DOCSLIB.ORG

A Study of Eccentric Orbit Circularization Using Low-Thrust Propulsion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Study of Eccentric Orbit Circularization Using Low-Thrust Propulsion A Study of Eccentric Orbit Circularization using Low-Thrust Propulsion by Brenton John Du®y B.S. in Aerospace Engineering, Dec. 2005, North Carolina State University A Thesis submitted to the Faculty of The School of Engineering and Applied Science of The George Washington University in partial satisfaction of the requirements for the degree of Master of Science May 18, 2008 Thesis directed by Dr. David F. Chichka Assistant Professor of Engineering and Applied Science The School of Engineering and Applied Science of The George Washington University certi¯es that Brenton John Du®y has passed the Final Examination for the degree of Master of Science as of May 9, 2008. This is the ¯nal and approved form of the thesis. A Study of Eccentric Orbit Circularization using Low-Thrust Propulsion Brenton John Du®y Thesis Review Committee: Dr. David F. Chichka Assistant Professor of Engineering and Applied Science Dr. James D. Lee Professor of Engineering and Applied Science Dr. Helmut Haberzettl Associate Professor of Physics °c Copyright by Brenton John Du®y 2008 Acknowledgements I would like to express my thanks to Dr. David F. Chichka for all of his support and insight during this study. Our discussions were invaluable toward the progression of the study and I am sincerely grateful for all of his help. I would also like to thank Dr. Chichka, Dr. James D. Lee and Dr. Helmut Haberzettl for serving on my thesis committee and providing their expertise in reviewing my work. iv Abstract This study considers the optimal control problem of circularizing an orbit via a con- tinuous and variable low-thrust maneuver. In addition, constraints are applied to avoid disrupting scienti¯c experiments being conducted at apoapsis. The dynamics are modeled using Lagrange's planetary equations for the classic orbit elements with control variables de¯ned by the speci¯c thrust components. The objective is to max- imize the raise in periapsis while minimizing the fuel consumption. Several variations of the problem are considered including an unconstrained maneuver, a terminal point constraint maneuver, and a multi-point constraint maneuver. The results show a trade-o® between adding energy by increasing the semi-major axis and circularizing by decreasing the eccentricity. In comparing the low-thrust performance to equivalent impulsive maneuvers, the results demonstrate that the low-thrust systems can provide improved performance over the impulsive maneuvers. In addition, the optimization of multi-orbit maneuvers has a strong dependence on the number of orbits whereby the optimal thrust pro¯le changes between the orbits. This can be mitigated with some loss in performance by adding interior apoapsis constraints to each orbit. v Table of Contents Acknowledgements iv Abstract v Table of Contents viii List of Figures xi List of Tables xii List of Symbols xiii 1 Introduction 1 1.1 Magnetospheric Multiscale Mission ................... 1 1.2 Orbital Mechanics ............................. 2 1.2.1 Classic Orbit Elements ...................... 2 1.2.2 Circularization .......................... 5 1.3 Low-Thrust Propulsion .......................... 7 1.4 Notation .................................. 8 2 Optimal Control Theory for Low-Thrust Propulsion 10 2.1 Problem Formulation ........................... 10 2.2 Necessary Conditions for Optimality . 16 2.2.1 First-Order Conditions ...................... 20 2.2.2 Second-Order Conditions ..................... 23 2.3 Control Law ................................ 24 vi 2.4 Numerical Solution to the Terminally Constrained TPBVP . 26 2.5 Interior Point Constraints ........................ 31 2.5.1 First-Order Necessary Conditions for a Single Orbit . 