DELFT UNIVERSITYOF TECHNOLOGY

FACULTY OF AEROSPACE ENGINEERING

MASTER’STHESIS

Formation Flying in the Sun-Earth/Moon Perturbed Restricted Three-Body Problem

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTEROF SCIENCE IN AEROSPACE ENGINEERING

Committee: Author: Prof. Dr. Ir. P.N.A.M VISSER Ingvar OUT Dr. Ir. J.A.A. van den IJSSEL Ir. B.T.C. ZANDBERGEN

April 18, 2017

Preface

This report describes the results of a Master’s thesis on formation flying in the Sun-Earth/Moon perturbed restricted three-body problem, performed in partial fulfillment of a Master of Science degree in Aerospace Engineering at Delft University of Technology.

Note that all simulations performed for the investigation of formation flying in the restricted three-body problem, are written entirely by the current author, with the exception of the Dormand-Prince numerical integrator, for which the TU Delft Astrodynamics Toolbox (TUDat) is used, comprising a set of C++ libraries written for a variety of astrodynamics applications. TUDat is developed and maintained by the astrodynamics and space missions research group at the faculty of Aerospace Engineering at Delft University of Technology.

This Master’s thesis has been performed in collaboration with Dr. Ir. J. van den IJssel, whose guidance throughout the process of my thesis was key in shaping my research, and whom my sincerest gratitude goes out to for her feedback and support.

iii

Contents

Preface iii

Summary vii

1 Introduction 1

2 The Circular Restricted Three-Body Problem 3 2.1 Equations of motion ...... 3 2.2 Jacobi’s integral ...... 4 2.3 Lagrange libration points ...... 5

3 The Perturbed Restricted Three-Body Problem 7 3.1 Elliptic restricted three-body problem ...... 7 3.1.1 Transformation to a rotating reference frame ...... 7 3.1.2 Normalized equations of motion ...... 8 3.2 Restricted four-body problem ...... 10 3.3 Solar radiation pressure ...... 13

4 The Nominal Halo Orbit 15 4.1 Differential correction method ...... 15 4.1.1 State transition matrix ...... 16 4.1.2 Numerical implementation ...... 17 4.2 Verification ...... 17 4.3 Halo orbits in the Sun-Earth/Moon CRTBP ...... 17 4.4 Eclipse avoidance ...... 18 4.5 Halo orbits in the Sun-Earth/Moon ERTBP ...... 21

5 Formation flying in the CRTBP 25 5.1 Formation configuration ...... 25 5.2 Equitime targeting method ...... 26 5.2.1 Approach ...... 26 5.2.2 Considations for (semi-)inertial formations ...... 27 5.2.3 Results ...... 28 5.3 Tangent targeting method ...... 34 5.3.1 Approach ...... 34 5.3.2 Results ...... 36 5.4 Linear approximation of the results ...... 38 5.4.1 Rotating formations ...... 39 5.4.2 Inertial formations ...... 41 5.5 Precision and integration accuracy ...... 42

6 Formation Flying in the ERTBP 45 6.1 Sensitivity to absolute station keeping ...... 45 6.2 Results ...... 46 6.3 Linear approximation of the results ...... 50 6.3.1 Rotating formations ...... 50 6.3.2 Inertial formations ...... 51 6.3.3 Difference between the CRTBP and ERTBP ...... 51

v CONTENTS

7 Formation Flying in the Restricted Four-Body Problem 57 7.1 Results ...... 57 7.2 Linear approximation of the results ...... 59

8 Formation Flying with Solar Radiation Pressure 63 8.1 Approach ...... 63 8.2 Results ...... 64

9 Conclusions and Recommendations 67

Bibliography 71

A Second Partial Derivatives of the Pseudo-Potential Function 73 A.1 CRTBP ...... 73 A.2 ERTBP ...... 73 A.3 RFBP ...... 74

B Unit Conversions 75 B.1 CRTBP ...... 75 B.2 ERTBP ...... 76

C Constants 79

vi Summary

Formation flying in an orbit about the Sun-Earth L2 point has a number of potential benefits as compared to Earth-orbiting formations, among which are the thermally stable environment for eclipse-free halo orbits, as well as the low force gradients associated with such orbits, allowing for very high relative accuracies to be maintained within a formation. These properties are particularly useful for deep-space observations, for which a formation can be used to create a virtual aperture much larger than is possible with a single conventional satellite. A mission that was planned to employ a two-spacecraft formation in a halo orbit about the Sun-Earth L2 point is the X-ray Evolving Universe Spectroscopy mission (XEUS), consisting of a mirror spacecraft and a detector spacecraft, with a nominal separation of 35 m being equal the focal length. As follows from Marchand and Howell (2005), relative gravitational accelerations associated with small formation separations as for XEUS, might allow for high relative accuracies, on the order of 1 cm, to be maintained using impulsive control every few days. In this study, the relative dynamics for a formation in a halo orbit about the the Sun-Earth L2 point have been investigated , considering the Circular Restricted Three-Body Problem (CRTBP) as well as perturbations due to the Earth’s eccentricity, the presence of the Moon, and Solar Radiation Pressure (SRP). The impact of the formation’s orientation, separation, and required relative accuracy on the time in between corrective maneuvers, or segment time, as well as the ∆V’s required, is investigated for an impulsive relative station keeping strategy. Also, the impact of using an inertially fixed formation, as opposed to a rotating formation is considered, given its relevance to deep-space observation missions such as XEUS.

Firstly, using an integration of the full non-linear system of equations of motion in the unperturbed CRTBP, it was shown that for a XEUS-like mission, with a nominal formation separation of 50 m and occupying a relatively small halo orbit, a relative position accuracy of 1 cm can be achieved with impulsive control for segment times of 0.6-1.7 days and ∆V’s of 0.6-1.4 µm/s, depending on the formation’s position, orientation and type, begin either inertial or rotating. Given the very slowly changing relative acceleration for formations in a halo orbit, the maximum relative position error is approximately proportional to the formation separation distance, as well as the square of the segment time. The ∆V’s are also nearly proportional to the formation separation distance, as well as the segment time. Furthermore, it has been shown that using an inertial formation as opposed to a rotating formation helps to increase the segment time for formations along the Sun-Earth line, by up to a maximum of nearly 8%, and decreases the ∆V by a maximum of 7%. A formation perpendicular to the Sun-Earth line, in the Sun-Earth orbital plane, suffers a decrease in segment time for inertial formations, by up to 18%, and an increase in ∆V by a maximum of 23%. The effects of perturbations to the CRTBP on formation flying have been investigated for the Sun- Earth/Moon system, where the main gravitational perturbation is caused by the elliptical orbit of the Earth about the Sun, described by the Elliptical Restricted Three-Body Problem (ERTBP). It was shown that the differences in relative station keeping between the CRTBP and ERTBP are mainly driven by the Earth’s true anomaly, showing differences ranging from approximately -2.6 % to 2.6 % in terms of segment time and ∆V, where the maximum increase in segment time occurs when the Earth is in apohelion and the maximum decrease when the Earth is in perihelion. Note that an increase in segment time corresponds to a decrease in ∆V, by approximately the same amount. The gravitational perturbation of the Moon has also been in- vestigated, showing a maximum decrease in segment time of up to 0.6 %. The last perturbation covered, is the Solar Radiation Pressure (SRP), which depending on the formation separation and spacecraft properties, can dominate the relative dynamics for a formation near the Sun-Earth L2 point. It was found that for a difference in area-to-mass ratio of 1/100 m2/kg, the relative acceleration due to SRP is already 1000 times larger than the relative gravitational acceleration for a 100 m rotating formation along the Sun-Earth line for a relatively small halo orbit. Even for spacecraft of equal dimensions and identical surface properties, a 1 degree difference in orientation of the surface normal with respect to the incident solar radiation can already cause a relative acceleration of 1µm/s for an area-to-mass ratio of 1/100 m2/kg. Hence, if one aims to achieve high relative accuracies of 1 cm for a 100 m formation, using impulsive control in the presence of

vii CONTENTS

SRP, segment times on the order of a day are only possible if the spacecraft attitudes are actively controlled, such as to cancel the relative accelerations due to SRP. Linear approximations for the maximum relative position error, segment times, and ∆V’s have been de- rived for formations in the CRTBP, ERTBP, as well as under the influence of the Moon’s perturbation and SRP, allowing for a quick and easy way to determine the aforementioned quantities for any point in the Sun- Earth/Moon system. Accuracies better than 1% can be achieved by the linear approximations for formations separations of up to 10,000 km, and segment times of up to 2 days, for the relatively small halo orbit in the Sun-Earth/Moon system considered. Even thought the focus of this study was on impulsive control, one can extend many of the results to the case of continuous control, by treating it as impulsive control in the limit of infinite maneuvers, which applies to the linear approximations in particular, for they become more accurate as the segment time decreases.

viii Chapter 1

Introduction

Most missions flown to date achieve their science objectives by employing (a) spacecraft near one primary celestial body. In these cases, the dynamics can be modeled by considering one main gravitational attractor, where any other influences can be considered small perturbations to the problem. As space missions become more ambitious, science objectives may impose the need of an orbit in a region where multiple celestial bodies exercise equally significant gravitational forces on a spacecraft. The most interesting kind of orbit of the aforementioned type involves flying about a so-called ’libration point’, which is a point where the placement of a spacecraft results in a fixed configuration with respect to two main attracting bodies, or primaries, whose movement is defined by the two-body problem. The first flown mission to utilize such an orbit was the International Sun-Earth Explorer-3 (ISEE-3), which was launched in 1978 and arrived in a region near the libration point located in between the Sun and the Earth approximately three months later. Here, it spent more than three and a half years in a three-dimensional, so-called ’halo’ orbit about the libration point, under the nearly equal influence of the Sun and the Earth’s gravitation. The nearly constant geometry of this orbit with respect to the Sun and the Earth allowed ISEE-3 to continuously observe interactions between solar wind and the Earth’s magnetosphere. The potential benefits of libration point orbits are not limited to missions requiring constant observation of one of the primaries. One other potential benefit of such an orbit comes from the nearly constant ori- entation with respect to the primaries, allowing for relatively easy shielding strategies, which is especially beneficial to missions with stringent thermal stability requirements. This applies to the second libration point in particular, which sees the Sun, the Earth, and the Moon in approximately the same direction at all time (see Section 2.3). Another attractive feature of libration point orbits is the gravitational stability1, given the low force gradients associated with regions near libration points. One mission that takes advantage of these conditions is the Laser Interferometer Space Antenna (LISA) pathfinder, a mission by the European Space Agency (ESA) that was launched in December 2015 and serves to demonstrate technologies needed to de- velop future spaceborne gravitational wave detectors (Rudolph et al., 2016). LISA pathfinder’s very stringent gravitational stability requirements make a libration point orbit a very suitable choice.

Formation flying about the second Sun-Earth L2 libration point will be the main focus of the current study. Orbits about libration points are ideal for maintaining very high relative accuracies within a formation, given the low force gradients in this region, which can be beneficial for deep-space observations. Using an accu- rately controlled formation, it is possible to achieve far higher focal lengths, and a larger ’virtual’ aperture radius, than can be achieved using a single conventional spacecraft. An example of a mission concept em- ploying such a formation near the Sun-Earth L2 point is the Stellar Imager Vision Mission (SI), a mission concept by the Goddard Space Flight Center which is aimed to be realized in the late 2020s. SI is to image stars by employing a formation of 20-30 mirror spacecraft, positioned along a virtual parabolic surface of 100-1000 m in diameter (depending on the angular size of the target to be observed), with a detector space- craft located at a focal lenght’s distance of 1-10 km (Carpenter et al., 2005). Another example of a formation flying mission about the Sun-Earth L2 libration point is the X-ray Evolving Universe Spectroscopy mission (XEUS), which was planned to employ a two-spacecraft formation, a detector and a mirror spacecraft, at a 35 m nominal separation, whose goal was to perform deep space x-ray observations in order to investigate black-hole formation and galaxy evolution on cosmic timescales. As follows from Howell and Marchand (2005), the difference in formation separation distance for the two aforementioned missions could result in very different formation control strategy requirements. Namely, it is

1The term gravitational stability is used here to indicate a nearly constant gravitational potential, and should not be confused with orbital stability, which collinear libration point orbits generally do not possess.

1 CHAPTER 1. INTRODUCTION shown by Howell and Marchand (2005) that for small formation separations of 10-100 m, the relative posi- tion could be controlled to within 1 cm accuracy with only one impulsive maneuver every day or two, whereas larger formation sizes on the order of 1000 km and above, will likely require continuous control to achieve the same relative accuracy. Apart from accuracy requirements, the control strategy for formation flying about the Sun-Earth L2 point could be dictated by minimum thrust requirements. Namely, continuous control for closely separated formations near the Sun-Earth L2 point is associated with potentially prohibitively small thrust requirements. This follows from Marchand and Howell (2005), where it is mentioned that for a for- mation separation distance of 10 m, and a spacecraft mass of 700 kg, continuous control requires thrust levels on the order of 1 nN, which likely eliminates it as an option, with current minimum thrust levels demonstrated in-flight being on the order of 1µN, as is the case for the Gaia mission (Milligan et al., 2016) as well as LISA pathfinder (Rudolph et al., 2016). Using impulsive control, hence allowing the formation to drift in between maneuvers, will lead to less stringent minimum thrust requirements, as might be necessary according to the previous example. In this case, the minimum deliverable impulse bit will impose a limit on the achievable accuracy in terms of relative position for a given formation in orbit about the Sun-Earth L2 point. This study will mainly focus on relatively small separation distances comparable to that of XEUS, in the range of 10-100 m. It should be mentioned that XEUS is currently not planned to be flown. However, given the potential of this type of mission for deep-space observations, it is an interesting study case that might benefit any similar future missions to be flown, and is hence considered a suitable reference mission. Given the small separation distance for XEUS, and based on the previous discussion, this study investigates the use of impulsive control for formations in orbit about the Sun-Earth L2 point, mainly in terms of ∆V requirements and time in between corrections, depending on the formation configuration. Note that a lot of research in the area of formation flying dynamics and control near the Sun-Earth libration points has already been performed by, among others, K.C. Howell, B. Marchand, A.M. Segerman, M.F. Zedd and H.J. Pernicka. This study aims to contribute to the existing body of knowledge, firstly by ”performing an extensive investigation of the formation control’s dependency on different formation configurations in orbit about the Sun-Earth L2 point”, and secondly by ”investigating the effect of perturbations on the formation dynamics, hence control requirements”. This sec- ond objective in particular, could yield useful insights, given that most research on formation flying about the Sun-Earth L2 point was performed using either the Circular Restricted Three-Body Problem (CRTBP), or a full ephemerides model, without distinguishing the contributions due to the Earth’s eccentricity, the Moon, and Solar Radiation Pressure (SRP).

This report will be structured as follows. Firstly, Chapter 2 will present a discussion on the CRTBP, forming the basis of dynamics about libration points. Next, Chapter 3 presents modifications to the CRTBP to account for perturbations caused by the eccentricity of the Earth, the Moon’s orbit about the Earth, and SRP, the former two of which are described by the Elliptic Restricted Three-Body Problem (ERTBP) and the Restricted Four-Body Problem (RFBP) respectively. Then, Chapter 4 presents a method for obtaining so-called ’halo’ orbits, being three-dimensional orbits of fixed dimensions about a libration point, which is the preferred nominal orbit for reasons mentioned later on. Chapter 5 presents the implementation of impulsive control strategies and investigates the results for formation flying in the CRTBP in terms of ∆V requirements and time in between maneuvers. Similar investigations for formation flying in the ERTBP and RFBP are presented in Chapters 6 and 7 repectively. Finally, the effect of SRP on the formation dynamics is investigated and presented in Chapter 8, after which Chapter 9 presents the conclusions and recommendations resulting from this study.

2 Chapter 2

The Circular Restricted Three-Body Prob- lem

The CRTBP describes the dynamics of a body of negligible mass under the influence of two massive bodies, or primaries, moving in circular orbits about each other. The CRTBP contains an interesting set of equilibrium points, existing in a rotating reference frame that is synchronous to the rotation of the primaries. Hence, placing a body of infinitesimal mass in one of these equilibrium points would result in a constant orientation with respect to the primaries, which can be beneficial to achieving certain science objectives, as mentioned in Chapter 1. This chapter provides a brief description of the CRTBP based on Wakker (2010). Firstly, Section 2.1 presents the equations of motion for the CRTBP. Then, Section 2.2 discusses Jacobi’s integral, which is an integral of motion for the CRTBP. Finally, the solution set of equilibrium points in the CRTBP, called the Lagrange libration points, are discussed in Section 2.3.

2.1 Equations of motion

The equation of motion for the CRTBP can be written with respect to an inertial reference frame by simply considering Newton’s gravitational law. This is given by Equation 2.1, where r is the position vector of the third body with respect to the barycenter, r1 and r2 are the relative position vectors from the primaries to the third body, m1 and m2 are the masses of the primaries, and G is the universal gravitational constant. The value of G, and all other relevant constants used in this study, are presented in Appendix C for reference.

2 d r m1 m2 = −G 3 r1 − G 3 r2 (2.1) dt r1 r2 A more useful expression is obtained by transforming the equations of motion to a rotating, normalized reference frame, such as the xyz-frame shown in Figure 2.1. Here, the origin coincides with the barycenter of the system, the x-axis is aligned with the line connecting the primaries P1 and P2, the xy-plane lies in the orbital plane, and the right-hand rule completes the coordinate system. In order to normalize the equations of motion, new units for mass, distance, and time are introduced. (m1 + m2) is taken as a unit of mass, such that the masses can be expressed as m1 = 1 − µ and m2 = µ, where µ is a mass parameter that satisfies 0 < µ ≤ 1/2, such that m1 corresponds to the mass of the larger primary. Note that literature is inconsistent in placement of the larger primary either to the left or the right of the origin, though this study follows the former approach, placing it on the left. Furthermore, a unit of length is taken to be equal to the distance between the primaries. Consequently, the position vectors with respect to the primaries can be written as r1 = (x + µ, y, z) and r2 = (x − 1 + µ, y, z) where x, y, and z are the coordinates of the third body. Finally, a unit of time is taken as (df/dt)−1, where df/dt is the orbital angular velocity of the primaries.

3 CHAPTER 2. THE CIRCULAR RESTRICTED THREE-BODY PROBLEM

P y 3

r2 r 1 r x

P P1 2 z

Figure 2.1: Rotating reference frame for the CRTBP with its origin at the barycenter. P1, P2, and P3 corre- spond to the larger primary, the smaller primary, and the third body respectively.

The resulting equations of motion in the rotating, normalized reference frame become: 1 − µ µ x¨ − 2y ˙ =x − 3 (x + µ) − 3 (x + µ − 1) r1 r2 1 − µ µ y¨ + 2x ˙ =y − 3 y − 3 y (2.2) r1 r2 1 − µ µ z¨ = − 3 z − 3 z r1 r2 A pseudo-potential function U can be defined as: 1 1 − µ µ U = (x2 + y2) + + (2.3) 2 r1 r2 such that Equation 2.2 can be written as: ∂U x¨ − 2y ˙ = ∂x ∂U y¨ + 2x ˙ = (2.4) ∂y ∂U z¨ = ∂z U is not an explicit function of time. Hence, it represents a conservative force field, that accounts for the gravitational and centrifugal force. The full derivation to the equations obtained in this section is presented in Wakker (2010). Also, a derivation of the equations of motion for the ERTBP is presented in Section 3.1. Naturally, the results for the ERTBP reduce to the CRTBP in the case of zero eccentricity.

2.2 Jacobi’s integral

There exists a single integral of motion for the CRTBP, called Jacobi’s integral. This integral can be derived by first multiplying the equations of motion, given by Equation 2.4, by x˙, y˙ and z˙ respectively, and then taking their sum, which yields: ∂U ∂U ∂U x˙x¨ +y ˙y¨ +z ˙z¨ =x ˙ +y ˙ +z ˙ (2.5) ∂x ∂y ∂z

4 2.3. LAGRANGE LIBRATION POINTS

L4 y 60°

P1 60° P2 L3 L1 L2 x

60° γ3 γ1 γ2

60°

L5

Figure 2.2: Positions of the Lagrange libration points.

Since U is only a function of spatial coordinates, the right-hand side of Equation 2.5 can simply be written as dU/dt. Now, integrating Equation 2.5 yields:

x˙ 2 +y ˙2 +z ˙2 = 2U − C (2.6) or, V 2 = 2U − C (2.7) Equation 2.7 is known as Jacobi’s integral, and it provides a relation between a body’s velocity and its position in the CRTBP. The integration constant C is known as Jacobi’s constant, which is determined by the position and velocity of the third body at a certain time. One possible application of Jacobi’s constant, which is also used in the current study, is to use it as a means of checking the accuracy of a numerical orbit propagator in the CRTBP.

2.3 Lagrange libration points

An interesting set of solutions that can be derived from Equation 2.2, exists in the form of five equilibrium points (see Wakker (2010) for the full derivation). These equilibrium point are called Lagrange libration points, and constitute points of zero velocity and acceleration in the rotating reference frame. Figure 2.2 shows the position of the libration points in the CRTBP. The set of libration points is contained within the xy-plane, consisting of three collinear libration points located along the x-axis, and two equilateral libration points, forming equilateral triangles with the primaries. The normalized distances of the collinear libration points with respect to the primaries are expressed as γ1, γ2, and γ3, and they depend on the mass parameter µ of the system. Since the current study is concerned with the L2 libration point in the Sun-Earth system, the rest of the discussion will be limited to the collinear libration points.

A first-order stability analysis as presented in, for instance, Szebehely (1967, Chap. 5), shows that the collinear libration points are generally unstable, such that a massless body placed in the vicinity of one of these points will move unboundedly far over time. However, the initial conditions can be chosen such that the motion about the collinear libration points in the xy-plane becomes a pure oscillation. Doing so poses the following constraints on the initial conditions: s x˙ 0 = y0 v (2.8) y˙0 = − svx0

5 CHAPTER 2. THE CIRCULAR RESTRICTED THREE-BODY PROBLEM where s and v are constants characteristic of the system considered. For the Sun-Earth/Moon1 system, the corresponding values are s = 2.087 and v = 3.229 (Wakker, 2010). The aforementioned analysis also shows that the motion in the z-direction is a pure oscillation that is completely decoupled from the motion in the xy-plane. In general, the periods in the xy-plane and the z- direction are not equal, resulting in a three-dimensional orbit about the collinear libration point, called a Lissajous orbit. For the collinear libration points in the Sun-Earth/Moon system, the difference in periods is quite small, such that the orbit can be viewed as a slowly changing elliptical path. Note that the aforementioned results are based on linearized theory, thus one can only speak of infinitesi- mal stability. In reality, second-order effects, as well as perturbations to the CRTBP, will influence the stability and motion about the collinear libration points. In fact, it can be shown that for sufficiently large amplitudes, second-order effects may induce coupling between the motion in the xy-plane and the z-direction, giving rise to a three-dimensional orbit of fixed geometry and size about one of the collinear libration points, a so-called halo orbit. Halo orbits come in two classes: Class 1 halo orbits have their apogee above the ecliptic, and Class 2 halo orbits have their apogee below the ecliptic. In the Sun-Earth/Moon system, halo orbits are found to be possible for minimum amplitudes of approximately 215,000 km, and 680,000 km in the x- and y-directions respectively (see Section 4.3). Note that Chapter 4 is dedicated to finding such halo orbits.

