ε
(φ− 2e
sin (θ − ξ)1 + e cos (θ − ξ)
φ
)= 0 ,
which implies that they exist only for circular orbits of the primaries (e = 0) or at the times of pericentre
passage (θ = ξ) or apocentre passage (θ = ξ + π) of m1 and m2.6 These steady states coincide with
the ones depicted in Figures 3.6a and 3.7a for the PCR3BP.
Since all the motions described so far under this category, and particularly the steady states (3.82),
depend upon η and δ being equal to 1/2, all of them are structurally unstable, as the slightest perturbation
in the masses disrupts such solutions. The steady states (3.82) for which θ− φ = ±π/2 actually belong
to the next category, by virtue of all four distances being equal to√ρ2 + ε2/2.
Category 2
We require that all four distances, ‖ρ13‖, ‖ρ14‖, ‖ρ23‖ and ‖ρ24‖, are equal. The case
ε = 0 and η =1
2, (3.83)
for which the dumbbell reduces to a single point mass and the primaries have equal masses, clearly falls
under this category. By (3.72)–(3.73), the equation of motion for this case is
w − 2esin (θ − ξ)
1 + e cos (θ − ξ)w = −ρ4 w
(ρ2/4 + w2)3/2
. (3.84)
Noting that the left-hand side of (3.84) equals ρ4d2w
dτ2by (3.40), we immediately recognise that this is in
fact the equation for the conventional SP in dimensionless coordinates. The SP is a structurally unstable
problem, since it requires that the primaries be equally massive bodies.
There are two cases that fall under this category for ε > 0. By the same reasoning that lead to
equation (3.61) in Section 3.4.1 or simply by symmetry arguments alone, one can easily conclude that
η = 1/2 and sinψ cos (θ − φ) = 0 are necessary conditions so that all four distances are equal. Further-
more, equating ‖ρ13‖2 to ‖ρ14‖2 and ‖ρ24‖2 in (3.73), one obtains the condition
δ2ε2 + 2 δ εw cosψ = (1− δ)2 ε2 − 2 (1− δ) εw cosψ ,
6Naturally, if e 6= 0 and the steady states (3.82) exist only for θ = ξ or θ = ξ + π, then these will be static, instead of steady,states.
60
or, equivalently,
ε2 (1− 2δ) = 2εw cosψ , (3.85)
where we have substituted the values for both η and sinψ cos (θ − φ). Equation (3.85) ascertains that the
quantity w cosψ is a constant of the motion when all four distances are equal, whose value is ε (1/2− δ).
The other cases under this category thus satisfy the following properties:
sinψ cos (θ − φ) = 0 and η =
1
2,
2εw cosψ = ε2 (1− 2δ) .
The first case is
ψ = 0, π and η =1
2and ± 2εw = ε2 (1− 2δ) , (3.86)
wherein the upper and lower signs correspond to the cases ψ = 0 and ψ = π, respectively. This case
gives an equilibrium solution. Indeed, from the first equation in (3.72) and from (3.73), one gets
w
(1− δ‖ρ13‖3
+δ
‖ρ14‖3
)= 0 ,
where ‖ρ13‖ = ‖ρ14‖ =√ρ2 + ε2/2. This has the unique solution w = 0, which, by (3.86), implies
δ = 1/2. Therefore we obtain the following steady state
w = 0; ψ = 0, π;
w = ψ = 0; η = δ = 1/2 ,(3.87)
for which the dumbbell is aligned along the coordinate axis that is orthogonal to the Lagrange plane, with
its centre of mass located at the barycentre of the primaries. Nothing can be said about synchronisation,
because ψ = 0, π are singular points of the coordinate system used. Steady state (3.87) coincides with
the one depicted in Figure 3.6b for the PCR3BP when e = 0. As this steady state only occurs for
η = δ = 1/2, we conclude that it is also structurally unstable.
The last case is
θ − φ = ±π/2 and η =1
2and 2εw cosψ = ε2 (1− 2δ) , (3.88)
whose equations of motion are, by (3.72)–(3.73),
w − 2esin (θ − ξ)
1 + e cos (θ − ξ)w = −ρ4 w
‖ρ13‖3
ε
(ψ sinψ cosψ − e sin (θ − ξ)
1 + e cos (θ − ξ)sin2 ψ
)= 0
ε
(ψ − 2e
sin (θ − ξ)1 + e cos (θ − ξ)
ψ − sinψ cosψ
)= 0 .
