on the stability of quasi-satellite orbits in the elliptic restricted three-body problem ·...
TRANSCRIPT
On the Stability of Quasi-Satellite Orbits in the EllipticRestricted Three-Body ProblemApplication to the Mars-Phobos System
Francisco da Silva Pais Cabral
Dissertação para a obtenção de Grau de Mestre em
Engenharia Aeroespacial
Júri
Presidente: Prof. Doutor Fernando José Parracho Lau
Orientador: Prof. Doutor Paulo Jorge Soares Gil
Vogal: Prof. Doutor João Manuel Gonçalves de Sousa Oliveira
Dezembro de 2011
Acknowledgments
The author wishes to acknowledge
His thesis coordinator, Prof. Paulo Gil, for, one, presenting this thesis opportunity in the author’s field of
interest and, second, for the essential orientation provided to the author that made this very same thesis
possible,
His professors, both in IST and TU Delft, for the acquired knowledge and transmitted passion in the most
diverse fields,
His university, IST, for providing the means to pursue the author’s academic and professional goals,
His colleagues for their support and availability to discuss each others’ works,
His friends for keeping the author sane.
iii
Abstract
In this thesis, the design of quasi-satellites orbits in the elliptic restricted three-body problem is ad-
dressed from a preliminary space mission design perspective. The stability of these orbits is studied
by an analytical and a numerical approach and findings are applied in the study of the Mars-Phobos
system. In the analytical approach, perturbation theories are applied to the solution of the unperturbed
Hill’s equations, obtaining the first-order approximate averaged differential equations on the osculating
elements. The stability of quasi-satellite orbits is analyzed from these equations and withdrawn conclu-
sions are confirmed numerically. We also use the fast Lyapunov indicator, a chaos detection technique,
to analyze the stability of the system. The study of fast Lyapunov indicator maps for scenarios of par-
ticular interest provides a better understanding on the characteristics of quasi-satellite orbits and their
stability. Both approaches are proven to be powerful tools for space mission design.
Keywords: Quasi-Satellite Orbits, Elliptic Restricted Three-Body Problem, Stability, Perturbation The-
ory, Fast Lyapunov Indicator, Mars-Phobos System.
v
Resumo
Nesta tese, o planeamento de quase-orbitas no contexto do problema restrito dos tres corpos elıptico
e estudado de uma perspectiva do planeamento preliminar de missoes espaciais. A estabilidade destas
orbitas e estudada atraves de uma aboradagem analıtica e de uma abordagem numerica e as descober-
tas sao aplicadas ao caso do sistema Marte-Fobos. Na abordagem analıtica, teorias de perturbacao
sao aplicadas a solucao das equacoes de Hill nao perturbadas, obtendo-se as equacoes diferenciais
medias aproximadas de primeira ordem nos elementos osculadores. A estabilidade de quase-orbitas
e analisada atraves destas equacoes e as conclusoes retiradas sao confirmadas numericamente. Us-
amos tambem o indicador rapido de Lyapunov, uma tecnica de deteccao de caos, para analisar a
estabilidade do sistema. O estudo dos mapas do indicador rapido de Lyapunov para cenarios de par-
ticular interesse vai-nos providenciar uma melhor compreensao das caracterısticas de quase-orbitas e
da sua estabilidade. Ambas as abordagens demonstram ser ferramentas poderosas no planeamento
de missoes espaciais.
Palavras Chave: Quase-Orbitas, Problema Restrito dos Tres Corpos Elıptico, Estabilidade, Teoria de
Perturbacao, Indicador Rapido de Lyapunov, Sistema Marte-Fobos.
vii
Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1. Introduction 1
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Quasi-Satellite Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5. Bibliographic Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.1. Orbits In The Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.2. Chaos Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6. Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2. Dynamics 9
2.1. Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1. Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2. Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3. The Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2. The N-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3. The Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4. Modifications of the Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . 17
2.4.1. The Spatial Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2. The Elliptic Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . 17
2.4.3. Change Of Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5. Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.1. Hamilton’s Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2. An Invariant Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
ix
2.6. Chaos Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.1. Lyapunov Characteristic Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.2. Fast Lyapunov Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3. QSO Solutions and Stability 27
3.1. Linearization of the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2. Unperturbed Hill’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1. Homogeneous Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2. General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.3. Stability Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.4. Constants of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.5. Constant Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3. Influence of the Second Primary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1. Region of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2. Approximate Solutions in the Osculating Elements . . . . . . . . . . . . . . . . . . 41
3.4. Application to the Mars-Phobos System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4. Numerical Exploration of QSOs 49
4.1. Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2. Implementation Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1. Validation Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2. Implementation Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3. Computational Parameters & Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1. Integration Method and Time-Step . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2. Orbit Escape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4. FLI Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4.1. Planar QSOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.2. Three-Dimensional QSOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.3. Velocity Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5. Conclusions 77
A. Programming Code 79
Bibliography 92
x
List of Tables
1.1. Mars & Phobos Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.1. Amplitudes for Values of the Mean Motion Ratio . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1. Validation Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2. MatLab & C Performance Comparison I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3. MatLab & C Performance Comparison II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4. Reference Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5. Parameter Computation With Chosen Time-Step . . . . . . . . . . . . . . . . . . . . . . . 55
4.6. Orbits - Mean Orbital Motion Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
xi
List of Figures
1.1. Hill Sphere and Region of Influence of Phobos . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1. Analysis of the behavior of xnp(f) and ynp(f) . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2. Analysis of the behavior of xnp(f) and ynp(f) with C3 = −eC2 . . . . . . . . . . . . . . . 33
3.3. Parametric plot of the stable solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4. Osculating Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5. Orbital Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1. FLI Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2. Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3. Position and Velocity Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4. FLI Map - x0 Vs. y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5. FLI Map - y0 Vs. x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6. Planar QSOs - Tangential Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.7. 2:1 Mean Motion Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.8. FLI Map - y0 Vs. y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.9. FLI Map & QSO - y0 Vs. y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.10.FLI Map - Initial True Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.11.FLI Map - Vertical Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.12.3D QSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.13.FLI Map - z0 Vs. y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.14.FLI Map - z0 Vs. z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.15.3D QSOs - Amplitude Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.16.3D QSOs - Large Amplitude Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.17.FLI Map - Velocities I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.18.FLI Map - Velocities II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
xiii
List of Acronyms
3BP Three-Body Problem
APLE Averaged Power Law Exponent
CI Chaos Indicator
ER3BP Elliptic Restricted Three-Body Problem
FLI Fast Lyapunov Indicator
GALI Generalized Alignment Index
LCE Lyapunov Characteristic Exponent
MEGNO Mean Exponential Growth factor of Nearby Orbits
mLCE maximum Lyapunov Characteristic Exponent
ODE Ordinary Differential Equation
PCR3BP Planar Circular Restricted Three-Body Problem
PSS Poincare Surface of Section
QSO Quasi-Satellite Orbit
R3BP Restricted Three-Body Problem
RKn n-order Runge-Kutta
RLI Relative Lyapunov Indicator
SALI Smaller ALignment Index
SER3BP Spatial Elliptic Restricted Three-Body Problem
S/C Spacecraft
SSN dynamical Spectra of Stretching Numbers
VOP Variation Of Parameters
xv
List of Symbols
a Orbit Semi-Major Axis
C (Modified) Jacobi Integral
ci Arbitrary Constants
Ci Integration Constant
d Distance
D Norm of the Deviation Vector
e Orbital Eccentricity
E 3BP Energy (as function of generalized coordinates)
E(θ, k) Incomplete Integral of the Second Kind
E(k) Complete Integral of the Second Kind
f Orbit True Anomaly
F (θ, k) Incomplete Integral of the First Kind
h Time-Step
H Hamiltonian
J2 Second Gravitational Moment
K(k) Complete Integral of the First Kind
L Lagrangian
L1,2 Lagrangian Points
m Mass
xvii
n Orbital Mean Motion
p Semi-Latus Rectum
P Orbital Period
pi Generalized Momenta
P (λ) Characteristic Polynomial
qi Generalized Coordinates
r1 Distance To First Primary
r2 Distance To Second Primary
RH Hill’s Sphere Radius
T Kinetic Energy
tL Lyapunov Time
u, v, w Perturbation Coordinates
U Potential Energy
V Velocity
W Wronskian Determinant
w(t) Deviation Vector
x(t) State Vector
x, y, z Cartesian Coordinates
Y(t) Fundamental Matrix
G Gravitational Constant
α, φ, δx, δy, γ, ψ Osculating Elements
∆V Impulse
λi Eigenvalues
µ Mass Parameter
∇ Vector Differential Operator
ω Orbital Angular Velocity
xviii
Ω Amended Potential
Φ Fundamental Matrix
ξ, η, ζ Pulsating Coordinates
χ1 Maximum LCE
xix
A perfection of means, and confusion of aims, seems
to be our main problem.
(Albert Einstein 1879 - 1955)
CH
AP
TE
R
1 Introduction
The objective of the present work is to study the stability of orbits around the Martian moon Phobos
and to analyze the motion of such orbits. The problem is complex as Keplerian orbits are not possible
due to the moon’s small mass which demands the account of Mars gravitational influence in a three-body
problem. The problem is addressed with both analytical and numerical approaches to identify sufficiently
stable orbits around the Martian moon. The consideration of Phobos’ orbital eccentricity increases the
complexity of the problem as the system becomes non-autonomous, i.e., time-dependent.
1.1. Motivation
The Martian moon Phobos is one of the most prominent candidates for future space exploration mis-
sions. The interests of a mission to this moon range from scientific to engineering purposes that make
it an appealing research subject. One of the major challenges in the design of such a mission is the
search for sufficiently stable orbits around the moon.
A mission to Phobos would answer many unresolved questions regarding the origin of the Martian
satellites, believed to be captured asteroids, as well as the formation process and origin of the Solar
system. If Phobos was formed by a material within the Solar System, a sample collection from this
moon would provide unprecedented information on its formation and evolution. This way, a sample
return mission to the moon has been a discussed option for space exploration programs, being already
part of the current Russian space program — a mission was scheduled to be launched on November,
2011 (Mukhin et al., 2000; Galimov, 2010). This mission failed to begin its interplanetary flight and is set
to crash on Earth during the present month.
Space exploration missions are complex, expensive, and time and mass-constrained, therefore, a
space station in the Solar system to refuel the spacecraft should overcome the mass and cost related
limitations. Phobos is a good candidate to host such resource space station, as the ∆V ’s required to
reach Phobos are smaller than those to reach Earth’s own moon, although the travel times are much
larger. This can be explained by the smaller required energy to break in order to orbit the Martian moon
when compared to our moon. This resource station would open the door to new possibilities for space
1
exploration programs (Wiesel, 1993).
The selection of a sufficiently stable orbit to circumnavigate Phobos is very important. Depending on
the mission, the spacecraft must orbit the moon enough time to update ephemerides, perform scientific
experiments, select a landing site, or prepare for a landing approach and the stability of such orbit will
increase the probability of mission success (Gil and Schwartz, 2010).
Some space missions, more or less successful, have already targeted Phobos. The first data on the
moon was obtained by the early NASA missions to Mars, Mariner 7 & 9, and the Russian mission Mars
5. The first mission to exclusively target Phobos, however, only took place in 1988 — the Russian probes
Phobos 1 & 2. This mission did not succeed as the control of both satellites was lost. The second probe,
however, managed to send back to Earth a number of high-definition images before being lost. Since
then, data about the Martian moons has only been collected by missions targeting Mars. However, there
is a Russian sample-return mission, Phobos-Grunt, that is set to be launched November 2011 after
successive delays. A successful mission to Phobos is yet to be accomplished, restating such a mission
as an important milestone to be overcome (Marov et al., 2004).
1.2. Problem Statement
The small mass of Phobos, when compared to Mars’ (about seven orders of magnitude larger), relin-
quishes any possibility of a Keplerian orbit as the region of influence of Phobos is below its own surface,
and its Hill sphere, with the Lagrange points L1,2 on its surface, is just a few kilometers above its surface
(see fig. 1.1) with radius (Hamilton and Burns, 1992)
RH ≈ a(1− e2)
(mPh
3(mM +mPh)
)1/3
(1.1)
with a being the semimajor axis, e the orbital eccentricity of Phobos, and mPh and mM the mass of
Phobos and Mars, respectively. Consequently, the problem should be treated as a three-body problem,
hereafter 3BP, with Mars, Phobos, and the spacecraft (S/C) (Gil and Schwartz, 2010).
Figure 1.1. Sketch of the ellipsoid representing Phobos as viewed from the North Pole looking down onthe equator. The minor axis is presented as a solid line. The interior and exterior dashed linesare, respectively, the region of influence and the Hill sphere containing the Lagrange pointsL1 and L2 (Gil and Schwartz, 2010).
The challenge goes further. The nonspherical gravitational potential of Mars and Phobos influences
the stability of possible orbits about the moon, specially the latter at close distances to the moon. The
2
inclusion of Phobos’ orbital eccentricity also poses a challenge as the problem becomes time-dependent.
The exploration of this time-dependence in moons orbiting with a small eccentricity, such as Phobos, is
one of the main objectives of this work.
The oblateness of Mars and Phobos’ irregular shape and rotation are, however, not considered — this
work is developed under the point mass approximation for both bodies. The solar radiation pressure
is also neglected. The relevant bulk and orbital parameters of both Mars and Phobos are presented in
table 1.1.
Mars Phobos
Bulk Parameters
Mass [kg] 6.4185× 1023 1.06× 1016
Equatorial Radius [km] 3396.2 13.4× 11.2× 9.2
µ [km/s2] 4.283× 104 7.1× 10−4
J2 1.96045× 10−3 0.100523
Orbital Parameters
Semimajor Axis [km] 2.2792× 108 9.3772× 103
Sidereal Orbit Period [day] 686.980 0.31891
Sidereal Rotation Period [day] 1.02595 0.31891
Orbit Inclination [deg] 1.850 1.08
Orbit Eccentricity 0.0935 0.0151
Obliquity to orbit [deg] 25.19 Tidally locked
Table 1.1. Mars and Phobos bulk and orbital parameters. Phobos radius is represented as a set of threeaxes: major, median and minor. The data was collected from (NASA, 2010; Gil and Schwartz,2010).
A specific type of orbits denominated quasi-satellite orbits, QSOs, can be found under certain initial
conditions. The search of sufficiently stable QSOs by analytical and numerical approaches is the main
subject of this work.
1.3. Quasi-Satellite Orbits
Keplerian orbits about Phobos are not possible. As sketched in fig. 1.1, the Hill sphere, region for
which a body is the main attracting force, is just above Phobos’ surface which prevents us from treating
this as a two-body problem. Mars influence has to be considered.
However, it is possible to find sufficiently stable orbits around Phobos in the setup of the 3BP. In
the case one of the primaries has a much larger mass than the other, m1 >> m2, a special type of
orbits assumes particular interest — the so-called quasi-satellite orbits (QSOs). They are a special type
of orbits as they are not closed periodic trajectories although they tend to occupy the same region in
space.
3
QSOs are also known as distant retrograde orbits, DROs, or instant satellite orbits. In this thesis
we chose to adopt the nomenclature quasi-satellite orbits. We would also like to emphasize that a
number of articles refer to QSOs as orbits near the 1:1 resonance but here this concept is extended to
all resonances.
All sufficiently stable QSOs are resonant orbits or close to these. Resonant orbits are any orbit that
forms with the second primary (in our case Phobos) a system that orbits the same primary (Mars) whose
orbital mean motions are in the ratio of small whole numbers (Peale, 1976). As the orbital amplitude of
a QSO increases, the orbit tends to a 1:1 resonance, often called quasi-synchronous orbits, whereas at
close distance sufficiently stable 2:1, 3:2, and 4:3 resonances can be found (Wiesel, 1993).
1.4. Stability
This thesis concerns about the stability of QSOs, but first it is necessary to define the concept of
stability. In the literature there are many definitions of stability and we adopt the one that best fit to help
us achieve our objective, i.e., find suitable orbits around Phobos for the purposes of mission design.
One of the most used definitions states that an orbit is stable if the distance to an initially nearby orbit
increases linearly with time, whereas it is chaotic if the distance to an initially nearby orbit increases
exponentially with time (Meyer et al., 2009). This concept is known as exponential divergence. In this
work we use chaos detection techniques to analyze the stability of QSOs that are based on this definition
of stability. However, apart from the use of the chaos indicator, this definition does not suit our purposes
as there are orbits that escape from Phobos and enter in a orbit around Mars that present a stable
nature.
The stability definition best fit to our purpose is one that can fulfill the orbit’s objective, i.e., orbit Phobos
during enough time to perform any reconnaissance, or scientific activities required for the mission without
colliding with the moon or escaping from orbit. This way, in our work, we define a sufficiently stable orbit
as follows.
A sufficiently stable orbit about Phobos is one that will orbit the moon for, at least, a period of
100 revolutions of the moon around Mars without colliding against it or get more than 1,000
km away from it.
The number of 100 revolutions of Phobos about Mars, about a month, is chosen. Other studies in the
literature use similar timespans (25 days in (Wiesel, 1993) and 30 days in (Gil and Schwartz, 2010)).
The upper limit of 1,000 kms is chosen following (Gil and Schwartz, 2010).
1.5. Bibliographic Review
A fair amount of research work has been performed and published in the topics concerning this thesis,
namely, orbits in the 3BP and their stability, and chaos detection techniques.
4
1.5.1. Orbits In The Three-Body Problem
The study of orbital stability in the three-body problem is one of the most researched subjects in
celestial mechanics. This problem concerns the study of the stability of moons in a Sun-planet system
or the stability of orbits in a planet-moon system. The systems Sun-Earth-Moon, Mars-Phobos-S/C, and
Jupiter-Europa-S/C are amongst the most interesting cases for their scientific interest.
One of the earliest contributions to this matter is due to George W. Hill and his studies on the lunar
theory (Hill, 1878). He considered the limiting case where the mass parameter of the second primary,
µ, tends to zero, and scaled his spatial dimensions by the factor µ1/3. The equations of motion under
this technique are known to this day as the Unperturbed Hill’s Equations. A variant of this method is
employed in this work in the analytical approach to our problem.
Another important contribution to this field was Poincare’s Methodes Nouvelles de la Mecanique
Celeste (english-translated version: (Poincare, 1993)) and Sur le probleme des trois corps et les equations
de la dynamique. Poincare’s sufaces of section (PSS) is a very used technique to analyze stability in
systems with two-degrees of freedom.
Victor Szebehely’s Theory of Orbits (Szebehely, 1967) is one of the most important books on the
3BP. His book collects all the major contributions on the field. Szebehely’s own contribution is the most
important to this thesis; his work on developing a coordinate system suitable for the Elliptic Restricted
Three Body-Problem (ER3BP), the pulsating coordinates, is of great help as the time-dependent terms
reduce to a factor on the problem’s potential.
Henon contributed to the research of the 3BP with his numerical study of periodic orbits and their
stability in the planar circular restricted three-body problem (PCR3BP). At first he studied the case of
primaries with equal masses (Henon, 1965a,b), extending his search to the Hill case, µ→ 0, afterwards
(Henon, 1969). He also expanded his study to the analysis of quasi-periodic orbits and their stability
(Henon, 1970). Finally, he performed vertical stability analysis for the case of primaries with equal
masses and for Hill’s case (Henon, 1973, 1974).
Between 1974 and 1977, Benest published four articles about the existence of retrograde orbits in the
PCR3BP and the effect of the mass parameter µ on their stability. Amongst another research topics, he
approached the 3D stability of planar orbits and the effect of the mass parameter in this (Benest, 1974,
1975, 1976, 1977a). He also contributed to the numerical exploration of stable periodic and non-periodic
orbits (Benest, 1977b).
In (de Broeck, 1989) quasi-synchronous orbits — QSOs near the 1:1 resonance — are studied. This
work was based in the simplified model of the planar circular restricted three-body problem and uses
the unperturbed Hill’s equations as a starting point applying then a perturbation to obtain the first-order
approximate equations. Theoretical conditions to obtain stability are derived and applied to the case of
Mars-Phobos. These conditions are verified through numerical integration of the equations of motion.
One of the methodologies used in our analytical approach is based on this work but our approach differs
in the used model — we consider the eccentricity of Phobos and extend our analysis to the three-
dimensional case.
The work developed in (Kogan, 1989) studies QSOs in the vicinity of the smaller primary separated
5
by distances greater than the radius of the Hill sphere in the context of the spatial circular restricted
three-body problem. The long-term evolution of QSOs is described and stability conditions are investi-
gated. This work also used the unperturbed Hill’s equations of motions as a starting point and described
their solutions in terms of the osculating elements. The variation of these elements, under the pertur-
bation of the second primary, is then derived through the method of variation of the arbitrary constants.
The differential equations on the osculating elements are averaged and, from these, stability conditions
are derived. This method is also adopted in our work but the eccentricity of the primaries’ orbits are
considered.
In (Wiesel, 1993) sufficiently stable QSOs are explored in Phobos and Deimos with a dynamics model
that includes Mars oblateness, the moons’ orbital eccentricity, its irregular shape, and rotation. The
cases of zero and nonzero eccentricity are studied and sufficiently stable resonant orbits are found
to exist only in the latter. Sensitivity to poorly known system parameters is explored, and accuracy
requirements for spacecraft insertion maneuvers are established. We emphasize the study of resonant
orbits in this work which require a more complete dynamics model as the one used by Wiesel. The
proximity to Phobos (or Deimos) of these orbits requires usually neglected perturbations as the moons’
irregular shapes and their moments of inertia to be considered.
