Thesis for Master of Science Astrodynamics
BALLISTIC CAPTURE TO MARS:
STABILITY ANALYSIS WITH ACTIVE CONTROL
AND ENERGY CRITERIA
Thesis supervisor
Prof. Francesco Topputo
Thesis co-advisor
Diogene Alessandro Dei Tos
Candidate
Edoardo Dell’Aglio
STABILITY ANALYSIS WITH ACTIVE CONTROL AND ENERGY CRITERIA
Master of Science Thesis supervisor Francesco Topputo Thesis co-advisor
Diogene Alessandro Dei Tos
Declaration
I hereby declare that this thesis is my original work, it has not been subject to plagiarism, and has not been presented for a degree in any other University.
Milan, 2016
Edoardo Dell’Aglio
This thesis comes from the simple approach developed in astrodynamics called ballistic
capture useful for analysing eventual advantages about space travels with very little fuel.
An option like this returns important consequences not only for the lighter propulsive system required, but also for the greater safety, payload and control authority achieved in a long period. In particular such application is given for Mars capture, referred to the specific case of a 3U CubeSat, a nano-satellite. Hence the present study grows transversely about the matter, developing many important and innovative points:
the impulsive burn manoeuvres, treated as composition of unperturbed motions. It represents the most immediate analysis about the low-energy transfers. In this way a nomenclature based on the variation of eccentricity is suggested, aimed to quantify in a rigorous and elegant way the impulse needed for the control.
the continuum nano-propulsion applied to this theory. This aspect required a com-plete new model, resulting from the standard differential problem adopted for elliptic restricted three body problem, modified for the active perturbation. Together with this, the Levi-Civita local regularization used to improve the numerical algorithm has been provided by the author.
some pioneering criteria about the energy, very important aspect applicable in all the fields of trajectory analysis. The concepts of Jacoby energy and Hill’s surfaces are exploited for reaching remarkable considerations on the integrated orbits. As some evidences are showed and verified, some precise definitions are made.
All the informations are kept general as much as possible. The proof is that usually critical cases are analysed in a coarse but reasonable way, instead of adopting a particularly precise and expensive numerical computation.
In the final part crucial aspects are listed fine to be studied and to be improved thereafter, and the general conclusions. In summary this thesis can represent a particular reflection of a wider topic, worthwhile tool for future interplanetary missions.
Questa tesi nasce dall’approccio chiamato cattura balistica, utile in astrodinamica per scoprire eventuali vantaggi circa i viaggi spaziali con poco propellente. Un approccio di
questo tipo porta importanti conseguenze non solo per il pi`u leggero sistema propulsivo
richiesto, ma anche per la maggiore sicurezza, carico pagante e robustezza del controllo nel lungo periodo. In particolar modo tale applicazione `e volta alla cattura su Marte, riferita al caso specifico del nanosatellite 3U CubeSat. Questo studio si ramifica quindi trasversalmente circa l’argomento, sviluppando molti ed inediti spunti di riflessione:
manovre a spinta impulsiva, intese come composizione di diverse traiettorie
natu-rali. Rappresenta l’analisi pi`u immediata rispetto ai trasferimenti a bassa energia,
e propone una nuova nomenclatura basata sulla variazione di eccentricit`a, atta a
quantificare rigorosamente l’impulso utile al controllo.
nano-propulsione in continua applicata a questa teoria. Questo punto richiede un modello completamente nuovo, ottenuto dalle equazioni differenziali standard del problema ristretto ellittico, modificate per la perturbazione attiva. Assieme a questo, l’autore fornisce una appropriata regolarizzazione locale Levi-Civita utile al miglio-ramento dell’algoritmo numerico.
alcuni pionieristici criteri energetici, aspetto molto importante per tutti i campi riguardanti l’analisi di traiettoria. I concetti di energia di Jacobi e superfici di Hill, sono sfruttati al fine di ottenere considerazioni rilevanti circa le orbite integrate. Vengono allora fornite alcune definizioni per sugellare alcune derivazioni dimostrate e verificate.
Tutte le informazioni mostrate sono mantenute generali quanto pi`u possibile. Infatti
soli-tamente sono studiati casi critici in modo non troppo approfondito e senza appesantire i contenuti con metodi numerici troppo onerosi.
In calce sono elencati degli aspetti cruciali da studiare e migliorare in seconda battuta con
considerazioni di carattere generale. In sintesi questa tesi pu`o rappresentare una
parti-colare riflessione rispetto ad un argomento pi`u vasto, strumento fruttifero per le missioni
Non nobis solum nati sumus.
(Not for ourselves alone are we born.) Marcus Tullius Cicero
Abstract III Sommario V Acknowledgments VII Contents XI List of Figures . . . XV List of Tables . . . XV Nomenclature . . . XVI 1 Introduction 1 1.1 History . . . 1 1.1.1 From astronomy ... . . 1 1.1.2 ... to Ballistic Capture . . . 2 1.2 Motivation . . . 3
1.3 State of the art . . . 4
1.4 Goals . . . 6
1.5 Organization of the work . . . 6
2 Dynamic model and algorithms 9 2.1 The n-Body Problem . . . 9
2.1.1 The 2BP case . . . 11
2.1.2 The Keplerian model . . . 12
2.1.3 The Restricted Three-Body Problem . . . 14
2.2 Exploited differential models . . . 17
2.2.1 The Elliptic Restricted Three Body Problem . . . 17
2.2.2 Regularization . . . 19
2.3 Construction of stable sets . . . 20
2.3.1 Initial conditions and constrains . . . 21
2.3.2 The energetic stability . . . 23
2.3.3 The geometric stability . . . 24
2.3.4 Procedure and notation . . . 25
2.4 Recurrent hypothesis and peculiarities . . . 25
2.4.2 Possible thrusting law logic . . . 28
3 Stability with Impulsive manoeuvres 31 3.1 Impulsive case analysis . . . 31
3.1.1 Preliminary computations . . . 32
3.1.2 Equivalence ∆e ←→ ∆V . . . 37
3.1.3 Validation . . . 41
3.2 Other results . . . 41
3.2.1 Eccentricity perturbation analysis . . . 43
3.2.2 Some considerations about trajectory shapes . . . 45
4 Stability with Low-Thrust manoeuvres 49 4.1 Low-Thrust equations . . . 49
4.1.1 Perturbed and inhomogeneous model . . . 50
4.1.2 Application of regularization . . . 55
4.1.3 Validation . . . 58
4.2 Results . . . 59
4.2.1 Preliminary computations . . . 59
4.2.2 Enhanced Stability Sets . . . 60
4.2.3 Typical trajectory trends . . . 62
4.2.4 Stability indexes distribution . . . 63
5 Study about energy for capture 67 5.1 Preliminary consideration . . . 67
5.1.1 The Hill surfaces . . . 69
5.2 General strong and weak stability . . . 72
5.2.1 Practical results . . . 73
5.3 Hyperbolic stability criterion . . . 75
5.3.1 Practical results . . . 75
6 Final remarks 77 6.1 Conclusions . . . 77
6.1.1 Impulsive manoeuvres . . . 77
6.1.2 Low-Thrust manoeuvres . . . 78
6.1.3 Energy criteria applications . . . 79
6.2 Prospective work . . . 79
Appendix A The Szebehely method 83 A.1 Time dependant equations . . . 83
A.2 Anomaly dependant equations . . . 84
A.3 Differential relation df /dt . . . 86
Appendix B The regularization procedure 87 B.1 Coordinates Transformation . . . 87
Appendix C The Oberth effect 93 C.1 Parabolic case . . . 94
Appendix D Smart ways to find Lagrangian points 95