(3.89)
61
Here ‖ρ13‖ =
√ρ2 + 4w2 + ε2 − (2w cosψ)
2/2 =
√ρ2 + ε2 + (2w sinψ)
2/2. As a result of the fact that
w cosψ is a constant of the motion for (3.88)–(3.89), the w equation in (3.89) becomes decoupled from
the equations for ψ. Hence, after solving the equation for w, one may readily obtain the solution for ψ
from the additional condition w cosψ = const. The value of this constant is fixed by δ and ε. Moreover,
since δ ∈ (0, 1), the value of w cosψ has to lie within the interval from −ε/2 to ε/2. We can get the
corresponding value of δ from (3.85), which is
δ =1
2− w cosψ
ε. (3.90)
The steady states (3.82) and (3.87) found above, with θ − φ = ±π/2, are fixed points of (3.89).
Category 3
As we have seen, it is necessary that sinψ cos (θ − φ) = 0 so that ‖ρ13‖ = ‖ρ23‖ and ‖ρ14‖ = ‖ρ24‖. By
inspection of (3.73), this in turn implies that η = 1/2, or, the masses of the primaries have to be equal.
On the other hand, the equality of the distances in category 3 is not sufficient to satisfy the second
equation of (3.74). The only way to satisfy this equation and the condition sinψ cos (θ − φ) = 0 is to have
ψ = 0, π. Accordingly, the only case under category 3 is the following:
ψ = 0, π and η =1
2, (3.91)
for which the equations of motion (3.72) become
w − 2esin (θ − ξ)
1 + e cos (θ − ξ)w = −ρ4
((1− δ‖ρ13‖3
+δ
‖ρ14‖3
)w
± (1− δ) δ ε
(1
‖ρ13‖3− 1
‖ρ14‖3
)), (3.92)
with ‖ρ13‖ = ‖ρ23‖ =√ρ2/4 + w2 + δ2 ε2 ± 2 δ εw and ‖ρ14‖ = ‖ρ24‖ =√
ρ2/4 + w2 + (1− δ)2 ε2 ∓ 2 (1− δ) εw . As before, the upper and lower signs correspond to the cases
ψ = 0 and ψ = π, respectively.
Equation (3.92) governs the motion of the centre of mass of a dumbbell lined up with theW axis along
that same axis. As this motion occurs only for equal masses of the primaries, it is structurally unstable.
The steady states (3.87) are fixed points of (3.92) which occur for equal masses of the dumbbell.
We conclude from this analysis that the SP with dumbbell satellite is a structurally unstable problem,
as all the possible motions and steady states require that the mass parameter η be equal to 1/2, that is,
they only exist for equal masses of the primaries.
62
3.5 Numerical analysis of the case under category 4 of the PCR3BP
with dumbbell satellite
We give some examples of orbits for the system of equations (3.70) together with (3.71), which studies
the planar oscillations of a rigid dumbbell satellite in the Lagrange plane of the primaries in a context of
spin-orbit interaction. These may be regarded as being only a small subset of all the possible orbits of
(3.70). We used parameters appropriate for the Pluto-Charon-Nix system, namely ε = 10−3 ([52, 42])
and η = 0.1 ([42]). The value of δ was chosen to be 0.8 throughout all the simulations. The equations
were integrated using a classic Runge-Kutta method of order 8 (Hairer et al. [53]).
In Figures 3.10–3.13, we show several special orbits of the system of equations (3.70). We were
able to numerically verify that most, if not all, initial conditions lead to either ejection of the dumbbell
from the system or collision with one of the primaries in a short amount of time (after a few revolutions
of the primaries). Specifically, if the dumbbell starts in a position outside of the unit circle centred at
the centre of mass of the primaries, that will always, according to our numerical simulations, result in
immediate ejection of the system. The orbits in Figures 3.10–3.13 are special in the sense that they
aren’t short-lived and they show more interesting behaviours of the dumbbell satellite. Nonetheless, all
of them will still lead to either a collison or ejection of the system after a certain amount of time. The fate
of the dumbbell satellite is a collision with one of the primaries in Figures 3.10, 3.11 and 3.13, and an
ejection from the system in Figure 3.12.
