N-Body Problem: Analytical and Numerical Approaches
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AAS 13-250 Hohmann Spiral Transfer with Inclination Change Performed
AAS 13-250 Hohmann Spiral Transfer With Inclination Change Performed By Low-Thrust System Steven Owens1 and Malcolm Macdonald2 This paper investigates the Hohmann Spiral Transfer (HST), an orbit transfer method previously developed by the authors incorporating both high and low- thrust propulsion systems, using the low-thrust system to perform an inclination change as well as orbit transfer. The HST is similar to the bi-elliptic transfer as the high-thrust system is first used to propel the spacecraft beyond the target where it is used again to circularize at an intermediate orbit. The low-thrust system is then activated and, while maintaining this orbit altitude, used to change the orbit inclination to suit the mission specification. The low-thrust system is then used again to reduce the spacecraft altitude by spiraling in-toward the target orbit. An analytical analysis of the HST utilizing the low-thrust system for the inclination change is performed which allows a critical specific impulse ratio to be derived determining the point at which the HST consumes the same amount of fuel as the Hohmann transfer. A critical ratio is found for both a circular and elliptical initial orbit. These equations are validated by a numerical approach before being compared to the HST utilizing the high-thrust system to perform the inclination change. An additional critical ratio comparing the HST utilizing the low-thrust system for the inclination change with its high-thrust counterpart is derived and by using these three critical ratios together, it can be determined when each transfer offers the lowest fuel mass consumption. -
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. -
Astrodynamics
Politecnico di Torino SEEDS SpacE Exploration and Development Systems Astrodynamics II Edition 2006 - 07 - Ver. 2.0.1 Author: Guido Colasurdo Dipartimento di Energetica Teacher: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Two–Body Orbital Mechanics 1 1.1 BirthofAstrodynamics: Kepler’sLaws. ......... 1 1.2 Newton’sLawsofMotion ............................ ... 2 1.3 Newton’s Law of Universal Gravitation . ......... 3 1.4 The n–BodyProblem ................................. 4 1.5 Equation of Motion in the Two-Body Problem . ....... 5 1.6 PotentialEnergy ................................. ... 6 1.7 ConstantsoftheMotion . .. .. .. .. .. .. .. .. .... 7 1.8 TrajectoryEquation .............................. .... 8 1.9 ConicSections ................................... 8 1.10 Relating Energy and Semi-major Axis . ........ 9 2 Two-Dimensional Analysis of Motion 11 2.1 ReferenceFrames................................. 11 2.2 Velocity and acceleration components . ......... 12 2.3 First-Order Scalar Equations of Motion . ......... 12 2.4 PerifocalReferenceFrame . ...... 13 2.5 FlightPathAngle ................................. 14 2.6 EllipticalOrbits................................ ..... 15 2.6.1 Geometry of an Elliptical Orbit . ..... 15 2.6.2 Period of an Elliptical Orbit . ..... 16 2.7 Time–of–Flight on the Elliptical Orbit . .......... 16 2.8 Extensiontohyperbolaandparabola. ........ 18 2.9 Circular and Escape Velocity, Hyperbolic Excess Speed . .............. 18 2.10 CosmicVelocities -
Optimisation of Propellant Consumption for Power Limited Rockets
Delft University of Technology Faculty Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Optimisation of Propellant Consumption for Power Limited Rockets. What Role do Power Limited Rockets have in Future Spaceflight Missions? (Dutch title: Optimaliseren van brandstofverbruik voor vermogen gelimiteerde raketten. De rol van deze raketten in toekomstige ruimtevlucht missies. ) A thesis submitted to the Delft Institute of Applied Mathematics as part to obtain the degree of BACHELOR OF SCIENCE in APPLIED MATHEMATICS by NATHALIE OUDHOF Delft, the Netherlands December 2017 Copyright c 2017 by Nathalie Oudhof. All rights reserved. BSc thesis APPLIED MATHEMATICS \ Optimisation of Propellant Consumption for Power Limited Rockets What Role do Power Limite Rockets have in Future Spaceflight Missions?" (Dutch title: \Optimaliseren van brandstofverbruik voor vermogen gelimiteerde raketten De rol van deze raketten in toekomstige ruimtevlucht missies.)" NATHALIE OUDHOF Delft University of Technology Supervisor Dr. P.M. Visser Other members of the committee Dr.ir. W.G.M. Groenevelt Drs. E.M. van Elderen 21 December, 2017 Delft Abstract In this thesis we look at the most cost-effective trajectory for power limited rockets, i.e. the trajectory which costs the least amount of propellant. First some background information as well as the differences between thrust limited and power limited rockets will be discussed. Then the optimal trajectory for thrust limited rockets, the Hohmann Transfer Orbit, will be explained. Using Optimal Control Theory, the optimal trajectory for power limited rockets can be found. Three trajectories will be discussed: Low Earth Orbit to Geostationary Earth Orbit, Earth to Mars and Earth to Saturn. After this we compare the propellant use of the thrust limited rockets for these trajectories with the power limited rockets. -