D.1 Stability analysis . . . 96 D.2 Quintic form solutions . . . 99
Appendix E Sets of Capture and related TOF 103
E.1 Stability sets and stability indexes . . . 104
E.2 Capture sets and stability indexes . . . 106
E.3 Low-thrust manoeuvres and stability indexes . . . 108
1.1 Hiten-Hagoromo mission: the qualitative trajectory followed for lunar
swing-by . . . 3
1.2 Hiten-Hagoromo probe . . . 3
1.3 CubeSat: the dimension of a nanosatellite . . . 4
1.4 SMART-1 mission: the orbiter . . . 5
1.5 Genesis mission: the probe . . . 5
1.6 GRAIL: the spacecrafts . . . 5
2.1 Geometry of the n-body problem in an inertial reference frame (X, Y, Z) . . 10
2.2 Keplerian conic: planar ellipse in (ˆe,p) reference frame . . . 14ˆ 2.3 The rotopulsating reference frame for ER3BP: polar coordinates . . . 18
2.4 Initial conditions analysed . . . 23
2.5 Particular of initial condition in P2-centred reference frame . . . 23
2.6 Qualitative picture about the loop describing the stability of the motion . . 24
2.7 Correcting an elliptic trajectory with low-thrust . . . 29
3.1 W1 with negative kepler energy (H2<0) at perihelion . . . 32
3.2 W1 with negative kepler energy (H2<0) at aphelion . . . 32
3.3 W1 at perihelion . . . 32
3.4 W1 at aphelion . . . 32
3.5 W1 for one loop with negative kepler energy (H2 <0) at perihelion . . . 33
3.6 W1 for one loop with negative kepler energy (H2 <0) at aphelion . . . 33
3.7 W1 for one loop at perihelion . . . 33
3.8 W1 for one loop at aphelion . . . 33
3.9 Trend of stable points percentage vs eccentricity for stability set W1(0π, e) . 34 3.10 Trend of stable points percentage vs eccentricity for stability set W1(1π, e) . 34 3.11 W−1 for high eccentricity at perihelion . . . 34
3.12 W−1 for low eccentricity at aphelion . . . 34
3.13 Capture set C1 −1 at perihelion with low eccentricity . . . 36
3.14 Capture set C1 −1 at aphelion with low eccentricity . . . 36
3.15 Capture set C1 −1 at perihelion with high eccentricity . . . 36
3.16 Capture set C1 −1 at aphelion with high eccentricity . . . 36
3.17 Trend of stable points percentage vs eccentricity for capture set C1 −1(0π, e) . 36 3.18 Trend of stable points percentage vs eccentricity for capture set C1 −1(1π, e) . 36 3.19 Capture set C1 −1 at perihelion with impulsive control of ∆e = 0.01 . . . 39
3.20 Capture set C1
−1 at perihelion with impulsive control of ∆e = 0.05 . . . 39
3.21 Capture set C1 −1 at aphelion with impulsive control of ∆e = 0.01 . . . 39
3.22 Capture set C1 −1 at aphelion with impulsive control of ∆e = 0.05 . . . 39
3.23 Trend of stable points percentage vs eccentricity for capture set C1 −1(0π, e, 0.01), i.e. perihelion capture . . . 40
3.24 Trend of stable points percentage vs eccentricity for capture set C1 −1(1π, e, 0.01), i.e. aphelion capture . . . 40
3.25 Trend of stable points percentage vs eccentricity for e = [0.9, 0.99] . . . 41
3.26 Trend of stable points percentage vs eccentricity for e = [0.9, 0.98] and ∆e= 0.01 . . . 41
3.27 Comparison example of stable trajectory in different references . . . 42
3.28 Comparison example of unstable trajectory in different references . . . 42
3.29 Maximum eccentricity field in P2-centred frame for W1(0π, e) . . . 44
3.30 Maximum eccentricity field in P2-centred frame for W1(1π, e) . . . 44
3.31 Maximum speed field in P2-centred frame for W1(0π, e) . . . 44
3.32 Maximum speed field in P2-centred frame for W1(1π, e) . . . 44
3.33 Example of stable trajectory (SR= 1.0029) . . . 46
3.34 Example of weird trajectory (SR= 0.5805) . . . 46
3.35 Example of acrobatic trajectory . . . 47
3.36 Example of completely unstable trajectory . . . 47
3.37 Stability indexes comparison with Keplerian period (W1(0π, 0.95)) . . . 47
3.38 Stability indexes comparison with Keplerian period (W1(1π, 0.05)) . . . 47
3.39 Stability indexes distribution for capture (C1 −1(0π, 0.95)) . . . 48
3.40 Stability indexes distribution for capture (C1 −1(1π, 0.05)) . . . 48
4.1 Angles definition for polar perturbed model . . . 55
4.2 Sample of stable trajectory in different references with low-thrust . . . 58
4.3 Sample of unstable trajectory in different references with low-thrust . . . . 58
4.4 W1(0, 0.5) with decelerating low-thrust . . . 59
4.5 W1(0, 0.5) with accelerating low-thrust . . . 59
4.6 Enhanced W1(0, 0.05) with low-thrust . . . 60
4.7 Enhanced W1(π, 0.05) with low-thrust . . . 60
4.8 Enhanced W1(0, 0.95) with low-thrust . . . 60
4.9 Enhanced W1(π, 0.95) with low-thrust . . . 60
4.10 Capture time field for low eccentricity at perihelion . . . 61
4.11 Capture time field for low eccentricity at aphelion . . . 61
4.12 Capture time field for high eccentricity at perihelion . . . 62
4.13 Capture time field for high eccentricity at aphelion . . . 62
4.14 Weird to acrobatic orbit within Moulton sphere . . . 63
4.15 Weird to acrobatic orbit post-capture behaviour . . . 63
4.16 Acrobatic to acrobatic orbit within Moulton sphere . . . 63
4.17 Acrobatic to acrobatic orbit post-capture behaviour . . . 63
4.18 Quasi-Keplerian orbit within Moulton sphere . . . 64
4.19 Quasi-Keplerian orbit post-capture behaviour . . . 64
4.21 Stability index distribution for low eccentricity at aphelion . . . 64
4.22 Stability index distribution for high eccentricity at perihelion . . . 65
4.23 Stability index distribution for high eccentricity at perihelion . . . 65
5.1 Hill’s curves example: opened neck in L1 . . . 70
5.2 Hill’s curves example: closed neck in L1 . . . 70
5.3 Hill’s curves example: critical case (singular curve on L1) . . . 70
5.4 The function of Jacobi’s energy limit Clim(f ) . . . 71
5.5 The regions behind by Hill’s curve: the P1, the P2 capture and the exterior regions . . . 73
5.6 Strong stability for quasi-Keplerian trajectory . . . 74
5.7 Strong stability for acrobatic trajectory . . . 74
5.8 Weak stability for quasi-Keplerian trajectory in unperturbed dynamics . . . 74
5.9 Weak stability for acrobatic trajectory in unperturbed dynamics . . . 74
5.10 Particular for quasi-Keplerian trajectory in unperturbed dynamics . . . 76
5.11 Particular for acrobatic trajectory in unperturbed dynamics . . . 76
5.12 Stability set for Mars perihelion . . . 76
5.13 Stability set for Mars aphelion . . . 76
D.1 Zero-velocity isolines for energy . . . 99
D.2 Particular in P2-centred neighbourhood and relative collinear points . . . . 99
D.3 Gradient curve for potential . . . 99
D.4 Quintic curve in roto-pulsating adimensional reference . . . 100
D.5 Positions of Lagrangian points with respect to the system barycentre in roto-pulsating reference . . . 100
D.6 Quintic curve in simply scaled adimensional reference . . . 102
D.7 Positions of Lagrangian points with respect to the system barycentre in scaled reference . . . 102
E.1 W1(0.5π, 0.05) in P2-centred inertial reference . . . 104
E.2 W1(0.5π, 0.05) Stability index distribution vs pericentre radius . . . 104
E.3 W1(0.5π, 0.5) in P2-centred inertial reference . . . 104
E.4 W1(0.5π, 0.5) Stability index distribution vs pericentre radius . . . 104
E.5 W1(0.5π, 0.95) in P2-centred inertial reference . . . 105
E.6 W1(0.5π, 0.95) Stability index distribution vs pericentre radius . . . 105
E.7 W1(1.5π, 0.05) in P2-centred inertial reference . . . 105
E.8 W1(1.5π, 0.05) Stability index distribution vs pericentre radius . . . 105
E.9 W1(1.5π, 0.5) in P2-centred inertial reference . . . 105
E.10 W1(1.5π, 0.5) Stability index distribution vs pericentre radius . . . 105
E.11 W1(1.5π, 0.95) in P2-centred inertial reference . . . 105