32 2.5.2 First-Order Necessary Conditions for Multiple Orbits . 38 2.5.3 Second-Order Necessary Conditions . 40 2.5.4 Numerical Solution to the Interior Constrained MPBVP . 40 2.6 Performance Measures .......................... 44 3 Algorithm Validation 45 3.1 Maximize Semi-Major Axis ........................ 46 3.2 Minimize Eccentricity ........................... 49 4 Optimizing the Circularization Maneuver 54 4.1 Unconstrained Apoapsis ......................... 54 4.1.1 Single Orbit ............................ 55 4.1.2 Multiple Orbits .......................... 59 4.2 Terminally Constrained Apoapsis .................... 65 4.2.1 Single Orbit ............................ 66 4.2.2 Multiple Orbits .......................... 70 4.3 Interior and Terminally Constrained Apoapsis . 76 4.3.1 Single Orbit ............................ 76 4.3.2 Multiple Orbits .......................... 79 5 Conclusions 86 5.1 Low-Thrust Maneuver vs. Impulsive Maneuver . 87 5.2 Dependence on Number of Orbits .................... 88 References 91 vii Appendix A Co-State Di®erential Equations 94 Appendix B State and Co-State Perturbation Equations 96 Appendix C TPBVP Second Variation M-Matrix 109 Appendix D MPBVP Second Variation M-Matrix 111 viii List of Figures 1.2.1 In-Plane Orbit Elements ........................ 3 1.2.2 Classic Orbit Elements ......................... 4 2.1.1 Speci¯c Thrust Components ...................... 11 3.1.1 Maximize Semi-Major Axis: Optimization Parameters . 47 3.1.2 Maximize Semi-Major Axis: Orbit Elements . 47 3.1.3 Maximize Semi-Major Axis: Speci¯c Thrust Pro¯le . 48 3.1.4 Maximize Semi-Major Axis: Thrust Direction . 49 3.2.1 Minimize Eccentricity: Optimization Parameters . 50 3.2.2 Minimize Eccentricity: Orbit Elements . 51 3.2.3 Minimize Eccentricity: Apsidal Radii . 51 3.2.4 Minimize Eccentricity: Speci¯c Thrust Pro¯le . 52 3.2.5 Minimize Eccentricity: Thrust Direction . 53 4.1.1 UCA Single Orbit Maneuver: Orbit Elements . 55 4.1.2 UCA Single Orbit Maneuver: Apsidal Radii . 56 4.1.3 UCA Single Orbit Maneuver: Thrust Pro¯le . 57 4.1.4 UCA Single Orbit Maneuver: Thrust Direction . 57 4.1.5 UCA Single Orbit Maneuver: Argument of Periapsis . 58 4.1.6 UCA Multi-Orbit Maneuver: Semi-Major Axis . 59 4.1.7 UCA Multi-Orbit Maneuver: Eccentricity . 60 4.1.8 UCA Multi-Orbit Maneuver: Periapsis Radius . 60 4.1.9 UCA Multi-Orbit Maneuver: Apoapsis Radius . 61 4.1.10 UCA Multi-Orbit Maneuver: Thrust Pro¯les . 61 4.1.11 UCA Multi-Orbit Maneuver: Total Thrust . 62 ix 4.1.12 UCA Multi-Orbit Maneuver: Thrust Direction . 62 4.1.13 UCA Multi-Orbit Maneuver: Argument of Periapsis . 63 4.1.14 UCA Multi-Orbit Maneuver: Trajectories . 64 4.2.1 TCA Single Orbit Maneuver: Orbit Elements . 67 4.2.2 TCA Single Orbit Maneuver: Apsidal Radii . 67 4.2.3 TCA Single Orbit Maneuver: Thrust Pro¯le . 68 4.2.4 TCA Single Orbit Maneuver: Thrust Direction . 69 4.2.5 TCA Single Orbit Maneuver: Argument of Periapsis . 69 4.2.6 TCA Multi-Orbit Maneuver: Semi-Major Axis . 70 4.2.7 TCA Multi-Orbit Maneuver: Eccentricity . 71 4.2.8 TCA Multi-Orbit Maneuver: Periapsis Radius . 71 4.2.9 TCA Multi-Orbit Maneuver: Apoapsis Radius . 72 4.2.10 TCA Multi-Orbit Maneuver: Thrust Pro¯les . 73 4.2.11 TCA Multi-Orbit Maneuver: Total Thrust . 73 4.2.12 TCA Multi-Orbit Maneuver: Thrust Direction . 74 4.2.13 TCA Multi-Orbit Maneuver: Argument of Periapsis . 74 4.2.14 TCA Multi-Orbit Maneuver: Trajectories . 75 4.3.1 ITCA Single Orbit Maneuver: Orbit Elements . 77 4.3.2 ITCA Single Orbit Maneuver: Apsidal Radii . 77 4.3.3 ITCA Single Orbit Maneuver: Thrust Pro¯le . 78 4.3.4 ITCA Single Orbit Maneuver: Thrust Direction . 78 4.3.5 ITCA Single Orbit Maneuver: Argument of Periapsis . 79 4.3.6 ITCA Multi-Orbit Maneuver: Semi-Major Axis . 80 4.3.7 ITCA Multi-Orbit Maneuver: Eccentricity . 80 4.3.8 ITCA Multi-Orbit Maneuver: Periapsis Radius . 