1In the CRTBP, the Sun-Earth/Moon system refers to an approximation of the three-body system, where the Earth-Moon system is modelled as a point mass, accounting for the second primary in the CRTBP.

6 Chapter 3

The Perturbed Restricted Three-Body Prob- lem

This chapter describes the perturbed restricted three-body problem, which accounts for perturbations to the CRTBP due to, for instance, the ellipticity of the primaries, gravitational perturbations by other massive bodies, and SRP. One of the main focuses of this study is to determine the effects of these perturbations on the motion and control effort required for a formation in the vicinity of the Sun-Earth L2 point. This chapter presents models for these perturbations that will later be implemented in a formation flying simulation. A modification to the CRTBP, where a non-zero eccentricity is allowed, is referred to as the ERTBP and is discussed in Section 3.1, where the corresponding equations of motion are derived. Furthermore, Section 3.2 presents the equations of motion for the so-called RFBP, a model which accounts for an additional massive body to the Restricted Three-Body Problem (RTBP)1. This section also presents a short discussion on the restricted five-body problem, mainly to justify not considering the effect of other major planets for formation flying in the Sun-Earth/Moon system. Finally, Section 3.3 discusses the effect of SRP.

3.1 Elliptic restricted three-body problem

This section present the full derivation of the equations of motion for the ERTBP, following a procedure sim- ilar to the derivation presented in Wakker (2010) for the CRTBP, though taking a more general approach by allowing for elliptical orbits of the primaries. Equations of motion for the ERTBP that are formally identical to Equation 2.4 can be obtained, with differences being contained in the pseudo-potential function. Note that a definition of the perturbation due to the ellipticity of the primaries is not completely straightforward, for differences in the normalized units of the CRTBP and ERTBP cause a dimensional orbit to have different representations in either problem, a further discussion of which is presented in Section 4.5. The transforma- tions to a rotating and a normalized reference frame in the ERTBP are presented in Sections 3.1.1 and 3.1.2 respectively.

3.1.1 Transformation to a rotating reference frame In the following derivation, take δ/δt and d/dt to be the time derivatives in the rotating and inertial reference frame respectively. Given an angular velocity df/dt of the rotating reference frame about the z-axis, the first derivative of the position vector can be transformed using: dr δr df = + × r (3.1) dt δt dt The same applies to the time derivative of the velocity vector, so that: d δr  δ2r df δr = + × (3.2) dt δt δt2 dt δt Differentiation of Equation 3.1 gives: d2r d δr  d2f df dr = + × r + × (3.3) dt2 dt δt dt2 dt dt

1RTBP is used here as a collective term to refer to either the CRTBP or ERTBP.

7 CHAPTER 3. THE PERTURBED RESTRICTED THREE-BODY PROBLEM

Combining Equations 2.1 and 3.1 through 3.3, yields the acceleration in the rotating reference frame: 2   2 δ r m1 m2 df δr df df d f 2 = −G 3 r1 + 3 r2 − 2 × − × × r − 2 × r (3.4) δt r1 r2 dt δt dt dt dt The second, third, and fourth term on the right-hand side of Equation 3.4 represent the Coriolis acceleration, the centrifugal acceleration, and the acceleration due to the non-uniform rotating of the reference frame respectively. Note that the remainder of this report will only be concerned with time derivatives in the rotating reference frame, which will be denoted simply by d/dt from this point on.

3.1.2 Normalized equations of motion The equations of motion in the ERTBP can be normalized by using normalized units for distance, time, and mass, similar to those used in the CRTBP, defined by: a(1 − e2) rˆ ≡ 1 + e cos f df −1 (3.5) tˆ≡ dt

mˆ ≡ m1 + m2 where a, e, and f are the semi-major axis, the eccentricity, and the true anomaly of the primaries’ orbits. Contrary to the CRTBP, the normalized units for distance and time are not constant in the ERTBP, due to the eccentricity of the primaries. Hence, for the ERTBP, the resulting reference frame is pulsating and non- uniformly rotating. Denoting the normalized variables by an asterisk, one can write: r =r ˆr∗; t = ttˆ ∗; m =mm ˆ ∗ (3.6) Note that the normalized time t∗ progresses at the same rate as the true anomaly, and is equal to the true anomaly if the time is initialized when the primaries are in pericenter2. The first and second derivative of the dimensional and normalized time are related through: d dt∗ d df d d2 d2f d df 2 d2 = = ; = + (3.7) dt dt dt∗ dt dt∗ dt2 dt2 dt∗ dt dt∗2 Now, using Equations 3.5 through 3.7, each terms in Equation 3.4 can be rewritten: d2r d  dr∗ drˆ  d2r∗ drˆ dr∗ d2rˆ = rˆ + r∗ =r ˆ + 2 + r∗ dt2 dt dt dt dt2 dt dt dt2 ! d2f dr∗ df 2 d2r∗ drˆ df dr∗ d2rˆ =r ˆ + + 2 + r∗ dt2 dt∗ dt dt∗2 dt dt dt∗ dt2 df dr df  dr∗ drˆ  2 × = 2 e × rˆ + r∗ dt dt dt z dt dt df  df dr∗ drˆ  = 2 e × rˆ + r∗ (3.8) dt z dt dt∗ dt df df df 2 × × r =r ˆ e × e × r∗ dt dt dt z z d2f d2f × r =r ˆ e × r∗ dt2 dt2 z     m1 m2 G(m1 + m2) 1 − µ ∗ µ ∗ G 3 r1 + 3 r2 = 2 ∗3 r1 + ∗3 r2 r1 r2 rˆ r1 r2 −1 2  df  ∗ ∗ Defining the reciprocal of the instantaneous angular velocity as a unit of time, such that t = dt t , with t being the normalized −1 R t  df  R t∗  df   df  ∗ ∗ time, and assuming the initial true anomaly to be zero, yields: f = 0 dt dt = 0 dt dt dt = t

8 3.1. ELLIPTIC RESTRICTED THREE-BODY PROBLEM

where again, µ is the mass parameter of the primaries, and ez is the unit vector normal to the orbital plane. Substituting Equation 3.8 into Equation 3.4 and collecting some terms, yields:

 d2f drˆ df  dr∗  d2f drˆ df  df 2 d2r∗ df 2 dr∗ rˆ + 2 + rˆ + 2 e × r∗ +r ˆ + 2ˆr e × = dt2 dt dt dt∗ dt2 dt dt z dt dt∗2 dt z dt∗ (3.9)   2  2 m1 + m2 1 − µ ∗ µ ∗ d rˆ ∗ df ∗ = −G 2 ∗3 r1 + ∗3 r2 − 2 r − rˆ ez × ez × r rˆ r1 r2 dt dt Now, Equation 3.9 can be rewritten by using the following properties of the two-body problem: i) Conservation of angular momentum: d  df  rˆ2 = 0 (3.10) dt dt or, d2f drˆ df rˆ + 2 = 0 (3.11) dt2 dt dt ii) The equation of motion: d2rˆ df 2 G(m + m ) − rˆ = − 1 2 (3.12) dt2 dt rˆ2

iii) The angular momentum integral:

 df 2 rˆ2 = a(1 − e2)G(m + m ) (3.13) dt 1 2

Substituting Equations 3.5, and 3.11 - 3.13 into Equation 3.9, and noting the x1 = x + µ and x2 = x − 1 + µ, the equations of motion for the ERTBP can be written as:   −1 1 − µ µ x¨ − 2y ˙ =(1 + e cos f) x − 3 (µ + x) − 3 (x − 1 + µ) r1 r2   −1 1 − µ µ y¨ + 2x ˙ =(1 + e cos f) y − 3 y − 3 y (3.14) r1 r2   −1 1 − µ µ z¨ + z =(1 + e cos f) z − 3 z − 3 z r1 r2 where x, y and z are the normalized coordinates. Note that for the asterisk notation is dropped for simplicity, since the remainder of this report will present equations in normalized form, unless mentioned otherwise. A more concise notation can be obtained by defining the pseudo-potential function:

1 1 − µ µ  ω = (1 + e cos f)−1 (x2 + y2 + z2) + + (3.15) 2 r1 r2 such that, ∂ω x¨ − 2y ˙ = ∂x ∂ω y¨ + 2x ˙ = (3.16) ∂y ∂ω z¨ + z = ∂z

9 CHAPTER 3. THE PERTURBED RESTRICTED THREE-BODY PROBLEM

P y 3

r rM 1 r r E r2 Moon Sun x ΔrM P Δr 2 Earth E P1 z

Figure 3.1: Configuration for the RFBP model.

Note that in some references an alternative notation is used, such as to obtain a formally identical notation for the ERTBP as the CRTBP, given by:

∂ω0 x¨ − 2y ˙ = ∂x ∂ω0 y¨ + 2x ˙ = (3.17) ∂y ∂ω0 z¨ = ∂z with 1 1 − µ µ  ω0 = (1 + e cos f)−1 (x2 + y2 − e cos fz2) + + (3.18) 2 r1 r2 Comparing Equations 2.2 and 3.14, the most pronounced difference between the normalized equations of motion in the CRTBP and ERTBP is the term (1 + e cos f)−1, which counterintuitively decreases the non- dimensional accelerations when the primaries are in pericenter. Note however, that this is a result of the normalization, whereas the dimensional accelerations are actually larger for the ERTBP when the primaries are in pericenter, as is shown in Section 6.3.3. This increase in dimensional acceleration is a consequence of the smaller unit of distance in the ERTBP as compared to the CRTBP when the primaries are in pericenter, hence closer proximity to the primaries for the same non-dimensional position, yielding larger gravitational accelerations. Furthermore, we find an extra z-term for the acceleration in z-direction, which is there only to account for the pulsating axes as introduced in the ERTBP. In the case of zero eccentricity, Equations 3.15 and 3.16 naturally reduce to equations 2.3 and 2.4, representing the CRTBP. It can readily be deduced that the collinear libration points have the same non- dimensional coordinates in the CRTBP and ERTBP. Note however, that their dimensional coordinates are not the same, given the difference in a unit of distance, yielding pulsating dimensional coordinates for the libration points in the ERTBP. Conversions from dimensional to non-dimensional units, and vice versa, are presented in Appendix B for both the CRTBP and ERTBP, where the latter generally requires slightly more effort, given the true-anomaly dependence of the normalized units.

3.2 Restricted four-body problem

The RFBP accounts for another massive body in addition to the RTBP. Specifically, this section considers the effect of the Moon in the Sun-Earth/Moon system. Whereas our implementation of the RTBP assumes the Earth and the Moon to coincide, modelling the RFBP requires Equation 3.14 to be adjusted to account for the distance of the Earth and the Moon from their barycenter, as shown in Figure 3.1. The resulting equations of

10 3.2. RESTRICTED FOUR-BODY PROBLEM motion in the Sun-Earth/Moon RFBP are given by:   −1 1 − µ µ(1 − µ¯) µµ¯ x¨ − 2y ˙ =(1 + e cos f) x − 3 (µ + x) − 3 (x − xE) − 3 (x − xM ) r1 rE rM   −1 1 − µ µ(1 − µ¯) µµ¯ y¨ + 2x ˙ =(1 + e cos f) y − 3 y − 3 (y − yE) − 3 (y − yM ) (3.19) r1 rE rM   −1 1 − µ µ(1 − µ¯) µµ¯ z¨ + z =(1 + e cos f) z − 3 z − 3 (z − zE) − 3 (z − zM ) r1 rE rM where xE and xM are the Earth and the Moon’s position along the x-axis, rE and rM are position vectors with respect to the Earth and the Moon, and µ¯ is the Moon-Earth mass parameter, defined by: m µ¯ = M (3.20) mE + mM where mE and mM are the mass of the Earth and the Moon. Note that in the Sun-Earth/Moon system, the mass parameter µ is given by: m + m µ = E M (3.21) mS + mE + mM To keep our notation consistent with the formulation of the ERTBP, one could introduce a pseudo-potential function for the RFBP, given by:

1 1 − µ µ(1 − µ¯) µµ¯  F = (1 + e cos f)−1 (x2 + y2 + z2) + + + (3.22) 2 r1 rE rM such that, ∂F x¨ − 2y ˙ = ∂x ∂F y¨ + 2x ˙ = (3.23) ∂y ∂F z¨ + z = ∂z The implementation of the RFBP presented in this section, is based on the assumption of a bi-elliptical model, where the Earth and the Moon are assumed to follow elliptical orbits about each other, and their barycenter moves in an elliptical orbit about the Sun. As such, the conditions associated with the ERTBP, Equations 3.10 through 3.13 in particular, are assumed to be valid still. In reality, the Sun (and other planets) will disrupt the ellipticity of the Earth and the Moon’s orbits about each other. As such, Equations 3.10 through 3.13 are no longer exactly satisfied. However, a bi-elliptical model has been implemented for the Earth-Moon L1 libration point by Ghorbani and Assadian (2013), and verified against a full ephemerides model, showing no significant differences. Although the current study does not provide a similar verification for the Sun-Earth L2 libration point, it is assumed to work equally well for this purpose. The validity of this assumption is strengthened by the fact that the eccentricity of the Moon is subject to large changes because of solar per- turbations3, whereas the eccentricity of the Sun-Earth orbit is far more constant. As follows from Ghorbani and Assadian (2013) and Segerman and Zedd (2006), the eccentricity of the primaries is the largest grav- itational perturbation to the CRTBP for both the Earth-Moon and Sun-Earth libration points. Therefore, if the bi-elliptical model gives good results for the Earth-Moon libration points, it can be expected to give good results for the Sun-Earth libration points.

3Whereas the Moon’s mean eccentricity is equal to 0.0549, the instantaneous eccentricity can vary by up to approximately 0.04 over the course of a synodic month. The Moon’s instantaneous eccentricity is shown to range from 0.0266 to 0.0762 from 2008 through 2010 in Espenak and Meeus (2009)

11 CHAPTER 3. THE PERTURBED RESTRICTED THREE-BODY PROBLEM

P y 3

r2 r1 r r5 x r r51 52 P2+P4 P1 r2

z P5

Figure 3.2: The restricted five-body problem configuration.

Restricted five-body problem In addition to the RFBP, one could consider the restricted five-body problem, including additional massive bodies, for instance, to account for other major planets in the solar system. Figure 3.2 is a representation of the five-body problem, showing a fifth body acting on the satellite as well as the primaries, where for simplicity the second and fourth body (corresponding to the Earth and the Moon for the Sun-Earth/Moon system) are assumed to coincide again. The dimensional perturbing acceleration of the fifth body on a satellite with respect to the primaries’ barycenter is given by:

 2    d r r5 r51 r52 2 = −Gm5 3 − (1 − µ) 3 − µ 3 (3.24) dt 5bp r5 r51 r52 where m5 is the mass of the fifth body. The first term on the right-hand side is the direct effect of the fifth body acting on the satellite, and the second and third term account for the indirect effect, being an acceleration of the primaries’ barycenter due to the fifth body. Although the indirect effect will not instantly affect the relative dynamics of a formation, it will do so over time, as it influences the orbits of the primaries. Note that this fifth body interaction with the primaries causes the bi-elliptical assumption for the Sun-Earth/Moon system to becomes less valid, however, to a much lesser extent than the effect of the Sun’s gravity gradients near the Earth-Moon system. Given the presumably small effect of fifth body perturbations due to major planets in the solar system, this study will not consider them. This is shown by Campagnola et al. (2008) for the Sun-Mercury system, where orbits obtained in the ERTBP very closely corresponds to orbits obtained in a full ephemerides model, indicating the insignificance of fifth body perturbations. This study assumes a similar result to apply to the Sun-Earth/Moon system, which can be justified by performing a quick check of the relative acceleration caused by Venus, presumably the biggest perturber on relative dynamics for a formation near the Sun-Earth L2 point. From Equation 2.2 (and Appendix B), it follows that a displacement of 10 m along the x-axis in the −12 2 Sun-Earth L2 libration point causes a relative acceleration of approximately 3.5 × 10 m/s in the CRTBP. In comparison, the relative acceleration caused by Venus when it is at its closest position to the Earth, as- suming it is aligned with the x-axis, is approximately 1.2 × 10−16 m/s2, nearly a factor 30, 000 smaller, hence considered negligible.

Note that a full ephemerides model would allow for a more accurate simulation of formation dynamics in the RFBP as well as the restricted-five body problem, given that it uses the actual positions of all relevant massive bodies, rather than depending on imperfect assumptions such as the bi-elliptical assumption. However, this would not allow for a means of distinguishing individual contributions from different perturbations, which is one of the main objectives of the current study, hence we will not employ such a full ephemerides model.

12 3.3. SOLAR RADIATION PRESSURE

3.3 Solar radiation pressure

The perturbing force due to SRP depends not only on the distance from the Sun and the sunlight’s angle of incidence, but is also dependent on the type of surface, which dictates the apportionment of forces due to reflection, absorption, and re-emission of sunlight, as well as the fraction of diffuse reflection as opposed to specular reflection. This determines the magnitude as well as the orientation of the resulting solar radiation force. Similar to Howell and Marchand (2005), this study assumes the solar radiation force to be normal to the surface, in which case the dimensional SRP acceleration can be written as:

d2r  kS A AU 2  0 2 ˆ 2 = 2 cos γη (3.25) dt srp msc r1 where k is a reflectance factor, being equal to 1 for a perfect absorber and equal to 2 for a perfect reflector, S0 is the solar energy flux at the mean Sun-Earth distance AU, c is the speed of light in a vacuum, A is the spacecraft’s effective cross-sectional area, ms is the spacecraft mass, γ is the angle of incidence of incoming solar radiation, and ηˆ is the surface normal, directed away from the Sun. Note that a more complete SRP model is presented by McInnes (1999, Equation 2.57a-b), where the de- viation of the SRP force’s orientation from the surface normal due to absorption is accounted for. This model shows that for a non-perfectly reflecting solar sail, small angles of incident solar radiation of up to 20 degrees keep the SRP force’s orientation within 2 degrees of the surface normal (Figure 2.10 McInnes, 1999). A deep space observation mission will likely require shielding from the Sun, presumably with a surface normal that is close to the incoming solar radiation. For example, thermal stability requirements for XEUS constrain the maximum deviation of the sunshields to within approximately 15 degrees of the incident solar radiation, as follows from Bavdaz et al. (2005). Do note that the model by McInnes (1999) applies to a solar sail with a relatively high reflectivity coefficient, where 88% of the incoming solar radiation is reflected. Other types of surfaces, solar panels in particular, might have far lower reflectivity coefficients, which could cause the SRP force’s orientation to deviate from the surface normal by more than the aforementioned 2 degrees, in which case a more complex SRP might be necessary.

In order to obtain the perturbing SRP acceleration in the normalized reference frame defined in Section 3.1, 2 ∗ it follows from Equation 3.5 that one has to divide through rˆ(df/dt) . Furthermore, using r1 =rr ˆ 1, where the asterisk again denotes non-dimensional units, one obtains:

d2r∗  1 kS A AU 2  0 2 ˆ ∗2 = 3 2 ∗2 cos γη (3.26) dt srp rˆ (df/dt) msc r1

Now, substituting Equations 3.5 and 3.13 into Equation 3.26, the SRP perturbation can be written as:

d2r∗  1 kS A AU 2  0 2 ˆ ∗2 = ∗2 cos γη (3.27) dt srp (1 + e cos f)G(m1 + m2) msc r1

13

Chapter 4

The Nominal Halo Orbit

Halo orbits are three-dimensional orbits of fixed geometry, about one of the collinear libration points. Halo orbits were first proposed by Farquhar (1966), as a means to maintain a constant line of communication with the far side of the Moon, using an orbital amplitude large enough such that the line of sight with the Earth is never occulted by the Moon. Similarly, one could place a satellite in a halo orbit about the Sun-Earth L2 point, again with an amplitude large enough to avoid any occultation of the Sun by the Earth. The stable thermal environment, as well as the low force gradients associated with such orbits, are ideal conditions for a deep-space observation mission employing a formation. This chapter closely follows the methods presented by Breakwell and Brown (1979) and Howell (1984) to find periodic halo orbits near the Sun-Earth L2 point. The result of this chapter is a selected halo orbit, in the CRTBP as well as the ERTBP, for which the investigation of formation flying is to be performed. Firstly, the differential correction method used for finding perdiodic halo orbits is presented in Section 4.1. Then, the implementation of this method, as performed in this study, is verified against Howell (1984) in Section 4.2, after which halo orbits about the Sun-Earth L2 point in the CRTBP are presented in Section 4.3. Furthermore, Section 4.4 briefly discusses requirements for eclipse avoidance. Finally, Section 4.5 provides a discussion on the selection of a nominal orbit in the ERTBP.

4.1 Differential correction method

The differential correction method presented here is based on Breakwell and Brown (1979) and Howell (1984). In order to find periodic halo orbits in the CRTBP, one can make use of the invariance of the system given by Equation 2.2 under a transformation from y to −y, and t to −t. Given this property, an orbit that crosses the xz-plane perpendicularly is symmetric with respect to this plane. Therefore, if a second perpendicular crossing to the xz-plane can be found at some later time T/2, the orbit is periodic with period T and of fixed geometry, i.e. a halo orbit. Hence, for an initial state:

T X0 = [x0, 0, z0, 0, y˙0, 0] (4.1) the sufficient condition for having a periodic halo orbit becomes: X(T/2) = [x, 0, z, 0, y,˙ 0]T (4.2) Given an initial state in the form of Equation 4.1, that does not belong to a halo orbit but is sufficiently close to one, Howell (1984) uses the iterative approach of differential corrections to adjust the initial state until a perpendicular crossing of the xz-plane at T/2 is found, and hence Equation 4.2 is satisfied. The required correction to the initial state, δX0, can be calculated from: ∂X(T/2) δX(T/2) = Φ(T/2, 0)δX + δ(T/2) (4.3) 0 ∂t where δX(T/2) is the deviation from the desired final state, Φ(T/2, 0) is the State Transition Matrix (STM), which is a linear mapping of the initial state onto the state at T/2, and T/2 is defined as the time when the orbit crosses the xz-plane again. Note that T/2 will change as the initial conditions are corrected, which is the reason for the inclusion of the second term on the right-hand side of Equation 4.3. Now, given that T/2 corresponds to a crossing of the xz-plane, one can write: ∂y(T/2) δy(T/2) = 0 = Φ δx + Φ δz + Φ δy˙ + δ(T/2) (4.4) 21 0 23 0 25 0 ∂t

15 CHAPTER 4. THE NOMINAL HALO ORBIT or, Φ δx + Φ δz + Φ δy˙ δ(T/2) = − 21 0 23 0 25 0 (4.5) y˙T/2 Using Equations 4.3 and 4.5, the required correction to the initial state, that yields a perpendicular crossing at T/2, can be found by keeping one of the non-zero initial conditions in Equation 4.1 fixed. For the case that x0 is held fixed, Equations 4.3 and 4.5 yield corrections to δz0 and δy˙0 for a desired change in δx˙ T/2 and δz˙T/2, following from:          δx˙ T/2 Φ43 Φ45 1 x¨T/2 δz0 = − (Φ23 Φ25) (4.6) δz˙T/2 Φ63 Φ65 y˙T/2 z¨T/2 δy˙0 or,       −1   δz0 Φ43 Φ45 1 x¨T/2 δx˙ T/2 = − (Φ23 Φ25) (4.7) δy˙0 Φ63 Φ65 y˙T/2 z¨T/2 δz˙T/2 Note that Equation 4.7 is only exact for a perfectly linear system, or infinitesimally small corrections. For the case of the CRTBP, one has to perform multiple iterations, or differential corrections, until a solution is found that satisfies Equations 4.1 and 4.2, to within a certain tolerance, where δx˙ T/2 and δz˙T/2 are close enough to zero.