A Strategy for the Eigenvector Perturbations of the Reynolds Stress Tensor in the Context of Uncertainty Quantification
Center for Turbulence Research 425 Proceedings of the Summer Program 2016 A strategy for the eigenvector perturbations of the Reynolds stress tensor in the context of uncertainty quantification By R. L. Thompsony, L. E. B. Sampaio, W. Edeling, A. A. Mishra AND G. Iaccarino In spite of increasing progress in high fidelity turbulent simulation avenues like Di- rect Numerical Simulation (DNS) and Large Eddy Simulation (LES), Reynolds-Averaged Navier-Stokes (RANS) models remain the predominant numerical recourse to study com- plex engineering turbulent flows. In this scenario, it is imperative to provide reliable estimates for the uncertainty in RANS predictions. In the recent past, an uncertainty estimation framework relying on perturbations to the modeled Reynolds stress tensor has been widely applied with satisfactory results. Many of these investigations focus on perturbing only the Reynolds stress eigenvalues in the Barycentric map, thus ensuring realizability. However, these restrictions apply to the eigenvalues of that tensor only, leav- ing the eigenvectors free of any limiting condition. In the present work, we propose the use of the Reynolds stress transport equation as a constraint for the eigenvector pertur- bations of the Reynolds stress anisotropy tensor once the eigenvalues of this tensor are perturbed. We apply this methodology to a convex channel and show that the ensuing eigenvector perturbations are a more accurate measure of uncertainty when compared with a pure eigenvalue perturbation. 1. Introduction Turbulent flows are present in a wide number of engineering design problems. Due to the disparate character of such flows, predictive methods must be robust, so as to be easily applicable for most of these cases, yet possessing a high degree of accuracy in each. -
Spacecraft Guidance Techniques for Maximizing Mission Success
Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2014 Spacecraft Guidance Techniques for Maximizing Mission Success Shane B. Robinson Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Mechanical Engineering Commons Recommended Citation Robinson, Shane B., "Spacecraft Guidance Techniques for Maximizing Mission Success" (2014). All Graduate Theses and Dissertations. 2175. https://digitalcommons.usu.edu/etd/2175 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. SPACECRAFT GUIDANCE TECHNIQUES FOR MAXIMIZING MISSION SUCCESS by Shane B. Robinson A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mechanical Engineering Approved: Dr. David K. Geller Dr. Jacob H. Gunther Major Professor Committee Member Dr. Warren F. Phillips Dr. Charles M. Swenson Committee Member Committee Member Dr. Stephen A. Whitmore Dr. Mark R. McLellan Committee Member Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2013 [This page intentionally left blank] iii Copyright c Shane B. Robinson 2013 All Rights Reserved [This page intentionally left blank] v Abstract Spacecraft Guidance Techniques for Maximizing Mission Success by Shane B. Robinson, Doctor of Philosophy Utah State University, 2013 Major Professor: Dr. David K. Geller Department: Mechanical and Aerospace Engineering Traditional spacecraft guidance techniques have the objective of deterministically min- imizing fuel consumption. -
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. -
Richard Dalbello Vice President, Legal and Government Affairs Intelsat General
Commercial Management of the Space Environment Richard DalBello Vice President, Legal and Government Affairs Intelsat General The commercial satellite industry has billions of dollars of assets in space and relies on this unique environment for the development and growth of its business. As a result, safety and the sustainment of the space environment are two of the satellite industry‘s highest priorities. In this paper I would like to provide a quick survey of past and current industry space traffic control practices and to discuss a few key initiatives that the industry is developing in this area. Background The commercial satellite industry has been providing essential space services for almost as long as humans have been exploring space. Over the decades, this industry has played an active role in developing technology, worked collaboratively to set standards, and partnered with government to develop successful international regulatory regimes. Success in both commercial and government space programs has meant that new demands are being placed on the space environment. This has resulted in orbital crowding, an increase in space debris, and greater demand for limited frequency resources. The successful management of these issues will require a strong partnership between government and industry and the careful, experience-based expansion of international law and diplomacy. Throughout the years, the satellite industry has never taken for granted the remarkable environment in which it works. Industry has invested heavily in technology and sought out the best and brightest minds to allow the full, but sustainable exploitation of the space environment. Where problems have arisen, such as space debris or electronic interference, industry has taken 1 the initiative to deploy new technologies and adopt new practices to minimize negative consequences. -