81 4.3.9 ITCA Multi-Orbit Maneuver: Apoapsis Radius . 81 x 4.3.10 ITCA Multi-Orbit Maneuver: Thrust Pro¯les . 82 4.3.11 ITCA Multi-Orbit Maneuver: Total Thrust . 83 4.3.12 ITCA Multi-Orbit Maneuver: Thrust Direction . 83 4.3.13 ITCA Multi-Orbit Maneuver: Argument of Periapsis . 84 4.3.14 ITCA Multi-Orbit Maneuver: Trajectories . 84 xi List of Tables 5.1 Variation in the Orbit Elements .................... 86 5.2 Performance Results ........................... 87 5.3 Normalized Performance Results .................... 89 xii List of Symbols (_ ) Total Time Derivative d=d( ), @=@( ) Total Derivative, Partial Derivative a Semi-Major Axis e Eccentricity 2 g0 Standard Sea Level Acceleration of Gravity (= 9:81m=s ) h Angular Momentum Vector i Orbit Inclination m, M Mass of Satellite, Mass of Central Body p n Mean Motion (= ¹=a3) q Number of Orbits for Multi-Orbit Maneuvers r Position Magnitude ra Apoapsis Radius rp Periapsis Radius rs Desired Scienti¯c Apoapsis Radius t; t0, tf , tn Time; Initial, Final, and Interior Times u Vector of Control Variables v Velocity Magnitude x; x0 Vector of State Variables; Prescribed Initial Conditions z Error Vector A, B, C, D Second Variation Sub-Matrices AU Canonical Distance Unit E Speci¯c Orbit Energy E; E0, Ef , En Eccentric Anomaly; Initial, Final, and Interior Eccentric Anomalies xiii F (x; u; t) Vector of System Dynamic Equations G Gravitational Constant H(x; u; ¸; t) Hamiltonian Function Isp Speci¯c Impulse J, J~ Performance Index, Augmented Performance Index L(x; u; t) Fuel Consumption Penalty Function M Error Matrix MR Mass Ratio of Final Mass over Initial Mass R Radial Speci¯c Thrust S Circumferential Speci¯c Thrust T Normal Speci¯c Thrust ± First Variation Operator ±2 Second Variation Operator ´n Vector of Interior Constraint Lagrange Multipliers for t = tn ¸ Co-States (Dynamic Equation Lagrange Multipliers) ¹ Gravitational Parameter (= G(M + m)) º Vector of Terminal Constraint Lagrange Multipliers º True Anomaly (only for Chapter 1) »n Vector of Interior Constraints for t = tn Á Objective Function à Vector of Terminal Constraints ! Argument of Periapsis ¢v Required Change in Velocity © Transition Matrix ­ Á + ºT à ­ Longitude of the Ascending Node (only for Chapter 1) xiv Chapter 1 - Introduction As the science of astrodynamics and celestial physics progresses, so too must the technology and engineering for designing our spacecraft of exploration.
Load more

Recommended publications
  • Appendix a Orbits
    Appendix A Orbits As discussed in the Introduction, a good ¯rst approximation for satellite motion is obtained by assuming the spacecraft is a point mass or spherical body moving in the gravitational ¯eld of a spherical planet. This leads to the classical two-body problem. Since we use the term body to refer to a spacecraft of ¯nite size (as in rigid body), it may be more appropriate to call this the two-particle problem, but I will use the term two-body problem in its classical sense. The basic elements of orbital dynamics are captured in Kepler's three laws which he published in the 17th century. His laws were for the orbital motion of the planets about the Sun, but are also applicable to the motion of satellites about planets. The three laws are: 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. The line joining the planet to the Sun sweeps out equal areas in equal times. 3. The square of the period of a planet is proportional to the cube of its mean distance to the sun. The ¯rst law applies to most spacecraft, but it is also possible for spacecraft to travel in parabolic and hyperbolic orbits, in which case the period is in¯nite and the 3rd law does not apply. However, the 2nd law applies to all two-body motion. Newton's 2nd law and his law of universal gravitation provide the tools for generalizing Kepler's laws to non-elliptical orbits, as well as for proving Kepler's laws.