4.1.1 State transition matrix The STM is a linear mapping between two states and hence calls for the linearized equations of motion, given by: X˙ (t) = A(t)X(t) (4.8) where X(t) = [x(t), y(t), z(t), x˙(t), y˙(t), z˙(t)]T is the state vector, and A(t) is a matrix containing the partial derivatives of the equations of motion, which can always be formed as long as the equations of motion are continuously differentiable. For the CRTBP, inspection of Equation 2.4 shows that A(t) is given by:  0 0 0 1 0 0  0 0 0 0 1 0    0 0 0 0 0 1 A(t) =   (4.9) Uxx Uxy Uxz 0 2 0   Uyx Uyy Uyz −2 0 0 Uzx Uyz Uzz 0 0 0

Here, the second derivatives, Ukl, can be obtained from Equation 2.3, and are given in Appendix A for reference. Now, the general solution to Equation 4.8 can be written as:

X(t) = Φ(t, t0)X(t0) (4.10) where Φ(t, t0) is a linear mapping of the state X(t0) onto X(t). Substituting Equation 4.10 into Equation 4.8 yields an equation for the derivative of the STM:

Φ(˙ t, t0) = A(t)Φ(t, t0) (4.11) which, along with the initial condition, Φ(t0, t0) = I6 (4.12) can be used to propagate the STM. If the time interval is small compared to the orbital period, A(t) can be approximated by A(t0). Note that the STM, along with the equations of motion, yields a system of 42 equations to be simultaneously integrated.

Note that the STM for one full orbit can be used to investigate the orbital stability, where the periodicity requirement for the eigenvalues of the STM is to have a modulus of 1. Such a stability analysis is performed by Howell (1984). For the current study, such a stability analysis will not be performed, since the perturbations to the CRTBP that are to be included will throw off this stability.

16 4.2. VERIFICATION

Table 4.1: Comparison of initial conditions for halo orbits about L2 with a mass parameter of 0.04, as found by Howell (1984) and the current author. Howell x0 1.057222 1.092791 1.140216 1.173414 1.220839 1.258203 z0 0.300720 0.309254 0.298898 0.272900 0.200987 0.050000 y˙0 -0.238026 -0.281140 -0.316028 -0.324710 -0.310434 -0.250410 Current author x0 1.057222 1.092791 1.140216 1.173414 1.220839 1.258203 z0 0.300721 0.309253 0.298897 0.272900 0.200986 0.050000 y˙0 -0.238026 -0.281140 -0.316028 -0.324710 -0.310434 -0.250409

4.1.2 Numerical implementation

In accordance with the procedure followed by Howell (1984), starting at X0, the state can be propagated until y changes sign, so that a second crossing with the xz-plane is found. In order to ensure that the fi- nal state is close enough to the xz-plane, the last integration step is repeated with a smaller stepsize until −11 |yT/2| < 10 , corresponding to an accuracy of approximately 1.5 m in dimensional units. According to −8 Howell (1984), the orbit can be considered periodic if at this point |x˙ T/2| and |z˙T/2| are smaller than 10 , or approximately 3 × 10−4 m/s in dimensional units. These assumptions are adopted in this study, where Equation 4.7 is used iteratively until periodic orbits are found. Once two periodic solutions have been found, the difference between their initial states can be used to guess the initial state of a third solution, assuming the difference in x0 to be proportional to the differences in z0 and y˙0.

Note that Howell (1984) used a Runge-Kutta Merson RK4(3)-5 integration procedure1, whereas the current author uses a Dormand-Prince (DOPRI)-method, being an RK8(7)-13 method, whose table of coefficients can be found in Montenbruck and Gill (2005). Furthermore, the absolute and relative integration tolerances were set to 10−14, yielding dimensional tolerances on the order of millimeters for the satellite’s position.

4.2 Verification

In order to verify the current author’s implementation of the differential correction algorithm presented in Section 4.1, it is used to reproduce halo orbits as found by Howell (1984). Table 4.1 presents a comparison of the initial conditions for halo orbits about the L2 point for a mass parameter of 0.04, as found by Howell (1984) and the current author. As can be seen, the initial conditions are very similar. Only in a few cases, the last digit is found to be off by one, which can be attributed to the different integration algorithms used.

4.3 Halo orbits in the Sun-Earth/Moon CRTBP

In order the find halo orbits about the Sun-Earth L2 point, the differential correction algorithm by Howell (1984) is applied for a mass parameter of 3.040328 × 10−6 (Wertz, 2009). Note that only class 1 halo orbits are considered here, but class 2 halo orbits are easily obtained by mirroring the obtained orbits in the xy- plane. As mentioned before, all orbits are obtained using a DOPRI integration scheme, and the absolute and relative tolerances are set to 10−14 here. Furthermore, the accuracy of the numerical propagator in the CRTBP is checked by keeping track of Jacobi’s constant throughout the integration, which is found to vary by less than 10−15 for all orbits found, indicating accurate results. Figure 4.1 shows a family of halo orbits about the Sun-Earth L2 point that has been found, where the Earth-Moon system is assumed a point mass, which constitutes the second primary denoted by the symbol ⊕.

1The regular notation for RK-methods is RKp(q) − s, where p and q are the order of the method and embedded method respectively, and s is the number of stages .

17 CHAPTER 4. THE NOMINAL HALO ORBIT

Table 4.2: Initial conditions, as well as half the orbital period T/2, corresponding to the halo orbits presented in Figure 4.1, for a mass parameter of 3.040328 × 10−6. x0 1.005221 1.006300 1.007300 1.008300 1.009300 1.010300 z0 0.012392 0.012342 0.012075 0.011394 0.010000 0.007563 y˙0 -0.011857 -0.013458 -0.014567 -0.015163 -0.014987 -0.013646 T/2 1.008808 1.125642 1.241498 1.354847 1.448933 1.511700

x0 1.011000 1.011240 1.011272 z0 0.004485 0.001831 0.000634 y˙0 -0.011270 -0.009434 -0.009011 T/2 1.540533 1.549662 1.551111

Furthermore, Table 4.2 presents the initial conditions corresponding to the halo orbits presented in Figure 4.1. Note that these orbits extend all the way up to the second primary, which in this case is located at approximately x = 1. Table 4.2 shows that the period increases for halo orbits located further away from the second primary. Each halo orbit about the L2 point in the CRTBP is uniquely defined by its maximum x- and z-position. Hence, the entire family of halo orbits found in the Sun-Earth/Moon system can be represented by a single graph of xmax and zmax values, as given by Figure 4.2. Halo orbits have been found with maximum z-values ranging from nearly 0 to approximately 12.4×10−3, or 1,860,000 km in dimensional units. The minimum amplitudes in x- and y-direction for which halo orbits start to exist are found to be equal to approximately 0.001430 and 0.004532 respectively, or 215,000 km and 680,000 km in dimensional units. At this amplitude, second-order effects become large enough to induce coupling between the in- and out-of-plane motion, causing their periods to become equal.

4.4 Eclipse avoidance

Note that an advantage of a three-dimensional halo orbit, as opposed to a more general Lissajous orbit, is that its fixed geometry allows one to choose an orbit that never enters the Earth’s shadow cone, which yields a more stable thermal environment as might be required for deep-space observation missions. In this section, we will look for an approximate minimum size requirement for halo orbits to never enter the Earth’s shadow cone. Figure 4.3 presents an exaggerated schematic of the Earth’s umbra and penumbra, based on Montenbruck and Gill (2005, Fig. 3.7). From figure 4.3, the condition for an eclipse-free position can be derived as:

p 2 2 y + z > RC (4.13) with, RC = d tan f (4.14) R d = s + E (4.15) x sin f and, R + R sin f = S E (4.16) rSE where y and z are coordinates in the rotating reference frame, as specified for the CRTBP in Section 2.1, RC is the radius of the Earth’s shadow cone at a distance sx, where sx is the x-coordinate of the satellite with respect to Earth, RS and RE are the Sun and the Earth radii respectively, and rSE is the distance between the Sun and the Earth. Note that the Earth’s atmosphere and oblateness are neglected here and all relevant constants used are presented in Appendix C. As can be seen from Figure 4.1, the minimum x- and z-position correspond to the point along the halo orbit that is closest to the x-axis, which is found to

18 4.4. ECLIPSE AVOIDANCE

(b) xy-projection (a) 3D-view 10

15 -3 z [-]× 10 5 10 5 -3 0 L2 0 ⊕ L2 -5 ⊕ y[-]× 10 10 5 -5 0 1.015 y [-]× 10-3 1.01 -5 1.005 1 x [-] -10 0.995 0.995 1 1.005 1.01 1.015 x[-]

(c) xz-projection (d) yz-projection 15 15

10 10

-3 5

-3 5

L z[-]× 10 0 ⊕ 2 z[-]× 10 0 ⊕ L2

-5 -5

-10 -10 -10 -5 0 5 10 0.995 1 1.005 1.01 1.015 y[-]× 10-3 x[-]

−6 Figure 4.1: Halo orbits about the Sun-Earth L2 point for a mass parameter of 3.040328 × 10 .

19 CHAPTER 4. THE NOMINAL HALO ORBIT

14

12

10 -3 8

[-] × 10 6 max z 4

2

0 1.005 1.006 1.007 1.008 1.009 1.01 1.011 1.012

xmax [-]

Figure 4.2: xmax-zmax profile for the family of halo orbits about the Sun-Earth L2 point.

Penumbra P3

RS s

x Sun f RE Umbra Umbra Earth

RS RC Penumbra

rSE d

sx

Figure 4.3: Schematic representation of the Earth’s umbra and penumbra.

20 4.5. HALO ORBITS IN THE SUN-EARTH/MOON ERTBP drive the minimum-size requirement for a halo orbit not to enter the Earth’s penumbra2. The smallest halo orbits have their minimum x-value at approximately 1.0084, at which points the Earth’s penumbra extends up to approximately 8.21 × 10−5 normal to the x-axis, or 12, 280 km in dimensional units, which is relatively small compared to the dimensions of a halo orbit in the Sun-Earth/Moon system. As halo orbits are located further to the left, their minimum z-value becomes greater in the absolute sense, and occur closer to the Earth, where the Earth’s penumbra becomes smaller in radius. Hence, selecting a halo orbit whose minimum z has an absolute value greater than 8.21 × 10−5 ensures no eclipse is ever entered. Note that choosing a larger halo orbit than is strictly required for avoiding the Earth’s shadow cone does not yield any benefits in terms of the thermal stability of the environment. In fact, the larger Sun-Earth angles associated with larger halo orbits will be detrimental to shielding strategies, and will also lead to larger variations in relative formation dynamics, as shown in Section 5.3.2, which might be undesirable. Figure 4.1 shows that the most leftward-located halo orbit found, is associated with a Sun-Earth viewing angle of approximately 90 degrees at perigee. Nonetheless, a slightly higher amplitude than is strictly required from an eclipse-avoidance standpoint would actually be beneficial for communications, by avoiding the line of sight to come too close to the Sun-Earth line, hence avoiding excessive background noise. This is reflected by the fact that a commonly chosen out-of-plane amplitude for a halo-orbit about the Sun-Earth L2 point is approximately equal to 250, 000 km, which is also the orbit suggested for XEUS by Chabot and Udrea (2006). This corresponds approximately to a halo orbit with x0 = 1.01124, which is the second smallest halo orbit in Figure 4.1, represented in red. This halo orbit has a minimum z-position of approximately −0.0014, hence stays well outside of the Earth’s shadow cone. For the remainder of this report we will mostly consider the same reference orbit corresponding to x0 = 1.01124. Note that in the simulations to be performed, the halo orbit is always initialized in apogee, hence in the xz-plane with a positive x0 and z0, in which case y˙0 is negative.

4.5 Halo orbits in the Sun-Earth/Moon ERTBP

Halo orbits found in the previous section do not naturally extend to the ERTBP, due to the non-autonomous nature of the equations of motion in the ERTBP, contrary to the CRTBP. As shown in, for instance, (Cam- pagnola et al., 2008), quasi-periodic halo orbits do exist in the elliptic problem, but unlike those found in the CRTBP, halo orbits can only exist in the ERTBP if their principal period has a resonance ratio with the orbital period of the primaries, given by T = 2πN/M where N is the number of primary revolutions, and M is the number of halo orbit revolutions. However, the purpose of the current study is not to establish the most natural nominal orbit in the ERTBP, but rather to compare the effects of perturbations on relative station keeping for a formation. In order to conduct a fair comparison of the results obtained in the CRTBP and the ERTBP, The nominal orbits for these cases should not show significant deviations. To this purpose, the nominal orbits found in the CRTBP are used in the ERTBP as well, which necessitates some amount of absolute station keeping to be performed in the latter case, such as to enforce the non-natural motion in the ERTBP. Note that, using this approach, the nominal orbits in the CRTBP and ERTBP coincide in terms of their non-dimensional coordinates. However, they do not coincide in dimensional space, due to the true anomaly dependence of units for distance and time in the ERTBP. The resulting nominal orbit in the ERTBP is pulsat- ing in dimensional space, in accordance with the normalization as presented in Section 3.1. This approach assures that the same distance fractions of the ’instantaneous’ dimensional halo orbit with respect to the primaries and the libration points are maintained, and is hence considered a fair comparison for studying the influence of eccentricity on the relative dynamics about the Sun-Earth L2 point by the current author.

In order to enforce the non-natural motion that constitutes a halo orbit in the ERTBP, a multiple-shooting method is used, where a number of impulsive maneuvers along the orbit ensures an intersection with the

2The radius of the Earth’s shadow cone is slightly larger at the halo orbit’s apogee position. However, the absolute difference between the minimum and maximum z-values is bigger than the increase in the Earth’s shadow cone radius between the perigee and apogee position, such that the minimum z-value drives the minimum-size requirement for a halo orbit not to enter the Earth’s shadow cone.

21 CHAPTER 4. THE NOMINAL HALO ORBIT

6 nominal actual 4

2

-3

0 y[-] × 10 -2

-4

-6 1 1.002 1.004 1.006 1.008 1.01 1.012 x[-]

Figure 4.4: The nominal orbit compared to the actual uncontrolled orbit for a satellite in the ERTBP, with initial conditions corresponding to x0 = 1.01124 as given in Table 4.2, and f0 = 0. nominal orbit at some later point in time. To demonstrate this method, first the uncontrolled motion that follows from placing a satellite in the ERTBP is considered, for the nominal orbit corresponding to x0 = 1.01124 as given in Table 4.2. A comparison of the nominal and the actual orbit over approximately 180 days is shown in Figure 4.4, where the Earth’s initial true anomaly, f0, is equal to zero. It can be seen that after approximately a quarter halo orbit, the actual orbit starts to rapidly drift away from the nominal orbit. Much like the differential correction method used for finding halo orbits in the CRTBP, one can the differential correction method to find the impulsive maneuver, or shoot, required to enforce a certain position at some later point in time in the ERTBP. The required change in initial velocity is related to the desired change in final position through:

   −1   ∆x ˙ 0 Φ1,4 Φ1,5 Φ1,6 ∆xf ∆y ˙0 ≈ Φ2,4 Φ2,5 Φ2,6 ∆yf  (4.17) ∆z ˙0 Φ3,4 Φ3,5 Φ3,6 ∆zf

Note that the STM can be obtained, using the method presented in Section 4.1.1, where the matrix A(t) follows from the linearized equations of motion in the ERTBP:

 0 0 0 1 0 0  0 0 0 0 1 0    0 0 0 0 0 1 A(t) =   (4.18) ωxx ωxy ωxz 0 2 0   ωyx ωyy ωyz −2 0 0 ωzx ωyz ωzz − 1 0 0 0

The second derivatives of ω are given in Appendix A for reference. Again, one has to apply a number of differential corrections until a final state is found to within a certain tolerance. Figure 4.5 shows the result of applying the multiple-shooting method in the Sun-Earth/Moon system for the same nominal orbit as considered before, corresponding to x0 = 1.01124, where four shoots are used, evenly spaced in terms of non-dimensional time, and indicated by a cross mark. Note that the differential correction method is applied until an intersection with the nominal orbit is found within a tolerance of 10−11,

22 4.5. HALO ORBITS IN THE SUN-EARTH/MOON ERTBP

6 nominal actual 4

2 -3

0 y [-] × 10 -2

-4

-6 1.004 1.006 1.008 1.01 1.012 1.014 1.016 x [-]

Figure 4.5: Multiple-shooting method applied to a satellite in the ERTBP, performing four shoots along the orbit represented by crosses, showing the nominal an actual orbit, with initial conditions corresponding to x0 = 1.01124 as given in Table 4.2, and f0 = 0.

Table 4.3: Total ∆V and maximum deviation from the nominal orbit against the number of shoots used for absolute station keeping, for a halo orbit corresponding to x0 = 1.01124 and f0 = 0. shoots 2 3 4 5 10 50 ∆V per orbit [m/s] 18.71 9.48 9.12 9.39 10.76 12.16 max ∆s [km] 18,435.47 4,016.46 1,997.40 1,238.33 321.05 13.46 corresponding to an accuracy of approximately 1.5 m in dimensional units. The nominal and actual orbit as presented in Figure 4.5 are found to nearly overlap. Table 4.3 shows the effect of the number of shoots per orbit on the ∆V budget over an entire halo orbit, as well as the maximum deviation from the nominal orbit, where the number of shoots are spread evenly over the halo orbit in terms of non-dimensional time. It can be seen that an optimum ∆V budget of approximately 9.12 m/s is obtained for four shoots per orbit. In the limit of continuous control, the ∆V budget becomes approximately 12.4 m/s, which isn’t a significant increase compared to the 9.12 m/s as required for a four-shoot strategy. However, the added complexity of performing continuous control should also be taken into consideration. Since this study is not particularly focused on the absolute station keeping required, but rather the relative station keeping in a formation, a four-shoot strategy will be used for simplicity. Note that the maximum deviation of approximately 1,997 km from the nominal orbit is assumed to be of insignificant influence on the relative dynamics within the formation, given it’s relatively small value compared to the halo orbit’s dimensions. A quick check to justify this assumption is performed in Section 6.1. Note that the required absolute station keeping depends on the initial true anomaly of the Earth, showing differences of up to approximately 1 m/s from the values presented in Table 4.3 depending on the initial true anomaly. However, the general conclusions will not change, and since absolute station keeping is not the primary focus of this study, we will not further consider this.

Finally, a quick check is performed to investigate the effect of the chosen nominal halo orbit on the multiple- shooting method. The results of the multiple-shooting method for a larger nominal halo orbit, corresponding to x0 = 1.011, hence slightly to the left of the halo orbit for x0 = 1.01124, are presented in Table 4.4. It can

23 CHAPTER 4. THE NOMINAL HALO ORBIT

Table 4.4: Total ∆V and maximum deviation from the nominal orbit against the number of shoots used for absolute station keeping, for a halo orbit corresponding to x0 = 1.011 and f0 = 0. shoots 2 3 4 5 10 50 ∆V per orbit [m/s] 18.45 11.63 11.32 11.92 13.79 15.61 max ∆s [km] 17,901.85 4,825.55 2,599.52 1,669.64 440.82 18.34

be seen that the absolute station keeping cost increases compared to the orbit for x0 = 1.01124. The orbit corresponding to x0 = 1.011 reaches an absolute station keeping ∆V budget of approximately 15.9 m/s in the limit of continuous control. The absolute station keeping cost can be found to continue to increase as halo orbits are located further to the left, hence closer to the second primary. Looking at Equations 2.2 and 3.14, this can be explained by the increasing differences in acceleration between the CRTBP and ERTBP as −1 the distance from the equilibrium point L2 increases, caused by the (1 + e cos f) term in the xy-plane, and an additional z-term in the z-direction.

24 Chapter 5

Formation flying in the CRTBP

Having selected a nominal halo orbit for a formation about the Sun-Earth L2 point, this chapter provides an investigation of the relative dynamics for a formation in the CRTBP, and the impulsive control required to keep the formation configuration fixed to within a certain tolerance. Impulsive control could become rele- vant for small formations yielding prohibitively small thrust requirements in the case of continuous control, or might be necessary due to mission constraint for deep space observation missions. This chapter will dis- cuss two strategies for impulsive control, a simple linear targeter performing corrections at constant time intervals, as presented in Howell and Marchand (2005), and a modification to this method taking advantage of the maximum allowable relative error within the formation throughout the orbit, which resembles discrete targeting approaches as presented in Pernicka et al. (2006) and Qi and Xu (2011). The two targeting ap- proaches will be referred to as the Equitime Targeting Method (ETM) and Tangent Targeting Method (TTM) respectively, in accordance with the terminology used in Qi and Xu (2011). Section 5.1 will define the formation configuration parameters used for the discrete targeting simulations. Then, Section 5.2 discusses the ETM method, providing results for different formation configurations in terms of the required ∆V’s, and the relative position error. Similarly, the TTM method is presented in Section 5.3, providing a comparison between the ETM and TTM, as well as numerical results for different formation configurations. Next, linear approximations to the resuls obtained are presented in Section 5.4, for easy reproducibility of the results, as well as providing a better understanding of the results obtained. Finally, some precision and integration accuracy considerations are presented in Section 5.5. Note that, unless mentioned otherwise, all results apply to a nominal halo orbit of x0 = 1.01124.