An analytical approach to QSOs in the elliptic restricted problem is presented in (Lidov and Vashkov’yak,
1993). This work obtained second-order approximate solutions but these, as stated by the authors, are
complex. Their method is restricted to QSOs with small inclinations.
The stability of quasi-synchronous orbits around Neptune and Uranus are explored numerically in
(Wiegert et al., 2000) for timespans as long as 109 years. It is found that orbits can only exist for such
long time periods if at low inclinations relative to their accompanying planet and over a restricted range
of heliocentric eccentricities.
In (Gil and Schwartz, 2010) a numerical study of QSOs from a preliminary mission design perspective
is performed. Their stability and impact on the design of a mission to Phobos (eclipses, observation
conditions, etc.) is addressed. QSOs with an inclination to Phobos’ orbital plane are also studied.
1.5.2. Chaos Indicators
The so-called chaos indicators (CIs) are numerical techniques that distinguish regular from chaotic
motion in a dynamical system. Their number has increased over the past years. They can be divided in
two categories: techniques based on the evaluation of the deviation vector (vector containing an initial
deviation from the initial position) and how it evolves, and techniques based on the analysis of the orbit
itself. The first are, in most cases, derivatives of the Lyapunov Characteristic Exponents, LCEs. In
chapter 2 we address their worth for our work.
In 1892, Lyapunov published a work on the stability of motion. This text was originally published
in Russian, meanwhile translated (Lyapunov, 1992), and still remains as one of the most important
references in the subject. Several methods and techniques received the name of Lyapunov as a tribute,
namely the LCEs, only developed during late 1970s following theoretical work performed in the end of
6
the 1960s (Skokos, 2010).
Henon & Heiles, in their pioneer work (Henon, M. and Heiles, 1964), launched the bases for the de-
velopment of CIs in the subsequent years. One of the first and most used techniques was the graphical
method of Poincare’s Surface of Section (PSS) developed for the analysis of 2D maps and of 2D Hamil-
tonian systems. However, this method is not suitable for the analysis of systems with more degrees
of freedom. This graphical technique was then extended for 3D maps and 3D Hamiltonian systems by
Froeschle (Froeschle, 1970, 1972). By this time, the analysis of the nature of trajectories in a system
was limited to systems with 3 or less degrees of freedom and, hence, there was a need for tools that
could work in systems with more degrees of freedom.
Theoretical background for the computation of the LCEs was developed in (Oseledec, 1968). This
work provided the basis for the seminal papers by Benettin et al. (Benettin et al., 1980a,b), where
a combination of the important theoretical and numerical results on LCEs can be found as well as a
developed explicit method for the computation of all LCEs.
Most of the CIs based on the analysis of the deviation vector derive from the LCEs. The most used
are based on the evolution of the deviation vector, and one may list, based on the evaluation of the
maximum LCE: the Fast Lyapunov Indicator (FLI) (Froeschle et al., 1997), the smaller alignment index,
SALI, the Generalized Alignment Index (GALI) (Skokos et al., 2007), the Mean Exponential Growth factor
of Nearby Orbits (MEGNO) (Cincotta and Simo, 2000), the Relative Lyapunov Indicator (RLI) (Sandor
et al., 2000), and the averaged power law exponent, APLE; and based on the spectrum of LCEs: the
stretching numbers, the helicity angles, the twist angles, and the study of the differences between such
spectrum (Skokos, 2010). A survey of some of these methods and respective comparison can be found
in (Maffione et al., 2011).
In the category of indicators based on the analysis of a particular orbit, one may list the frequency
map analysis of Laskar, the ”0–1” test, the method of the low-frequency spectral analysis, the ”patterns
method”, the recurrence plots technique, and the information entropy index (Skokos, 2010).
Since the first published work on the FLIs by Froeschle, its definition has evolved until more modern
definitions (for instance (Fouchard et al., 2002)). This CI has been used to study the order of periodic
orbits and distinguish resonant from non-resonant orbits (Lega and Froeschle, 2001).
In (Villac and Aiello, 2005; Villac and Lara, 2005; Villac, 2008) the FLI maps, or stability maps, were
introduced for the study of stability regions near the libration points in the restricted three-body problem.
These maps are presented as design tools for preliminary trajectory design associated with planetary
satellite orbiter missions and they are used to find stability regions in the Jupiter-Europa system. In
(Villac and Lara, 2005) unstable periodic orbits are studied as a mean to perform thrust-free, dynamical
transfers between stability regions.
1.6. Thesis Overview
Mathematical and physical tools that are used in our work are reviewed in chapter 2. The 3BP is
also reviewed. The Hamiltonian formulation of mechanics is introduced and applied in the context of
7
the Spatial Elliptic Restricted Three-Body Problem (SER3BP) in order to obtain the equations of motion.
The variational equations in the Hamiltonian formulation are presented and the CIs of interest — the
LCEs and the FLIs — are introduced at the end of the chapter.
In chapter 3 we address the elliptic problem from the analytical point of view for the Hill’s limiting case,
µ → 0. Perturbation theories, by two different methods, are applied to the solutions of the unperturbed
equations leading to the derivation of stability conditions and approximate solutions. These are applied
to the Mars-Phobos system.
In the numerical approach in the fourth chapter we use the FLI to identify sufficiently stable QSOs.
The study of these orbits is performed with the analysis of the so-called FLI maps where the variation of
the FLI value over a set of initial conditions is addressed.
This work ends with the conclusions where achievements and recommendations for future work are
discussed.
8
Before we take to sea we walk on land, Before we
create we must understand.
(Joseph-Louis Lagrange 1736 - 1813)
CH
AP
TE
R
2 Dynamics
In this chapter we review important mathematical and physical concepts relevant to this work. Clas-
sical mechanics are reviewed (from references (Sussman and Wisdom, 2000; Meyer et al., 2009)) as
they will be required to solve the addressed problem. The Hamiltonian of our problem is derived and the
equations of motions and the variational equations are obtained from the Hamiltonian. Furthermore, the
theory behind CIs is addressed as well as the method for the computation of the CIs of interest.
2.1. Classical Mechanics
This review of the classical mechanics is a survey from the most relevant concepts to this work from
reference (Sussman and Wisdom, 2000).
In the variational formulation the equations of motion are formulated in terms of the difference of the
kinetic and potential energies. Neither of these energies depend on how the positions and velocities are
specified; the difference is characteristic of the system as a whole. Thus, there is a liberty of choice
regarding on how the system is chosen to be described, in contrast with the Newtonian formulation
where there is a particle-by-particle inherent description.
The variational formulation has numerous advantages over the Newtonian formulation. The equations
of motion for those parameters that describe the state of the system are derived in the same way re-
gardless of the choice of those parameters: the method of formulation does not depend on the choice
of the coordinate system. If there are positional constraints among the particles of a system the New-
tonian formulation requires that the forces maintaining these constraints to be considered, whereas in
the variational formulation the constraints can be built into the coordinates. Furthermore, the variational
formulation reveals the association of conserved laws with symmetries.
Considering a mechanical system composed by point masses, extended bodies can be thought as
composed by a large number of these particles with spatial relationships between them. Specifying
the position of all the constituent particles of a system specifies the configuration of the system. The
existence of constraints between parts of the system, such as those that determine the shape of an
extended body, means that the constituent particles cannot assume all possible positions. The set of
9
all configurations of the system that can be assumed is called the configuration space of the system.
The dimension of the configuration space is the smallest number of parameters that have to be given to
completely specify a configuration. The dimension of the configuration space is also called the number
of degrees of freedom of the system. Strictly speaking, these are not the same but for most systems
they are identical.
In order to talk about specific configurations a set of parameters is required to label the configurations.
The parameters that are used to specify the configuration of the system are called the generalized coor-
dinates. The number of coordinates does not need to be the same as the dimension of the configuration
space, though there must be at least that many. More parameters than necessary may be defined, but
then the parameters will be subject to constraints that restrict the system to possible configurations, i.e.,
to elements of the configuration space. Hence, sets of coordinates with the same dimension as the con-
figuration space are easier to work with because there are no explicit constraints among the coordinates
to deal with. The set of generalized coordinates are represented by the n-tuple q(t) = (q1(t), . . . , qn(t))
and the set of the rate of change of the generalized coordinates, or generalized velocities, are repre-
sented by the n-tuple q(t) = (q1(t), . . . , qn(t)). Slightly different generalized coordinates are used in the
Hamiltonian formulation.
2.1.1. Lagrangian Mechanics
The Euler-Lagrange equations or just the Lagrange equations, use the principle of stationary action
to compute the motions of mechanical systems, and to relate the variational and Newtonian formulation
of mechanics.
If L is a Lagrangian for a system that depends on time, coordinates, and velocities, and if q is a
coordinate path thend
dt
∂L
∂qi− ∂L
∂qi= 0 (2.1)
with i = 1, . . . , n. These Lagrange equations form a system of second-order ordinary differential equa-
tions that must be satisfied.
In order to use the Lagrange equations to compute the evolution of a mechanical system a suitable
Lagrangian must be found. There is no general way to construct a Lagrangian for every system, but there
is an important class of systems for which it is possible to identify Lagrangians in a straightforward way
in terms of the kinetic energy, T , and the potential energy, U . The key idea is to construct a Lagrangian
L such that Lagrange’s equations are Newton’s equations ~F = m~a. The reader may refer to (Sussman
and Wisdom, 2000) for this deduction which leads to the Lagrangian
L(t,q, q) = T (t,q, q)− U(t,q) (2.2)
It is important to emphasize that Lagrangians are not in a one-to-one relationship with physical systems
— many Lagrangians can be used to describe the same physical system.
A quantity that is a function of the state of the system that is constant along a solution path is called a
10
conserved quantity, a constant of motion, or, following historical practice, an integral of motion. In fact,
system symmetries are associated with integrals of motion. For instance, linear momentum is conserved
in a system with translational symmetry; angular momentum is conserved if there is rotational symmetry;
and energy is conserved if the system does not depend on the origin of time.
A generalized coordinate that does not appear explicitly in the Lagrangian is called a cyclic coordinate.
The generalized momentum component conjugate to any cyclic coordinate is a constant of motion. An
example is a particle in a central force field where the Lagrangian, expressed in polar coordinates, does
not depend on the polar coordinate θ and the angular momentum is conserved.
Another important integral of motion is the energy. If the Lagrangian L does not depend explicitly on
time, energy is conserved; systems with no explicit time-dependence are called autonomous systems.
2.1.2. Hamiltonian Mechanics
The formulation of mechanics with generalized coordinates and momenta as dynamical state variables
is called the Hamiltonian formulation. The Hamiltonian and the Lagrangian formulations of mechanics
are equivalent, but each presents a different point of view. The Lagrangian formulation is especially use-
ful in the initial formulation of a system; the Hamiltonian formulation is especially useful in understanding
the evolution, particularly when there are symmetries and conserved quantities.
For each continuous symmetry of a mechanical system there is a conserved quantity. If the gener-
alized coordinates can be chosen to reflect a symmetry, the conjugate momentum is conserved; such
conserved quantities allow the deduction of important properties of the motion. The Hamiltonian formu-
lation is motivated by the desire to focus attention on the momenta.
The momenta can be rewritten in terms of the coordinates and the velocities, so, locally, the velocities
can be solved in terms of the coordinates and momenta. For a given mechanical system and given
coordinates, the momenta and the velocities can be deduced from one another, thus, the dynamical
state of the system can be represented in terms of the coordinates and momenta just as well as with the
coordinates and the velocities. This formulation of the equations governing the evolution of the system
has the advantage that if some of the momenta are conserved, the remaining equations are immediately
simplified. The relation between the generalized velocities and the momenta, deduced with the so-called
Legendre transformation, is
pi = − ∂L∂qi
. (2.3)
The Hamiltonian formulation of dynamics provides much more than the stated goal of expressing
the system derivative in terms of potentially conserved quantities. The Hamiltonian formulation is a
convenient framework in which the possible motions may be placed and understood. It makes possible
to see the range of stable resonance motions, and the range of states reached by chaotic trajectories,
and discover other unsuspected possible motions. The Hamiltonian formulation leads to many additional
insights.
11
The Hamilton’s equations are
dqidt
=∂H
∂piand
dpidt
= −∂H∂qi
(2.4)
where the momenta state variables are represented by the tuple p(t) = (p1(t), . . . , pn(t)). The first
equation in (2.4) is just a restatement of the relation between the velocities and the momenta whereas
the second equation characterized the evolution of the dynamical system.
The Hamiltonian is obtained from the Lagrangian by the Legendre Transformation
H = pq− L (2.5)
which can be proven to be, for conservative systems,
H = pq− L = T + U (2.6)
In an analogous way, conserved quantities can also be found in the Hamiltonian as they were in the
Lagrangian but now with an advantage: if a coordinate does not appear in the Hamiltonian the dimension
of the system of coupled equations that are remaining is reduced by two — the coordinate does not
appear and the conjugate momentum is constant.
2.1.3. The Variational Equations
The initial deviation vector contains the displacements of an initial state of a system. The stability of a
system can be analyzed by the evolution of these initial displacements. The time evolution of orbits and
deviation vectors is addressed in (Skokos, 2010) and reviewed here.
The Hamilton’s equations of a system with N degrees of freedom can be written in matrix notation as
x = f(x) =
[∂H
∂p−∂H∂q
]T≡ J∇xH(t,x) (2.7)
where x(t) = (q1(t), . . . , qn(t), p1(t), . . . , pn(t)), and ∇x is the vector differential operator in respect to
the state of the system x, where
J =
0N IN
−IN 0N
(2.8)
with IN being the N ×N identity matrix and 0N the N ×N matrix with all its elements equal to zero. The
solution of (2.7) is formally written as
x(t) = Φt(x(0)). (2.9)
where Φt is the fundamental matrix of solutions of (2.7) and maps the evolution of x(0) to x(t).
Now, let us see how can we determine the evolution of the deviation vector. We denote by ∇xΦt the
12
linear mapping of the evolution of the deviation vector w(t)
w(t) = ∇xΦtw(0) (2.10)
where w(0) and w(t) are deviation vectors with respect to the orbit x(t) at times t = 0 and t > 0,
respectively.
An initial deviation vector w(0) = (δq1(0), . . . , δqn(0), δp1(0), . . . , δpn(0)) from an orbit x(t) evolves
according to the so-called variational equations
w =∂f
∂x(x(t)) ·w = J∇2
xH(t,x) ·w =: A(t) ·w (2.11)
where ∇2zH(t, z) is the Hessian matrix of the Hamiltonian computed on the reference orbit x(t), i.e.,
∇2xH(t,x)i,j =
∂H
∂xi∂xj
∣∣∣∣Φt(x(0))
i, j = 1, 2, . . . , 2N (2.12)
Note that equation (2.11) represents a set of linear differential equations with respect to w, having
time-dependent coefficients, since matrix A(t) depends on the particular reference orbit, which is a
function of time t. The solution of (2.11) can be written as
w(t) = Y(t) ·w(0) (2.13)
where Y(t) is denominated the fundamental matrix of solutions of (2.11), satisfying the following equa-
tion
Y(t) =∂f
∂x(x(t)) ·Y(t) = A(t) ·Y(t) , Y(0) = I2N . (2.14)
The analysis of the deviation vector is the basis for most CIs, taking advantage of the concept — intro-
duced by Lyapunov — of exponential divergence, i.e., that initially nearby chaotic orbits separate roughly
exponentially with time, whereas nearby regular orbits separate roughly linearly with time.
2.2. The N-Body Problem
Following (Meyer et al., 2009), the N gravitating bodies problem is defined.
Consider N point masses moving in a Newtonian reference system, R3, with the only force acting on
them being their mutual gravitational attraction. Let the i-th particle have position vector qi and mass
mi > 0.
Newton’s second law states that the mass times the acceleration of the i-th particle, miqi, is equal to
the sum of the forces acting on the particle. Additionally, Newton’s law of gravity says that the magnitude
of the force on particle i due to particle j is proportional to the product of the masses and inversely
proportional to the square of the distance between them, Gmimj/ ‖qj − qi‖2 (G is the gravitational
constant). The direction of this force is along a unit vector from particle i to particle j, (qj−qi)/ ‖qj − qi‖.
13
This way, the forces acting on particle i are
miqi =
N∑j=1,i6=j
Gmimj(qj − qi)
‖qj − qi‖3= − ∂U
∂qi(2.15)
from which the potential energy U is derived
U = −∑
1≤i<j≤N
Gmimj
‖qj − qi‖. (2.16)
The kinetic energy of the system is given by the sum of the products of the mass and the squared
velocity of each particle
T =
N∑i=1
‖pi‖2
2mi
(=
1
2
N∑i=1
mi ‖qi‖2)
(2.17)
The Hamiltonian of the system is the sum of the kinetic and the potential energies
H = T + U =
N∑i=1
‖pi‖2
2mi−
∑1≤i<j≤N
Gmimj
‖qj − qi‖(2.18)
where the correct conjugate of position qi is momentum pi.
2.3. The Restricted Three-Body Problem
Following reference (Szebehely, 1967), we define the classic restricted three-body problem (R3BP)
first formulated by Euler in his memoir as follows.
Two bodies revolve around their center of mass in circular orbits under the influence
of their mutual gravitational attraction and a third body (attracted by the previous two
but not influencing their motion) moves in the plane defined by the two revolving
bodies. The restricted problem of three bodies is to describe the motion of this third
body.
The two revolving bodies are called the primaries with masses, m1 and m2 and internal mass distribu-
tions such that can be considered as point masses. The third body does not influence the motion of the
primaries as its mass is much smaller, m3 << m1,m2, assumption valid for our case: the description of
a spacecraft’s motion in the Mars-Phobos system.
The problem is often described in a synodic reference frame rotating with origin at the barycenter of
the two primaries and rotating with angular velocity equal to the mean motion of the primaries. The
problem becomes time-independent when described in this reference frame.
Furthermore, dimensionless variables can be introduced for the problem to be described by only one
parameter
t = nt∗ r12 = l = 1 1− µ =m1
M=a
lµ =
m2
M=b
l(2.19)
14
where t and t∗ are the dimensionless and the dimensional time, respectively; n is the orbital mean
motion; M is the total mass of the system; r12 is the distance between the primaries; and a and b are
the distances from the second mass and the first mass to the origin of the reference frame, respectively.
In this setup, the orbital mean motion, n, and the reference rotating speed, ω, equal one. The first
body of mass 1 − µ is fixed in the reference frame at (µ, 0), and the second body of mass µ is fixed
in the reference frame at (µ − 1, 0). The R3BP becomes dependent of only one parameter: the mass
parameter µ.
Let us begin by defining the R3BP for the planar case. The Lagrangian and Hamiltonian of the problem
described in a synodic frame can be derived by first considering the Lagrangian of the problem in an
inertial frame
LI =1
2(x2I + y2
I )− UI(xI , yI , t) (2.20)
where qI = (xI , yI) is the vector with the third body coordinates in the inertial reference frame, and
UI(xI , yI , t) is the dimensionless time-dependent potential
UI(xI , yI , t) = −1− µr1I
− µ
r2I
(2.21)
with r21I = (xI − µ cos(t))2 + (yI − µ sin(t))2 and r2
2I = (xI + (1− µ) cos(t))2 + (yI + (1−µ) sin(t))2 being
the distance from the third body to the first and second body, respectively.
The coordinate conjugate momenta in the inertial frame pI = (pxI , pyI ) are, from (2.3)
piI =∂LI∂qiI
= qiI (2.22)
and the non-autonomous Hamiltonian of the R3BP in the inertial frame is obtained with (2.5)
HI = pI qI − LI
= p2I −
1
2p2I + UI(xI , yI , t)
=1
2(p2xI + p2
yI ) + UI(xI , yI , t)
(2.23)
The time-dependence is eliminated if the problem is described in a synodic coordinate system. In
order to find the problem’s description in a synodic frame we can perform a coordinate change through
the relation between the velocities in the inertial frame and the velocities in the rotating frame
qI = qR + ω × qR, (2.24)
Remember that ω = (0, 0, ω) = (0, 0, 1). The velocities in the two different frames are related by
x
y
0
I
=
x
y
0
R
+
0
0
1
×x
y
0
R
⇒
xI = xR − yRyI = yR + xR
. (2.25)
15
The change of the velocities in the Lagrangian gives the Lagrangian in the synodic or rotating frame
LR =1
2((xR − yR)2 + (yR + xR)2)− UR(xR, yR) (2.26)
with the time-independent potential
UR(xR, yR) = −1− µr1R
− µ
r2R
(2.27)
where r21R = (xR − µ)2 + y2
R and r22R = (xR + 1− µ)2 + y2
R are the distances from the third body to the
first and second body, respectively.
The conjugate momenta for the synodic frame are easily derived
piR =∂LR∂qiR
⇒
pxR = xR − yRpyR = yR + xR
⇒
xR = pxR + yR
yR = pyR − xR. (2.28)
Note that, in the rotating frame, piR 6= qiR .