In each Figure, we represent the path of the centre of mass of the dumbbell satellite in the inertial
reference frame S by a solid line. The coordinates (U, V ) are dimensionless, cartesian coordinates
associated to frame S. The dashed lines indicate the orbits traced out by the primaries, which are
depicted by black circles at the time of stop of the simulation. Also shown in the Figures is the temporal
evolution of the angular difference θ−φ, which characterises the rotation of the dumbbell. In Figure 3.10,
the dumbbell starts close to one of the fixed points (3.51), with u(0) = u1(η = 0.1, ε = 10−3) ≈ 0.609,
v(0) = 0, v(0) = 0.01, φ(0) = π/2 and φ(0) = 0.99, although the mass parameter δ differs from 1/2,
as required in the fixed point, and it starts moving along a circular arc. However, soon after that the
trajectory of the centre of mass appears to become chaotic, starting to rotate around the outer of the
primaries and eventually colliding with it. This information is corroborated by the graph of the time
evolution of θ − φ, which shows that the coordinate suddenly begins to rotate around all possible values
after a start in which it remained almost constant.
In Figures 3.11 and 3.12, the dumbbell starts close to another of the fixed points (3.51), with u(0) =
u2(η = 0.1, ε = 10−3) ≈ −1.042, v(0) = 0, φ(0) = −π/2, and φ(0) = 0.9 in the case of Figure 3.11
and φ(0) = 0 in the case of Figure 3.12. In both cases it starts by tracing out a quasi-circular orbit in
configuration space, very close to the fixed point orbit, then becomes attracted to the inner, heavier,
primary, revolving around this body, and finally it ends up leaving the region in-between the primaries
(Figures 3.11a, 3.12a). In the case of Figure 3.12, the dumbbell permanently leaves this region and is
eventually ejected from the system (Figure 3.12c). In the orbit of Figure 3.11, it remains in the system,
describing large ellipses around the primaries (Figure 3.11b), until it finally returns to the starting region,
63
-1.0 -0.5 0.5 1.0U
-1.0
-0.5
0.5
1.0
V
(a)
0.5 1.0 1.5 2.0 2.5 3.0θ
1
2
3
4
5
6
θ-ϕ(θ)
(b)
Figure 3.10: Collisional solution of (3.70) for η = 0.1, δ = 0.8, ε = 10−3 and initial conditions close toone of the fixed points (3.51): u(0) ≈ 0.609, u(0) = 0, v(0) = 0, v(0) = 0.01, φ(0) = π/2 and φ(0) = 0.99.(a) The motion of the centre of mass of the dumbbell in configuration space, represented by a solid line,appears to be chaotic. After about 11 revolutions of the primaries, depicted by black circles moving overthe dashed lines, around their barycentre, the dumbbell eventually collides with the lighter one. (b) Timeevolution of the angular difference θ − φ between θ = 0 and θ = π, with information about the rotationof the dumbbell. The coordinate suddenly begins at θ ≈ 1.8 to rotate around all possible values, after astart in which it remained almost constant.
only to collide with the lightest body (Figure 3.11c). The graphs of θ − φ in Figures 3.11d, 3.12b show
similar behaviour to the right portion of the graph on Figure 3.10b.
In Figure 3.13, the dumbbell satellite does not start near any fixed point. Its initial position in the
Lagrange plane is u(0) = −0.81, v(0) = 0.03, and the initial value of the azimuthal angle φ is φ(0) =
π/7.1. All the initial velocities are zero. The course of the centre of mass of the dumbbell (Figures 3.13a–
3.13c) is similar to the one just described on the basis of Figure 3.11. The time evolution of the angular
difference θ − φ in Figure 3.13d appears to be chaotic.
64
-1.0 -0.5 0.5 1.0U
-1.0
-0.5
0.5
1.0
1.5
V
(a)
-60 -50 -40 -30 -20 -10U
-5
5
10
15
V
(b)
-1.0 -0.5 0.5 1.0U
-1.5
-1.0
-0.5
0.5
1.0
V
(c)
5 10 15 20 25 30θ
1
2
3
4
5
6
θ-ϕ(θ)
(d)
Figure 3.11: Collisional solution of (3.70) for η = 0.1, δ = 0.8, ε = 10−3 and initial conditions closeto another of the fixed points (3.51): u(0) ≈ −1.042, u(0) = 0, v(0) = 0, v(0) = 0, φ(0) = −π/2 andφ(0) = 0.9. (a) Initial phase of the motion of the centre of mass of the dumbbell in configuration space,between θ = 0 and θ = 65. (b) Whole trajectory of the dumbbell in configuration space, with the durationof more than 420 revolutions of the primaries (θ = 2640), corresponding in the case of the Nix-Pluto-Charon system to more than 7 years. (c) Ending phase of the motion of the centre of mass of thedumbbell in configuration space, between θ = 2612 and θ = 2640. (d) Time evolution of the angulardifference θ − φ between θ = 0 and θ = 34, with information about the rotation of the dumbbell. Noevidence of periodic motion seems to exist. The coordinate rotates around all possible values.