5.1 Formation configuration

The effect of different formation configurations on the achievable accuracy and control effort for discrete formation control is to be investigated. The formation configuration is defined by its nominal separation and orientation, as depicted in Figure 5.1, where the chief spacecraft follows a ’natural’ halo orbit1, and the deputy spacecraft is to perform all relative station keeping maneuvers. Here, the xyz-coordinate axes are defined by the rotating reference frame as specified for the RTBP, hence the xy-plane being coplanar to the orbital plane of the primaries, and the x-axis being parallel to the line connecting the primaries. Note that the x0y0z0-axes simply represent a shift of the origin to the chief spacecraft. The formation orientation is defined by the angles α and β, the former of which is the angle of the formation’s xy-projection with the x-axis, and the latter presents the formation’s angle with the z-axis. Furthermore, s represents the formation’s separation distance. Another distinction for formation types can be made between rotating and inertial formations, the former having its orientation fixed in the reference frame specified for the RTBP and the latter having its orientation fixed in an inertial reference frame2. For a rotating formation, the nominal position of the deputy with respect to the chief is readily calculated using Equation 5.1 for any point in time.

xdeputy =xchief + s sin β cos α

ydeputy =ychief + s sin β sin α (5.1)

zdeputy =zchief + s cos β

1Note that the halo orbit is no longer natural when perturbations are included. 2Note the potentially confusing terminology of rotating and inertial formations, since rotating formations are actually fixed in the RTBP reference frame, and inertial formations have an apparent rotation.

25 CHAPTER 5. FORMATION FLYING IN THE CRTBP

z' z β Deputy s Chief y' x' α Sun y

x Earth

Figure 5.1: Formation configuration defined in the RTBP reference frame.

For inertial formations, the positive rotation of the primaries about the z-axis causes α to decrease at the rate of the angular velocity of the primaries. Hence, for an inertial formation, the deputy’s position with respect to the chief’s position at some point in time in the RTBP can be calculated from:

xdeputy =xchief + s sin β cos (α − ∆f)

ydeputy =ychief + s sin β sin (α − ∆f) (5.2)

zdeputy =zchief + s cos β where ∆f represents the true anomaly traversed by the primaries since initialization. Note that for deep space observations, as were to be performed by XEUS, inertial formations are required, given the inertial observation target. However, an investigation of rotating formations allows one to gain more insight into formation dynamics in the CRTBP, so we will consider both types of formations in the anal- yses to follow. In the following discussions we will offen refer to semi-inertial formations, which are inertial formations whose orientation is often reinitialized throughout the orbit, such as to allow for a reasonable comparison with rotating formations. A more detailed discussion on this is presented in Section 5.2.2.

5.2 Equitime targeting method

The Equitime Targeting Method (ETM) is a formation control strategy that uses impulsive maneuvers at the beginning of equal time intervals to target a nominal orbit. Section 5.2.1 describes the approach, which is similar to implementations as performed by, among others, Howell and Marchand (2005) and Qi and Xu (2011). Then, Section 5.2.2 presents some important considerations with respect to the comparison of results obtained for rotating and (semi-)inertial formations. Finally, Section 5.2.3 presents the results for different formation configurations in terms of the maximum relative position error and ∆V requirements.

5.2.1 Approach The ETM method, which uses impulsive maneuvers at constant time intervals in order to keep the relative position in a spacecraft formation within a certain tolerance, is illustrated by Figure 5.2. Here, the nominal deputy orbit is defined by a constant offset s from the natural chief orbit, such that the deputy orbit is generally unnatural. In order to enforce the deputy to follow the unnatural orbit, the orbit is split up into equal time segments, at the start of which the deputy performs an impulsive maneuver, ∆Vi, such as to enforce an intersection with the nominal orbit at the end of the segment. In figure 5.2, the subscripts indicate the corresponding segment, and the relative velocities of the deputy are given by δVi, where the superscripts

26 5.2. EQUITIME TARGETING METHOD

ΔV + nominal deputy orbit 3 δV3 deputy orbit

- + δV3 ΔV2 δV2 s

chief orbit δV- + 2 ΔV1 δV1 s

- δV1

s

Figure 5.2: Illustration of a simple linear targeting method for formation control.

− and + indicate the state before and after the impulsive maneuver is performed, respectively. Note that, in order to find the required ∆V, again a differential correction algorithm according to Equation 4.17 is used.

5.2.2 Considations for (semi-)inertial formations Note that, as mentioned before, a formation flying missions such as XEUS will have inertial observation tar- gets, hence requiring a formation with an inertially fixed orientation. According to Equations 5.1 and 5.2, initially identically oriented formations will rotate with respect to each other as time progresses. Although the changing orientation of inertial formations is not really considered in comparisons between rotating and inertial formations by, for example, Qi and Xu (2011), whether this leads to a fair comparison is arguable. To illustrate the ambiguity of such a comparison, consider initially identically oriented formations along the y-axis. As time progresses, the orientation of the inertial formation deviates with respect to the orientation of the rotating formation, where after approximately 3 months (in the Sun-Earth system), they are perpen- dicular to each other and the inertial formation is now approximately aligned with the Sun-Earth line, which is undesirable for any deep-space observations, and will also be shown to yield less desirable conditions for formation station keeping in Section 5.2.3. To allow for a more straightforward comparison, we will define a so-called semi-inertial formation, which is an inertial formation whose orientation is re-initialized at the beginning of each segment, such that the orientation of a semi-inertial formation does not deviate too much from a rotating formation throughout the halo orbit. Note that re-initializing the semi-inertial formation at the beginning of each segment does lead to discontinuities in the formation orientation, which is not a very realistic situation. However, one should interpret the semi-inertial formation to represent the situation of an inertial formation that happens to coincide with the specified orientation in the RTBP frame at the considered point in time. To illustrate the impact of considering a semi-inertial formation as opposed to an inertial formation, the relative position error along an entire orbit using the ETM method is compared to a rotating formation for both cases, and is presented in Figure 5.3. Note that results as displayed in Figure 5.3 will be more elaborately discussed in Section 5.2.3. Figure 5.3 shows that the differences between a rotating and inertial formation with the same initial orientation, start to show large deviations halfway along the orbit. The reason for this is that at this point the inertial formation will be approximately oriented along the x-axis, which will be shown to be the least favorable orientation in terms of formation control in Section 5.2.3. The semi-inertial formation on the other hand, maintains a similar orientation compared to the rotating formation, yielding smaller differences in terms of relative position, which can almost fully be attributed to the extra effort required to keep the formation inertially oriented, rather than just a difference in orientation, yielding a more fair comparison. Note that re-initializing a semi-inertial formation at the beginning of each segment is somewhat compli- cated by the fact that the required ∆V depends on the relative velocity, which is uniquely defined by the state at the beginning of the previous segment. Hence, in order to obtain ∆V values that are representative of an

27 CHAPTER 5. FORMATION FLYING IN THE CRTBP

(a) (b) 60 30

50 25

40 20

30 15

20 10 position error [cm] position error [cm]

10 5 rotating rotating inertial semi-inertial 0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 5.3: Relative position error throughout approximately one halo orbit using the ETM method, applied to a (a) rotating and inertial formation at a nominal separation of 50 m, initially oriented along the y-axis performing an impulsive maneuver every 5 days, and similarly for (b) a rotating and semi-inertial formation. inertial formation, each re-initialization of the formation orientation requires performing the ETM method for one segment backwards in time, so as to obtain a relative velocity that is uniquely defined by a previous maneuver and not dependent on an arbitrary re-initialization. Note that this also applies to the initial state of a rotating formation, where such a backwards step is only required once at initialization. Figure 5.4 shows how the initial ∆V is an obvious outlier for the case that such a backwards differential correction step is not performed for a rotating formation. In literature such as Qi and Xu (2011) and Howell and Marchand (2005), such a backwards differential correction step is not performed, leading to an arbitrary initialization of the relative velocity in the formation, and consequently a jump in the required ∆V after the first correc- tive maneuver is performed, much like the one displayed for the non-corrected approach in Figure 5.4. For rotating formations, this will not have a significant impact on most of the results, since it only affects one ∆V value out of many. However, given the re-initialization of a semi-inertial formation at the beginning of every segment, it is strictly necessary to perform this backwards step in order to get any sensible data on ∆V’s at all. In this study, all arbitrary initialization dependencies are removed by applying such a backwards differential correction step whenever it is needed.

5.2.3 Results Results of the ETM method in terms of ∆V’s and the relative position error are presented here for rotating and semi-inertial formations, discussing their dependence on time in between maneuvers, or segment time, the formation separation distance, and the formation orientation.

Segment time Figure 5.5 shows the maximum relative position error and the required ∆V’s for formation control of rotating and semi-inertial formations at a nominal separation of 50m along the y-axis, using the ETM method, for segment times of every 1, 2, 3, 4, and 5 days. Note that every error arc in Figure 5.5 (a) and (b) corresponds to one segment, at the beginning of which an impulsive maneuver is performed, with the maximum error occurring approximately halfway along the segment.

28 5.2. EQUITIME TARGETING METHOD

4.5

4

3.5

3

2.5

Δ V [µm/s] 2

1.5

1 with correction without correction 0.5 0 20 40 60 80 100 120 140 160 180 time [days]

Figure 5.4: ∆V’s for a 50 m rotating formation using the ETM method for 5 days in between maneuvers, with and without a backwards differential correction step performed, which removes the dependency on arbitrary initialization conditions.

It can be seen that the maximum error in between maneuvers is approximately equal to the square of the segment time, for both rotating and semi-inertial formations. An expression is derived by Qi and Xu (2011) that indeed shows that in the limit of the segment time approaching zero, this is exactly the case for the CRTBP. This relation can more intuitively be deduced by realizing that for the segment time approaching zero, the relative acceleration approaches a constant, hence yielding a parabolic error curve. The reason that this relation still holds fairly well for the finite segment times considered, is that they cover relatively short time spans with respect to the orbital period of roughly 180 days, such that relatively small distances are covered, hence variations in force gradients are small. Furthermore, the maximum position error of the deputy is relatively small compared to the formation distance, also yielding relatively small changes in the relative acceleration. As a result, the relative acceleration can be approximated to be of constant magnitude throughout one segment, yielding a quadratic relationship between segment time and maximum position error. Another observation that can be made from Figure 5.5 is that the ∆V’s required are approximately pro- portional to the segment times. This again makes sense, because sustaining a nearly constant acceleration for a longer period of time, requires a proportional ∆V increase to dump the built up velocity. Furthermore, Figure 5.5 shows that the maximum relative position errors, as well as the ∆V’s required, have a peak at around 90 days, roughly corresponding to the halo orbit’s perigee position3. The larger maximum error, and larger required ∆V here, can be attributed to the closer proximity to the Earth, leading to larger gravity gradients, hence more rapidly varying dynamics along any direction. Finally, it can be seen that semi-inertial formations require more control and yield higher relative errors than a rotating formation for the case considered. This however, is not generally applicable but rather depends on the nominal formation orientation. This is more elaborately discussed and explained in Section 5.4. 3Recall that the halo orbit is initialized in apogee position for all simulations performed.

29 CHAPTER 5. FORMATION FLYING IN THE CRTBP

(a) rotating (b) semi-inertial 25 30 1 day 1 day 2 days 2 days 25 20 3 days 3 days 4 days 4 days 5 days 20 5 days 15 15 10 10 position error [cm] position error [cm]

5 5

0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

(c) rotating (d) semi-inertial 5 6 1 day 1 day 2 days 2 days 5 4 3 days 3 days 4 days 4 days 5 days 4 5 days 3 3

Δ V [µm/s] 2 Δ V [µm/s] 2

1 1

0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 5.5: Relative position error for a 50 m (a) rotating formation and a (b) semi-inertial formation, oriented along the y-axis, using the ETM method for different segment times. The corresponding ∆V’s are presented in (c) and (d) respectively.

30 5.2. EQUITIME TARGETING METHOD

Formation separation Figure 5.6 shows the maximum relative position error and the required ∆V’s for formation control of rotating and semi-inertial formations at nominal separations of 10, 50, and 100 m along the y-axis, for a segment time of 2 days. It can be seen that the maximum relative position error is approximately proportional to the formation separation distance. The same proportional relation can be found for the case of a semi- inertial formation, as well as the ∆V’s required for both types of formations. The proportionality of the maximum position error with the formation separation distance can be explained by the nearly constant force gradient over the relatively small formation separation distances as compared to the distance from either of the primaries. As a result, the relative acceleration approximately scales with the formation separation distance, hence also the maximum relative position error and ∆V.

Formation orientation Figure 5.7 shows the maximum relative position error and the required ∆V’s for 50 m rotating and semi- inertial formations along the x-, y-, and z-axes, for a segment time of 2 days. It can be seen that the least favorable orientation is along the x-axis, which yields the largest maximum position errors, as well as the largest ∆V’s. This is to be expected, since the x-axis is most closely aligned with the line of sight to either primary, in which direction the gravity gradient is most significant. It should be noted that for much larger halo orbits, such as the largest halo orbit presented in Figure 4.1, the most favorable formation orientation will vary much more significantly throughout the orbit, mostly according to the different orientation with respect to the Earth. Furthermore, such a large halo orbit will have it’s perigee at an x-value nearly similar to that of the Earth itself, resulting in a Sun-Earth view angle of nearly 90 degrees, negating the usual benefits of easy shielding strategies for smaller halo orbits. The worst formation orientation in terms of relative acceleration for the nominal halo orbit is calculated in Section 5.4 for a couple of points along the halo orbit considered, which shows it to be closely aligned with the x-axis for each point. Finally, it can be seen that a translation of the results from rotating formations to semi-inertial formations is not straightforward, but rather depends on the formation’s orientation. A formation along the x-axis seems to benefit from the semi-inertial formation’s rotation with respect to the RTBP frame, showing smaller position errors and ∆V’s, whereas a formation along the y-axis shows larger maximum position errors for semi-inertial formations, and a formation along the rotating z-axis naturally remains unaffected, since such a formation is parallel to the axis of rotation. The position errors for semi-inertial formations along the y- and z-axes actually become very similar, which is most easily explained by considering an inertial reference frame where the Sun-Earth line coincides with the x-axis for the point in time considered. In this case, the relative accelerations for inertial formations in y- and z-direction are affected only by gravity gradients from the Sun and the Earth, which have a similar effect in either of these direction for small values of y and z and become even identical if the formation is located on the x-axis (for infinitesimally small separation distances, though still nearly identical for the relatively small separation distances considered in this study). A more analytical explanation to the relation between rotating and semi-inertial formation is given in Section 5.4.2, where linear expressions are derived for, among other, the maximum relative position error as a function of absolute position and formation configuration.

Minimum ∆V Looking at the results presented by Figures 5.5 through 5.7, something that stands out which has not been mentioned before, is how small the required ∆V’s are, even falling well below 1 µm/s for formation sizes of 10 m. One should keep in mind that very small thrust levels are generally associated with higher implementation inaccuracies, the effect of which is not investigated in the current study. Furthermore, for small satellites at small separations, the ∆V requirements might impose prohibitively small thrust requirements. In order to check the feasibility of implementing ∆V’s on the order of 1 µm/s and below, one can look at similar missions that also have strict thrust requirements. One such mission, currently occupying a Lissajous orbits about the Sun-Earth L2 point, is Gaia. Gaia uses cold gas thrusters for fine attitude adjustments, complying with strict accuracy requirements, yielding thrust levels in the range of 1 µN - 500 µN, a thrust resolution smaller than a

31 CHAPTER 5. FORMATION FLYING IN THE CRTBP

(a) rotating (b) semi-inertial 8 10 10 m 10 m 7 50 m 9 50 m 100 m 8 100 m 6 7 5 6 4 5

3 4

position error [cm] position error [cm] 3 2 2 1 1 0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

(c) rotating (d) semi-inertial 4 5 10 m 10 m 50 m 50 m 100 m 4 100 m 3

3 2

Δ V [µm/s] Δ V [µm/s] 2

1 1

0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 5.6: Relative position error for (a) rotating formations and (b) semi-inertial formations at different separation distances, oriented along the y-axis, using the ETM method for a segment time of 2 days. The corresponding ∆V’s are presented in (c) and (d) respectively.

32 5.2. EQUITIME TARGETING METHOD

(a) rotating (b) semi-inertial 10 10 along x along x 9 along y 9 along y 8 along z 8 along z 7 7 6 6 5 5 4 4

position error [cm] 3 position error [cm] 3 2 2 1 1 0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

(c) rotating (d) semi-inertial 6 5 along x along x along y along y 5 along z 4 along z

4 3 3

Δ V [µm/s] Δ V [µm/s] 2 2

1 1

0 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 5.7: Relative position error for (a) rotating and (b) semi-inertial formations at a separation of 50 m, oriented along the x-, y-, and z-axes, using the ETM method for a segment time of 2 days. The corresponding ∆V’s are presented in (c) and (d) respectively.

33 CHAPTER 5. FORMATION FLYING IN THE CRTBP

µN, and response times lower than 300 msec, as defined in Noci et al. (2009). Note that the LISA pathfinder uses identical cold gas thrusters, currently following a Lissajous orbit about the Sun-Earth L1 point, also using thrust levels on the order of µN’s (Rudolph et al., 2016). Based on these missions, it follows that low ∆V’s on the order of 1 µm/s are quite feasible if one considers spacecraft with masses on the order of 1000 kg or more, as was the case for the concept design of XEUS (Bavdaz et al., 2004). This would lead to minimum required specific impulses on the order of 10−3 Ns. The problem would become more challenging if smaller spacecraft were considered, for instance, with a mass of 1 kg, in which case the minimum impulse bit that Gaia’s cold gas thrusters are capable of would become too large to deliver the ∆V’s presented in Figures 5.5 through 5.7. Although we will no longer consider any practical issues with the implementation of resulting ∆V’s, it is important to keep these possible limitations in mind. Given the state of propulsion technology at some future point in time, the linear approximations to be derived in this report, can be useful for determining a limit on the possibilities of formation control for a given spacecraft mass. Namely, the resulting minimum ∆V could be used to approximate the minimum formation size for which a certain upper bound on the relative position error can be achieved (see Equation 5.10 through 5.12). It should be realized that the above discussion applies to the unperturbed CRTBP, whereas in reality, perturbations, solar radiation in particular, can cause the relative accelerations to be much higher. Conse- quently, the required ∆V’s will be higher, alleviating the minimum thrust requirements, as further discussed in Chapter 8.

5.3 Tangent targeting method

Looking at the ETM method, for a certain allowed maximum error, or error corridor, the segment time is driven by a single point along the orbit where the relative position error reaches a global maximum, occurring approximately when the formation is in perigee. Subsequently, every other segment is overachieving in terms of relative accuracy requirements. An adjustment to the ETM that addresses this issue is the TTM method, which aims for a maximum relative position error that is tangent to the allowed error corridor for every segment along the orbit. This can be achieved by varying the segment time, effectively changing the independent variable from segment time to the maximum allowed relative position error. This will increase the segment time for the less dynamically sensitive regions along the orbit, which could be beneficial for observations, and will be shown to be beneficial in terms of ∆V requirements. The TTM approach and numerical results obtained, are presented in Section 5.3.1 and 5.3.2 respectively.

5.3.1 Approach In order to implement a TTM algorithm, one can use the apparent quadratic relation between segment time and maximum error as observed in Figure 5.5. Note that the validity of this relation is backed up by the linear approximation presented in Section 5.4. An iterative method can be used that corrects the segment time according to: rε ∆t+ = ∆t− allowed (5.3) ε− + − where ∆t and ∆t are the segment times corresponding to the new and previous iteration, and εallowed and ε− are the maximum relative position error allowed and the error found for the previous iteration respectively. Equation 5.3 can be used until the maximum relative position error along the segment is within acceptable limits, which in this work is specified to be within a 0.1% range of the desired maximum relative position error. Slightly different approaches to the TTM method are taken by Pernicka et al. (2006) and Qi and Xu (2011), where the latter presents a more sophisticated expression to correct for the segment time. However, all simulations performed in this study for formation separations up to 1000 m, have shown to need no more than a few iterations using Equation 5.3, where most of the time even just a single iteration is enough to correct the segment time.

34 5.3. TANGENT TARGETING METHOD

(a) (b) 1 14 ETM ETM TTM TTM 12 0.8

10 0.6 8

0.4 Δ V [µm/s] 6 position error [m] position error

0.2 4

0 2 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 5.8: comparison between the ETM and TTM method, for a 100 m rotating formation along y, for a 1 m error corridor.

To illustrate the benefits of using the TTM method as opposed to the ETM method, Figure 5.8 presents a comparison between the ETM and TTM methods for a rotating formation, separated 100 m along the y-axis, and an error corridor of 1 m. It can be seen that the benefit of using the TTM method as opposed to the ETM is twofold: firstly, the maximum segment time increases, allowing for longer observation times without any thruster disturbances, and secondly, the minimum required ∆V increases, which can be beneficial, given the possibly prohibitively small thrust requirements. The formation configuration considered in Figure 5.8 is chosen because it provides a means of verifying the results to Qi and Xu (2011), where numerical data for a formation in a similar halo orbit is presented, approximately corresponding to x0 = 1.01124. The results obtained by Qi and Xu (2011) and the current author are compared and presented in Table 5.1. Firstly, it can be seen that data with respect to the segment time and number of maneuvers is found to be quite similar, with small differences occuring most likely due to the non-identical halo orbits considered. Secondly, it can be seen that some significant differences do occur in the ∆V’s. This can be explained by the non-inclusion of a backwards integration step to remove the initial ∆V’s dependency on the arbitrary formation initialization used in Qi and Xu (2011). As such, the minimum ∆V found by Qi and Xu (2011) actually correspond to the first corrective maneuver, which completely depends on the arbitraty initiliazation of the formation’s relative position and velocity, hence is not a very representative result. Also note that the accuracy of the results obtained in the current study are backed up by the linear approximations presented in Section 5.4. For completeness, Table 5.2 presents the benefits of the TTM method over the ETM method for a semi-inertial formation. Similar to rotating formations, Table 5.2 shows an increase in minimum ∆V and maximum segment time for semi-inertial formations using the TTM as compared to the ETM, and a decrease in number of maneuvers, while the total ∆V does not change significantly, especially considering how small the total ∆V budget is. It should be noted that for inertial formations, as opposed to semi-inertial formation, the benefits of using the TTM method could become even more pronounced, depending on the formation orientation and the amount of time during which the formation is to maintain its inertially fixed orientation. This follows from Figure 5.3, showing more strongly varying dynamics throughout the orbit, which the TTM method is optimized for.