Perturbation Theory in Celestial Mechanics
Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion. -
Session 6:Analytical Approximations for Low Thrust Maneuvers
Session 6: Analytical Approximations for Low Thrust Maneuvers As mentioned in the previous lecture, solving non-Keplerian problems in general requires the use of perturbation methods and many are only solvable through numerical integration. However, there are a few examples of low-thrust space propulsion maneuvers for which we can find approximate analytical expressions. In this lecture, we explore a couple of these maneuvers, both of which are useful because of their precision and practical value: i) climb or descent from a circular orbit with continuous thrust, and ii) in-orbit repositioning, or walking. Spiral Climb/Descent We start by writing the equations of motion in polar coordinates, 2 d2r �dθ � µ − r + = a (1) 2 2 r dt dt r 2 d θ 2 dr dθ aθ + = (2) 2 dt r dt dt r We assume continuous thrust in the angular direction, therefore ar = 0. If the acceleration force along θ is small, then we can safely assume the orbit will remain nearly circular and the semi-major axis will be just slightly different after one orbital period. Of course, small and slightly are vague words. To make the analysis rigorous, we need to be more precise. Let us say that for this approximation to be valid, the angular acceleration has to be much smaller than the corresponding centrifugal or gravitational forces (the last two terms in the LHS of Eq. (1)) and that the radial acceleration (the first term in the LHS in the same equation) is negligible. Given these assumptions, from Eq. (1), dθ r µ d2θ 3 r µ dr ≈ ! ≈ − (3) 3 2 5 dt r dt 2 r dt Substituting into Eq. -
New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion
New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion Michelle Chernick∗ and Simone D'Amicoy Aeronautics and Astronautics, Stanford University, Stanford, California, 94305, USA This paper addresses the fuel-optimal guidance and control of the relative motion for formation-flying and rendezvous using impulsive maneuvers. To meet the requirements of future multi-satellite missions, closed-form solutions of the inverse relative dynamics are sought in arbitrary orbits. Time constraints dictated by mission operations and relevant perturbations acting on the formation are taken into account by splitting the optimal recon- figuration in a guidance (long-term) and control (short-term) layer. Both problems are cast in relative orbit element space which allows the simple inclusion of secular and long-periodic perturbations through a state transition matrix and the translation of the fuel-optimal optimization into a minimum-length path-planning problem. Due to the proper choice of state variables, both guidance and control problems can be solved (semi-)analytically leading to optimal, predictable maneuvering schemes for simple on-board implementation. Besides generalizing previous work, this paper finds four new in-plane and out-of-plane (semi-)analytical solutions to the optimal control problem in the cases of unperturbed ec- centric and perturbed near-circular orbits. A general delta-v lower bound is formulated which provides insight into the optimality of the control solutions, and a strong analogy between elliptic Hohmann transfers and formation-flying control is established. Finally, the functionality, performance, and benefits of the new impulsive maneuvering schemes are rigorously assessed through numerical integration of the equations of motion and a systematic comparison with primer vector optimal control. -
Coordinate Systems in Geodesy
COORDINATE SYSTEMS IN GEODESY E. J. KRAKIWSKY D. E. WELLS May 1971 TECHNICALLECTURE NOTES REPORT NO.NO. 21716 COORDINATE SYSTElVIS IN GEODESY E.J. Krakiwsky D.E. \Vells Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton, N .B. Canada E3B 5A3 May 1971 Latest Reprinting January 1998 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. TABLE OF CONTENTS page LIST OF ILLUSTRATIONS iv LIST OF TABLES . vi l. INTRODUCTION l 1.1 Poles~ Planes and -~es 4 1.2 Universal and Sidereal Time 6 1.3 Coordinate Systems in Geodesy . 7 2. TERRESTRIAL COORDINATE SYSTEMS 9 2.1 Terrestrial Geocentric Systems • . 9 2.1.1 Polar Motion and Irregular Rotation of the Earth • . • • . • • • • . 10 2.1.2 Average and Instantaneous Terrestrial Systems • 12 2.1. 3 Geodetic Systems • • • • • • • • • • . 1 17 2.2 Relationship between Cartesian and Curvilinear Coordinates • • • • • • • . • • 19 2.2.1 Cartesian and Curvilinear Coordinates of a Point on the Reference Ellipsoid • • • • • 19 2.2.2 The Position Vector in Terms of the Geodetic Latitude • • • • • • • • • • • • • • • • • • • 22 2.2.3 Th~ Position Vector in Terms of the Geocentric and Reduced Latitudes . • • • • • • • • • • • 27 2.2.4 Relationships between Geodetic, Geocentric and Reduced Latitudes • . • • • • • • • • • • 28 2.2.5 The Position Vector of a Point Above the Reference Ellipsoid . • • . • • • • • • . .• 28 2.2.6 Transformation from Average Terrestrial Cartesian to Geodetic Coordinates • 31 2.3 Geodetic Datums 33 2.3.1 Datum Position Parameters .