Prof. Francesco Topputo Co-Advisor

SPACE TRAJECTORY OPTIMISATION IN HIGH FIDELITY MODELS

Industrial Engineering Faculty Department of Aerospace Science and Technology Master of Science in Space Engineering

Advisor: Prof. Francesco Topputo Graduation Thesis of: Erind Veruari Co-advisor: 837650 Diogene A. Dei Tos, MSc

Academic Year 2015-2016

To my grandparents: even if fate kept us distant, I know you have me close to your heart, and I have you close to mine. Sommario

Il campo della progettazione ed ottimizzazione di traiettorie spaziali procede di pari passo con l’evoluzione del mondo scientifico e tecnologico. Le richieste in questo ambito prevedono trasferimenti che abbiano un alto livello di accuratezza e, al contempo, un basso costo in termini di propellente a bordo. Un esempio esplicativo è rappresentato dal numero crescente di satelliti a bassissima autorità di controllo in orbita (cubesats), il cui studio per missioni interplanetarie si sta intensificando. Tra le varie strategie di progettazione di missione, quelle che sfruttano la dinamica del problema dei tre corpi offrono una serie di soluzioni a basso costo con caratteristiche stimolanti. Tuttavia, il loro utilizzo in modelli reali del sistema solare presenta grandi discrepanze. Il seguente lavoro prende spunto da questa divergenza, muovendosi in due direzioni, una teorica ed una pratica. Quella teorica prevede la riscrittura delle equazioni del moto del problema a tre corpi inserendo le perturbazioni dovute alle azioni gravitazionali degli altri pianeti, così come l’effetto della pressione della radiazione solare. Le equazioni, ottenute a partire dal for- malismo Lagrangiano, vengono poi ruotate in un sistema di riferimento roto-pulsante, nel quale si mantengono le caratteristiche delle orbite progettate in un modello a tre corpi. Esso permette, inoltre, un facile confronto tra le orbite rifinite e quelle di partenza. Dal punto di vista pratico, questo lavoro si occupa di creare alcuni algoritmi che siano in grado di propagare le traiettorie e poi ottimizzarle. Innanzitutto, vengono sviluppati alcuni strumenti numerici che permettano la soluzione del problema ad n-corpi. Per questi viene presentata una strategia di validazione tramite software open-source, i quali sono stati utilizzati nell’ambito della progettazione di missioni già in volo. In secondo luogo, si sfruttano orbite progettate in modelli meno accurati come soluzioni di partenza per la soluzione di un problema non lineare vincolato al contorno, che rappresenta il metodo di ottimizzazione della soluzione. Questi strumenti vengono, infine, applicati al caso del satellite LISA Path Finder (LPF) nell’ambito dell’estensione di durata della missione. Con questo specifico esempio si è voluto testare la capacità dell’algoritmo di ottenere soluzioni convergenti anche in quella regione altamente instabile rappresentata dai punti Lagrangiani del sistema dinamico.

iv Abstract

The design and the optimization of space trajectories goes side by side with the evolution of the scientific and technological world. In this field, there are requirements of transfers with a high-level of accuracy and, at the same time, of low costs in terms of on-board propellant. The high number of ultra-low thrust orbiting satellites (cubesats), whose study for interplanetary mission is growing, gives an instructive example of this situation. Among the different mission design strategies, the ones that exploit restricted three- body problem (R3BP) dynamics provide many low cost solutions with challenging characteristics. However, their inclusion in real solar system models results in high discrepancies. This work starts from this incongruence, and takes a twin-track approach, analyzing it from a theoretical and a practical point of view. From a theoretical point of view,taking into consideration the gravitational effects of a set of n-bodies and adding the perturbing effect of the solar radiation pressure. The equations have been computed exploiting the Lagrangian formalism. Afterwards, they have been rotated in a roto-pulsating reference frame (RPF), where the features of the orbits designed in R3BP are preserved. Moreover, RPF allows to easily compare guess and refined solutions. This work aims at creating some algorithms, which can be used for trajectory propagation and optimization in practical terms. First of all, numerical tools which solve the n-body problem are established. Then, they are validated using open-source softwares, which have been adopted in the design process of already flown missions. In the second place, formerly designed orbits in less accurate models are considered as guesses for the solution of the non-linear constrained boundary value problem, which represents the optimization strategy for the trajectory. Finally, these tools are applied to the case of the mission extension for the LISA Path Finder (LPF) satellite. Using this practical example, the converging properties of the algorithm have been tested, specifically in the highly unstable region that is represented by the Lagrangian points of the dynamic system.

v vi Contents

Sommario iv

Abstract v

1 Introduction 1 1.1 Context ...... 1 1.2 Problem deﬁnition ...... 2 1.3 State of the art ...... 2 1.4 Motivation ...... 3 1.5 Research question ...... 4 1.6 Structure of the thesis ...... 4

2 Models in inertial frames 7 2.1 Reference frames ...... 7 2.1.1 Inertial reference frames ...... 7 2.1.2 J2000 and EME2000 ...... 8 2.2 n-body problem ...... 10 2.2.1 The general n-body problem ...... 10 2.2.2 The restricted n-body problem and the Lagrangian formulation . 12 2.2.3 Planeto-centred equations of motion ...... 15 2.3 Gravitational perturbation ...... 17 2.4 Non-gravitational perturbations ...... 21 2.4.1 SRP ...... 23 2.5 Equations of motion ...... 24

3 Models in rotating frames 27 3.1 Definition of RPF ...... 27 3.2 Rotation of the equations into RPF ...... 32 3.3 Alternative derivation of the equation of motion ...... 35 3.4 Perturbations ...... 38 3.4.1 Gravitational effects ...... 39 3.4.2 Non-gravitational effects ...... 41 3.5 Logics of the models ...... 42 3.6 Special cases ...... 44 3.6.1 The Restricted 2 Body Problem ...... 44 3.6.2 Restricted 3 Body Problem ...... 46

vii 4 Validation of the models 51 4.1 Integration scheme ...... 51 4.1.1 Runge–Kutta–Fehlberg methods and ODE78 ...... 52 4.2 JPL’s SPICE ...... 54 4.3 GMAT ...... 57 4.4 Validation examples ...... 60 4.4.1 Validation with SPICE ...... 61 4.4.2 Validation with GMAT ...... 72

5 Impulsive trajectory optimization 79 5.1 Introduction to a multiple shooting strategy ...... 79 5.2 The variational equations ...... 82 5.3 Implementation ...... 85 5.3.1 Objective Function ...... 86 5.3.2 Equality Linear Constraints ...... 87 5.3.3 Inequality Linear Constraints ...... 87 5.3.4 Equality Non-Linear Constraints ...... 88 5.3.5 Inequality Non-Linear Constraints ...... 94

6 Application to the LISA PathFinder mission extension 97 6.1 Presentation of the LPF mission ...... 97 6.1.1 Saddle Point and LPF mission extension ...... 100 6.2 Guess Solutions ...... 104 6.3 Optimisation in RPF n-body problem ...... 108 6.4 Analysis of the results ...... 113 6.4.1 Direct solution ...... 113 6.4.2 Indirect solution ...... 114 6.4.3 Divergence and change of class ...... 116

7 Conclusions 119 7.1 Summary of the results ...... 119 7.2 Future work ...... 120

Acknowledgements 123

Bibliography 125

viii List of Figures

2.1 Representation of J2000 and EME2000 reference frame ...... 9 2.2 Geometrical representation of forces on general n-body model ...... 11 2.3 Change of origin: from SSB to Planet ...... 16 2.4 Change of origin: from SSB to Planet ...... 18

3.1 Representation of the roto-pulsating frame ...... 28 3.2 Logics for writing the equations of motion in RPF ...... 44

4.1 GMAT interface ...... 57 4.2 RPF trajectory for asteroids: Ceres - Fortuna - Hermione - Hektor . . . . 63 4.3 RPF trajectory for asteroids: Amor - Apollo - Einstein - Aten ...... 66 4.4 Relative error for 1862 Apollo in long integration period ...... 67 4.5 Relative error for the ﬁrst 6 asteroids in long integration period . . . . . 68 4.6 Relative error for the second 6 asteroids in long integration period . . . . 69 4.7 RPF trajectory for asteroids: Eureka - Damocles - Chaos - Atira . . . . . 70 4.8 Errors on diﬀerent primaries for long integration period ...... 71 4.9 Trajectory evolution ...... 75 4.10 Particular of error trend ...... 76 4.11 Trajectory evolution for J002E3 ...... 77

5.1 Multiple shooting strategy ...... 80 5.2 Sparsity pattern of equality linear contraint matrix ...... 87 5.3 Sparsity pattern of equality linear constraint matrix - particular . . . . . 88 5.4 Sparsity pattern of equality non-linear constraint matrix ...... 94

6.1 LPF technolgy, courtesy of ESA ...... 98 6.2 xy motion of LPF in RPF ...... 99 6.3 LPF manifold extension ...... 99 6.4 SP trajectory ...... 100 6.5 Intersection of LPF and SP trajectory ...... 101 6.6 Miss distance depending on departure date ...... 102 6.7 Time evolution for miss distance - Minimum miss distance ...... 103 6.8 Time evolution for miss distance - Maximum miss distance ...... 103 6.9 Synodic trajectory for minimum and maximum miss distance cases . . . 104 6.10 Shooting from 4-body to n-body: 2 manoeuvre, 5 segments ...... 105 6.11 Shooting from 4-body to n-body in diﬀerent cases ...... 106 6.12 Shooting from 4-body to n-body in diﬀerent cases ...... 107 6.13 Periodic solution for the phasing problem ...... 108

ix 6.14 Shooting from 4-body to n-body with diﬀerent number of segments . . . 110 6.15 Shooting from 4-body to n-body: 3 manouevres, 98 segments ...... 111 6.16 Converged direct solution: 6 manoeuvres, 9 segments ...... 114 6.17 Converged indirect solution: 2 manoeuvres, 21 segments ...... 115 6.18 Converged indirect solution: 2 manoeuvres, 19 segments ...... 116 6.19 Converged indirect solution: 2 manoeuvres, 19 segments ...... 117 6.20 Converged indirect solution: 2 manoeuvres, 19 segments ...... 118

x List of Tables

4.1 Butcher’s tableau ...... 53 4.2 Name and properties of chosen asteroids ...... 62 4.3 Validation periods ...... 62 4.4 Name and properties of chosen dwarf planet ...... 62 4.5 Attractors and their gravitational parameters ...... 63 4.6 Absolute and relative maximum positions errors ...... 65 4.7 Absolute and relative maximum velocity errors ...... 65 4.8 Absolute errors for the position in Earth RPF ...... 69 4.9 Postion absolute errors in Jupiter RPF ...... 71 4.10 Mininum max. absolute errors related to the primary ...... 72 4.11 Physical and integration properties ...... 73 4.12 Initial conditions for the integrators ...... 75 4.13 Max relative and absolute errors after 100 days integration ...... 75 4.14 Initial conditions for the integrators ...... 76 4.15 Max relative and absolute errors after 100 days integration ...... 77 4.16 Comparison between the integrators ...... 77

6.1 Percentage of trajectories with miss distance smaller than the tolerance . 102 6.2 Minimum and maximum miss distance ...... 102 6.3 Values for the main parameters ...... 109 6.4 Direct trajectory properties ...... 113 6.5 Comparison between diﬀerent reﬁned solutions ...... 114 6.6 Indirect trajectory properties ...... 115 6.7 Properties of non-converging solution ...... 117

xi Chapter 1

Introduction

1.1 Context

The evolution of space exploration has highly changed throughout the centuries. It began as a fascinating study of the stars and their motion. It then developed as a philosophical question dethroning the humankind as the centre of the universe. Nowadays, it goes on in the search of new frontiers to reach, and new forms of life to unveil. Science has played the main role in these discoveries. Astronomy led the knowledge of the surrounding universe to enter in the history, whereas the orbital mechanics gave a physical meaning to the motion of celestial bodies. Astrodynamics extended those notions to man-made objects, whose presence has become a natural part even of daily life, through communication satellites and navigation systems. Space trajectory design is one of the branches of the astrodynamics that has mostly evolved in the last twenty years, due to an outstanding interest in space mission and their implication to scientific progress. The design of a trajectory considers the gravitational effects of celestial bodies on orbiting vehicles. The aim is to achieve a transfer whose requirements are met. However, new scientific frontiers need more demanding ones, and this translates into the development of higher fidelity models. The classic approach to the trajectory design has consolidate bases. The formulation of the problem dates back to Sir Isaac Newton and his universal gravitational law. The approximated formulation, called Two-Body Problem (2BP), has an analytical solution. It can be efficiently used for the trajectory design, where only one main body is considered and the artificial object describes conics around it. In the case of a transfer between different primaries the conics can be patched together. The simplification of the model permits the generation of meaningful solutions. Nevertheless, these solutions must be corrected because of the gravitational effects of other influencing celestial bodies. The perturbation of these attractors are accounted in more complex models, where the solutions are refined. New possibilities emerge as the trajectories are directly designed in more refined models, like the Restricted Three Body Problem (R3BP). It takes advantage of the idea of co-rotating bodies with respect to a common barycentre. Then it considers the spacecraft as a point mass in relative motion. In this way it is easier to understand the mutual interaction between celestial bodies and the probe. Mathematically speaking the problem has not an analytical solution, but it is possible to find one integral of motion. In this

1 Introduction model new trajectory families are discovered, bringing to new opportunities for low cost transfers. Also the trajectory design in a 3-body model needs a refinement process. Nowadays, there are new technologies, like cubesats, that greatly sense the effect of the perturbations. Even scientific purposes are revealing interesting features that are not defined into a combination of only three bodies. Moreover, given that the requirements on new mission are becoming more stringent and rigorous in terms of fidelity and precision, the current trend moves forward into the improvement of all those tools that allow a deeper comprehension of the trajectory design in multi body dynamics.

1.2 Problem deﬁnition

The problem discussed in this work connects to the R3BP model as a starting point for every analysis in the trajectory design branch. It considers the highly non linear context of the model and adds new material of discussion analysing the effects caused by the simultaneous action of many gravitational attractions. In particular, the contribution of all the planets of the solar system s considered, including our Moon. This is an essential step toward a high fidelity model where the celestial bodies will not be seen as perturbations, but as opportunities to enhance the possibilities of a lower cost transfer. As a subsequent step, other perturbations will be considered, such as the solar radiation pressure and the inhomogeneity of gravitation field. Even if their influence could be pointless with respect to the gravitational effect of the main bodies in the short term, they are required for mid/long missions, especially for interplanetary transfers or close gravity assist manoeuvres. The meeting point of this analysis with real missions stays in the generation of feasible trajectories where the control is almost zero. The direct transformation of the control capability into fuel costs makes the problem of extreme interest and applicability.

1.3 State of the art

The literature concerning the problem of two body has comprehensive development: a basic introduction could be found in Curtis [10] and a more detailed as well as more dated one in Chobotov [9]. The perturbation analysis is present in Vallado [47] and for useful numerical simulations it is possible to look at Battin [4]. Instead, the R3BP has been extensively examined and expertly presented in Szebehely [42]. However, the problem of orbital mechanics dates back to great names: Newton, La- grange, Euler, Poincaré [34]. And since this section can not cover the vast amount of astrodynamics publications, it will be a brief collection of some authors and missions related to the main topics. This work starts as a continuation of a MSc thesis by Dei Tos [11] at Politecnico di Milano and a following article with Topputo [12]. The necessity of a high fidelity model appears in many different fields of astrodynamics: from the necessity of a study on ballistic capture, as in Hyeraci [24], to the resonant motion of the planets in Topputo [43].

2 1.4 Motivation

The groundwork has been laid from Belbruno [5], whereas Gomez [20] and the Barcelona Group are leading it forward. The missions that express the power of the new dynamics given from a three-body model are those directed to Lagrangian Points, which are mathematical points of equilibrium that do not exist into a model with a unique primary. The Herschel Space Observatory targeted the L2 point in the Sun-Earth system 1. It pointed out, in the infra-red range, cold and/or dusty objects in space. From past missions also Lisa Path Finder 2 deserves a particular remark, since it will be presented in this work. The launch of the James Webb Space Telescope 3 is expected in 2018 and it should take the inheritance of the Hubble Telescope. Even this will be targeted toward the Sun-Earth L2 point. The equilibrium points L4 and L5 are acquiring new interest as they are stability points where information of million ages are hidden. The Lucy mission [32] is headed to the Sun-Jupiter L4 and L5 points, for research and analysis on Trojan Asteroids.

1.4 Motivation

As already seen in the previous section, since R3BP has given life to new fields and missions, one could wonder why it is important to deepen the argument and add new elements. This was true until some years ago but now the technology is reaching levels where the presence of "only" three body cannot be accepted any more. The following examples may explain why. The cubesats, that are mini-satellites with an edge of 10 cm, are gathering attention due to the possibility they give in accessing space with repeatable modules with a very low cost. Given the very small dimensions, these objects cannot combine high performances with high control capability. Therefore, a good knowledge of the a priori motion is needed in order to efficiently use them: the solar radiation pressure or outer planets will induce accelerations that should not be classified as perturbation but should benefit their motion. The research on gravitational waves is producing interesting results with experiments like LIGO 4 and VIRGO 5. However, the space is an ideal place for such experiments. It is, indeed, necessary to find a point where the gravitational effects of all celestial bodies are shut down. Therefore in the perspective of a more realistic mission design, new degrees of complexity shall be added to the problem. Nevertheless, this will repay creating higher reliability solutions and reducing costs even more: new choices will derive from greater possibilities. Finally, even the orbit detection and the guidance and navigation control for spacecraft will get benefits: a more reliable knowledge of the designed trajectory means also a more robust transfer, where the communication with the spacecraft can be relaxed.

1http://sci.esa.int/herschel/ 2http://sci.esa.int/lisa-pathﬁnder/ 3https://www.jwst.nasa.gov/ 4https://science.jpl.nasa.gov/projects/LIGO/ 5https://www.ego-gw.it/

3 Introduction

1.5 Research question

The thesis will follow diﬀerent paths and therefore it will try to answer several questions. The main one is:

can we design space trajectories directly in high ﬁdelity models?

Some sub-questions can help to answer it. The first one is if it possible to validate an n-body model with currently used tools for space mission. It will be searched if it is possible to build an integrator that can be successfully compared with programs such as SPICE of the Jet Propulsion Laboratory at the Caltech, or with open source programs such as GMAT. Once the integrator is validated, the real trajectory design part will be set. At this point many different strategies can be followed: is it reasonable to design the trajectory directly into the n-body model? Or is it better to design it into a less refined model and then optimize it into the high fidelity one? Are these approaches always converging or does this happen only for elementary trajectories?

1.6 Structure of the thesis

A brief description of the space trajectory design problem has already been given in this introductive chapter. Some motivations have been presented on why it is interesting to focus on high ﬁdelity models exploitation. Then it has been formulated a research question that brings into focus the real applications of the model. The rest of the work has been organized as follows.

Chapter 2 deals with all the reference frames that have been considered in the thesis, giving an overview and describing the differences between them. Next, the equations of motion for the n-body problem in an inertial frame are derived: the procedure will exploit the lagrangian formalism for rigid bodies. After an accurate analysis of these equations perturbations are inserted into the system. The gravitational ones are included firstly as a transformation of the lagrangian of the system: mainly are considered those dealing with the inhomogeneity of the gravitational field for celestial bodies. Then the non gravitational perturbations are added, focusing mainly on the solar radiation pressure.

In Chapter 3 the main frame of this work is presented and analysed: the roto-pulsating frame. After a description of its properties the equations of motion are rotated into this frame. To prove the correctness of this logic, an alternative derivation of the equations of motion is given. Also in this frame the adequately rotated perturbations are added to the equations. A comparison between the diﬀerent logics is analysed. Eventually, some considerations are taken in order to examine if it is possible to reconstruct special cases (R2BP or R3BP) from the general one (RNBP).

The validation of the models is performed in Chapter 4. Here, the integration scheme is introduced. It follows a description of the two validation sources SPICE and GMAT. Some examples within the two programs are given showing the validity of the high ﬁdelity

4 1.6 Structure of the thesis

model integrator.

Chapter 5 concerns the optimization process of an impulsive trajectory. The multiple shooting strategy is hereby presented. In this approach the trajectory is subdivided in diﬀerent legs and each leg in separate arcs, with the aim of linking together the arcs, without discontinuities between them. To fulﬁl this task, the equations of motion are supplied with the variation of a particular matrix, called state transition matrix. The new equations are named variational equations and have important consequences for the evolution of the dynamical system. To carry on an optimization process an objective function is needed and it is linked with the velocity variation, which is connected with fuel consumption through the Tsiolkovsky equation. The smoothness of the trajectory is granted only thanks to some constraints, that are inserted into the dynamical problem through constraint equations. The implementation process is the one described.

Chapter 6 copes with the application of trajectory optimization to a real case: the possible mission extension for Lisa Path Finder. The chapter starts with a description of the LPF mission and its proposal for the mission extension. In fact, the satellite should pass through a particular point called Saddle Point. For this case the initial solutions already exist, but considered in a 4-body model. Starting from those guesses the trajectories are optimized into a full roto-pulsating n-body model and the results are investigated.

The work concludes with Chapter 7 where a brief summary of the results is given. In the meanwhile the limits of the followed strategies are reconsidered and some hints for future works are given.

5 Introduction

6 Chapter 2

Models in inertial frames

This chapter lays the bases for the overall work presented in the thesis. The main topic is the modelling of the n-body problem in inertial frames. Considering the mutual gravitational interaction of n celestial bodies, the celestial mechanics is that branch of the science that tries to define their motion. The astrodynamics is a branch of the celestial mechanics that tries to define the motion of a man-made object in space influenced by the gravitational fields of celestial bodies. The motion is subject to rigorous equations, that once assembled create a model, called n-body model. The motion of a body cannot be absolutely defined, but has to be specified with respect to other bodies, or a set of particular points, that delimit a frame of reference.

2.1 Reference frames

Before moving forward with the mathematical description of the n-body problem it will be necessary a foreword regarding the concept of reference frame. In fact this is a fundamental notion for this work and will be examined in many aspects. To give a definition for a frame of reference it can be said that it is a set of reference points which allow the description of the motion of an object through a coordinate system. Therefore, the reference frame can be seen as a physical concept related to the state of motion of an observer. According to Salençon [37] it can be also defined as the set of all points in the Euclidean space with the rigid body motion of the observer. For a formal definition of coordinate system the reader is referred to a manual of topology or algebra, such as Pontryagin [35] or Sernesi [39]. Here it will be expressed as the mathematical tool used to represent the physical quantities in the reference frame. This means that it is required to attach a coordinate system to a reference frame in order to have an idea of how and how much the motion of an object is changing. Then, it is possible to chose the coordinate system for sake of convenience: a spherical or cylindrical representation can be preferred to a cartesian one if the motion follows curved trajectories.

2.1.1 Inertial reference frames A reference frame is called inertial when the ﬁrst principle of dynamics is valid: a free point mass, i.e. where no force acts, or the resultant of the forces is zero, will persist in

7 Models in inertial frames a state of inertia or rectilinear motion. It means that an observer in inertial frame will measure for the point mass a null acceleration for every time instant. Inertial reference frames are extremely useful since the equations of motion do not change between two of them. If two frames are inertial, the only thing changing between them is the velocity. However, by deﬁnition, this velocity diﬀerence is constant:

VIF 1(t) = VIF 2(t) + VIF 12 (2.1) where VIF 1(t) and VIF 2(t) are, respectively, the absolute velocities of the ﬁrst and second inertial frame, while VIF 12 is their relative velocity. Taking the derivative of the previous equation it is possible to prove the fact that the acceleration of the two frames are equal:

AIF 1(t) = AIF 2(t) (2.2) where AIF 1(t) and AIF 2(t) represent the accelerations of the inertial frames. The second law of dynamics grants that the forces, F (t), acting on a body of a certain mass, m, depend only on the accelerations, A(t), acting on it:

F (t) = mA(t) (2.3)

Since in the two inertial frames the accelerations are equal, the forces will be equal, hence the equations of motion will be the same. Undoubtedly, this short proof cannot represent an exhaustive description of the properties of an inertial reference frame, which can be found for instance in Landau [29].

2.1.2 J2000 and EME2000 The previous description of the inertial frames is indispensable since in the following work at least two of them are used. The main difficulty for choosing a reference system in space is that it is complex to find an origin that is inertial. Usually, the system of Sun and stars is considered inertial. Therefore, according to what preciously said, all the frames that are in constant rectilinear motion in the Sun-stars frame will be inertial. The complexity is related to the definition of the orientation of the coordinate system that must be attached to it. To solve this problem it is necessary to define some constant direction in space. For this reason the equatorial coordinates are used as explained in Hoehenkerk [23]. The steps to define the celestial frame are related to the definition of the axes.

1. The z-axis is aligned with the polar axis.

2. The x-axis is the result of the intersection between the celestial equator with the ecliptic. The axis points toward the intersection of the sun with the celestial equator on its south to north journey through the sky in the spring and is also known as vernal equinox.