    [Show full text]
  • Elliptical Orbits
    1 Ellipse-geometry 1.1 Parameterization • Functional characterization:(a: semi major axis, b ≤ a: semi minor axis) x2 y 2 b p + = 1 ⇐⇒ y(x) = · ± a2 − x2 (1) a b a • Parameterization in cartesian coordinates, which follows directly from Eq. (1): x a · cos t = with 0 ≤ t < 2π (2) y b · sin t – The origin (0, 0) is the center of the ellipse and the auxilliary circle with radius a. √ – The focal points are located at (±a · e, 0) with the eccentricity e = a2 − b2/a. • Parameterization in polar coordinates:(p: parameter, 0 ≤ < 1: eccentricity) p r(ϕ) = (3) 1 + e cos ϕ – The origin (0, 0) is the right focal point of the ellipse. – The major axis is given by 2a = r(0) − r(π), thus a = p/(1 − e2), the center is therefore at − pe/(1 − e2), 0. – ϕ = 0 corresponds to the periapsis (the point closest to the focal point; which is also called perigee/perihelion/periastron in case of an orbit around the Earth/sun/star). The relation between t and ϕ of the parameterizations in Eqs. (2) and (3) is the following: t r1 − e ϕ tan = · tan (4) 2 1 + e 2 1.2 Area of an elliptic sector As an ellipse is a circle with radius a scaled by a factor b/a in y-direction (Eq. 1), the area of an elliptic sector PFS (Fig. ??) is just this fraction of the area PFQ in the auxiliary circle. b t 2 1 APFS = · · πa − · ae · a sin t a 2π 2 (5) 1 = (t − e sin t) · a b 2 The area of the full ellipse (t = 2π) is then, of course, Aellipse = π a b (6) Figure 1: Ellipse and auxilliary circle.
    [Show full text]
  • Kepler's Equation—C.E. Mungan, Fall 2004 a Satellite Is Orbiting a Body
    Kepler’s Equation—C.E. Mungan, Fall 2004 A satellite is orbiting a body on an ellipse. Denote the position of the satellite relative to the body by plane polar coordinates (r,) . Find the time t that it requires the satellite to move from periapsis (position of closest approach) to angular position . Express your answer in terms of the eccentricity e of the orbit and the period T for a full revolution. According to Kepler’s second law, the ratio of the shaded area A to the total area ab of the ellipse (with semi-major axis a and semi-minor axis b) is the respective ratio of transit times, A t = . (1) ab T But we can divide the shaded area into differential triangles, of base r and height rd , so that A = 1 r2d (2) 2 0 where the polar equation of a Keplerian orbit is c r = (3) 1+ ecos with c the semilatus rectum (distance vertically from the origin on the diagram above to the ellipse). It is not hard to show from the geometrical properties of an ellipse that b = a 1 e2 and c = a(1 e2 ) . (4) Substituting (3) into (2), and then (2) and (4) into (1) gives T (1 e2 )3/2 t = M where M d . (5) 2 2 0 (1 + ecos) In the literature, M is called the mean anomaly. This is an integral solution to the problem. It turns out however that this integral can be evaluated analytically using the following clever change of variables, e + cos cos E = .
    [Show full text]
  • 2. Orbital Mechanics MAE 342 2016
    2/12/20 Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler’s Equation Position and velocity in orbit Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE342.html 1 1 Orbits 101 Satellites Escape and Capture (Comets, Meteorites) 2 2 1 2/12/20 Two-Body Orbits are Conic Sections 3 3 Classical Orbital Elements Dimension and Time a : Semi-major axis e : Eccentricity t p : Time of perigee passage Orientation Ω :Longitude of the Ascending/Descending Node i : Inclination of the Orbital Plane ω: Argument of Perigee 4 4 2 2/12/20 Orientation of an Elliptical Orbit First Point of Aries 5 5 Orbits 102 (2-Body Problem) • e.g., – Sun and Earth or – Earth and Moon or – Earth and Satellite • Circular orbit: radius and velocity are constant • Low Earth orbit: 17,000 mph = 24,000 ft/s = 7.3 km/s • Super-circular velocities – Earth to Moon: 24,550 mph = 36,000 ft/s = 11.1 km/s – Escape: 25,000 mph = 36,600 ft/s = 11.3 km/s • Near escape velocity, small changes have huge influence on apogee 6 6 3 2/12/20 Newton’s 2nd Law § Particle of fixed mass (also called a point mass) acted upon by a force changes velocity with § acceleration proportional to and in direction of force § Inertial reference frame § Ratio of force to acceleration is the mass of the particle: F = m a d dv(t) ⎣⎡mv(t)⎦⎤ = m = ma(t) = F ⎡ ⎤ dt dt vx (t) ⎡ f ⎤ ⎢ ⎥ x ⎡ ⎤ d ⎢ ⎥ fx f ⎢ ⎥ m ⎢ vy (t) ⎥ = ⎢ y ⎥ F = fy = force vector dt
    [Show full text]
  • Synopsis of Euler's Paper E105