65
-1.0 -0.5 0.5 1.0U
-1.0
-0.5
0.5
1.0
V
(a)
5 10 15 20 25 30θ
1
2
3
4
5
6
θ-ϕ(θ)
(b)
-1 1 2U
-10
-8
-6
-4
-2
V
(c)
Figure 3.12: Ejection solution of (3.70) for η = 0.1, δ = 0.8, ε = 10−3 and initial conditions close to one ofthe fixed points (3.51): u(0) ≈ −1.042, u(0) = 0, v(0) = 0, v(0) = 0, φ(0) = −π/2 and φ(0) = 0. (a) Initialphase of the motion of the centre of mass of the dumbbell in configuration space, between θ = 0 andθ = 45, time at which the dumbbell starts the ejection (about 7 revolutions of the primaries have passed).(b) Time evolution of the angular difference θ − φ between θ = 0 and θ = 34, with information about therotation of the dumbbell. No evidence of periodic motion seems to exist. The coordinate rotates aroundall possible values. (c) Trajectory of the dumbbell in configuration space up until θ = 60. The dumbbellhas been ejected from the system.
66
-1.0 -0.5 0.5 1.0U
-1.0
-0.5
0.5
1.0
V
(a)
5 10 15 20 25 30U
-5
5
10
15
V
(b)
-1.5 -1.0 -0.5 0.5 1.0U
-1.0
-0.5
0.5
1.0
V
(c)
5 10 15 20 25 30θ
1
2
3
4
5
6
θ-ϕ(θ)
(d)
Figure 3.13: Collisional solution of (3.70) for η = 0.1, δ = 0.8, ε = 10−3 and initial conditions u(0) =−0.81, u(0) = 0, v(0) = 0.03, v(0) = 0, φ(0) = π/7.1 and φ(0) = 0. (a) Initial phase of the motion of thecentre of mass of the dumbbell in configuration space, between θ = 0 and θ = 45. (b) Whole trajectoryof the dumbbell in configuration space, with the duration of more than 194 revolutions of the primaries(θ = 1220), corresponding in the case of the Nix-Pluto-Charon system to more than 3 years. (c) Endingphase of the motion of the centre of mass of the dumbbell in configuration space, between θ = 1208 andθ = 1220. (d) Time evolution of the angular difference θ− φ between θ = 0 and θ = 34, showing signs ofchaotic rotation.
67
68
Chapter 4
Conclusions
In this thesis we proposed to study the problem of spin-orbit interaction using an approach other than
the Averaging Theory, or the Tidal Theory, which rely on poorly determined parameters related to the
internal structure of celestial bodies. Taking that into account, we decided to consider the problem
where a satellite, modelled as a dumbbell, revolves around a point mass, interacting with it through
the gravitational force. This is what we called the Keplerian Dumbbell (KD) system. Despite not being
new, an analysis without approximations of the full dynamics of the KD was lacking. We derived the
exact equations of motion for this system, and then found and analysed its steady states or stationary
orbits. We showed that all the steady states of the KD system are unstable to small variations on the
initial conditions and that the KD is a structurally unstable problem for equal masses of the dumbbell.
For the case where the two masses of the dumbbell are equal and the motion of the centre of mass
of the dumbbell is planar, the steady states are Lyapunov unstable. For the case where the dumbbell
is aligned with the direction connecting its centre of mass to the centre of mass of the KD system and
the motion of the centre of mass of the dumbbell is planar, we have proved that for a sufficiently large
trajectory radius, these steady states are also Lyapunov unstable. Numerical analysis of the eigenvalues
of the Hessian matrix of the effective Hamiltonian associated to these steady states suggests that they
are always unstable, irrespective of the radius of the trajectory. The effective Hamiltonian is at least a
1−saddle near the steady states.