8 2.1 Reference frames

3. The y-axis is the vector orthogonal to both.

However, since it is diﬃcult to measure directly the position of the vernal equinox, it is necessary to rely on star catalogues. In a certain epoch the position will be computed indirectly and after that the reference system can be used. For the J2000 frame the epoch was January 1, 2000, basing on the ﬁfth Fundamental-Katalog, FK5 [18]. Since this frame is earth-based it must be said that it is almost inertial, because it must consider Earth rotation and a series of other factors as the direction variation of the polar axis due to precession and nutation motion: their formulation will be omitted here, but can be found in Seidelmann [38]. What is missing is an origin for the system and usually the Solar System Barycentre (SSB) is chosen. It is also possible to chose the Earth as centre for the reference system, and in this thesis work this frame will take the name EME2000, or Earth Mean Equator in J2000. The two references are just translated one with respect to the other, therefore the position vector of an object can be written as:

R(t) = RE(t) + r(t) (2.4) where R(t) and r(t) represents the position of the object with respect to SSB and Earth, respectively. RE is the position of the Earth with respect to the SSB. However it is possible to say that the passage from the ﬁrst to the second is a passage from an inertial to a non inertial reference frame.

Figure 2.1: Representation of J2000 and EME2000 reference frame

9 Models in inertial frames

Taking the derivative of the previous equation:

V (t) = VE(t) + v(t) (2.5) where V (t) and v(t) are the velocities of the object with respect to SSB and Earth, respectively. VE is the velocity of Earth with respect to SSB. This means that the two reference frames are not moving of mutual constant rectilinear motion. It will be converted also in a new acceleration term that will modify the equation of motion with an apparent term:

A(t) = AE(t) + a(t) (2.6) where A(t) and a(t) are the accelerations of the spacecraft with respect to SSB and Earth, respectively, while AE is the velocity of Earth with respect to SSB. In practice, the new term is due to the fact that the Earth is rotating around the solar system barycentre. So this is what we meant for a quasi-inertial frame: in fact the two frames J2000 and EME2000 are inertial with respect to distant stars (the rotation of EME is insigniﬁcant due to the very high distance), but they are not strictly inertial one with respect to the other.

2.2 n-body problem

The general n-body problem is the starting case of the orbital mechanics. It studies the motion of n gravitational sources, due to their mutual interaction. Since the primordial formulation of the dynamics, this problem has been one of the most studied: Newton built its dynamics together with the universal gravitational law. Even if the general relativity has changed the concepts of motion and mechanics, the classical formulation of the problem still represents the more direct way to give a physical meaning to a problem that can appear as a strictly mathematical problem. To get a strong connection between the physics that lays behind the n-body problem and the mathematics that will be needed to ﬁnd some solution, the following section will put forward its classical expression. After that the Lagrangian formalism will be used to deﬁne the restrict n-body problem.

2.2.1 The general n-body problem The most complete description for the motion of n bodies subjected to the mutual gravitational interaction is the n-body problem. Each body of mass mi, i = 1, .., n, depends on the masses mj of the other n-1 bodies. The Newton’s law for general gravitation expresses the force acting on the i-th body, Fi, as:

j=n j=n X X Rj − Ri Fi = Fji = Gmjmi i = 1, .., n (2.7) kR − R k3 j=1 j=1 j i j6=i j6=i

10 2.2 n-body problem

Figure 2.2: Geometrical representation of forces on general n-body model

In fact, equation (2.7) represents a set of 3n equations, where the Ri is the position T vector of the i-th body, whose cartesian coordinates can be written as Ri = [Xi Yi Zi] .G is the gravitational constant and Fji is the force acting on the i-th body due to the gravitational eﬀect fo the j-th body. The latter is directly proportional to the product of the masses of the two bodies and inversely proportional to the square of their distance, that is the expression of the universal gravitational law. From the already cited second law of dynamics it is possible to relate the acceleration ai acting on a body with the forces that are generating the motion. This is due to the fact that the inertial mass and the gravitational mass are equal, so it can be written as:

¨ Fi = miai = miRi (2.8) ¨ where Ri is the diﬀerential expression of the same acceleration. In the assumption that the forces acting on the bodies are only the gravitational ones, it is possible to combine equations (2.7) and (2.8) writing:

j=n ¨ X Rj − Ri miRi = Gmjmi i = 1, .., n (2.9) kR − R k3 j=1 j i j6=i

Equation (2.9) represents again a set of 3n diﬀerential second order equations (that can be converted always in a set of 6n ﬁrst order equations) where the acceleration of the i-th body is related only to the relative position of that body with respect to the

11 Models in inertial frames other attractive sources. The problem can be further simpliﬁed noting that if masses are constant and nonnull the equations set (2.9) can be written as:

j=n ¨ X Rj − Ri Ri = µj ∀i = 1, .., n (2.10) kR − R k3 j=1 j i j6=i

In equation (2.10), µj represents the gravitational parameter of a body, which is by deﬁnition the product of its mass with the gravitational constant. Since the problem is highly non-linear, solutions are hard to ﬁnd. However, from Picard-Lindelöf [30] theorem, it is possible to prove that solutions exist and are unique. It is possible to see this problem as an initial value problem, or a Cauchy problem, that can be rewritten in the state space form:

x˙ = f(x) (2.11)

As remarked previously any second order diﬀerential equation can be formulated as a system of ﬁrst order equations, and in the considered case it is possible to set xi = ˙ T [Ri, Ri] Given a set of 6n initial condition at a time t0, the solution is granted to be existent and unique if the function f and its gradient ∂f/∂x are continuous in a manifold containing t0 and x0. In the n-body problem the only discontinuity can be given from the denominator of equation (2.10). Since the only way for this denominator to be 0 should be Rj = Ri, and this cannot happen except in the case of collisions, the right hand side is continuous. The same reasoning can be done for the gradient. Regarding the discontinuity point, some strategies can be adopted to analytically remove it, by regularizing the equations of motion. In this way it is possible to have a continuous right hand side in the whole domain, according to the Levi–Civita transformation [40].

2.2.2 The restricted n-body problem and the Lagrangian formulation Solving the general n-body problem is a very interesting matter of study, however it is more related to astronomy or orbital mechanics. Astrodynamics is involved in finding the motion of an artificial object under the gravitational effect of the other celestial bodies. The mass of this object is much smaller than those of the planets, therefore, the force that it exerts on them is negligible. So the problem in converted from a general formulation into a restricted one. The motion of the planets can be seen as independent from the motion of the satellite, then their equations can be decoupled from the rest of the system. If those are independently solved then the problem transforms into finding a solution for the motion of the satellite. To give an expression for the restricted n-body problem, in this work it has been followed an approach that considers a Lagrangian formalism for rigid bodies, which has been mathematically postulated by Lagrange in 1811. All planets are assumed as point sources.

12 2.2 n-body problem

Then this way of writing the equations of motion gives the possibility of modifying them easily if non inertial reference frames are considered, or even if new perturbations are inserted. On the other hand writing the equations of motion for non inertial frames, exploiting D’Alembert principle for dynamic equilibrium or cardinal equations can exhibit huge complexities, especially on the arrangement of apparent forces.

To express the Lagrangian formalism it is necessary to deﬁne a new function, called Lagrangian:

L = L (q, q˙, t) (2.12)

where L is the Lagrangian function, q and q˙ are the new coordinates and their derivatives, while t is the time variable. The Lagrangian of a system is a function that can describe its motion, because according to the principle of the least action (or Hamilton’s principle) a physical system in motion between two points chooses always the path that minimizes the integral of the Lagrangian for the whole trajectory [21]. It expresses a motion where it is possible to deﬁne an action functional S(q) in a time interval [t1,t2] that has a minimum:

Z t2 S(q) = L (q(t), q˙(t), t) dt (2.13) t1 From this deﬁnition it is also possible to write the equations of motion, since it must be true that:

d ∂L ∂L − = 0 (2.14) dt ∂q˙ ∂q The opportunity given from this formalism is that a suitable choice for the coordinates q can bring to a simpler way of mathematical formulation for the problem.

In this procedure the Lagrangian free coordinates q coincide with the physical ones for position R. It is then possible to express the Lagrangian as the diﬀerence between the kinetic,T , and the potential energy,V , that in the more general way can be written as:

L (R(t), R˙ (t), t) = T (R(t), R˙ (t), t) − V (R(t), R˙ (t), t) (2.15)

The next step is to deﬁne the components of the Lagrangian for the system, starting from the kinetic energy. This reads:

1 T = m R˙ · R˙ (2.16) 2 As always the kinetic energy depends directly on the mass and on the square of the velocity of the object. In this work the choice has been to keep, when possible, a vectorial

13 Models in inertial frames

notation for all quantities, in order to have a better understanding, even when those are derived or manipulated.

The deﬁnition of the potential energy includes all the gravitational eﬀects on the body:

X µj V = −m (2.17) kR − Rjk j∈S

The used notation is as similar as possible with the one used for the general n-body problem: the µj represents the gravitational parameter of the j-th attractor, while Rj is its position vector. Therefore the quantity expressed in equation (2.17) is the gravitational potential seen as the summation of all the inﬂuences on the object. The expressions j ∈ S stays for a general model where all the desired bodies are considered, so S is a set of attractors. Redeﬁning the Lagrangian the result is:

1 X µj L (R, R˙ ) = mR˙ · R˙ + m (2.18) 2 kR − Rjk j∈S

In order to get the equations of motion for the dynamic system, stating that the action is minimized along a trajectory, the minimum equation can be expressed as:

d ∂L ∂L − = 0 (2.19) dt ∂R˙ ∂R At this point the power of the Lagrangian formalism appears and shows how a physical problem has been translated into a mathematical expression. Everything from this point on can be automated. The ﬁrst step will be to compute the diﬀerent derivatives and then everything can be combined together.

 d ∂L  = mR¨  dt ∂R˙ (2.20) ∂L X R − Rj   = −m µj 3  ∂R kR − Rjk  j∈S

The complete set of equations describing the motion of the restricted problem is:

¨ X R − Rj R + µj 3 = 0 (2.21) kR − Rjk j∈S

Some little notes should be added as a comment for equation (2.21). In appearance it is very similar to what expressed in eq. (2.10), however the meaning is different. In fact equation (2.10) is a set of 3n second order differential equations, so what is written is just one of the equations of the system. In eq. (2.21) the second order differential equations

14 2.2 n-body problem

are only 3, because there is no need to solve for the motion of the planets. This fact has turned the problem to nonautonomous, because there is a hidden cariable in that equation, or:

Rj = Rj(t) (2.22)

The motion of the planets is now directly dependent upon time, which enters in this way into the equation of motion. Another important note is the frame in which these equations are written. Every step of this procedure is true if an inertial frame is chosen, in this case the J2000, and the results will be expressed as positions with respect to solar system barycentre (SSB).

2.2.3 Planeto-centred equations of motion

The equations of motion previously written define the evolution of a system referred to the solar system barycentre. However, this is not always the case of interest. Often it is more interesting to consider the dynamics of a satellite with respect to a planet, therefore, it would be convenient to have a system that can directly be integrated in that frame. In this case the results will be relative to the planet and it would be easier to have a direct view on how the quantities of the problem depend on the gravitational effect of that source with respect to others. Also, this procedure can yield to more accurate results since numerical cancellations may occur when (2.21) is integrated in the proximity of a celestial body in the set S. A further evolution of this strategy can be found in Amato [1]: the regularized equations of motion are integrated in a planeto-centred frame, that changes the primary if a close encounter happens in order to increase numerical efficiency and accuracy. It is possible to prove that applying a coordinate transformation, the result does not change. This can be done before or after writing the equations of motion. In this work it has been chosen to apply the transformation after the equations of motion have been written. Recalling equation (2.21) it is possible to see how the dynamical system depends on position and acceleration. To write the equation of motion then, it is necessary to apply a coordinate transformation acting on position, that will be directly transformed into a condition on acceleration:

R = RP + r (2.23)

In eq. (2.23) R and RP represent the position vectors respectively of satellite and planet, while r is the relative positions, that will be the new coordinate, as shown in ﬁg. 2.3. Taking the derivatives of eq. (2.23) will bring to new relations:

˙ ˙ R = RP + r˙ (2.24) ¨ ¨ R = RP + r¨ (2.25)

15 Models in inertial frames

Figure 2.3: Change of origin: from SSB to Planet

Plugging eq. (2.25) and (2.23) into (2.21) it is possible to get that:

X r + RP − Rj ¨ ¨ r + RP + µj 3 = 0 (2.26) kr + RP − Rjk j∈S

Even if the reference system is changing, there is no modiﬁcation for the set of planets S where the gravitational forces are considered. Physically all the planets belonging to S that had an eﬀect over the spacecraft when the equations were written in SSB, will still attract it, even if the centre of the reference is changing. Now, eq. (2.23) refers not only to the spacecraft, but to every body in the system, therefore the last equation, can be rewritten in a more suitable way:

X r − rj ¨ ¨ r + RP + µj 3 = 0 (2.27) kr − rjk j∈S

where rj = Rj − RP are the relative positions of the planets with respect to the one taken as origin. In fact this is quite evident, since a change of reference cannot change relative positions or distances. However, it is possible to see another term, that represents the acceleration of the planet with respect to SSB. To express this term, let’s just recall eq. (2.10):

¨ X RP − Rk X rk RP = − µk 3 = µk 3 (2.28) kRP − Rkk krkk k∈S2 k∈S2

16 2.3 Gravitational perturbation

where S2 is a subset f S, that comprehends all the planets but the one chosen as origin for the frame. The last equation has been expressed both in SSB and in a planeto-centred version. Plugging it into eq. (2.27) the complete equations of motion can be written:

X rk X r − rj ¨ r + µk 3 + µj 3 = 0 (2.29) krkk kr − rjk k∈S2 j∈S It is worth to note that in the set S also the planet with respect to which the equations are written is included. However, for this body is true that rp = 0, hence, the complete equations can also be expressed as:

r X r − rk rk ¨ r + µP 3 + µk 3 + 3 = 0 (2.30) krk kr − rkk krkk k∈S2 From eq. (2.30) it is clear that if the summation term were neglected, what’s remaining is nothing else than the classical diﬀerential formulation of the two body problem, whose solution can be given in an analytical fashion.

2.3 Gravitational perturbation

In the n-body problem presented before, the attractors have been considered as point masses. But in real world this is not true; actually all bodies have a distributed mass. It is possible to prove that if this mass were distributed with a spherical symmetry the potential would be exactly the same. Since this is not always possible there is the need to deﬁne a new potential that accounts for the oblateness of the celestial bodies. These procedure has many references in literature: here it will be followed the strategy given in Vallado [47].

To deﬁne the new potential it is possible to see it as the summation of the potential due to inﬁnite point masses:

X mq U = −G (2.31) ρ q q

Each point has a mass of value mq and it is at a distance ρq from the point where the potential is evaluated. Passing from a discrete formulation to a continuous, it is possible to transform the summation into an integral over the surface Q:

Z 1 U = −G dm (2.32) Q ρq

Calling r the distance of a point P from the centre of mass, rq the distance of the inﬁnitesimal mass, the vectorial relation is true (ﬁg. 2.4):

r = rq + ρq (2.33)

17 Models in inertial frames

Figure 2.4: Change of origin: from SSB to Planet

that scalarly can be written as:

2 2 2 ρq = r + rq − 2rrq cos Λ (2.34) The rules of cosines have been used to express the modulus of the vectors, combined with the knowledge of the angle between r and rq. Λ is called also ground angle and is expressed as:

r · rq cos Λ = (2.35) rrq If the ratio α is deﬁned as:

rq α = (2.36) r it is possible to write: √ 2 ρq = r 1 − 2α cos Λ + α (2.37) Plugging the last equation into eq. (2.32) it is possible to obtain the new formulation for the potential:

G Z 1 U = − √ m 2 d (2.38) r Q 1 − 2α cos Λ + α

18 2.3 Gravitational perturbation

Since it will always be α < 1 and cos Λ < 1 it is possible to expand the denominator using the binomial theorem:

∞ 1 X l √ = α Pl[cos Λ] 2 (2.39) 1 − 2α cos Λ + α l=0

This equivalence is due to the fact that integrating the quantity over the square root is much more complex than expressing it as a summation of polynomials. The expression in eq. (2.39) represents the classical deﬁnition of Legendre polynomials, and according to Rodrigues’ formula [36] it has the following expression:

γ = cos Λ   l 1 X (−1)j(2l − 2j)! (2.40) P [γ] = γl−2j  l 2l j!(l − j)!(l − 2j)!  j=0

This argument has been deeply studied in literature, therefore for the next equations there will not be mathematical proofs. In fact, it will be shown only those expression needed for the acceleration model. For further information the reader can refer to Vallado [47]. The ground angle Λ is a complex angle to be measured. Moreover, from this notation there is the necessity to decouple the eﬀect of the central bodies, from those coming from the position of the point where the potential is computed. To do this, it is possible to write an equivalence between ground angle and latitude and longitude of the point mass (φq and λq respectively) and of the potential point (φP and λP respectively). From spherical trigonometry:

cos Λ = sin Φq sin ΦP + cos Φq cos Φp cos(λq − λP ) (2.41)

Note that this equivalence permits a new formulation of the potential through the so called associated Legendre polynomials, and the harmonics coeﬃcients:

∞ l l µ X X RB U = − Pln[sin Φp] Cln cos(nλP ) + Sln sin(nλP ) (2.42) r r l=0 n=0

In eq. (2.42) RB is the radius of the body, Pln the associated Legendre polynomials, Cln and Sln the harmonics coefficients. It is possible to see how the meaning of that expression is that the gravitational potential can be seen as a superposition of many different potentials, which come from the harmonics coefficients which represent the different combinations:

- Zonal harmonics: are the harmonics due to the n = 0 terms. They represents symmetrical ﬁelds about the polar axis, as simple bands of latitude, where the function is simply increasing or decreasing.

19 Models in inertial frames

- Sectorial harmonics: are the harmonics due to the n = l terms. They represent bands of longitudes where the function increases or decreases. - Tesseral harmonics: are all the harmonics where n 6= l 6= 0. Those are a combination of the previous two shapes and are represented by tiles. It is possible to prove that the ﬁrst term of the summation is zero ad the 0th order term is represented by the µ/r ratio. In this work it has been chosen to approximate eq. (2.42) by considering only the ﬁrst term. It is the only relevant term because the trajectory of the spacecraft is far from the planet.

2 ! µ RB U = − 1 − J2 P20[sin Φp] (2.43) r r where it has been called C20 = −J2. Expressing the associated Legendre polynomials of second order, the equation (2.43) can be rewritten as:

2 2 µ µJ2R 3r U = − 1 − B z − 1 (2.44) r 2r3 r2

where rz is the third component of the relative distance. The expression of eq. (2.44) can be found in literature (Vallado [?]) and gives some indications on how to compute the potential for non homogeneous bodies. Moreover, this expression has to be considered inside the set of equations of motion, therefore distances must be written with respect to the frame in which the dynamics is deﬁned. Remembering that the attractors belong to a set, the gravitational potential can be written as:

2 2 2 ! X µj X µjJ2jRBj X 3µjJ2jRBj rk V = U = − + 3 − 5 (2.45) kR − Rjk 2kR − Rjk 2 kR − Rjk j∈S j∈S j∈S

It is worth noting that rz is a scalar values. However, to maintain a vectorial notation, it can be rewritten as a product between vectors:

T rz = Iz (R − Rj) = [0 0 rk] (2.46) where:

0 0 0 Iz = 0 0 0 0 0 1

In this way rz is a vector that has only zeros, but for the third component. The scalar quantity can therefore be expressed as:

2 T T rk = rk rk = (R − Rj) Iz (R − Rj) (2.47)

20 2.4 Non-gravitational perturbations

since

T Iz Iz = Iz

From the Lagrangian equations it is needed to compute the gradient of the potential. To ease the derivation of the high number of terms, it can be split into three parts: dV V = Ω1 + Ω2 + Ω3 =⇒ = ∇V = ∇Ω1 + ∇Ω2 + ∇Ω3 dR

X µj ∇Ω1 = 3 (R − Rj) (2.48a) kR − Rjk j∈S

2 X 3µjJ2jRBj ∇Ω2 = 5 (R − Rj) (2.48b) kR − Rjk j∈S

2 T X µjJ2jRplj 5(R − Rj) Iz(R − Rj) ∇Ω3 = 5 2Iz(R − Rj) − 2 (R − Rj) (2.48c) kR − Rjk kR − Rjk j∈S

Thus the equations of motion can be simply written as:

R¨ + ∇V = 0 (2.49)

Comparing the last equations with those derived from the simplified potential model of eq. (2.21), it can be seen how the terms acting are always those due to a gradient of a potential. Moreover, it is possible to notice the power of a notation with Legendre polynomials. This permits not only a more complete physical representation, but also an immediate comprehension of how simplified models are only specific cases.

2.4 Non-gravitational perturbations

Every time non conservative elements are inserted in a problem, it becomes hard to use potentials and energies, therefore the concept of work will come at hand for this problem. In the case of non conservative forces, according to Lagrange [28], the Lagrangian formalism can be written as in eq. (2.50):

d ∂L ∂L − = Q (2.50) dt ∂q˙T ∂qT where Q are the generalized non conservative forces. According to this formulation, it is necessary to add a term due to the generalized forces acting on a system to the previously presented equations of Euler-Lagrange, on the right hand side. These represent the variation of the generalized virtual coordinates

21 Models in inertial frames

δq that generate the same virtual work on the system. To better explain this concept, the virtual work δW as the work done by a generic force Fi over a virtual displacement δRi shall be introduced:

N X ∂Ri δW = F T δq (2.51) i ∂q i=1 where δq is the virtual displacement of the free coordinates and it has been computed using the derivation rules for a change of coordinates. Now, by deﬁnition it should be that the virtual work done by these force and the generalized forces must be the same, therefore holds:

δW = QT δq (2.52) Thus, only if the two right hand sides are equal the virtual work can be equal:

N X ∂Ri QT δq = F T δq (2.53) i ∂q i=1 The last equation, transformed into a vectorial relation, shows the equivalence between the generalized forces and the real ones:

N T X ∂Ri Q = Fi (2.54) ∂q i=1 Spending only a few words on the last expression, it is possible to say that the generalized forces can be seen as conversion due to a change of coordinates for the real forces. In the case when the two coordinates are the same, as in the case under consideration, the last equation simpliﬁes up to see that the partial derivative is converted into the identity matrix and the generalized forces coincide with the summation of the real forces.

∂Ri = I (2.55a) ∂q

N X Q = Fi (2.55b) i=1 where I is the identity matrix. Finally eq. (2.50)can be written in a more suitable way as: N d ∂L ∂L X − = Fi (2.56) dt ∂q˙T ∂qT i=1 Eq. (2.56) concludes the ﬁrst part of this section and shows the mathematical procedure to follow in order to insert the non conservative forces into the equations of motion. The next step will be to give an example of this kind of forces that enter into the n-body model.

22 2.4 Non-gravitational perturbations

2.4.1 SRP The most known and studied is the so called Solar Radiation Pressure perturbation. This is a perturbation due to the solar ﬂux interacting with the spacecraft: in fact it is possible to think of it as a change in momentum by the photons coming from the Sun and the satellite in its motion. For this reason this interaction can be seen as a push of the spacecraft by the solar radiation, as if a pressure is exerted on it. A very simple proof can be obtained considering the power passing from the photons to the spacecraft. It is known that the power is conserved (unless dispersion as heat) and the two expression for the power are:

( Pi = ΨASC (2.57) Po = F c

This means that the input power Pi into the system is due to a solar ﬂux of photons Ψ that encounters an area of spacecraft ASC which can be pushed. The output power Po come as a consequence of the momentum exchanged by the photons with the spacecraft: they apply a force F moving at light speed c. In any case, if the two powers are equal, the following expression can be written:

Ψ F = ASC (2.58) c The force due to the radiation pressure can be seen as the one created by a pressure pSR acting on a surface:

F = pSRASC (2.59)

Thus it is possible to relate the solar radiation pressure to the ﬂux of photons incoming from the Sun and the speed of light:

Ψ pSR = (2.60) c The solar ﬂux is a quantity that depends only on the distance from the Sun and it deceases with the square of that distance:

2 d0 Ψ = Ψ0 (2.61) d

where Ψ is the solar flux at the distance d, while Ψ0 at d0. So if the solar flux is known at a given distance d0, it is known in every other point of the space. In fact it is already known that a distance of 1 AU the solar flux is equal to a quantity of 1368 W/m2. This quantity has a variation depending on solar cycles and solar activities, which can be found explained in a detailed way in Hargreaves [22], but for this work it will be assumed constant.

23 Models in inertial frames

So, what has been computed is just the modulus of the force. Every force being a vector, it is also needed a direction for a full description of the physical quantity. In this case it should be considered the shape of the satellite, and in particular its surfaces. For a good approximation it can be considered the spacecraft as spherical, and the force as pushing it away from the Sun:

2 Ψ0ASC d0 R − Rs F = (2.62) c d kR − Rsk

Eventually, a correction to the last formula should be made. In fact the area that faces the solar radiation pressure should be corrected by a reflectivity coefficient. This means that, depending on the material of the satellite it can react in different ways. The coefficient cr can have a value from 0 to 2.A 0 value means that the object is translucent to the incoming radiation, and there is no interaction with it. A value equal to 1 means that all the incoming radiation is absorbed (as in the case of a black body) and the total force is completely transmitted to the system. If the reflectivity coefficient were 2 it would mean that all the radiation is reflected and twice the force is transmitted to the satellite (i.e. flat mirror perpendicular to light source). However, one last note could be added: usually, in astrodynamics is not possible to think of objects that do not absorb light. Therefore the coefficient of reflectivity is only the part bigger than 1 that should be added to the absorbed component of solar pressure. In this case cr would be a number between 0 (black body) and 1 (flat mirror).