The Hamiltonian in the rotating frame is
HR = pRqR − LR
=
pxR
pyR
· pxR + yR
pyR − xR
− 1
2(p2xR + p2
yR) + UR(xR, yR)
=1
2(p2xR + p2
yR) + pxRyR − pyRxR + U(xR, yR)
. (2.29)
In the literature it is common practice to express the Hamiltonian as
HR =1
2
[(pxR + yR)2 + (pyR − xR)2
]− ΩR(xR, yR) (2.30)
where ΩR(xR, yR) is the so-called amended potential
ΩR(xR, yR) =1
2(x2R + y2
R)− UR(xR, yR) +1
2µ(1− µ) (2.31)
The first term in the amended potential is related to the centrifugal force in the synodic frame, and the
last term is an added constant (which will not affect the equations of motion) for a more symmetric form
of the potential (Szebehely, 1967).
The Hamiltonian is time-independent and, thus, presents an integral of motion related with the con-
stant energy of the third body. This integral can be represented by the so-called Jacobi Integral (Szebe-
hely, 1967),
C(x0R , y0R , x0R , y0R) = 2ΩR(xR, yR)− v2R = −2ER (2.32)
where v2R = (x2
R + y2R) is the spacecraft’s velocity, and E is the Hamiltonian expressed as a function of
(xR, yR, xR, yR). The Jacobi Integral depends only on the initial conditions of the system.
16
2.4. Modifications of the Restricted Three-Body Problem
We have defined the restricted three-body problem for the planar case. However, the circumstances
may require modifications to this formulation. Two common modifications are the extensions to the three-
dimensional case, and the more complex extension to the elliptic case. We are interested in both. Let
us begin with the simpler case of the spatial restricted three-body-problem.
2.4.1. The Spatial Restricted Three-Body Problem
Notice that the change from the sidereal to the synodic reference does not affect the third coordinate
as the rotation is performed about the z-axis, hence, the conjugate momentum of the third coordinate is
pz = qz.
For the sake of simplicity, we omit the index on the coordinates as from now on they all refer to the
synodic reference frame.
The Hamiltonian of the three-dimensional case is
H =1
2
[(px + y)2 + (py − x)2 + p2
z
]+
1
2z2 − Ω(x, y, z) (2.33)
where the extra term, 12z
2, is added to obtain the amended potential Ω in the symmetric form (see
(Szebehely, 1967))
Ω(x, y, z) =1
2(x2 + y2 + z2) +
1− µr1
+µ
r2+
1
2µ(1− µ). (2.34)
The Jacobi integral of motion in the three-dimensional case is obtained following the same derivation
technique as for the planar case
C(x0, y0, z0, x0, y0, z0) = 2Ω′(x, y, z)− v2 = −2E (2.35)
with v2 = x2 + y2 + z2 as the spacecraft’s velocity in the synodic reference frame, and Ω′ = Ω− 1/2z2.
2.4.2. The Elliptic Restricted Three-Body Problem
We have already derived the Hamiltonian for the case when the primaries describe circular orbits
around their barycenter as described in the formulation of the R3BP. A more general case is when the
primaries describe elliptic orbits about their barycenter. This is known as the elliptic restricted three-body
problem (ER3BP). Let us start by deriving the Hamiltonian for the elliptic planar problem.
One of the consequences of dealing with the elliptic case is the loss of possession of the Jacobi
integral. The distance between the primaries is not constant anymore and the Hamiltonian becomes
time-dependent. As we have seen, time-dependent Hamiltonians do not possess the integral of energy
and, thus, neither the Jacobi integral. However, we derive an invariant relation for this case.
When the primaries move on elliptic orbits, the introduction of a non-uniformly rotating and pulsating
coordinate system (from (Szebehely, 1967)) results in fixed locations for the primaries. Such a pulsating
17
coordinate system is introduced by using the variable distance between the primaries as the length unit
of the system by which distances are normalized. This way, dimensionless variables are introduced by
using the distance between the primaries,
r =p
1 + e cos f, p = a(1− e2) (2.36)
where p is the semi-latus rectum, a and e are the semi-major axis and the eccentricity of either primary
around the other, and f is the true anomaly of m1 (and m2).
A coordinate system which rotates with the variable angular velocity f is also introduced. This angular
motion is given bydf
dt∗=G1/2(m1 +m2)1/2
a3/2(1− e2)3/2(1 + e cos f)
2, (2.37)
where t∗ is the dimensional time. This equation follows from the principle of the conservation of the
angular momentum of the two primaries,
df
dt∗r2 =
[a(1− e2)G(m1 +m2)
]1/2. (2.38)
The coordinates are transformed to dimensionless pulsating coordinates by
ξ =x
r=x(1 + e cos f)
a(1− e2)η =
y
r=y(1 + e cos f)
a(1− e2)(2.39)
with (ξ, η) being the pulsating coordinates which allow for the first and second primaries, Mars and
Phobos, to continue fixed at the positions (µ, 0) and (µ − 1, 0). Note that both primaries orbit around
each other with the same eccentricity, semi-major axis, and true anomaly (Szebehely, 1967).
The true anomaly as the independent variable may be introduced by the equation
d
dt∗=
df
dt∗d
df(2.40)
which, after the change of the independent variable, t→ f , gives the Hamiltonian (Szebehely, 1967)
H =1
2
[(pξ + η)2 + (pη − ξ)2
]− Ωe(ξ, η) (2.41)
with
Ωe =Ω
1 + e cos f=
1
1 + e cos f
[1
2(ξ2 + η2) +
1− µr1
+µ
r2+
1
2µ(1− µ)
](2.42)
where r21 = (ξ−µ)2 +η2 and r2
2 = (ξ−µ+1)2 +η2 are the distances to the primaries located at (µ,0) and
(µ− 1,0), respectively. The normalized angular velocity of the synodic pulsating reference frame equals
the unity.
The extension to the three-dimensional case is straight-forward, but note that while the third coor-
dinate does not take place in the transformation involving the rotation around the z-axis, it is made
dimensionless by the variable distance between the primaries and also assumes a pulsating character,
18
i.e.,
ζ =z
r=z(1 + e cos f)
a(1− e2)(2.43)
resulting in the Hamiltonian
H =1
2
[(pξ + η)2 + (pη − ξ)2 + p2
ζ
]+
1
2ζ2 − Ωe(ξ, η, ζ, f). (2.44)
with
Ωe =Ω
1 + e cos f=
1
1 + e cos f
[1
2(ξ2 + η2 + ζ2) +
1− µr1
+µ
r2+
1
2µ(1− µ)
](2.45)
where r21 = (ξ−µ)2 +η2 + ζ2 and r2
2 = (ξ−µ+1)2 +η2 + ζ2 are the distances to the primaries located at
(µ,0,0) and (µ− 1,0,0), respectively. In the elliptic three-dimensional case the Hamiltonian is sometimes
expressed in the literature as (Szebehely, 1967)
H =1
2
[(pξ + η)2 + (pη − ξ)2 + p2
ζ
]− Ω′e(ξ, η, ζ, f). (2.46)
with
Ω′e = Ωe −1
2ζ2 =
Ω′
1 + e cos f=
1
1 + e cos f
[1
2(ξ2 + η2 − eζ2 cos f) +
1− µr1
+µ
r2+
1
2µ(1− µ)
](2.47)
2.4.3. Change Of Origin
In the present work, we are interested in the motion in the vicinity of Phobos, hence, an origin change
is performed to center the reference frame on the second primary. The study of the problem in this new
reference frame has some advantages for the analytical treatment of the problem, as the coordinates
(ξ, η, ζ) are small when compared to the length unit. The origin shift is translated by the coordinate
change
ξ → ξ + µ− 1 (2.48)
and by the transformed Hamiltonian
H =1
2
[(pξ + η)2 + (pη − ξ − µ+ 1)2 + p2
ζ
]+
1
2ζ2 − Ωe(ξ, η, ζ, f). (2.49)
with
Ωe =1
1 + e cos f
[1
2((ξ + µ− 1)2 + η2 + ζ2) +
1− µr1
+µ
r2+
1
2µ(1− µ)
](2.50)
where r21 = (ξ − 1)2 + η2 + ζ2 and r2
2 = ξ2 + η2 + ζ2 are the distances to the primaries now located at
(1, 0, 0) and (0, 0, 0), respectively.
We work in a synodic reference frame centered in Phobos with the x-axis pointing towards Mars, the
z-axis with the direction of the angular momentum of Phobos, and the y-axis completes the right-handed
orthogonal system.
19
2.5. Equations of Motion
Let us now derive the equations of motion to be used in the analytical and numerical approaches.
Finally, an invariant relation based on the Jacobi Integral is derived.
2.5.1. Hamilton’s Equations of Motion
After finding the Hamiltonian of the problem in a convenient reference frame, the Hamilton’s equations
of motion are obtained from (2.4). For the sake of simplicity, we now change the pulsating coordinates
notation back, (ξ, η, ζ)→ (x, y, z),
x =∂H
∂px= px + y px = −∂H
∂x= py − x− µ+ 1 + Ωex
y =∂H
∂py= py − x− µ+ 1 py = −∂H
∂y= −px − y + Ωey
z =∂H
∂pz= pz pz = −∂H
∂z= −z + Ωez
(2.51)
with
Ωex =∂Ωe∂x
=1
1 + e cos f
[x+ µ− 1− (1− µ)(x− 1)
r31
− µx
r32
]
Ωey =∂Ωe∂y
=1
1 + e cos f
[y − (1− µ)y
r31
− µy
r32
]
Ωez =∂Ωe∂z
=1
1 + e cos f
[z − (1− µ)z
r31
− µz
r32
](2.52)
The set of 6 first-order equations in Hamilton’s formulation can be combined into a set of 3 second-
order equations. This will be particularly useful for the analytical treatment of the problem. This way,
x− 2y = Ωex
y + 2x = Ωey
z + z = Ωez
(2.53)
These equations are in accordance with the equations found in (Szebehely, 1967).
2.5.2. An Invariant Relation
We have already seen that in the formulation of the E3BP the system does not possess the Jacobi
integral, which can be explained by the introduction of the independent variable in the Hamiltonian
20
causing it to lose the energy constant. However, it is possible to derive an invariant relation for this
problem (Szebehely, 1967).
If we multiply each equation in (2.53) by x, y, and z, respectively, we get
xx− 2yx = Ωex x
yy + 2yx = Ωey y
zz = Ωez z − zz
(2.54)
If we sum equations (2.54), integrate, and use the relations∫uudf = 1
2 u, u = x, y, z∫zzdf = 1
2z2
(2.55)
we obtain
V 2 = x2 + y2 + z2 = 2
∫ f
0
(Ωex x+ Ωey y + Ωez z
)df − z2 (2.56)
In the circular problem, which is described by autonomous equations, the integral on the right side
of equation (2.56) is immediately deduced to be 2Ω − C. In this case, however, the potential depends
explicitly on the normalized time f ,
dΩe = Ωexdx+ Ωeydy + Ωezdz + Ωef df. (2.57)
where
Ωef =∂Ωe∂f
=Ωee sin f
(1 + e cos f)2 (2.58)
Hence, the right hand side of equation 2.56 can be written as
2
∫ f
0
(Ωex x+ Ωey y + Ωez z
)df − z2 = 2Ωe − 2
∫ f
0
Ωef df − C − z2 (2.59)
and equation 2.56 becomes
V 2 = 2Ωe − 2e
∫ f
0
Ωe sin f
(1 + e cos f)2 df − C − z
2. (2.60)
Considering an orbit for a short time, meaning that we select the time to start the motion at, e.g., f = 0
and we are interested only in that part of the trajectory which takes place between f = 0 and f = ε,
where ε is a sufficiently small positive quantity. Since f is the true anomaly, this restriction amounts
to considering a sufficiently small time interval, during which the primaries describe sufficiently small
arcs. The second term on the right of (2.60) contains in this case the product of the eccentricity and ε,
consequently it is smaller than the term 2Ωe. This way, equation (2.60) can be approximated by
V 2 = 2Ωe − z2 − C = 2Ω′e − C (2.61)
21
or
V 2 = 2Ω′
1 + e cos f− C. (2.62)
In this case, the zero velocity curves can be computed by
2Ω′ − C∗ = 0, (2.63)
however, at every instant these curves have to be computed as they directly depend on the independent
variable f . The variation of their shape is governed by
C∗ = C (1 + e cos f) . (2.64)
These curves are known as the pulsating curves of zero velocity.
2.6. Chaos Indicators
The Chaos Indicators, CIs, are numerical techniques to accomplish one of the most important aspects
in the study of the behavior of dynamical systems: the differentiation of trajectories of regular and chaotic
nature. Such task is of difficult realization as the difference between two trajectories of different nature
can be very subtle (Maffione et al., 2011). This was the motivation behind the development of CIs.
There are two main types of CIs: those which are based on the analysis of the deviation vector (and
related to the concept of exponential divergence), and those which are based on the analysis of the
orbit itself. Under the first category, one of the first and most used CIs are the Lyapunov Characteristic
Exponents, LCEs. These can be studied through the maximum LCE, mLCE, or through their spectra
(the distribution of all the LCEs). The method may take a huge amount of time to determine the nature
of a trajectory, especially in orbits that remain regular for a long period of time before presenting any
chaotic behavior. The mLCE, however, does much more than to determine the nature of a trajectory,
it also quantifies the notion of chaosticity by providing a timescale for the studied dynamical system,
namely the Lyapunov time (Skokos, 2010).
In (Maffione et al., 2011), a comparison of some methods based on the mLCE is performed. In this
article, the FLI, the Lyapunov indicator, LI, the MEGNO, the SALI, the Dynamical Spectra of stretching
numbers, SSN, and the corresponding spectral distance, and the RLI are compared in terms of robust-
ness, speed of convergence, final values, and behavior under complex scenarios. In our work we are
interested in the test of robustness (i.e., capacity to distinguish chaos) and speed of convergence. The
first is important to have confidence in our results and the second because we compute the value of
the CI for a large set of initials conditions which can reveal to be time-consuming, hence, we need an
indicator that takes as less time for its computation as possible. The FLI showed the better relation
between these two tests and, thus, it was selected as the CI to be used in our work.
22
2.6.1. Lyapunov Characteristic Exponents
Following (Skokos, 2010) we introduce the theoretical background on the LCEs and the numerical
methods for their computation. We also review the computation method of the mLCE, which is useful to
define the FLI computation method.
The knowledge of the spectrum of the LCEs provides the basic information on the behavior of a
dynamical system. The LCEs are asymptotic measures characterizing the average rate of growth (or
shrinking) of small perturbations to the solutions of a dynamical system. The value of the mLCE is an
indicator of the chaotic or regular nature of orbits, while the whole spectrum of LCEs is related to the
underlying dynamics of a system.
The computation of the mLCE, χ1, of a trajectory allows us to characterize its nature as regular or
chaotic. For regular orbits we have χ1 = 0 whereas chaotic orbits have χ1 > 0, implying exponential
divergence. Furthermore, the mLCE has also the ability to quantify the orbit’s chaosticity. It defines
a specific timescale for the considered dynamical system as the inverse of the mLCE, the so-called
Lyapunov time,
tL =1
χ1(2.65)
which gives an estimate of the time needed for a dynamical system to become chaotic. It measures
the time needed for nearby orbits of the system to diverge by e (Neper’s number, do not confuse with
eccentricity).
The evaluation of the mLCE of an orbit with initial condition x(0) requires the evaluation of the orbit’s
time evolution and of the deviation vector’s time evolution with initial condition w(0), i.e., the Hamilton’s
equations of motion and the variational equations must be solved simultaneously.
In order to prevent the increase of the deviation vector to extreme large values causing numerical
overflow, one may fix a small time interval τ and define the mLCE for time t = kτ, k = 1, 2, . . .. First, let
us recall equation 2.10
w(t) = ∇xΦtw(0)
and have the initial deviation vector w(0) with norm
D0 = ‖w(0)‖ (2.66)
We denote by
w((i− 1)τ) =∇x(0)Φ
(i−1)τw(0)∥∥∇x(0)Φ(i−1)τw(0)∥∥D0, (2.67)
the deviation vector at the point Φ(i−1)τ (x(0)) having the same direction with w((i− 1)τ) and norm D0,
and by Di its norm after its evolution for τ time units
Di =∥∥∇Φ(i−1)τ (x(0))Φ
τ w((i− 1)τ)∥∥ . (2.68)
We define now the local coefficient of expansion of the deviation vector, αi, for a time interval τ when
23
the orbit evolves from Φ(i−1)τ (x(0)) to Φiτ (x(0)) as
ln
∥∥∇x(0)Φiτw(0)
∥∥∥∥∇x(0)Φ(i−1)τw(0)∥∥ = ln
‖w(iτ)‖‖w((i− 1)τ)‖
= lnDi
D0= lnαi. (2.69)
The value lnαi/τ is also called stretching number.
The mLCE is then computed by
χ1 = limk→∞
1
kτ
k∑i=1
lnDi
D0= limk→∞
1
kτ
k∑i=1
lnαi (2.70)
2.6.2. Fast Lyapunov Indicator
The Fast Lyapunov Indicator, FLI, was introduced in (Froeschle et al., 1997) motivated by the need to
have a quicker method to distinguish between chaotic and regular orbits. In some of the FLI definitions,
it can also distinguish resonant from non-resonant motion (Skokos, 2010).
The main difference of the FLI to the evaluation of the mLCE is that the FLI registers the current value
of the norm of the deviation vector whereas the mLCE computes the limit value, t → ∞, of the mean
of the stretching numbers. By dropping the time average requirement of the stretching numbers, FLI
succeeds in determining the nature of orbits faster than the computation of the mLCE (Skokos, 2010).
Since the initial definition of the FLI by Froeschle et al. there has been an evolution of this definition. In
their pioneer article, they used and tested the FLI definitions Ψ1, Ψ2, and Ψ3 and later on, in (Froeschle
and Lega, 2000), the new Ψ4 was developed
Ψ1 =1
‖w1(t)‖n; Ψ2 =
1∏nj=1 ‖wj(t)‖
; Ψ3 =1
supj ‖wj(t)‖n; Ψ4 = supt≤tf ln ‖w(t)‖ (2.71)
where wj is a basis of deviation vectors and w a randomly chosen deviation vector.
In our work we use an adapted FLI definition from (Villac and Aiello, 2005)
FLI = supt≤tf
supi
ln ‖wi(t)‖ (2.72)
or in the notation used in the definition of the mLCE
FLI = supt≤tf
supi
ln ‖αi(t)‖ (2.73)
where tf is the final time of integration, and wi(t) is a basis of n deviation vectors with initial conditions
wi(0) = (w1(0),w2(0), . . . ,wn(0)) = In (2.74)
and αi(t) the respective expansion coefficients with n = 6. The FLI is the largest logarithmic variation
between two consecutive steps in all six coordinates. This definition was developed not to distinguish
resonant motion but to depend as little as possible on the choice of the initial deviation vectors basis.
24
This definition provides only one value per set of initial conditions, making it possible to construct FLI
maps where chaotic and regular regions are easily distinguishable.
There is still one issue in the definition of the FLI: the normalization. It is stated and proved in (Skokos,
2010) that, although for different chosen norms the value of the FLI changes, its capacity to distinguish
regular from chaotic motion remains intact, i.e., the norm choice only affects the FLI value quantitatively
but not qualitatively. In the computation of the FLI we use the norm
‖w‖ =
√1
r
(w2x + w2
y + w2z
)+
1
p
(w2px + w2
py + w2pz
)(2.75)
where r and p represent the Euclidean norms of the position and momenta of the third body state at the
current time t, respectively.
25
It is far better to foresee even without certainty than
not to foresee at all.
(Henri Poincare 1854 - 1912)
CH
AP
TE
R
3 QSO Solutions and Stability
In this chapter we analyze the stability of QSOs in the elliptic restricted three-body problem. The
starting point are the solutions of Hill’s problem to which a perturbation theory is then considered in two
different approaches. In the first, perturbations are added to the solutions of Hill’s case. The averaging of
these perturbations provides a set of differential equations on the perturbation coordinates from which
it is possible to withdraw conclusions on the stability region for QSOs. In the second approach, the
method of variation of the arbitrary constants is applied to the solutions of Hill’s problem defined in
osculating coordinates. Approximate solutions in these coordinates are obtained. The deduced stability
considerations and approximate solutions are then applied to the Mars-Phobos system.
3.1. Linearization of the Equations of Motion
We consider the motion in the vicinity of the second primary. In this case, the equations of motion
(2.53)
x− 2y =1
1 + e cos f
[x+ µ− 1− (1− µ)(x− 1)
r31
− µx
r32
]
y + 2x =1
1 + e cos f
[y − (1− µ)y
r31
− µy
r32
]
z + z =1
1 + e cos f
[z − (1− µ)z
r31
− µz
r32
](3.1)
can be simplified since the distance from the moon to Mars equals the unity and therefore x, y, z << 1,
i.e., we can linearize the equations by neglecting the second-order terms. The Mars-influenced potential
27
terms can be expanded in a Taylor Series in x, y, z about zero up to the first order to obtain
(1− µ)(x− 1)
((x− 1)2 + y2 + z2)3/2= (1− µ)(x− 1)((x− 1)2 +
70
y2 +0
z2)−3/2
≈ (1− µ)(x− 1)(>0
x2 − 2x+ 1)−3/2
≈ (1− µ)(x− 1)(1− 2x)−3/2
≈ (1− µ)(x− 1)(1 + 3x)
≈ (1− µ)(* 0
3x2 − 2x− 1)
≈ −(1− µ)(1 + 2x)
(1− µ)y
((x− 1)2 + y2 + z2)3/2= (1− µ)y((x− 1)2 +
70
y2 +0
z2)−3/2
≈ y(>0
x2 − 2x+ 1)−3/2
≈ (1− µ)y(1− 2x)−3/2
≈ (1− µ)y(1 + 3x)
≈ (1− µ)(y +*
03xy)
≈ (1− µ)y
(1− µ)z
((x− 1)2 + y2 + z2)3/2= (1− µ)z((x− 1)2 + y2 + z2)−3/2
≈ (1− µ)z
(3.2)
Substituting in (3.1), we obtain
x− 2y − 1
1 + e cos f[x− 1− (1− µ)(−2x− 1)] = − 1
1 + e cos f
(µx
r32
)y + 2x− 1
1 + e cos f[y − (1− µ)y] = − 1
1 + e cos f
(µy
r32
)z + z − 1
1 + e cos f[z − (1− µ)z] = − 1
1 + e cos f
(µz
r32
)(3.3)
If we assume that the mass of the second primary is much smaller than the mass of the first, i.e., µ << 1,
the equations (3.3) can be simplified even more. The assumption that the third body is in the vicinity of
the second primary gives r2 << 1 (with x, y, z << 1) and, this way, the terms that are multiplied by µ
and are not divided by r32 can be neglected because they are much smaller than the remaining terms.