E.12 W1(1.5π, 0.95) Stability index distribution vs pericentre radius . . . 105
E.13 C1 −1(0π, 0.50) in P2-centred inertial reference . . . 106
E.14 C1 −1(1π, 0.50) in P2-centred inertial reference . . . 106
E.15 C1 −1(0π, 0.50, 0.01) in P2-centred inertial reference . . . 106
E.16 C1 −1(1π, 0.50, 0.01) in P2-centred inertial reference . . . 106
E.17 C1 −1(0π, 0.50, 0.05) in P2-centred inertial reference . . . 106
E.20 C1
−1(0π, 0.05, 0.05) in P2-centred inertial reference . . . 107
E.21 C1 −1(1π, 0.95, 0.01) in P2-centred inertial reference . . . 107
E.22 C1 −1(1π, 0.95, 0.05) in P2-centred inertial reference . . . 107
E.23 W1(0π, 0.50) in P2-centred inertial reference with low-thrust . . . 108
E.24 W1(1π, 0.50) in P2-centred inertial reference with low-thrust . . . 108
E.25 Capture time field for W1(0π, 0.50) with low-thrust . . . 108
E.26 Capture time field for W1(1π, 0.50) with low-thrust . . . 108
E.27 Low-thrust control for W1(0π, 0.50) stability index distribution . . . 108
E.28 Low-thrust control for W1(1π, 0.50) stability index distribution . . . 108
E.29 W1(0.5π, 0.05) in P2-centred inertial reference with low-thrust . . . 109
E.30 W1(1.5π, 0.05) in P2-centred inertial reference with low-thrust . . . 109
E.31 Capture time field for W1(0.5π, 0.05) with low-thrust . . . 109
E.32 Capture time field for W1(1.5π, 0.05) with low-thrust . . . 109
E.33 Low-thrust control for W1(0.5π, 0.05) stability index distribution . . . 109
E.34 Low-thrust control for W1(1.5π, 0.05) stability index distribution . . . 109
E.35 W1(0.5π, 0.50) in P2-centred inertial reference with low-thrust . . . 110
E.36 W1(1.5π, 0.50) in P2-centred inertial reference with low-thrust . . . 110
E.37 Capture time field for W1(0.5π, 0.50) with low-thrust . . . 110
E.38 Capture time field for W1(1.5π, 0.50) with low-thrust . . . 110
E.39 Low-thrust control for W1(0.5π, 0.50) stability index distribution . . . 110
E.40 Low-thrust control for W1(1.5π, 0.50) stability index distribution . . . 110
E.41 W1(0.5π, 0.95) in P2-centred inertial reference with low-thrust . . . 111
E.42 W1(1.5π, 0.95) in P2-centred inertial reference with low-thrust . . . 111
E.43 Capture time field for W1(0.5π, 0.95) with low-thrust . . . 111
E.44 Capture time field for W1(1.5π, 0.95) with low-thrust . . . 111
E.45 Low-thrust control for W1(0.5π, 0.95) stability index distribution . . . 111
E.46 Low-thrust control for W1(1.5π, 0.95) stability index distribution . . . 111
List of Tables
2.1 Sensible data of planets in the Solar System . . . 232.2 Sample of thrusters available for future nanospacecraft . . . 27
3.1 ∆V comparison with ∆e = 0.01 for varying pericentre radii for Mars . . . . 38
3.2 ∆V comparison with ∆e = 0.05 for varying pericentre radii for Mars . . . . 38
a Generic scalar variable
A Position or location in Cartesian plane
a Vectorial variable referred to unperturbed model
¯
a Vectorial variable referred to active forces
ˆ
a Vector direction
|a| Vector modulus
Symbol Units Description
a [km] Semi-major axis of osculating orbit
a [m/s2] Active acceleration in 2-body problem
ap [km] Semi-major axis of P1 – P2 elliptic motion
C [km2/s2] Jacobi energy
γ [rad] Flight path angle (clock-wise)
e [−] Eccentricity of osculating orbit
ep [−] Eccentricity of P1 – P2 elliptic motion
E [km2/s2] Keplerian energy
f [−] Adimensional true anomaly (1/2π)
F1 Focus of Keplerian conic
¯
F [−]3 Vectorial adimensional force
¯
F [−]3 Vectorial Adimensional force in roto-pulsating equation
g [m/s2] Earth gravitational acceleration
G [N m2/kg2] Universal gravitational constant
h [km] Periapsis altitude
h [km2/s]3 Angular momentum vector
H [−] Adimensional Keplerian energy in roto-pulsating reference
I [−] Integral term in elliptic dynamic motion equations
Isp [s] Specific Impulse
J Jacobi integral
Li Lagrangian points
mi [kg] Mass referred to i-body
µ [−] Adimensional mass factor
µi [km3/s2] Planetary constant referred to i-body
N [−] Number of initial conditions
O Barycentre of the system /
P1 Larger Primary
P2 Smaller Primary
P3 Spacecraft
ri [−] Adimensional position of spacecraft relative to i-primary
ρi [km] Dimensional position of spacecraft relative to i-primary
rLC [−] Levi-Civita Radius RH [km] Hill radius Rp [−] Adimensional pericentre R Rotation matrix S [−] Stability Index t [s] Physical time τ [−] Fictitious time
T s Dimensional period of the Keplerian orbit
¯
T [−]3 Vectorial Adimensional force in roto-pulsating equation
U [N/kg] Potential function for dimensional, inertial reference frame
vi [km/s] Dimensional speed relative to i-primary
V [−] Regularized potential
Vi [−] Adimensional speed referred to i-primary
θi [rad] Angle between x-axis and P3 relative to i-primary
ω [−] Adimensional potential of elliptic problem
Ω [−] Adimensional potential of circular problem
˙
( ) Time derivative
( )0 Fictitious time derivative /
Planetary anomaly derivative
(X, Y, Z) [km]3 Dimensional variables in inertial reference frame
(x, y, z) [−]3 Adimensional variables in inertial reference frame /
Adimensional variables in roto-pulsating reference frame
(η, ξ, ζ) [−]3 Adimensional variables in roto-pulsating reference frame
(u, v, w) [−]3 Regularization variables
Acronyms
BC Ballistic Capture
BCs Boundary Conditions
CRnBP Circular Restricted n-Body Problem
ERnBP Elliptic Restricted n-Body Problem
IC Initial Conditions
ODE Ordinary Differential Equation
OE Oberth Effect
PIE Planetary Interchange Escape
SSB Solar System Barycentric reference
TOF Time Of Flight
Introduction
Mankind is not a circle with a single center but an ellipse with two focal points of which facts are one and ideas the other.
Victor Hugo
This first chapter will explain, in the easiest way possible, where this thesis comes from, what are the goals and the ideas that guided the reasoning throughout the work. From the beginning up to new proposals that hopefully will be fundamental to improve in future, concerning thrusting criteria and energetic considerations.
1.1
History
1.1.1 From astronomy ...
In early times, the only way to understand what was the logic behind the celestial me-chanics was the observation with naked eye. Then the facts were some lights in black sky moving along strange trajectories and the ideas about this phenomenon were many. As ancient civilization in Mesopotamia, Greece, India, China, Egypt and Central America developed, astronomical observatories were assembled. Even then seemed important to start exploring the nature of the Universe and, in this way, some preliminary interpreta-tions grew up. The first (and most known) of these was to believe that the Earth was the center of the Universe with the Sun, the Moon and the stars rotating around it.
The Geocentric model, or the Ptolemaic theory - from Claudius Ptolemy (AD 100 - c.170), has been the greatest idea, undisputed and confirmed for thousands of years. However wrong it was, it represented the first hypothesis of a new kind of science called astronomy. Since then, the “law (or culture) of the stars” (from Greek etymology), was improved by the greatest mind of mankind.
Nicolaus Copernicus (1473 - 1543) was the first proposing a new Heliocentic model of the Solar System. During the Renaissance this hypothesis was confirmed by the observations of Tycho Brahe (1546 - 1601) and Galileo Galilei (1564 - 1642), and developed with the rudimentary mathematical model of Johann Kepler (1571 - 1630).