35 CHAPTER 5. FORMATION FLYING IN THE CRTBP

Table 5.1: Comparison of the ETM and TTM method for a 100 m, rotating formation along the y-axis and an error corridor of 1 m, as obtained by Qi and Xu (2011) as well as the current author. The results are representative of one full halo orbit corresponding to x0 = 1.01124. minimum total min segment maximum number of ma- ∆V[µm/s] ∆V[µm/s] time [days] segment time neuvers [days] Current author ETM 4.98 219.83 7.16 7.16 26 TTM 8.00 221.36 7.16 11.42 21 Qi and Xu (2011) ETM 5.08 208 7.19 7.19 25 TTM 6.14 218 7.19 11.39 21

Table 5.2: Comparison of the ETM and TTM method for a 100 m, semi-inertial formation along the y-axis and an error corridor of 1 m, over one full halo orbit corresponding to x0 = 1.01124. minimum total min segment maximum number of ma- ∆V[µm/s] ∆V[µm/s] time [days] segment time neuvers [days] ETM 6.77 259.07 6.53 6.53 28 TTM 9.78 263.09 6.53 9.47 23

5.3.2 Results For the remainder of this report, we will mostly consider the TTM method, given its benefits as compared to the ETM method, and hence will likely be the preferred method of choice for any practical applications. Given this shift, from this point onwards, most results will be given in terms of the segment time as a function of the position in orbit, as opposed to the relative position error throughout the orbit. Note that for the relatively small formation separations considered in this study, hence slowly varying relative accelerations, the qualitative results for maximum relative position error observed in Section 5.2.3 can be extended to the variation of the segment time along a halo orbit, where an increase in maximum position error between two positions along the halo orbit now implies a decrease in segment time by approximately the square root of that factor. Other than this, the conclusions presented in Section 5.2.3 remain the same, hence we will not reproduce Figures 5.5 through 5.7 for the TTM method. Numeric results are presented in Tables 5.3 and 5.4, where results for the ∆V’s and segment times are presented for different formation sizes and orientations, using the TTM method with an error corridor of 1 cm, over one full halo orbit, for rotating and semi-inertial formations. Note that the error corridor is chosen based on benchmark missions for formation flying about the Sun-Earth L2 point, suggested by Bristow et al. (2000) and Carpenter et al. (2003).

Although we have already decided on a specific halo orbit, corresponding to x0 = 1.01124, and do not gain any practical benefits from using a larger halo orbit for purposes considered in this study, we will still perform one quick check on the effect of increasing the size of the halo orbit to satisfy our curiosity. Figure 5.9 shows the segment times and ∆V’s for a 50 m rotating formation along y, for the relatively small reference halo orbit 4 considered, and a much larger halo orbit, corresponding to x0 = 1.01124 and x0 = 1.0083 respectively . Note that the larger halo orbit has a smaller period, so that it traverses more than one orbit during the timespan presented in Figure 5.9. As expected, figure 5.9 shows larger variations in both segment time and ∆V for the larger halo orbit, due to the larger variation in position with respect to the primaries, which also causes the worst orientation in terms of segment time to change more prominantly throughout the orbit. Hence, formation keeping becomes more dependent on the position along the orbit. Furthermore, for comparison,

4 Recall that moving the initial position x0 to the left, initially increases the size of the halo orbit, as can be seen from Figure 4.1

36 5.3. TANGENT TARGETING METHOD

Table 5.3: Numerical results for rotating formations of different separations and orientations using the TTM method with an error corridor of 1 cm, over one full halo orbit corresponding to x0 = 1.01124. orien- minimum average ∆V total minimum maximum average seg- tation ∆V[µm/s] [µm/s] ∆V[µm/s] segment segment ment time time [days] time [days] [days] 10 m formation x 0.465 0.524 52.898 1.424 1.989 1.794 y 0.253 0.338 21.947 2.266 3.650 2.820 z 0.310 0.362 25.375 2.056 2.988 2.600 50 m formation x 1.041 1.172 263.696 0.637 0.890 0.802 y 0.566 0.757 109.060 1.013 1.634 1.258 z 0.693 0.811 126.497 0.920 1.336 1.162 100 m formation x 1.472 1.658 527.106 0.450 0.629 0.567 y 0.801 1.072 217.622 0.717 1.156 0.889 z 0.980 1.147 252.385 0.650 0.945 0.822

Table 5.4: Numerical results for semi-inertial formations of different separations and orientations using the TTM method with an error corridor of 1 cm, over one full halo orbit corresponding to x0 = 1.01124. orien- minimum average ∆V total minimum maximum average seg- tation ∆V[µm/s] [µm/s] ∆V[µm/s] segment segment ment time time [days] time [days] [days] 10 m formation x 0.432 0.495 47.064 1.480 2.146 1.904 y 0.310 0.367 26.056 2.076 2.991 2.566 z 0.310 0.362 25.375 2.056 2.988 2.600 50 m formation x 0.966 1.109 235.016 0.662 0.959 0.851 y 0.692 0.822 129.886 0.929 1.337 1.147 z 0.693 0.811 126.497 0.920 1.336 1.162 100 m formation x 1.367 1.568 470.327 0.468 0.678 0.602 y 0.979 1.163 259.367 0.657 0.946 0.811 z 0.980 1.147 252.385 0.650 0.945 0.822

37 CHAPTER 5. FORMATION FLYING IN THE CRTBP

(a) (b) 2.4 3 x large x large 2.2 x small x small y large y large y small 2.5 y small 2 z large z large z small z small 1.8 2 1.6 1.4 1.5 1.2

1 Δ V [µm/s] 1

segment time[days] 0.8

0.6 0.5 0.4 0.2 0 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 5.9: (a) Segment times and (b) ∆V’s for the TTM method for a small and large halo orbit, corre- sponding to x0 = 1.01124 and x0 = 1.008300 respectively (see Table 4.2 for more initial conditions). A 50 m, rotating formation with an error corridor of 1 cm is considered.

Table 5.5: Numerical results for 50 m rotating formations of different orientations using the TTM method with an error corridor of 1 cm, for a large halo orbit corresponding to x0 = 1.0083 over one full orbit. orien- minimum average ∆V total minimum maximum average seg- tation ∆V[µm/s] [µm/s] ∆V[µm/s] segment segment ment time time [days] time [days] [days] x 0.730 1.150 190.872 0.422 1.267 0.955 y 0.410 1.357 227.959 0.416 2.257 0.945 z 0.501 1.389 230.649 0.317 1.848 0.955

Table 5.5 shows numeric results in similar format to Table 5.3, for the larger halo orbit considered, which shows an increase in average ∆V for any orientation, as well as larger variation in both ∆V and segment times. A formation along x does, however, show a smaller ∆V budget for the larger orbit, because its orientation no longer aligns closely with the worst orientation throughout the entire orbit. This can deduced from Figure 4.1, by realizing that the worst formation orientation in terms of relative accelerations is close to the line of sight to the Earth. Do note that the ∆V budget is given for one halo orbit, the period of which is smaller for the larger halo orbit considered, as shown in Table 4.2.

5.4 Linear approximation of the results

The results obtained so far, indicating a nearly constant relative acceleration throughout a segment, as well as a nearly constant force gradient over the relatively small formation separations considered, suggest that a linear approximation to the equations of motion would provide an accurate way of obtaining the required ∆V’s and segment time, or maximum relative position error, at a specific point along the halo orbit for a given formation configuration. Linear approximations can also help to clarify or reinforce the results obtained from the simulation. Such expressions are derived for rotating and inertial formations in Sections 5.4.1 and 5.4.2 respectively.

38 5.4. LINEAR APPROXIMATION OF THE RESULTS

5.4.1 Rotating formations A linear approximation to the relative acceleration at a specific point along the orbit can be obtained by considering the linear variational equation of motion for the CRTBP, derived from Equation 4.8, given by:

δX˙ = A(t)δX (5.4) where δ denotes a differences from the nominal state, in our case that of the deputy with respect to the chief. Note that for rotating formations, hence fixed in the RTBP reference frame, the relative velocity can be assumed to be equal to zero, especially for small formation sizes, such that Equation 5.4 reduces to:   Uxx Uxy Uxz δr¨ = Uyx Uyy Uyz δr (5.5) Uzx Uzy Uzz where δr¨ and δr are the relative acceleration and relative position of the deputy with respect to the chief respectively. Note that expressions for the second partial derivatives are given in Appendix A for reference. Now, assuming the relative acceleration to be nearly constant throughout the duration of one segment, thus creating a parabolic error arc in time, the relative position error along a segment can be approximated by: 1 ε ≈ V¯ t − at¯ 2 (5.6) 0 2 where V¯0 indicates the initial velocity and a¯ the constant acceleration. Given that the position error is zero at the beginning and the end of a segment: 1 V¯ ≈ a¯∆t (5.7) 0 2 thus, 1 1 ε ≈ at¯ ∆t − at¯ 2 (5.8) 2 2 where ∆t is the segment time. Now, the maximum relative position error along the segment, occuring at ∆t/2 is given by: 1 ε ≈ a¯∆t2 (5.9) max 8 Substituting Equation 5.5 into Equation 5.9, the maximum error can be approximated for any point along the orbit, using:   Uxx Uxy Uxz 1 2 εmax ≈ Uyx Uyy Uyz δr ∆t (5.10) 8 Uzx Uzy Uzz Furthermore, from Equation 5.7 the required ∆V ’s can be approximated by:

∆V ≈ 2 ∗ V0   Uxx Uxy Uxz (5.11) ≈ Uyx Uyy Uyz δr ∆t

Uzx Uzy Uzz and finally, rewriting Equation 5.10 for ∆t gives an approximation of the maximum segment time for a given error corridor: v u   −1 u Uxx Uxy Uxz u ∆t ≈ 8 Uyx Uyy Uyz δr εmax (5.12) t Uzx Uzy Uzz Note that the results of Equations 5.10 and 5.12 are non-dimensional, though are readily converted to dimen- sional units using Appendix B. Furthermore, these equations are exact only for infitesimally small formation

39 CHAPTER 5. FORMATION FLYING IN THE CRTBP

Table 5.6: Percentage differences between the linear approximations given by Equations 5.10 and 5.11, and the integration of the full non-linear system, using the ETM, for different formation separations and segment times ∆t. A rotating formation oriented along the x-axis is considered, approximately at perigee position. Accuracies for the maximum relative position error and ∆V are shown as ”ε; ∆V”. X XXX separation XXX 100 m 10 km 1,000 km 10,000 km ∆t XXX 1 day 0.04; 0.05 0.05; 0.05 0.14; 0.14 0.96; 0.96 2 days 0.18; 0.17 0.18; 0.17 0.27; 0.26 1.10; 1.09 5 days 1.12; 1.00 1.12; 1.00 1.21; 1.09 2.03; 1.91 10 days 4.40; 3.83 4.40; 3.83 4.48; 3.94 5.26; 4.71 separations and segment times, though the constant nature of the relative dynamics suggest that these ap- proximations are quite accurate for relatively small formation separations and segment times. In order to get some indication of the accuracy of Equations 5.10 and 5.11, they are compared to results obtained from integration of the full non-linear system. This is done and presented in Table 5.6, where the differences are presented in percentages for a formation along the x-axis, using the ETM method, at the point along the halo orbit where the maximum relative position occurs (near perigee), so as to represent a near-to-worst case scenario where the dynamics most rapidly change, hence the assumptions for the linear approximations most easily break up. It can be seen from Table 5.6 that the linear approximation formulas are very accurate for relatively small formation sizes and small segment times. Moreover, the accuracy is not very sensitive to the formation size, where accuracies of around 1% can still be obtained for a formation size as large as 10,000 km at a segment time of 1 or 2 days, and naturally an even better result will be obtained for more favorable orientations. The results are more sensitive to a change in the segment time. This because large segment times will lead to much larger changes in absolute position than the deputy’s separation from the chief, hence more significantly deteriorating the assumption of constant acceleration than an increase in formation size.

Finally, we will quickly discuss the worst orientation in terms of relative acceleration, which in the accuracy analysis before, was assumed to be along the x-axis. In order to find the actual orientation along which the relative acceleration increases most rapidly, we can take the eigenvector of the mapping matrix in Equation 5.5 corresponding to the largest eigenvalue, for a specific point along the orbit. This will give the exact orientation yielding the worst relative acceleration for infinitesimally small formation separations, and still an accurate approximation for small finite formation separations, considering the small impact of formation size on the relative acceleration as follows from Table 5.6. The worst orientations in terms of relative accel- eration are checked for four positions along the halo orbit, corresponding to the apogee position, minimum y-position, perigee position, and maximum y-position, given by:

−0.995 −0.935 −0.990 −0.935

vapo =  0.000  ; vymin =  0.352  ; vper =  0.000  ; vymax = −0.351 (5.13) −0.100 −0.019 0.139 −0.019

The worst orientation is indeed closest to the x-axis for any position along the halo orbit. As expected, there is a small tilt away from the x-axis towards the primaries for any point along the halo orbit as well. At the perigee position, for which the accuracy analysis in Table 5.6 is performed, there is a small tilt in positive z-direction, due to the positive relative z-position of the Earth with respect to the formation, as can be seen in Figure 4.1. Therefore, the percentage differences presented in Table 5.6 could become slightly larger still. However, the difference in relative accelerations between the x-direction and the worst orientation found, would be small5. Hence, Table 5.6 still gives a good indication of the obtainable accuracy using the linear approximations presented.

5The worst orientation at perigee position, as given by Equation 5.13, is tilted approximately 0.14 rad from the x-axis. Hence, the relative acceleration for a formation in the x-direction is no more than 1% smaller than the relative acceleration for a formation oriented along the worst orientation, having a negligible effect on the percentage differences obtained in Table 5.6.

40 5.4. LINEAR APPROXIMATION OF THE RESULTS

5.4.2 Inertial formations Similar to rotating formations, we can formulate linear approximations for inertial formations. We again turn to Equation 5.4, though in this case the relative velocity cannot be assumed equal to zero. Namely, given the counterclockwise rotation of the primaries, an inertial formation requires a clockwise rotation of equal magnitude with respect to the RTBP reference frame. The finite relative velocity gives rise to an apparent relative coriolis acceleration. From Equation 4.9, the linearized relative acceleration is given by:

U U U 0 −2 0 xx xy xz δr δr¨ = U U U 2 0 0 (5.14)  yx yy yz  δr˙ Uzx Uzy Uzz 0 0 0

This can be simplified by realizing that a non-dimensional unit of time is equal to the inverse of the rotation rate of the primaries. Given that the required rotation rate of an inertial formation is equal to the rotation rate of the primaries, and opposite in direction, the necessary rotation rate of an inertial formation in non- dimensional units is equal to unity, and in clockwise direction, so that:

δx˙  δy  = (5.15) δy˙ −δx hence, we can write:   Uxx − 2 Uxy Uxz δr¨ =  Uyx Uyy − 2 Uyz δr (5.16) Uzx Uzy Uzz Note that Equation 5.16 gives the total relative acceleration, which is not necessarily equal to the perturbing acceleration. Namely, one has to account for the centripetal acceleration that is required to maintain the clockwise rotation in an inertial formation with respect to the RTBP frame. Again, given that the non- dimensional rotation rate is equal to unity, a simple expression can be obtained for the required centripetal acceleration, given by: δx¨ δx = − (5.17) δy¨ δy Subtracting the required centripetal acceleration from the total relative acceleration yields the perturbing acceleration:   Uxx − 1 Uxy Uxz δr¨ =  Uyx Uyy − 1 Uyz δr (5.18) Uzx Uzy Uzz Now, we can formulate expressions similar to Equations 5.10 through 5.12 for the case of inertial formations, given by:   Uxx − 1 Uxy Uxz 1 2 εmax ≈  Uyx Uyy − 1 Uyz δr ∆t (5.19) 8 Uzx Uzy Uzz   Uxx − 1 Uxy Uxz

∆V ≈  Uyx Uyy − 1 Uyz δr ∆t (5.20)

Uzx Uzy Uzz v u   −1 u Uxx − 1 Uxy Uxz u ∆t ≈ 8  Uyx Uyy − 1 Uyz δr εmax (5.21) t Uzx Uzy Uzz A similar comparison between the linear approximations and results obtained from the simulation has been performed for an inertial formation, as presented in Table 5.7. Table 5.7 shows that the accuracy of a linear approximation for inertial formations is slightly lower than for rotating formation, which is to be expected,

41 CHAPTER 5. FORMATION FLYING IN THE CRTBP

Table 5.7: Percentage differences between the linear approximations given by Equations 5.19 and 5.20, and the integration of the full non-linear system, using the ETM method, for different formation separations and segment times ∆t. An inertial formation oriented along the x-axis is considered, approximately at perigee position. Accuracies for the maximum relative position error and ∆V are shown as ”ε; ∆V”. X XXX separation XXX 100 m 10 km 1,000 km 10,000 km ∆t XXX 1 day 0.06; 0.06 0.06; 0.06 0.16; 0.16 1.05; 1.05 2 days 0.18; 0.55 0.18; 0.55 0.28; 0.65 1.17; 1.55 5 days 1.13; 3.34 1.13; 3.34 1.23; 3.44 2.11; 4.32 10 days 4.42; 12.41 4.42; 12.42 4.52; 12.51 5.37; 13.37 given the change in orientation during a time segment, causing a change in relative acceleration. The differ- ence in accuracy is more pronounced for the ∆V approximation, which can be explained by the fact that the required ∆V depends on the acceleration built up over both the previous and the following segment, whereas the maximum error depends only on the acceleration profile over the segment to follow. The dependency of ∆V’s on the acceleration profile over a longer time interval, causes the larger inaccuracies, given the in- creased change in orientation for an inertial formation.

In order to explain the differences between rotating and semi-inertial formations, as shown in Figure 5.7, we can look at the mapping matrix presented in Equation 5.4, for different positions along the halo orbit. Taking the same four point along the halo orbit as used in determining the worst orientation, we obtain:

6.887 0 0.986   6.571 −3.285 0.195 

Aapo =  0 −2.024 0  ; Aymin = −3.285 −0.938 −0.101 0.986 0 −2.864 0.195 −0.101 −3.633 (5.22)  13.060 0 −2.707 6.571 3.285 0.195 

Aper =  0 −5.267 0  ; Aymax = 3.285 −0.938 0.101  −2.707 0 −5.793 0.195 0.101 −3.633

Note that the mapping matrices’ first, second and third columns represent the resulting acceleration due to a displacement along the x-, y-, and z-axes respectively. Now, given that inertial formations have a correcting term -1 in the top left entry, we find that for any position along the halo orbit, the norm of the first column will be smaller, hence the perturbing acceleration due to a displacement along the x-axis is smaller for inertial formations than for rotating formation. Similarly, due to the correcting -1 term in the middle entry, we see that the norm of the second column becomes larger. Hence, the perturbing acceleration due to a displace- ment along the y-axis becomes larger for inertial formations. These finding are in agreement with Figure 5.7.

5.5 Precision and integration accuracy

To conclude this chapter, we will briefly discuss the accuracy limitations of the numerical integrator used for the simulations, imposed by double precision limits. This is especially the case for formation flying in the Sun-Earth RTBP, where the relative position accuracy for formations can become very small with respect to the Sun-Earth distance. The error corridor of 1 cm used in this study, is approximately a factor 1.5 × 1013 smaller than the Sun-Earth distance. Now, given that we wish to find an error arc that does not overshoot, or undershoot, this error corridor by more than 0.1 percent, a position difference of 0.01 mm needs to be clearly distuingishable for the numerical methods used. If we keep the origin fixed at the Sun-Earth barycenter, we cannot achieve such accuracy using double precision, for it would at the very least require the 17th significant digit to be defined. In order to circumvent this problem, we shifted the origin to coincide with the Earth, in which case the ratio of absolute position, reaching values on the order of 1.5 × 109 m, and the 0.01 mm

42 5.5. PRECISION AND INTEGRATION ACCURACY accuracy requirement requires at least the 15th significant digit to be defined, which is on the edge of what one can achieve using double precision. Note that another solution would be to use a higher precision format, which was not desirable for the current study, given the program’s dependencies on software packages that necessarily require double precision variables, and hence, would have to be rewritten if one opts for using a higher precision format, being a time consuming process.

43

Chapter 6

Formation Flying in the ERTBP

In this chapter we will shift the problem to that of a formation in the ERTBP and analyze its implications on formation control. As mentioned in Section 4.5, impulsive station keeping maneuvers are to be performed in order to enforce the generally unnatural motion in the ERTBP. Section 6.1 presents a brief sensitivity analysis for the formation control’s dependence on the resulting deviation from the specified nominal orbit. Section 6.2 presents results of the integration of the full system of non-linear equations of motion in the ERTBP. Similarly to the CRTBP, a linear approximation of the results is presented in Section 6.3, giving insight into the results obtained and allowing for them to be easily reproduced.

6.1 Sensitivity to absolute station keeping

As discussed in Section 4.5, a stable unnatural halo orbit is enforced by performing discrete station keeping maneuvers along the orbit. For the halo orbit considered, the optimum number of impulsive maneuvers in terms of ∆V was found to be four, yielding a maximum deviation from the nominal orbit of approximately 2000 km, which is relatively small compared to the halo orbit’s dimensions. Now, in order to verify that a maximum deviation of this magnitude will not significantly affect the relative dynamics, the 4-shoot absolute station keeping strategy will be compared to a 50-shoot strategy, where the latter can be considered to approach the nominal orbit, having a maximum deviation of approximately 13.5 km from the nominal orbit as given by Table 4.3. Hence, the differences in relative formation control between 4- and 50-shoot strategies can be assumed to be equal to the differences between a 4-shoot strategy and a formation that exactly follows the nominal halo orbit. Table 6.1 presents results on ∆V’s and segment times for the TTM method, applied to a 100 m rotating formation along the x-axis, with a 1 cm error corridor, for a 4- and 50-shoot strategy. The results are representative of one entire halo orbit, with the Earth initialized at a true anomaly of 3π/2, so that the formation reaches its perigee position approximately when the Earth is in perihelion, in which case a change in the formation’s position presumably has the most pronounced effect. It can be seen from Table 6.1 that the number of shoots, or maximum deviation from the nominal orbit, does not significantly impact the relative dynamics. Note that these results also indicate that formation control in the RTBP is not significantly impacted by a difference in absolute position due to, for instance, tracking errors, which may be on the order of 1 km as assumed by Segerman and Zedd (2006).

Table 6.1: Results for segment times and ∆V’s of the TTM method for a rotating, 100 m formation along the x-axis in the ERTBP, initialized at a true anomaly of 3π/2 of the Earth, for 4- and 50-shoot absolute station keeping strategies. number of minimum average total minimum maximum average shoots ∆V[µm/s] ∆V[µm/s] ∆V[µm/s] segment segment segment time[days] time[days] time[days] 4 1.486 1.690 535.583 0.439 0.623 0.557 50 1.487 1.688 535.249 0.439 0.623 0.557

45 CHAPTER 6. FORMATION FLYING IN THE ERTBP

6.2 Results

This section presents results obtained from integration of the full system of non-linear equations of motion in the ERTBP. Results for formation flying in the ERTBP will be graphically presented for different orientations and different initial true anomalies of the Earth, whereas results for different formation separations and time in between maneuvers are omitted, given that, following similar reasoning as for the CRTBP, they obey the same relationships where the maximum error is proportional to formation separation and the square of the segment time; or rather, the segment time is proportional to the square root of the error corridor, as well as the square root of the formation separation.