The result are the linearized equations of motion
x− 2y − 3x
1 + e cos f= − 1
1 + e cos f
(µx
r32
)y + 2x = − 1
1 + e cos f
(µy
r32
)z + z = − 1
1 + e cos f
(µz
r32
)(3.4)
28
Equations (3.4) are the common form of the equations of motion for the spatial elliptic restricted three-
body problem to study orbits in the vicinity of the second primary with dimensionless mass µ << 1.
3.2. Unperturbed Hill’s Equations
In Hill’s approach, valid for µ << 1, we start by neglecting the influence of the second primary on the
equations of motion, µ = 0, and by finding their solutions. The linearized equations of motion for the
unperturbed problem are
x− 2y − 3x
1 + e cos f= 0
y + 2x = 0
z + z = 0
(3.5)
There is no explicit method of finding the solution of a set of differential equations with time-dependent
coefficients. This illustrates the difficulty of handling the elliptic case. However, a workaround is possible.
Let us start by solving the equation in z as it is independent from the other equations and has constant
coefficients. The solution to this equation and its derivative are easily obtained
znp(f) = Cz1 cos f + Cz2 sin f = z0np cos f + z0np sin f
znp(f) = −Cz1 sin f + Cz2 cos f = −z0np sin f + z0np cos f
(3.6)
where Cz1 and Cz2 are constants of integration that are solved from the initial conditions z(0) = z0 and
z(0) = z0. The index np denominates the non-perturbed case.
The other two differential equations can be manipulated to obtain one differential equation on x alone.x− 2y − 3x
1 + e cos f= 0∫
y =
∫− 2x
⇒
x− 2(−2x+ C3)− 3x
1 + e cos f= 0
y = −2x+ C3
⇒
x+ (4− 3
1 + e cos f)x = 2C3
y = −2x+ C3
(3.7)
Once we solve the first equation the solution on y will be obtained by direct integration. In order to
solve the equation on x we will start by considering the homogeneous case C3 = 0 and, only then,
derive the particular solution.
29
3.2.1. Homogeneous Solution
In order to find the homogeneous solution for x we must solve the second-order differential equation
x+ (4− 3
1 + e cos f)x = 0 (3.8)
The linear homogeneous second-order differential equation has two independent solutions. It is trivial
to verify that
x1(f) = (1 + e cos f) sin f, (3.9)
obtained with the help of the program Mathematica (Wolfram Research, Inc., 2011), is a solution of
equation (3.8).
The second solution returned by Mathematica was in a form too complex to be useful. We can obtain,
instead, the second solution from the first. We will start by looking for a second solution, x2, that satisfies
x2 = x1v (Zwillinger, 1997). In this case, the second derivative of the second solution is
x2 = x1v + 2x1v + x1v (3.10)
and, as x1 is a solution of (3.8), we have
x1 = −(
4− 3
1 + e cos f
)x1 (3.11)
These relations, when applied to the homogeneous equation (3.8) for the independent solution x2,
yield
x2 +
(4− 3
1 + e cos f
)x2 = 0
x1v + 2x1v + x1v +
(4− 3
1 + e cos f
)x1v = 0
−(
4− 3
1 + e cos f
)x1v + 2x1v + x1v +
(4− 3
1 + e cos f
)x1v = 0
2x1v + x1v = 0
(3.12)
We now introduce the variable change u = v and substitute the first solution and its derivative in (3.12)
to obtain
2x1u+ x1u = 0
(1 + e cos f) sin fu+ 2[(1 + e cos f) cos f − e sin2 f
]u = 0
(3.13)
Another variable change s = cos f with derivative
d
df= − sin f
d
ds⇒ d
dfu(f) = − sin f
d
dsu(s) = −
√1− s2
d
dsu(s) (3.14)
30
is applied to 3.13
2[(1 + e s)− e(1− s2)
]u(s)− (1 + e s)(1− s2)u(s) = 0 ⇔
⇔ u(s)
u(s)=
2[(1 + e s)s− e(1− s2)
](1 + e s)(1− s2)
=2s
1− s2− 2e
1 + e s
(3.15)
Integrating both sides of the equation to find the solution of u
∫u(s)
u(s)ds =
∫2s
1− s2ds−
∫2e
1 + e sds ⇔
⇔ log u = − log(s2 − 1)− 2 log(1 + e s) + Ca ⇔
⇔ u = Ca1
(1− s2)(1 + e s)2
(3.16)
The factor v can now be derived from u
v = u = Ca1
(1− s2)(1 + e s)2= Ca
1
sin2 f(1 + e cos f)2⇔
⇔ v =
∫Ca
1
sin2 f(1 + e cos f)df + Cb
(3.17)
and Mathematica is used to obtain
v = Cb + Ca
−6e2 arctan
((e− 1) tan ( f2 )√
1− e2
)(1− e2)5/2
−cot ( f2 )
2(1 + e)2+
e3 sin f
(1− e2)2(1 + e cos f)+
tan ( f2 )
2(1− e)2
(3.18)
Let us now denote the homogeneous solution as xh. The homogeneous general solution is a linear
combination of the two independent solutions
xh = Cx1x1 + Cx2x2 = x1(Cx1 + Cx2v) = x1(Cx1 + Cx2 Cb + Ca [. . .])
xh = (1 + e cos f) sin f
C1 + C2
−6e2 arctan
((e− 1) tan ( f2 )√
1− e2
)(1− e2)5/2
−
−cot ( f2 )
2(1 + e)2+
e3 sin f
(1− e2)2(1 + e cos f)+
tan ( f2 )
2(1− e)2
](3.19)
where C1 and C2 are chosen as constants from the combination of Cx1, Cx2
, Ca, and Cb. Their value
can be derived from the initial conditions.
31
3.2.2. General Solution
The general solution of the non-homogeneous differential equation (3.8) is the sum of the general
solution of the corresponding homogeneous equation (3.19) with the particular solution of the non-
homogeneous equation. In differential equations of the form
g2(f)x(f) + g1(f)x(f) + g0(f)x(f) = h(f) (3.20)
the general solution can be obtained by (Polyanin and Zaitsev, 1995)
x = xh + x2
∫x1
h
g2
df
W− x1
∫x2
h
g2
df
W(3.21)
where W = x1x2 − x2x1 is the Wronskian determinant. Substitution of the first and second solutions
reveals the Wronskian determinant to be 1.
The general solution of the non-homogeneous differential equation on x is found through equation
(3.21)
xnp = (1 + e cos f) sin f
−
6e(C3 + eC2) arctan
√1− e tan(f2
)√
1 + e
(1− e2)5/2
+
+(e2 − 1)
[C1(e2 − 1)(1 + e cos f) +
[C3(e2 − 2)− eC2 + (C2 + eC3) cos f
]csc f
](e2 − 1)2(1 + e cos f)
+e[C2(1 + 2e2)− eC3(e2 − 4)
]sin f
(e2 − 1)2(1 + e cos f)
(3.22)
now denoted xnp in an allusion to being the solution for the non-perturbed Hill’s equations 3.5.
Let us now recall the equation on ynp
ynp = −2xnp + C3 (3.23)
Its solution, after the computation of xnp can be obtained immediately
ynp = −2
∫xnp df + C3 f + C4
ynp =1
2(1− e2)5/2
−12(C3 + eC2) arctan
√1− e tan(f2
)√
1 + e
(1 + e cos f)2+
+√
1− e2[(2C4 + eC1)(1− e2)2 + 4C1(1− e2)2 cos f + eC1(1− e2)2 cos (2f)+
+2(eC3(5− 2e2) + C2(2 + e2)) sin f + e(C2 + 2e2C2 − eC3(e2 − 4)) sin (2f)]
(3.24)
32
3.2.3. Stability Considerations
Let us now analyze the behavior of the solutions xnp and ynp.
(a) xnp(f) (b) ynp(f)
Figure 3.1. Functions xnp(f) and ynp(f) with the following parameters: e = 0.25, C1 = −0.3, C2 = 0.2,C3 = 0.1, C4 = −0.2
In figure 3.1 it can be observed that the functions xnp(f) and ynp(f) present a non-differentiability on
the first solution and a discontinuity on the second solution for f = π. These irregularities are caused by
the term
arctan
√1− e tan(f2
)√
1 + e
The tangent function is not defined in f = kπ, k = 1, ..., n which provokes the anomalies in the
analyzed functions. Furthermore, as the inverse tangent function ranges from −π/2 to π/2, an inverse
tangent with a tangent function as an argument, suffers an offset every f = kπ, k = 1, ..., n. This offset
can be eliminated adding π by continuity and it becomes clear that this term has a secular effect on
ynp, leading to an unstable solution. There is, however, a family of stable orbits - the ones for which the
troublesome term vanishes. This way, the first stability condition for orbits around the second primary is
(C3 + eC2) arctan
√1− e tan(f2
)√
1 + e
= 0⇒ C3 + eC2 = 0⇒ C3 = −eC2 (3.25)
(a) xnp(f) (b) ynp(f)
Figure 3.2. Functions xnp(f) and ynp(f) with the following parameters: e = 0.25, C1 = −0.3, C2 = 0.2,C3 = −0.05, C4 = −0.2
33
The behavior of these functions, under the stability condition C3 = −eC2 , is presented in figure 3.2.
The functions are now continuous and do not possess any secular terms and their analytical expressions
are simplified with the application of the stability condition (3.25)xnp(f) = (1 + e cos f)(C1 sin f − C2 cos f)
ynp(f) = 12 [(e+ 4 cos f + e cos (2f))C1 + 2C2(2 + e cos f) sin f + 2C4]
(3.26)
Figure 3.3. Parametric plot of the solutions of the unperturbed equations under the stability condition,C3 = −eC2, with the following parameters: e = 0.25, C1 = −0.3, C2 = 0.2, C3 = −0.05,C4 = −0.2
In figure 3.3, the orbit for the previously used parameters under the stability condition is presented.
These constants can be computed from the non-perturbed initial conditions and depend on the initial
true anomaly f0. Although there is not a strict way to distinguish the non-perturbed initial conditions,
let us define the perturbed conditions x0 as the sum of the non-perturbed initial conditions x0np and the
initial conditions caused by a perturbation x0p with the latter being much smaller, i.e.,
x0 = x0np + x0p
x0np >> x0p
(3.27)
The practical meaning of the above definition is that the stability condition (3.25) allows for small
displacements on the initial conditions that affect C2 and C3.
34
3.2.4. Constants of Integration
The value of the constants of integration can be determined from the initial conditions. The expres-
sions for these are extensive as in the non-autonomous system we can have f0 6= 0 which may not be
time transformed, f → f − f0, to set f0 = 0 as it would be possible in the autonomous circular case. We
present here the computation of the integration constants for two initial cases of interest: the passage of
the second primary at the perigee (f0 = 0) and apogee (f0 = π).
f0 = 0 f0 = π
Constants
C1 =x0np
1 + eC1 = − 3(2− e)eπ
(1− e2)3/2(1 + e)x0np −
x0np
1− e− 3eπ
(1− e2)1/2(1 + e)2y0np
C2 = (3 + e)x0np + (2 + e)y0np C2 = −(3− e)x0np − (2− e)y0np
C3 = 2x0np + y0np C3 = 2x0np + y0np
C4 = y0np −2 + e
1 + ex0np C4 = − 3(2− e)π
(1 + e)5/2(1− e)3/2x0np + y0np −
2− e1− e
x0np −3π
(1 + e)5/2(1− e)1/2y0np
Stab. Cond. y0np = −2 + e
1 + ex0np y0np = −2− e
1− ex0np
(3.28)
For both values of the initial true anomaly the stability condition demands that the initial velocity in y
has an opposite sign of the initial position in x resulting in a retrograde orbit. This is the origin of the
alternative denomination DRO, distant retrograde orbits, for QSOs.
The stability conditions in dimensional coordinates are
y0np
∣∣f0=0
= −2 + e
1 + en x0np y0np
∣∣f0=π
= −2− e1− e
n x0np(3.29)
for f0 = 0 and f0 = π, respectively, where n is the orbital mean motion of the second primary.
For the z-coordinate, which solution is independent from the other two coordinates, the computation
of the constants is a trivial task.
3.2.5. Constant Transformation
Experimentation shows that the constants C1 and C2 influence mainly the amplitude of the orbit
whereas C4 influences the position of the origin of the orbit. This suggests that a change to some
variation of ‘polar’ constants would be advantageous.
The constants C1, C2, and C4 can be transformed to a set of three alternative equivalent constants α,
φ, and δy by C1 = −α sinφ
C2 = −α cosφ
C4 = δy
⇒
φ = arctan
(C1
C2
)α = (C2
1 + C22 )1/2
C4 = δy
(3.30)
35
The solutions for Hill’s case (equations (3.26)) can now be rewritten using the new constants
xnp(f) = (1 + e cos f)(C1 sin f − C2 cos f) =
= (1 + e cos f)(α cosφ cos f − α sinφ sin f) =
= (1 + e cos f)α cos (f + φ) =
= α cos (f + φ) + αe cos f cos (f + φ)
= α(1 + e cos f) cos(f + φ)
(3.31)
ynp(f) = 12 [(e+ 4 cos f + e cos (2f))C1 + 2C2(2 + e cos f) sin f + 2C4] =
= 2(C1 cos f + C2 sin f) + 12e(C1 cos (2f) + 2C2 cos f sin f) + C4 + 1
2eC1 =
= 2(C1 cos f + C2 sin f) + 12e(C1 cos (2f) + C2 sin (2f)) + C4 + 1
2eC1 =
= −2α(sinφ cos f + cosφ sin f)− 12eα(sinφ cos 2f + cosφ sin 2f) + C4 + 1
2eC1 =
= −2α sin (f + φ)− 12αe sin (2f + φ)− 1
2eα sinφ+ C4
= −α(2 + e cos f) sin(f + φ) + δy
(3.32)
A similar transformation is easily performed for the z-coordinate
z0np = γ cosψ
z0np = −γ sinψ⇒
ψ = arctan
(−z0np
z0np
)γ = (z2
0np + z20np)1/2
(3.33)
znp(f) = z0np cos f − z0np sin f =
= γ cosψ cos f − γ sinψ sin f =
= γ cos (f + ψ)
(3.34)
These constants are known as osculating elements1 and are represented in fig. 3.4. The orbit’s projec-
tion on the x-y plane resembles an ellipse (distorted by the eccentric terms) with semi-axes α and 2α
along the x and y direction, respectively. The point oscillates in the z-direction with amplitude γ. The
angles φ and ψ define, respectively, the motion phase of the point in the projection of the orbit in the x-y
plane and the motion phase in the oscillations along the z direction. The point travels in the orbit in a
retrograde direction with period 2π (in the unperturbed case). The center of the orbit is shifted along the
x and y directions by δx (zero for the unperturbed case) and δy, respectively. Finally, the contour of the
orbit lies in a plane which intersects the x-y plane along a line. This line has an inclination of β = ψ − φ
to the x-axis.
1Although osculating elements are nothing more than the elements chosen to describe an osculating orbit, i.e., an orbit describedby neglecting its perturbations, we follow the nomenclature adopted in (Kogan, 1989) and address the elements defined infigure 3.4 as osculating elements.
36
Figure 3.4. Geometric representation of the osculating elements. O is the origin of the Cartesian refer-ence frame and C is the orbit’s center. Figure adapted from (Gil and Schwartz, 2010).
3.3. Influence of the Second Primary
We now consider the influence of the second primary in the ER3BP. We find a region of stability for
which sufficiently stable quasi-synchronous orbits can be found and we derive approximate solutions for
QSOs described by the osculating elements defined in (Kogan, 1989).
3.3.1. Region of Stability
After studying Hill’s case, we now study the stability of QSOs when the influence of the second primary
is considered, µ 6= 0. For this purpose we describe the QSO perturbed solutions as the non-perturbed
solutions (equations (3.31), (3.32), and (3.34)) plus a perturbation (variables u, v, w)
x(f) = xnp(f) + xp(f) = α(1 + e cos f) cos(f + φ) + u
y(f) = ynp(f) + yp(f) = −α(2 + e cos f) sin(f + φ) + δy + v
z(f) = znp(f) + zp(f) = γ cos (f + ψ) + w
(3.35)
We assume that α and φ are constant. The first assumption is valid as α does not have a secular
variation2 but the second assumption limits our analysis to quasi-synchronous orbits.
We want to study the stability of the perturbed problem (µ 6= 0) in the perturbation variables u, v, w.
2The secular variation of α is found to be zero in the next section.
37
Substituting the perturbed solutions (3.35) in the equations of motions (3.4) we obtain
u− 2v − 3u
1 + e cos f= − µ
1 + e cos f
(x
(x2 + y2 + z2)3/2
)= fx(u, v, w)
v + 2u = − µ
1 + e cos f
(y
(x2 + y2 + z2)3/2
)= fy(u, v, w)
w + w = − µ
1 + e cos f
(z
(x2 + y2 + z2)3/2
)= fz(u, v, w)
(3.36)
with
fx(u, v, w) = − µ
1 + e cos f
(α cos (f + φ) + αe cos f cos (f + φ) + u
((α cos (f + φ) + αe cos f cos (f + φ) + u)2 + (−2α sin (f + φ)− 12αe sin (2f + φ) + δy + v)2 + (γ cos (f + ψ) + w)2)3/2
)
fy(u, v, w) = − µ
1 + e cos f
( −2α sin (f + φ)− 12αe sin (2f + φ) + δy + v
((α cos (f + φ) + αe cos f cos (f + φ) + u)2 + (−2α sin (f + φ)− 12αe sin (2f + φ) + δy + v)2 + (γ cos (f + ψ) + w)2)3/2
)
fz(u, v, w) = − µ
1 + e cos f
(γ cos (f + ψ) + w
((α cos (f + φ) + αe cos f cos (f + φ) + u)2 + (−2α sin (f + φ)− 12αe sin (2f + φ) + δy + v)2 + (γ cos (f + ψ) + w)2)3/2
)(3.37)
In order to continue our analysis, we have to make two assumptions. First, the displacement of the
orbit in y, δy, is assumed to be much smaller than the amplitude of the orbit in the x-y plane, α, i.e.,
δy << α; second, the amplitude of the orbit in z, γ, is assumed to be much smaller than the amplitude
of the orbit in the x-y plane, α, i.e., γ << α. These assumptions limit the cases for which the solution
is valid but they will allow for a complete analysis of the motion of QSO’s. Expanding fx, fy, and fz in a
Taylor series up to the first order on the variables u, v, w, e, and γ we obtain
fx(u, v, w) ≈ µ
[−α cos θ
d3+α2(2 cos2 θ − 4 sin2 θ)
d5u− 6α2 sin θ cos θ
d5v
+3α3 cos θ(cos (θ − φ) cos2 θ + 2 cos (θ − φ) sin2 θ)
d5e
]
fy(u, v, w) ≈ µ
[2α sin θ
d3− 6α2 sin θ cos θ
d5u− α2(cos2 θ − 8 sin2 θ)
d5v
−α3 sin θ(7 cos (θ − φ) cos2 θ + 16 cos (θ − φ) sin2 θ)
d5e
]
fz(u, v, w) ≈ µ
[− 1
d3
]
(3.38)
where d is the unperturbed non-eccentric distance given by
d = α(cos2 θ + 4 sin2 θ
)1/2(3.39)
with θ = f + φ being the angle of the third body position vector with the positive x semi-axis.
The resulting differential equations have periodic coefficients which can not be solved analytically.
Floquet’s Theory (Floquet, 1883) is often used to study the stability of systems described by differential
equations with periodic coefficients. However, the analytical application of this theory to sixth-order
systems (or fourth-order since the system is independent on w and w) is not possible.
38
Nevertheless, enough insight into the stability properties of the dynamical system is provided by the
average of the periodic terms in (3.36) over their period (de Broeck, 1989). These terms have period
θ = 2π with θ measuring the angle to the positive x semi-axis; this angle measures the completion of a
revolution of the third body around the second primary, θ = f + φ. The resulting averaged differential
equations provide insight in the secular and long-term effects of the problem dynamics. The average of
a quantity a over a period T is defined as
a =1
T
∫ T
0
a(t) dt (3.40)
In (Arnold et al., 1978) one can find a good description of this approach
”We note that this principle [of averaging] is neither a theorem, an axiom, nor a definition,
but rather a physical proposition, i.e., a vaguely formulated and, strictly speaking, untrue
assertion. Such assertions are often fruitful sources of mathematical theorems.”