2 Ψ0d0 R − Rs F = (1 + cr)ASC 3 (2.63) c kR − Rsk

2.5 Equations of motion

Finally, in this section, the complete equations of motion will be summarized in the cases of free and perturbed dynamics. They will be written in the J2000 frame, taking as origin the Solar System Barycentre.

Unperturbed equations of motion:

 R¨ + ∇V = 0  X µj (2.64) ∇V = 3 (R − Rj)  kR − Rjk  j∈S

24 2.5 Equations of motion

Perturbed equations of motion: J2

R¨ + ∇V = 0  ∇V = ∇Ω + ∇Ω + ∇Ω  1 2 3  µ  X j ∇Ω1 = 3 (R − Rj)  kR − Rjk  j∈S 3µ J R2 (2.65)  X j 2j Bj ∇Ω2 = 5 (R − Rj)  kR − Rjk  j∈S   2 T  µjJ2jR 5(R − R ) I (R − R )  X plj j z j ∇Ω3 = 5 2Iz(R − Rj) − 2 (R − Rj)  kR − Rjk kR − Rjk  j∈S

Perturbed equations of motion: J2 + SRP The most complete set of equations of motion will be written, considering both the eﬀects of the inhomogeneous gravitational ﬁeld and the SRP. It is importante to note that in this term will appear also the mass of the spacecraft mSC , to get the acceleration aSRP that the force described by eq. (2.63) applies to the system.

 ¨ R + ∇V = aSRP  ∇V = ∇Ω1 + ∇Ω2 + ∇Ω3   X µj ∇Ω = (R − R )  1 3 j  kR − Rjk  j∈S  2  X 3µjJ2jRBj ∇Ω = (R − R ) 2 5 j (2.66) kR − Rjk  j∈S   2 T  X µjJ2jRplj 5(R − Rj) Iz(R − Rj) ∇Ω = 2I (R − R ) − (R − R )  3 5 z j 2 j  kR − Rjk kR − Rjk  j∈S   A Ψ d2 R − R  SC 0 0 s aSRP = (1 + cr) 3  mSC c kR − Rsk

25 Models in inertial frames

26 Chapter 3

Models in rotating frames

Inertial frames represent a comfortable reference to write equations of motion. No apparent forces appear and according to Einstein’s relativity they are the frames where the equations of motion appear in their simplest form [14]. This means that the comprehension of the mathematical terms is straightforward and immediate. However, these frames aren’t always the most suitable choice. Indeed, in many cases it is very convenient to have a frame that rotates. Simply thinking of a pendulum, it is much easier to represent the tension in a frame that follows the rotation of the mass, or even a satellite rotating around the Earth, which rotates around the Solar System Barycentre.

3.1 Deﬁnition of RPF

The ﬁrst historical steps into formulating rotating frames for astrodynamics were taken into the restricted three-body problem. In this problem two main bodies are taken into account, whose rotation is relative to their common barycentre. If a set of coordinates is chosen such that the x-axis is aligned, at every time, t, with the conjunction of the two bodies, and the z-axis with the relative angular momentum, the rotating frame will follow the motion. Moreover, in this frame, the two bodies will be always ﬁxed in the chosen points of the x-axis. The frame will be then properly adimensionalized and will be ready to be used. Exploiting this basic idea, already followed from Langrange and Euler, in this work it has been developed further, according to the works of the Barcelona group [19]. It has been called roto-pulsating frame because:

- roto: it is a frame that rotates with two main bodies called primaries. The origin of the frame will coincide with the barycentre of the two primaries.

- pulsating: the distance between the primaries is not ﬁxed, but variable in time, according to the ephemeris. It will be adimensionalized accordingly, such that in the new frame their relative distance will be always the unit length.

Following these deﬁnitions it is possible to write one equation that converts position

27 Models in rotating frames

Figure 3.1: Representation of the roto-pulsating frame and times, according to the procedure followed in Dei Tos [11]:

R(t) = b(t) + k(t)C(t)ρ(τ) (3.1a)

τ = ω(t − t0) (3.1b)

The description of the variables in eq. (3.1), depicted in ﬁg. 3.1 follows: - R(t) represents the position vector of the satellite in the inertial frame of reference J2000. It depends on the dimensional time, t - b(t) represents the position vector of the barycentre of the primaries in the inertial frame of reference J2000. Its presence in the equation (3.1a) shows that the frame has been translated from the SSB to the barycentre of the primaries. It depends on dimensional time, because it is a function of the position of the primaries. In fact, from the deﬁnition of barycentre it is possible to say:

mpRp + msRs b(t) = (3.2) mp + ms where m and R represent masses and positions, while the subscripts P and S stay for primary body or secondary body. Usually is taken as primary body the one with higher mass. It is assumed that the position of the primary and the secondary are known: this information comes from the ephemeris. Their masses are assumed to be constants.

28 3.1 Definition of RPF

- k(t) is the adimensionalization factor for the distances. It is the present distance of the primaries, that in the new frame is equal to one. This ca be proved to be true if k(t) is chosen as the modulus of the relative distance between the primaries, bmrP(t).

k(t) = kRp(t) − Rs(t)k = krP(t)k (3.3) The position in the inertial frame f the primaries can be expressed as:

( Rp(t) = b(t) + k(t)C(t)ρp(τ)

Rs(t) = b(t) + k(t)C(t)ρs(τ)

Taking the diﬀerence of one with the respect to the other it is possible to see that:

Rp(t) − Rs(t) = k(t)C(t) ρp(τ) − ρp(τ)

Comparing the norms of the two sides of the equation results that:

kR (t) − R (t)k ρ (τ) − ρ (τ) = p s = 1 p p k(t) where the term related to the rotation of the frame doesn’t aﬀect its modulus, as it will be proved later. - C(t) introduces the rotation into the frame. It is an orthogonal matrix or direct cosine matrix whose task is to rotate the J2000 frame into the RPF frame. The C(t) matrix can be computed as formed by three unit vectors, which are aligned with the desired directions:

 r (t)  P e1(t) =  k(t)  C(t) = e1(t) e2(t) e3(t) =⇒ e2(t) = e3(t) ∧ e1(t) (3.4)   rP(t) ∧ vP(t) e3(t) =  hP(t)

drP(t) where vP(t) = . dt Given this orthonormal deﬁnition of the direct cosine matrix, it will have a very useful property, i.e. the transpose of the matrix coincides with its inverse:

CT = C−1 (3.5) In the previous expressions it has been introduced also the relative velocity of the 1 primaries vP and the magnitude of their relative angular momentum hP deﬁned as : 1From now on ∧ represents the cross product

29 Models in rotating frames

hP(t) = krP(t) ∧ vP(t)k (3.6)

- ρ(τ) is the non dimensional position vector in the RPF frame. In order to have it fully non dimensional it should also depend on a non dimensional time, such as τ.

- τ is the non dimensional time introduced into the model to make the new variable fully adimensional. The adimensional time is obtained from a dimensional time considering the relative angular velocity of the planets around the barycentre.

- ω is the relative angular velocity of the planets with respect to their common barycentre. It is used in the RPF as a non-dimensionalizing factor. In this work this quantity has been chosen as constant and has been computed according to Dei Tos [11] as the ratio between 2π and the orbital relative revolution period between the primaries, T :

2π ω = (3.7) T

The advantage of a similar choice is that the adimensional time is scaled accordingly to the revolution period, such that at for the value t = T corresponds, τ = T . The computational of the orbital period is something amply described in literature, for instance in Bradley [6], where it is related with the mass of the primaries and the mean value of the semi-major axis a˜ for a suﬃciently high interval of time to remove secular perturbations, as:

s a˜3 T = 2π (3.8) G(mp + ms)

- t0 is the starting dimensional time, needed to express the relations in the new adimensional time.

At this point, since all the introduced quantities have been presented, the RPF can be described as a frame that introduces a ﬁrst translation from SSB to the system and then it starts rotating as the planets do. To keep the distance between the primaries ﬁxed it can be imagined as a pulsating motion for the frame: it tightens and stretches following the positions of the main bodies. To continue and rotate the equations into RPF there is the need to take some derivatives. As already mentioned, in non-inertial frames there are some apparent forces. To understand their meaning, eq. (3.1) is derived in time.

R˙ = b˙ + k˙ Cρ + kCρ˙ + kCρ0τ˙ (3.9a) τ˙ = ω (3.9b)

30 3.1 Definition of RPF

Taking the ﬁrst derivative of the position vector, the derivatives of the translational motion, b˙, will appear (it can be seen as velocity of the RPF with respect to the inertial frame) and that of the rotational motion. In particular this will have three terms linked to k˙ , the rotation of the frame, C˙ , and the relative velocity, ρ0. Focusing on the relative velocity on the RPF, it has been said that it depends on the adimensional time and no longer on the dimensional one. In this way it is possible to explain the presence of the τ˙. Following the chain rule:

dρ dρ dτ = = ρ0τ˙ (3.10) dt dτ dt Note that all time dependencies have been dropped to ease the notation.

R¨ = b¨ + k¨C + 2k˙ C˙ + kC¨ ρ + 2 k˙ C + kC˙ ρ0τ˙ + kCρ00τ˙ 2 (3.11a) τ¨ = 0 (3.11b) The second derivative represents the inertial acceleration of the spacecraft and its conversion. It is possible to see how many components arise. Given that all these parameters will be used, it is interesting to have an overview of also their derivation procedure.

The adimensionalization factor k:

rP · vP k˙ = (3.12a) k ˙ k(vP · vP + rP · aP) − krP · vP k¨ = (3.12b) k2

The ﬁrst derivative of the rotation matrix:

 kv − k˙ r e˙ = P P  1 k2  e˙2 = e˙3 ∧ e1 + e3 ∧ e˙1 (3.13)  h (r ∧ a ) − h˙ (r ∧ v ) e˙ = P P P P P P  3 2 hP The second derivative of the rotation matrix:

 (2k˙ 2 − kk¨)r − 2kk˙ v + k2a e¨ = P P P  1 k3  e¨2 = e¨3 ∧ e1 + 2(e˙3 ∧ e˙1) + e3 ∧ e¨1 (3.14)  (2h˙ 2 − h h¨ )r ∧ v − 2h h˙ r ∧ a + h2 (r ∧ j + v ∧ a ) e¨ = P P P P P P P P P P P P P P  3 3 hP

31 Models in rotating frames

The angular momentum hP:

˙ (rP ∧ vP) · (rP ∧ aP) hP = (3.15a) hP 2 ˙ ¨ hP (rP ∧ aP) + (rP ∧ vP) · (vP ∧ aP + rP ∧ jP) − h(rP ∧ vP) · (rP ∧ aP) hP = 2 hP (3.15b) All the relative computed values can be expressed as the diﬀerence of the absolute value of the secondary and the primary:

r = R − R  P s p  vP = Vs − Vp (3.16a) aP = As − Ap  jP = Js − Jp From the ephemeris only the value of position and velocity is known. The other terms represent the second and third time derivatives of the position and they have to be computed according to new formulas:

dvP daP aP = jP = dt dt where aP is the relative acceleration, while jP the relative jerk. The absolute values for these can be computed according to the universal gravitational law:

X R − Rj A = − µj 3 (3.17) kR − Rjk j∈S

The absolute acceleration for the primary body AP will be computed according to eq. (3.17), choosing as position RP. The same for the secondary body. To obtain a formulation for the jerk, it si necessary to derive the acceleration in eq. (3.17) with respect to time.

X V − Vj (R − Rj) · (V − Vj) J = − µj 3 − 5 (R − Rj) (3.18) kR − Rjk kR − Rjk j∈S The absolute jerk for the primaries can then be computed applying a substitution to the general position R and velocity V their speciﬁc values.

3.2 Rotation of the equations into RPF

The main quantities that have an important role into the description of the roto pulsating frame have been introduced. The next step will be to exploit these quantities and build the equations of motion in this frame. As already said, in this frame there are two important beneﬁts:

32 3.2 Rotation of the equations into RPF

- The equations of motion result directly adimensionalized due to the conversion from one reference to the other. This limits computational errors, since quantities which can be of diﬀerent magnitudes are opportunely scaled.

- The solutions designed in the R3BP ca be easily interpreted and reﬁned in this frame, that represents the natural extension of R3BP dynamics.

- It is easy to check if the reﬁned solution retained the features of the R3BP one.

The equations of motion can be rotated following different procedures, which should bring to the same results. In this section a first procedure will apply the transformation directly on the differential equations, which can be the simplest way. In a second moment the equations of motion will be derived for a system where the transformation is applied before the statement of the Lagrangian. Following the structure of the previous Chapter the free dynamics is expressed and then the equations are modified due to the perturbation terms.

Plugging (3.1a) and (3.11a) into (2.21) comes out:

2 00 ˙ ˙ 0 ¨ ˙ ˙ ¨ ¨ X C(ρ − ρj) τ˙ kCρ + 2τ ˙kC + 2τk ˙ C ρ + kC + 2kC + kC ρ+b+ µj 2 3 = 0 (3.19) k kρ − ρjk j∈S

where the adimensional position of the attractors ρj are computed inverting (3.1a):

CT (R − b) ρj = (3.20) k

Is worth noting the development of the relative position terms. It can be seen how they are now independent from b:

R − Rj = kC(ρ − ρj)

Regarding the distance, this should not depend on the reference system, since they are presented as relative quantities. It can easily proved that:

n n n kR − Rjk = k kρ − ρjk

It can be seen that the rotation matrix does not appear and this is a direct consequence of the orthonormality property. In fact:

33 Models in rotating frames

n n q kR − Rjk = (kC(ρ − ρj)) · (kC(ρ − ρj)) q n T = (kC(ρ − ρj)) (kC(ρ − ρj)) q n 2 T T = k (ρ − ρj) C C(ρ − ρj) q n 2 T −1 = k (ρ − ρj) C C(ρ − ρj) q n T = k (ρ − ρj) (ρ − ρj)

n n = k kρ − ρjk

In order to allow implementation, eq. (3.19) should be written in the canonical form:

! ! 1 2k˙ 1 k¨ k˙ ρ00 + I + 2CT C˙ ρ0 + I + 2 CT C˙ + CT C¨ ρ τ˙ k τ˙ 2 k k ! 1 1 T ¨ 1 X ρ − ρj + 2 C b + 3 µj 3 = 0 (3.21) τ˙ k k kρ − ρjk j∈S where I is the identity matrix. It is possible to note that all the products of the direct cosine matrix with its transpose have disappeared, as expected. Eq. (3.21) diﬀers from eq. (2.21) not only on the number of terms in the expression, which come out after the rotation, but mostly on the domain of the integration time. The independent variable ρ is in fact dependent on τ. Its derivatives are then taken with respect to this variable. However, it is possible to see in equation (3.21) many terms which depend on time t and have also derivatives taken with respect to it. The dissimilarity is mandatory for the equation of motion of this system and is originated from two diﬀerent requirements:

- On one side the independent variable will be the one which is integrated. The numerical procedures introduce numerical errors and those are even bigger when considering very small or very big numbers. Since the characteristic distances of the system are in the order of the astronomical units, the magnitude is in the order of 1010 − 1011 m. This translates into very high characteristic times: to have meaningful trajectories for such a system, hundreds of days are required, therefore in the order of 106 −107 s. Adimensionalizing this number will reduce the numerical error introduced in the integration procedure. - on the other side, there are some terms that depend on the real time, or rather on the epoch. As already said, to have a high ﬁdelity model, it is needed to get the

34 3.3 Alternative derivation of the equation of motion

position of the celestial bodies from their ephemeris, which are dependent on the considered epoch time. All those variable then cannot depend on the integration time. Indeed, they have to be computed on-line, but their dependence should be left to the dimensional time, which makes them quantities with a precise physical meaning. This makes the problem non-autonomous.

3.3 Alternative derivation of the equation of motion

An useful exercise to better comprehend the terms of the equations of motion is to try to obtain them in a different way. In the previous section these were derived in an inertial frame, then they were rotated into RPF at the level of the differential equations. Taking into consideration operating the transformation before the Lagrangian definition, all the quantities shall be expressed with respect to the same independent variable, the non dimensional variable. In order to make the exercise mathematically complete:

R(τ) = b(τ) + k(τ)C(τ)ρ(τ) (3.22)

Since the Lagrangian formulation has many time dependent variables, the chain rule for a generic quantity follows:

∼ (t) =∼ (τ)

d ∼ d ∼ dτ ∼˙ = = =∼0 τ˙ dt dτ dt

d∼˙ d(∼0 τ˙) d ∼0 dτ˙ dτ d ∼0 dτ˙ ∼¨ = = =τ ˙ + ∼0 =τ ˙ + ∼0 =τ ˙ 2 ∼00 + ∼0 τ¨ dt dt dt dt dt dτ dt

According to its deﬁnition, the Lagrangian is:

L = T − V where the relations depending on τ hold:

R0 = b0 + k0Cρ + kC0ρ + kCρ0 (3.23)

mpRp + msRs b = (3.24) mp + ms

35 Models in rotating frames

In the new frame, the kinetic energy can be written, exploiting eq. (3.23), as follows:

1 T = R˙ · R˙ 2 τ˙ 2 = R0 · R0 2 τ˙ 2 = b0 + k0Cρ + kC0ρ + kCρ0)T (b0 + k0Cρ + kC0ρ + kCρ0 2 (3.25) τ˙ 2 = (b0T b0 + k02ρT ρ + k2ρT C0T C0ρ + k2ρ0T ρ0 2 + 2k0b0T Cρ + 2kb0T C0ρ + 2kb0T Cρ0 + 2kk0ρT CT C0ρ + 2kk0ρT ρ0 + 2k2ρT C0T Cρ0) Furthermore, the potential energy can also be deﬁned in the new frame: X µj V = − kR − Rjk j∈S X µj (3.26) = − kkC(ρ − ρj)k j∈S It is possible to see that both T and V have been deﬁned according to their expression in the inertial frame, and after that the rotated quantities have been introduced. This is due to the fact that the Lagrangian formulation requires certain quantities that are related with physical properties of the system. At this point it is possible to express the Lagrangian as dependent on the two energies, which have been completely rotated into the new frame:

L (ρ, ρ0, τ) = T (ρ, ρ0, τ) − V (ρ, τ) (3.27) The equations of motion derive from the Lagrange formalism, which can be written as:

d ∂L ∂L − = Q = 0 dt ∂ρ˙ ∂ρ Once again it must be noted that the previous equation is considered for inertial frames, where the independent variable is time t, depending on ρ and no more on R . Since in this case the variable does not depend on t, but on τ, those equations can be rotated following the chain rule:

dτ d ∂L dt ∂L − = dt dτ ∂ρ0 dτ ∂ρ d ∂L 1 ∂L τ˙ − = (3.28) dτ ∂ρ0 τ˙ ∂ρ d ∂L τ¨ ∂L ∂L − − = 0 dτ ∂ρ0 τ˙ 2 ∂ρ0 ∂ρ

36 3.3 Alternative derivation of the equation of motion

New terms have appeared on the formulation of the problem and it will be necessary to compute them all:

∂L =τ ˙ 2 k2ρ0 + kCT b0 + kk0ρ + k2CT C0ρ ∂ρ0

d ∂L = 2¨τ k2ρ0 + kCT b0 + kk0ρ + k2CT C0ρ dt ∂ρ0 +τ ˙ 2(2kk0ρ0 + k2ρ00 + k0CT b0 + kC0T b0 + kCT b00 + k02ρ + kk00ρ + kk0ρ0 + 2kk0CT C0ρ + k2C0T C0ρ + k2CT C00ρ + k2CT C0ρ0)

τ¨ ∂L =τ ¨ k2ρ0 + kCT b0 + kk0ρ + k2CT C0ρ τ˙ 2 ∂ρ0

∂L 0 = k02ρ + k2C0T C0ρ + k0CT b0 + kCT b0 + kk0ρ0 + k2C0T Cρ0 − ∇V ∂ρ

Plugging all the terms into (3.28) it is then possible to arrive at the equations of motion, which are completely dependent on the adimensional time τ:

τ˙ 2k2 ρ00 + τk¨ 2 + 2τ ˙ 2kk0 I + 2τ ˙ 2k2CT C0 ρ0 + τkk¨ 0 +τ ˙ 2kk00 I + τk¨ 2 + 2τ ˙ 2kk0 CT C0 +τ ˙ 2k2CT C00 ρ +τk ¨ CT b0 +τ ˙ 2kCT b00 + ∇V = 0 (3.29)

It has already been said that it is not convenient to have all the terms depending on τ, therefore some of them should be rotated back into the time dependent form, according to the chain rule: d ∼ ∼˙ ∼0= = dτ τ˙

d2 ∼ d d ∼ d ∼˙ 1 d ∼˙ ∼¨ ∼˙ τ¨ ∼00= = = = = − dτ 2 dτ dτ dτ τ˙ τ˙ dt τ˙ τ˙ 2 τ˙ τ˙ 2 Plugging the new terms into the equations of motion and simplifying all the terms it is possible to obtain:

37 Models in rotating frames

h i τ˙ 2k2ρ00 + τk¨ 2 + 2τk ˙ k˙ I + 2τk ˙ 2CT C˙ ρ0 h i + kk¨I + 2kk˙ CT C˙ + k2CT C¨ ρ + kCT b¨ + ∇V = 0 (3.30)

The last equation is almost similar to eq. (3.21), but for some few terms. To get the exact expression it must be noted that:

τ¨ =ω ˙ = 0

X µj ρ − ρj ∇V = 3 k kρ − ρjk j∈S

Operating this last transformation the equation (3.30) can be expressed as:

! ! 1 2k˙ 1 k¨ k˙ ρ00 + I + 2CT C˙ ρ0 + I + 2 CT C˙ + CT C¨ ρ τ˙ k τ˙ 2 k k ! 1 1 T ¨ 1 X ρ − ρj + 2 C b + 3 µj 3 = 0 (3.31) τ˙ k k kρ − ρjk j∈S

This shows not only that the two equations are equal, but also proves that according to Lagrangian formalism, it is not important where a transformation is introduced, until every following step is taken accordingly.

3.4 Perturbations

The free dynamic equations in RPF were retrieved after a transformation procedure from the equations in the inertial frame. The same can be done for the perturbed equation, introducing both gravitational and non-gravitational effects. Also in this case there are two possibilities. Firstly, it is possible to write the equations of motion in inertial frame and then convert them into a rotating one, applying the transformation directly on the differential equation. This procedure has as reference eq. (3.19). The method will be used to introduce non gravitational effects. In the second case it is possible to rotate directly the Lagrangian into RPF and then obtain the equations of motion. The reference equation for this case is (3.30) and it will be used for the gravitational effects.

38 3.4 Perturbations

3.4.1 Gravitational effects In a previous Chapter an approximated way to introduce the inhomogeneity of the gravitational field of non perfectly spherical bodies was presented. It was shown how the J2 perturbation was the only considered, mainly because it was three order of magnitudes greater than the others. The procedure consisted into creating a new potential, which accounted for the non homogeneous gravitational field. Eq. (2.45) referred to the new potential, with both components. In this Section that equation will be rotated and its gradient will be computed, in order to write the full equations of motion. The first step is to write the gravitational potential in RPF. To do this, the rotation equations are taken and then plugged into the definition of the potential:

2 2 2 ! X µj X µjJ2jRBj X 3µjJ2jRBj rz V = − + 3 3 − 5 5 kkρ − ρjk 2k kρ − ρjk 2 k kρ − ρjk j∈S j∈S j∈S

Particular attention must be paid to the third component of the relative position vector. It was already deﬁned with the aid of an auxiliary matrix Iz. In this case it must be added also the rotation part into synodic frame:

rz = kIzC (ρ − ρj) (3.32)

Therefore, the needed term can be expressed directly, exploiting the properties of transpose matrices:

2 T 2 T T T 2 T rz = rz rz = k (ρ − ρj) C Iz IzC (ρ − ρj) = k (ρ − ρj) M (ρ − ρj)

The four matrices have been condensed in only one matrix M, which takes into account their product due to the transpose. Due to the property of those matrices, being C orthogonal and Iz symmetric, it is possible to say that M is symmetric, too. This comes into a complete synodic deﬁnition for the gravitational ﬁeld, as:

2 2 ! X µj X µjJ2jRBj X T 3µjJ2jRBj V = − + 3 3 − (ρ − ρj) M (ρ − ρj) 3 5 kkρ − ρjk 2k kρ − ρjk 2k kρ − ρjk j∈S j∈S j∈S (3.33)

To derive the gradient, which is needed for the formulation of the equations, the potential is split into three components.

V = Ω1 + Ω2 + Ω3

X µj Ω1 = − kkρ − ρjk j∈S

39 Models in rotating frames

2 X µjJ2jRBj Ω2 = − 3 3 2k kρ − ρjk j∈S

2 X T 3µjJ2jRBj Ω3 = (ρ − ρj) M (ρ − ρj) 3 5 2k kρ − ρjk j∈S

Applying the gradient operator means to take the derivatives with respect to the variables (it is also deﬁned as the vector of the partial derivatives). It can be seen that:

∂V ∂V ∂V T ∇V = ∂ρ1 ∂ρ2 ∂ρ3

Since this is a linear operator, it is possible to say that:

∇V = ∇Ω1 + ∇Ω2 + ∇Ω3 (3.35)

Computing the gradients of the components it is possible to see, as in the previous cases, that the ﬁrst term is related to the spherical homogeneous ﬁelds. The other two terms instead are the terms related to the zonal harmonics expressed in the new frame.