As all the steady states are Lyapunov unstable, we expect the KD to exhibit chaos. Only the steady
states 4 and 5, in which the dumbbell is aligned with the direction connecting its centre of mass to the
centre of mass of the KD, exist for different masses of the dumbbell. Interestingly, some of the unstable
steady states are the Eulerian solutions of the General Three-Body Problem. In this way, we provided a
link between the KD system and this problem.
In the limit when the length of the dumbbell goes to zero, the Kepler problem is recovered. Neverthe-
less, the stability of the system changes abruptly in this limit, since the fixed point of the Kepler problem
is stable of the centre type.
The dumbbell model was then incorporated into the framework of the Restricted Three-Body Problem
(R3BP). Our goal was to model the dynamics of the satellite Nix under the gravitational influence of Pluto
69
and Charon. We derived the exact equations of motion of the R3BP with dumbbell satellite and focused
our attention on two special cases, one in which the dumbbell’s centre of mass is constrained to move
in the Lagrange plane — the Planar Circular Restricted Three-Body Problem (PCR3BP) with dumbbell
satellite — and the other in which it is constrained to move along an axis orthogonal to this plane and
that passes through the barycentre of the primaries — the Sitnikov Problem (SP).
Necessary conditions for these two motions to occur were obtained and, based on that, the steady
states of the system were found. Some of these steady states are the direct analogues or extensions
of the Lagrangian points in the (conventional) CR3BP ([44, 3]). While in the conventional CR3BP there
are three equilibrium points over the “x” axis, when the satellite has a dumbbell shape, there may be
1, 2, 3, 4 or 5 equilibrium points over this axis. On the other hand, there are still only two “equilateral”,
symmetric, equilibrium points in the problem with dumbbell satellite, as in the case of the conventional
CR3BP.
We have determined that these steady states only occur for special configurations of the dumbbell,
namely, when it lies on the Lagrange plane, making either a right angle, or being aligned, with the line that
connects the primaries, or when it is aligned orthogonally to this plane. Moreover, the new equilateral
equilibrium points only exist for configurations in which the dumbbell either lies on the Lagrange plane,
aligned with the line that joins the primaries, or is aligned orthogonally to the plane. And they don’t occur
for different masses of the primaries, unless the dumbbell is aligned vertically to the plane.
Finally, these equilibrium points only exist for equal masses of the dumbbell, therefore we conclude
that the R3BP with dumbbell satellite is a structurally unstable problem, much like the KD system. The
stability of these fixed points will be studied in a future paper.
The special case of the PCR3BP with dumbbell satellite in which the spin axis of the dumbbell is or-
thogonal to the Lagrange plane was also studied numerically, with values of the parameters appropriate
for the Pluto-Charon-Nix system. We verified that most, if not all, initial conditions will lead to either a
collision with a primary or ejection of the dumbbell from the system in finite time. This could mean that
Nix can be just passing by our solar system right now. For instance, we obtained a solution in which Nix
would eventually collide with Charon in a little over 7 years. Some trajectories show definitely signs of
chaoticity or, at least, of non periodicity.
Lastly, we point out that, unlike the conventional CR3BP, the CR3BP with dumbbell satellite doesn’t
in general possess any invariant. We derived sufficient conditions for the conservation of an effective
Hamiltonian in the CR3BP with dumbbell satellite. For instance, whenever the motion of the dumbbell
is such that it is either always perpendicularly oriented to the Lagrange plane or confined to move and
rotate in the mediating plane of the line that connects the primaries, there will be conservation of an
effective Hamiltonian.
In the future we plan to extend the analysis of the dumbbell rigid body into a body constituted by
multiple dumbbells, each aligned along one of the coordinate axes. We also wish to study the inter-
action between two dumbbell rigid bodies, in order to investigate other types of synchronisations, and,
ultimately, to replace the dumbbell by an axisymmetric body.
70
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74
Appendix A
Fixed Points and Steady States of
Differential Equations
We follow the work of Dilao [54] in this appendix.
A.1 Differential equations as dynamical systems
Let S be a dynamical system characterised by a finite number of state variables, x =(x1, x2, . . . , xn
).
The set of all possible values of the state variables(x1, x2, . . . , xn
)is called the phase space of the
system S .