We will compute the averaged terms as functions of the so-called elliptic integrals. The elliptic integrals
of the first and second kind, F (θ, k) and E(θ, k), respectively, with module k and amplitude θ are defined
as (Gradshteyn and Ryzhik, 1994)
F (θ, k) =
∫ θ
0
1
∆(k)dθ, E(θ, k) =
∫ θ
0
∆(k) dθ, (3.41)
with ∆(k) = ∆ = (1 − k2 sin2 θ)1/2 and 0 ≤ k < 1. When θ = π/2, they are said to be complete elliptic
integrals of the first and second kind and are represented as K(k) and E(k), respectively. The following
properties (Gradshteyn and Ryzhik, 1994) are important in our derivation of the averaged terms
F (2π, k) = 4K(k)
E(2π, k) = 4E(k)
K(ı kk′ ) = k′K(k)
E(ı kk′ ) =1
k′E(k)
(3.42)
where k′ = (1−k2)1/2, and i is the imaginary unit. For instance, the averaged distance d can be obtained
in terms of the elliptic integrals
d =1
2π
∫ 2π
0
α(cos2 θ + 4 sin2 θ
)1/2dθ
=α
2π
∫ 2π
0
(1 + 3 sin2 θ
)1/2dθ
=2α
πE(ı√
3)
=4α
πE
(√3
2
)
=4α
πE
(3.43)
39
where α is the averaged amplitude of the orbit3. E and K are the complete elliptic integrals with k =√
3/2, and k′ = 1/2. Also note that, for symmetry reasons
∫ 2π
0
sin θ
∆3dθ = 0∫ 2π
0
cos θ
∆3dθ = 0∫ 2π
0
sin θ cos θ
∆5dθ = 0
(3.44)
which cancels some of the averaged terms. The relations in equations (3.42) are used to obtain the
values of the averaged terms. In the more complex terms, the program Mathematica was used to
compute the averaged terms.
Averaging equations (3.36) we obtain
u− 2v − 3u√1− e2
= fx(u, v, w)
v + 2u = fy(u, v, w)
w + w = fz(u, v, w)
(3.45)
with
fx(u, v, w) =µ
α3A1 u+B1 =
µ
3πα3(4E −K)u+
5µ e
36πα2(10E −K) cos φ
fy(u, v, w) =µ
α3A2 v +B2 =
µ
3πα3(K − E)v +
7µ e
9πα2(10K − 7E) sin φ
fz(u, v, w) = − µ
α3A3 w = − µ
πα3E w
, A1, A2, A3, B1, B2 > 0
(3.46)
Again, the system is independent on its third coordinate, w, which forms a stable second-order system
(it has purely imaginary eigenvalues). This way, the study of the stability of this system is reduced to the
study of the fourth-order system
d
df
u
v
u
v
=
0 0 1 0
0 0 0 1
3√1−e2 +
µ
α3A1 0 0 2
0µ
α3A2 −2 0
u
v
u
v
(3.47)
with characteristic polynomial
P (λ) = λ4 +
(4− 3√
1− e2− µ
α3(A1 +A2)
)λ2 +
µ2
α6A1A2 +
3√1− e2
µ
α3A2 (3.48)
3α does not depend on the independent variable f because ˙α = 0. This is proved in the next section.
40
which can be reduced to a quadratic polynomial by the variable change κ = λ2
P (κ) = aκ2 + bκ+ c = κ2 +
(4− 3√
1− e2− µ
α3(A1 +A2)
)κ+
µ2
α6A1A2 +
3√1− e2
µ
α3A2 (3.49)
For the system (3.47) to be stable we need the four roots of the characteristic polynomial in (3.48) to
have non-positive real part which can only be achieved if all of the four roots are pure imaginaries. From
the variable change, this translates in having two real negative roots in (3.49). These roots are given by
κ1,2 =−b±
√b2 − 4c
2(3.50)
with a = 1. The roots of (3.48) are real negative for
κ1 + κ2 < 0
κ1κ2 > 0
b2 − 4c > 0
⇒
b > 0
c > 0
b2 − 4c > 0
⇒
4− 3√1− e2
− µ
α3(A1 +A2) > 0
µ2
α6A1A2 +
3√1− e2
µ
α3A2 > 0(
4− 3√1− e2
− µ
α3(A1 +A2)
)2
− 4
(µ2
α6A1A2 +
3√1− e2
µ
α3A2
)> 0
(3.51)
Notice that the second condition is always satisfied since A1, A2 > 0. The conditions are then reduced
to two4 α >
(µ(A1 +A2)
4− 3√1−e2
)1/3
(4− 3√
1− e2− µ
α3(A1 +A2)
)2
− 4
(µ2
α6A1A2 +
3√1− e2
µ
α3A2
)> 0
(3.52)
For given values of e and µ a minimum value of the amplitude for which it is possible to find sufficiently
stable orbits is defined by conditions (3.52). The maximum limit is defined by the validity of the approxi-
mation x, y, z << 1 which can be studied numerically. Remember that this region of stability is valid for
quasi-synchronous orbits. It is possible to find sufficiently stable orbits below this limit with a different
periodicity (Wiesel, 1993). These, however, are not the main focus of our work.
3.3.2. Approximate Solutions in the Osculating Elements
We now derive approximate solutions for QSOs in the ER3BP. The method of variation of the arbitrary
constants is used to obtain the approximate solutions in the osculating elements.
We assume that the osculating elements in the unperturbed solutions (equations (3.31), (3.32), and
(3.34)) vary in the perturbed case. The differential equations that define the variation of the osculating
elements can be derived with the method of variation of arbitrary constants, or, the method of variation
of parameters (VOP). In many modern textbooks the VOP is addressed as a method to solve inhomoge-
neous linear differential equations which does not illustrate the generality of this approach (Efroimsky,
2002).
4The second condition in (3.52) generates a large expression which does not provide any additional insight. The expression issolved numerically in the application to the Mars-Phobos system.
41
The solutions for Hill’s unperturbed case are
x = g1(c, f) = α(1 + e cos f) cos (f + φ) + δx x = h1(c, f) = −α(sin (f + φ) + e sin (2f + φ))
y = g2(c, f) = −α(2 + e cos f) sin (f + φ) + δy y = h2(c, f) = −α(2 cos (f + φ) + e cos (2f + φ))
z = g3(c, f) = γ cos (f + ψ) z = h3(c, f) = −γ sin (f + ψ)
(3.53)
or, in short form
r = g(c, f) r = h(c, f) (3.54)
with the function hi being the derivatives in respect to the independent variable f
h =
(∂g
∂f
)c=const.
(3.55)
and ci the osculating elements
c =
αφδxδyγψ
(3.56)
with δx = 0 for the unperturbed case.
The method of variation of the arbitrary constants used to derive the differential equations that describe
the variation of c can be found in (Danby, 1962). The equations of motion (3.4) can be written as
xi = fi(x, f) + pi(x, f) (3.57)
with |f | >> |p|. Let the unperturbed system
xi = fi(x, f) (3.58)
have general solution
xi = Xi(c1, c2, . . . , cn, f) = Xi(c, f) (3.59)
with derivative
xi =∂Xi
∂f+
n∑j=1
∂Xi
∂cjcj (3.60)
Substituting in equation (3.57) we get
∂Xi
∂f+
n∑j=1
∂Xi
∂cjcj = fi(xi, f) + pi(xi, f) (3.61)
but we know that Xi is the general solution of the unperturbed system defined in (3.58), i.e., Xi =
42
fi(xi, f). This way, we haven∑j=1
∂Xi
∂cjcj = pi(xi, f) (3.62)
The variations of the osculating elements in equation (3.62) can be obtained in matrix form (Schaub
and Junkins, 2003)
c = [L]−1
[∂R
∂c
]T(3.63)
where [L] is the anti-symmetric matrix defined by the Lagrangian brackets
Lij =
(∂g
∂ci
)T (∂h
∂cj
)−(∂h
∂ci
)T (∂g
∂cj
)(3.64)
with Lij = −Lji, Lii = 0, and g and h defined by (3.53). R is the perturbation function written as a
function of the osculating elements
R =1
1 + e cos f
µ
(g21 + g2
2 + g23)1/2
(3.65)
This way, the differential equations on ci are obtained
α = µα[3 + 2e cos f − cos (2(f + φ))] e sin f + 2 [2 sin (f + φ) + e sin (2f + φ)] δxα + 2 [cos (f + φ) + e cos (2f + φ)]
δyα
2r32(1 + e cos f)(2 + e2 + 3e cos f)
φ = µ4 + e [6 cos f + sin f sin (2(f + φ)) + e+ e cos (2f)] + 2 [2 cos (f + φ) + e cos (2f + φ)] δxα + 2 [sin (f + φ)− e sin (2f + φ)]
δyα
2r32(1 + e cos f)(2 + e2 + 3e cos f)
δx = µα(1 + e cos f)(2 sin (fφ) + e sinφ)− e sin f δxα − (1 + e cos f)
δyα
r32(2 + e2 + 3e cos f)
δy = µα(2 + e cos f)(cos (f + φ) + e cosφ) + (2 + e cos f) δxα − e sin f
δyα
r32(1 + e cos f)(2 + e2 + 3e cos f)
γ = µγcos (f + ψ) sin (f + ψ)
r32(1 + e cos f)
ψ = µcos2 (f + ψ)
r32(1 + e cos f)
(3.66)
These equations are too complex to conclude on the stability of the system. Nevertheless, if we
assume e << 1, and the amplitude of the motion in the z coordinate and the displacements of the
’ellipse’ origin, δx and δy, to be much smaller than the amplitude of the motion in the x-y plane, i.e.,
δx/α, δy/α, γ/α << 1, we can expand equations (3.66) in a Taylor series up to the first order about zero
43
on these quantities. By doing so we obtain the approximated differential equations
α = µ
[3 sin f − cos (2(f + φ)) sin f
4d3α+
sin (f + φ)
d3δx +
cos (f + φ)
2d3δy
]
φ = µ
[1
d3+
sin f sin (2(f + φ))− 4 cos f
4d3e− 3
cos f cos2 (f + φ) + 2 cos f sin2 (f + φ)
d5eα2
+cos (f + φ)(3 sin2 (f + φ)− 2)
d5αδx −
sin (f + φ)(3 sin2 (f + φ)− 11)
d5αδy
]
δx = µ
[sin (f + φ)
d3α− cos (f + φ) sin f
2d3eα− 3 sin (f + φ)(cos f cos2 (f + φ) + 2 cos f sin2 (f + φ))
d5eα3
−3 cos (f + φ) sin (f + φ)
d5α2δx +
9 sin2 (f + φ)− 1
2d5α2δy
]
δy = µ
[2 cos(f + φ)
d3α
cosφ− 3 cos (2f + φ)
2d3eα− 6 cos (f + φ)(cos f cos2 (f + φ) + 2 cos f sin2 (f + φ))
d5eα3
+4(2− 3 cos2 (f + φ))
d5α2δx +
12 cos (f + φ) sin (f + φ)
d5α2δy
]
γ = µcos (f + ψ) sin (f + ψ)
d3γ
ψ = µ
[cos2 (f + ψ)
d3− cos2 (f + ψ)(4 cos f cos2 (f + φ) + 10 cos f sin2 (f + φ))
d5eα2
−3cos (f + φ) cos2 (f + ψ)
d5αδx + 6
sin (f + φ) cos2 (f + ψ)
d5αδy
](3.67)
with d as defined by (3.39).
We can not solve equations (3.67) analytically and these are too complex for any interpretation. How-
ever, we can average their periodic coefficients which provides enough insight to assess the stability of
the system. This way, we get the averaged differential equations on the osculating elements
˙α = 0 ˙φ =µ
πα3E
˙δx =µ
6πα3(K − E)δy −
µ
9πα2e(5K − 8E) sin φ ˙δy = − 2µ
3πα3(4E −K)δx −
µ
9πα2e(70E − 19K) cos φ
˙γ = − µ
6πα3γ(5E − 2K) sin (2β) ˙ψ =
µ
6πα3(3E + (5E − 2K) cos (2β))
˙β = ˙ψ − ˙φ =µ
6πα3(−3E + (5E − 2K) cos (2β))
(3.68)
where K and E are complete elliptic integrals of module k =√
3/2. The differential equation on φ is
independent from the other osculating elements and its solution is easily derived
φ =µ
πα3E f + φ0 (3.69)
44
After substitution of φ, the system composed by δx and δy becomes separable from the other variables
and can be represented as a second-order system
d
df
[δxδy
]=
[0 a−b 0
][δxδy
]+
[−c sin φ−d cos φ
], a, b, c, d > 0 (3.70)
The solution of a non-homogeneous system of differential equations with constants coefficients, x =
Ax + u(f), can be written as (Russell, 2007)
x(f) = M(f)M(f0)−1x0 +
∫ f
f0
M(f)M(s)−1u(s) ds (3.71)
where, for a second-order system
M(f) =[v1e
λ1f v2eλ2f]
(3.72)
with λ1,2 and v1,2 being the eigenvalues and eigenvectors, respectively, of matrix A, and x0 the system’s
initial conditions. The solution of the system is of the form
δx =
(δx0
+˙φc+ ad
ab− ˙φ2cos φ0
)cos (√abf) +
√a
b
(δy0 −
bc+ ˙φd
ab− ˙φ2sin φ0
)sin (√abf)
−˙φc+ ad
ab− ˙φ2cos φ
δy =
(δy0 −
bc+ ˙φd
ab− ˙φ2sin φ0
)cos (√abf)−
√b
a
(δx0
+˙φc+ ad
ab− ˙φ2cos φ0
)sin (√abf)
+bc+ ˙φd
ab− ˙φ2sin φ
(3.73)
where c and d are the only factors depending on the orbital eccentricity of the second primary. The
system (generalized in (3.70)) has characteristic polynomial
P (λ) = λ2 + a b, a, b > 0 (3.74)
and, as the eigenvalues λ1,2 are a pure imaginary conjugate pair, the system is stable. Solutions (3.73)
are composed by two distinct periodic motions. The first has an amplitude composed by both terms
that depend and not depend on the orbital eccentricity. This motion has frequency ω1 =√a b (from the
argument of the trigonometric functions) and period
P1 =2π√a b
=6π2
(5EK − 4E2 −K2)1/2
α3
µ≈ 37.1483
α3
µ(3.75)
the other distinguishable motion has an amplitude that depends on the orbital eccentricity of the second
primary and has the same frequency and period as φ — ω2 = ˙φ and
P2 =2π˙φ
=2π2
E
α3
µ≈ 16.2992
α3
µ(3.76)
45
The system (3.68) is also independent on the variable β and its solution is
β = − arctan
(√m− nm+ n
tan(√
m2 − n2(f + Cβ)))
(3.77)
with m = 3µE/(6πα3) and n = 3µ(5E − 2K)/(6πα3). The frequency and period of this oscillation are
obtained in the same fashion, ωβ =√m2 − n2 and
Pβ =2π√
m2 − n2=
6π2
(5EK − 4E2 −K2)1/2
α3
µ= P1 ≈ 37.1483
α3
µ(3.78)
After substitution of β the solution on γ is obtained
γ = Cγ(m− n cos (2β)
)−1/2 (3.79)
which oscillates two times faster than β and, thus, Pγ = Pβ/2.
The relation between the orbital mean motions of the QSO, nQSO, and of the second primary, n, in their
orbits around the first primary can also be obtained. The QSO has orbital mean motion nQSO = 1 + ˙φ
and n is 1.nQSOn
= 1 + ˙φ = 1 +µE
πα3≈ 1 + 0.385491
µ
α3(3.80)
The second primary, located on the origin of the reference frame, acts as a restoring force on the
the third-body. This force varies with the inverse of the distance between these two bodies. If the third-
body gets too far from the origin, there is a chance that this restoring force will not be strong enough to
maintain the third-body in orbit around the second primary.
The inclination of the QSO influences the distance of the third body to the second primary as the
motions in the z direction and on the x-y plane are independent. Consequently, the inclination of the
QSO also influences the restoring capability of the second primary. A critical value for the ratio γ/α, for
which this restoring capability vanishes, can be derived.
The motion of the third-body in the x-y plane is nearly elliptic and the distance to the second primary
is maximum when it crosses the y-axis in y = ±2α ± δy. The motion in the z-direction has also to
be considered. The maximum distance between the second and third bodies is achieved when the
third-body achieves the maximum height (in absolute value) z = ±γ in the same point that achieves the
maximum distance to the second primary in the x-y plane, y = ±2α±δy. This is the worst-case scenario
for the analysis of the restoring capability of the second primary with the inclination of the QSO and it is
defined by an angle β = ±π/2 (angle between the intersection line of the QSO plane with the primaries’
orbital plane and the positive x semi-axis).
If we give up on the assumption that the quantity q = γ/α is small, the system composed by δx and δy
46
in (3.70) maintains the same form but now with
a =4Kq − (q4 + 3q2 + 4)Eqπ(q2 + 3)(q2 + 4)3/2
µ
α3
b =4(q2 + 4)Eq − 4Kq
π(q2 + 3)(q2 + 4)1/2
µ
α3
(3.81)
where the elliptic integrals Kq and Eq have module k =√
(q2 + 3)/(q2 + 4). The period of the main
motion in this case is
P =2π√ab
=2π2(q2 + 3)(q2 + 4)
[(4Kq − (q4 + 3q2 + 4)Eq)(4(q2 + 4)Eq − 4Kq)]1/2
α3
µ(3.82)
We now want to compute the critical ratio qc for which separation occurs. Analytically, the period is
infinite for ejected orbits, hence, the quantity qc can be computed numerically by finding the root of the
denominator in (3.82)
P |q=qc =∞→[(4Kq − (q4
c + 3q2c + 4)Eq)(4(q2
c + 4)Eq − 4Kq)]1/2
= 0→ qc = 0.961073 (3.83)
This conclusion is based on the averaged equations of motion where the peaks of periodic effects are
neglected. Thus, it is expected that separation occurs before the value found for qc. The computed value
is merely a statement that QSOs with q > qc will suffer orbit separation. The numerical exploration of the
problem will tell us how accurate is this prediction.
3.4. Application to the Mars-Phobos System
We now apply the results to the case of a QSO around Phobos. All the parameters computed here
are for the case of initial true anomaly f0 = 0. Recall the numerical values of the relevant parameters:
m1 = 6.4185× 1023 kg
m2 = 1.06× 1016 kg
µ =m2
m1 +m2= 1.65148× 10−8
a = 9377.2 km
e = 0.0151
P = 0.31891 days = 27553.8 s
n =2π
P= 2.288933× 10−4 rad/s
(3.84)
and the relation between the normalized amplitude, α, and the dimensional amplitude α∗
α =α∗(1 + e cos f)
a(1− e2)(3.85)
47
which for f0 = 0 results in α∗ = αa(1 − e). We omit, for the sake of simplicity, the superscript ∗ as all
values computed in this section are dimensional. This way, the stability conditions for quasi-synchronous
orbits are, from equations (3.52) α > 17.3774 km
α > 29.4262 km(3.86)
with the second condition prevailing over the first.
The periods of the osculating elements are, as a function of the dimensional amplitude (in kilometers)
P1 = Pβ = 37.1483
(α3
µa3(1− e)3n
)= 13.1961α3
P2 = 16.2992
(α3
µa3(1− e)3n
)= 5.49413α3
Pγ =37.1483
2
(α3
µa3(1− e)3n
)= 6.59806α3
(3.87)
The relation between the orbital mean motions of the QSO and Phobos’ can also be studied through
equation (3.80). In figure 3.5 it is presented how this ratio changes with the amplitude. Moreover, in table
3.1 the amplitudes of QSOs with ratios represented by two consecutive small integers are presented.
From (Wiesel, 1993), it is known that sufficiently stable resonant orbits exist for these values of the ratio
nQSO/n. Note that these non-synchronous orbits are not restricted by the conditions (3.86).
Figure 3.5. The change of the ratio nQSO/n with the amplitude is plotted. The values for which it isexpected to find resonances 1:1, 2:1, 3:2, and 4:3 are marked.
nQSO/n Amplitude [km]
2:1 17.117
3:2 21.5661
4:3 24.687
Table 3.1. Amplitudes for values of the ratio nQSO/n represented by a fraction of two small consecutiveintegers.
48
All truths are easy to understand once they are dis-
covered; the point is to discover them.
(Galileo Galilei 1564 - 1642)
CH
AP
TE
R
4 Numerical Exploration of QSOs
In this chapter we integrate numerically the equations of motion and the variational equations to obtain
the evolution of both the orbit and the deviation vector. The solutions obtained numerically are the basis
for the study of the stability of the system with the FLI maps.
4.1. Numerical Integration
Numerical integration is the numerical analysis of the solutions of ordinary differential equations
(ODEs). This approach is particularly useful when the analytical solution of the set of ordinary dif-
ferential equations is not possible, too complex, or just too time-consuming. As a numerical method, the
technique fails to provide the same insight into a problem behavior as an analytical approach. Further-
more, it incorporates in the computed solution an error that depends on the specific numerical integration
method used as well as on the computational parameters such as the time step.