X µj ∇Ω1 = 3 (ρ − ρj) (3.36) kkρ − ρjk j∈S

2 X 3µjJ2jRplj ∇Ω2 = 3 5 (ρ − ρj) (3.37) 2k kρ − ρjk j∈S

2 T X 3µjJ2jRplj 5(ρ − ρj) M(ρ − ρj) ∇Ω3 = 3 5 2M(ρ − ρj) − 2 (ρ − ρj) (3.38) 2k kρ − ρjk kρ − ρjk j∈S

The equations of motion can therefore be simply written as an overwriting of the new potential into eq. (3.30):

! ! 1 2k˙ 1 k¨ k˙ ρ00 + I + 2CT C˙ ρ0 + I + 2 CT C˙ + CT C¨ ρ τ˙ k τ˙ 2 k k 1 1 1 + CT b¨ + ∇V = 0 (3.39) τ˙ 2 k k2

40 3.4 Perturbations

3.4.2 Non-gravitational eﬀects

To introduce non-gravitational eﬀects, the considered approach directly converts the diﬀerential equations from inertial to RPF. This is the simplest way, otherwise new derivatives would appear, to justify the change of coordinates. The dynamic system’s equations of motion due to a series of non conservative forces can be written as:

N ¨ X mR + m∇V = Fi (3.40) i=1

In the ﬁrst part of the equation the masses of the system appear also, because there is no zero in the right-hand side. In fact, for the free dynamics, the mass of the system is not important. However when non conservative forces are acting on it, they create accelerations that do depend on this mass. The forces appear in the right-hand side of the equations. In fact, in the inertial frame, they coincide with the generalized forces required from the Lagrangian formalism. Firstly, the equations should be subdivided by the mass, in order to obtain an equation for dynamic equilibrium, and then they should be converted side by side:

N ¨ 1 X R + ∇V = Fi (3.41) m i=1

Written in this form it is easier to see how the left-hand side has already been rotated once and the result is eq. (3.19). The deﬁnition for the forces of eq. (2.63) can be used to convert the right-hand side: the considered force is the one produced by the pressure of solar radiation. The conversion in roto pulsating frame is just a matter of algebra:

2 SF0d0 R − Rs F = (1 + cr)ASC 3 c kR − Rsk (3.42) 2 SF0d0 C(ρ − ρs) = (1 + cr)ASC 2 3 c k kρ − ρsk

Combining together all the equations it is possible to obtain a complete formulation for the solar radiation pressure perturbation into RPF:

! ! 1 2k˙ 1 k¨ k˙ ρ00 + I + 2CT C˙ ρ0 + I + 2 CT C˙ + CT C¨ ρ τ˙ k τ˙ 2 k k

! 2 1 1 T ¨ 1 X ρ − ρj ASC Ψ0d0 (ρ − ρs) + 2 C b + 3 µj 3 = (1 + cr) 3 3 (3.43) τ˙ k k kρ − ρjk mSC c τk˙ kρ − ρsk j∈S

41 Models in rotating frames

3.5 Logics of the models

The high number of equations that have been presented can be somehow confusing, since many of them are very similar, but with minor changes that give the description of the diﬀerent procedures that have been used. To summarize the main concepts and focus on their application, in this section will be given a brief description of the main characters and a logical scheme of their interaction.

- LJ2000 represents the lagrangian of the N-body problem written in the inertial frame J2000. This quantity plays a fundamental role, since it is the beginning of every procedure which permits the deﬁnition of the equations of motion describing the problem.

1 ˙ ˙ X µj LJ2000 = mR · R + (3.44) 2 kR − Rjk j∈S

- LRPF is the lagrangian of the dynamic system written in the roto pulsating frame. It can be obatined applying a conversion to LJ2000 and it can be used to obtain the equations of motion in RPF.

2 τ˙ 0 0 0 0 T 0 0 0 0 LRPF = b + k Cρ + kC ρ + kCρ ) (b + k Cρ + kC ρ + kCρ + 2 X µj (3.45) + kkC(ρ − ρj)k j∈S

- EoMJ2000 are the equations of motion of the N-body problem written in the inertial frame J2000. The equations of motion permit the understanding of the dynamical evolution of the system. They can be obtained in diﬀerent ways: in this work, their derivation from the equations of Euler–Lagrange through the application of the Lagrangian formalism has already been described.

N ¨ X mR + m∇V = Fi (3.46) i=1

- EoMRP F are the equations of motion in roto pulsating frame. They are already adimensionalized and have a more flexible capability in refining and understanding R3BP orbits. They can be obtained by a conversion of EoMJ2000 or exploiting the modified Lagrangian formalism to LRPF .

! ! 1 2k˙ 1 k¨ k˙ ρ00 + I + 2CT C˙ ρ0 + I + 2 CT C˙ + CT C¨ ρ τ˙ k τ˙ 2 k k N T 1 1 T ¨ 1 1 T X ∂Ri + C b + ∇V = C Fi (3.47) τ˙ 2 k k2 mτ˙ 2k2 ∂ρ i=1

42 3.5 Logics of the models

In order to connect between them the quantities previously described there are some mathematical operators that can be used.

- The Lagrangian formalism permits to obtain the equations of motion in J2000 once the LJ200 lagrangian of the system has been deﬁned.

d ∂L ∂L − = Q (3.48) dt ∂R˙ ∂R

- The modified Lagrangian formalism allows to obtain the equations of motion in RPF once the LRPF Lagrangian of the system has been defined. It is a modified version of the previous formalism since it includes a new independent variable with respect to which the equations should be written.

d ∂L τ¨ ∂L ∂L − − = Q (3.49) dτ ∂ρ0 τ˙ 2 ∂ρ0 ∂ρ

- The conversion procedure considers the transformation of a position vector in the inertial frame into a new one in RPF.

R(t) = b(t) + k(t)C(t)ρ(τ) (3.50a)

τ = ω(t − t0) (3.50b)

In the following ﬁgure the whole scheme has been included. There are two possible paths that can be followed. The steps of the ﬁrst are the following:

1. Write the Lagrangian in the J2000 frame according to eq. (3.44).

2. Use the Lagrangian formalism of eq. (3.48) to obtain the equations of motion in J2000, as written in eq. (3.46).

3. Use the conversion procedure of eq. (3.50) to rotate the equations of motion in RPF, which result in eq. (3.47).

Alternatively, diﬀerent steps can be taken, but obtaining the same result.

1. Write the Lagrangian in the J2000 frame according to eq. (3.44).

2. Use the conversion procedure of eq. (3.50) to rotate the Lagrangian into the one expressed in RPF frame, as presented in 3.45.

3. Use the Lagrangian formalism of eq. (3.49) to obtain the equations of motion in RPF, as written in eq. (3.47).

43 Models in rotating frames

Figure 3.2: Logics for writing the equations of motion in RPF

3.6 Special cases

The equations of motion of the N-body problem describe a dynamic system of many bodies. Then it should be possible to lead them back to the equations relative to a minor number of celestial bodies. In this section it will be shown how it is possible to get the restricted two or three body problem from the already presented equations. Those problems will be brieﬂy characterized, but without a profound description, which can be found in literature.

3.6.1 The Restricted 2 Body Problem

The Restricted 2 Body Problem (R2BP) is one of the ﬁrst dynamic problems that have been formulated, and it is fundamental in astrodynamics, since it is the only one that has an analytical solution. Referring to eq. (2.30) it is possible to see its classical formulation, simply neglecting the terms due to the presence of the other bodies. The result is the set of diﬀerential equations:

r r¨ + µP = 0 (3.51) krk3

This is a second order diﬀerential equation. It has been written in a relative fashion, therefore it is possible to see how the main body is the ﬁxed body, around which orbits

44 3.6 Special cases the second body.

For this problem it is possible to deﬁne some constants of motion, i.e. quantities that do not change in time. One of them is the mechanical energy. Physically this is true since there are only conservative forces acting on the system. Multiplying eq. (3.51) scalarly for the velocity of the spacecraft:

r˙ · r r˙ · r¨ + µ = 0 P krk3

Manipulating the equation, it is possible to obtain:

d r˙ · r˙ µP − = 0 dt 2 r that means the quantity inside brackets is constant and it is immediate to see that is a sum of kinetic and potential energy, therefore it is called total energy:

r˙ · r˙ µP E = − (3.52) 2 r To obtain another constant of the motion, it is possible to multiply vectorially eq. (3.51) for the relative position:

r ∧ r r ∧ r¨ + µ = 0 P krk3

In this case only the ﬁrst term will be nonnull, therefore the equation can be:

d(r ∧ r˙) = 0 dt The constant terms that have just appeared deﬁne an already considered quantity, known as angular momentum of the system:

h = r ∧ r˙ (3.53)

The quantity determines the rotation axis of the spacecraft in its motion around the primary. Since it is constant, it is possible to say that in this model, the satellites lay always on the same plane.

It is possible to deﬁne an other constant of the motion, called eccentricity vector. It can be obtained multiplying eq. (3.51) for the angular momentum:

d r˙ ∧ h r de − = = 0 (3.54) dt µp r dt

45 Models in rotating frames

It is possible to verify that this vector lays always on the plane deﬁned by the angular momentum vector and it is usually used as a reference vector for the analytical solution of the R2BP:

h/µp r = (3.55) 1 + e cos θ

where θ = acos(r · e) describes the angle between the position vector and the eccentricity vector. Eq. (3.55) describes the motion of the orbiting body, as following a conical path. Depending on the value of the eccentricity, the trajectory will be a diﬀerent conic:

- e = 0 circular orbit

- 0 < e < 1 elliptical orbit

- e = 1 parabolic orbit

- e > 1 hyperbolic orbit

3.6.2 Restricted 3 Body Problem The three body problem can be analyzed following the same path. The first step will be to define the equations in the SSB model. They will consider only the effect of two bodies that act on the spacecraft:

¨ R − R1 R − R2 R = −µ1 3 − µ2 3 (3.56) kR − R1k kR − R2k

The meaning of the equation is that the acceleration on a body in space depends on the mass of the attractor and their relative position with the body. At this point it is possible to rotate the frame of reference using a direct cosine matrix. In fact, if a rotation around the z-axis of an angle θ is supposed, the rotation relation can be written as:

R = T ρˆ

while the direct cosine matrix can be explicated as:

cos θ − sin θ 0 T = sin θ cos θ 0 0 0 1

At this point it is possible to rotate directly the equation of motion to obtain the equation into the synodic frame. This is the deﬁnition for a frame that rotates of a certain angular velocity with respect to an axis. A wise choice of all these elements can yield to a problem of a wider comprehension. In fact if the angular velocity is the relative

46 3.6 Special cases

one between the primaries, those can be thought as ﬁxed with respect to the frame. The rotated equations are:

ρˆ − ρˆ ρˆ − ρˆ ˆ¨ ˙ ˆ˙ ¨ ˆ 1 2 T ρ + 2T ρ + T ρ = −µ1T 3 − µ2T 3 (3.57) kρˆ − ρˆ1k kρˆ − ρˆ2k As already mentioned it is necessary to have some adimensionalization factors, because non dimensional equations are easier to be integrated and less prone to errors. Choosing the mean distance between primaries L as characteristic length and the relative angular velocity ω to get characteristic times, brings the following relations:

ρˆ = Lρ

τ = ωt Exploiting the previous relations it is possible to get directly the non dimensional equations, if plugged into eq. (3.57).

2 00 ˙ 0 ¨ µ1 T (ρ − ρ1) µ2 T (ρ − ρ2) ω T ρ + 2ωT ρ + T ρ = − 3 3 − 3 3 (3.58) L kρ − ρ1k L kρ − ρ2k Some very useful properties are hereby listed. A relative gravitational parameter is deﬁned: it can represent in a proper way the position of a primary with respect to their common :

m µ = 2 m1 + m2 In the synodic frame the position of the primaries is quite easy to be found, since they lay on the x-axis, with a non dimensional distance that depends only on their mass. This comes as a consequence of the choice of their barycentre as origin of the system:

1 1 m ρ = 1 0 = (1 − µ) 0 1 m + m     1 2 0 0

1 1 −m ρ = 2 0 = −µ 0 2 m + m     1 2 0 0 Briefly recalling the equations of motion for RPF, it is possible to see some analogies, that in fact will result in a simplification of some terms. Neglecting the effects of all bodies but primaries the equations are:

τ˙ 2kCρ00 + 2τ ˙k˙ C + 2τk ˙ C˙ ρ0 + k¨C + 2k˙ C˙ + kC¨ ρ + b¨ (3.59) C(ρ − ρ1) C(ρ − ρ2) = −µ1 2 3 − µ2 2 3 k kρ − ρ1k k kρ − ρ2k

47 Models in rotating frames

Considering the previous equations it is necessary to determine how all the terms in RPF are related with the terms in synodic frame. The deﬁnition of the non dimensional time has not changed between the two frames, therefore, the ﬁrst derivative will correspond to the angular velocity:

τ˙ = ω

The characteristic length of the two frames is diﬀerent. In R3BP it was chosen almost intuitively as the distance between primaries, but it is possible to prove that this choice correspond to the deﬁnition of k in RPF:

k = krk = kR2 − R1k

In the synodic frame the two distances are simply: cos θ m R = 1 L sin θ 2 m + m   1 2 0

cos θ m R = − 2 L sin θ 1 m + m   1 2 0

Therefore, the relative distance is just:

cos θ r = L sin θ 0

In this case it is possible to explicit the non-dimensionalizing factor:

k = L

k˙ = 0 k¨ = 0 The distance of the origin from the barycentre is null and this can be proved:

cos θ m1R + m R −m m + m m b¨ = 1 2 2 = 1 2 1 2 sin θ = 0 m + m m + m   1 2 1 2 0

In this case the eﬀects of all the other planets have been neglected, therefore it was correct to foresee that the two origins will coincide.

48 3.6 Special cases

The last step is to build the rotation matrix. In RPF it has an exact procedure and it will be followed here to prove that the direct cosine matrices of the two frames coincide. C is deﬁned as:

C = [e1 e2 e3]

Before writing the expressions for the basis vectors it is necessary to deﬁne the relative velocity and the relative angular momentum in the new frame:

− sin θ v = Lθ˙  cos θ  0

 0  r ∧ v =  0  L2θ˙

At this point it is possible to compute the three vectors. According to eq. (3.4):

cos θ 0 − sin θ r r ∧ v e = = sin θ e = = 0 e = e ∧ e = cos θ 1 krk   3 kr ∧ vk   2 3 1   0 1 0

Plugging them all together it is possible to write:

cos θ − sin θ 0 C = sin θ cos θ 0 0 0 1

Then eq. (3.59) can be written as:

2 00 ˙ 0 ¨ T (ρ − ρ1) T (ρ − ρ2) ω LT ρ + 2ωLT ρ + LT ρ = −µ1 2 3 − µ2 2 3 (3.60) L kρ − ρ1k L kρ − ρ2k

The division of eq. (3.60) by L gets as result eq. (3.57).

49 Models in rotating frames

50 Chapter 4

Validation of the models

The procedure of writing the equations of motion for a model is just the first step to get a full description of the evaluation of the state. The previously presented differential equations are far from having an analytical solution, therefore numerical procedures must be invoked. In this way it is possible to get a solution, which will be correct only up to a certain level of accuracy, due to the fact that numerical errors are present. So the capability of integrating the differential equations does not necessary give a correct result: this should be compared, where possible, against an already existing one, which is reliable. This process is called validation, where the main quantities are the achieved solution, the reference solution and the error between them. Firstly, in this chapter, it will be given a description of the numerical scheme used for the integration of the equations of motion. Then it will be explained how to obtain reference solutions and which programs are used in this work. Finally the results of the process are shown, proving that the results are reliable enough.

4.1 Integration scheme

An integration scheme is a procedure aiming at the numerical solution for an initial value problem (IVP). The n-body problem represents an IVP, therefore, it will be necessary to adopt an integrator to solve it. The basical formulation of the problem can be seen as described by the Cauchy problem:

( x˙ = f(x, t) (4.1) x(t0) = x0

In the most generic case eq. (4.1) is a non-linear problem along with its initial conditions (hence the name). The domain where to search for the solution is the totality of the real numbers, having the time t as independent variable:

n n+1 n x ∈ R f : R → R

The solution of this problem is not possible unless a certain approximation is introduced in its expression and the Taylor series expansion is the most suitable tool to do

51 Validation of the models

that. It can express a suﬃciently regular function as an inﬁnite summation of terms, computed at every single point together with its derivatives:

2 ∗ ∗ dxi 1 dxi 2 xi(t + h) = xi(t ) + h + h + ··· (4.2) dt t∗ 2! dt2 t∗

This means that to compute the function xi after a small time step, the function value at the time t∗ are needed, along with its derivatives. Recalling eq. (4.1), it is possible to rewrite the previous equation as:

∗ ∗ 2 ∗ ∗ ∗ ∗ dfi(x(t ), t ) h xi(t + h) = xi(t ) + fi(x(t ), t )h + + ··· (4.3) dt 2! The derivative of the function has been substituted by the right-hand side of the differential equations. One of the most interesting things to note is that at a certain point the summation has to be truncated, because mathematically it is possible to sum infinite terms, but numerically it is impossible. The number of terms considered define the order of the integration scheme, and consequently its accuracy. The Taylor expansion of order n is:

n ∗ ∗ n−1 n ∗ ∗ X dfi(x(t ), t ) h n+1 xi(t + h) = xi(t ) + + O(h ) (4.4) tn−1 n! j=1 d

The Taylor series has been truncated at order n, and all other terms of superior order are not accounted: therefore, the method is called of order n. This is the basis for every integration scheme and also for the one that has been used for this thesis work. In literature it is possible to get a deeper walk-through for every concept that hereby has been just slightly introduced; see Atkinson [3] or Butcher [7].

4.1.1 Runge–Kutta–Fehlberg methods and ODE78 In this work one of the Runge–Kutta methods will be used, therefore it results necessary to give a brief overview on them. Considering equation (4.4) there are some disturbing terms represented by the derivatives of the function. In order to achieve an optimal situation, it would be necessary to identify a method to compute those terms without taking the derivatives. There are many way to do this and at this point the integration schemes are divided into two main groups. A ﬁrst group consists into the so called single step methods: there are all those schemes ∗ ∗ where xi(t + h) can be found knowing exclusively the value of the precedent node xi(t ). It is possible to see how the derivatives are a direct consequence of the function evaluation process. Calling xk th function at the current step and xk+1 the one at the next step, the generic Runge–Kutta formula is:

s X xk+1 = xk + h cjf(zj, tk + αjh) (4.5) j=1

52 4.1 Integration scheme

This step is usually called corrector step, because corrects the s predicted values zj using the coeﬃcients cj. The most general expression for the predicted values is:

s X zj = xk + h βjif(zi, tk + αih) (4.6) i=1

It is clear that this formula is implicit, since the zj predicted value depends on itself. Runge–Kutta integrators are usually used as explicit methods, therefore eq. (4.6) can be expressed as:

j X zj = xk + h βjif(zi, tk + αih) (4.7) i=1 This is called a Runge–Kutta method of order s+1, since there are present s predictors and 1 corrector. The computation of the coefficients is a cumbersome process: every predictor must be plugged in the next and finally all of them must be plugged into the corrector. At that point it is possible to get the Taylor expansion of the order s + 1 and compare the terms. The result will be a system of equations that once solved can give the value of the coefficients. The result is expressed in a fairly readable way called Butcher’s tableau:

(a) Implicit RK Method (b) Explicit RK Method

α1 β11 β12 ··· β1s 0 α2 β21 β22 ··· β2s α2 β21 β22 ...... αs βs1 βs2 ··· βss αs βs1 βs2 ··· βs s−1 c1 c2 ··· cs c1 c2 ··· cs−1 cs Table 4.1: Butcher’s tableau

The method should stop when a certain error has been reached. A method of s order has a truncation error that depends on the s + 1 order term, therefore to compute it, the derivatives related to this term are needed. This means other function evaluations: a high accuracy method becomes easily numerically heavy. Fehlberg [15] exploited the fact that some coeﬃcients from Butcher’s tableau can be arbitrarily chosen. Along with this, is possible to exploit some function evaluations already taken to compute the previous step: in this ways, these methods can decrease the overall computational cost. The new methods are called Runge–Kutta-Fehlberg. Fehlberg’s work has been later improved by Dormand and Prince [13], whose work is the reference in propagators used in MATLAB®or Octave. The integration scheme exploited in this work is a RKF method of 7-th order, which is veriﬁed with an 8-th order RK, exploiting Dormand and Prince formulas. This function is open source and free to use, thanks to the work of Mr. Govorukhin V.N. To use this method 13 function evaluations are needed and it guarantees a truncation error of O(h9).

53 Validation of the models

4.2 JPL’s SPICE

The validation process of a model needs some reference solutions. When analytical solutions are available, they are preferred, otherwise specific software can be used. The first of them to be considered is SPICE. This tool has been created at the Jet Propulsion Laboratory (JPL) from the Navi- gation and Ancillary Information Facility (NAIF). This group leads the design and implementation of the SPICE ancillary information system. It has been designed to assist NASA scientists in planning and interpreting scientific observations from space-borne instruments, to help engineers design missions, plan scientific observations, analyse science data and conduct various engineering functions associated with flight projects. The acronym hides in it the meaning and the functionalities of SPICE:

- S: Spacecraft

- P: Planet

- I: Instrument

- C: Camera matrix

- E: Events

It does not only include ephemeris information for planets, bodies and spacecraft, but also data for on board instruments, their relative and absolute position and attitude, the frame orientation from a local to a global level and even particular events as eclipses or science plans. It appears evident that its functionalities are far beyond what needed for this work. However, every attempt of building a propagator in space missions must compare its results with SPICE, being it the U.S. Planetary Data System’s standard for archiving ancillary data and its usage is highly recommended by the International Planetary Data Alliance, even if not a formal requirement. For the ﬁnal user, the main characteristic of SPICE is not its completeness and the totality of data it can provide. Indeed, it is much more useful the modularity work that NAIF has carried on into programming this tool: every needed information comes into a special data format, which can be obtained from NAIF’s website 1 or independently built.

- SPK: File containing spacecraft, planet or small bodies ephemeris, or more generally, location of any target body, given as a function of time.

- PCK: File containing physical, dynamical and cartographic constants for target bodies, such as size and shape speciﬁcations, and orientation of the spin axis.

- LSK: Leap seconds kernels are used in converting time tags between various time measurement systems.

1https://naif.jpl.nasa.gov/naif/ downloaded April 29 2016

54 4.2 JPL’s SPICE

Here only the main data format have been presented, however, it is possible to ﬁnd one for every functionality of SPICE.

The classical usage that have been done in this thesis consists of ephemeris acquisition, and it will be better explained through an example. This has been achieved downloading the proper SPK file and associating it with a PCK and a LSK. The state vector data are contained as binaries in the SPK, while the LSK permits a full conversion between different epochs. The PCK gives, instead, the value of the gravitational parameter for the planet. In the SPK file are embedded not only the binary data, but also an identifier (ID) that associates a number with the correct state vectors. It can be easier to understand it with a practical example, that has been taken from a MATLAB® routine:

SV = cspice_spkezr(target_id_code,et,frame,corrections,obs_id_code)

This is the main routine that has extensively been used for all the work. Analysing term by term:

- SV: it is the output of the routine provided by SPICE. It contains the state vector of a target body at a certain time. According to the presented model this information can be used in two ways. Firstly, it is possible to get the positions of the planets, which are needed in the computation of the potential. The problem is non- autonomous and it is necessary to introduce the time variable. In the second case it is possible to use this function for the validation of the model. Choosing small bodies, like asteroids or comets, their motion can be propagated. At the same time, the state vector at time et0 will represent the initial condition for the propagator to be validated.

- cspice_spkezr: this is one of the many functions that SPICE supplies to the end user. This one can compute the state vector of a body, given some inputs. However there are many others that can convert times or vectors from one frame to the other, can compute position or Keplerian elements. They can help ﬁnding properties of the bodies, computing eclipses or intersecting the ﬁeld of view of a camera with the surface of a planet.

- target_id_code: it represents the identifier that SPICE needs for the operation of browsing through the SPK file, recover the binary file and then build the state vector. For major bodies, like planets the string is build with their names, whereas, for other bodies the string is composed by their identifying number. Note that SPK kernels can be also built from the user and it is necessary a proper numerical code. Once the SPK file has been built, a string name can be associated to the id, through another SPICE function:

NAME = cspice_boddef(SPK,ID)

- et: SPICE functions work with ephemeris time, that from 2006, accordingly to the IAU deﬁnition, is the same as the barycentric dynamical time (TDB), i.e. a

55 Validation of the models

relativistic coordinate time scale which equals the Julian date at instant 1977-01-01 00:00:00.000: SPICE gives many routines to convert from one time format to the other, including Julian dates, modiﬁed Julian dates, barycentric coordinate time and international atomic time.

- frame: in SPICE there are many embedded frames that can be used, and this tool can easily convert from one to the other. From the others the J2000 inertial frame is included and this greatly helped the building of RPF, since no other rotation was needed, except from what has already been presented. There are also topocentric frames, body-ﬁxed frames or instrument frames. It is even possible to built one’s own frame and set and identiﬁer that can be renamed in a second moment.

- corrections: they are adjustments made to state vectors and time dependent reference frames to accurately reflect the apparent state and attitude of a target object as seen from a specified observer at a specified time, as opposed to the actual state. Actual states are called geometrical states, and these are needed for potential definition. But sometimes it is necessary to point a camera or an antenna toward the planet, therefore, the aberration introduced into the instrument must be corrected. Since this work has considered only gravitational potentials, this parameter have been always neglected.