Different orientations Figure 6.1 presents the segment times and ∆V’s for a 50 m formation at different orientations, for both rotating and semi-inertial formations, in the CRTBP and the ERTBP for an initial Earth true anomaly f0 = 0. Note that we have defined the segment time at a certain point along the orbit as the segment time corresponding to the segment whose maximum error occurs at the point considered, rather than a segment that starts at the point considered. Also recall that halo orbits are initialized in apogee position for all simulations. A first observation that follows from inspection of Figure 6.1, is that the segment time for formation flying in the ERTBP, as compared to the CRTBP, is slightly smaller at the start of the orbit and slightly larger after one full halo orbit. This can be explained by the fact that the Earth is initially in perihelion, whereas after one halo orbit the Earth will be near apohelion, corresponding to smaller and larger units of length in the ERTBP respectively. Consequently the formation will be slightly closer or further away from the Earth in dimensional units1, leading to larger gravity gradients, hence larger relative accelerations, when the Earth is in perihelion, and vice versa when the Earth is in apohelion. The differences in segment times between the CRTBP and ERTBP are more quantifiably shown in Section 6.3.3. Figure 6.1 also shows that the global maximum of the segment time for a formation in the ERTBP for f0 = 0 occurs slightly sooner than for a formation in the CRTBP. This can be explained by the faster progression of non-dimensional time in the ERTBP compared to the CRTBP when the Earth is near perihelion, such that the formation reaches it’s apogee position slightly sooner in Figure 6.1. Furthermore, the ∆V’s required, show predictable behavior, where formation keeping requires slightly larger ∆V’s near perihelion and slightly smaller ∆V’s near apohelion, where differences between the CRTBP and ERTBP are approximately equal to the inverse of the factor by which the segment time increases. Note that using the ERTBP normalized reference frame has more implications on the formation dynamics. Namely, the differences in segment times and ∆V’s do not only depend on the Earth’s true anomaly, but also on the formation’s orientation and position along the halo orbit. The exact dependencies for infinitesimally small segment times and formation separations are given by linear approximations presented in Section 6.3.

Different initial true anomalies Figure 6.2 presents the segment times and ∆V’s for a 50 m formation along the y-axis, for both rotating and semi-inertial formations, in the CRTBP and the ERTBP for different initial true anomalies of the Earth. It can be seen that differences in segment time and ∆V between the ERTBP and CRTBP show approximately the opposite result when starting at a true anomaly of π as opposed to starting at a true anomaly of 0. This due to the mirrored true anomaly profile they experience because the halo orbital period is almost equal to half a year, or π in non-dimensional time. The largest differences between the ERTBP and CRTBP in segment time and ∆V occur when the Earth is either near perihelion or apohelion. This, along with the fact that the segment time is at a minimum when the formation is near perigee, and the ∆V being minimum near apogee, can be used to initialize the formation at a specific epoch in accordance with one’s desired strategy. Namely, initializing the halo orbit in apogee when the Earth is in perihelion or apohelion, assures that the minimum segment time does not get significantly smaller, for in this case the worst true anomaly of the Earth and position along the halo orbit in

1Recall that the nominal orbits in the CRTBP and ERTBP are identical when represented in their non-dimensional form.

46 6.2. RESULTS

(a) rotating (b) semi-inertial 1.8 1.8 x ERTBP x ERTBP x CRTBP x CRTBP y ERTBP y ERTBP 1.6 y CRTBP 1.6 y CRTBP z ERTBP z ERTBP z CRTBP z CRTBP

1.4 1.4

1.2 1.2

1 1 segment time[days] segment time[days]

0.8 0.8

0.6 0.6 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

(c) rotating (d) semi-inertial

x ERTBP x ERTBP x CRTBP x CRTBP 1.4 y ERTBP 1.4 y ERTBP y CRTBP y CRTBP z ERTBP z ERTBP z CRTBP z CRTBP 1.2 1.2

1 1 Δ V [µm/s] Δ V [µm/s]

0.8 0.8

0.6 0.6

0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 6.1: Segment times for (a) rotating and (b) semi-inertial 50 m formations in the CRTBP and ERTBP, oriented along the x-, y-, and z-axes, with an error corridor of 1 cm and f0 = 0. The corresponding ∆V’s in the rotating and semi-inertial frame are presented in (c) and (d) respectively.

47 CHAPTER 6. FORMATION FLYING IN THE ERTBP

(a) rotating (b) semi-inertial 1.7 1.4 CRTBP CRTBP 0 ERTBP 1.35 0 ERTBP 1.6 π/2 ERTBP π/2 ERTBP π ERTBP π ERTBP 3π/2 ERTBP 1.3 3π/2 ERTBP 1.5 1.25 1.4 1.2 1.3 1.15

1.2 1.1

segment time[days] segment time[days] 1.05 1.1 1 1 0.95 0.9 0.9 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

(c) rotating (d) semi-inertial 0.95 1.05 CRTBP CRTBP 0 ERTBP 0 ERTBP 0.9 π/2 ERTBP 1 π/2 ERTBP π ERTBP π ERTBP 3π/2 ERTBP 3π/2 ERTBP 0.85 0.95

0.8 0.9

0.75 0.85

Δ V [µm/s] 0.7 Δ V [µm/s] 0.8

0.65 0.75

0.6 0.7

0.55 0.65 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 6.2: Segment times for (a) rotating and (b) semi-inertial 50 m formations along the y axis in the CRTBP and ERTBP, at initial true anomalies of 0, π/2, π, and 3π/2, for an error corridor of 1 cm. The corresponding ∆V’s in the rotating and semi-inertial frame are presented in (c) and (d) respectively.

48 6.2. RESULTS

Table 6.2: Segment times and ∆V’s for a 100 m formation along the z-axis, with a 1 cm error corridor, over one full halo orbit for different initial true anomalies of the Earth. initial true minimum average ∆V total ∆V minimum maximum average seg- anomaly ∆V[µm/s] [µm/s] [µm/s] segment segment ment time [rad] time [days] time [days] [days] CRTBP - 0.980 1.147 252.385 0.650 0.945 0.822 ERTBP 0 0.956 1.148 252.581 0.650 0.968 0.821 π/2 0.977 1.127 249.085 0.667 0.947 0.835 π 0.956 1.147 252.376 0.650 0.968 0.822 3π/2 0.981 1.168 255.876 0.634 0.945 0.808 terms of minimum segment time do not occur simultaneously. Rather, the formation will reach perigee when the Earth is at a true anomaly of approximately π/2, 3π/2 and so on, in which case differences between the ERTBP and CRTBP are small, as is also quantifiably shown in Section 6.3.3. This initialization also results in a global maximum segment time, occuring once a year when the apogee position of the formation and apohelion position of the Earth occur simultaneously. Following the previously mentioned strategy of leaving the minimum segment time relatively unaffected, inevitably causes the minimum ∆V to decrease, occuring when the formation is in apogee and the Earth is in apohelion. Since this could be undesirable due to possible minimum thrust constraints as discussed in Section 5.2.3, one might instead wish to aim for the minimum ∆V to be relatively unaffected. This can be achieved by initializing the formation in apogee when the Earth is at a true anomaly of either π/2 or 3π/2. Note that the influence of the Sun-Earth eccentricity on a formation near L2 is small, on the order of only a few percent, rendering the choice for any specific strategy to benefit from the eccentricity somewhat irrelevant. Moreover, the difference between the halo orbital period and half the orbital period of the pri- maries will cause an initial alignment of the halo orbit and the primaries’ orbit to drift, further diminishing the potential benefits from any specific initialization strategy. For instance, the small difference between the halo orbital period and half the primaries’ orbital period will cause an initial alignment of the halo orbit’s perigee position with the Earth’s perihelion, to drift to an alignment of the halo orbit reaching perigee when the Earth is at a true anomaly of π/2 over the course of approximately 9 years. For systems with larger eccentricities, as well as a resonance period for the halo orbit and primaries’ orbit, the benefits from a proper initialization of the problem using one of the strategies mentioned before, could become more significant, though in the Sun-Earth/Moon system, the benefits of any specific initialization can be considered irrelevant due to the aforementioned reasons.

Numerical results To conclude the section on simulation results, Table 6.2 shows the ∆V’s and segment times for a 100 m formation along z, at different initial true anomalies in the ERTBP. Other orientations will not be considered for now, though Section 6.3 shows that similar results would apply to formations along x or y. Similar conclusions can be drawn from Table 6.2 as from Figure 6.2, showing a decrease in minimum ∆V by approximately 2.5% for an initial true anomaly of 0 or π, whereas it does not change significantly for initial true anomalies of π/2 or 3π/2. Similarly, an increase and decrease of about 2.5% in minimum segment time is observed for initial true anomalies of π/2 ans 3π/2 respectively, whereas initial true anomalies of 0 or π leave the minimum segment time relatively unaffected. Do again keep in mind that the halo orbit considered has an orbital period of approximately half a year, such that a halo orbit starting when the Earth is in perigee will be followed by a halo orbit that starts when the Earth is near apogee.

49 CHAPTER 6. FORMATION FLYING IN THE ERTBP

6.3 Linear approximation of the results

Similarly to the linear approximations derived for the CRTBP, one can derive linear approximations for a formation in the ERTBP. This section presents such expressions for rotating and inertial formations. Finally, the linear approximation is used to present a more in-depth look on the differences between the CRTBP and ERTBP.

6.3.1 Rotating formations The linear variational equation of motion in the ERTBP is given by:

δx     δy δx¨ ωxx ωxy ωxz 0 2 0   δz δy¨ = ωyx ωyy ωyz −2 0 0   (6.1) δx˙ δz¨ ωzx ωzy ωzz − 1 0 0 0   δy˙ δz˙ or from Appendix A:           δx¨ Uxx Uxy Uxz δx 0 2 0 δx˙ −1 δy¨ = (1 + e cos f) Uyx Uyy Uyz  δy + −2 0 0 δy˙ (6.2) δz¨ Uzx Uzy Uzz − e cos f δz 0 0 0 δz˙

Note that, for a nominally fixed dimensional separation, the non-dimensional separation in the ERTBP is pulsating, given the pulsating unit of distance in the normalized reference frame. The resulting apparent relative velocity in non-dimensional space is given by:

d 1 d 1 + e cos f  −e sin f −e sin f δr˙ ∗ = δr = δr = rδˆ r∗ = r∗ (6.3) df rˆ df a(1 − e2) a(1 − e2) 1 + e cos f where the asterisk denotes the non-dimensional separation. Moreover, a nominally fixed dimensional sepa- ration imposes the need for an apparent relative acceleration in non-dimensional space, given by:

d2 1 d2 1 + e cos f) −e cos f −e cos f δr¨∗ = δr = δr = rδˆ r∗ = r∗ (6.4) df 2 rˆ df 2 a(1 − e2) a(1 − e2) 1 + e cos f

Substituting Equation 6.3 into Equation 6.1 and subtracting Equation 6.4 yields the perturbing acceleration in non-dimensional space, given by Equation 6.5 or 6.6.

   e cos f 2e sin f    δx¨ ωxx + 1+e cos f ωxy − 1+e cos f ωxz δx δy¨ = ω + 2e sin f ω + e cos f ω  δy (6.5)    yx 1+e cos f yy 1+e cos f yz    δz¨ e cos f δz ωzx ωzy ωzz − 1 + 1+e cos f       δx¨ Uxx + e cos f Uxy − 2e sin f Uxz δx −1 δy¨ = (1 + e cos f) Uyx + 2e sin f Uyy + e cos f Uyz δy (6.6) δz¨ Uzx Uzy Uzz δz Similar to the linear approximations presented in Section 5.4, the non-dimensional maximum error and ∆V for a formation in the ERTBP can now be approximated by:   Uxx + e cos f Uxy − 2e sin f Uxz 1 −1 2 εmax ≈ (1 + e cos f) Uyx + 2e sin f Uyy + e cos f Uyz δr ∆t (6.7) 8 Uzx Uzy Uzz

50 6.3. LINEAR APPROXIMATION OF THE RESULTS

Table 6.3: Percentage differences between the linear approximations given by Equations 6.7 and 6.8, com- pared to the integration of the full non-linear system, for different formation separations and segment times ∆t. A rotating formation oriented along the x-axis is considered, starting at a true anomaly of 3π/2. The table shows the accuracy of ε; ∆V. X XXX separation XXX 100 m 10 km 1,000 km 10,000 km ∆t XXX 1 day 0.05; 0.06 0.05; 0.06 0.14; 0.15 0.97; 0.98 2 days 0.19; 0.20 0.19; 0.20 0.28; 0.29 1.11; 1.12 5 days 1.17; 2.24 1.17; 2.24 1.26; 2.33 2.07; 3.12 10 days 4.50; 9.22 4.50; 9.22 4.58; 9.29 5.30; 9.95

  Uxx + e cos f Uxy − 2e sin f Uxz −1 ∆V ≈ (1 + e cos f) Uyx + 2e sin f Uyy + e cos f Uyz δr ∆t (6.8)

Uzx Uzy Uzz v u   −1 u Uxx + e cos f Uxy − 2e sin f Uxz u ∆t ≈ 8(1 + e cos f) Uyx + 2e sin f Uyy + e cos f Uyz δr εmax (6.9) t Uzx Uzy Uzz Table 6.3 presents an accuracy analysis for Equations 6.7 and 6.8, for what can again be considered as the approximate worst-case scenario, representing a formation along the x-axis near perigee position, which was initialized at a true anomaly of 3π/2, such that the perigee position in the halo orbit occurs approximately when the Earth is at perihelion. The inaccuracies of the linear approximations in the ERTBP are slightly higher than for the CRTBP, due to the explicit time dependence of the pseudo-potential function. Furthermore, the changing pseudo-potential function over time has a larger effect on the accuracy of the ∆V than the accuracy of the segment time, for similar reasons as inertial formations, as mentioned in Section 5.4.2.

6.3.2 Inertial formations Converting the linear approximation from rotating formations to inertial formations is identical to the steps performed for the CRTBP. Hence, the perturbing acceleration for inertial formations in the ERTBP is obtained by subtracting 1 from the top left and middle entry of the mapping matrix in Equation 6.5. The resulting linear approximations for the maximum error, ∆V, and segment time become:   Uxx − 1 Uxy − 2e sin f Uxz 1 −1 2 εmax ≈ (1 + e cos f) Uyx + 2e sin f Uyy − 1 Uyz δr ∆t (6.10) 8 Uzx Uzy Uzz   Uxx − 1 Uxy − 2e sin f Uxz −1 ∆V ≈ (1 + e cos f) Uyx + 2e sin f Uyy − 1 Uyz δr ∆t (6.11)

Uzx Uzy Uzz v u   −1 u Uxx − 1 Uxy − 2e sin f Uxz u ∆t ≈ 8(1 + e cos f) Uyx + 2e sin f Uyy − 1 Uyz δr εmax (6.12) t Uzx Uzy Uzz An accuracy analysis of the linear approximations given by Equations 6.10 and 6.11 is presented in Table 6.4, showing slightly higher inaccuracies for inertial formations than for rotating formations.

6.3.3 Difference between the CRTBP and ERTBP The linear approximations of the segment time for rotating and inertial formations in the CRTBP and ERTBP, given by Equations 5.12, 5.21, 6.9 and 6.12, show that the difference in segment time due to the eccentricity

51 CHAPTER 6. FORMATION FLYING IN THE ERTBP

Table 6.4: Percentage differences between the linear approximations given by Equations 6.10 and 6.11, compared to the integration of the full non-linear system, for different formation separations and segment times ∆t. An inertial formation oriented along the x-axis is considered, starting at a true anomaly of 3π/2. The table shows the accuracy of ε/∆V. X XXX separation XXX 100 m 10 km 1,000 km 10,000 km ∆t XXX 1 day 0.06; 0.07 0.06; 0.07 0.16; 0.17 1.06; 1.07 2 days 0.24; 0.26 0.24; 0.26 0.34; 0.36 1.23; 1.25 5 days 1.30; 2.66 1.30; 2.67 1.39; 2.75 2.26; 3.60 10 days 4.76; 10.70 4.77; 10.70 4.85; 11.77 5.63; 11.47

2π 3 -2.4 -2.4 -1.8 -1.8 -1.2 -1.2 2 -0.6 -0.6 0 0 0.6 0.6 1.2 1.2 1 1.8 1.8 2.4 2.4 1π 0

f [rad] f 2.4 2.4 1.8 1.8 1.2 1.2 -1 0.6 0.6 0 0 -0.6 -0.6 -1.2 -1.2 -2 -1.8 -1.8 -2.4 -2.4 0 -3 0 0.5T 1.0T 1.5T 2.0T θ [T]

Figure 6.3: Percentage differences in segment time between the CRTBP and ERTBP as a function of the true anomaly of the primaries and the position along the halo orbit, for a formation along the z-axis. depends on the true anomaly as well as the formation’s orientation and position. The difference is most straightforward for a formation along the z-direction, for which the true anomaly does not appear in the mapping matrix, such that the difference in segment time between a formation in the CRTBP and ERTBP is uniquely determined by the true anomaly of the primaries. For this orientation,√ the linear approximations show that the non-dimensional segment time differs only by the term 1 + e cos f. Taking the different normalizations of the CRTBP and ERTBP into account2, one can obtain the difference in dimensional segment time (using Appendix B), yielding: s (1 − e2)3 ∆t = ∆t (6.13) ERT BP,dimensional (1 + e cos f)3 CRT BP,dimensional

It follows from Equation 6.13 that the largest influence on differences in segment times between the CRTBP and ERTBP, for a formation in z-direction, is the difference in progression of non-dimensional time. Figure 6.3 shows the percentage differences in segment time between the CRTBP and ERTBP as a function of the true anomaly of the primaries and the position along the halo orbit, for a formation along the z-axis. Here,

2Note that the different normalizations of the CRTBP and ERTBP also explain the counterintuitive results presented in Section 3.1, showing the non-dimensional accelerations to be smaller when the primaries are in pericenter.

52 6.3. LINEAR APPROXIMATION OF THE RESULTS the position along the halo orbit is represented by a phase angle θ, where T is the halo orbital period. Note that diagonal, dashed lines represent orbital paths of the formation. Although a contour plot is not very relevant for the case of a formation along the z-axis, given its independence on the position in orbit, this is not the case for formations in any other direction, for which differences in formation keeping do depend on the formation’s orientation and position along the halo orbit, due to the true anomaly appearing in the mapping matrices for Equations 6.9 and 6.12. Figure 6.4 shows the percentage differences in segment time between the CRTBP and ERTBP as a function of the true anomaly of the primaries and the position along the halo orbit, for rotating and inertial formations along the x- and y-axes. It can be seen from Figure 6.4 that formations along x and y show similar differences in segment time to a formation along z, hence are still mostly influenced by the different rates of time progression, which is given by Equation 6.13. However, small deviations from Equation 6.13 are caused by the appearance of the true anomaly in the mapping matrices for Equations 6.9 and 6.12. In fact, Figure 6.4 shows pockets of maximum increase or decrease at points where a certain true anomaly and a certain position along the halo orbit occur simultaneously. Somewhat interesting to note is that these pockets of increased difference could be avoided by proper initialization of the formation. However, the difference between the halo orbital period and half the period of the primaries would cause the initial intended alignment to drift over several orbits, negating any possible practical benefits of such an approach in the Sun-Earth/Moon system. Note that these pockets of maximum increase and decrease occur for all formations with an orientation component in the xy-plane, even though the number of contours used in Figure 6.4 is too small for these pockets to be visible for other formations types than a rotating formation along y. Figures 6.3 and 6.4 can be produced for differences in ∆V as well. However, as follows from Equations 6.8 and 6.9, these differences would be the exact inverse of the differences in segment time. Therefore, such figures are not presented here.

A more in-depth analysis of Figure 6.4 can be given by considering the true anomaly dependent terms in- troduced in the mapping matrices for Equations 6.9 and 6.12 for the ERTBP, as well as the second partial derivatives as given by Equation 5.22 for different points along the halo orbit. Firstly, the e cos f term intro- duced in the ERTBP, representing a relative acceleration along the formation orientation due to the pulsating axes, is positive when the Earth is near perihelion and negative when it is near apohelion. Given the negative value for Uyy for any point along the halo orbit, the e cos f term decreases the magnitude of the relative acceleration for a formation along y when the Earth is near perihelion and vice versa when the Earth is near apohelion, hence increases and decreases the segment times respectively. The opposite result applies to a formation along x, due to the positive values for Uxx along the halo orbit, causing a decrease in segment time when the Earth is near perihelion, and an increase when the Earth is near apohelion. Hence the e cos f term increases the differences in segment times between the CRTBP and ERTBP for a formation along x, and vice versa for a formation along y. Furthermore, for a formation along y, the effect of the e cos f term is most significant when the formation is in apogee, given the smallest norm of the second column given in Equation 5.22, clearly separating the pockets of maximum difference for rotating formation as seen in Figure 6.4 for θ = 0, 1T, 2T , and to a lesser extent at θ = 0.5T, 1.5T , corresponding to the formation’s perigee position. Note that for the halo orbit considered, the relative effect of the true anomaly dependent terms is smaller for formations along x, as well as inertial formations, due to the generally larger relative accelerations associated with such formations, causing Figure 6.4 (b), (c), and (d) to more closely resemble Figure 6.3. A similar analysis can be performed for the 2e sin f term, or the apparent coriolis acceleration caused by the apparent velocity due to the pulsating axes in the ERTBP. It can be found that, when the Earth is near a true anomaly of π/2, the 2e sin f term causes an decrease in segment times for negative y-values (the first half of the halo orbit), and an increase for positive y-values. The opposite is true for when the Earth is near a true anomaly of 3π/2. This causes the observable wave for percentage differences in y-direction presented in Figure 6.4. The effect of the 2e sin f term in x-direction is opposite to that in y-direction, causing a wave in opposite direction in Figure 6.4, that is less pronounced than in y-direction due to aforementioned reasons.