Runge-Kutta methods are a commonly used class of numerical integrators; they are regarded as the
horsepower of celestial mechanics. In this work we are going to implement and test two variants of
the Range-Kutta method: the commonly used explicit fourth-order Runge-Kutta, RK4, with four function
evaluations, and an explicit eight-order Runge-Kutta method with thirteen function evaluations, adapted
from (Prince and Dormand, 1981). This paper defines an adaptive method with RK7 and RK8 (seventh
and eighth-order Runge-Kutta), however, we need the step size h to remain constant for the evaluation
of the deviation vector evolution in the computation of the FLI value. This way, only the higher-order
method is adapted to our problem.
4.2. Implementation Validation
The integration methods RK4 and RK8 were first implemented in the programming language MatLab.
However, the program proved to be fairly slow as the CPU times required for the computation of the FLI
Maps were unaffordable. Consequently, the methods were implemented in C, a language that delivers
faster CPU times when compared to MatLab. In this section, we validate both implementations with
49
the exact solution of a Keplerian orbit, i.e., a satellite orbiting a body of gravitational parameter µ (do not
confuse with the mass parameter of the 3BP). After the validation of both implementations, a comparison
between the performance of the RK8 method, implemented in MatLab and C, is also presented.
4.2.1. Validation Test
The implementation of both Runge-Kutta methods, RK4 and RK8, are validated in C. For this purpose,
we test a Keplerian orbit in the two-body problem. The equations of motion are
d
dt
x
y
z
x
y
z
=
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
− µr3x 0 0 0 0 0
0 − µr3y 0 0 0 0
0 0 − µr3z 0 0 0
x
y
z
x
y
z
(4.1)
with initial conditions
X0 =
rp0
0
0
Vp0
(4.2)
and the parameters for an elliptic orbit
µ = GMM P =2π
nn =
µ
a3Vp =
õ
(2
rp− 1
a
)Va =
õ
(2
ra− 1
a
)(4.3)
with rp = a(1−e), ra = a(1+e), and e = 0.5. Mars is used as the attracting body and the semi-major axis
a is the semi-major axis of Phobos’ orbit. The equations are integrated over 100.5 periods with initial
conditions x0 = rp and y0 = Vp. It is expected that the satellite meet the final conditions x0 = −ra and
y0 = −Va. The final positions and velocities normalized with the apogee radius and velocity, respectively,
are presented for both methods and for different time-steps in table 4.1.
The results for both methods strongly suggest that the error decreases and the CPU time increases
with the decrease of the size of the time-step. Furthermore, the implementations of both methods are
validated as, with the right step-sizes, the results meet the final state of the system, known beforehand.
4.2.2. Implementation Language
The implementations of the RK8 method in MatLab and C are tested and compared. The implemen-
tation in C is found to be much faster in the computation of the FLI. As an interpreting language MatLab
does not possess the same computational speed as the more basic language C.
The comparison of the two implementations is carried out with a sample orbit in our problem. The
50
RK4 RK8
Step P/20 P/200 P/500 P/20 P/200 P/500
xf [ra] 160.665957 -0.999564 -0.999999 -0.814849 -1.000000 -1.000000
yf [ra] -507.420687 0.017827 -0.000196 -0.395417 0.000000 0.000000
zf [ra] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
xf [Va] 0.704791 0.035159 0.000379 0.874789 0.000000 0.000000
yf [Va] -2.222066 -0.999795 -1.000001 -0.802449 -1.000000 -1.000000
zf [Va] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
CPU [s] 0.000 0.05 0.11 0.02 0.22 0.55
Table 4.1. Validation test results for a Keplerian orbit using RK4 and RK8 with different time-steps.
state of the system, xf , the fundamental matrix of the deviation vector evolution at the final time,
Y(ff ) = ∇xΦff (defined in (2.13)), the FLI, and the computational times are compared at the end
of the integration period of 100 revolutions of Phobos about Mars. These results were obtained with a
small time-step of π/125 and are presented in tables 4.2 and 4.3.
MatLab C
xf [km] 24.8497 24.8496
yf [km] 152.1574 152.1578
zf [km] 0.0000 0.0000
xf [km/s] -0.0157 -0.0157
yf [km/s] -0.0057 -0.0057
zf [km/s] 0.0000 0.0000
dmin [km] 74.6405 74.6138
dmax [km] 249.3773 249.3668
FLI 0.1213 0.1237
CPU [s] 86.7898 4.64
Table 4.2. Comparison of the final position, FLI, minimum and maximum distance and computational timeof a sample orbit with y0 = −100 km and x0 = −20 m/s with MatLab and C implementationsof RK8.
The comparison of the results obtained using the two different implementations indicate that there is
a slight difference on the values of the fundamental matrix of the evolution of the deviation vector that
results in a small difference in the FLI value. The final result of the FLI is not important, only its behavior
for different orbits, i.e., its capacity to distinguish chaosticity. Further analysis has to be performed to
assess if both implementations demonstrate the same capacity to characterize the stability of orbits
despite the difference in the absolute value of the FLI.
This assessment is performed by analyzing the FLI values for different orbits along the y-axis from
-20 km to -300 for both implementations. The results are shown on figure 4.1 where a time-step of
51
Language Y(ff ) [×1059]
MatLab
0.3835 0.0104 0 0.0387 0.3597 0
-3.5181 -0.0955 0 -0.3551 -3.2998 0
0 0 0.0000 0 0 0.0000
3.6375 0.0987 0 0.3671 3.4117 0
-0.2385 -0.0065 0 -0.0241 -0.2237 0
0 0 0.0000 0 0 0.000
C
0.3631 0.0121 0 0.0387 0.3402 0
-3.4673 -0.1156 0 -0.3695 -3.2491 0
0 0 0.0000 0 0 0.0000
3.5976 0.1199 0 0.3833 3.3711 0
-0.2169 -0.0072 0 -0.0231 -0.2033 0
0 0 0.0000 0 0 0.000
Table 4.3. Comparison of the final fundamental matrix of the evolution of the deviation vector of a sampleorbit with y0 = −100 km and x0 = −20 m/s with MatLab and C implementations of RK8.
h = π/125 is used. Orbits that escape from Phobos’ vicinity (more than 1000 km) are assigned a value
between 1 and 2, depending on how long they are able to stay near the moon. The reasoning behind
this assignment is discussed later.
(a) Full View (b) Zoom-In
Figure 4.1. FLI Behavior computed by the method implemented in MatLab and in C.
Although there is a small difference in the absolute value of the FLI, the results presented in figure
4.1 suggest that the FLI keeps its behavior in both implementations. There is a small bump in the
FLI behavior in the C implementation but this can be neglected since it does not affect its capability to
identify the chaotic nature of orbits. It is important to refer that the data for this graph took more than 118
minutes to compute in the MatLab implementation against less than 2 minutes in the C implementations
illustrating the reason why we use the latter.
52
4.3. Computational Parameters & Methodology
We now choose the method to use, RK4 or RK8, the time-step, and discuss how the FLI value for
orbits that escape Phobos (that do not meet the stability requirements defined in chapter 1) is defined.
The goal is to choose a method with a proper time-step that allows the smallest CPU times possible
meeting defined performance thresholds.
4.3.1. Integration Method and Time-Step
In order to analyze the performance of both methods for different time-steps we set reference values
(table 4.4) computed with the RK8 method with a very small step-size to serve as a basis of comparison.
There is not a reference value for the FLI since its value depends on the time-step size and, hence, it is
not possible to compare it for different time-step sizes.
Method RK8
Time-Step π/2000
y0 -100 km
x0 -20 m/s
xf 24.849701 km
yf 152.157434 km
zf 0.0 km
xf -15.745171 m/s
yf -5.729053 m/s
zf 0.0 m/s
dmin 74.639542 km
dmax 249.380685 km
Table 4.4. Parameters computed for an orbit with initial conditions y0 = −100 km and x0 = −20 m/s witha time step of π/2000.
The analysis of the performance of both methods, and its variation with the time-step size, is carried
out by analyzing the norms of the error vectors and the differences in the minimum and maximum
distances. Th error vector is defined as the difference vector between the reference vectors and the
obtained vectors for each step-size. The FLI values and the CPU times for different time-steps are also
computed for both methods. The results are presented in figures 4.2.
The results suggest that the RK8 outperforms the RK4 in every parameter with the exception of the
CPU time. However, the RK8 method converges faster so we define performance thresholds to assess
which of the methods has smaller CPU times meeting these thresholds.
Note that the analysis of the error in the computation of the minimum and maximum distances is a little
more complex — with lesser steps, less points are analyzed and there is a higher probability of missing
the points where the minimum and maximum distances are achieved by a fair distance. The result
53
(a) Position Error (b) Velocity Error
(c) Minimum Distance Error (d) Maximum Distance Error
(e) FLI (f) CPU Time
Figure 4.2. Performance comparison between RK4 and RK8 for different time-steps. The errors arecomputed in relation to the reference values previously computed. Note that the performancevalues are evaluated as functions of the number of integration steps per period 2π and notdirectly as functions of the time-step.
is the observed periodicities of these errors but these can not be minimized since different orbits will
present different periodicities depending on the location of the points where the minimum and maximum
distances to the second primary are achieved. These parameters are not critical as we only use them to
assess if an orbit collides against the moon (d < 15 km) or if it escapes from the QSO (d > 1000) km and
not to perform any analysis on its stability. Nevertheless, these errors are fairly small and the minimum
and maximum distance achieved can be used for orbit design purposes.
We define the thresholds as a maximum position error of 1 m and a maximum velocity error of 0.001
m/s. Figures 4.2(a) and 4.2(b) present a plot of the error in these parameters with the number of steps
per period for both methods. The RK4 needs almost 200 steps per period to meet the same performance
than the RK8 with 20 steps per period. Figure 4.2(f) suggests that for these numbers of steps per period
54
the RK8 method presents smaller CPU times and, consequently, it is chosen as the integration method
to be used for the computation of the FLI maps.
(a) Position Error (Zoom-In) (b) Velocity Error (Zoom-In)
Figure 4.3. The zoom-in of both the position and velocity errors allows us to choose a proper step-size tomeet the defined thresholds.
Figure 4.3 presents a zoom-in of the errors in the position and velocity for a more careful choice of the
number of steps per period. The results suggest that 16 steps per period is enough for the RK8 to meet
the defined thresholds. This way, we use a time-step h = π/8. In table 4.5 the used reference values
and the computed values with the selected time-step are presented. Remember that the FLI values are
not comparable when computed with different time-steps.
Method RK8
Time-Step π/2000 π/8
y0 [km] -100
x0 [m/s] -20
xf [km] 24.849701 24.849769
yf [km] 152.157434 152.157531
zf [km] 0.0 0.0
xf [m/s] -15.745171 -15.745201
yf [m/s] -5.729053 -5.729068
zf [m/s] 0.0 0.0
dmin [km] 74.639542 74.658947
dmax [km] 249.380685 248.344936
FLI 0.007763 1.447331
CPU Time [s] 73.57 0.29
Table 4.5. Parameter computation with chosen time-step and respective comparison with the referencevalues.
55
4.3.2. Orbit Escape
The two objectives of the numerical exploration are to obtain the solution of orbits and to assess which
of these are sufficiently stable by the study of the deviation vector.
A probe that is ejected from orbit is not necessarily unstable (from the point of view of the concept
of exponential divergence, measured by the FLI). In fact, when a probe escapes from Phobos’ vicinity
it can enter in a Keplerian orbit around Mars which is more stable than a QSO. In this sense, the FLI is
not suited to distinguish orbits that escape from Phobos’ vicinity and are not stable following our criteria
defined in chapter 1.
With the purpose of separate the ejected orbits from the remainder, we will define the FLI of an orbit
that escaped Phobos in our algorithm as
FLIesc = 20− 10fescff
(4.4)
where fesc is the normalized time when the third body escaped from Phobos and ff is the total time
of integration, i.e., 100 revolutions of Phobos around Mars. This way, the range of FLI values 10–20 is
reserved for the assessment of ejected orbits and in this case the FLI measures the time that a probe
remained in Phobos’ vicinity without colliding against it. This provides the information if an orbit can be
used for a smaller timescale or if it escapes or crashed Phobos too soon to be used for any mission
purposes.
4.4. FLI Maps
The representation of the values of the FLI over a set of initial conditions generates a FLI map (or
stability map) where regions filled with sufficiently stable orbits can be identified to provide insight into
the characteristics of these regions.
We study the problem of the ER3BP which is defined by 6 parameters plus the origin of the normalized
time f0. The FLI maps represent the FLI value as a function of two varying parameters, hence, the set of
initial conditions needs to have 5 fixed initial parameters. A complete study of the stability of the system
is not possible but it is possible to study the stability in a plane of two varying parameters. This approach
provides enough insight into the stability properties of the system to help in the choice of an entry point,
i.e., the point at which a probe is maneuvered to enter in a sufficiently stable QSO.
Following (Villac and Aiello, 2005), we introduce the modified Jacobi integral1 defined in 2.5.2
C = 2Ω′
1 + e cos f− V 2 (4.5)
where V = (x2 + y2 + z2)1/2 is the spacecraft velocity and Ω′ is the amended potential
Ω′ =1
2((x+ µ− 1)2 + y2 − e z2 cos f) +
1− µr1
+µ
r2+
1
2µ(1− µ) (4.6)
1Strictly speaking, this modification of the Jacobi integral is not an integral of motion in the elliptic case but, as it possesses aninvariant relation (which is constant for a constant value of f ), we call it modified Jacobi integral for the sake of simplicity.
56
with r1 and r2 being the distance of the spacecraft to Mars and Phobos, respectively. The greatest
influence in the amended potential Ω′ is Mars gravitational potential and this is almost constant in the
vicinity of Phobos which causes the modified Jacobi integral variation to be predominated by the variation
of the spacecraft’s velocity and by the variation of the true anomaly f . The characterization of FLI maps
as a function of the modified Jacobi integral provides insight into the sensibility of the stability regions to
the initial conditions.
Orbits are integrated over 100 periods of Phobos around Mars and the FLI value is computed in
respect to this time-span. The dark regions of the FLI maps correspond to regions of mostly regular
motion whereas the near white regions correspond to orbits that escaped or collided in the moon. The
orange regions correspond to orbits that were able to remain stable for some time before escaping or
crashing the moon.
We study the particular case of quasi-synchronous orbits. Resonant orbits are not considered since
in our model of the dynamics we did not considered the perturbations that lock QSOs at resonances
represented by two small whole integers, namely, Phobos’ irregular shape and rotation. Nevertheless,
the relation between the mean motions of the QSO and Phobos is addressed.
4.4.1. Planar QSOs
We begin with the simpler case of planar QSOs before extending the study to three-dimensional orbits.
x0 Vs. y0
Let us begin with the stability analysis of planar QSOs with the variation of x0 and y0. This analysis
is performed for the case of initial true anomaly f0 = 0, i.e., Phobos’ perigee passage. The other initial
parameters y0, z0, x0, z0 are set to zero.
The orbital amplitude in the x-y plane, α, is best measured in the passage of the orbit by y0 = 0
because experimentation suggests that the displacement of the QSO in the x direction is much smaller
than in the y direction, i.e., δx << δy. The analysis of the QSO stability with the distance to the moon
on the x-axis provides a much better estimation of the value of the orbital amplitude α than the same
analysis when carried out in the y-axis.
The results presented in figure 4.4 suggest an almost linear relation between x0 and y0 for sufficiently
stable QSOs. These FLI maps also suggest that amplitudes as large as 400 km are possible although
the stability in y0 seems to decrease with the increase of the amplitude α ≈ x0.
The zoom-in of the FLI map in 4.4(b) suggests that the minimum amplitude for sufficiently stable
quasi-synchronous orbits is in the interval 35–40 km, a value not far from the value of 29.43 km that we
estimated in the analytical approach of our problem.
y0 Vs. x0
We now analyze the stability of QSOs with the variation of y0 and x0. The initial true anomaly and the
other initial parameters are set to zero. We are interested in the comparison with the previous case to
57
(a) Full View
(b) (Zoom-In)
Figure 4.4. FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions negative x0 andpositive y0. The remainder of the initial parameters are set to zero.
assess which of the two cases presents as a better candidate for the transfer to a QSO or for the escape
from a QSO.
The FLI map on the analyzed parameters y0 and x0 (fig. 4.5) suggests that the stability region in this
case is larger (regarding the analyzed parameters) than the previous case where x0 and y0 were varied
and, thus, the QSO stability is less sensitive to the variation of the initial velocity when maintaining the
original direction. For now, no conclusions can be made regarding the sensitivity of the QSO stability to
changes in the direction of the velocity.
Note that the maximum distance for which sufficiently stable QSOs are found is, roughly, 270 km, a
value that is not in accordance with the value obtained in the previous analysis since, from our analytical
approach, it is predicted that a QSO reaches larger distances in the y-axis than in the x-axis. The
previous case found sufficiently stable at larger distance in the x-axis. A possible explanation is that
large amplitude QSOs require a velocity in the y-axis to remain stable. A hypothesis that is analyzed
later with the variation of this velocity.
The light-color areas in the FLI map are due to orbits that escaped or collided against Phobos. Chaotic
58
Figure 4.5. FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions negative y0 andnegative x0. The remainder of the initial parameters are set to zero.
regions are known to coexist with stability regions near the libration points (Villac and Aiello, 2005) but
these chaotic regions were not found with the resolution used for the computation of the FLI maps.
However, we are not interested in the study of the stability of the region near the libration points, just
above Phobos’ surface, due to the restrictions of our dynamics model which neglected perturbations
which are considerable at such close distances.
(a) x0 = −20 m/s (b) x0 = −50 m/s
Figure 4.6. Trajectories of two planar QSOs with entry point at y0 = −100 km and initial tangential velocityx0 = −20 m/s and x0 = −50 m/s.
From the FLI map in figure 4.5, two sample QSOs are presented in figure 4.6 with initial position
y0 = −100 km and with initial tangential velocity x0 = −20 m/s and x0 = −50 m/s, respectively. Note
that the maximum distance of the QSO change with the initial velocity. The sample QSOs suggest that a
larger initial velocity at the same initial distance results in a QSO with a larger amplitude α and a larger
variation of the displacement of its origin δy.
59
Relation Between the Mean Motion of the QSO and Phobos
The dynamics model developed in our work does not take into account the oblateness of Mars, the
solar radiation pressure, the irregular shape of Phobos or the rotation of the moon. One of the conse-
quences of neglecting these perturbations is the nonexistence of resonant orbits, i.e., the orbits do not
lock in the resonances 2:1, 3:2, 4:3 — ratio of the QSO’s with Phobos’ mean motion — as studied in
(Wiesel, 1993). These type of QSOs are particularly interesting for their proximity to the moon.
Nevertheless, the distances at which orbits with a relation between its mean motion and the moon’s
mean motion formed by two small integers could be found as discussed in section 3.3.2. We present in
table 4.6 the distances found for orbits with these relations of the mean motion found analytically as well
as the ones found numerically with our model. The obtained values are in accordance with some of the
resonant orbits found in (Wiesel, 1993) although there is not enough data to assess if these intervals
change in a more realistic model. An example of a QSO with a mean orbital motion that is twice the
mean orbital motion of Phobos is presented in 4.7.
ωQSO/ωPhNumerical Results Analytical Results
x0 [km] y0 [m/s] Amplitude [km]
2:1 [−20,−17] [11.7, 11.9] 17.117
3:2 [−26,−24] [13.4, 13.7] 21.5661
4:3 [−31,−29] [15.0, 15.3] 24.687
Table 4.6. Comparison between the prediction for the resonant orbits amplitude and the intervals wherethey where found by numerical exploration.
Figure 4.7. Trajectory of a QSO with a mean orbital motion that is twice the mean orbital motion of Phoboswith initial conditions y0 = −22.1 km and x0 = −8.6 m/s.
y0 Vs. y0
We now study the stability of QSOs with the variation of the parameters y0 and y0. This analysis is
performed in Phobos’ perigee passage, f0 = 0, with x0, z0, and z0 set to zero. The initial velocity x0 is
derived from a defined value of the initial value of the modified Jacobi integral. From equation (4.5), the
60
initial modified Jacobi integral for f0 = 0 is
C0 = 2Ω′0
1 + e− V 2
0 (4.7)
where V0 = (x20 + y2
0 + z20)1/2 is the spacecraft initial velocity and Ω′0 is the initial amended potential
Ω′0 =1
2((x0 + µ− 1)2 + y2
0 − e z20) +
1− µr10
+µ
r20
+1
2µ(1− µ) (4.8)
with r10and r20
being the initial distance of the spacecraft to Mars and Phobos, respectively.
We compute the range of C0 for which sufficiently stable QSOs were found in figure 4.5 which gives
the interval [2.9546; 2.9554]. With these values in mind, FLI maps are now plotted for three different
values of C0: 2.9547, 2.9550, and 2.9553. Note that smaller values of C0 imply larger velocities and, thus,
larger amplitudes. The modified Jacobi integral depends on the energy of the spacecraft by C = −2E.
In the analytical approach we estimated stability conditions for QSOs depending on f0 (equations
3.29). These conditions did not need to be fulfilled strictly since they referred to the ’unperturbed initial
conditions’. Recall this condition for f0 = 0 in dimensional coordinates
y0 ≈ −2 + e
1 + en x0 (4.9)
For an initial condition x0 = 0, sufficiently stable orbits are predicted to exist only for small values of
y0. No stability conditions were imposed on the other coordinates.