- obs_id_code: it is an identifier for the observer state. Presenting J2000 and EME2000 frames, it was said that they were the same, but for the origin, that was fixed for the former in SSB and for the latter in Earth Barycentre. SPICE features the same characteristics: this identifier can be related to where the origin of the system should be.

The kernels for main celestial bodies do already exist and have been made public from the JPL itself into main ﬁles called DExxx.bsp. The most recent one at the time of this work is DE432.bsp, that has been used to call the state vectors and that has been described in two technical reports by Folkner [16] and [17]. However, for small bodies there are no ﬁles, and the user should built them on his own. JPL gives the possibility to use other tools to build them, mainly exploiting the Horizons On-Line Ephemeris System. It provides access to key solar system data and dynamic production of highly accurate ephemerides for solar system objects, including asteroids, comets, natural satellites, all planets, the Sun, more than 60 selected spacecrafts, and dynamical points such as Earth- Sun L1, L2, L3, L4, L5. The access is granted via a prompt-based interface 2, where the results can be directly downloaded as SPK binaries.

The main limitations in using SPICE derive from the fact that it is impossible to inves- tigate on the right-hand side of its dynamics model. Moreover, to produce an ephemeris, observational data containing measurement errors are combined with dynamical models containing modelling imprecisions. A best ﬁt is developed to statistically minimize those errors. The resulting ephemeris has an associated uncertainty that ﬂuctuates. To validate against this tool means that it is impossible to compare the same models: the one of SPICE will always be more accurate including terms that are not accounted in the 2telnet://ssd.jpl.nasa.gov:6775

56 4.3 GMAT present model and it is impossible to remove them to have a validation in a lower level of accuracy. This does not mean that SPICE should be dropped as a validating tool, but that it should work alongside with another tool which can adapt its accuracy level to the one of this model.

4.3 GMAT

The second tool exploited for the validation of the model is the General Mission Analysis Tool (GMAT). It is an open source trajectory design and optimization system developed by NASA and private industry 3. Its open source development makes it an optimal instrument to maximize technology transfer between users, and permits anyone to develop and validate new algorithms, in order to enable them to quickly transition into the high fidelity core. GMAT can be used to model and optimize spacecraft trajectories in different flight regimes ranging from low Earth orbit to lunar, interplanetary, and other deep space missions. The main interesting characteristic is that it contains initial value solvers (propagators) and efficiently propagates spacecraft motion. Other worth to note aspects can be the support of constrained and unconstrained trajectory optimization which can be directly connected with MATLAB®. GMAT usage requires almost no effort thanks to its graphical user interface (GUI) and its custom scripting language modelled after the syntax used in MATLAB®system. All of the system elements can be expressed through either interface, and users can convert between the two in either direction, even if the script syntax presents some more options to modify. It is also permitted to call GMAT scripts directly from MATLAB®, but only if both programs work in a 32-bit architecture.

Figure 4.1: GMAT interface

3https://gmat.gsfc.nasa.gov/ downloaded on March 9 2017

57 Validation of the models

Considering a case mission in GMAT, ﬁg. 4.1 is how it appears to the user. The upper part is dedicated to the commands for running the program. Instead, the left side is used for all the components related to the mission.

The ﬁrst step is to create a spacecraft. The high functionalities of GMAT require also many diﬀerent inputs with respect to a common MATLAB®script, where the only input are the Initial Condition and the time epoch. It would take too long to describe all of them, however, the most important will be listed, in order to have a full reproducing test. Since many data have a relation with SPICE, when possible it will be shown how to get a conversion from one program to the other.

- ORBIT DEFINITION: To describe the orbit it is necessary to give information on the epoch, coordinate system and state.

The epoch can be given in modiﬁed Julian date or in Gregorian date. A function can be used to obtain the latter from SPICE et epoch: GD = cspice_et2utc(ET,’C’,3) The ’C’ represents the output required format, while the number 3 represents the number of decimal digits for the seconds in Gregorian date that GMAT needs.

The coordinate system can be chosen between Mean Earth J2000 Ecliptic or Equa- torial or even Earth ﬁxed. The most compatible with SPICE is the Mean Earth J2000 Equatorial frame, called simply EME2000, the same introduced in Chapter 2.

The state vector for the initial conditions is accepted in a diﬀerent set of coordinates, but the more interesting are the cartesian coordinates (in km and km/s) and the Keplerian elements. Since from SPICE it is always possible to get the state in cartesian coordinate, this is the preferred choice for GMAT, too.

- BALLISTIC COEFFICIENTS: Then, the elements that influence the motion of the spacecraft are considered, mainly those related to perturbations. The main elements are the dry mass of the spacecraft (it is also possible to compute the propellant after a manoeuvre), the drag and reflectivity coefficients, so as drag and SRP area, which do not coincide, accordingly with the chosen model.

- OTHER: The full potential of GMAT appears thanks to the many functions the user can exploit. It is possible to add study on the attitude, add tanks or power system to the spacecraft, or model the actuators. This is not strictly connected with the purpose of this work, therefore, they will not be further described.

Once the spacecraft model is set, it is necessary to deﬁne the model of the forces together with the numerical integrator. This is the most interesting subject of the analysis, because it is the validating part of the program, the one that will be used as a comparison for the previously derived model.

58 4.3 GMAT

- INTEGRATOR: Defines the integration scheme to use with the differential equations that describe the motion of the spacecraft. There are many different options that exploit single step procedures, as RungeKutta89 or PrinceDormand78, or multiple step procedures, as the AdamBashforthMoulton. To be consistent with the ODE78 that is used in MATLAB®, the PrinceDormand78 integrator has been chosen,that is an adaptive step, eighth order Runge–Kutta integrator with seventh order error control. The accuracy has been set to 2.5e − 014. - FORCE MODEL: This is the main element of GMAT used as a validator. In the force model it is possible to include all the terms that will act on the spacecraft through a force applied to it. The great opportunity given by the program is to choose which force to consider and which to discard. This means that the right hand side is built by the user according to his necessities. GMAT supports numerous force models such as point mass and spherical harmonic gravity models, atmospheric drag, solar radiation pressure, tide models, and relativistic corrections.

In GMAT there is a difference between primary bodies and point masses. Bodies modelled as point masses use the gravitational parameter defined on the body (i.e. Earth.Mu) in the equations of motion. Bodies defined as primary bodies use the constants defined on the potential file in the equations of motion. Moreover, a primary body is a celestial body that is modelled with a complex force model which may include a spherical harmonic gravity model, tides, or drag. In the integration process it cannot appear as both a primary and a point mass, otherwise there would be a double force acting on the spacecraft. Currently only one body can be set as primary and it coincides with the central body of the integration. For this body is possible to set the degree and the order for the spherical harmonic gravity model. An atmospheric drag model can be used for a primary body, for instance in simulat- ing ballistic capture procedures. GMAT supports many density models for Earth including Jacchia-Roberts and various MSISE models. Density models for non- Earth bodies, the Mars-GRAM model for example, can be included using custom plug-in components. Since no consideration has been done for what regards the atmospheric drag, this parameter is set to NONE. The solar radiation pressure has been used, exploiting a simple cannonball model, whereas a more detailed and high fidelity SRP modelling is possible. The relativistic corrections or tide have not been included since they have not been considered in the modelling of the equations of motion. One last note is that also GMAT shall solve the non autonomous equations of motion, because it needs the position of the planets on certain epochs. To perform this task it exploits the ephemeris of the planets in an SPK format, using JPL’s SPICE data. This means that it is very useful for a validation process because in this way, using the same SPK, the position of the attractors in space is the same. They can be set from the SO- LAR SYSTEM folder in the Resources Tree.

To plan a mission with GMAT it is necessary to deﬁne some MISION SEQUNCES from the Mission Tab and they permit to deﬁne what the spacecraft shall do in between,

59 Validation of the models

thinking of ballistic motion or of the application of a ∆V . Once the simulation has been run it is possible to obtain a report with the needed data, usually the State Vector and the integration time, and then it can be compared with the result to be validated, looking for the maximum absolute and relative error.

4.4 Validation examples

In the next paragraphs some of the examples that have been used to validate the model will follow. The numerical integration has been performed and the results are compared with those taken from GMAT or SPICE. The detailed procedure will be described later on, however, the results must be compared and a certain value should be chosen which will become an indicator of the goodness of the integrator. The two elements that will be used in this work are the relative and the absolute errors, which are two values that need to be minimized. The integration absolute error can be deﬁned as the discrepancy between an exact value and the approximated value. Considering the results from SPICE and GMAT as exact, the solution of the model integration can be compared to them and their diﬀerence will be the absolute error.

(t) = kX(t) − Xˆ (t)k (4.8)

It is worth noting that the error is a time dependant quantity, since at every time epoch there is a different value for the considered quantity. The vector X represents the approximated value, that can be either the position or the velocity, while Xˆ is the exact value. The relative error is defined in two different ways according to the norm of the exact value:

kX − Xˆ k  if kXˆ k ≥ 1 r(t) = kXˆ k (4.9)  kX − Xˆ k if kXˆ k < 1

While the absolute error is an indicator of how much the approximated value diﬀers from the exact one, the relative error gives the same measure, however, it scales the values of the problem. Saying that the absolute error is of 100 m, gives a precise quantity for it, but it is not compared with the physical sizes of the problem. If a plane committed such an error in the landing phase, braking 100 m further or earlier it would be very dangerous because the track is in the order of some km, since it is making an error of 5/10%. If the spacecraft position is known with an error of 100 m, the result is much better, because the order of the physical quantities is in the order of the million of km, giving a relative error of 10-7/10-8.

60 4.4 Validation examples

4.4.1 Validation with SPICE Usually a validation is done taking a huge amount of computations and then statistical considerations are taken. For this work there was not the possibility to follow this path, mainly due to the time and the computation power such a procedure requires. Therefore, the work follows the path already trodden in Dei Tos [11]. The idea is to consider many bodies, and their keplerian elements should diﬀer as much as possible one to the other. In this way, many diﬀerent cases can be analysed, in the attempt of proving the integrator under the more varying hypotheses. The preferred celestial bodies accounted for this duty are asteroids. It is worth noting that the validation against asteroid trajectories is not something unusual in literature, given the particular paths they follow. An interesting

example can be the case study of 2006 RH120, presented by Urrutxua [46], where the temporary capture of this asteroid in 2006-2007 by the Earth was simulated and validated against GMAT and JPL47 ephemeris. In the Solar System, asteroids can be found in diﬀerent regions. According to the JPL’s Small Body Database 4 they can be subdivided into diﬀerent classes, according to their semimajor axis a, perihelion q, aphelion Q or eccentricity e: 1. Atira: An asteroid orbit contained entirely within the orbit of the Earth (Q < 0.983 AU). Also known as an Interior Earth Object. 2. Aten: Near-Earth asteroid orbits similar to that of 2062 Aten (a < 1.0 AU; Q > 0.983 AU). 3. Apollo: Near-Earth asteroid orbits which cross the Earth’s orbit similar to that of 1862 Apollo (a > 1.0 AU; q < 1.017 AU). 4. Amor: Near-Earth asteroid orbits similar to that of 1221 Amor (1.017 AU < q < 1.3 AU). 5. Mars-crosser: Asteroids that cross the orbit of Mars constrained by (1.3 AU < q < 1.666 AU; a < 3.2 AU). 6. Inner Main-belt Asteroid: Asteroids with orbital elements constrained by (a < 2.0 AU; q > 1.666 AU). 7. Main-belt Asteroid: Asteroids with orbital elements constrained by (2.0 AU < a < 3.2 AU; q > 1.666 AU). 8. Outer Main-belt Asteroid: Asteroids with orbital elements constrained by (3.2 AU < a < 4.6 AU). 9. Jupter Trojan: Asteroids trapped in Jupiter’s L4/L5 Lagrange points (4.6 AU < a < 5.5 AU; e < 0.3). 10. Centaur: Objects with orbits between Jupiter and Neptune (5.5 AU < a < 30.1 AU). 11. Trans Neptunian Object: Objects with orbits outside Neptune (a > 30.1 AU).

4http://ssd.jpl.nasa.gov/sbdb_query.cgi

61 Validation of the models

Table 4.2: Name and properties of chosen asteroids Class Name SPK ID a [AU] e [-] Atira 163693 Atira 2163693 0.741 0.322 Aten 2062 Aten 2002062 0.967 0.183 Apollo 1862 Apollo 2001862 1.479 0.560 Amor 1221 Amor 2001221 1.919 0.436 Mars-Crosser 5261 Eureka 2005261 1.524 0.648 Inner Main-belt 2001 Einstein 2002001 1.933 0.986 Main-belt 19 Fortuna 2000019 2.442 0.159 Outer Main-belt 121 Hermione 2000121 3.447 0.134 Trojan 624 Hektor 2000624 5.255 0.024 Centaur 5335 Damocles 2005335 11.826 0.867 Trans Neptunian 19521 Chaos 2019521 45.650 0.103

For each of them it has been generated an SPK ﬁle containing the binaries data from 1900-Jan-01 00:00:00.000 up to 2101-Jan-01 00:00:00.000. They are needed because the validation process will be subdivided into three periods of diﬀerent lengths, as expressed in table 4.3:

Table 4.3: Validation periods Period Starting date Ending date Time span Short 2016 - Aug - 31 2031 - Aug - 31 15 years Medium 2016 - Aug - 31 2091 - Aug - 31 75 years Long 1900 - Jan - 01 2100 - Jan - 02 200 years

There is another class of objects that has very peculiar characteristics and that has been included in the validation analysis. Dwarf planets are in direct orbit of the Sun, and are massive enough for their gravity to crush into a hydrostatic equilibrium shape (usually a spheroid), but they have not cleared the neighbourhood of other materials around their orbit [25]. For these bodies the assumption of restricted body still persists (they are not massive enough to inﬂuence other attractors). However, they are very interesting tests being the most massive celestial objects in solar system and so less perturbed by solar pressure or other forces, like the one due to Yarkovski eﬀect. There is a great number of dwarf planets in the Solar System: one of them, Pluto, has been declassed to this category in 2006. Even so, in this work Pluto and his satellite Charon, have been put between the attractors. For this reason as a dwarf planet it has been chosen Ceres, the largest object in the asteroid main belt.

Table 4.4: Name and properties of chosen dwarf planet Class Name SPK ID a [AU] e [-] Dwarf Planet 1 Ceres 2000001 2.767 0.076

In order to complete the integration there are also other information needed, i.e. the ephemerides of the planets. They are other characters of the validation process and they have been used as attractors to generate the gravitational potential. Their position is retrieved directly from SPICE and their characteristics are described in tab. 4.5.

62 4.4 Validation examples

Table 4.5: Attractors and their gravitational parameters Name SPK ID µ [km2/s3] Sun 10 1.3271e+11 Mercury 1 2.2032e+04 Venus 2 3.2486e+05 Earth body 399 3.9860e+05 Moon 301 4.9028e+03 Mars 4 4.2828e+04 Jupiter 5 1.2671e+08 Saturn 6 3.7941e+07 Uranus 7 5.7945e+06 Neptune 8 6.8365e+06 Pluto 9 9.7700e+02

A full list of the asteroids in every class can be found in Jpl’s small-bodies database, however, here it has been chosen one representative asteroid for each class. The ephemerides data are obtained by an SPK ﬁle generated at JPL’s, called de430.bsp5. They are generated by ﬁtting numerically integrated orbits of the Moon and planets to observations.

Figure 4.2: RPF trajectory for asteroids: Ceres - Fortuna - Hermione - Hektor

5Download at https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/on March 13 2017

63 Validation of the models

The positions and velocities of the Sun, Earth, Moon, and planets, along with the orientation of the Moon, result from a numerically integrated dynamical model. They are stored as Chebyshev polynomial coeﬃcients ﬁt in 32-day-long segments. Moreover, perturbations from 343 asteroids have been included in the dynamical model. The asteroid orbits were iteratively integrated with the positions of the planets, the Sun, and the Moon. The covered timespan goes from 1549-DEC-21 00:00:00.000 to 2650-JAN-25 00:00:00.000.

SSB Integrator Since the integration results are generally of diﬀerent step-size (due to the chosen integration tehcnique), the following procedure has been used for every asteroid. The pseudo-code has been written to permit the reproducibility of the work by the reader:

1. Choose the time epoch and compute the state vector with SPICE.

SV0 = cspice_spkezr(’2000001’,et0,’J2000’,’NONE’,’SSB);

In this example it is computed the state vector of Ceres at time epoch et0, in the J2000 reference frame, with origin at Solar System Barycentre without any correction.

2. Integrate the motion of the body exploiting the integration scheme (eq. (2.49) ) and the built model for diﬀerential equations:

options = odeset(’Reltol’,2.5e-14,’AbsTol’,2.5e-14);

[et,SV] = ode78(@SSB_model,[et0 etf],SV0,options,param); R = SV(:,1:3); V = SV(:,4:6);

Firstly, the options for the DormandPrince78 integrator scheme should be set, then the model is passed with all the parameters, mainly the gravitational parameter of the planets. It needs a set of initial conditions to be solved, as every diﬀerential equation, and since the problem is non-autonomous, it needs also the real time and not only a time duration.

3. Compute the position of the body according to SPICE, having the precise time step.

SVS = cspice_spkezr(’2000001’,et’,’J2000’,’NONE’,’SSB); RS = SVS(:,1:3); VS = SVS(:,4:6);

Note that SPICE can take multiple dates as input and gives multpile states as output, and this form drastically reduces the time of validation. The same thing could have been done with Horizons, but it would have required a dedicated interpolation technique and it would have been slower.

64 4.4 Validation examples

4. Compute the absolute and relative errors. Only the ones related to position are shown.

= norm(R - RS); r = norm(R - RS)./norm(R);

It is worth noting that in this case the position is always a number greater than 1, since the distances between the planets are in the order of the AU.

Following this procedure and repeating it for every asteroid and for every time period, it is possible to ﬁnd the maximum of the errors. They will be shown in table 4.6 and 4.7, the position and the velocity errors respectively.

Table 4.6: Absolute and relative maximum positions errors Absolute error [km] Relative error [-] Asteroid Short Mid Long Short Mid Long ATIRA 1.0810e+03 2.3344e+03 8.4860e+03 1.4440e-05 3.0987e-05 1.1251e-04 ATEN 6.8228e+02 1.5140e+03 1.0796e+04 5.7790e-06 1.2718e-05 9.0785e-05 APOLLO 9.5720e+02 1.3588e+05 3.8187e+07 9.8962e-06 1.4063e-03 3.8844e-01 EUREKA 4.4740e+02 2.3593e+03 5.4727e+03 2.0971e-06 1.1064e-05 2.5546e-05 AMOR 4.1079e+02 1.0101e+03 1.5285e+05 2.5302e-06 6.2335e-06 9.4909e-04 EINSTEIN 5.1068e+02 2.6969e+03 5.3680e+03 1.9597e-06 1.0229e-05 2.0691e-05 FORTUNA 5.8116e+02 3.0421e+03 9.3763e+03 1.8175e-06 9.8549e-06 3.0382e-05 CERES 1.4974e+02 5.1144e+02 2.7572e+03 3.8537e-07 1.3355e-06 7.0004e-06 HERMIONE 1.4474e+02 5.4385e+02 2.8659e+04 3.1803e-07 1.2168e-06 6.3593e-05 HEKTOR 4.3598e+01 1.1146e+02 1.4935e+02 5.6277e-08 1.4391e-07 1.8571e-07 DAMOCLES 2.0017e+01 4.6602e+02 9.4001e+02 8.4784e-08 1.9752e-06 2.7860e-06 CHAOS 4.9854e-01 2.9510e+00 3.6209e+01 8.1258e-11 4.4581e-10 5.3730e-09

Table 4.7: Absolute and relative maximum velocity errors Absolute error [m/s] Relative error [-] Asteroid Short Mid Long Short Mid Long ATIRA 4.4887e-01 7.9269e-01 3.1588e+00 9.2903e-06 1.6403e-05 6.5355e-05 ATEN 1.6003e-01 3.4259e-01 3.0131e+00 4.3926e-06 9.3950e-06 8.2634e-05 APOLLO 2.6859e-01 4.1704e+01 1.1608e+04 5.8109e-06 9.0189e-04 2.5243e-01 EUREKA 4.9802e-02 2.6059e-01 6.0091e-01 1.9358e-06 1.0123e-05 2.3325e-05 AMOR 5.3929e-02 1.2980e-01 2.2419e+01 1.5749e-06 3.7877e-06 6.5407e-04 EINSTEIN 4.0917e-02 2.1452e-01 4.2857e-01 1.7304e-06 9.1587e-06 1.8120e-05 FORTUNA 3.4731e-02 1.8725e-01 5.7804e-01 1.5927e-06 8.3826e-06 2.5872e-05 CERES 6.9597e-03 2.3560e-02 1.2847e-01 3.6511e-07 1.2189e-06 6.6597e-06 HERMIONE 4.9403e-03 2.0689e-02 1.0297e+00 2.7329e-07 1.1182e-06 5.6338e-05 HEKTOR 7.4776e-04 1.7686e-03 2.2675e-03 5.9057e-08 1.3371e-07 1.7738e-07 DAMOCLES 1.1388e-03 3.5583e-02 4.5237e-02 3.5973e-08 1.0991e-06 1.3953e-06 CHAOS 1.8231e-06 2.8234e-06 2.1869e-05 3.7537e-10 6.0759e-10 4.8025e-09

65 Validation of the models

From the tables of the maximum errors it is evident how they have an increasing trend. This is a consequence of the accumulation of the error during the integration process. And it is possible to see that, for a relatively long time, these errors will exceed the tolerance that have been set to 1e−5; this value can be interpreted as an error of 0.1 m or 1000 km.

Figure 4.3: RPF trajectory for asteroids: Amor - Apollo - Einstein - Aten

The asteroids have been listed in order of increasing semi-major axis, but there is no evident relation between this parameter and the maximum error. Probably the error is related both to numerical issues and to divergences in the model.As regards the asteroid Apollo, such a divergence can be caused by a gravity assist with Mars’ moon, for which the present model does not account. Looking more in the detail to the trend of the error (fig. 4.4) it is possible to give importance to this theory: It is possible to see that this error increases dramatically in a relative short period of time, making the error diverge quickly. Usually this is an effect of the presence of another massive body which brings an unexpected gravity assist. Fig. 4.5 and 4.6 show the comparison between all the trends. It is possible to note how all of them have a particular pattern, which does not repeat for two different asteroids. However, a common aspect is the superposition of different

66 4.4 Validation examples

Figure 4.4: Trend of the relative error for 1862 Apollo in long integration period sinusoidal waves that bring a certain regularity into the error function. They can be related to many aspects present in the force model of SPICE, for instance the sun cycle, the Yarkovski eﬀect, solar pressure. In fact it is not known which values SPICE uses to model these aspects.

RPF Integrator The integration of the model in inertial frame is very useful and sometimes it attains a lower error. However, it is not always useful and possible, mainly because sometimes there is the need to use adimensional quantities, like it will be shown in the next chapter. Here follows the pseudo-code procedure which permits to use this kind of integrator and after some of the attained results will be analysed.

1. Choose the time epoch and compute the state vector with SPICE. Convert it into the synodic reference frame.

SV0 = cspice_spkezr(’2000001’,et0,’J2000’,’NONE’,’SSB); [XF0,τ 0] = ssb2rpf(SV0,et0,param);

In this example it is computed the state vector of Ceres at time epoch et0, in the J2000 reference frame, with origin at Solar System Barycentre without any correction. It is then converted into the roto-pulsating reference frame, exploiting a self built routine, which needs the inertial state and the ephemeris time, in input. The routine needs a param. structure with the gravitational parameters of the planets, an explicit declaration of the primaries and the angular velocity of the system.

2. Integrate the motion of the body exploiting the integration scheme and the built model for diﬀerential equations (eq. (3.31)):

67 Validation of the models

Figure 4.5: Trend of the relative error for the ﬁrst 6 asteroids in long integration period

options = odeset(’Reltol’,2.5e-14,’AbsTol’,2.5e-14);

[τ,XF] = ode78(@RPF_model,[τ 0 τ f],XF0,options,param); ρ = XF(:,1:3); η = SV(:,4:6);

The ﬁrst step is to set the options for the DormandPrince78 integrator scheme, then the model is passed with all the parameters.The initial conditions shall be set and, since the problem is non-autonomous, the integration time is also needed. In the model, the integration time will be converted into ephemeris time, because an initial epoch must be speciﬁed.

3. Compute the position of the body according to SPICE, having the precise time step converted into time epoch

et = tau2et(τ,param); SVS = cspice_spkezr(’2000001’,et’,’J2000’,’NONE’,’SSB); RS = S(:,1:3); VS = SVS(:,4:6); SV = rpf2ssb(XF,τ,param); R = SV(:,1:3); V = SV(:,4:6);

The best choice for the validation of the RPF integrator is to consider the error in the SSB frame, because it gives a direct measure of how much the distance or the velocity have been missed. To do this there is the necessity to convert the synodic state vector into an inertial one and after that comparing it with SPICE.

4. Compute the absolute and relative errors; only position errors are shown.

68 4.4 Validation examples

Figure 4.6: Trend of the relative error for the second 6 asteroids in long integration period

= norm(R - RS); r = norm(R - RS)./norm(R);

Also for this case the errors are considered: the absolute and the relative. They are deﬁned in the same way for comparison necessities.