53 CHAPTER 6. FORMATION FLYING IN THE ERTBP

(a) y rotating (b) y inertial 2π 3 2π 3 -2.4 -1.8 -2.4 -2.4 -2.4 -2.4 -1.8 -1.2 -1.8 -1.8 -1.8 2 -1.2 -1.8 2 -0.6 -1.2 -1.2 -0.6 -1.2 -1.2 0 -0.6 0 -0.6 0.6 -0.6 0 -0.6 0 0 0.6 0.6 0 1.2 0.6 0.6 1.2 0.6 1.2 1 1.8 1.2 1.2 1 1.8 1.2 1.8 1.8 1.8 1.8 2.4 2.4 2.4 2.4 2.4 1π 0 1π 0 2.4 2.4 2.4

f [rad] f 1.8 [rad] f 2.4 1.8 2.4 1.2 1.8 1.8 1.8 0.6 -1 1.2 1.2 1.8 -1 1.2 1.2 0.6 0.6 1.2 0 0.6 0.6 0 0 0.6 -0.6 0 0 -0.6 0 -1.2 -0.6 -0.6 -1.2 -0.6 -0.6 -1.2 -2 -1.8 -1.2 -1.2 -2 -1.8 -1.2 -1.8 -1.8 -1.8 -1.8 -2.4 -2.4 -2.4 -2.4 0 -3 0 -2.4 -3 0 0.5T 1.0T 1.5T 2.0T 0 0.5T 1.0T 1.5T 2.0T θ [T] θ [T]

(c) x rotating (d) x inertial 2π 3 2π 3 -2.4 -2.4 -2.4 -2.4 -2.4 -2.4 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -1.2 -1.2 -1.2 2 -1.2 -1.2 -1.2 2 -0.6 -0.6 -0.6 -0.6 -0.6 -0.6 0 0 0 0 0 0.6 0.6 0.6 0.6 0.6 0.6 1.2 1.2 1 1.2 1.2 1.2 1 1.8 1.8 1.8 1.8 1.8 2.4 2.4 2.4 2.4 2.4 2.4 1π 0 1π 0 2.4 2.4 2.4 f [rad] f 2.4 2.4 2.4 [rad] f 1.8 1.8 1.8 1.8 1.8 1.2 1.2 -1 1.2 1.2 1.2 -1 0.6 0.6 0.6 0.6 0.6 0.6 0 0 0 0 0 -0.6 -0.6 -0.6 -0.6 -0.6 -0.6 -1.2 -1.2 -1.2 -2 -1.2 -1.2 -1.2 -2 -1.8 -1.8 -1.8 -1.8 -1.8 -1.8 -2.4 -2.4 -2.4 -2.4 -2.4 -2.4 0 -3 0 -3 0 0.5T 1.0T 1.5T 2.0T 0 0.5T 1.0T 1.5T 2.0T θ [T] θ [T]

Figure 6.4: Percentage differences in segment time between the CRTBP and ERTBP as a function of the true anomaly of the primaries and the position along the halo orbit, for formations along the x- and y-axes.

54 6.3. LINEAR APPROXIMATION OF THE RESULTS

Table 6.5: Maximum percentage differences in segment time between the ERTBP and CRTBP for rotating and inertial formations along different orientations, as follows from the linearized equations of motion. orientation rotating inertial x 2.64 2.52 y 2.49 2.55 z 2.52 2.52

To conclude, the maximum differences in segment time between the CRTBP and ERTBP corresponding to Figures 6.3 and 6.4 are given in Table 6.5. Note that the maximum difference of segment times in the ERTBP with respect to the CRTBP is always positive. However, the maximum decrease is only slightly smaller in magnitude (no more than 0.03%) than the maximum increase for the cases considered in Table 6.5.

55

Chapter 7

Formation Flying in the Restricted Four- Body Problem

This chapter discusses the second gravitational deviation from the CRTBP in the Sun-Earth/Moon system, caused by the Moon’s gravitation. The effect of the Moon can be simulated by adjusting the state derivative model according to the equations of motion presented in Section 3.2. Note that the Moon is assumed to follow a constant elliptical orbit about the Earth, that is defined using JPL orbital parameters for April 10, 2000, which are given in Appendix C for reference. The state derivative model used for the simulations, includes a calculation of the Moon’s position, performing a Keplerian to Cartesian state conversion, as well as a correction for the rotation of the RTBP reference frame. Section 7.1 will present results obtained from the integration of the full non-linear system, after which results of the linear approximation are presented in Section 7.2.

7.1 Results

Figure 7.1 presents the segment times and ∆V’s for rotating, 50 m formations at different orientations, in the CRTBP as well as the RFBP, where the Moon is initialized at a true anomaly of zero. It can be seen that gravitational perturbations due to the Moon are very small, and hardly noticeable in Figure 7.1. Upon close inspection, one can distinguish an oscillation of results for the RFBP about the CRTBP for a formation in y-direction as it is near perigee, where the effect of the Moon will become most pronounced. In order to more clearly demonstrate the effect of a third massive body in the bi-elliptical model used, Figure 7.2 presents an exaggerated case, where the Moon is assumed to have a mass parameter µ¯ = 0.1, as opposed to the actual µ¯ = 0.012150579. Figure 7.2 more clearly shows an oscillation of the RFBP segment times and ∆V’s about those in the CRTBP. We will refer to positive and negative differences between the RFBP and the CRTBP as seen in Figure 7.2 as hills and troughs respectively. It can be seen that the hills and troughs become more pronounced when the formation is near perigee, due to the closer proximity to the Moon’s orbit. Considering the segment time, the troughs correspond to epochs when the Earth and Moon are close to alignment with the x-axis, where a deeper trough corresponds to the Moon having a positive displacement along x and a subsequent more shallow trough corresponds to the Earth having a positive displacement along x, the former causing the largest change in relative acceleration. The hills, being less pronounced than the troughs, correspond to epochs when the Earth and Moon are close to alignment with the y-axis, during which the difference of the overall gravitation from the Earth-Moon system with respect to a coinciding Earth-Moon system become negative. Troughs in the segment time correspond to a larger relative acceleration, and hence become hills in the ∆V graph. Given the small effect of the Moon on formation flying in the Sun-Earth/Moon system, it is not considered relevant to present more graphical or numerical data based on the simulations performed, but rather the remainder of conclusions related to formation flying will be based on the linear approximation presented in Section 7.2. The absolute station keeping required to account for the Moon’s gravitational perturbation to the CRTBP in the Sun-Earth/Moon system, using a 4-shoot strategy, is approximately 1.24 m/s over an entire halo orbit, if the Moon is initialized at a true anomaly of zero 1. Note that this ∆V budget is approximately 7 times smaller than the absolute station keeping required to account for the perturbation due to the Earth’s eccentricity, being 9.12 m/s for the Earth’s initial true anomaly being equal to zero, as presented in Table

1Similar to the station keeping required in the ERTBP, the absolute station keeping to account for the Moon’s gravitation is slightly different for different initializations of the Earth-Moon system, showing differences of up to approximately 0.1 m/s.

57 CHAPTER 7. FORMATION FLYING IN THE RESTRICTED FOUR-BODY PROBLEM

(a) (b) 1.7 1.5 x CRTBP 1.6 x FBP y CRTBP 1.4 y FBP 1.5 z CRTBP 1.3 z FBP 1.4 1.2 1.3 1.1 1.2 1 1.1

Δ V [µm/s] 0.9 1

segment time[days] 0.8 0.9 x CRTBP 0.7 x FBP 0.8 y CRTBP y FBP 0.7 0.6 z CRTBP z FBP 0.6 0.5 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 7.1: (a) Segment times and (b) ∆V’s for 50 m rotating formations, oriented along the x-, y-, and z-axes in the CRTBP as well as the RFBP, for an error corridor of 1 cm and the Moon’s initial true anomaly equal to zero.

(a) (b) 1.8 1.5 x CRTBP x FBP y CRTBP 1.4 1.6 y FBP z CRTBP 1.3 z FBP 1.2 1.4 1.1 1.2 1

Δ V [µm/s] 0.9 1

segment time[days] 0.8 x CRTBP 0.7 x FBP 0.8 y CRTBP y FBP 0.6 z CRTBP z FBP 0.6 0.5 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 time [days] time [days]

Figure 7.2: (a) Segment times and (b) ∆V’s for 50 m rotating formations, oriented along the x-, y-, and z-axes in the CRTBP as well as the RFBP, for an error corridor of 1 cm and the Moon’s initial true anomaly equal to zero. The Moon’s mass parameter is increased to 0.1.

58 7.2. LINEAR APPROXIMATION OF THE RESULTS

4.3. Do note that superimposing the RFBP onto the ERTBP, hence accounting for the Moon’s orbit and the Earth’s eccentricity simultaneously, can result in situations where the Moon’s gravitational perturbation either adds to, or decreases the gravitational perturbation due to the Earth’s eccentricity. Hence, depending on the initialization of the Sun-Earth/Moon system, the absolute station keeping ∆V budget for the RFBP superimposed onto the ERTBP, can be either larger or smaller than the 9.12 m/s required for the ERTBP alone. We will not further consider the absolute station keeping requirements, as it is not the focus of this study.

7.2 Linear approximation of the results

Adjusting the linear approximations for the RFBP according to the equations of motion given by Equation 3.19 is straightforward and can be achieved by simply changing the second partial derivatives appearing in Equations 6.7 - 6.9 and 6.10 - 6.12 by the second partial derivatives corresponding to the pseudo-potential function F , given by Equation 3.22, such that linear approximations to the maximum relative position error, ∆V, and segment time in the RFBP for rotating formations become:   Fxx + e cos f Fxy − 2e sin f Fxz 1 −1 2 εmax ≈ (1 + e cos f) Fyx + 2e sin f Fyy + e cos f Fyz δr ∆t (7.1) 8 Fzx Fzy Fzz   Fxx + e cos f Fxy − 2e sin f Fxz −1 ∆V ≈ (1 + e cos f) Fyx + 2e sin f Fyy + e cos f Fyz δr ∆t (7.2)

Fzx Fzy Fzz v u   −1 u Fxx + e cos f Fxy − 2e sin f Fxz u ∆t ≈ 8(1 + e cos f) Fyx + 2e sin f Fyy + e cos f Fyz δr εmax (7.3) t Fzx Fzy Fzz The second partial derivatives of F are presented in Appendix A for reference. Note that linear approxima- tions for inertial formations are again obtained by adding −1 to the top-left and middle entry of the mapping matrix in Equations 7.1 through 7.3. As for the CRTBP and ERTBP, an accuracy analysis can be performed for the linear approximations in the RFBP. This is presented in Table 7.1, showing the percentage differences between the linear approximations and the integration of the full non-linear system, for the same conditions as used for Table 6.3 in the ERTBP. Note that the RFBP is superimposed onto the ERTBP to obtain a worst case scenario. Namely, the closer proximity of the formation to the Earth-Moon system when the Earth is in perihelion will increase the Moon’s perturbation. As expected, The accuracy of the linear approximations becomes slightly worse after adding the RFBP to the ERTBP, due to more rapidly varying relative accelerations.

In order to map the effect of the Moon on a formation about the Sun-Earth L2 point, one could use the linear approximation presented by Equations 7.3 for a simplified Moon model, assuming a circular Moon orbit, that lies within the xy-plane. Under these assumptions, Figure 7.3 shows the percentage differences in segment time between the RFBP and the CRTBP for rotating and inertial formations along the x-, y-, and z-axes, depending on the position along the halo orbit and the Moon’s position, which is specified by its angle α with the x-axis. Looking at Figure 7.3, the biggest differences naturally occur when the formation is at closest proximity to the Moon, occurring when the formation is in perigee, and the Moon has a positive displacement along the x-axis, with an α of 0 or 2π, causing a decrease in segment time. There is a smaller decrease in segment time when the Moon is at an angle π, since the positive displacement of the Earth along the x-axis still increases the relative acceleration at the formation’s position, but less so than when the Moon has a positive displacement along x. Furthermore, the first half of the halo orbit corresponds to negative y-values, whereas the Moon has positive y-values for 0 < α < π, hence the distance between the Moon and the formation is larger than for the second half of the halo orbit, causing slightly smaller and

59 CHAPTER 7. FORMATION FLYING IN THE RESTRICTED FOUR-BODY PROBLEM

(a) x rotating (b) x inertial 2π 0.2 2π 0.2

0.1 0.1

0 0

-0.1 -0.1 1π 1π

α [rad] -0.2 α [rad] -0.2

-0.3 -0.3

-0.4 -0.4

0 -0.5 0π -0.5 0 0.5T 1.0T 0. 0.5T 1.0T θ [T] θ [T]

(c) y rotating (d) y inertial 2π 0.3 2π 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 1π -0.2 1π

α [rad] α [rad] -0.2 -0.3 -0.3 -0.4 -0.5 -0.4 -0.6 -0.5 0 -0.7 0 -0.6 0 0.5T 1.0T 0 0.5T 1.0T θ [T] θ [T]

(e) z 2π 0.1

0

-0.1

-0.2 1π

α [rad] -0.3

-0.4

-0.5

0 -0.6 0 0.5T 1.0T θ [T]

Figure 7.3: Percentage differences between the RFBP and the CRTBP for rotating and inertial formations along the x-, y-, and z-axes, where the Moon is assumed to follow a circular orbit that lies in the ecliptic.

60 7.2. LINEAR APPROXIMATION OF THE RESULTS

Table 7.1: Percentage differences between the linear approximations given by Equations 7.1 and 7.2, com- pared to the integration of the full non-linear system, for different formation separations and segment times ∆t. Rotating and inertial formations oriented along the x-axis are considered, with the Earth’s initial true anomaly equal to 3π/2 and the Moon’s initial true anomaly equal to zero. The table shows the accuracy of ε; ∆XV. XXX separation XXX 100 m 10 km 1,000 km 10,000 km ∆t XXX Rotating 1 day 0.12; 0.16 0.13; 0.16 0.22;0.25 1.06; 1.09 2 days 0.26; 0.48 0.27; 0.48 0.36; 0.59 1.20; 1.41 5 days 1.26; 2.14 1.26; 2.14 1.36; 2.23 2.18; 3.07 10 days 4.80; 10.25 4.80; 10.25 4.89; 10.34 5.59; 11.15 Inertial 1 day 0.14; 0.19 0.14; 0.19 0.24; 0.29 1.15; 1.20 2 days 0.28; 0.56 0.28; 0.56 0.38; 0.66 1.29; 1.57 5 days 1.40; 2.57 1.40; 2.57 1.50; 2.67 2.40; 3.57 10 days 5.09;12.08 5.09;12.08 5.19;12.18 6.06; 13.06

Table 7.2: Maximum percentage differences between the RFBP and the CRTBP, as well as the ERTBP with the Earth’s true anomaly equal to zero, for rotating and inertial formations along different orientations. CRTBP ERTBP orientation rotating inertial rotating inertial x -0.46 -0.49 -0.48 -0.52 y -0.62 -0.52 -0.65 -0.55 z -0.53 -0.53 -0.56 -0.56 larger relative accelerations respectively, giving rise to the observable wave in Figure 7.3 for any formation. A similar conclusion holds when the Moon is located at π < α < 2π, in which case the first half of the halo orbit corresponds to slightly smaller segment times than the second half. Finally, Table 7.2 presents the maximum differences in segment time for rotating and inertial formations along different orientations, between the RFBP and the CRTBP, as well as between the RFBP and ERTBP for an Earth true anomaly of zero, representing a worst case scenario. As mentioned before, we indeed observe a slight increase in the maximum differences when the RFBP is superimposed onto the ERTBP compared to the CRTBP. Note that the maximum effect of the Moon on formations near the Sun-Earth L2 point is about four times smaller than the effect of the Sun-Earth ellipticity, as shown in Table 6.5.

61

Chapter 8

Formation Flying with Solar Radiation Pres- sure

This chapter presents a brief discussion on SRP and its effect on the relative formation dynamics. The SRP will be modelled using the normalized perturbing acceleration given by Equation 3.27, where the surface reflectivity constant is assumed equal to 1.6. Note that only the linearized equations of motion are considered, rather than a simulation of the full non-linear system, as explained in Section 8.1. The results of SRP in terms of its effect on the required formation control is presented in Section 8.2.

8.1 Approach

The analysis of SRP will be limited to a formation whose surfaces are normal to the incoming solar radiation, where the same optical properties are assumed for the chief and deputy. Also note that no eclipses are entered, so that the SRP acceleration becomes fairly constant, changing only with distance from the Sun, which is a relatively small variation throughout the course of an orbit, let alone over the course of one segment. Also neglecting the difference in absolute position between the chief and deputy, as well as the angle between r1 (the position vector with respect to the Sun) and the x-axis, being smaller than 0.3 deg for any point along the halo orbit considered, allows for a simple implementation of the SRP that yields an acceleration directed entirely along the x-axis and varies only with the chief and deputy’s difference in area-to-mass ratio and the magnitude of r1, where the latter causes a variation of no more than 1% in SRP acceleration throughout the relatively small halo orbit considered. The relative acceleration due to SRP can readily be added to the variational equation of motion. The linearized acceleration for a rotating formation in the CRTBP under the influence of SRP becomes:     Uxx Uxy Uxz δr¨SRP δr¨ = Uyx Uyy Uyz δr +  0  (8.1) Uzx Uzy Uzz 0

Note that Equation 8.1 can replace the acceleration in Equations 5.10 through 5.12 to obtain linear approx- imations of the maximum relative position error, ∆V, and segment time in the CRTBP under the influence of SRP. Checking the magnitude of relative accelerations with and without SRP, it becomes apparent that for non-identical satellites, the SRP quickly dominates the relative dynamics for small formation separations. Table 8.1 shows a comparison of the relative accelerations due to gravitation in the CRTBP (as obtained from Equation 5.5) and SRP for a rotating formation along x that is in perigee, for different formation separations and area-to-mass ratio differences between the deputy and the chief. Looking at Table 8.1, it can be seen that the solar radiation pressure has a significant impact on the relative acceleration for a formation of non- identical satellites, showing that only a difference of 1 mm2 in spacecraft effective area per kg mass already causes a relative acceleration larger than the gravitational acceleration for a 10 m formation in the CRTBP. Also recall that the SRP model assumes a surface that is normal to the incoming solar radiation, whereas in reality, small angles of the surface normal with respect to the incoming solar radiation could already cause large relative accelerations. Namely, an area-to-mass ratio1 of 1/100 m2/kg causes an acceleration compo- nent on the order of 1µm/s2 normal to the x-axis for a 1 deg tilt from the x-axis, being near the same order of

1Note that Wertz (2009) specifies 1/200 m2kg−1, 1/65 m2kg−1, and 1/20 m2kg−1 to correspond to high, moderate, and low density satellites respectively.

63 CHAPTER 8. FORMATION FLYING WITH SOLAR RADIATION PRESSURE

Table 8.1: Comparison of the relative acceleration magnitude due to gravitation, ag, and SRP, asrp, for a rotating formation along x in the CRTBP while being in perigee for a halo orbit corresponding to x0 = 1.01124. Differences in area-to-mass ratios, ∆A/m, and formation separation distances are considered. separation 10 m 10 km 1,000 km 10,000 km 2 −12 −9 −7 −6 ag [m/s ] 5.29 ×10 5.29 ×10 5.29 ×10 5.29 ×10 ∆A/m [m2/kg] 1/100 1/103 1/104 1/106 2 −8 −9 −10 −12 asrp [m/s ] 7.13 ×10 7.13 ×10 7.13 ×10 7.13 ×10 magnitude as the relative gravitational acceleration for a 10 km formation, already decreasing the maximum segment time to a little over 2 hours. Based on the above discussion, it can be concluded that impulsive control for small formation separations on the order of 100 m will yield small segment times in the presence of SRP for a formation of non-identical spacecraft requiring high relative accuracies. If not because of a difference in area-to-mass ratio, a different orientation of the surface normal with respect to the incoming solar radiation can cause the relative acceler- ation due to SRP to dictate the formation dynamics. Using impulsive control to achieve relative accuracies of 1 cm, and segment times on the order of a day for a 100 m formation in the halo orbit considered, requires active attitude control of individual spacecraft to cancel the relative acceleration due to SRP (or identical spherical spacecraft with homogeneous surface properties). Given the fact that the implementation of the SRP model considered in this chapter causes a very constant relative acceleration throughout the halo orbit, the following section presents results based on the variational equation of motion, given by Equation 8.1, rather than the full non-linear system of equations of motion.

8.2 Results

As was shown in the previous section, SRP can easily become the dominant source of relative accelerations within a formation for the halo orbit considered. In order to show the effect of SRP on a formation where the relative gravitational accelerations are of the same order as the relative accelerations due to SRP, Figure 8.1 presents the segment time for 1000 km formations along the x-, y-, and z-axes in the CRTBP under the influence of SRP, with area-to-mass ratios of 1/50 and 1/100 m2/kg for the deputy and chief respectively. Looking at the effect of SRP for a formation along x, it can be seen that the SRP decreases the time in between maneuvers, hence posing more strict requirements on relative formation keeping. This because the relative gravitational acceleration increases in x-direction for a positive x-displacement as follows from the positive sign of Uxx (see Equation 5.22). The higher area-to-mass ratio of the Deputy also causes a positive relative acceleration along x, so the SRP is additive to the CRTBP in x-direction for the entire halo orbit in terms of relative acceleration. Note that if the deputy has a lower area-to-mass ratio than the chief, the SRP cancels part of the gravitational acceleration in x, yielding larger segment times, assuming the SRP acceleration is smaller than the gravitational acceleration. The effect of SRP on a formation along the y-axis can be explained by considering the effect of a displace- ment along y on the relative gravitational acceleration in x, as contained in the Uxy partial derivative. Uxy is negative for negative y-values, hence, for most of the first half of the halo orbit the solar radiation pressure cancels part of the relative gravitational acceleration, yielding larger segment times, and vice versa for the second half of the orbit. Finally, considering the Uxz partial derivative, which is positive for positive z-values and vice verca, one finds that the SRP increases the relative gravitational acceleration if the formation is above the xy-plane, which it is for the largest part of the halo orbit, and cancels part of the relative acceleration when the forma- tion is below the xy-plane.

In order to obtain a quick estimate on the absolute station keeping required due to the SRP for a formation in the halo orbit considered, the SRP acceleration can be integrated over one halo orbit, amounting to

64 8.2. RESULTS

(a) rotating 22 x x with SRP 20 y y with SRP z 18 z with SRP

16

14

12

10 time per maneuver[minutes] 8

6 0 20 40 60 80 100 120 140 160 180 time [days]

Figure 8.1: Comparison of segment times for rotating formations in the CRTBP along the x-, y-, and z-axes with a 1000 km separation, and area-to-mass ratios of 1/50 and 1/100 m2kg−1 for the Deputy and Chief respectively, using the TTM method with a 1 cm error corridor. approximately 1.1 m/s for an area-to-mass ratio of 1/50 m2/kg. Note that this is on the same order as the station keeping required to counteract the Moon’s gravitation, and approximately 8 times smaller than that required to account for the Earth’s eccentricity as given in Table 4.3.