Thanks to Sir Isaac Newton (1642 - 1727) the mightiest breakthrough of Modern Astron-omy knew its acme: the universal theory of gravitation formulated in its Philosophiae
Naturalis Principia Matematica gave a solid mathematical foundation to astronomy,
re-lating the motion of planets in terms of geometrical variables. This gravitational model was the evidence to the three laws of planetary motion by Kepler. The same algebra was improved by a simpler notation and simplified by Gottfried Leibniz (1646 - 1716) and finally completed in Pierre-Simon Laplace’s (1749 - 1827) masterpiece five volumes work,
M´ecanique C´eleste, completely published in 1825.
With all these achievements, the attention of the scientific community was partly drawn to the study of the motion of an artificial body put within the gravitational web formed by the celestial bodies, the branch of science known as Astrodynamics. The dynamical model used by both Kepler and Newton in their researches took into account only one attractor at a time, describing a closed form time independent analytical trajectory. It is a sore approximation, valid for two isolated bodies, that is reliable, intuitive, adopted with restrictions in most of classical mechanics qualitative computations.
A more accurate dynamical model is the Restricted Three-Body Problem, often referred to RTBP (or R3BP) in this work, that deals with two main attractors and a non-massive artificial object. The first contribution to this approximation was made by Leonhard Euler (1707 - 1783) in 1722 as aid to his lunar theories, and later by Giuseppe Lodovico La-grangia (1736 - 1813), thanks to his knowledge in Analytical Mechanics. Although a great deal of effort has been put into this dynamical model, no closed-form solution has been found yet. Euler had the brilliant idea of using a rotating reference frame instead of an inertial one, so simplifying considerably the equations of motion; Lagrangia discovered 5 equilibrium points (called Lagrangian points); Carl Jacobi (1804 - 1851) found an integral of motion named after him; and George William Hill (1838 - 1904) depicted the region of coherent motion, starting from energetic considerations. The extraordinary piece of work
L´es M´ethodes Nouvelles de la M´ecanique C´eleste by Henri Poincar´e (1854 - 1912)
estab-lished the fundamentals of the Dynamical System Theory, which he used later as tool to analyse qualitatively the dynamic of the R3BP. Finally, further simplification to solve the resulting differential problem where introduced by Tullio Levi Civita (1873 - 1941) with
his local regularization, fundamental for the algorithms implemented in computers1. All
these contributions are fundamental for the math adopted in the upcoming analysis.
1.1.2 ... to Ballistic Capture
In 1991, the Japanese mission Hiten-Hagoromo, represented the first occasion to complete successfully a low energy transfer, worth to specify, for unforeseen reasons. The Hiten spacecraft was launched in 1990 to the Moon and was supposed to be placed in an highly elliptical Earth orbit with an apogee of 476000 km, which would swing past the Moon. However, the injection took place with a ∆V deficit of 50 m/s, resulting in an apogee of only 290000 km. Anyway on the first lunar swing-by, Hiten released a small orbiter called Hagoromo, into lunar orbit. The transmitter on Hagoromo failed, but its orbit was
visually confirmed from Earth2.
1
Details given in the following chapter and in the appendix.
In order to save the mission, Edward Belbruno and James Miller helped to salvage the operation by developing a so-called ballistic capture trajectory that would enable the main Hiten probe to itself enter lunar orbit. Belbruno had been working on numerically modelling low-energy trajectories, and heard of the probe’s problems. He developed a trajectory for the main probe to enter lunar orbit and on June 21, 1990, sent an unsolicited proposal to the Japanese space agency. They responded favourably, and later implemented a version of the proposal.
The trajectory Belbruno and Miller developed for Hiten used Weak Stability Boundary
Theory and required only a small perturbation to the elliptical swing-by orbit, sufficiently
small to be achievable by the spacecraft’s thrusters. This course would result in the probe being captured into temporary lunar orbit with no costs (called ballistic transfer), but required five months instead of the usual three days for a Hohmann transfer.
Figure 1.1: Hiten-Hagoromo mission: the qualitative trajectory followed for lunar swing-by
Figure 1.2: Hiten-Hagoromo probe
This was the first time a satellite had used low energy transfer to transfer to a Moon orbit. On October 2, 1991, Hiten was captured temporarily into lunar orbit. In an unconventional way: an unpredictable event was turned into an opportunity.
That was the occasion to see the orbital transfers from a new point of view. For the first time it was not important the time of flight taken into account by the standard bi-elliptic manoeuvre, but the cost and the safety. Other missions were then repeated with the same philosophy ever since.
1.2
Motivation
The most intuitive way to solve a problem is not necessary the best, a fortiori in un-expected. As matter of fact, this new method appears to have great potential also for interplanetary and deep space transfers. Finding a guide line to search optimal solution in this sense, could become the way for undertaking space travels with potentially great improvements in terms of safety, cost (in terms of payload and/or propellant), launch window and reasonable TOF.
Anyway, the analysis is not simple and, in some way can appear stimulant. In fact, all the provided models do not have a closed and simple analytical form. All need a numerical approach performed by a computer. The algorithms in this sense must respect the best trade off between computational cost and accuracy of the solutions, in this case more than in others similar (e. g. thermal study, structural analysis). It is worth to mention that even small errors in the initial part of the integration, typically of truncation, can lead to complete different results from the exact solution at the end of integration. The results must be checked every time and are sometimes not trivial. This is the reason why a smart, well-imposed numerical approach probably represents the most efficient way to solve these kind of problems up to date.
The practical example taken into account is the CubeSat3, a nanosatellite. The small
di-mension and inertia of this system can be a good opportunity to understand what results can be obtained by scaling the problem for bigger spacecraft. All the propulsive control are introduced in the model, compatible with the real thrusters available on the market.
Figure 1.3: CubeSat: the dimension of a nanosatellite
1.3
State of the art
It is not possible to give in few lines or pages the complete bibliography behind these algorithms and the general astrodynamics, even if some authors have been already men-tioned and others may be found at the end of the present work. The thesis is principally guided from relevant literature and different contributions about this topic, listed in the Bibliography.
The material that started this work was firstly provided by professor Topputo, supervisor of the thesis. The ground explaining what WSB, Ballistic Capture with energetic and geometric considerations are, was given by different papers and some bibliographical ref-erences, mostly written by Belbruno (the already mentioned creator of the theory), the same Topputo, Hyeraci and Dei Tos.
Since the first implementations on computers, two things were very clear: the difficulty of organizing in a good way all the variables with acceptable tolerances and the errors that could come out from the model, independently how correct it was. It became of primary importance to know the limit of the transfers, the strengths and the possible improvable points and, in a wider sense, the technology readiness level. For this reason it was necessary for the author to get some information about all the other successful missions adopting low-energy transfers, after the Hiten-Hagoromo lucky event. Hereby some accomplished ones are listed:
SMART-1: Swedish-designed European Space Agency satellite that orbited around the Moon, propelled by a solar-powered Hall effect thruster (Snecma PPS-1350-G) using xenon as propellant. Goals: testing new propulsive system, collection of spec-tral emissions (Sun and deep space) and collection of images. Terminated in 2006 with a programmed lunar impact.
Genesis: NASA return probe that collected a sample of solar wind and returning back for analysis. Electrostatic propulsion with hydrogen. Crash-landed in Utah in 2004, after a design flaw prevented the deployment of its drogue parachute. This mis-sion did not require a capture on a massive body but simply exploited the dynamics
in the neighbourhood of Lagrangian points L1 and L2.
GRAIL (Gravity Recovery and Interior Laboratory): American lunar science mis-sion in NASA’s Discovery Program which used high-quality gravitational field map-ping of the Moon to determine its interior structure. Two probes, chemical
propul-sion with pressurized hydrazine (N2H4). Exploited ballistic capture.
Figure 1.4: SMART-1 mission: the orbiter
Figure 1.5: Genesis mission: the probe
Figure 1.6: GRAIL: the space-crafts
Missions’ modules that adopted the low-energy transfer strategy.