    1 Synopsis of Euler’s paper E105 -- Memoire sur la plus grande equation des planetes (Memoir on the Maximum value of an Equation of the Planets) Compiled by Thomas J Osler and Jasen Andrew Scaramazza Mathematics Department Rowan University Glassboro, NJ 08028 [email protected] Preface The following summary of E 105 was constructed by abbreviating the collection of Notes. Thus, there is considerable repetition in these two items. We hope that the reader can profit by reading this synopsis before tackling Euler’s paper itself. I. Planetary Motion as viewed from the earth vs the sun ` Euler discusses the fact that planets observed from the earth exhibit a very irregular motion. In general, they move from west to east along the ecliptic. At times however, the motion slows to a stop and the planet even appears to reverse direction and move from east to west. We call this retrograde motion. After some time the planet stops again and resumes its west to east journey. However, if we observe the planet from the stand point of an observer on the sun, this retrograde motion will not occur, and only a west to east path of the planet is seen. II. The aphelion and the perihelion From the sun, (point O in figure 1) the planet (point P ) is seen to move on an elliptical orbit with the sun at one focus. When the planet is farthest from the sun, we say it is at the “aphelion” (point A ), and at the perihelion when it is closest. The time for the planet to move from aphelion to perihelion and back is called the period.
    [Show full text]
  • Analytical Low-Thrust Trajectory Design Using the Simplified General Perturbations Model J
    Analytical Low-Thrust Trajectory Design using the Simplified General Perturbations model J. G. P. de Jong November 2018 - Technische Universiteit Delft - Master Thesis Analytical Low-Thrust Trajectory Design using the Simplified General Perturbations model by J. G. P. de Jong to obtain the degree of Master of Science at the Delft University of Technology. to be defended publicly on Thursday December 20, 2018 at 13:00. Student number: 4001532 Project duration: November 29, 2017 - November 26, 2018 Supervisor: Ir. R. Noomen Thesis committee: Dr. Ir. E.J.O. Schrama TU Delft Ir. R. Noomen TU Delft Dr. S. Speretta TU Delft November 26, 2018 An electronic version of this thesis is available at http://repository.tudelft.nl/. Frontpage picture: NASA. Preface Ever since I was a little girl, I knew I was going to study in Delft. Which study exactly remained unknown until the day before my high school graduation. There I was, reading a flyer about aerospace engineering and suddenly I realized: I was going to study aerospace engineering. During the bachelor it soon became clear that space is the best part of the word aerospace and thus the space flight master was chosen. Looking back this should have been clear already years ago: all those books about space I have read when growing up... After quite some time I have come to the end of my studies. Especially the last years were not an easy journey, but I pulled through and made it to the end. This was not possible without a lot of people and I would like this opportunity to thank them here.
    [Show full text]
  • Observed Binaries with Compact Objects
    Chapter 9 Observed binaries with compact objects This chapter provides an overview of observed types of binaries in which one or both stars are compact objects (white dwarfs, neutron stars or black holes). In many of these systems an important energy source is accretion onto the compact object. 9.1 Accretion power When mass falls on an object of mass M and with radius R, at a rate M˙ , a luminosity Lacc may be produced: GMM˙ L = (9.1) acc R The smaller the radius, the more energy can be released. For a white dwarf with M ≈ 0.7 M⊙ and −4 2 R ≈ 0.01 R⊙, this yields Lacc ≈ 1.5 × 10 Mc˙ . For a neutron star with M ≈ 1.4 M⊙ and R ≈ 10 km, we 2 2 2 find Lacc ≈ 0.2Mc˙ and for a black hole with Scwarzschild radius R ∼ 2GM/c we find Lacc ∼ 0.5Mc˙ , i.e. in the ideal case a sizable fraction of the rest mass may be released as energy. This accretion process is thus potentially much more efficient than nuclear fusion. However, the accretion luminosity should not be able to exceed the Eddington luminosity, 4πcGM L ≤ L = (9.2) acc Edd κ 2 where κ is the opacity, which may be taken as the electron scattering opacity, κes = 0.2(1 + X) cm /g for a hydrogen mass fraction X. Thus the compact accreting star may not be able to accrete more than a fraction of the mass that is transferred onto it by its companion. By equating Lacc to LEdd we obtain the maximum accretion rate, 4πcR M˙ = (9.3) Edd κ The remainder of the mass may be lost from the binary system, in the form of a wind or jets blown from the vicinity of the compact star, or it may accumulate in the accretor’s Roche lobe or in a common envelope around the system.