65

Chapter 9

Conclusions and Recommendations

The investigation of formation flying about the Sun-Earth L2 point in the unperturbed CRTBP has provided a clear mapping of the station keeping requirements in a halo orbit using impulsive control, showing the depen- dencies of the time in between corrective maneuvers, or so-called segment time, and ∆V on the formation’s orientation, separation, and type, being either inertial or rotating along with the RTBP reference frame, as well as the maximum allowed relative position error, or so-called error corridor. Using an integration of the full non-linear system of equations of motion in the unperturbed CRTBP, it was shown that for a XEUS-like mission, with a nominal formation separation of 50 m and occupying a relatively small halo orbit1, a relative position accuracy of 1 cm can be achieved with impulsive control for segment times of 0.6-1.7 days and ∆V’s of 0.6-1.4 µm/s, depending on the formation’s position, orientation and type. For the relatively small formation separations and segment times considered, the maximum relative position error is approximately proportional to the formation separation distance and the square of the segment time, and the ∆V is approxi- mately proportional to the segment time and formation separation distance. Furthermore, it has been shown that using an inertial formation, as opposed to a rotating formation, increases segment times by up to 8% and decreases required ∆V’s by a maximum of 7% for a formation along x, depending on the position along the halo orbit considered. Contrary to a formation along x, an inertial formation along y shows a decrease in segment times and an increase in the required ∆V’s, by a maximum of 18% and 23% respectively, when compared to a rotating formation. Three perturbations to the CRTBP and their effect on formation flying in the Sun-Earth/Moon system have been investigated, being the eccentricity of Earth’s orbit about the Sun, the gravitation of the Moon, and SRP. The largest gravitational perturbation was shown to be caused by the Earth’s eccentricity. Using linear approximations, the differences in segment time and ∆V between the CRTBP and ERTBP were shown to be mainly driven by the Earth’s true anomaly, showing differences in segment time ranging from approximately -2.6 % to 2.6%, where the maximum increase occurs when the Earth is near apohelion and the maximum decrease when the Earth is near perihelion. Differences in ∆V are approximately equal to differences in segment time, but opposite in sign. For formation orientations with a component in x- or y- direction, the differences in segment time are also slightly dependent on the formation’s orientation and position along the halo orbit, due to the apparent acceleration in the non-dimensional reference system for the ERTBP, caused by the pulsating axes, changing the total percentage difference by up to 0.4%2. A second gravitational perturbation to the CRTBP that has been investigated is the Moon’s gravitation. Assuming a circular Moon orbit that lies in the ecliptic, linear approximations were used to show the differ- ences in segment time and ∆V between the CRTBP and RFBP as a function of the Moon’s position and the position along the halo orbit. A maximum difference in segment time and ∆V of up to approximately 0.6% was found, being about four times smaller than the maximum difference due to the eccentricity of the Earth. The last perturbation covered, is the SRP, which was shown to be the most dominant cause of relative accelerations for formations of non-identical spacecraft at small separations. It was shown that for a differ- ence in area to mass ratio of 1/100 m2/kg, the relative acceleration due to SRP is already over 1000 times larger than the relative gravitational acceleration for a 100 m rotating formation along x. For such a case, using impulsive control to achieve a 1 cm relative accuracy would require segment times of approximately 17 minutes, mostly negating the benefits of impulsive control as opposed to continuous control. Even for spacecraft of equal dimensions and surface properties, a difference in orientation of the surface normal with

1 The orbit proposed for XEUS by Chabot and Udrea (2006), is close to the halo orbit corresponding to x0 = 1.01124 as given in Table 4.2. 2The maximum 0.4% percentage difference caused by the pulsating axes of the non-dimensional reference system occurs for a rotating formation along y that is near apogee when the Earth is near perihelion or apohelion, as explained in Section 6.3.3.

67 CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS respect to the incident solar radiation can cause large relative accelerations, already on the order of 1 µm/s for a 1 degree tilt from the incoming solar radiation. Hence, if one aims to achieve high relative accuracies of 1 cm for a 100 m formation, using impulsive control in the presence of SRP, segment times on the order of a day are only possible if the spacecraft attitudes are actively controlled, such as to cancel the relative accelerations due to SRP. Linear approximations were derived for the maximum relative position error, ∆V, and segment time in the CRTBP, ERTBP, and RFBP. These linear approximations are useful for a quick and easy way to determine the aforementioned quantities for any point in the Sun-Earth/Moon system. Given the nearly constant relative acceleration for relatively small segment times compared to the halo orbital period, these approximations have shown to be quite accurate for the halo orbit considered in the Sun-Earth/Moon system, still having an accuracy of around 1% for formation separations as high as 10,000 km, and a segment time of 2 days. Moreover, the linear approximations give insight into the relative dynamics, and helped us to understand influences due to, for instance, different formations orientations, inertial versus rotating formations, and the eccentricity of the primaries. Even though the focus of this study was on impulsive control, one can extend many of the results to continuous control, by treating it as impulsive control in the limit of infinite maneuvers, which applies to the linear approximations in particular, for they become more accurate as the segment time decreases.

Based on certain obstacles, or ideas, encountered during the course of this study, of few recommendations for future research are given here. Firstly, quite early on, the scope of this study was limited to a formation in a halo orbit of certain dimensions in the CRTBP. This selected nominal orbit was extended to the perturbed problem, with little attention payed to the absolute station keeping cost. Given that the absolute station keeping yields a far higher ∆V budget than relative station keeping for formations in a halo orbit about the Sun-Earth L2 point, further research would benefit from optimizing the nominal halo orbit in terms of the required absolute station keeping. Secondly, the ∆V’s required for relative station keeping were simulated without inaccuracies, whereas in reality thrust is always associated with a certain implementation error. This is especially the case for the small thrust levels required for formations near the Sun-Earth L2 point. Hence, future research would benefit from adding a sensitivity analysis to such thrust implementation errors. Furthermore, an impulsive control strategy was investigated. This choice was based on Howell and Marchand (2005), mentioning that such a discrete targeting approach can be effective for small formations in a halo orbit in the Sun-Earth/Moon system, for which continuous control could yield prohibitively small thrust requirements. Whereas this study draws a similar conclusion when considering the relative gravitational accelerations, it was shown that SRP can have a very significant impact on the relative acceleration for a formation about the Sun-Earth L2 point, causing the segment time for an impulsive control strategy to become very small, being less than half an hour for a difference in area-to-mass ratio of 1/50 m2kg−1, for any kind of formation. The LISA pathfinder mission has already demonstrated that continuous control can be used to constantly counteract the solar radiation pressure. Similarly, continuous control could be used for a formation in a halo orbit about the Sun-Earth L2 point, to simultaneously perform absolute and relative station keeping of a formation, which would be an interesting topic for future studies. The SRP model used in the simulations performed, was greatly simplified by assuming a spacecraft surface that is normal to the incoming solar radiation, and the formation distance from the x-axis was neglected, such that the solar radiation force was directed entirely along the x-axis. Given that the SRP can easily be the biggest contributor to relative accelerations within a formation, a more realistic SRP model would likely be essential for accurate simulations corresponding to a well defined mission, taking into account the shape of the spacecraft, its surface properties, and its absolute position. One could also consider using the SRP to ones advantage, as a means of control to perform relative and/or absolute station keeping, the former of which seems very feasible, given that a 1 cm2 effective area difference per kg already causes a relative acceleration due to SRP whose magnitude is of the same order as the relative gravitational acceleration for formations separations up to 100 km. The simulations were performed, assuming a bi-elliptical model, whereas in reality the eccentricity of the Earth about the Sun, and more significantly, the eccentricity of the Moon about the Earth, will show

68 variations over time. In order to simulate the dynamics within a formation about the Sun-Earth L2 point most accurately, one should employ a full ephemerides model, taking into account the exact positions of all major bodies in the solar system. This does however, do away with the possibility of distinguishing the individual perturbations considered in this study, hence was not performed here. Finally, the double precision limits should be taken into account, which becomes especially relevant for simulations of a formation near the Sun-Earth L2 point, given the high contrast of relative and absolute positions. These limits were not restrictive for the simulations performed in this study, though they will likely become so if larger relative accuracies, smaller than 1 cm, are required, which might necessitate the use of a higher precision format.

69

Bibliography

Bavdaz, M., Lumb, D., and Peacock, T. XEUS mission reference design. In Proceedings of SPIE - The Interna- tional Society for Optical Engineering, volume 5488, pages 530–538, 2004. Bavdaz, M., Lumb, D., Gerlach, L., Parmar, A., and Peacock, A. The status of the XEUS X-ray observatory mission. In Proceedings of SPIE - The International Society for Optical Engineering, volume 5898, pages 1–9, 2005. Breakwell, J. V. and Brown, J. V. The ’halo’ family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem. Celestial Mechanics, 20:389–404, 1979.

Bristow, J., Folta, D., and Hartman, K. A formation flying technology vision. In Space 2000 Conference and Exposition, 2000. Campagnola, S., Lot, M., and Newton, P. Subregions of motion and elliptic halo orbits in the elliptic restricted three-body problem. volume 130 PART 2, pages 1541–1555, 2008.

Carpenter, K. G. et al. SI - the stellar imager, vision mission study report. 2005. URL http://hires.gsfc. nasa.gov/~si/. Carpenter, R. J., Leitner, J. A., Folta, D. C., and Burns, R. D. Benchmark problems for spacecraft formation flying missions. In AIAA Guidance, Navigation, and Control Conference and Exhibit, 2003. Chabot, T. and Udrea, B. XEUS mission guidance navigation and control. In Collection of Technical Papers - AIAA Guidance, Navigation, and Control Conference 2006, volume 6, pages 4110–4128, 2006. De Pater, I. and Lissauer, J. J. Planetary Sciences. Cambridge university press, 2001. Espenak, F. and Meeus, J. Five millenium catalog of solar eclipses. (TP-2009-214714), 2009. Farquhar, R. W. Station-keeping in the vicinity of collinear libration points with an application to a lunar communications problem. AAS Science and Technology Series: Spaceflight Mechanics Specialist Symposium, 11:519–535, 1966. Ghorbani, M. and Assadian, N. Optimal station-keeping near Earth-Moon collinear libration points using continuous and impulsive maneuvers. Advances in Space Research, 52(12):2067–2079, 2013. Howell, K. C. Three-dimensional, periodic, ’halo’ orbits. Celestial Mechanics, 32:53–71, 1984.

Howell, K. C. and Marchand, B. G. Natural and non-natural spacecraft formations near the L1 and L2 libration points in the Sun-Earth/Moon ephemeris system. Dynamical Systems, 20:149–173, 2005. Marchand, B. C. and Howell, K. C. Aspherical formations near the libration points in the Sun-Earth/Moon ephemeris system. In Advances in the Astronautical Sciences, volume 119, pages 835–859, 2005.

McInnes, C. R. Solar Sailing: Technology, Dynamics, and Mission Applications. Springer-Verlag Berlin Heidel- berg, 1999. Milligan, D., Rudolph, A., Whitehead, G., Loureiro, T., Serpell, E., di Marco, F., Marie, J., and Ecale, E. ESA’s billion star surveyor flight operations experience from Gaia’s first 1.5 years. Acta Astronautica, 127:394 – 403, 2016. ISSN 0094-5765.

Montenbruck, O. and Gill, E. Satellite Orbits: Models, Methods, and Applications. Springer, 2005.

71 BIBLIOGRAPHY

Noci, G., Matticari, G., Siciliano, P., Fallerini, L., Boschini, L., and Vettorello, V. Cold gas micro propulsion system for scientific satellites fine pointing: Review of development and qualification activities at Thales Alenia Space Italia. 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, 2009.

Pernicka, H. J., A., C. B., and N., B. S. Spacecraft formation flight about libration points using impulsive maneuvering. 29(5):1122–1130, 2006. doi: 10.2514/1.18813. Qi, R. and Xu, S. Impulsive control strategy for formation flight in the vicinity of the libration points. In 62nd International Astronautical Congress 2011, IAC 2011, volume 6, pages 4874–4884, 2011.

Rudolph, A., Harrison, I., Renk, F., Firre, D., Delhaise, F., Garcia Marirrodriga, C., McNamara, P., Johlander, B., Caleno, M., Grzymisch, J., Cordero, F., Wealthy, D., and Olive, J. LISA pathfinder mission operations concept and launch and early orbit phase in-orbit experience. 2016.

Segerman, A. M. and Zedd, M. F. Perturbed equations of motion for formation flight near the sun-earth L2 point. volume 123 II, pages 1381–1400, 2006.

Szebehely, V. Theory of orbits, the restricted problem of three bodies. Academic Press, inc., 1967. Tapley, B. D., Schutz, B. E., and Born, G. H. Statistical Orbit Determination. Elsevier Academic Press, 2004. Wakker, K. F. Astrodynamics-I reader AE4874, TU Delft, 2010. Wertz, J. R. Space Missions Analysis and Design. Microcosm Press and Kluwer Academic Pubilshers, 2005.

Wertz, J. R. Orbit and Constellation, Mission and Design. Microcosm Press and Springer, 2009.

72 Appendix A

Second Partial Derivatives of the Pseudo- Potential Function

A.1 CRTBP

The second partial derivatives for the pseudo-potential function U, defined by Equation 2.3, are:   1 − µ µ 1 − µ 2 µ 2 Uxx =1 − 3 − 3 + 3 5 (x + µ) + 5 (x − 1 + µ) r1 r2 r1 r2   1 − µ µ 2 1 − µ µ Uyy =1 − 3 − 3 + 3y 5 + 5 r1 r2 r1 r2 1 − µ µ 1 − µ µ  U = − − + 3z2 + zz r3 r3 r5 r5 1 2 1 2 (A.1) 1 − µ µ  Uxy =Uyx = 3y 5 (x + µ) + 5 (x − 1 + µ) r1 r2 1 − µ µ  Uxz =Uzx = 3z 5 (x + µ) + 5 (x − 1 + µ) r1 r2 1 − µ µ  Uyz =Uzy = 3yz 5 + 5 r1 r2

A.2 ERTBP

The second partial derivatives for the pseudo-potential function ω, defined by Equation 3.15, are:

−1 ωxx =(1 + e cos f) Uxx −1 ωxy =(1 + e cos f) Uxy −1 ωxz =(1 + e cos f) Uxz −1 (A.2) ωyy =(1 + e cos f) Uyy −1 ωyz =(1 + e cos f) Uyz −1 ωzz =(1 + e cos f) (1 + Uzz)

73 APPENDIX A. SECOND PARTIAL DERIVATIVES OF THE PSEUDO-POTENTIAL FUNCTION

A.3 RFBP

The second partial derivatives for the pseudo-potential function F , defined by Equation 3.22, are:   1 − µ µ(1 − µ¯) µµ¯ 1 − µ 2 µ(1 − µ¯) 2 µµ¯ 2 Fxx =1 − 3 − 3 − 3 + 3 5 (x + µ) + 5 (x − xE) + 5 (x − xM ) r1 rE rM r1 rE rM   1 − µ µ(1 − µ¯) µµ¯ 1 − µ 2 µ(1 − µ¯) 2 µµ¯ 2 Fyy =1 − 3 − 3 − 3 + 3 5 y + 5 (y − yE) + 5 (y − yM ) r1 rE rM r1 rE rM 1 − µ µ(1 − µ¯) µµ¯ 1 − µ µ(1 − µ¯) µµ¯  F = − − − + 3 z2 + (z − z )2 + (z − z )2 zz r3 r3 r3 r5 r5 E r5 M 1 E M 1 E M (A.3) 1 − µ µ(1 − µ¯) µµ¯  Fxy =Uyx = 3 5 (x + µ)y + 5 (x − xE)(y − yE) + 5 (x − xM )(y − yM ) r1 rE rM 1 − µ µ(1 − µ¯) µµ¯  Fxz =Uzx = 3 5 (x + µ)z + 5 (x − xE)(z − zE) + 5 (x − xM )(z − zM ) r1 rE rM 1 − µ µ(1 − µ¯) µµ¯  Fyz =Uzy = 3 5 yz + 5 (y − yE)(z − zE) + 5 (y − yM )(z − zM ) r1 rE rM

74 Appendix B

Unit Conversions

Expressions for unit conversions from a dimensional to a non-dimensional system, and vice versa, are pre- sented for the CRTBP and ERTBP in Sections B.1 and B.2 respectively. All conversions in the CRTBP are one- to-one, whereas conversion in the ERTBP are not, so that conversions to dimensional and non-dimensional units have to be treated separately.

B.1 CRTBP

Note that, from Section 2.1, the dimensional and non-dimensional quantities for position and time are related through:

t =ttˆ ∗ (B.1) r =ˆrr∗ where the dimensionless quantities are denoted by an asterisk, and the normalized units for position and time are given by:

df −1 tˆ= dt (B.2) rˆ =a

Time Given the uniform rotation of the primaries, the unit conversion for time in the CRTBP is given by:

−1 r df  a3 t = t∗ = t∗ (B.3) dt GM assuming t and t∗ are both initialized at zero.

Position Given the constant distance between the primaries, the unit conversion for position in the CRTBP is given by:

r = ar∗ (B.4)

Velocity Again, due to the constant distance between the primaries, the unit conversion for velocity in the CRTBP is given by: r dr dt∗ d(r∗rˆ) df  dr∗ GM dr∗ = = a = (B.5) dt dt dt∗ dt dt∗ a dt∗

75 APPENDIX B. UNIT CONVERSIONS

Acceleration From Equation 3.8, for a uniform rotation and constant distance between the primaries, one obtains the unit conversion for acceleration in the CRTBP: d2r df 2 d2r∗ GM d2r∗ =r ˆ = (B.6) dt2 dt dt∗2 a2 dt∗2

B.2 ERTBP

Unit conversions in the ERTBP are slightly more involved, given the non-uniform rotation of the primaries, as well as the pulsating distance between them. The normalized units for position and time are given by:

df −1 tˆ= dt (B.7) a(1 − e2) rˆ = 1 + e cos f Note that conversions of accelerations in the ERTBP are never performed in this study, hence are not presented here.

Conversions to dimensional units Time A conversion to dimensional units for time is performed here by calculating the time since last pericenter passage form the true anomaly. First, the eccentric anomaly can be calculated from the true anomaly using Equation B.8 (Tapley et al., 2004). e + cos f cos E = (B.8) 1 + e cos f after which the dimensional time, t, can be obtained form Kepler’s equation:

E − e sin E = n(t − tp) (B.9) where t − tp is the time since last pericenter passage, and n is the mean motion. Note that Equation B.8 yields an eccentric anomaly in the range of 0 - 180 deg. Hence, for a true anomaly larger than 180 deg, 2(π − E) should be added to the eccentric anomaly. To obtain a certain timespan in dimensional units one has to calculate the time since last pericenter for both epochs and subtract.

Note that for relatively small timespans, a conversion to dimensional units can be performed by assuming the progression of time to be constant:

df −1 ∆t ≈ ∆t∗ dt s (B.10) a3(1 − e2)3 ≈ ∆t∗ GM (1 + e cos f)4

Position The unit conversion for position in the ERTBP is given by: a(1 − e2) r = r∗ (B.11) 1 + e cos f

76 B.2. ERTBP

Velocity The unit conversion to a dimensional system for velocity in the ERTBP is given by: dr d(r∗rˆ) dr∗ drˆ df  dr∗ drˆ = =r ˆ + r∗ =r ˆ + r∗ (B.12) dt dt dt dt dt dt∗ dt drˆ df d  a(1 − e2)  df e sin f = =r ˆ (B.13) dt dt df 1 + e cos f dt 1 + e cos f dr df dr∗ e sin f  =r ˆ + r∗ (B.14) dt dt dt∗ 1 + e cos f Note that the angular momentum in the two-body problem is given by: df p H =r ˆ2 = GMa(1 − e2) (B.15) dt Hence, s dr GM  dr∗  = (1 + e cos f) + e sin fr∗ (B.16) dt a(1 − e2) dt∗ Note that the velocity conversion in the ERTBP is not a simple one-to-one relationship as is the case for the CRTBP, but rather depends on both the true anomaly and the position at the time considered. The second term on the right-hand side of Equation B.16 accounts for the pulsating axes of the non-dimensional system. In the current study, we will mostly be concerned with impulsive ∆V ’s for station keeping, which will not be affected by the pulsation of the axes, hence: s GM ∆V = (1 + e cos f) ∆V ∗ (B.17) a(1 − e2)

Conversions to dimensionless units Time A conversion to dimensionless units for time is performed here by calculating the true anomaly form the time since last pericenter passage. The eccentric anomaly can be calculated from the mean anomaly, or last pericenter passage, using Kepler’s Equation:

M = n(t − tp) = E − e sin E (B.18) A Newton-Rhapson method can be used to iteratively solve for E (Montenbruck and Gill, 2005):

Ei − e sin Ei − M Ei+1 = Ei − (B.19) 1 − e cos Ei Once an eccentric anomaly with high enough accuracy is found, the true anomaly can be calculated using Equation B.20 (Tapley et al., 2004): cos E − e cos f = (B.20) 1 − e cos E Again, in case of an eccentric anomaly in the range of 180-360 deg, the true anomaly should be adjusted by 2(π − f).

For small timespans a conversion to dimensionless units can be performed by assuming the progression of time to be constant: df  ∆t∗ ≈ ∆t dt s (B.21) GM (1 + e cos f)4 ≈ ∆t a3(1 − e2)3

77 APPENDIX B. UNIT CONVERSIONS

Velocity The unit conversion to a dimensionless system for velocity in the ERTBP is given by:

dr∗ d(r/rˆ) df −1 d(r/rˆ) df −1 1 dr d(1/rˆ)  = = = + r (B.22) dt∗ dt∗ dt dt dt rˆ dt dt

Now, using d(1/rˆ) df d 1 + e cos f  df e sin f = = − (B.23) dt dt df a(1 − e2) dt a(1 − e2) and df p H =r ˆ2 = GMa(1 − e2) (B.24) dt one has r dr∗ a(1 − e2) dr e sin f = (1 + e cos f)−1 − r (B.25) dt∗ GM dt a(1 − e2) Again, calculating an instantaneous ∆V is not affected by the pulsation of the system, hence: r a(1 − e2) ∆V ∗ = (1 + e cos f)−1 ∆V (B.26) GM

78 Appendix C

Constants

Constants that are used for the simulations performed, are given by Table C.1.

Table C.1: Solar system constants. Quantity Value Units Source c; speed of light in a vacuum 299,792,458 [m/s] Wertz (2009) GM (Moon) 4.902798882 ×1012 [m3/s2] Wertz (2005) GM (Earth) 3.98600441 ×1014 [m3/s2] Wertz (2009) GM (Sun) 1.327178 ×1020 [m3/s2] Wertz (2009) AU 149,597,870,660 [m] Wertz (2009) Earth equatorial radius 6,378,136 [m] Wertz (2009) 2 S0; Solar flux at 1AU 1,367 [W/m ] Wertz (2009) Solar radius 6.96000×108 [m] De Pater and Lissauer (2001) e; eccentricity (Earth) 0.016708617 [-] De Pater and Lissauer (2001) a; semi-major axis (Moon) 381,055,426 [m] Ghorbani and Assadian (2013) e; eccentricity (Moon) 0.033544 [-] Ghorbani and Assadian (2013) i; inclination (Moon) 5.04246 [deg] Ghorbani and Assadian (2013) Ω; RAAN (Moon) 119.9538 [deg] Ghorbani and Assadian (2013) θ; argument of perigee (Moon) 314.4519 [deg] Ghorbani and Assadian (2013)

79