From the knowledge of previous missions it appeared necessary to study a good model for the low thrust case. This can represent a particular case of modified dynamic of the problem under control and/or under any perturbation. The fundamental contributes then become the bibliography of Victor Szebehely (1921-1997) about the CR3BP and ER3BP (respectively Circular and Elliptic problem). With the relative local regularizations that it is possible to perform either near to the primary (e.g. Sun) and the secondary body (e.g. Mars) to avoid singularities in the solution. Some passages required a more sophisticated
approach, based on ideas described in Celletti’s paper about the regularization for this kind of problems. The work required a lot of re-elaboration of already fixed discoveries. A path made of inhomogeneous differential problems, methods to avoid singularities, algorithmic precautions to minimize truncation errors and patience.
1.4
Goals
This work aim to better understand the concept of ballistic capture and analyse how it can be influenced by any kind of perturbation. For this reason some criteria and new models are introduced, as we are talking about such innovative theory. These methods can represent the occasion to improve the interplanetary transfers in terms of payload and security. In fact it’s not excluded that such strategy, with the incoming considerations, could be applied also for eventual manned missions in the future. Then the fundamental goal is to search for the necessary conditions in order to accomplish successfully ballistic capture, for a generic active control.
After such conditions are defined, the author will try to formulate new criteria, based on energetic consideration aimed to understand what link can exists between the initial condition of the body under study and the orbit it is going to tread.
1.5
Organization of the work
The organization of the work follows a precise logic, described hereby:
Dynamic model and algorithms: The argumentation is introduced by the treatment of the adopted models. From the n-Body Problem, through the ER3BP in synodic reference frame up to the regularization modified for the peculiar case of low-thrust, and not always adopted in computations. Finally the complete description of the ballistic capture, with hypothesis, limitations, constrains and eventual modifications with some preliminary and interesting results.
Stability with Impulsive manoeuvres: The impulsive case analysis starts from the natural trajectory behaviour and presents modifications only in the grid initial conditions. This chapter aims to understand how Keplerian parameters involved in the model can influence Stability Sets and Capture Sets (to be defined concepts). Then the results are shown with useful plots to fully comprehend how much an impulsive manoeuvre can mod-ify the ballistic capture and what conditions can be favourable to the goal.
Stability with Low-Thrust manoeuvres: This a critical case, because a totally new model is adopted. As some results can appear not so trivial, but reasonable, a verification in parallel with a Cartesian model is performed. Then it is showed how the possible area of capture can increase with a proper constant acceleration, and the benefits it can lead to. To fix the ideas the shape of the trajectories involved and the relative TOFs are showed. Weak and Strong Stability criterion: After the chapters about the generic active control we have a report about a criterion as reasonable from the theoretical point of
view as useful to the implementation of a control law for a generic low-energy transfer. Consequently the analysis focuses on the concept of Hill surfaces, highlighting then the peculiarities of the criterion and the not negligible differences between ER3BP and CR3BP. Final remarks: The conclusion represents a final overview about the covered topics with some considerations and a parallelism between the two thrust strategies is possible to adopt. Final remarks and observations are objectively discussed and prospective work is suggested.
Dynamic model and algorithms
In order to solve this differential equation you look at it till a solution occurs to you.
Gy¨orgy P´olya
The theory, the equations, the reference frames and the models behind the computer pro-cessing are given hereby. But firstly is worth to recall some basic definitions recurrent throughout the thesis.
Gravitational problems are known as general, if the motion of the body in analysis in-fluences the motion of the other celestial bodies, or as restricted if it does not. Then, dynamical models can be classified as coherent, if the motion of the primaries is mutually influenced by their gravitational forces, or non-coherent (or incoherent) if their motion arises from approximated functions of time, a priori assigned. Finally, the differential system describing the gravitation field of the problem can be defined as autonomous, if it does not depend explicitly on the time variable, or non-autonomous, if time appears explicitly in the dynamical equations.
2.1
The n-Body Problem
The law zero in astrodynamics is the Newton’s law of Universal Gravitation. It represents the base upon which we can build all the rest of the discussion. The first form of this model, was comprehensive of only two bodies in bi-dimensional case. However we will skip that version and we start directly from the well-known vectorial form for n-bodies. Given a free space with n point masses, the model describes the motion of mass particle with index i interacting with the others n − 1 gravitational fields. The dynamics of the mass
particles mk with a position Rk= [Xk, Yk, Zk]T ∈ R3 with k = 1, . . . , n is governed by the
following relation: Fjk = n X j=1,6=k Gmjmk Rj− Rk |Rj− Rk|3 ∀k = 1, . . . , n (2.1)
where:
G= 6.6741011 N m2/kg2 is the universal gravitational constant
|Rj− Rk| = Rjk is the euclidean distance between bodies j and k
Applying the hypothesis of constant masses for the celestial bodies and for the artificial satellite whose dynamics we are interested in, the equation may be written as:
mkR¨k= n X j=1,6=k Gmjmk R3jk (Rj− Rk) ∀k = 1, . . . , n (2.2)
The equation 2.2 above is written in inertial reference frame and represents a set of 3n ordinary second order differential equations, or equivalently a set of 6n first order ordinary differential equations. As the gravitational force is conservative, it admits a potential energy written as:
U = n X j=1,6=k Gmjmk Rjk (2.3)
U is a function of the 3n variables Rk. Provided Rjk > 0, it has been demonstrated18
that U is a smooth1 and well-defined function of its 3n variables. In the case at hand,
Rjk comes from a vectorial norm, and it can be either positive or null. In this last case
collision occurs and this function become singular2. However, considering the constraints
imposed by the real bodies (i.e. fortunately in this case planets all have a limiting radius) we avoid all kinds of numerical problem in upcoming calculations (→ regularization).
m1 mk mj ms X Y Z R1 Rk Rj Fk1 Fj1 F1k Fjk F1j Fkj F1 Fj Fk
Figure 2.1: Geometry of the n-body problem in an inertial reference frame (X, Y, Z)
1In mathematical analysis, the smoothness of a function is a property measured by the number of
derivatives it has which are continuous. A smooth function is then a function that has derivatives of all orders everywhere in its domain → no singularities occur.
2
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is “special” (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. A point of an algebraic variety which is not singular is said to be regular.
Considering now the gradient operator like ∇k = [∂/∂xk, ∂/∂yk, ∂/∂zk]T with respect to
the vector Rk, the set of differential equations governing the motion of n mass
parti-cles moving in a Cartesian three-dimensional space (→ f (X, Y, Z)) under their mutual gravitational influence, might be written as:
mkR¨k = ∇kU ∀k = 1, . . . , n (2.4)
The n celestial bodies have been considered point masses so far, that is their real shape is neglected. Their total mass is concentrated at the barycentre and the gravitational forces, exerted and felt, act on the same point. This assumption corresponds to a perfect spherical
celestial body with central symmetric internal mass distribution. It can be proved38 that
the gravitational potential of a sphere with symmetric mass distribution coincides with the potential of a point placed at the sphere barycentre whose mass is the total sphere mass. The oblateness of a celestial body can give rise to incongruence from the expected orbit, and is regarded as a perturbation in classic astrodynamics (a.k.a. Non-uniform gravity
field17).
In order to cast the initial value problem in a more compact state-space form, position and
velocity vectors are grouped into x = (R1, Rk, ..., Rn, ˙R1, ˙Rk, ..., ˙Rn)T ∈ R6n, for which
the relation holds:
˙x = f (x) (2.5)
where:
f = [V1, Vk, ..., Vn, ∇1U/m1, ∇kU/mk, ..., ∇nU/mn]T
Note that the vector field f does not directly depends on the epoch, that is the gravita-tional dynamics cast onto an inertial frame of reference leads to an autonomous dynamical
problem. Again it can be proved3that thanks to the smoothness property of the potential
energy, the initial value problem admits a unique solution once a set of proper 6n initial conditions are given.
