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

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Formation Flying in the Sun-Earth/Moon Perturbed Restricted Three-Body Problem DELFT UNIVERSITY OF TECHNOLOGY FACULTY OF AEROSPACE ENGINEERING MASTER’S THESIS Formation Flying in the Sun-Earth/Moon Perturbed Restricted Three-Body Problem IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF 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.
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