Table 4.8: Absolute errors for the position in Earth RPF Absolute error [km] Asteroid Short Mid Long ATIRA 4.8795e+02 1.6761e+03 5.3638e+03 ATEN 2.4276e+02 2.1451e+03 5.9988e+03 APOLLO 7.5017e+03 2.2595e+06 2.5591e+08 EUREKA 2.0038e+03 1.0914e+04 2.7079e+04 AMOR 8.0498e+03 3.1768e+04 9.2867e+05 EINSTEIN 5.0502e+03 2.7241e+04 7.7882e+04 FORTUNA 9.0448e+03 5.7991e+04 1.3252e+05 CERES 1.0864e+04 5.8224e+04 1.6321e+05 HERMIONE 2.2213e+04 1.3110e+05 3.7005e+05 HEKTOR 4.3412e+04 1.0864e+05 1.2360e+05 DAMOCLES 1.0640e+06 7.2270e+06 2.9962e+07 CHAOS 8.2742e+05 2.5085e+07 1.6245e+08

69 Validation of the models

The RPF integrations need a very important parameter that brings a high variation to the results: the primaries couple. While the first primary is always the sun, the second one can change between the other attractors. Theoretically the results should be identical, however, the computational errors make them vary greatly. This verifies mainly because of the scaling factor k, which is computed starting from the relative position of the primaries: the acceleration that is needed to compute its second derivatives, considers the other bodies only as perturbation. Their contribution to k is, thus, only partial. In tab. 4.8 the integration results of the Earth synodic reference frame are shown. For the first two asteroids there is an improvement of the absolute error, more or less of a factor 3, whereas for all the other bodies the error increases, considerably, of a factor 105or106 in the cases of Damocles and Chaos, that coincide with the most distant bodies from the secondary.

Figure 4.7: RPF trajectory for asteroids: Eureka - Damocles - Chaos - Atira

Searching for an improvement, Jupiter has been considered as the secondary. The results of the integration are shown in tab. 4.9. It can be seen that in this case some of the results present an improvement, mainly those with a similar distance from Sun with Jupiter, while some other circumstances

70 4.4 Validation examples

Table 4.9: Postion absolute errors in Jupiter RPF Absolute error [km] Asteroid Short Mid Long ATIRA 1.0767e+03 2.3319e+03 8.4767e+03 ATEN 6.8140e+02 1.5122e+03 1.0790e+04 APOLLO 9.6087e+02 1.3744e+05 3.5314e+07 EUREKA 4.3284e+02 2.2885e+03 5.5209e+03 AMOR 4.2227e+02 1.0301e+03 1.5210e+05 EINSTEIN 4.7308e+02 2.4879e+03 5.1902e+03 FORTUNA 5.8069e+02 3.1645e+03 8.8784e+03 CERES 1.3013e+02 5.5053e+02 1.3248e+03 HERMIONE 9.7972e+01 7.7885e+02 2.9493e+04 HEKTOR 4.9228e+00 1.0170e+01 3.9591e+01 DAMOCLES 9.7172e+02 6.7015e+03 2.2855e+04 CHAOS 7.8505e+02 2.4602e+04 1.5704e+05

provide poor results: it is the case of the firstntwonasteroids, Given these results with a very high order of difference, it was chosen to compute the errors for every asteroid in the synodic frame of every planet. This brought to 324 validation processes, with a totality of 1296 computed errors. To show every single number would be extremely long, therefore the next figures and tables will focus only on some of them.

Figure 4.8: Errors on diﬀerent primaries for long integration period

Figure 4.8 shows in a semi-logarithmic scale how does the error for each asteroid change, varying the secondary. It is worth noting that every asteroid has a diﬀerent

71 Validation of the models optimal primary choice: for that couple of bodies the maximum position error is the minimum. The primaries are ordered according to their SPICE ID, therefore in increasing distance from the Sun. It seems that the error starts decreasing if a more distant secondary is chosen. However, this trend stops, revealing that a minimum can be found and after that, the error starts rising again.

It would be interesting to look at the resulting trajectories. They will be shown in their projection onto the XY-plane. It is clear that synodic trajectories diﬀer enough from the classic trajectory in inertial frame: they are no more the classical heliocentric ellipses. In the ﬁgures it have been chosen the best primary for the medium range integrations.

Table 4.10: Mininum max. absolute errors related to the primary Absolute error [km] Asteroid Short IDS Mid IDS Long IDS ATIRA 1.3013e+02 399 5.5053e+02 399 1.3248e+03 2 ATEN 5.8069e+02 399 3.1645e+03 4 8.8489e+03 399 APOLLO 9.7972e+01 5 5.8159e+02 5 2.9493e+04 6 EUREKA 4.9228e+00 4 1.0170e+01 5 3.9591e+01 8 AMOR 4.2227e+02 5 1.0301e+03 5 3.8230e+04 4 EINSTEIN 9.6087e+02 5 1.3744e+05 5 3.5281e+07 8 FORTUNA 4.7308e+02 5 2.4879e+03 5 5.1555e+03 7 CERES 2.4276e+02 5 1.3787e+03 5 5.9988e+03 5 HERMIONE 9.7032e+01 5 2.2885e+03 7 5.4854e+03 5 HEKTOR 1.7538e+01 5 3.6378e+02 5 9.2935e+02 5 DAMOCLES 7.5845e+00 9 3.1151e+01 7 1.1825e+02 9 CHAOS 4.8795e+02 8 1.6761e+03 7 3.2536e+03 7

4.4.2 Validation with GMAT The second tool that has been used to carry on some validations is GMAT. It differs from SPICE mainly due to the fact that it has the possibility to vary the validating object’s properties. For this reason it is more prone to the validation of artificial bodies, which geometry and physical quantity are known. Moreover, it is possible to use GMAT as validator with the Solar Radiation Pressure effect. There is no need for high integration time to simulate the motion of a spacecraft, because it is, usually, taken under control with navigation sensors. This means that a window of some days can work better that one of many years, and the results are equally good. For this, in the validation process with GMAT two elements have been chosen:

1. Lisa Path Finder is a spacecraft which will be further described in the following chapters. It orbits around the L1 Lagrangian point of the Sun-Earth system in a Lissajous orbit [44].

2. J002E3 is an artificial object which periodically enters in the Earth’s Sphere of Influence, with an approximately 40-year cycle between heliocentric and geocentric orbit. It has been identified to be the S-IVB third stage of the Apollo 12 Saturn

72 4.4 Validation examples

V rocket based on spectrographic evidence consistent with the paint used on the rockets [26].

These particular bodies have been chosen, because one of the intents of this work is to validate the model, even in some of the more delicate cases of astrodynamics: their dynamics is impossible to be detected in the R2BP and even with the R3BP it is necessary to consider at least the perturbation of Moon and Jupiter. In tab. 4.2 and 4.6 the physical properties are listed. The SRP model that has been used is the simple cannonball model. The reﬂectivity coeﬃcient, the mass and the SRP area are the only elements that are needed.

(a) LPF (b) J002E3 Par. Val. Par. Val. Par. Val. Par. Val.

et0 31 - AUG - 2016 m 500 [kg] et0 01 - MAR - 2003 m 115 [ton] et0 09 - DEC - 2016 ASRP 3.7 [m2] et0 01 - JAN - 2004 ASRP 184.5 [m2] ∆t 100 [days] cr 1.08 ∆t 306 [days] cr 1.2 ω 1.9910e − 07 [1/s] IDS 399 ω 1.9910e − 07 [1/s] IDS 399

Table 4.11: Physical and integration properties

When it comes to artificial objects SPICE becomes a complex tool, because the only way to obtain SPK files is to get them from the Navigation Team of the considered vehicle and the model can be extremely different. Sometimes, when the objects were non-cooperative bodies, as in the case of J002E3, some observations were made and the state vector data were written in Horizons 6, a NASA on-line tool. In this case the data are limited to less than one year, therefore they are good for generating initial conditions but not good enough to validate the results. Nonetheless, GMAT is a propagator that can be properly used for this aim. Similarly to what already presented for SPICE, it is hereby describe the validation procedure with GMAT:

1. Choose the time epoch and select the initial conditions.

This is a non trivial step within the process. It is not possible to rely on SPICE where there are no SPK ﬁles, therefore, they should be checked from Horizons or from literature.

2. Convert the initial conditions in the desired reference frame for integration.

Usually, from Horizons the initial conditions come as J2000 state vector with respect to whichever body is needed, and from literature they come in terms of keplerian elements. GMAT needs them as relative vectors from Earth: 6http://ssd.jpl.nasa.gov/horizons.cgi

73 Validation of the models

SV0 = read_hor(et0,ID,’SSB’); SVE = cspice_spkezr(’EARTH’,et0,’J2000’,’NONE’,’SSB’); SVG = SV0 - SVE; XF0 = ssb2syn(SV0,et0,param);

In this pseudo-code example, it has been shown how to use the data read from Horizons, which include the initial condition of an ID body with respect to SSB. Note that it has been generated an initial state vector, for GMAT and for the RPF integrator.

3. Integrate SVG in GMAT: modify the spacecraft orbit and then choose the integration time.

Once the integration has ended, the results are taken in a ﬁle which can be read by MATLAB®and compared with RPF integrator.

4. Integrate XF0 in RPF integrator.

[τ,XF] = ode78(@RPF_model,[τ 0 τ f],XF0,options,param);

This step il similar to what has already been presented for SPICE validation: the RPF_model is used to integrate the set of intial conditions in the non-dimensional time range, according to the parameters listed in param, such as the primaries, the initial epoch and the properties of the spacecraft.

5. Compare the results.

It is worth noting that the step-size of the two integrators is diﬀerent, therefore is not possible to compare directly the results step by step. Thus, it is needed an interpolation procedure that converts the result in XF at the same output epochs from GMAT. This can directly be done by a routine in MATLAB®.

LPF integration Firstly, the initial conditions are chosen. The values represent the state vector of LPF on its Lissajous orbit. The initial values for the RPF shall be in an adimensional form, while those for GMAT are needed as relative vectors from Earth, expressed in km and km/s. In tab. 4.11(a) and 4.11(b) the components are expressed with 16 decimals, leaving the possiblity to reproduce the results.

74 4.4 Validation examples

(a) RPF I.C. ρ [-] η [-] 9.916193557083384e-01 -3.904469880950060e-05 -2.045624054718153e-04 -9.607987372078436e-03 -2.964007069075622e-04 3.863319892365587e-05

(b) GMAT I.C.

rG [km] vG [km/s] -1.182965986483097e+06 -1.949592578461381e-01 4.302466721291989e+05 -4.587288328904613e-01 1.377367809387483e+05 -1.974434715210052e-01

Table 4.12: Initial conditions for the integrators

Fig. 4.9 shows the evolution of the trajectories in both frames, the relative with respect to Earth and the roto-pulsating. The two motions seem quite diﬀerent, but some common aspects can be identiﬁed. In both is quite clear the near passage to Earth and the change in direction after this gravity assist. In any case the errors, after a 100 days integration are in the order of 1 km.

(a) Relative frame (b) RPF Figure 4.9: Trajectory evolution

Table 4.13: Max relative and absolute errors after 100 days integration Typology Position Error Velocity Error NO SRP - abs. 1.0912e+00 [km] 1.0913e-01 [m/s] SRP - abs 1.0804e+00 [km] 1.0602e-01 [m/s] NO SRP - rel. 4.0052e-05 [-] 2.0376e-05 [-] SRP - rel. 3.9156e-05 [-] 1.9919e-05 [-]

The error is computed as the norm of the diﬀerence at every time step, however it is possible to compute also the error between every component. In ﬁg. 4.10 this error

75 Validation of the models

Figure 4.10: Particular of error trend has been plotted and it is interesting to note how it looses smoothnes at a certain point, mainly the y and Z components. This can be an eﬀect related to numerical issues. They greatly increase near a gravity assist manoeuvre where the integration step lowers and many of them are required.

J002E3 integration The integration of this body brings interesting validating results. It is an artiﬁcial body that orbits periodically around Earth entering its sphere of in- ﬂuence and after a period of time it escapes again to re-enter in orbit after some years. Therefore, the motion can be split into two main branches: an heliocentric one and a geocentric one.

(a) RPF I.C. ρ [-] η [-] 1.000127953056645e+00 1.322272821733515e-02 -5.080991860952158e-03 -7.657098686329858e-03 -2.857791380694205e-05 1.567705720673701e-03

(b) GMAT I.C.

rG [km] vG [km/s] 2.400020070012808e+05 -4.332355526822358e-01 6.568165611345470e+05 3.456191821416965e-01 2.801320032509528e+05 2.003197238255616e-01

Table 4.14: Initial conditions for the integrators

76 4.4 Validation examples

Table 4.15: Max relative and absolute errors after 100 days integration Typology Position Error Velocity Error NO SRP - abs. 4.1950e+01 [km] 6.9167e-03 [m/s] SRP - abs 4.7942e+01 [km] 7.9094e-03 [m/s] NO SRP - rel. 3.0188e-06 [-] 4.0023e-06 [-] SRP - rel. 3.4060e-06 [-] 4.5058e-06 [-]

The data from Horizons are available only for some months of a geocentric period. This has been exploited for an integration with GMAT and RPF. The results are listed in tab. 4.15. It is possible to see how the absolute errors are increased: the position error is now in the order of tenth of km, however the relative error has decreased. This can be easily explained saying that this is mainly an accumulation error. From fig. 4.11(a) it is possible to recognize a gravity assist: the numerical error that establishes here is almost impossible to be reduced and it amplifies with time. However, since the objects escapes from Earth’s sphere of influence, fig. 4.11(b), the absolute distance increases, therefore the relative error decreases.

(a) RPF trajectory (b) RPF trajectory - particular Figure 4.11: Trajectory evolution for J002E3

Finally a comparison between the three results at the ﬁnal time has been taken. This is the worst condition for the errors, even if some non-linearity in it have appeared. In tab. 4.16 is shown how much the absolute error changes between the integrators.

Table 4.16: Comparison between the integrators Typology RPF vs GMAT RPF vs SPICE GMAT vs SPICE NO SRP - abs. 3.6573e+01 [km] 1.9985e+05 [km] 1.9989e+05 [km] SRP - abs 4.2083e+01 [km] 1.5241e+05 [km] 1.5245e+05 [km]

It can be noted how the error between RPF and GMAT is in the order of some km, while between this and SPICE it becomes in the order of the hundreds of thousands of km. This is related to the very complex force model used by SPICE, that RPF cannot

77 Validation of the models replicate, however a very interesting aspect is the fact that as soon as the solar eﬀect was introduced, the error lowered of almost the 25%. A summary of the concepts presented in this chapter can be denoted by the last table. Until the RPF model is compared with other model of the same accuracy order, as GMAT, it can present a good level of reliability, even if it cannot still be compared with highly accurate tools, like SPICE.

78 Chapter 5

Impulsive trajectory optimization

This chapter deals with the problem of optimising trajectories directly within the roto- pulsating frame. At this point, only ballistic trajectories have been presented; however, in many circumstances the need arises to limit the trajectory to follow certain constraints. An example could be the near or the exact passage through a region in space: the spacecraft should pass across that area at a particular epoch. The targeting of time- dependent boundary conditions can be achieved by means of different strategies. The first one consists in exploiting the initial conditions of the spacecraft as a degree of freedom to reach the desired point: they can be slightly varied in order to see the effects produced on the ballistic evolution of the trajectory. This procedure is called exploration of the solution space and brings an optimality in terms of costs for the mission, once a cost function is specified. However, this logic has some limitations, which are defined from the unattainability in reaching exterior points to that space. Another weakness of this approach is that it is an heuristic one, which means high time and computational power for the solution definition is required. The second procedure, which is the one followed in this work, consists in introducing control capabilities along the nominal solution, i.e. the S/C can perform manoeuvres. In the following, impulsive manoeuvres are considered. This kind of manoeuvre impulsively changes the velocity vector of the spacecraft, without modifying its position. If the location where the velocity vector is discontinuous coincides exclusively with the initial point, the scheme is called single shooting: it can be compared to an engine that shoots only once, changing the momentum of the S/C. It is termed single shooting because only a single numerically integrated trajectory arc or segment is involved in the two point boundary value problem (TPBVP). The strategy can be extended to an approach with different discontinuity points, where many integrated arcs and segments are involved. It is called multiple shooting and represents the most general case, where the further can be thought as a mere simplification of the latter. For a deeper insight of the problem the reader can refer to Keller [27] or Stoer [41]. This is the reason why only the multiple shooting strategy has been presented.

5.1 Introduction to a multiple shooting strategy

The multiple shooting strategy represents a method of solving boundary value problems for diﬀerential equations,as presented in Stoer and Bulirsch [41]. A shooting tech-

79 Impulsive trajectory optimization

Figure 5.1: Multiple shooting strategy

nique points at finding some unknown parameters, mainly the initial conditions for the ODE. It starts guessing them and solving the differential equations: at this points the constraints at the boundaries are checked. If they are not satisfied, the parameters are varied with an iterative procedure, which changes the result of the shot. A multiple shooting consists in subdividing the trajectory into intervals: for each of them an initial- value problem is defined. The solutions are then adjusted in successive iterations until the boundary conditions and continuity properties at the ends of the intervals are satisfied. In this way, a boundary value problem can be successfully converted into a set of initial value problems. Hence, the multiple shooting technique is more reliable and efficient from a simple shooting, where only the degrees of freedom related with the initial conditions are accounted for.

From the ﬁgure 5.1 it is possible to understand how the multiple shooting strategy works. It represents an approach where the trajectory is subdivided in nodes, arcs and segments. A node is a point where the state is taken as an unknown variable. A segment is a portion of the trajectory between two nodes, while an arc is a section between two impulsive manoeuvres, where velocity can have a discontinuity. Every node represents an initial condition point for the dynamic system, which can be integrated according to the diﬀerential equations:

x˙ = f(x, τ) (5.1)

The nodes are also called free variables because they represent the degrees of freedom of the problem. Meanwhile, they should satisfy also some constraints: from one side the continuity of the trajectory, and on the other the peculiar constraints of the boundary value problem, which will be detailed properly in the next sections. At this point, it is possible to combine all the free variables in a vector y:

80 5.1 Introduction to a multiple shooting strategy

    x1 ρ1  .  η  .   1     .   xi   .   .     .   ρN   .     η   xN   N  X     i = [1 : N] y = =  τ  = τ1 T  1    j = [1 : n ] (5.2)  .   .  m  .   .       τ   τj,f   j,f    τ  τj+1,i  j+1,i    .   .   .   .  τN τN where xi represents the state of the i-th node, τ1 the initial time, τj,f the time on the final node of an arc, τj+1,i the time on the initial node of the next arc, and τN the final time, i.e. the time at the initial and final node. The vectors X and T represent the sub-vectors that include all those elements. The total number of nodes will be equal to:

N = (na)(ns + 1) (5.3)

where na is the number of arcs and ns is the number of segments. The number of manoeuvres can be found directly from the number of arcs, since the following relation holds:

na = nm + 1 (5.4)

To solve a boundary value problem, a solution must be sought that fulﬁls the set of m constraint equations:

C(y) = 0 (5.5)

The solutions of this problem can be approached with several algorithms, for instance by means of the MATLAB®embedded fmincon routine. However, it is possible to give a general idea of how these algorithms work, without entering too much into detail. In fact, they can be thought as particular expressions of the main algorithm used to find the zeros of non-linear functions, e.g. the Newton’s algorithm. It states that it is always possible to find the zeros of a function if some conditions are verified:

- The function shall admit a zero in the interval, i.e. it is possible to apply the zero theorem on it.

81 Impulsive trajectory optimization

- The function shall be continuous and derivable in the interval.

This theorem exploits the expansion in Taylor series to build the iterative procedure:

∂C(y0) C(y) ≈ C(y0) + (y − y0) (5.6) ∂y0

Exploiting the expression of eq. (5.5), it is possible to rewrite the last equation as:

−1 ∂C(y0) y ≈ y0 + C(y0) (5.7) ∂y0 which can be set as an equation only if an accuracy error is accepted, hence, iteratively solved:

∂C(yj)−1 yj+1 = yj − C(yj) (5.8) ∂yj

Equation 5.8 represent a set of n equations that solve the non-linear Newton problem of ﬁnding a zero. The matrix term is the inverse of the Jacobian of the constraints of size n × m which can be computed exploiting the pseudo-inverse relations:

" #−1 ∂C(yj)−1 ∂C(yj)T ∂C(yj) ∂C(yj)T = (5.9) ∂yj ∂yj ∂yj ∂yj

Finally, it is worth noting that the procedure should go on until a certain tolerance is met, according to eq. (5.10):

kC(yj+1)k < (5.10)

This procedure represents the starting mathematical tool to solve an optimisation problem. In the next sections these equations will be written for applications inside the N-body problem. It will be necessary to derive a new set of equations, called the variational equations and afterwards, their results will be used to derive constraints and objective functions.

5.2 The variational equations

In most direct optimization methods, differential information is of paramount importance for optimality, convergence of the solution, and numerical efficiency. Indeed, fast and precise information on how the system responds to changes on initial conditions and time directly influences terms in eq. (5.9). This is a sensitivity analysis of how the final

82 5.2 The variational equations integrated state can have a variation caused by the initial state. The equations that map this kind of transformation are called variational equations. They are related to the state transition matrix, which considers in every term the derivative of a coordinate of the ﬁnal state with respect to one of the initial state. It is possible to associate a set of partial derivatives with the variational equations:

∂x δx(τ) = δx(τ0) (5.11) ∂x0

The matrix written in this equation is by deﬁnition the state transition matrix:

∂x Φ = (5.12) ∂x0

To compute its value it is possible to rely on the variational equations. They can be derived directly from the deﬁnition of state transition matrix:

d ∂x ∂ dx ∂ ∂ ∂f ∂x = = x˙ = f(τ, x) = dτ ∂x0 ∂x0 dτ ∂x0 ∂x0 ∂x ∂x0 where the relation of eq. (5.1) has been used. The gradient of the right-hand side is usually referred as state propagation matrix (SPM):

∂f A(τ, x) = (5.13) ∂x

Hence, it is possible to write the ﬁnal formulation of the ﬁrst order variational equations:

Φ˙ = A(τ, x)Φ (5.14)

Eq. (5.14) represents a set of n × n differential equations, where n is the order of the differential system, involving the state transition matrix. It is evident that they cannot be integrated independently from the equations of motion, since the SPM is a time dependent function of the right-hand side. The initial conditions to be added along the integration process can be found recalling the definition of Φ. If it is the matrix that at every time step maps the dependence of the current state with respect to the initial time, at the starting epoch it cannot be else than the identity matrix.

Φ(τ0, τ0) = I6×6 (5.15)

83 Impulsive trajectory optimization

To proceed forward with computations it is necessary to give an analytical formulation for the gradient of the right-hand side in the N-body problem. Following eq. (5.13) it can be computed taking the derivatives of f with respect to the state. The model should be ﬁrstly expressed in the state space form, where the adimensional velocity is taken as a new state and called η, and the position ρ:

 0 ρ = f1 = η  " # " #  2 k˙ 1 k¨ k˙ η0 = f = − I + CT C˙ η − I + 2 CT C˙ + CT C¨ ρ 2 τ˙ k 3×3 τ˙ 2 k 3×3 k (5.16)   1 1 1 SP ρ − ρ  − CT b¨ + ∇U + 0 s  2 2 2 3 3 τ˙ k k τ˙ k kρ − ρsk

In the last equations all the perturbing elements have been included, and all the parameters for the solar pressure computation have been condensed into the only element SP0:

2 ASC Ψ0d0 SP0 = (1 + cr) (5.17) mSC c The gradient matrix can be then expressed with four sub-matrices:

∂f1 ∂f1   ∂ρ ∂η  A =   (5.18) ∂f2 ∂f2  ∂ρ ∂η

The dependence of the right-hand side on the position ρ and the velocity η, has been split. Therefore, the gradient will be composed of the derivatives of f with respect to them. The four components of the SPM are at this point presented, without focusing too much into the procedure of how they have been derived:

∂f 1 = 0 ∂ρ 3×3

∂f 1 = I ∂η 3×3

84 5.3 Implementation

 " # ∂f 2 k˙ 1 ∂∇U  2 = − I + CT C˙ −  ∂ρ τ˙ k 3×3 τ˙ 2k2 ∂ρ    SP I 3(ρ − ρ )(ρ − ρ )T  + 0 3×3 − s s  2 3 3 5  τ˙ k kρ − ρsk kρ − ρsk   ∂∇U 1 ∂∇Ω 1 ∂∇Ω 3 ∂∇Ω  = 1 + 2 + 3  3 3  ∂ρ k ∂ρ 2k ∂ρ 2k ∂ρ   ∂∇Ω I 3(ρ − ρ )(ρ − ρ )T  1 X 3×3 j j  = µj 3 − 5  ∂ρ kρ − ρjk kρ − ρjk  j∈S (5.19)  T ∂∇Ω2 X I3×3 5(ρ − ρj)(ρ − ρj)  2  = 3µjJ2jRplj 5 − 7  ∂ρ kρ − ρjk kρ − ρjk  j∈S    T ∂∇Ω3 X 2MI3×3 10M(ρ − ρj)(ρ − ρj)  = µ J R2 − +  j 2j plj 5 7  ∂ρ kρ − ρjk kρ − ρjk  j∈S  T T  5(ρ − ρj) M(ρ − ρj)I3×3 10(ρ − ρj)(ρ − ρj) M  − − +  kρ − ρ k7 kρ − ρ k7  j j  T  T (ρ − ρj)(ρ − ρj)  +35(ρ − ρj) M(ρ − ρj) 9 kρ − ρjk

" # ∂f 2 k˙ 2 = − I + CT C˙ ∂η τ˙ k 3×3

The variational equations have a very precise meaning in the deﬁnition of the model, however, they are generally used to compute the state transition matrix, which is a very powerful tool in the optimization [48].