2.1.1 The 2BP case
The system of equations 2.4 represents the complete dynamical description of the n-body problem, where all the position vectors of the celestial bodies are unknown and the acceleration of each mass particle depends on all the remaining ones. This is the general n-body problem. However, in the case we are going to analyse the hypothesis of restricted dynamics can be applied with accurate results. The artificial object moves in the vectorial field created by the n celestial bodies (or primary/secondary bodies), or alternatively it does not affect the motion of the n main attractors. It is quite clear that the mass of the artificial object under study has to be significantly less than the other objects masses for
this assumption to be valid. Considering the Figure 2.1 it means ms m1. For this case,
system of differential equations3 hold: d2r1 dt2 = −Gm2 r1− r2 |r1− r2|3 d2r 2 dt2 = −Gm1 r2− r1 |r2− r1|3 d2rs dt2 = −Gm1 rs− r1 |rs− r1|3 − Gm2 rs− r2 |rs− r2|3 (2.6) where:
ri = Ri+ O refers to the Cartesian coordinates in primaries barycentric reference
Another significant simplification is obtained if the trajectories of the primaries are proper time-dependent functions, obtaining thus a coherent model, like the R3BP cases. This model will be soon studied.
Given S, set of celestial bodies, the motion of an artificial satellite is represented by a system of three second order differential equations. This yields to:
¨ R=X j∈S µj Rj− R |Rj− R|3 (2.7) where:
µj = Gmj [km3/s2] is the planetary constant of planet j
R= [X, Y, Z]T is the position of our spacecraft for a non-inertial reference frame
2.1.2 The Keplerian model
Given the basic model, we enter more in the specific with the Keplerian theory. The fundamental idea is to find an analytical way to uniquely define the shape of trajectory
for two bodies. The equations describing the dynamics of the particle masses m1 and m2
read: ¨ R1= −µ2 R1− R2 |R1− R2|3 (2.8) ¨ R2= −µ1 R2− R1 |R2− R1|3 (2.9) Subtracting the first from the second equation, the relative dynamics is obtained:
¨r = −µ2b
r
|r|3 (2.10)
(2.11) where:
r= R2− R1 is the relative vector of position
µ2b= µ2+ µ1= G(m2+ m1) is the planetary constant gravitation4
The second order differential equation describing the relative motion of m2 about m1,
and vice versa, has a closed form solution. It is interesting to notice that this does not mean that the complete 2BP has been solved because inertial information of one of the mass particle is yet to be found. The last equation possesses integrals of motion that greatly simplify the mathematical treatment of the problem, illustrated hereafter.
Detailed mathematical proofs of the following conservation laws and of the basic
princi-ples underlying the Kepler problem can be found in literature16. The following statements
hold:
- Balance for conservation of energy: d dt | ˙r|2 2 − µ2b |r| = 0 → E = |˙r| 2 2 − µ2b |r| (2.12)
- Balance for conservation of angular momentum: dh
dt =
r ∧˙r
dt = 0 (2.13)
where:
h is the angular momentum vector
- The trajectory is a conic respecting the following relation:
r(θ) = p
1 + e cos θ (2.14)
where:
pis the semi-latus rectum
θ(f if referred to primaries) is the true anomaly
e(ep if referred to primaries) is the eccentricity
For instance e = |e| is the norm of a time invariant vector (→ de/dt = 0). Confirmed by the well-known three Keplerian laws:
1. The orbit of each planet is an ellipse, with the Sun at a focus.
The law explains itself considering the orbits the masses must follow are conic sec-tions.
2. The line joining the planet to the Sun sweeps out equal areas in equal times. Considering |h| = h The aerial velocity is a constant for the two-body problem:
dA
dt = h/2 (2.15)
4
Usually if m2 m1 → µ2b = Gm1. This means that the motion of the mass particle m1 is not
s F1 eˆ Apse line ˆ p r θ
Figure 2.2: Keplerian conic: planar ellipse in (ˆe,p) reference frameˆ
3. The square of the period of a planet is proportional to the cube of its mean distance from the Sun.
The lapse of time required by the mass particle to complete a revolution around its focus is the orbital period T. Using the second law for the entire closed orbit and manipulating the results, the period formula reads:
T2
a3 =
4π
Gm1
= constant (2.16)
From these relation it is easy to get a general angular velocity for a closed system of two primaries, like:
n=r µ2b
a3 (2.17)
also known as mean motion. As we will see, this quantity is fundamental in definition of a convenient reference frame for the dynamic analysis.
2.1.3 The Restricted Three-Body Problem
This model is a better approximation of the real dynamics and plays, in this thesis, the role of simplified reference model. Starting from the Cartesian equations 2.6 and following Keplerian theory on what concerns the two primaries identified by indexes 1 and 2, it is possible to create such simpler model. Formally, the restricted hypothesis requires the mass
of the third body to be much smaller than the masses of the primaries, ms m2 < m1.
This means that the cluster formed by the primaries actually moves according to Kepler’s
laws and msdoes not influence their motion. The main consequence is that the barycentre
of the system coincides with the m1, m2pair’s and it moves obviously in an inertial fashion;
in this way it is very useful for a very convenient reference frame. Hereafter the motion of
the primaries will be treated as a known function of time, R1 = R1(t) and R2 = R2(t).
four-body problem33. In fact, if the specified functions respect Newton gravitational equations
(i.e., ephemeris data), then the problem is still coherent. The third mass particle moves
in the gravitational vectorial field created by the primaries. Note that, if initially ms has
no position and velocity components out of the primaries motion plane, it can but orbit in the same plane.
Going deeper in the topic we can distinguish between the circular and the elliptic problem.
The first case is the simpler, good for very small planetary eccentricity (ep ∼ 0). As the
primaries are revolving in circular orbits on the plane (X, Y ) at constant angular speed ω, the resulting problem is called Circular Restricted Three Body Problem (or CR3BP). Defining a and b the orbital radii of the primary and secondary body from barycentre, respectively, and the angle between the X reference axis and the vector from the origin to
the smaller primary m2as θ = n(t−t0), the circular motion yields R1 = −a[cos θ, sin θ, 0]T
and R2= b[cos θ, sin θ, 0]T. The explicit dynamics of R = [X, Y, Z]T ∈ R3:
¨ X= −G m1 X+ a cos θ |r1|3 + m2 X − bcos θ |r2|3 ¨ Y = −G m1 Y + a cos θ |r1|3 + m2 Y − bcos θ |r2|3 ¨ Z= −G m1 Z |r1|3 + m2 Z |r2|3 (2.18) with: |r1| =p(X + a cos θ)2+ (Y + a sin θ)2+ Z2 |r2| =p(X − b cos θ)2+ (Y − b sin θ)2+ Z2
The dynamics written in the sidereal inertial frame gives birth to a non-autonomous set of differential equations whose closed-form solution has not been found yet.
The system of differential equations 2.18 describing the motion of a particle subjected to the gravitational attraction of two primary bodies can be expressed in a more convenient way, which transforms the set of equations in an autonomous set. The basic idea is to find a reference frame that results in a time-independent force function. Euler first proposed the synodic frame of reference, this one is again centred at the primaries’ barycentre but it is rotating so as to maintain the primary bodies at a fixed position in space.