The FLI maps presented in figure 4.8 suggest that, for low and medium amplitude orbits, the velocity
on y can not deviate much from zero. The FLI maps also suggest that by adding a positive velocity in
the y direction in large amplitude QSOs it is possible to obtain sufficiently stable orbits as far as 900 km.
An example of a large amplitude QSO is presented in figure 4.9. These QSOs extend over a very large
region and, thus, are good candidates for transfer QSOs to reach small amplitude QSOs.
The study of the FLI maps in figures 4.8 and 4.5 suggest that the stability condition in (4.9) only holds
for initial distances up to 270 km in the y-axis which is an indication of the validity of the first order
approximations performed in the analytical approach.
In conjunction with the FLI map presented in figure 4.5, the present analysis also suggests that QSOs,
when in the negative y semi-axis for f0 = 0, are more sensitive to changes in the velocity in the y direction
than in the x direction.
Furthermore, it is important to discuss the effect of the modified Jacobi integral value, C0, which
depends on the initial conditions and are a measure of the energy of the spacecraft. Figure 4.8(c)
suggests that for large values of C0 (and thus smaller velocities) the maximum distance a QSO can
achieve is smaller.
61
(a) C0 = 2.9547
(b) C0 = 2.9550
(c) C0 = 2.9553
Figure 4.8. FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions y0 and y0. Theinitial parameters x0, z0 and z0 are set to zero, x0 is computed from the initial modified Jacobiintegral C0
62
(a) FLI Map (C0 = 2.9547) (b) QSO (y0 = 900 km)
Figure 4.9. FLI Map of 2D QSOs with large amplitudes in the Mars-Phobos system with initial conditionsx0 = 0, C0 = 2.5547 and tangential and radial velocity. Sample QSO trajectory with y0 = 900km, x0 = −55 m/s, and y0 = 15 m/s
Initial True Anomaly
Up until now we have been analyzing the stability of QSOs for an initial true anomaly f0 = 0 cor-
responding to Phobos’ perigee passage. However, we are interested in analyzing how the initial true
anomaly influences the stability of QSOs.
We repeat the analysis of the variation of y0 with x0 for three different values of the initial anomaly:
0, π, and π/2. The first value has already been analyzed and serves as a basis of comparison. The
second value corresponds to Phobos’ apogee passage and is also a case of interest due to mission
design purposes. The third value was chosen to analyze a less ordinary case.
The FLI maps in figure 4.10 suggest that the initial true anomaly plays an important role in the stability
of QSOs. The stability regions (in the varying parameters y0 and x0) are larger for the cases f0 = 0
and f0 = π which suggests that might be easier, from a stability point of view, to enter in a QSO during
Phobos passage in the perigee or apogee. The stability region in the latter appears to be slightly larger.
63
(a) f0 = 0
(b) f0 = π
(c) f0 = π2
Figure 4.10. FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions y0 and x0. Theinitial parameters x0, z0, y0, and z0 are set to zero. The FLI maps are presented for differentvalues of the initial true anomaly f0
64
4.4.2. Three-Dimensional QSOs
The search for sufficiently stable QSOs is now extend to the search for 3D QSOs.
y0 Vs. z0
Let us now analyze the stability of three-dimensional QSOs varying the initial vertical velocity and the
initial position in the negative y semi-axis. The velocity on the x axis is computed with the modified Jacobi
integral and the remainder of the initial conditions are set to zero, including the initial true anomaly.
The amplitude of a QSO in the z direction, γ, is in dimensional coordinates
γ =
√z2
0 +z2
0
n2(4.10)
where n is the mean orbital motion. Consequently, the larger the initial vertical velocity, the larger γ will
be.
The FLI maps in figure 4.11 suggest that the maximum γ (or maximum initial vertical velocity) in-
creases with the increase of the amplitude up to the point where the maximum distance of the orbit
reaches values for which the restoring force exercised by Phobos is not strong enough to keep the
probe in orbit. After this point, the maximum γ starts to decrease with the amplitude of the orbit to not
surpass this maximum distance for which Phobos can still keep the probe in orbit. These regions are
smaller, the larger the value of the modified Jacobi integral is. It is, however, for these large values of C0
that the larger values of the ratio γ/α are achieved.
In figure 4.12, we present an example of a QSO for the limiting case where a value of q = γ/α ≈
65/70 = 0.93 is achieved, a value close to our prediction of q∗ = 0.96 by the analysis of the equations
of motion in section 3.3.2. Notice that the angle between the line of intersection of the plane of the orbit
with the x-y plane and the positive x semi-axis, β, varies over its whole domain, from 0 to 2π, hence, the
worst-case scenario defined in section 3.3.2, β = π/2, is met.
65
(a) C0 = 2.9547
(b) C0 = 2.9550
(c) C0 = 2.9553
Figure 4.11. FLI Maps for 3D QSOs in the Mars-Phobos system with initial conditions x0 = 0 and tan-gential and vertical velocity. The stability regions decrease with the increase of the modifiedJacobi integral.
66
(a) x vs. y (b) x vs. z
(c) y vs. z
Figure 4.12. Trajectory of a 3D QSO with initial conditions y0 = −100 km, x0 = −15 m/s, and z0 = 14.5m/s (C0 = 2.9553). The orbit is represented in the three planes: x-y, x-z, and y-z.
67
z0 Vs. y0
Whereas the previous three-dimensional QSOs were obtained by the means of an initial vertical ve-
locity, the injection into 3D QSOs can also be performed with an initial height z0. The entry in a 3D QSO
by the means of an initial height has implications on its stability that are analyzed in this section.
We are interested in the analysis of the variation of the initial height with the variation of the velocity
y0, parameter that so far appears to be the most sensible to initial variations. The initial distance y0 is
set to 100 km, x0 is derived from the modified Jacobi integral and the remainder of the parameters are
set to zero.
The first remark upon observation of the FLI maps on figure 4.13 is that after a small value of the initial
height, positive or negative (the map is symmetric in the z coordinate), the range of values of y0 starts to
decrease until the end of the stability region near 100 km. This suggests that three-dimensional QSOs
with large amplitudes in the z direction are more sensible to variations of velocity in the y direction.
68
(a) C0 = 2.9547
(b) C0 = 2.9550
(c) C0 = 2.9553
Figure 4.13. FLI Maps for 3D QSOs in the Mars-Phobos system with initial conditions x0 = 0, y0 = 100km, z = 0, and with z0 and y0 as varying parameters. The velocity on x is derived from C0.
69
z0 Vs. z0
There are three distinct initial situations that result in a three-dimensional QSO: the presence of an
initial vertical velocity, an initial height, or the conjunction of both. We analyze in which of these larger
inclinations can be obtained. In figure 4.14, the FLI maps for the variation of the initial height and
initial vertical velocity are plotted where one can distinguish the different relations between positive and
negative velocities and heights. Notice that for C0 = 2.9553, the stability region is smaller and the relation
between initial vertical velocity and initial height is symmetric.
We are interested in analyzing which of the described initial situations lead to larger values of the ratio
between the amplitudes of the motion in the x-y plane and in the z direction, q = γ/α. In figure 4.15
sample QSOs of the limiting cases (larger q’s found) for each initial situation that lead to 3D QSOs are
presented in the x-z plane. The plotted trajectories indicate that similar values are obtained for all three
cases although the case with initial vertical speed and no initial height seems to present slightly larger
q’s.
Three-dimensional QSOs present a different behavior for large values of the amplitude in the x-y
plane, α. Figure 4.16 presents an example of an orbit of large amplitude α that along other similar
orbits suggest that these orbits present an almost constant angle β, unlike small amplitude QSOs. As
discussed in section 3.3.2, the ratio q is more restricted for β = ±π/2 and it is expected to find larger
values of q for β = 0 or β = π.
Figure 4.16(c) supports the hypothesis that, when analyzing the different initial situations that lead to
3D QSOs, the maximum q is achieved for an angle β close to 0 corresponding to an orbit with both initial
height and initial vertical velocity. Note that this maximum value qmax ≈ 0.75 is farther from the predicted
value predicted of 0.96 than the value of qmax obtained for small amplitude QSOs.
In the case with no initial vertical velocity for which β ≈ π/2 is obtained, which is expected to be the
worst-case scenario, the result is qmax ≈ 0.25, a much smaller value. This supports the hypothesis that
larger values of the ratio q = γ/α are obtained for angles β near zero or π. In QSOs with small amplitude
α the QSO plane rotates and, in a QSO orbiting for a long enough period, the angle β assumes values
in its whole domain (0 to 2π) and this analysis does not apply to these orbits.
From the definition of the osculating elements in section 3.2.4, recall that β = ψ − φ and notice that
in the negative y semi-negative axis we have φ = −π/2 whereas in the x axis φ = π or φ = 0. When
the spacecraft is injected in a QSO in the x-axis, or in any initial position other than x0 = 0, β might not
present the same behavior and this analysis must be repeated. For small amplitude orbits, β varies in
its whole domain and such analysis is not required.
Despite the advantage of obtaining large values of the ratio q with β = 0, this angle also implies that
when z = 0, we get y ≈ ±2α ± δy. In contrast, for β = π/2 we get x ≈ ±α ± δx when z = 0, which
represents a smaller minimum distance. In space mission design, it might be in the designer’s best
interest to have the spacecraft pass as close as possible to the moon, thus, the choice of the angle β
represents a trade-off between maximum QSO inclination and minimum distance to the moon.
70
(a) C0 = 2.9547
(b) C0 = 2.9550
(c) C0 = 2.9553
Figure 4.14. FLI Maps for 3D QSOs in the Mars-Phobos system with initial conditions x0 = 0, y0 = 100km, z = 0, and with z0 and y0 as varying parameters. The velocity on x is derived from C0.
71
(a) z0 = 0 km, z = 14.5 m/s (b) z0 = 50 km, z = 0 m/s
(c) z0 = 30 km, z = 10 m/s
Figure 4.15. Trajectories of 3D QSOs in the x-z plane with initial conditions y0 = −100 km, y0 = 0 m/s,and x0 computed from the invariant relation (C0 = 2.9553).
(a) z0 = 0 km, z = 22 m/s (b) z0 = 57 km, z = 0 m/s
(c) z0 = 50 km, z = 32 m/s
Figure 4.16. Trajectories of 3D QSOs in the x-z plane with initial conditions y0 = −100 km, y0 = 0 m/s,and x0 computed from the invariant relation (C0 = 2.9547).
72
4.4.3. Velocity Maps
Velocity maps are FLI maps plotted as function of two velocities where the third velocity is usually
derived from the (modified) Jacobi integral (Villac, 2008). These maps let us characterize the stability
robustness of a QSO (which represents its capability to survive model perturbations) as well as the
required ∆V ’s to escape from the QSO. One can understand better the trade-off between these two
characteristics by imagining, at any given point in the QSO, the three-dimensional stability region com-
posed by the velocities on the three coordinates — the larger this stability region the more stable the
QSO is but also the more ∆V is required to escape the stability region and thus escape from the QSO.
We can not analyze this stability region in three dimensions but we can analyze two-dimensional ’slices’
with the FLI value — these are the velocity maps.
In figure 4.17 the velocity maps for a QSO with initial position (x0, y0, z0) = (0,−100, 0) km for dif-
ferent values of the modified Jacobi constant, C0, are presented. The three cases present the same
robustness in y0 but present different robustness in z0 although the ∆V margin is still large enough for
the three values to account for velocity error during the injection. In a QSO with a large initial vertical
velocity (positive or negative), one might consider injecting the spacecraft with C0 = 2.9547, however,
this corresponds to larger amplitudes and, thus, a more restricted range of the orbit inclination.
Figure 4.18 suggests that the velocities stability region at (x0, y0, z0) = (100, 0, 0) km has different
characteristics. In these velocity maps one can observe how the initial value of the modified Jacobi
integral influences the stability robustness on both directions. The choice of the size of the stability
region implies a trade-off between robustness and required ∆V to escape from the QSO, although
the velocities being addressed here should pose no significant impact for a spacecraft designed for an
interplanetary mission.
Notice that the used values for C0 are much closer together than the ones analyzed in the position
(x0, y0, z0) = (0,−100, 0) km. This suggests that transfers to QSOs in the x axis are far more sensible
to initial conditions than when performed in the y axis (i.e., injections in the y-axis are less sensible to
errors in the initial energy), making the latter a safer option.
The study of all of the computed FLI maps up to this point suggest that QSOs are more sensible to
errors in the velocity components than in the position components. This emphasizes the importance of
the study of the velocity maps.
73
(a) C0 = 2.9547
(b) C0 = 2.9550
(c) C0 = 2.9553
Figure 4.17. FLI map for the velocities on y and z, the velocity on x is derived from C0. Initial position ony0 = −100 km, x0 = z0 = 0.
74
(a) C0 = 2.95526
(b) C0 = 2.95527
(c) C0 = 2.95528
Figure 4.18. FLI map for the velocities on x and z, the velocity on y is derived from C0. Initial position onx0 = 100 km, y0 = z0 = 0.
75
Now this is not the end. It is not even the beginning
of the end. But it is, perhaps, the end of the beginning.
(Sir Winston Churchill 1874 - 1965)
CH
AP
TE
R
5 Conclusions
This work focused on the use of analytical and numerical techniques to obtain and assess the stability
of the so-called quasi-satellite orbits in the elliptic restricted three-body problem. The problem is complex
due to its time-dependence forming a non-autonomous system.
The non-autonomous equations of motion for Hill’s problem, derived from the Hamiltonian formulation,
were solved to obtain the unperturbed solutions. These serve as the basis for the perturbation theories
applied in two different formulations, each providing its own motion and stability considerations from
the analysis of the approximate averaged equations of motion for µ << 1, e << 1, and x, y, z << 1.
These considerations regard important parameters of quasi-satellite orbits such as the amplitude, both
in the moon’s orbital plane and in a direction perpendicular to this plane; periodicity, of the orbit in the
both mentioned planes, of the displacement of the orbit’s center, and of the QSO plane angle β; and of
the achievable orbit inclinations measured by the ratio of amplitudes q. These parameters are of great
significance for the mission design.
The fast Lyapunov indicator, a chaos indicator based on the analysis of the evolution of the deviation
vector, was used to compute the FLI maps to study stability regions in which sufficiently stable QSOs
are found. Chaotic regions were not found in the studied regions due to the limitations of the FLI tech-
nique. The study of the FLI maps over sets of initial conditions provided insight about stability regions
in the Mars-Phobos system and their relation with the osculating elements was addressed. The stability
relations estimated in the analytical approach were confirmed with some certainty for QSOs with small
amplitudes. Furthermore, velocity maps were studied to analyze the stability robustness and required
∆V ’s to escape from a QSO.
The present work has some restrictions regarding both the analytical and numerical approaches which
pose interesting subjects for future work. The validity of the developed analytical technique is limited by
the assumptions µ << 1 and e << 1. The estimated stability considerations seem to hold only for a
range of values of the amplitude.
A study of how the change of µ influences the stability regions of QSOs would be an asset and is yet
to be performed. The study of systems with larger values for the eccentricity e poses an even greater
challenge as in such systems the time-dependent terms have a greater influence. The successful study
77
of these systems would represent a useful accomplishment. The use of second-order approximations
for the analytical approach could also extend the validity of the stability considerations to QSOs with
larger amplitudes.
A study of resonant orbits is also required as these represent the lowest altitude stable orbits for
Phobos and are particularly interesting for approach or close observation orbits. Such study would
imply the utilization of a more accurate dynamics model. In the system Mars-Phobos it is possible that
in distances close to the libration points, just above the moon’s surface, chaotic regions can be found.
The study of FLI maps could be an asset to find these chaotic regions. A future study on transfer orbits
between QSOs and respective optimization would also be an asset.
Any future works should look to apply their conclusions to systems of interest such as the systems
Jupiter-Europa, or Saturn-Titan. These moons are known to be potential targets of future missions and
the study of their orbital mechanics is an advantage for mission designers.
78
AP
PE
ND
IX
A Programming Code
This code was developed in the Linux OS Ubuntu and compiled with the gcc compiler (version 4.6.2)
using the compilation options -g -lm -Wall -std=c99.