Many systems in Nature may be modelled by differential equations, like, for instance, the demo-
graphic evolution of a certain country, or the phase transition of liquid helium from the normal to the
superfluid phase. Let f(x, t) =(f1(x, t), . . . , fn(x, t)
): Rn × R → Rn be a continuous function defined
on an open set U of Rn × R, and consider the system of ordinary differential equations
dx
dt= f(x, t), (A.1)
describing system S . Here t ∈ R, and Rn is the phase space of S and of (A.1). An application
φ(t; t0,x0) : I ⊂ R→ Rn is said to be a solution of (A.1) if it satisfies
dφ
dt= f(φ(t; t0,x0), t),
x0 being the value of the solution at the time instant t0. If, additionally, the application φ is such that
φ(t; t0,x0) = φ(t+ T ; t0,x0),
for all t ∈ I, where T > 0 is a positive constant, then it is called a periodic solution or periodic orbit of
(A.1). The image of the solution φ(t; t0,x0) in the phase space of (A.1) is called a phase curve of the
differential equation.
75
The system of differential equations (A.1) defines a vector field or a phase flow in the phase space,
whose components along the directions x1, . . . , xn are the functions f1(x, t), . . . , fn(x, t), respectively.
The image of this vector field at a point x∗ ∈ Rn, that is, the vector f(x∗, t) =(f1(x∗, t), . . . , fn(x∗, t)
),
is tangent to the phase curve associated to φ(t; t0,x0) at x∗. Conversely, a vector field always deter-
mines, or is always associated to, a differential equation. This property makes it possible to qualitatively
construct the solutions of (A.1) in the phase space, at least in dimensions 1, 2 and 3. The function f
may not depend on time. In that case, the system (A.1) is called autonomous and is invariant for time
translations. We now define the concept of fixed point of a differential equation:
Definition 1. A point x∗ ∈ Rn is said to be a fixed point for the phase flow of the differential equation
(A.1) if f(x∗, t) = 0.
If x∗ is a fixed point of (A.1), thendx
dt= 0 at that point, and thus φ(t; t0,x
∗) = x∗ for all t ∈ R. This
means that there is no phase flow at x∗. The fixed points are thus particular solutions of a differential
equation. The fixed point solutions are also known by the names equilibrium points or steady points.
Nonetheless, not all equilibrium solutions or steady states are fixed points, even though all fixed points
may be classified as equilibrium solutions or steady states. For instance, the periodic orbits of (A.1)
correspond to steady state configurations of the dynamical system S . Due to the coordinates adopted
throughout Chapters 2 and 3, the fixed points we determine there are in fact periodic orbits, and thus
steady states, of the KD system and of the R3BP with dumbbell satellite.
The existence of fixed points and periodic orbits of a differential equation constrains the topology of
the phase curves of that equation in phase space. Through the knowledge of the vector field associated
to the equation and its fixed points, one can determine, at least qualitatively, all topologies of the phase
curves. That is why it is so crucial to obtain the fixed points of a differential equation. Furthermore, fixed
points are, in general, the only easily obtainable solutions of a nonlinear differential equation.
A.2 Stability of the fixed points
Let us analyse the behaviour of the solutions of a differential equation in the neighbourhood of the fixed
points of the associated vector field.
Definition 2. (Lyapunov stability). A fixed point x∗ of a differential equation is said to be Lyapunov stable
if:
1. There exists a neighbourhood U (x∗) of x∗ such that φ(t; t0,x0), with x0 ∈ U (x∗), is defined for
all t ≥ 0.
2. For all sufficiently small neighbourhoods V (x∗) ⊂ U (x∗), there exists a neighbourhood V1(x∗) ⊂
V (x∗) such that, for all t ≥ 0, every solution φ(t; t0,x0) with origin in V1(x∗) is in V (x∗) (Fig-
ure A.1a).
Additionally, if limt→∞ φ(t; t0,x0) = x∗ and x0 ∈ V1(x∗), then x∗ is asymptotically stable (Figure A.1b).
If a fixed point is not Lyapunov stable, then it is (Lyapunov) unstable.
76
U (x∗)
x∗
V (x∗)
V1(x∗)
x0
(a)
U (x∗)
x∗
V (x∗)
(b)
Figure A.1: (a) Lyapunov stable fixed point. (b) Asymptotically stable fixed point.