For the assumptions made so far the synodic frame rotates at the very same primaries angular speed, n. The rotation matrix for the 3 dimensional case is written like:
R = cos θ − sin θ 0 sin θ cos θ 0 0 0 1
Let ρ be the third body position vector with respect to the synodic frame, the rotation
simply yields R = Rρ. It is easy to see that ρ1 = [−a, 0, 0]T, ρ2 = [b, 0, 0]T and ρ =
[x, y, z]T. By a simple inspection of the geometry and considering barycentric properties,
between them, like: a= m2 Ml b= m1 Ml (2.19) where:
M = m1+ m2 is the sum of the masses
l= a + b is the mutual distance
From equation 2.7 the dynamics in the synodic reference becomes: ¨ ρ+ 2RTR˙Tρ˙+ RTRρ = −G¨ m1 ρ − ρ1 |ρ − ρ1|+ m2 ρ − ρ2 |ρ − ρ2| (2.20) where the properties of a rotation cosine angle matrix have been used, namely the orthonor-mality and the invariance with respect to the vectorial norm. The first time derivative of the rotation matrix R is:
˙ R = −n sin θ −n cos θ 0 ncos θ −n sin θ 0 0 0 0
and the second one is:
¨ R = −n2cos θ n2sin θ 0 −n2sin θ −n2cos θ 0 0 0 0
Developing with some passages the equation 2.20 it is possible to write the explicit form for all the components:
¨ x −2n ˙y = n2x − G m1 x+ a ρ31 + m2 x − b ρ32 (2.21) ¨ y+ 2n ˙x = n2y − G m1 y ρ31 + m2 y ρ32 (2.22) ¨ z= −G m1 z ρ31 + m2 z ρ32 (2.23)
with ρ1 and ρ2 norm of the distances between spacecraft and primaries:
ρ1 =p(x + a)2+ y2+ z2 (2.24)
ρ2 =p(x − b)2+ y2+ z2 (2.25)
Note that the differential equations now do not depend directly on time and the set is hence autonomous. As it is showed in appendix (concerning specifically the ER3BP, taken from the work of Szebehely), the restricted problem depends on only one parameter by a proper adimensionalisation. This procedure is such that the distance between the primaries, the angular speed and the sum of their masses is set to a unity value. Then the problem
changes as follows:
µ= m2
m1+ m2
τ = nt
Associated to the selection of position vectors for primaries ρ1 = [−µ, 0, 0]T and ρ2 =
[1 − µ, 0, 0]T, yielding to the classic 3D problem:
¨ x −2 ˙y = Ωx ¨ y+ 2 ˙x = Ωy ¨ z= Ωz (2.26)
where the 3BP potential function can be expressed as:
Ω= 1 2(x 2+ y2) +1 − µ ρ1 + µ ρ2 +1 2µ(1 − µ) (2.27)
The constant term depending on µ only has been added for energetic considerations forth
discussed in Chapter 5. Terms ρ1 and ρ2 are the scalar distances between the third mass
and the primaries:
ρ1 =p(x + µ)2+ y2+ z2
ρ2 =p(x − 1 + µ)2+ y2+ z2
The equations depend only on the mass parameter µ and have time as independent vari-able.
2.2
Exploited differential models
In the following the basic systems already used in the scientific literature are given, com-prehending the polar equations of the ER3BP and the local regularizations. The notation is kept as simpler as possible in this preliminary section. The complete demonstrations and the explications about how to get to these algorithms are left in Appendices A and B.
2.2.1 The Elliptic Restricted Three Body Problem
The planar motion of a massless particle P3, is studied under the gravitational field
gener-ated by the mutual elliptic motion of two primaries, P1, P2, of masses m1, m2, respectively.
The mass parameter of the system is µ = m2/(m1+ m2). It is assumed that P3 moves in
the same plane as P1, P2, under the dynamics: (
x00− 2y0 = ωx
y00+ 2x0 = ωy
(2.28)
The equations of motions above35 are written in a non-uniformly rotating, barycentric,
µ,0), respectively. Moreover, the coordinate frame isotropically pulsates as the P1 – P2 distance, assumed to be the unit distance, varies according to the mutual position of the two primaries on their orbits. The primes in equations 2.28 represent differentiation with respect to f , the true anomaly of the two primaries. In this problem it is the independent variable and plays the role of the time. f is equal to zero when P1, P2 are at their periapse, as both primaries orbit their barycentre in similarly oriented ellipses having
common eccentricity ep. Normalizing the period of P1, P2 to 2π, the dependence of true
anomaly on time reads:
df dt = (1 + epcos f )2 (1 − e2 p)3/2 (2.29) The right side terms of system 2.28 is equivalent to:
ω= Ω
1 + epcos f
(2.30) with Ω already defined in 2.27 and ω angular velocity for elliptic problem.
P3 P2(1 − µ, 0) P1(−µ, 0) O x y r2 r1 θ1 θ2
Figure 2.3: The rotopulsating reference frame for ER3BP: polar coordinates
From numerical point of view one of the most suitable equations relative to this specific reference are the ones expressed in polar coordinates. The simple transformation is made
carrying out (xi, yi) → (ri, θi) relative for both P1 neighbourhood:
r001 − r1θ102− 2r1θ01= 1 1 + epcos f r1 1 − µ r23 −1 − µ r12 + µ cos θ1 1 r23 − 1 r1θ100+ 2r10θ01+ 2r01= µsin θ1 1 + epcos f 1 − 1 r23 (2.31)
and P2: r200− r2θ202− 2r2θ02= 1 1 + epcos f r2 1 −1 − µ r3 2 − µ r2 2 + + (1 − µ) cos θ2 1 − 1 r3 1 r2θ002+ 2r 0 2θ 0 2+ 2r 0 2= (1 − µ) sin θ2 1 + epcos f 1 r3 1 − 1 (2.32)
These equations in polar coordinates are properly integrated for every trajectory rising from the initial conditions grid that will be shown later.
2.2.2 Regularization
Computing stable sets involves integrating many thousands of orbits, and some of them can turn out to be collisions of P3 with either P1 or P2. In such cases, the numerical
integration of equations 2.31 and 2.32 are singular or ill posed when respectively r1 → 0
or r2 → 0. This causes the integrator either to fail or to meet the integration tolerance
at the cost of a prohibitive reduction of the integration step-size. It is therefore necessary to regularize the equations of motion to both avoid such singularities and improve the efficiency of the numerical integration. Levi-Civita local transformation is convenient to use.
Two different transformations are used32:
x+ iy = w2− µ (2.33)
in a neighborhood of P1 and
x+ iy = w2+ 1 − µ (2.34)
around P2, with w = u + iv and i =√1. For both transformations, the regularized vector
field yields: d2u dτ2 − 8 dv dτ(u 2+ v2) = ∂ ∂u4Vi(u 2+ v2) − 8uI d2v dτ2 + 8 du dτ(u 2+ v2) = ∂ ∂v4Vi(u 2+ v2) − 8uI (2.35) where: V1 = 1 1 + epcos f 1 2(u 2+ v2)2− 2µ(u2− v2) + µ2 + 1 − µ r1,1 + µ r2,1 +1 2µ(1 − µ) −c 2 (2.36) V2 = 1 1 + epcos f 1 2(u 2+ v2)2+ 2(1 − µ)(u2− v2) + (1 − µ)2 + 1 − µ r1,2 + µ r2,2 +1 2µ(1 − µ) −c 2 (2.37)
with: r1,1=p(u2+ v2)2− µ [2(u2− v2) − µ] (2.38) r1,2=p(u2+ v2)2− (1 − µ) [2(u2− v2) + 1 − µ] (2.39) r2,1=p(u2+ v2)2− (1 − µ) [2(u2− v2) + 1 − µ] (2.40) r2,2=p(u2+ v2)2− µ [2(u2− v2) − µ] (2.41) and where: I = e Z Ω (1 + e cos f )2df = e Z Ω (1 + e cos f )24(u 2+ v2)dτ (2.42)
Notice that the new independent variable is the fictitious time τ , substituting the true anomaly of the equations expressed in physical variables. Using this kind of transformation
means to choose arbitrarily a radius, for sake of simplicity called Levi-Civita radius (rLC),
within which the local regularization applies. This limit is fixed for this adimensional
problem as = 10−3 (≈ 200000 km for the Sun – Mars case). Whenever the computation
is not considered too slow (i.e. for short times of integration), this approach is completely discarded.
2.3
Construction of stable sets
From now on, all the treatment will be discussed for the specific case of the Sun – Mars sys-tem, conserving the opportune generalities. Instead of using the usual approach, studying the dynamics about the equilibria, this method focusses on the region about the primary, not requiring global extension of invariant manifolds. The stability definitions given in a few pages can be easily extended to any n-body vector field. This allows using such procedure within more refined models taking into account fourth-body perturbations and
planetary eccentricities33. In addition, stable sets can be constructed by holding fixed
orbital parameters (i.e., eccentricity), so matching possible prescribed mission constraints. However, the stable sets method is not devoid of drawbacks, as the following list illustrates: Stable sets are constructed by sampling the physical space and integrating hundreds of orbits. In general, this process is more computationally intensive than flowing and manipulating invariant manifold sets. The number of orbits to be integrated increases with the accuracy required for the representation of the stable sets. While the invariant manifolds allows us to explain free transport phenomena in
the circular restricted three-body problem, the stable sets seem to have much less information from a dynamical system perspective. They are used as black-box tools
to locate feasible capture orbits. Nevertheless, recent studies5 show that the weak
stability boundaries and the Lyapunov stable manifolds overlap for certain energy levels.
