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Space Flight Dynamics

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Names: Kluever, Craig A. (Craig Allan), author.Title: Space flight dynamics / by Craig A. Kluever.Description: First edition. | Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |Identifiers: LCCN 2017042818 (print) | LCCN 2017054455 (ebook) | ISBN 9781119157908 (pdf) | ISBN 9781119157847 (epub) | ISBN 9781119157823 (cloth)Subjects: LCSH: Astrodynamics. | Space flight.Classification: LCC TL1050 (ebook) | LCC TL1050 .K555 2018 (print) | DDC 629.4/1–dc23LC record available at https://lccn.loc.gov/2017042818

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Preface

This textbook is intended for an introductory course in space flight dynamics. Such a course is typically required for undergraduates majoring in aerospace engineering. It is also frequently offered as an elective in mechanical and aerospace engineering curricula. Whether taken for required or elective credit, this course is usually taken in the junior or senior year, after the student has completed work in university physics, rigid‐body dynamics, and differential equations. A brief survey of university catalogs shows that titles for these courses include Orbital Mechanics, Astrodynamics, Astronautics, and Space Flight Dynamics. The principal topic covered in essentially all courses is two‐body orbital motion, which involves orbit determination, orbital flight time, and orbital maneuvers. A secondary topic that appears in many of these courses is spacecraft attitude dynamics and attitude control, which involves analyzing and controlling a satellite’s rotational motion about its center of mass. A number of space flight courses also cover topics such as orbital rendezvous, launch trajectories, rocket propulsion, low‐thrust transfers, and atmospheric entry flight mechanics. The primary goal of this textbook is to provide a comprehensive yet concise treatment of all of the topics that can comprise a space flight dynamics course. To my knowledge, a single space flight textbook that covers all the topics mentioned above does not exist.

A secondary goal of this textbook is to demonstrate concepts using real engineering examples derived from actual space missions. It has been my experience that undergraduate students remain engaged in a course when they solve “real‐world” problems instead of academic “textbook” examples. A third goal is to produce a readable textbook with a conversational style inspired by my textbook‐author role model, John D. Anderson, Jr. Space Flight Dynamics is a distillation of 20 years of course notes and strategies for teaching space flight in the Mechanical and Aerospace Engineering Department at the University of Missouri‐Columbia.

Chapter 1 is a brief historical overview of the important figures and events that have shaped space flight. Chapter 2 provides the foundation of this textbook with a treatment of orbital mechanics. Here we are able to obtain analytical expressions for the orbital motion of a small body (such as a satellite) relative to a large gravitational body (such as a planet). Chapter 3 extends these concepts with a discussion of orbit determination, that is, the process of completely characterizing a satellite’s orbit. In Chapter 4 we present Kepler’s time‐of‐flight equations which allow us to predict a satellite’s orbital position at a future (or past) time. We also discuss Lambert’s problem: the process of determining an orbit that passes through two points in space separated by a particular flight time.

Chapter 5 introduces orbital perturbations that arise from the non‐spherical shape of the attracting body, third‐body gravity forces, and atmospheric drag. Perturbations cause the satellite’s motion to deviate from the analytical solutions we obtained for the two‐body motion studied in Chapters 2– 4 . We also introduce the restricted three‐body problem where gravitational forces from two primary bodies (such as the Earth and moon) simultaneously influence the satellite’s motion.

Chapter 6 presents fundamentals of rocket propulsion and launch trajectories. This chapter serves as a key transitional link to subsequent chapters that involve orbital maneuvers. Chapter 6 shows that burning a given quantity of rocket propellant corresponds to a change in orbital velocity, or Δv. The next four chapters involve orbital maneuvers, where the performance metric is typically the Δv increment. Chapter 7 discusses orbital changes achieved by so‐called impulsive maneuvers where a rocket thrust force produces a velocity change in a relatively short time. Chapter 8 treats relative motion and orbital rendezvous, where a satellite moves in proximity to a desired orbital location or another orbiting satellite. In Chapter 9, we discuss low‐thrust orbit transfers where an electric propulsion system provides a continuous but small perturbing thrust force that slowly changes the orbit over time. Interplanetary trajectories are treated in Chapter 10. Here we analyze a space mission by piecing together three flight segments: a planetary departure phase, an interplanetary cruise phase between planets, and a planetary arrival phase.