In the definition of Lyapunov stability, the condition limt→∞ φ(t; t0,x0) = x∗ is not a sufficient condi-
tion for the asymptotic stability of the fixed point x∗. In fact, the solution φ(t; t0,x0) may tend to the fixed
point x∗ at infinity without being bounded from above for finite values of t, Birkhoff and Rota [55].
The analysis of the stability of a fixed point may be a difficult problem for nonlinear equations. In
some cases though, we may apply the stability criterium formulated in the following theorem:
Theorem 3. (Lyapunov). Let x∗ ∈ Rn be an isolated fixed point of the differential equationdx
dt= f(x)
and V : V (x∗) → R be a differentiable function in V (x∗) − {x∗}. Moreover, suppose that V (x∗) = 0
and that V (x) > 0 for x 6= x∗. Then, if V is such thatdV
dt≤ 0 in V (x∗) − {x∗}, the fixed point x∗ is
Lyapunov stable. If V is such thatdV
dt< 0 in V (x∗) − {x∗}, the fixed point x∗ is asymptotically stable.
The function V is called the Lyapunov function.
We note that the Lyapunov theorem is valid only for autonomous differential equations. The Lyapunov
function, if it exists, has a simple geometrical meaning: if x∗ is a Lyapunov stable fixed point, the
Lyapunov function has a local minimum at x∗, and the vector field, restricted to the level sets of V ,
V (x) = z, points inward to these level sets, that is to say, it makes either an obtuse or a right angle with
the direction defined by the gradient of V at each point (Figure A.2).
In addition to the Lyapunov stability criterium, one may also study the stability of a fixed point of a
differential equation by linearising the associated vector field around the fixed point. If this approach is
taken, a matrix equation is obtained, and, due to the dependence of the solutions of a linear system of
differential equations on the eigenvalues of this matrix, the stability of the fixed point of the linear system
may be inferred from these eigenvalues. Specifically, for the two-dimensional case, the fixed point of the
linear system is Lyapunov stable if the real part of both eigenvalues is negative, or, this being equal to
zero, the imaginary parts are both nonvanishing. If the real part of one of the eigenvalues is positive, the
fixed point of the linear system is Lyapunov unstable.
Definition 3. Fixed points of differential equations with pure imaginary eigenvalues are said to be elliptic
fixed points or centres. Fixed points with eigenvalues outside of the imaginary axis of the complex plane
77
(a)
V −1(c)
(ii)
(i)
(iii)
(b)
Figure A.2: (a) Graph of a Lyapunov function V : R2 → R, depicted in the neighbourhood of a stablefixed point at (x, y) = (0, 0). Also illustrated is a plane that intersects the Lyapunov function at constantz, V (x, y) = c. (b) Level set of a Lyapunov function, projected onto the (x, y) plane. This projection is the
graph of the function V −1(c). Three phase vectors are sketched over the level set V −1(c): (i)dV
dt> 0;
(ii)dV
dt= 0; (iii)
dV
dt< 0. The sign of the derivative is determined by the angle between the vector field
and the direction of the gradient of V , which is always normal to the level set.
are called hyperbolic fixed points.
The above analysis is valid only in a neighbourhood of the fixed point. Moreover, if the nonlinear
equation has multiple fixed points, the linear analysis of the stability has to be done around each fixed
point. The following theorem states the conditions under which the stability of the fixed points of the
linear and of the nonlinear systems are the same:
Theorem 4. (Hartman-Grobman [56]). If the matrix of the linear approximation of a nonlinear system of
differential equations around a fixed point x∗, or Jacobian matrix, has no pure imaginary eigenvalues or
eigenvalues equal to zero, then there is an homeomorphism h, defined in an open neighbourhood U of
x∗, which takes orbits of the nonlinear system into orbits of the linear system. The homeomorphism h
preserves the sense of the orbits and may be chosen such that it also preserves the parameterisation
of time1.
This means that in the neighbourhood of the hyperbolic fixed points of a nonlinear system, the phase
curves of the nonlinear system and of the local, linear systems are topologically equivalent. For instance,
Figure A.3 shows that the orbits in phase space of a nonlinear system of differential equations in R2,
around an hyperbolic fixed point of the saddle type (one positive and one negative eigenvalue), and the
orbits of the associated linear system are homeomorphic to one another. The Hartman-Grobman theo-
rem justifies the qualitative construction of the orbits of the nonlinear system in phase space, discussed
in Section A.1.
We end this appendix with the definition of structural stability:
1An homeomorphism preserves the sense of the orbits if its Jacobian matrix has positive determinant.