# inc lude <s t d i o . h>
# inc lude <math . h>
# inc lude <t ime . h>
# def ine DIM 6
/ / S t ruc tu re w i th the o r b i t data
typedef s t r u c t Trajectory
long double FLI , d_min , d_max ;
long double x [ DIM ] , phi [ DIM ] [ DIM ] ;
TRAJ ;
/ / Funct ion to conver t from dimensional to pu l s a t i n g coord ina tes
vo id todimless ( long double ∗v , long double a , long double e , long double f , long double n , long double mu←
) ;
/ / Funct ion to conver t from p u l s a t i n g to dimensional coord ina tes
vo id todim ( long double ∗v , long double a , long double e , long double f , long double n , long double mu ) ;
/ / Funct ion t h a t m u l t i p l i e s matr ices − v=v1∗v2
void multm ( long double v [ DIM ] [ DIM ] , long double v1 [ DIM ] [ DIM ] , long double v2 [ DIM ] [ DIM ] ) ;
/ / RHS f u n c t i o n o f the equat ions o f motion
vo id motion ( long double ∗dx , long double ∗v , long double f , long double mu , long double e ) ;
/ / RHS f u n c t i o n o f the v a r i a t i o n a l equat ions
vo id variation ( long double dphi [ DIM ] [ DIM ] , long double phi [ DIM ] [ DIM ] , long double ∗x , long double f , long←
double mu , long double e ) ;
/ / I n t e g r a t i o n f u c n t i o n wi th Runge−Kut ta 8 method
void rk8 ( TRAJ ∗orbit , long double f_0 , long double f_f , long double h , long double mu , long double e , ←
long double n , long double sma ) ;
79
/ / Main Funct ion
i n t main ( )
/ / Var iab les d e f i n i t i o n
FILE ∗fp ;
char ∗file_name= ” Resul ts . t x t ” ;
long double f_0 , f_f , h , pi , mu , M_ph , M_m , e , a , n , G ;
i n t i , j ;
TRAJ orbit , ∗ptr_orbit ;
clock_t start , end ;
double cpu_time_used ;
/ / Compute S t a r t i n g Time
start = clock ( ) ;
/ / Po in te r to o r b i t data
ptr_orbit=&orbit ;
/ / Constants
M_m = 6.4185∗pow (10 ,23) ; / / mass of Mars [ kg ]
M_ph = 1.06∗pow (10 ,16) ; / / mass of Phobos [ kg ]
mu = M_ph / ( M_m+M_ph ) ; / / mass parameter
e = 0.0151; / / Phobos ’ s e c c e n t r i c i t y
a = 9377.2∗pow (10 ,3 ) ; / / Semi−major ax is [m]
G = 6.67384∗pow (10 ,−11) ;
n = sqrt ( G∗(M_m+M_ph ) / pow (a , 3 ) ) ; / / mean motion
pi = acos (−1.0) ; / / constant p i
/ / I n i t i a l cond i t i ons
orbit . x [ 0 ] = 0 ;
orbit . x [ 1 ] = −100;
orbit . x [ 2 ] = 0 ;
orbit . x [ 3 ] = −0.02;
orbit . x [ 4 ] = 0 ;
orbit . x [ 5 ] = 0 ;
/ / I n i t i a l i z a t i o n o f o r b i t parameters
orbit . d_min = sqrt ( pow ( orbit . x [ 0 ] , 2 ) +pow ( orbit . x [ 1 ] , 2 ) +pow ( orbit . x [ 2 ] , 2 ) ) ;
orbit . d_max = sqrt ( pow ( orbit . x [ 0 ] , 2 ) +pow ( orbit . x [ 1 ] , 2 ) +pow ( orbit . x [ 2 ] , 2 ) ) ;
orbit . FLI=0;
f o r ( i=0; i<DIM ; i++)
f o r ( j=0; j<DIM ; j++)
i f ( i==j )
orbit . phi [ i ] [ j ] = 1 ;
e lse
orbit . phi [ i ] [ j ] = 0 ;
/ / Computat ional parameters
h = pi / 8 ; / / s tep
f_0 = 0; / / i n i t i a l t r ue anomaly
f_f = 2∗pi∗100; / / f i n a l t r ue anomaly
80
/ / Opens F i l e
fp=fopen ( file_name , ”w” ) ;
/ / FILE opening v e r i f i c a t i o n
i f ( fp==NULL )
printf ( ”\nImpossib le to open f i l e %s\n ” , file_name ) ;
e lse
printf ( ”\nF i l e %s s ucce ss fu l l y open !\n ” , file_name ) ;
/ / F i r s t L ine on F i l e
fprintf ( fp , ” (∗ This f i l e conta ins data formated f o r Mathematica∗)\n ” ) ;
/ / I n i t i a l cond i t i ons i n p u l s a t i n g coord ina tes
todimless ( ptr_orbit−>x , a , e , f_0 , n , mu ) ;
/ / O rb i t I n t e g r a t i o n
rk8 ( ptr_orbit , f_0 , f_f , h , mu , e , n , a ) ;
/ / Resul ts
todim ( orbit . x , a , e , f_f , n , mu ) ;
fprintf ( fp , ”\n%Lf , %Lf , %Lf , %Lf , %Lf , %Lf\n ” , orbit . x [ 0 ] , orbit . x [ 1 ] , orbit . x [ 2 ] , orbit . x [ 3 ] , orbit . x [ 4 ] ,←
orbit . x [ 5 ] ) ;
printf ( ”\nMaximum Distance : %Lf km\nMinimum Distance : %Lf km\nFLI : %Lf\n ” , orbit . d_max , orbit . d_min , orbit .←
FLI ) ;
/ / F i n a l i z e s f i l e
fprintf ( fp , ” ” ) ;
/ / Closes f i l e
fclose ( fp ) ;
/ / End of the program
printf ( ”\nEnd of the program !\n ” ) ;
/ / Computation o f used cpu t ime
end = clock ( ) ;
cpu_time_used = ( ( double ) ( end − start ) ) / CLOCKS_PER_SEC ;
printf ( ”\nCPU Time : %f\n ” , cpu_time_used ) ;
r e t u r n 0 ;
/ / Funct ion to conver t from dimensional to pu l s a t i n g coord ina tes
vo id todimless ( long double ∗v , long double a , long double e , long double f , long double n , long double mu←
)
i n t i ;
long double aux [ DIM ] ;
f o r ( i=0; i<DIM ; i++)
aux [ i ]=v [ i ] ;
v [ 0 ] = aux [ 0 ]∗ (1+ e∗cos ( f ) ) / ( a∗(1−pow (e , 2 ) ) )∗pow (10 ,3 ) ;
v [ 1 ] = aux [ 1 ]∗ (1+ e∗cos ( f ) ) / ( a∗(1−pow (e , 2 ) ) )∗pow (10 ,3 ) ;
v [ 2 ] = aux [ 2 ]∗ (1+ e∗cos ( f ) ) / ( a∗(1−pow (e , 2 ) ) )∗pow (10 ,3 ) ;
v [ 3 ] = ( aux [3]−aux [ 1 ]∗ n ) ∗(1+e∗cos ( f ) ) / ( a∗(1−pow (e , 2 ) )∗n )∗pow (10 ,3 ) ;
81
v [ 4 ] = ( aux [ 4 ] + aux [ 0 ]∗ n ) ∗(1+e∗cos ( f ) ) / ( a∗(1−pow (e , 2 ) )∗n )∗pow (10 ,3 )+mu−1;
v [ 5 ] = aux [ 5 ]∗ (1+ e∗cos ( f ) ) / ( a∗(1−pow (e , 2 ) )∗n )∗pow (10 ,3 ) ;
/ / Funct ion to conver t from p u l s a t i n g to dimensional coord ina tes
vo id todim ( long double ∗v , long double a , long double e , long double f , long double n , long double mu )
i n t i ;
long double aux [ DIM ] ;
f o r ( i=0; i<DIM ; i++)
aux [ i ]=v [ i ] ;
v [ 0 ] = aux [ 0 ] / ( 1 + e∗cos ( f ) ) ∗(a∗(1−pow (e , 2 ) ) ) / pow (10 ,3 ) ;
v [ 1 ] = aux [ 1 ] / ( 1 + e∗cos ( f ) ) ∗(a∗(1−pow (e , 2 ) ) ) / pow (10 ,3 ) ;
v [ 2 ] = aux [ 2 ] / ( 1 + e∗cos ( f ) ) ∗(a∗(1−pow (e , 2 ) ) ) / pow (10 ,3 ) ;
v [ 3 ] = ( aux [ 3 ] ) / (1+ e∗cos ( f ) ) ∗(a∗(1−pow (e , 2 ) )∗n ) / pow (10 ,3 )+aux [ 1 ]∗ n ;
v [ 4 ] = ( aux [4]−mu+1) / (1+ e∗cos ( f ) ) ∗(a∗(1−pow (e , 2 ) )∗n ) / pow (10 ,3 )−aux [ 0 ]∗ n ;
v [ 5 ] = aux [ 5 ] / ( 1 + e∗cos ( f ) ) ∗(a∗(1−pow (e , 2 ) )∗n ) / pow (10 ,3 ) ;
/ / Funct ion t h a t m u l t i p l i e s matr ices − v=v1∗v2
void multm ( long double v [ DIM ] [ DIM ] , long double v1 [ DIM ] [ DIM ] , long double v2 [ DIM ] [ DIM ] )
i n t i , j , ij ;
f o r ( i=0; i<DIM ; i++)
f o r ( j=0; j<DIM ; j++)
v [ i ] [ j ] = 0 . 0 ;
f o r ( i=0; i<DIM ; i++)
f o r ( j=0; j<DIM ; j++)
f o r ( ij=0; ij<DIM ; ij++)
v [ i ] [ j ]=v [ i ] [ j ]+v1 [ i ] [ ij ]∗v2 [ ij ] [ j ] ;
/ / RHS f u n c t i o n o f the equat ions o f motion
vo id motion ( long double ∗dx , long double ∗v , long double f , long double mu , long double e )
long double r1 , r2 ;
/ / Distance to the f i r s t pr imary
r1=sqrt ( pow ( ( v [0]−1) ,2 )+pow ( v [ 1 ] , 2 ) +pow ( v [ 2 ] , 2 ) ) ;
/ / Distance to the second pr imary
r2=sqrt ( pow ( v [ 0 ] , 2 ) +pow ( v [ 1 ] , 2 ) +pow ( v [ 2 ] , 2 ) ) ;
dx [ 0 ] = v [ 3 ] + v [ 1 ] ;
dx [ 1 ] = v [4]−v [0]−mu+1;
dx [ 2 ] = v [ 5 ] ;
82
dx [ 3 ] = v [4]−v [0]−mu+1+1/(1+e∗cos ( f ) ) ∗(v [ 0 ] + mu−1−(v [0]−1)∗(1−mu ) / pow ( r1 , 3 )−v [ 0 ]∗ mu / pow ( r2 , 3 ) ) ;
dx [4]=−v [3]−v [ 1 ] + 1 / ( 1 + e∗cos ( f ) ) ∗(v [1]−v [1]∗(1−mu ) / pow ( r1 , 3 )−v [ 1 ]∗ mu / pow ( r2 , 3 ) ) ;
dx [ 5 ] = 1 / ( 1 + e∗cos ( f ) )∗(−e∗v [ 2 ]∗ cos ( f )−v [2]∗(1−mu ) / pow ( r1 , 3 )−v [ 2 ]∗ mu / pow ( r2 , 3 ) ) ;
/ / RHS f u n c t i o n o f the v a r i a t i o n a l equat ions
vo id variation ( long double dphi [ DIM ] [ DIM ] , long double phi [ DIM ] [ DIM ] , long double ∗x , long double f , long←
double mu , long double e )
long double r1 , r2 , dh [ DIM ] [ DIM ] = 0.0 , J [ DIM ] [ DIM ] = 0.0 , A [ DIM ] [ DIM ] ;
/ / Distance to the f i r s t pr imary
r1=sqrt ( pow ( ( x [0]−1) ,2 )+pow ( x [ 1 ] , 2 ) +pow ( x [ 2 ] , 2 ) ) ;
/ / Distance to the second pr imary
r2=sqrt ( pow ( x [ 0 ] , 2 ) +pow ( x [ 1 ] , 2 ) +pow ( x [ 2 ] , 2 ) ) ;
/ / Hessian Mat r i x o f the Hami l ton ian
dh [ 0 ] [ 0 ] = 1−1/(1+e∗cos ( f ) ) ∗(1+(3∗pow ( ( x [0]−1) ,2 ) / pow ( r1 , 5 )−1/pow ( r1 , 3 ) )∗(1−mu ) +(3∗ (pow ( x [ 0 ] , 2 ) / pow (←
r2 , 5 )−1/pow ( r2 , 3 ) )∗mu ) ) ;
dh [ 0 ] [ 1 ] = −1/(1+e∗cos ( f ) ) ∗ (3∗ (x [0]−1)∗x [1]∗(1−mu ) / pow ( r1 , 5 ) +3∗x [ 0 ]∗ x [ 1 ]∗ mu / pow ( r2 , 5 ) ) ;
dh [ 1 ] [ 0 ] = dh [ 0 ] [ 1 ] ;
dh [ 0 ] [ 2 ] = −1/(1+e∗cos ( f ) ) ∗ (3∗ (x [0]−1)∗x [2]∗(1−mu ) / pow ( r1 , 5 ) +3∗x [ 0 ]∗ x [ 2 ]∗ mu / pow ( r2 , 5 ) ) ;
dh [ 2 ] [ 0 ] = dh [ 0 ] [ 2 ] ;
dh [ 1 ] [ 1 ] = 1−1/(1+e∗cos ( f ) ) ∗(1+(3∗pow ( x [ 1 ] , 2 ) / pow ( r1 , 5 )−1/pow ( r1 , 3 ) )∗(1−mu ) +(3∗ (pow ( x [ 1 ] , 2 ) / pow ( r2←
, 5 )−1/pow ( r2 , 3 ) )∗mu ) ) ;
dh [ 1 ] [ 2 ] = −1/(1+e∗cos ( f ) ) ∗(3∗x [ 1 ]∗ x [2]∗(1−mu ) / pow ( r1 , 5 ) +3∗x [ 1 ]∗ x [ 2 ]∗ mu / pow ( r2 , 5 ) ) ;
dh [ 2 ] [ 1 ] = dh [ 1 ] [ 2 ] ;
dh [ 2 ] [ 2 ] = 1/ (1+e∗cos ( f ) ) ∗(1+(3∗pow ( x [ 2 ] , 2 ) / pow ( r1 , 5 )−1/pow ( r1 , 3 ) )∗(1−mu ) +(3∗ (pow ( x [ 2 ] , 2 ) / pow ( r2 , 5 )←
−1/pow ( r2 , 3 ) )∗mu )−e∗cos ( f ) ) ;
dh [ 0 ] [ 4 ] = −1;
dh [ 4 ] [ 0 ] = dh [ 0 ] [ 4 ] ;
dh [ 1 ] [ 3 ] = 1 ;
dh [ 3 ] [ 1 ] = dh [ 1 ] [ 3 ] ;
dh [ 3 ] [ 3 ] = 1 ;
dh [ 4 ] [ 4 ] = 1 ;
dh [ 5 ] [ 5 ] = 1 ;
J [ 0 ] [ 3 ] = 1 ;
J [ 1 ] [ 4 ] = 1 ;
J [ 2 ] [ 5 ] = 1 ;
J [ 3 ] [ 0 ] = −1;
J [ 4 ] [ 1 ] = −1;
J [ 5 ] [ 2 ] = −1;
multm (A , J , dh ) ;
multm ( dphi , A , phi ) ;
/ / I n t e g r a t i o n f u c n t i o n wi th Runge−Kut ta 8 method
void rk8 ( TRAJ ∗orbit , long double f_0 , long double f_f , long double h , long double mu , long double e , ←
long double n , long double sma )
/ / Method c o e f f i c i e n t s
long double a1 [ 1 3 ] [ 1 2 ] = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
83
1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
1 , 0 , 3 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
5 , 0 , −75, 75 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
3 , 0 , 0 , 3 , 3 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
29443841, 0 , 0 , 77736538, −28693883, 23124283, 0 , 0 , 0 , 0 , 0 , 0 ,
16016141, 0 , 0 , 61564180, 22789713, 545815736, −180193667, 0 , 0 , 0 , 0 , 0 ,
39632708, 0 , 0 , −433636366, −421739975, 100302831, 790204164, 800635310, 0 , 0 , 0 , 0 ,
246121993, 0 , 0 , −37695042795, −309121744, −12992083, 6005943493, 393006217, 123872331, 0 , 0 ,←
0 ,
−1028468189, 0 , 0 , 8478235783, 1311729495, −10304129995, −48777925059, 15336726248, ←
−45442868181, 3065993473, 0 , 0 ,
185892177, 0 , 0 , −3185094517, −477755414, −703635378, 5731566787, 5232866602, −4093664535, ←
3962137247, 65686358, 0 ,
403863854, 0 , 0 , −5068492393, −411421997, 652783627, 11173962825, −13158990841, 3936647629, ←
−160528059, 248638103, 0;
long double a2 [ 1 3 ] [ 1 2 ] = 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
18 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
48 , 16 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
32 , 0 , 32 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
16 , 0 , 64 , 64 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
80 , 0 , 0 , 16 , 20 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
614563906, 0 , 0 , 692538347, 1125000000, 1800000000, 0 , 0 , 0 , 0 , 0 , 0 ,
946692911, 0 , 0 , 158732637, 633445777, 2771057229, 1043307555, 0 , 0 , 0 , 0 , 0 ,
573591083, 0 , 0 , 683701615, 2616292301, 723423059, 839813087, 3783071287, 0 , 0 , 0 , 0 ,
1340847787, 0 , 0 , 15268766246, 1061227803, 490766935, 2108947869, 1396673457, 1001029789, 0 , ←
0 , 0 ,
846180014, 0 , 0 , 508512852, 1432422823, 1701304382, 3047939560, 1032824649, 3398467696, ←
597172653, 0 , 0 ,
718116043, 0 , 0 , 667107341, 1098053517, 230739211, 1027545527, 850066563, 808688257, ←
1805957418, 487910083, 0 ,
491063109, 0 , 0 , 434740067, 543043805, 914296604, 925320556, 6184727034, 1978049680, ←
685178525, 1413531060, 0;
long double c1 [13]=0 , 1 , 1 , 1 , 5 , 3 , 59 , 93 , 5490023248, 13 , 1201146811, 1 , 1 ;
long double c2 [13]=0 , 18 , 12 , 8 , 16 , 8 , 400 , 200 , 9719169821, 20 , 1299019798, 1 , 1 ;
long double b1 [13]=14005451 , 0 , 0 , 0 , 0 , −59238493, 181606767, 561292985, −1041891430, ←
760417239, 118820643, −528747749, 1 ;
long double b2 [13]=335480064 , 0 , 0 , 0 , 0 , 1068277825, 758867731, 797845732, 1371343529, 1151165299,←
751138087, 2220607170, 4 ;
i n t i , j , l , m ;
long double f , dx [ DIM ] , dphi [ DIM ] [ DIM ] , kx [ DIM ] [ 1 3 ] , kphi [ DIM ] [ DIM ] [ 1 3 ] , aux_x [ DIM ] , aux_phi [ DIM ] [ DIM ] , ←
D1 [ DIM ] , D0 [ DIM ] , r , p , d , alpha [ DIM ] , W0 [ DIM ] [ DIM ]=0.0 , W1 [ DIM ] [ DIM ] , pi , x1 , x2 , freq=0;
long double dim_x [ DIM ] , a [ 1 3 ] [ 1 2 ] , b [ 1 3 ] , c [ 1 3 ] ;
pi = acos (−1.0) ;
x1=orbit−>x [ 0 ] ;
/ / Method c o e f f c i e n t s
f o r ( i=0; i<13; i++)
f o r ( j=0; j<12; j++)
i f ( a2 [ i ] [ j ] != 0 .0 )
a [ i ] [ j ]=a1 [ i ] [ j ] / a2 [ i ] [ j ] ;
e lse
a [ i ] [ j ] = 0 . 0 ;
84
/∗ i f ( a [ i ] [ j ] ! = 0 )
f p r i n t f ( fp , ” a(%d,%d ) = %Lf\n ” , i +1 , j +1 ,a [ i ] [ j ] ) ; ∗ /
i f ( b2 [ i ] ! = 0 )
b [ i ]=b1 [ i ] / b2 [ i ] ;
e lse
b [ i ] = 0 ;
i f ( c2 [ i ] ! = 0 )
c [ i ]=c1 [ i ] / c2 [ i ] ;
e lse
c [ i ] = 0 ;
r = sqrt ( pow ( orbit−>x [ 0 ] , 2 ) +pow ( orbit−>x [ 1 ] , 2 ) +pow ( orbit−>x [ 2 ] , 2 ) ) ;
p = sqrt ( pow ( orbit−>x [ 3 ] , 2 ) +pow ( orbit−>x [ 4 ] , 2 ) +pow ( orbit−>x [ 5 ] , 2 ) ) ;
f o r ( i=0; i<3; i++)
W0 [ i ] [ i ] = 1 ;
D0 [ i ] = sqrt ( 1 / r ) ;
f o r ( i=3; i<DIM ; i++)
W0 [ i ] [ i ] = 1 ;
D0 [ i ] = sqrt ( 1 / p ) ;
/ / I n t e g r a t i o n over t ime
f o r ( f=f_0 ; f < f_f−0.1∗h ; f=f+h )
/ / Computation o f k ’ s
f o r ( i=1; i<=13; i++)
f o r ( l=0; l<DIM ; l++)
aux_x [ l ] = 0 ;
f o r ( m=0; m<DIM ; m++)
aux_phi [ l ] [ m ] = 0 ;
f o r ( l=0; l<DIM ; l++)
f o r ( m=0; m<DIM ; m++)
f o r ( j=1;j<=i−1 ; j++)
i f ( m==0)
aux_x [ l ]= aux_x [ l ]+a [ i−1][j−1]∗kx [ l ] [ j−1];
aux_phi [ l ] [ m ]= aux_phi [ l ] [ m ]+a [ i−1][j−1]∗kphi [ l ] [ m ] [ j−1];
aux_phi [ l ] [ m ]= orbit−>phi [ l ] [ m ]+h∗aux_phi [ l ] [ m ] ;
aux_x [ l ]= orbit−>x [ l ]+h∗aux_x [ l ] ;
motion ( dx , aux_x , f+c [ i−1]∗h , mu , e ) ;
variation ( dphi , aux_phi , aux_x , f+c [ i−1] , mu , e ) ;
f o r ( l=0; l<DIM ; l++)
kx [ l ] [ i−1]=dx [ l ] ;
f o r ( m=0; m<DIM ; m++)
kphi [ l ] [ m ] [ i−1]=dphi [ l ] [ m ] ;
85
/ / Computation x ( n+1)
f o r ( l=0; l<DIM ; l++)
aux_x [ l ] = 0 ;
f o r ( m=0; m<DIM ; m++)
aux_phi [ l ] [ m ] = 0 ;
f o r ( l=0; l<DIM ; l++)
f o r ( m=0; m<DIM ; m++)
f o r ( i=1;i<=13 ; i++)
i f ( m==0)
aux_x [ l ]= aux_x [ l ]+b [ i−1]∗kx [ l ] [ i−1];
aux_phi [ l ] [ m ]= aux_phi [ l ] [ m ]+b [ i−1]∗kphi [ l ] [ m ] [ i−1];
orbit−>phi [ l ] [ m ]= orbit−>phi [ l ] [ m ]+h∗aux_phi [ l ] [ m ] ;
orbit−>x [ l ]= orbit−>x [ l ]+h∗aux_x [ l ] ;
x2=orbit−>x [ 0 ] ;
i f ( x1 < 0 && x2 > 0)
freq++;
x1 = x2 ;
/ / Convert ion to dimensional coord ina tes
f o r ( i=0; i<DIM ; i++)
dim_x [ i ]= orbit−>x [ i ] ;
todim ( dim_x , sma , e , f+h , n , mu ) ;
/ / Computation o f rad ius and t o t a l momentum
d = sqrt ( pow ( dim_x [ 0 ] , 2 ) +pow ( dim_x [ 1 ] , 2 ) +pow ( dim_x [ 2 ] , 2 ) ) ;
/ / p = s q r t (pow( dim x [ 3 ] , 2 ) +pow( dim x [ 4 ] , 2 ) +pow( dim x [ 5 ] , 2 ) ) ;
r = sqrt ( pow ( orbit−>x [ 0 ] , 2 ) +pow ( orbit−>x [ 1 ] , 2 ) +pow ( orbit−>x [ 2 ] , 2 ) ) ;
p = sqrt ( pow ( orbit−>x [ 3 ] , 2 ) +pow ( orbit−>x [ 4 ] , 2 ) +pow ( orbit−>x [ 5 ] , 2 ) ) ;
/ / Update maximum and minimum dis tances
i f ( d > orbit−>d_max )
orbit−>d_max = d ;
i f ( d < orbit−>d_min )
orbit−>d_min = d ;
/ / S t a b i l i t y c o n d i t i o n to cont inue i n t e g r a t i o n
i f ( d > 1000 | | d < 15)
orbit−>FLI = 20 − ( f+h ) / ( f_f ) ∗10;
r e t u r n ;
/ / Computation o f bas is o f d e v i a t i o n vec to rs
multm ( W1 , orbit−>phi , W0 ) ;
/ / Computation o f D1 f o r each d e v i a t i o n vector , and computat ion o f alpha wi th l a r g e s t D1
86
f o r ( i=0; i<DIM ; i++)
D1 [ i ]= sqrt ( ( pow ( W1 [ 0 ] [ i ] , 2 ) +pow ( W1 [ 1 ] [ i ] , 2 ) +pow ( W1 [ 2 ] [ i ] , 2 ) ) / r+(pow ( W1 [ 3 ] [ i ] , 2 ) +pow ( W1 [ 4 ] [ i ] , 2 ) +pow (←
W1 [ 5 ] [ i ] , 2 ) ) / p ) ;
alpha [ i ] = D1 [ i ] / D0 [ i ] ;
D0 [ i ]=D1 [ i ] ;
/ / Update FLI
i f ( log ( alpha [ i ] ) > orbit−>FLI )
orbit−>FLI = log ( alpha [ i ] ) ;
printf ( ”\nOrb i t Frequency : %Lf\n ” , freq /100) ;
Num Exp.c
87
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