In this context it is performed a modification on the concept of stable set to design bal-listic capture orbits. In particular, stable sets are first derived and then manipulated in order to design orbits with prescribed orbital parameters and stability number. The latter indicates the number of turns performed about the smaller primary after capture. The dynamical model used is the planar, elliptic restricted three-body problem with the Sun and an inner planet as primaries. The scope is designing ballistic capture orbits upon arrival in interplanetary transfers.
Hence the so-called backward stability is also introduced, associated to stable sets and suit-ably manipulated to automatically derive regions that support weak capture. In essence it’s clear that also the unstable trajectories are fundamental during the approach of a secondary massive body (e.g. Mars). In fact the backward instability, combined with the forward stability, is worth to define the capture. This concept will be better explained later. Particularly, in the case of capture at Mars the derived solutions can be connected
to low-thrust interplanetary legs obtained through a shape-based approach31.
2.3.1 Initial conditions and constrains
Generating stable sets means finding the portion of configuration space (defined with r2,
θ2) that gives rise to stable orbits for fixed values of (e, f0), with e osculating eccentricity
and f0 initial planetary true anomaly. In practice, this involves defining a grid of initial
conditions. This is achieved by setting the periapsis altitudes and polar angles like:
ri = rL+ (i − 1) rU− rL Nr− 1 for i = 1 : Nr θj = θj+ (j − 1) θU− θL Nθ− 1 for j = 1 : Nθ where:
rL is the lower limit of r2 taken as the radius of Mars
rU is the upper limit of r2 taken as the Hill’s sphere radius (RH)5
θLand θU are included in the [0, 2π] interval for θ2
The Hill’s sphere is defined as that region of space in which the gravitational force of a secondary massive body for R3BP is more influencing than the primary body to the
dynamics of the minor body6. Mathematically this radius is approximated as:
RH ≈ ap(1 + ep)3
r µ2
3µ1
(2.43) where:
ap refers to the semi-major axis of Mars orbit around the Sun
5Notice that this limit can be increased accordingly to the study the reader is meant to deepen. All the
results given are valid for a small region close to Mars but nothing is excluded that this approach can be extended to larger domains. In this thesis only equispaced grids are used.
6
In more precise terms, the Hill sphere approximates the gravitational SOI of a smaller body in the face of perturbations from a more massive body. It was defined by the American astronomer George William
ep refers to the planetary eccentricity
µi = Gmi is the standard gravitational parameter
The total number of initial conditions is given by the product N = NrNθ. Once the
initial position is fixed the problem needs the initial speed to return an unique solution. The Keplerian pericentre velocity is chosen, selecting a local eccentricity for each numerical start. In absolute value we have:
|vi| =r µ
ri
(1 + e) (2.44)
That is an adimensional quantity as usual. As the polar variables are adopted for the Sun-Mars dynamics, the starting condition for the numerical algorithm must be converted in such coordinates, respecting the roto-pulsating reference frame. For instance we have:
r2(f0) = ri r20(f0) = − riepsin f0 1 + epcos f0 θ2(f0) = θj θ20(f0) = s µ(1 + e) r3i(1 + epcos f0) − 1 (2.45)
relative to the equations 2.32, and
r1(f0) = q r2 i + 2ricos θj+ 1 r10(f0) = r02(f0) cos(θj− θ1(f0)) − r2(f0)θ02(f0) sin(θj− θ1(f0)) θ1(f0) = arctan risin θj 1 + ricos θj θ10(f0) = r20(f0) r1(f0) sin(θj− θ1(f0)) + r2(f0)θ20(f0) r1(f0) cos(θj− θ1(f0)) (2.46)
relative to the dynamics with the Sun in 4.37.
Of course the integration must be stopped every time a crash with a planet occurs. The general limits for R3BP are given by:
r1(f ) ≤ R1
r2(f ) ≤ R2
where:
Ri is the limit radius of the planet approximated as an uniform sphere
This approach can be repeated for every couple of primaries respecting the R3BP hy-pothesis. By way of example, some values are given in the Table 2.1 in the following.
x 0.994 0.996 0.998 1 1.002 1.004 1.006 y ×10-3 -4 -3 -2 -1 0 1 2 3 4
Adimentional positions for ICs Qualitative velocities for ICs Hill Sphere
Figure 2.4: Initial conditions analysed
x 0.997 0.998 0.999 1 1.001 1.002 1.003 y ×10-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Adimentional positions for ICs Qualitative velocities for ICs Hill Sphere
Figure 2.5: Particular of initial condition in P2-centred reference frame
In this case is represented a 561 points polar equispaced grid. The initial conditions are taken Keplerian. The closer we are to P2, the higher will be the pericentre speed. The velocities relative for the points in the grid are qualitative, drew only to let the reader better understand what we are taking into account.
Planet Mass Radius Semi-major axis e Revolution period
[kg] [km] [km] [JY] Earth 5.97219 1024 6378 149.59 106 0.0167 1.000 Venus 4.8690 1024 6051 108.21 106 0.0067 0.615 Mars 6.4191 1023 3393 227.92 106 0.0934 1.880 Mercury 3.33 1023 2439 57.91 106 0.2056 0.241 Sun 1.989 1030 695000 - - -Jupiter 1.8987 1027 71492 778.57 106 0.0483 11.862
Table 2.1: Sensible data of planets in the Solar System
2.3.2 The energetic stability
If we consider the Keplerian energy per unit of mass of P3 with respect to P2, we get:
H2(f ) = 1 2v 2 2(f ) − µ r2(1 + epcos f ) (2.47)
where v2is the speed of P3 relative to P2-centred roto-pulsating reference frame. Adopting
the polar coordinates returns:
v22(f ) = r2epsin f 1 + epcos f + r20 2 + r22(1 + θ02)2 (2.48)
It’s clear H2 depends on the current value of planetary anomaly. For a given state
x = (r2, θ2, r2, θ2), the Kepler energy relative to P2 (i.e. Mars) varies even in case the
highlights the fundamental role of the independent variable in evaluating the ballistic cap-ture condition.
We will talk about energetic stability only if H2(x) < 0 verified. Notice that this criterion
was first formulated for an unperturbed model of the two body dynamics. Since the theory we can say we are treating such different case and consequently that this concept cannot fit properly the incoming analysis. A most rigorous prove is given later in Chapter 4 about the effects of low-thrust.
2.3.3 The geometric stability
Since the first publications about WSB and BC6, 20, the energy principle was requested to
be satisfied with the geometric criterion at the same time.
Defined a radial segment l(θ) comprehensive of all the initial conditions r(θ, e, f0), for a
fixed value of true anomaly, as already mentioned, P3 is initially set at the periapsis of an osculating prograde ellipse around P2. In this case the motion is said to be n-stable if the mass P3 leaves l(θ), makes n complete revolutions about P2 (n ≥ 1), and returns to l(θ) on a point with negative Kepler energy with respect to P2, without making a complete revolution around P1 along this trajectory. The motion is otherwise said to be n-unstable. Figure 2.6 summarizes what is requested by definition.
P2 P1 ξ l(θ2) stable unstable θ2
Figure 2.6: Qualitative picture about the loop describing the stability of the motion
A full turn of P3 around P2 is given by the following statement:
|θ2(fi) − θ2(f0)| = 2π (2.49)
not respecting together the loop around P1 obtained by:
![Figure 3.25: Trend of stable points percentage vs eccentricity for e = [0.9, 0.99] Eccentricity (e)0.90.910.920.930.94 0.95 0.96 0.97 0.98Stableorbitpercentage[%]0.050.10.150.2C1−1(f0, e) with: e = [0.9, 0.98]C1−1(0.00π, e, 0.01)C1−1(0.25π, e, 0.01)C1−1(0.](https://thumb-eu.123doks.com/thumbv2/123dokorg/7513928.105461/60.892.166.775.332.499/figure-trend-stable-points-percentage-eccentricity-eccentricity-stableorbitpercentage.webp)