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U
h
Figure A.3: Orbits of a nonlinear dynamical system in phase space (left), and corresponding approximatelinear system (right), in a neighbourhood U of an hyperbolic fixed point. The Hartman-Grobman theoremguarantees the existence of the homeomorphism h.
Definition 4. (Structural stability). A dynamical system is said to be structurally stable if any infinitesi-
mally small perturbation in the equations that define the phase flow doesn’t change the topology of the
phase curves.
Other concepts of stability include, for instance, Lagrange or Poisson stability. The motion of a point
x is Poisson stable if it returns infinitely many times to positions arbitrarily close to x. Poincare [47]
established that there are an infinite number of such motions in the R3BP. The trajectory of a point x
is Lagrange stable if it is contained in a totally bounded set2. Interestingly, both of these concepts were
actually introduced by Poincare in [47]. We shall not delve further into them.
2A totally bounded set is a set that can be covered by finitely many open balls of radius ε, for every ε > 0.
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Appendix B
Particular Solutions of the General
Three-Body Problem
In the Three-Body Problem, three bodies of arbitrary masses move in three-dimensional space under
their mutual gravitational interactions, according to Newton’s law of universal gravitation. When there
are no restrictions on the masses, this system is referred to as the General Three-Body Problem. In this
appendix we make a brief review of some of the special, periodic solutions to this problem.
B.1 Eulerian solutions
These special solutions were discovered by Euler [46] in 1767, who considered an initial straight line
configuration for three arbitrary masses. Euler proved that this initial straight line configuration is main-
tained for future times if the ratio of the distances between the masses has a certain value that depends
on them, and if suitable initial conditions are chosen. Furthermore, he showed that this ratio is also
conserved during the motion and that the line that joins the three masses rotates about their centre
of mass. This in turn results in all the masses travelling along ellipses (Figure B.1a). On the other
hand, for the special case where all the masses are equal, two of the masses rotate around the third
one, in identical ellipses or in a circle, always remaining in phase opposition relative to the central mass
(Figures B.1b–B.1c).
The Eulerian solutions are unstable to small deviations from the initial conditions. As the masses
may be ordered in three different ways along the line (if they are different), there are three solutions, one
for each ordering of the masses, corresponding to the case depicted in Figure B.1a.
B.2 Lagrangian solutions
In 1772, Lagrange [44] found a second class of periodic orbits of the General Three-Body Problem, now
known as Lagrangian solutions. In these solutions the three masses are initially positioned at the vertices
of an equilateral triangle. For suitable initial conditions, the masses travel along ellipses around their
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(a) (b)
(c)
Figure B.1: Eulerian solutions of the General Three-Body Problem: the three masses move in ellipsesaround their common centre of mass, maintaining at all times a collinear configuration represented bythe dashed line. In (a), all the masses are different. In (b) and (c), all masses are equal, and both theouter masses are at the same distance from the central mass. In (c), the outer masses move along thesame circular orbit. This configuration is obtained for suitable initial velocities of these masses.
centre of mass in configuration space, while retaining this special configuration. The triangle changes its
size and orientation periodically, but it remains equilateral for as long as the masses move. In Figure B.2
we depict two Lagrangian solutions for the case of equal masses.
(a) (b)
Figure B.2: Lagrangian solutions for the case of equal masses: the three masses are initially located atthe vertices of an equilateral triangle, which is represented by a dashed line, and this configuration ismaintained for future times. In (a), the masses move along identical ellipses. In (b), the masses travelalong the same circular orbit. This configuration is obtained for suitable initial velocities of the threemasses.
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Unlike the Eulerian solutions, Lagrangian solutions have regions of stability under certain conditions
[57]. By virtue of the limitation on the mutual positions of the masses in the Eulerian and Lagrangian
solutions, these are called particular solutions.
B.3 Other periodic solutions
More recently, new periodic solutions have been discovered, such as the famous figure-eight solution for
equal masses, in which the three masses trace the same figure-eight orbit, the intersection point being
the centre of mass. This solution was numerically discovered in 1993 by Moore [58] and independently
rediscovered in 2000 by Chenciner and Montgomery [59], who also rigorously proved its existence in
2001, [60].
Suvakov and Dmitrasinovic [61] have also found 13 new periodic, exotic solutions in the planar case
of the General Three-Body Problem.
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