5.3 Implementation

Even if a general procedure has been given, there are many algorithms that are already embedded in MATLAB® with the purpose of solving the optimization problem. The active-set algorithm is the used one and hereby the needed elements will be presented.

An optimization problem can be always reduced to a minimization process where an objective function is minimized while some equality and inequality constraints should be satisﬁed. It can be stated as:

85 Impulsive trajectory optimization

Minimize the function: J(y) subject to

equality linear constraints: Aey = be

inequality linear constraints: Aiy ≤ bi

equality non-linear constraints: Ce(y) = 0

inequality non-linear constraints: Ci(y) ≤ 0

5.3.1 Objective Function

Usually, in astrodynamics the quantity that better expresses mission costs is the amount of ∆V : through the Tsiolkovsky equation (5.20) it can be related directly to the on-board propellant consumption.

m0 ∆V = ISP g0 ln (5.20) mf where ISP is the speciﬁc impulse of the propellant, while g0 is the standard gravity at sea level, and m are the masses. The velocity variation is intended in an inertial frame, therefore it should be related somehow with the adimensional variable of the N-body problem.

∆V = kV2 − V1k ˙ ˙ = k(b2 − b1) + (kC + kC)(ρ2 − ρ1) + kτ˙C(η2 − η1)k

= kkτ˙C(η2 − η1)k (5.21)

= kτ˙kη2 − η1k = kτ˙∆η

From the previous equation it is possible to relate the variation of the dimensional velocity directly with the adimensional one. This is possible because the ∆V is thought as impulsive, therefore, applied at a certain epoch. Since the adimensional parameters b, k and C depend only on the epoch they are equal for both states, before and after the impulsive manoeuvre. As a deﬁnition the position cannot change for continuity constraints, therefore the ∆V is directly related to ∆η. Then, for numerical and physical reasons, the objective function is chosen as the square value of the adimensional velocity variation:

nm nm X 2 X T J(y) = ∆ηj = (ηj+1 − ηj) (ηj+1 − ηj) (5.22) j=1 j=1

The last expression has been conveniently expressed in vectorial notation, because optimization algorithms require also the gradient of the objective function. Since the

86 5.3 Implementation

adimensional velocities are elements of the variable vector, their derivatives should be computed and this formulation permits an easier derivation:

∂J = −2(ηj+1 − ηj) (5.23a) ∂ηj

∂J = 2(ηj+1 − ηj) (5.23b) ∂ηj+1

The resulting Jacobian matrix is sparse (just two indices are present in eqs. (5.23a) and (5.23b)), favouring numerical eﬃciency. For example, considering 6 manoeuvres and 9 segments there are 36 non zero elements in correspondence to the values of the adimensional manoeuvring velocities.

5.3.2 Equality Linear Constraints

In the variable vector all the nodes of an arc and all the arcs of the trajectory are included. The last node of the i-th arc and the ﬁrst node of the (i+1)-th arc are the nodes where the manoeuvre takes place. In the variable vector there are also the epochs where every arc starts and ends, therefore, the equality linear constraint equations can be subdivided into two main groups. From one side those where it is deﬁned that the position at manoeuvring node must be that same, and in the other side the one that says that the epochs of manoeuvring nodes must be the same.

Figure 5.2: Sparsity pattern of equality linear contraint matrix

This can be easily shown in fig. 5.2 where the first six groups (that are coupled) are related to the position constraints, while the last group is related to the time constraints. The zoomed fig. 5.3 presents the latter elements, and shows how they are coupled, according to the equations:

ρj,f − ρj+1,i = 0 τj,f − τj+1,i = 0 j = 1, ..., nm (5.24)

5.3.3 Inequality Linear Constraints

The inequality linear constraint are of three kinds, representing three diﬀerent conditions. In this work the manoeuvres have been considered as impulsive, however, the

87 Impulsive trajectory optimization

Figure 5.3: Sparsity pattern of equality linear constraint matrix - particular

engine needs a ﬁnite time to perform the manoeuvre. Therefore, the solution is constrained to remain on an arc higher than 2 days. This is done to ensure the actual feasibility of the manoeuvre.

τj,f − τj,i ≤ 172800ω0 j = [1 : nm] (5.25) At the same time it is needed that the ﬁnal epoch should be smaller than a given epoch and the initial epoch greater than 0:

τN,f ≤ etf ω0 (5.26)

τ1,i ≥ 0 (5.27)

5.3.4 Equality Non-Linear Constraints Just a few mathematical considerations will be introduced to get an easier implementation. The state vector for the i-th node can be subdivided into two parts: position and velocity. (For the sake of simpliﬁcation the i index will span all the nodes of the trajectory, while the j index only the manoeuvre nodes. When diﬀerently required it will be expressively presented.)

ρi xi = (5.28) ηi

88 5.3 Implementation

This means, that those sub-vectors can be written in a matricial notation as:

( ρi = Sρxi Sρ = I3×3 03×3 (5.29) ηi = Sηxi Sη = 03×3 I3×3

Finally, it is called ϕ(xi, τi, τi+1) the ﬂow of the solution. This means that, given the initial condition of the state xi, this value represents the evolution of the integrated trajectory from the time τi up to the time τi+1. For a matter of notation simpliﬁcation it will be called ϕi.

Z τi+1 ϕ(xi, τi, τi+1) = xi + f(x(t), t) dt (5.30) τi

The presentation of the constraint equation will follow:

• Position Constraints for Internal Nodes The following constraint equations must formulate the condition for which the evaluation of the flow at the final time must coincide with the following node. This is the condition that introduces the continuity on positions: the flow at the final time represent the position at final time after integration and it must coincide with the position of the next node. These are N − 1 equations because there is no condition of this kind for the last node.

i Cρ = Sρϕi − Sρxi+1 = 0 ∀i = [1 : N − 1] (5.31)

• Velocity Constraints for Internal Nodes The following constraint is very similar to the previous one: it states that also the velocity must be a continuous quantity over the trajectory, but only where there are no impulsive manoeuvres. By deﬁnition arcs are trajectory segments between which there is a ∆V , hence no velocity constraint should be on those nodes.

i Cη = Sηϕi − Sηxi+1 = 0 ∀i = [1 : N − 1] \ j = [1 : nm] (5.32)

• Constraints for External Nodes Constraints for external nodes are solely related to conditions on the problem boundaries, that can be expressed as two general manifolds of variables and other parameters.

89 Impulsive trajectory optimization

C(x(τ1,i), x(τN,f )) = 0 (5.33)

where C is a general non linear relation that includes the state at boundaries. Since it will be diﬀerent for every boundary value problem, it must be considered case by case. In this work it can be specialised for the case of the initial state to be equal to a given vector and the ﬁnal position equal to another vector:

( 0 Cx = x1 − x0 = 0 N (5.34) Cρ = SρxN − ρf = 0

It should be noted that in this case the relation has turned from non-linear to linear, being the latter a special case of the further.

The gradient of the equality non-linear constraints possesses a cumbersome mathematical expression, which, however, will bring incredible computational time savings in respect of using finite differences. At this point it is possible to define the matrix of derivatives for the constraint equations. it will be a huge matrix of [(6N −3na +6)×(6N +2na)]. According to the equation (5.2) the gradient of the constraint equations can be obtained taking the derivative with respect to the two main sub-vectors, which can be respectively divided in other sub-elements.

 i i  ∂Cρ ∂Cρ  ∂X ∂T     i i   ∂Cη ∂Cη    dC(y)  ∂X ∂T  =   (5.35) dy  0 0   ∂Cx ∂Cx     ∂X ∂T   N N  ∂Cρ ∂Cρ  ∂X ∂T

Generally, the gradient will be a sparse matrix, full of zeros and identity sub-matrices. Here some examples will follow:

i ∂Cρ - : it is a [3(N − 1) × 3N] matrix that is built of two kind of non-null elements: ∂X

i ∂Cρ ∂Sρϕi i = = SρΦ ∂xi ∂xi

90 5.3 Implementation

i ∂Cρ ∂Sρxi+1 = − = −SρI6×6 ∂xi+1 ∂xi+1

From the ﬁrst component it is possible to recognize the STM, that have been previously presented. Since the derivatives with respect to the initial conditions are taken, this is by deﬁnition the matrix that describes their evolution. The overall matrix can be seen as: 1 SρΦ −SρI6×6       .. ..   . .    ∂Ci   ρ  i  =  SρΦ −SρI6×6  ∂X        ......      N−1 SρΦ −SρI6×6

where all the other elements are null because their only dependence is from those nodes that appear in the constraint equations. As a matter of fact, the derivative of the state of one node, with respect to the derivative of the state of another, is null because they are independent by deﬁnition. This is also compliant with typical forms non linear programming methods. i ∂Cη - : it is a [(3(N − 1) − 3na) × 3N] matrix that is composed of two elements. ∂X

i ∂Cη ∂Sηϕi i = = SηΦ ∂xi ∂xi i ∂Cη ∂Sηxi+1 = − = −SηI6×6 ∂xi+1 ∂xi+1

Also in this case all the other elements are null because derivatives of independent variables. ∂C0 - x : it is [6 × 6(N − 1)] sub-matrix, which is deﬁned as: ∂X

0 ( ∂C ∂(x1 − x0) I6×6 if i = 1 x = = ∂xi ∂xi 06×6 if i 6= 1

91 Impulsive trajectory optimization

N ∂Cρ - : it is [3 × 6(N − 1)] sub-matrix, that can be computed as: ∂X

N ( ∂Cρ ∂(Sρx1 − ρf ) Sρ if i = N = = ∂xi ∂xi 03×6 if i 6= N

Time derivatives are a bit more complicated to be proved analytically, but a derivation can be found in Oshima [33] and Whitley-Ocampo [48]. In any case it is possible to recognize that every state depends only on the time element that describes its node (τk) and the following (τk+1). However this is strictly related with the elements that deﬁne the boundaries for the arc.

τj,f − τj,i ∆τ = , τk = τj,i + (k − 1)∆τ (5.36) ns The previous relation brings as consequence four fundamental time derivatives:

∂τ n − k + 1 ∂τ k − 1 k = s , k = ∂τj,i ns ∂τj,f ns ∂τ n − k ∂τ k k+1 = s , k = ∂τj,i ns ∂τj,f ns

Particular attention must be paid to the derivative of the ﬂow with respect to τk. To prove it, the derivative deﬁnition is used:

∂ϕi ϕ(xi, τk + δτk, τk+1) − ϕ(xi, τk, τk+1) = lim (5.37) ∂τk δτk→0 δτk It is worth noting that in the previous equation the perturbed ﬂow is obtained by varying of a quantity δτk the initial time. However, the same result is obtained computing the ﬂow starting from updated initial conditions, in the following relation:

0 ϕ(xi, τk + δτk, τk+1) = ϕ(xi + δxi, τk, τk+1) (5.38) 0 0 It has been called δxi the diﬀerence between the new supposed initial condition xi and the real initial condition xi. Hence, the derivative can be written in a diﬀerent way:

0 ∂ϕi ϕ(xi + δx , τk, τk+1) − ϕ(xi, τk, τk+1) = lim i (5.39) ∂τk δτk→0 δτk which can be related now to the state transition matrix, since the variations are small:

∂ϕ Φ(τ , τ )(x + δx − x ) i = lim k k+1 i i i ∂τk δτk→0 δτk δxi = Φ(τk, τk+1) lim δτk→0 δτk

92 5.3 Implementation

At this point, it is necessary do derive the diﬀerential. The relation between the initial condition vectors is:

0 xi = xi + x˙ iδτk (5.40)

hence,

0 δxi = xi − xi = −x˙ iδτk (5.41)

Plugging the last equation into the derivative of the ﬂow, it is possible to obtain:

∂ϕi −x˙ iδτk = Φ(τk, τk+1) lim ∂τk δτk→0 δτk

= −Φ(τk, τk+1)x˙ i

Remembering that for deﬁnition the derivative of the state is equal to the right-hand side, the time derivative of the ﬂow can be written as:

∂ϕi = −Φ(τk, τk+1)f(xi, τk) (5.42) ∂τk At this point, it is possible to take the derivatives with respect to the elements of the variable vector

∂Ci ∂S (ϕ(x , τ , τ ) − x ) ρ = ρ i k k+1 i+1 ∂τj,i ∂τj,i ∂ϕi τk ∂ϕi τk+1 = Sρ + ∂τk τj,i ∂τk+1 τj,i ns + 1 − k i ns − k = Sρ − Φ f(xi, τk) + f(ϕi, tk+1) ns ns

∂Ci ∂S (ϕ(x , τ , τ ) − x ) ρ = ρ i k k+1 i+1 ∂τj,f ∂τj,f ∂ϕi τk ∂ϕi τk+1 = Sρ + ∂τk τj,f ∂τk+1 τj,f k − 1 i k = Sρ − Φ f(xi, τk) + f(ϕi, tk+1) ns ns

The state transition matrix appears again, showing the dependence of the overall procedure on this operator. This is an innovative approach to the problem, which permits a high reduction of the time for the computation of the terms: in fact, it just needs the evaluation of the function

93 Impulsive trajectory optimization to compute a derivative, taking inspiration from the Taylor expansion series scheme, that has been shown when introducing the integration scheme.

Figure 5.4: Sparsity pattern of equality non-linear constraint matrix

From ﬁg. 5.4 it is possible to note the high number of zeros that are present in the gradient of these equations. In particular there are 3414 non zero elements out of 167958, a number that represent slightly more than the 2%. The non zero elements are all those that have presented, focusing on the diagonal with the state transition matrices and the identity matrices, while in the right there are all the elements related with time derivatives. They sometimes present a step because there are the manoeuvre nodes, which do not depend on the constraint equations.

5.3.5 Inequality Non-Linear Constraints The inequality non-linear constraints are related to the control authority the problem must have. This constraint is present showing that the maximum ∆V can not overcome a certain value.

n Xm ∆Vj ≤ ∆Vmax (5.43) j=1

94 5.3 Implementation

Note that the previous relation is written in an inertial frame and not in synodic. In order to obtain that formula, it should be used:

n Xm kτ˙∆ηj ≤ ∆Vmax (5.44) j=1

Once again for computational reasons it is better to use squared values to put the problem in a closer-to-convex form, exactly as for the objective function. Therefore, the constraint equation can be written as:

nm X 2 2 2 2 k τ˙ ∆ηj − ∆Vmax ≤ 0 (5.45) j=1

Another set of constraint equations derive from the necessity of translating impulsive manoeuvres into ﬁnite burn manoeuvres. To accomplish this, it is necessary that the time spent on an arc should be greater than a certain value, function of the burn magnitude, the maximum available thrust, and the mass of the S/C.:

mS/C k∆ηj τj,f − τj,i = ω0 j = [1 : nm] (5.46) Tmax From numerical simulations it is possible to show that this is an over-estimation of the real time to spread the manoeuvre into a ﬁnite burn.

95 Impulsive trajectory optimization

96 Chapter 6

Application to the LISA PathFinder mission extension

Many different tools have been presented in this work, according to a theoretical view point. This chapter focuses on showing how these methods can be applied to a real case study. Firstly, the LPF mission will be described [2], its main tasks [8] and the opportunities for the end of life disposal. Afterwards, it will be shown different possible trajectories in a 4-body problem, which are prone to a refinement process in the n-body problem. The 4-body solutions will represent the guesses for the optimisation strategy. Finally, an analysis of the results will be shown and its convergence properties.

6.1 Presentation of the LPF mission

LISA (Laser Interferometer Space Antenna) PathFinder is an ESA mission that paves the way for the direct observation of gravitational waves. It is a precursor for a more complex mission, LISA, which will employ three spacecraft in a constellation ﬂight to directly test the general relativity. The aim of LPF is to prove the feasibility of this second mission, testing in ﬂight the components of the spacecraft. The idea behind the test is to put two small masses in a near-perfect gravitational free-fall, control and measure their motion with very high accuracy. The main problem is related with the high number of perturbations that act on a spacecraft orbiting in space. To overrun this obstacles many innovative systems have been employed.

- The two test masses are cube-shaped with a weight of about two kilograms, each one ﬂoating in its own vacuum container;

- A gold-platinum alloy covers the masses, ensuring that magnetic forces do not have any eﬀect;

- Through ultraviolet radiation it is built a contact-less discharge system, which pre- vents any electrostatic charging of the masses;

- The test masses are protected against the launch vibrations through a caging and a grabbing mechanism;

97 Application to the LISA PathFinder mission extension

- The laser interferometer will measure the position and orientation of the two test masses relative to the spacecraft and to each other with a precision of approximately 10 picometers;

- The actual position of the satellite is controlled through cold-gas micronewton thrusters, which are also used to delete perturbing acceleration of solar radiation pressure.

Figure 6.1: LPF technolgy, courtesy of ESA

LPF was successfully launched on December 3 2015 and started its operations on March 1 2016. The desired level of precision was already obtained within the first day of LISA Pathfinder’s scientific operations, and in the following months the performances of the experiment improved up to a precision five times better than expected. Given these results, the mission, which end of operations was expected for December 7 2016, extended its duration for other 6 months, up to May 31 2017.

In the setting of this work, Lisa PathFinder gains high interest mainly for its particular location in space. Using multiple apogee burns, LPF reached a Lissajous orbit around Sun–Earth Lagrangian point L1. It is an unstable orbit, which requires periodically small manoeuvres for station keeping. In fig. 6.2 it is plotted the trajectory of LPF from September 20 2016 to May 31 2017. The state vector has been supplied, as a kernel file for SPICE users, directly from ESA’s SPK file and then it has been rotated in RPF, by means of equation (3.1) and (3.9).

98 6.1 Presentation of the LPF mission

Figure 6.2: xy motion of LPF in RPF

As already said, the parking orbit of LPF is unstable, therefore, some correction manoeuvres are needed for station keeping. A ﬁrst interesting analysis is related to how the trajectory would change if the free dynamics of the spacecraft were analysed. To do this, the motion of LPF is forward propagated with a time step of one day in the timespan of the plotted orbit. Without station keeping the trajectory of LPF closely resemble that of a halo unstable manifold. This phenomenon is shown in Fig 6.3.

Figure 6.3: LPF manifold extension

99 Application to the LISA PathFinder mission extension

6.1.1 Saddle Point and LPF mission extension

The end of life for LPF is a topic on discussion. Indeed, the S/C has still a very low amount of propellant available for further scientific tasks. Hence, this case represents a very good application example for the optimisation techniques hereby presented. It can be seen as a boundary value problem, where the initial conditions should lay still on the starting orbit, while the final condition must be chosen. The place where the summation of all the gravitational effects is zero and it is called Saddle Point. From the scientific point of view this would be a very interesting point in space to visit since it would provide, for instance, a very good testing scenario for the Modified Newtonian dynamics (MOND), as proposed by Milgrom [31]. For the experiments that are flying on-board Lisa PathFinder, to cross a region where no planetary gravitation is felt, it would mean a more direct contact with those gravitational waves that are studied. The experiment is justified because evidence is mounting on the validity of MOND gravitational theory at certain time and space scales, that was firstly proposed by Trenkel [45]. This relation can be expressed as:

¨ X RSP − Rj RSP = − µj 3 = 0 (6.1) kRSP − Rjk j∈S

This equation doesn’t have a close form solution, since the positions of the planets depend on their ephemerides. However, it is possible to ﬁnd a numerical result for eq. (6.1). A Newton based algorithm can be exploited, where:

X R − Rj(t) f(R, t) = µj 3 (6.2) kR − Rj(t)k j∈S

(a) SP: in plane motion (b) SP: out of plane motion Figure 6.4: SP trajectory

100 6.1 Presentation of the LPF mission

Now, referring to eq. (5.7), for every time t it is possible to expand that function in Taylor series. Then, guessing the ﬁrst solution, it is possible to write the iterative procedure:

∂f(Rk , t)−1 Rk+1 = Rk − SP f(Rk , t) (6.3) SP SP ∂R SP The resulting trajectory is presented in fig. 6.4, where it has been plotted its motion between the same previously presented epochs. In the figure its synodic representation is depicted: the trajectory is computed in an inertial frame and then it has been rotated. Both the in-plane and the out-of.plane motion have been presented. It is worth noting how the baseline shape of the in-plane motion is quasi symmetric with respect to the x-axis. The combination of the Saddle Point trajectory, together with the manifold, shows the number of ballistic trajectories that would pass through this region (fig. 6.5).

Figure 6.5: Intersection of LPF and SP trajectory

Few close SP passages occur (see Table 6.1) if compared to the overall attempts. Table 6.1 summarizes the percentage of trajectories that have a miss distance smaller than a certain tolerance. The values have been computed with a time step of 1 hour in the same epoch intervals (20 Sep 2016 to 31 May 2017).

101 Application to the LISA PathFinder mission extension

Table 6.1: Percentage of trajectories with miss distance smaller than the tolerance Toll. [km] Number of trj. [%] 1e3 0 1e4 1.63 5e4 6.11 1e5 13.54

At this point it is worth considering how does the miss distance varies with the departure date. This is a very precious information, since it can provide an insight on departure windows for the spacecraft. In other words, if the miss distance is a small value, it would be easier for the optimisation algorithm to converge without changing too much the variables. Moreover, the expected cost for such a mission would be lower, since the trajectory approaches ballistically the target.

Figure 6.6: Miss distance depending on departure date

Given the fact that the Halo parking orbit is a periodic orbit, the expected error should present an harmonic motion. This can be appreciated in ﬁg. 6.6. The error, in km has a sinusoidal behaviour, with a period of around 172 days. In fact, it is strictly related to the orbital period of the parking orbit, which is of 174 days, but cannot be exactly the same due to the eﬀect of the Moon.

Table 6.2: Minimum and maximum miss distance Dep. Date Miss Distance [km] 2016 DEC 13 2571 2016 SEP 27 782140

102 6.1 Presentation of the LPF mission

Figure 6.7: Time evolution for miss distance - Minimum miss distance

The consequence of this result is that there are departure windows where the error is at the minimum, that coincide with a particular region of the halo. The trajectories related to these departure dates, together with their miss distance time variation, are shown in ﬁgures 6.7, 6.8 and 6.9.

Figure 6.8: Time evolution for miss distance - Maximum miss distance

From ﬁg. 6.9 it appears more evident one aspect that was diﬃcult to see in the manifold picture: even if no manoeuvres are included, the spacecraft orbits in the Halo region for at least half a period. This shows also the low grade of instability of the parking orbit.

103 Application to the LISA PathFinder mission extension

Figure 6.9: Synodic trajectory for minimum and maximum miss distance cases

6.2 Guess Solutions

The ballistic approach to the problem gives direct information on its features, in this case the miss distance. However, it has to be thought simply as the initial step of an optimisation procedure, since the performances of the trajectories are very low: in the best of the cases, choosing wisely the departure date, it is possible to miss the target by few thousand kilometres, which are equivalent to 10e-8 nondimensional units. Now, even if in space these are very good results, considering that the distances are in the order of the AU, for an ultra-precise mission as LPF this cannot be the end point. It is necessary to manoeuvre the spacecraft, in order to reach the SP with higher precision and fidelity. It becomes a boundary value problem, that can be handled with the optimisation techniques previously described based on multiple shooting strategies. Every algorithm that deals with optimisation processes, needs as starting point a first guess: a solution which is near enough to the desired local optimum, otherwise solution convergence is ill-behaved. For the problem at hand there are then two paths to follow. An exploration of the solutions space through direct methods and evolutionary algorithms directly in the n-body frame, or the same but in a less refined frame. The advantage of the latter would be in getting the best guesses for the algorithm, however their computation would require too much time. Instead, the first, could produce solutions that have to be refined in the high fidelity model, but their computation is faster and a higher number

104 6.2 Guess Solutions can be simulated. Here, the second path has been chosen, optimizing in the n-body problem trajectories which have been designed in a 4-body problem, where the motion of the Moon have been projected into the same rotation plane of the primaries, Sun–Earth. The description of how these trajectories have been obtained passes beyond the scope of this work, which have relied for this aspect upon the eﬀort of [44].

Figure 6.10: Shooting from 4-body to n-body: 2 manoeuvre, 5 segments

In any case, it is interesting to analyse some of their features. The starting point of every optimisation process is the vector of variables: it includes the states at every node and the time at manoeuvre nodes. From this vector it is possible to fully recover the overall state of the trajectory in the 4-body problem. It is worth noting that in this model there is no need for a real epoch, since the only perturbing effect is the gravitational action of the Moon. Its motion has been assumed analytically as uniformly circular around the Earth and co-planar with the primaries, therefore, the only needed parameters is the lunar angle. This aspect is not of minor relevance, because the presence of the Moon influences highly the trajectory of the spacecraft. Fig. 6.10 depicts a classic scenario for an optimisation problem. The dashed black line is the trajectory of the spacecraft in the 4-body model, while the red arcs represent trajectories in the n-body problem. The markers are, in fact, the nodes of the solution. Knowing the state at those nodes it is possible to propagate forward the solution and analyse how much it diverges. The final state of a red segment is what it has been called flow of the integration and its discrepancy from the next node is the error (or defect vector) between the two models. The presented solution is just one of the many that have been considered, but this is particularly interesting since it shows clearly the main

105 Application to the LISA PathFinder mission extension aspects of the process:

(a) Indirect solution: 1 manouevre, 8 seg- (b) Direct solution: 3 manoeuvres, 4 segments ments Figure 6.11: Shooting from 4-body to n-body in diﬀerent cases

- The trajectory involves nm manoeuvres, that in the picture are represented with red markers (nm = 2). In the trajectory three diﬀerent arc are recognizable.