    [Show full text]
  • 1 CHAPTER 10 COMPUTATION of an EPHEMERIS 10.1 Introduction
    1 CHAPTER 10 COMPUTATION OF AN EPHEMERIS 10.1 Introduction The entire enterprise of determining the orbits of planets, asteroids and comets is quite a large one, involving several stages. New asteroids and comets have to be searched for and discovered. Known bodies have to be found, which may be relatively easy if they have been frequently observed, or rather more difficult if they have not been observed for several years. Once located, images have to be obtained, and these have to be measured and the measurements converted to usable data, namely right ascension and declination. From the available observations, the orbit of the body has to be determined; in particular we have to determine the orbital elements , a set of parameters that describe the orbit. For a new body, one determines preliminary elements from the initial few observations that have been obtained. As more observations are accumulated, so will the calculated preliminary elements. After all observations (at least for a single opposition) have been obtained and no further observations are expected at that opposition, a definitive orbit can be computed. Whether one uses the preliminary orbit or the definitive orbit, one then has to compute an ephemeris (plural: ephemerides ); that is to say a day-to-day prediction of its position (right ascension and declination) in the sky. Calculating an ephemeris from the orbital elements is the subject of this chapter. Determining the orbital elements from the observations is a rather more difficult calculation, and will be the subject of a later chapter. 10.2 Elements of an Elliptic Orbit Six numbers are necessary and sufficient to describe an elliptic orbit in three dimensions.
    [Show full text]
  • Low-Thrust Maneuvers for the Efficient Correction of Orbital Elements
    Low-Thrust Maneuvers for the Efficient Correction of Orbital Elements IEPC-2011-102 Presented at the 32nd International Electric Propulsion Conference, Wiesbaden • Germany September 11 – 15, 2011 A. Ruggiero 1 P. Pergola2 Alta, Pisa, 56100, Italy S. Marcuccio3 and M. Andrenucci4 University of Pisa, Pisa, 56100, Italy In this study, low-thrust strategies aimed at the modification of specific orbital elements are investigated and presented. Closed loop guidance laws, steering the initial value of a given orbital element to a target value, are derived, implemented and tested. The thrust laws have been defined analyzing the Gauss form of the Lagrange Planetary Equations isolating the relevant contributions to change specific orbital elements. In particular, considering the classical two body dynamics with the inclusion of the thrust acceleration, both optimal and near-optimal thrusting strategies have been obtained. In the study these thrust laws are described and special focus is given both to the thrusting time and to the total velocity increment required to perform specific or combined orbital parameter changes. The results, obtained by direct numerical simulations of the presented control laws, are compared with analytical approximation and, as study cases, some low-thrust transfers are also simulated. A direct transfer between the standard geosynchronous transfer orbit and the geosynchronous orbit is implemented together with generic demonstrations of the single strategies. Nomenclature a = semi-major axis e = eccentricity i = inclination Ω = right ascension of the ascending node ν = true anomaly E = eccentric anomaly ω = argument of perigee t = time ∆V = velocity increment = spacecraft acceleration f α = in-plane thrust angle 1 Research Engineer, Alta, Pisa; M.Sc.; [email protected].