- Every arc on the trajectory is composed of ns segments (ns = 5), which means ns + 1 nodes per arc. The nodes on the arc cannot be only ns because the defect vector must be computed also at the ﬁnal node.

- The integrated segments follow very nicely the 4-body trajectory for almost all its evolution. However, there is a particular point where the two segments diverge. This is related with lunar attraction and its diﬀerent position in the two models. As a matter of fact, until the spacecraft is far from the Moon, the two trajectories are similar. But, when it approaches the Moon and performs a lunar gravity assist (LGA) the methodology produces diverging solutions. This translates into an unexpected out-of-plane motion, that is associated to large defect vector.

As it can be seen from figure 6.11 and 6.12 there are different sets of complex or simple solutions that do reach the target in the 4-body problem, which need to be refined in the n-body model. It will be necessary to understand if those solutions exist also in the more refined models and to see if convergence is attainable in one of them.

Initial epoch and Moon phase The n-body model used for the propagation of the trajectories is represented by non- autonomous differential equations. This means that the time variable appears directly in the problem, also in the 4-body model, where the position of the third attractor, the Moon, is represented through an angle, called Moon phase α, that still depends on time. Therefore, to use guesses that come from this model, it is necessary to find a way to convert this angle into an epoch. There are at least two ways to tackle the problem. The first one considers the motion of the Moon and the fact that it is periodic. Then, due to this characteristic it is possible

106 6.2 Guess Solutions

(a) Indirect solution: 1 manouevre, 8 segments (b) Indirect solution: 3 manoeuvres, 4 segments Figure 6.12: Shooting from 4-body to n-body in diﬀerent cases to say that at every period it will return to the same state. If the epoch of a certain phase were known, for instance α = 0, the problem resolves into the mere knowledge of the number of revolutions n.

α(t) t = t0 + nωM + (6.4) 2πωM Of course this approach introduces a certain approximation in the problem and it presents some inaccuracies: for instance, the lunar angular velocity is not perfectly constant. To overcome this drawback, another approach can be followed. It is possible to see how the two primaries, and the Moon, being three different points in space, can uniquely define a plane. In this plane the Sun–Earth vector would define the x-axis, while, the Moon phase can be computed a posteriori by solving:

cos α(t) = rSE(t) · rEM (t) (6.5)

being rSE and rEM respectively the Sun–Earth and Earth-Moon connecting vector in the rotated frame. It is worth pointing out that this frame has no physical meaning, but it can be used as a mathematical tool, because at every epoch it can relate the ephemeris position with the 4-body dynamics. Since the equation is periodic, it has different solutions: an example is presented in fig. 6.13. To find the conversion relation through the second procedure, it is necessary to invert eq. (6.5), which can be done through a bisection method or a Newton method. In the second case a guess solution is needed, which can be computed, for instance, exploiting eq. (6.4). Particular attention must be paid to the limit cases, when the Moon is aligned with the primaries, because in this case the plane cannot be defined. However, this is a condition that can be easily verified using a control on the linear dependence of the mutual position expressed in inertial frame.

107 Application to the LISA PathFinder mission extension

Figure 6.13: Periodic solution for the phasing problem

6.3 Optimisation in RPF n-body problem

In addition to the guess solution, the optimisation ﬂow needs some diﬀerent parameters and functions, some of which have already been presented (objective function and constraint function), but some other have to be listed yet. LPF is a real spacecraft with some geometrical and physical properties which are linked with the n-body model and they are needed for the integration of th system. Moreover, the application of a real case gives a more concrete dimension to the tools that have been presented: it is easier for a user to understand the logics and the practical operation in its implementation on a calculator.

1. Parameters initialization

The first step is to initialize all those parameters that will be needed in the optimisation process. In this work they are defined as a structure with different fields, which can be easily passed through functions. Gravitational model parameters: pl_id: Vector of ID numbers for all the planets to be used as attractors mu: Vector of gravitational parameters µ for all the planets RADII: Vector of radii values for all the planets - only for inhomogeneous gravity field J2: Vector of J2 values for all the planets - use this only for inhomogeneous gravity field om0: Mean relative motion of the primaries idPR: ID number of primary body idSEC: ID number of secondary body

108 6.3 Optimisation in RPF n-body problem

Table 6.3: Values for the main parameters Unit of Parameter Value Measurement idPR 10 [-] idSEC 399 [-] om0 1.99098666e-07 [1/s] m_sc 500 [kg] A_sc 3.7 [m2] c_r 1.08 [-] AU 1.49597870e+11 [m] c 2.99792458e+08 [m/s] SF0 1371 [W/m2] ET0 31 AUG 2016 [s] ETf 31 DEC 2020 [s]

Spacecraft related parameters: m_sc: Mass of the spacecraft A_sc: Area of the spacecraft associated to the cannonball model for SRP c_r: Reﬂectivity coeﬃcient

SRP related parameters: AU: Astronomical unit c: Speed of light SF0: Solar ﬂux at 1 AU

Time related parameters: ET0: Initial epoch - Time variables can be computed as days past ET0 ETf: Final epoch - Time variables can not be greater than this value

The values for these parameters have been summarized in tab. 6.3. The only missing data are those related with the celestial bodies, which are all the planets of the solar system.

2. Vector of variables deﬁnition

The vector of variables must be converted from a synodic 4-body problem to a synodic n-body problem. The vector of variables will be composed of the state at nodes, and of the times at initial and ﬁnal arc nodes. Since the primaries are the same, there is no need in the conversion of the state variables, because the frame has been kept unchanged. In order to have better convergence properties the time variables cannot be accepted as epochs, intended as seconds past a certain date (usually Jan 01 2000). These are too large numbers that easily introduce numerical error in the process. The natural solution would then be to chose the adimensional time, however this brings

diﬃculties in interpreting the results. This is the reason why days past ET0 have

109 Application to the LISA PathFinder mission extension been chosen here as time variables, according to the relation:

t − ET0 τy = (6.6) 86400

Sometimes, especially when optimizing complex solutions, it is very difficult to attain convergence. It happens because of the large duration of a segment (in the order of tenth of days) that triggers the error between the two models to increase indefinitely. In order to counter these difficulties, the number of nodes per arc is increased, i.e. the problem has more but shorter segments. This finer discretisa- tion benefits the goodness of the initial guess as well as the solution convergence behaviour, albeit significantly increasing the problem dimension (i.e., the number of multiple shooting variables).

(a) Sample solution: 3 manouevres, 4 seg- (b) Sample solution: 3 manouevres, 33 segments ments Figure 6.14: Shooting from 4-body to n-body with diﬀerent number of segments

The procedure consists simply into evaluating the 4-body solution in more points, that are obtained by a linear partition of the time interval for the permanence on an arc. Therefore, the same trajectory can be described in very different ways and this helps refining it. As an example, a complex 3-manoeuvre solution has been selected, which can be seen in fig. 6.14. From the 4 body problem it had only 4 segments per node. They have been augmented firstly in order to have a maximum time step of 6 days between nodes (getting 33 segments per arc) and then of 2 days (98 segments per arc). Figure 6.14 presents the result of this node fitting. The trajectory is of course more accurate than the one with only 4 segments; however there are still some inaccuracies in the central region, where the Earth/Moon close passages are more frequent. This problem seems almost overcome in the solution with 98 segments of fig.6.15, at the cost of placing many nodes. The challenge is to trade-off the minimum number of segments that still brings the solution into convergence and the integration CPU time.

110 6.3 Optimisation in RPF n-body problem

Figure 6.15: Shooting from 4-body to n-body: 3 manouevres, 98 segments

3. Objective and constraints functions

The following step is to describe the optimisation problem, therefore, to deﬁne which is the objective function to be minimized with respect to the constraints of the boundary problem. For this particular case very minor diﬀerences with respect to what previously described have been made. The objective function is the sum of the adimensional velocities squared:

nm nm X 2 X T J(y) = ∆ηj = (ηj+1 − ηj) (ηj+1 − ηj) j=1 j=1

The constraints are exactly the same described in the previous chapter. However, two minor comments are worth of note.

Since the times in the variables vector τy are diﬀerent from the independent time variables τ, when deriving the constraint equations upon time, this fact must be considered.

∂C(y) ∂C(y) ∂τ = (6.7) ∂τy ∂τ ∂τy

While the ﬁrst term has been extensively analysed, some attention must be paid to the second. To derive it eq. (6.6) should be combined together with the relation of adimensional time:

111 Application to the LISA PathFinder mission extension

τ = ω(t − ET )  0 ∂τ t − ET0 =⇒ = 86400ω (6.8) τ = ∂τy  y 86400

One last note is that the gradient of both objective and constraint function haven been checked against MATLAB®’s central finite differences. In the worst cases, the relative error is in the order of 10-5 adimensional unit. However, the analytical gradient evaluation is from 20 to 100 time faster than the one computed with central finite differences.

4. Optimization options selection

All the parameters and functions are included in the call of MATLAB®’s routine fmincon, which embeds different algorithms for optimisation with non linear constraints. The one that have been used here, is the active-set algorithm. The basic idea is the following: firstly, the algorithm finds the minimum of the unconstrained problem. Then it moves along solution spaces or sets, where the constrains are called active if they are higher than a certain tolerance. The algorithm proceeds from the most constrained equation to the least one and it updates the solution step by step, until the last one where all the constrains are satisfied. Only at this point the minimum of the objective function can be found, which also satisfies the constraints. The options are defined in a structure which is passed to fmincon. There are specified the tolerances, the type of algorithm, where to find the gradients; in fact it is checked only once to verify if the user defined gradient is compatible with the one computed by central finite differences.

1 options = optimset('fmincon'); 2 options.Display ='iter'; 3 options.MaxFunEvals = 1e5; 4 options.MaxIter = 3e3; 5 options.TolFun = 1e-12; 6 options.TolX = 1e-12; 7 options.TolCon = 1e-12; 8 options.Algorithm ='active-set'; 9 options.ScaleProblem ='obj-and-constr'; 10 options.GradConstr ='on'; 11 options.GradObj ='on'; 12 % options.DerivativeCheck='on'; 13 options.FinDiffType ='central'; 14 options.OutputFcn = @StopFcn; 15 16 [y,fobjval,exitflag,out] = ... fmincon(fobj,y0,A,b,Aeq,beq,lb,ub,fcon,options);

112 6.4 Analysis of the results

6.4 Analysis of the results

This section concerns the results of the optimisation process and has the aim of showing how and if the guess solutions converged. The different typologies of guesses showed a different behaviour during the optimisation. Here they will be compared and some explanations on their trend will be given. For a further numerical simplification, which does not reduce by any meaning the utility of the analysis, another assumption has been made. As starting point has been chosen the one given from the initial conditions of the 4-body guess. It is not exactly coincident with the position of LPF at the same epoch, but this can be easily retrieved by a posteriori minimal corrections to the orbit. This assumption allows having the same initial conditions, with the possibility to choose over different epochs. The computations have been made using the parallel computational tool of MATLAB®8˙ contemporary processes have been run on Copernicus, the workstation at the Aerospace Department of Politecnico di Milano. A typical output file has been shown in fig.

6.4.1 Direct solution

It is called a direct solution, a trajectory followed by the spacecraft, which departs from its parking orbit and then targets directly the Saddle Point, without performing any Moon or Earth close passage. They can be addressed as the easiest solutions, because they exploit the manifold of the halo and, with minor manoeuvres they reduce the miss distance. Tab. 6.4 summarizes the diﬀerent properties of the two trajectories and many interesting aspects can be observed. Firstly, the number of nodes have been increased in order to have at the maximum 2.5 days between nodes. This threshold has been chosen after many attempts and it brought to a good compromise between optimisation time and convergence steps: the higher the number of nodes, the higher will be the computation time, but it will require less steps to converge.

Table 6.4: Direct trajectory properties NoM NoS ∆ V ∆ t Miss Distance MODEL [-] [-] [m/s] [days] [m] 4B 6 5 9.59 118.41 1.72e+01 NB 6 6 15.87 121.63 2.36e-04

From ﬁg. 6.16 it is possible to see how the reﬁned trajectory has maintained its original baseline shape. The general idea of this kind of solution is to follow the halo orbit and, by consequential manoeuvres, to correct the path directly to the Saddle Point. The fact that the objective function, the ∆V, has increased shouldn’t surprise. The guess solution is a trajectory in a coarser model and the starting point of the algorithm is not feasible. This is related to the perturbation included by all the other planets and greatly from the solar pressure. The starting point of the minimization can be thought as that unconstrained solution, which the algorithm needs for convergence.

113 Application to the LISA PathFinder mission extension

Figure 6.16: Converged direct solution: 6 manoeuvres, 9 segments

The time of ﬂight increase is just a consequence of the diﬀerent trajectory. However, the most interesting quantity is the miss distance from the Saddle Point. The best ballistic trajectory obtained after the study of the manifold, could provide a miss distance of two thousands kilometres. The new solution has a precision that is 10 order of magnitudes greater: it has a miss distance in the order of the tenth of millimetres. Moreover, its error is also 5 order of magnitude lower with respect to the same value in the 4-body model. And this result can be obtained paying simply few m/s of ∆V.

6.4.2 Indirect solution The indirect solution presents a trajectory that does not target directly the SP. After the departure from the parking orbit, the spacecraft follows a path that brings it behind the Earth, where it follows an elliptical orbit to get back to the Saddle Point. This trajectory has only one passage behind the secondary body, but this lengthens its path to the target of 60 days.

Table 6.5: Comparison between diﬀerent reﬁned solutions NoM NoS ∆ V ∆ t Miss Distance Iteration Steps MODEL [-] [-] [m/s] [days] [m] [-] 4B 2 5 4.33 188.82 1.72e-01 - NB 2 21 26.64 188.31 3.68e-04 308 NB 6 9 47.29 187.23 5.50e-05 104

114 6.4 Analysis of the results

From one side it has been tried to converge to the solution using two manoeuvres and 21 nodes. The convergence increased the objective function, but permitted a miss distance of 3 orders lower than the guessing one. This result is an index of the capability of the tool to reﬁne even more complex trajectories.

Figure 6.17: Converged indirect solution: 2 manoeuvres, 21 segments

At the same time, an other attempt with 6 manoeuvre and 9 segments has been made. The result converged even better, with an error of few tens of micrometres, but with a request of ∆V that doubles with respect to the previous solution. A new interesting result, which has to be considered, is the number of iteration steps needed to these solutions to be reﬁned. This is an index representing how much time will be required to the algorithm to attain convergence, because it is related with the number of function evaluations: the higher it is, the slower the optimisation. From tab. 6.5 it can be seen that the solutions with 2 manoeuvres and 21 segments (equal to 66 nodes) needed 308 steps, while to the 6 manoeuvres trajectory (70 nodes), 104 steps proved suﬃcient.

Table 6.6: Indirect trajectory properties NoM NoS ∆ V ∆ t Miss Distance MODEL [-] [-] [m/s] [days] [m] 4B 2 5 1.06 169.58 2.35e-01 NB 2 19 23.61 169.58 1.17e-04

It can be interpreted as a consequence of what previously described: due to the non- linearity of the problem, a higher number of nodes is not necessarily symptom of a slower convergence. Actually, the added 4 manoeuvre nodes have given to the algorithm the possibility to exploit at best the degrees of freedom. On the other side, the doubling

115 Application to the LISA PathFinder mission extension of the objective function is an aspect that must be handled with care: the performance request to the algorithm asks for a relax in terms of optimality. This brings the user of this kind of tool in the position of trading-off between the possibility of fast refining many solutions or looking with higher care for the minimum in a smaller set. Finally, tab. 6.6 refers to the properties of another indirect class of solution, that has been plotted in fig. 6.18. It clearly shows the non-linearity aspets of the problem. The sudden changes in the trajectory are typical of the unstable Lagrangian region, which has been perturbed even more from the lunar presence.

Figure 6.18: Converged indirect solution: 2 manoeuvres, 19 segments

6.4.3 Divergence and change of class

The proposed trajectories present a relatively simple pattern, which can be obtained easily because the integration time is small (e.g. less than 200 days). However, when more complex trajectories are used as guess for the optimisation, the convergence is not guaranteed. From one side this is related to the higher problem dimensions. From the other side, lunar gravity assists further increases the problem non-linearity. This does not necessary result in. A change in solution class might occur. Figure 6.19 displays a clear example. The initial guess was a trajectory which had 5 different elliptic orbits around Earth. The refinement process didn’t manage to follow the orbit, therefore it found another local minimum, but with a completely different trajectory. Actually, from indirect the solution became direct. In tab. 6.7 the properties of this case are listed.

116 6.4 Analysis of the results

Figure 6.19: Converged indirect solution: 2 manoeuvres, 19 segments The most interesting term to be analysed is the time of flight. In fact, it is the element that underlines the change of class of the solution. For all the presented refinements, the new orbit has a variation from the guess of a few days, sometimes even hours. In this case the solution has changed greatly: the duration is 70 days lower, more than two months. The reasons for this to happen are to be searched in the dynamical properties of the region where LPF orbits. Lagrangian regions of the collinear points feature unstable motion in the linear sense. 4-body and n-body representation are slightly different, but in that environment, differences in the 6-th or 7-th digits (non dimensional units) can bring to an evolution of the trajectory that is completely changed. Moreover, it must be added that the numerical problem is highly non linear and this would introduce new and significant distinctions.

Table 6.7: Properties of non-converging solution NoM NoS ∆ V ∆ t Miss Distance MODEL [-] [-] [m/s] [days] [m] 4B 1 8 5.10 336.15 2.54e+04 NB 1 34 18.23 259.25 1.00e-04

A final example is related to the lunar gravity assist. In fig. 6.20 it is shown the result of this refinement process. The guess solution presents clearly a gravity assist, that does not appear in the final trajectory. Actually, it is exactly that gravity assist that has forced the algorithm to find a new feasible solution. As already said, the position of the Moon in the two models is different, and a solution that presents the LGA in the same region is not feasible.

117 Application to the LISA PathFinder mission extension

Figure 6.20: Converged indirect solution: 2 manoeuvres, 19 segments

118 Chapter 7

Conclusions

7.1 Summary of the results

Innovative and high fidelity trajectories are nowadays matter of outstanding interest. On the one hand, there is the necessity of testing new physical theories, where space represents an environment almost completely free from Earth limitations (such as drag or gravity). On the other hand, the introduction of new technologies where the effect of perturbation is non-negligible, requires the introduction of more complete models, in order to follow the path of these objects. To reach this goal, the perturbations have been included since the early phase of mission design, in order to exploit their effects and not necessarily correct them. The procedure that this works has followed is to write the dynamical problem of an object in space including directly the effects of all the main attractors of the solar system. The n-body problem has been written in an inertial frame of reference, which can be also planeto-centred. Once the equations of motion have been mathematically derived, it has been a topic of interest the introduction of other main perturbations. It is widely known that the attractors are not perfectly spherical bodies and this is the reason why the first harmonics of the inhomogeneous gravitational field has been added. Moreover, the solar radiation pressure has a great influence over small bodies, which, under this effect, change greatly their path. The choice of the Lagrangian formulation for the problem made it more versatile and prone to the rotation in more comprehensive reference frames. The inertial definition of the problem gives a preferential view point for the physical interpretation. However, there are different strategies that can give a better mathematical formulation, hence, a new frame has been introduced and described. It rotates with respect to the inertial, following the motion of two primary bodies, inspired by the representation of the 3-body problem. The equations of motion have been rotated and adimensionalized in the roto-pulsating frame. Two ways of rotating them are shown, giving new validity to the work. In the first one, the equations of Euler-Lagrange are derived from the inertial Lagrangian and then rotated in the new frame, while in the second, the Lagrangian is rotated in RPF and the equations are derived. Thanks to the Lagrangian formulation, the introduction of both gravitational and non-gravitational effects has been easily achieved. Finally, it has been proved that this formulation can be related to the 2 and 3-body problem, essentially removing the other attractors. In order to show the validity of the theoretical models derived, it is necessary to find

119 Conclusions numerical solutions for them, which can be obtained using a calculator and an integration scheme. But this is not sufficient to say that the models are validated, because there is the necessity to have a comparison between what have been computed and a real solution. Since there are no analytical closed form for these differential equations, the results from two already validated tools (SPICE and GMAT) have been used as references, comparing the state vectors of different solar system asteroids. The accuracy level of the model is comparable with GMAT, but SPICE includes too many other perturbation effects that in the long period can introduce an error for the more unstable situations. This has been considered an acceptable starting point for the real scope of this work, i.e. the design and optimization of space trajectories, feasible solutions in very precise models. It is then introduced the multiple shooting strategy, which is method to solve boundary value problems. Its importance is connected with the possibility of designing trajectories which satisfy different constraints. In fact, to reach this goal some new theoretical features have been included in the work, such as the variational equations and the state transition matrix. The latter gives the possibility of an analytic computation of the gradient of the constraint equations, making the numerical integration many times faster. The practical implementation of the multiple shooting involves the definition of the constraints and of the objective function. Though it is possible to give generic formulation for those functions, the example of a real case study illustrates better the potentialities of the methods. Hence, the Lisa Path Finder mission has been chosen to test the numerical tool, due to its permanence in an unstable region and the real necessity of an orbit design for the mission extension. It was then possible to design trajectories which start from the parking orbit of LPF and reach the region of interest with extremely high precision (millimetres).

7.2 Future work

This work presents the evolution of a MSc thesis [11], where the perturbation eﬀects have been introduces and the validation procedure has been carried on. Similarly, here some recommendations for future add-ons could be given. They are addressed according to the three main topics that have characterized the work.

Modelling The high fidelity models presented include N-bodies, their first harmonics for the inhomogeneous gravitational field and the effects of solar radiation pressure. Actually, other contributions can be added to obtain a better physical representation. Firstly the number of harmonics could be increased, including also others of superior order. This would bring benefits in terms of orbit design, especially around a planet, where they influence greatly the spacecraft. Then, other non-gravitational perturbations should be added. One is represented by the relativistic corrections which have important effects on the long term propagation. In the meanwhile, also the Yarkovski effect due to thermal radiation would introduce interesting advantages in the orbit propagation. It is known that this perturbation causes a deviation on the long period due to a temperature variation on the different sides of

120 7.2 Future work the object. Finally, one crucial aspect in the model is given by the manoeuvres, which have been presented as impulsive. However, this is not true for real engines, which need a ﬁnite time of propulsion.

Simulation and validation Although the numerical integration of every differential equation will have errors with respect to reference solution, as seen when comparing with SPICE and GMAT, those can be reduced if the simulation is robust enough. To reach this goal, a statistical number of integrations must be performed and then the varying parameters analysed. Moreover, particular attention must be paid to the so called gravity assist events. When the spacecraft passes near to a planet, the influence of the latter changes considerably the trajectory of the further. This effect should be mapped accurately enough also from the integrator.

Optimization At the end, the utility of a tool for space trajectory optimization is given from the possibility of obtaining converged solutions, their number and the time needed to obtain them. Improving one of them, having the others at the same performance level, would bring many beneﬁts. A solution does not depend only from the initial guesses, but also from the formulation of the problem and even from its interaction with the used algorithms.

121 Conclusions

122 Acknowledgements

Usually, a MSc thesis has mainly an author. However, the characters that composed it are much more, even if sometimes play hidden roles. I would like to express them my deepest gratitude.

I am grateful to my advisor, prof. Francesco Topputo, who gave me the possibility to get involved in the astrodynamics matter; his advices guided me wisely in the understanding of what orbit design means in real life. But more, he managed in transferring his passion to me, showing the beauty of engineering through the exactness of mathematics.

I would like to thank Diogene Dei Tos for his continuous support, in this continent and the other. Taking the inheritance of his thesis helped me to start this work, his enthusiasm and advices encouraged me to ﬁnish it.

Quest’ultimo anno non sarebbe potuto essere così fervido senza la presenza dei miei compagni dell’ILIAS team. Con qualcuno ci si conosceva già, altri si sono aggiunti in corso d’opera. Condividere con loro la passione per lo spazio, e assieme affrontare le difficoltà degli ultimi corsi ha reso magnifico questo ultimo periodo in università.

Ai miei "Shama-amici" che sanno trovare la parola giusta per rendere più lieve ogni situazione: un grazie per ogni volta che mi fate cogliere il piacere semplice di stare insieme. In particolare Beltra, che sa toccare le corde giuste per allontanare stanchezzaed apatia.

Vorrei poi ringraziare i miei fratelli con cui condivido la casa e "il gruppo": il cammino insieme è una delle più belle occasioni di crescita che mi è stata donata. A Robi, amico e guida, per tutte le parole che mi hanno mostrato la via.

Ai miei genitori che mi hanno appoggiato incondizionatamente in questi anni non posso dire che grazie: per aver creduto in me, avermi sostenuto e insegnato ad essere prima ﬁglio e poi uomo. Vorrei ringraziare di cuore la mia sorellina traduttrice, che sa come farmi sentire orgoglioso per ogni cosa che faccio. Ai miei nonni, perché se anche il destino ci ha voluto tenere lontani, so che nel vostro cuore io ci sono e io porto voi nel mio.

Inﬁne, il mio grazie più grande per Veronica, che mi ha saputo accompagnare con amore in questi anni. Per il suo camminare al mio ﬁanco sostenendoci a vicenda e sorridendo insieme alla scoperta di quanta luce c’è.

123 Conclusions

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