    [Show full text]
  • Computing GPS Satellite Velocity and Acceleration from the Broadcast
    Computing GPS Satellite Velocity and Acceleration from the Broadcast Navigation Message Blair F. Thompson , Steven W. Lewis , Steven A. Brown Lt. Colonel, 42d Combat Training Squadron, Peterson Air Force Base, Colorado Todd M. Scott Command Chief Master Sergeant, 310th Space Wing, Schriever Air Force Base, Colorado ABSTRACT We present an extension to the Global Positioning System (GPS) broadcast navigation message user equations for computing GPS space vehicle (SV) velocity and acceleration. Although similar extensions have been published (e.g., Remondi,1 Zhang J.,2 Zhang W.3), the extension presented herein includes a distinct kinematic method for computing SV acceleration which significantly reduces the complexity of the equations and improves the mean magnitude results by approximately one order of magnitude by including oblate Earth perturbation effects. Additionally, detailed anal- yses and validation results using multiple days of precise ephemeris data and multiple broadcast navigation messages are presented. Improvements in the equations for computing SV position are also included, removing ambiguity and redundancy in the existing user equations. The recommended changes make the user equations more complete and more suitable for implementation in a wide variety of programming languages employed by GPS users. Furthermore, relativistic SV clock error rate computation is enabled by the recommended equations. A complete, stand-alone table of the equations in the format and notation of the GPS interface specification4 is provided, along with benchmark test cases to simplify implementation and verification. 1 j INTRODUCTION Basic positioning of a Global Positioning System (GPS) receiver requires accurate modeling of the location of the antenna phase center of four or more orbiting space vehicles (SV) in view.
    [Show full text]
  • Refining Exoplanet Ephemerides and Transit Observing Strategies
    PUBLICATIONS OF THE ASTRONOMICAL SOCIETY OF THE PACIFIC, 121:1386–1394, 2009 December © 2009. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A. Refining Exoplanet Ephemerides and Transit Observing Strategies STEPHEN R. KANE,1 SUVRATH MAHADEVAN,2 KASPAR VON BRAUN,1 GREGORY LAUGHLIN,3 AND DAVID R. CIARDI1 Received 2009 August 27; accepted 2009 September 27; published 2009 November 16 ABSTRACT. Transiting planet discoveries have yielded a plethora of information regarding the internal structure and atmospheres of extrasolar planets. These discoveries have been restricted to the low-periastron distance regime due to the bias inherent in the geometric transit probability. Monitoring known radial velocity planets at predicted transit times is a proven method of detecting transits, and presents an avenue through which to explore the mass- radius relationship of exoplanets in new regions of period/periastron space. Here we describe transit window cal- culations for known radial velocity planets, techniques for refining their transit ephemerides, target selection criteria, and observational methods for obtaining maximum coverage of transit windows. These methods are currently being implemented by the Transit Ephemeris Refinement and Monitoring Survey (TERMS). 1. INTRODUCTION stellar radiation. Recent observations of HD 80606b by Gillon (2009) and Pont et al. (2009) suggest a spin-orbit misalignment Planet formation theories thus far extract much of their caused by a Kozai mechanism; a suggestion which could be information from the known transiting exoplanets, which are largely in the short-period regime. This is because transit sur- investigated in terms of period and eccentricity dependencies veys that have provided the bulk of the transiting planet discov- if more long-period transiting planet were known.
    [Show full text]
  • Lecture 8 Motion in Central Force Fields
    Lecture 8 Motion in central force fields II. 8.1 Orbit in a central force field In Lecture 7 we have seen the basic procedure for solving the equation of motion of a body in a central force field. We can obtain r(t) and #(t) as the function of time by integrating (see Eq. (7.28)): dr dt q 2 Æ 2 ¡E V(r) ` ¢ m ¡ ¡ 2mr2 Let us now derive a differential equation for the shape of the orbit, r(#), ignoring how the body moves along the orbit in time. Using the conservation of angular momentum, Eq. (7.26a), we can relate the time derivative to the derivative with respect to #: d ` d mr2#˙ ` ` dt mr2 d# (8.1) Æ ) Æ , dt Æ mr2 d# Rewriting the equation of motion for r, Eq. (7.21), in terms of #- derivatives we get ` d µ ` dr ¶ `2 F(r) (8.2) r2 d# mr2 d# ¡ mr3 Æ Introducing the variable 1 du u , dr ´ r Æ¡ u2 1 equation (8.2) will take the simpler quasilinear form d2u m d ³ 1 ´ u V (8.3) d#2 Å Æ¡`2 du u Eq. (7.28) can also be written in terms of # and u: Z r dr # q #0 (8.4) Æ r0 r2 2mE 2mV 1 Å `2 ¡ `2 ¡ r2 Z u du # q #0 (8.5) Æ¡ u0 2mE 2mV u2 Å `2 ¡ `2 ¡ n 1 n Power law potentials of the form V(r) ar Å F(r) r have partic- ular importance. The gravitational potentialÆ and) Hooke’s/ law are both n 1 power laws.
    [Show full text]