Chapter 11 introduces atmospheric entry or the flight mechanics of a spacecraft as it moves from orbital motion to flight through a planetary atmosphere. Here we develop analytical solutions for entry flight both with and without an aerodynamic lift force.

Chapters 2– 11 involve particle dynamics, where we treat the satellite as a point mass. The last two chapters involve analyzing and controlling the rotational motion of a satellite about its center of mass. Chapter 12 presents attitude dynamics, or the analysis of a satellite’s rotational motion. Topics in Chapter 12 include rotational motion in the absence of external torques, spin stability, and the effect of disturbance torques on rotational motion. Chapter 13 presents an introduction to attitude control. Here we primarily focus on controlling a satellite’s angular orientation by using feedback and attitude control mechanisms such as reaction wheels and thruster jets.

Numerous examples are provided at key locations throughout Chapters 2–13 in order to illustrate the topic discussed by the particular section. Chapters 2–13 also contain end‐of‐chapter problems that are grouped into three categories: (1) conceptual problems; (2) MATLAB problems; and (3) mission applications. Many of the example and end‐of‐chapter problems illustrate concepts in space flight by presenting scenarios involving contemporary and historical space missions.

Appendix A presents the physical constants for celestial bodies. Appendix B provides a brief review of vectors and their operations and Appendix C is a review of particle kinematics with respect to inertial and rotating coordinate frames.

My intent was to write a comprehensive yet concise textbook on space flight dynamics. A survey of 35 space flight courses offered by US aerospace engineering programs shows that nearly half (17/35) are “orbits only” courses that focus on orbital mechanics, orbit determination, and orbital transfers. The remaining (18/35) courses include a mix of orbital motion and attitude dynamics and control. In addition, more than one‐third (13/35) of the surveyed courses cover rocket performance and atmospheric entry. Few existing space flight textbooks adequately cover all of these topics. I believe that this textbook has the breadth and depth so that it can serve all of these diverse space flight courses.

Several people have contributed to the production of this textbook. Many reviewers provided valuable suggestions for improving this textbook and they are listed here:

Jonathan Black, Virginia Polytechnic Institute and State University

Craig McLaughlin, University of Kansas

Eric Monda, United Launch Alliance

Erwin Mooij, Delft University of Technology

Henry Pernicka, Missouri University of Science and Technology

David Spencer, The Pennsylvania State University

Srinivas Rao Vadali, Texas A&M University

Ming Xin, University of Missouri‐Columbia

I am grateful for Jonathan Jennings’ help with figures and illustrations. Finally, I would like to thank my wife Nancy M. West for her patience, encouragement, and skilled editorial work throughout this project. This book is dedicated to her.

University of Missouri‐Columbia, May 2017

Craig A. Kluever

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1Historical Overview

1.1 Introduction

Before we begin our technical discussion of space flight dynamics, this first chapter will provide a condensed historical overview of the principle contributors and events associated with the development of what we now commonly refer to as space flight. We may define space flight as sending a human‐made satellite or spacecraft to an Earth orbit or to another celestial body such as the moon, an asteroid, or a planet. Of course, our present ability to launch and operate satellites in orbit depends on knowledge of the physical laws that govern orbital motion. This brief chapter presents the major developments in astronomy, celestial mechanics, and space flight in chronological order so that we can gain some historical perspective.

1.2 Early Modern Period

The fields of astronomy and celestial mechanics (the study of the motion of planets and their moons) have attracted the attention of the great scientific and mathematical minds. We may define the early modern period by the years spanning roughly 1500–1800. This time frame begins with the late Middle Ages and includes the Renaissance and Age of Discovery. Figure 1.1 shows a timeline of the important figures in the development of celestial mechanics during the early modern period. The astute reader will, of course, recognize these illuminous figures for their contributions to mathematics (Newton, Euler, Lagrange, Laplace, Gauss), physics (Newton, Galileo), dynamics (Kepler, Newton, Euler, Lagrange), and statistics (Gauss). We will briefly describe each figure’s contribution to astronomy and celestial mechanics.

Figure 1.1 Timeline of significant figures in the Early Modern Period.

The first major figure is Nicolaus Copernicus (1473–1543), a Polish astronomer and mathematician who developed a solar‐system model with the sun as the central body. Galileo Galilei (1546–1642) was an Italian astronomer and mathematician who defended Copernicus’ sun‐centered (or “heliocentric”) solar system. Because of his heliocentric view, Galileo was put on trial by the Roman Inquisition for heresy and spent the remainder of his life under house arrest.

Johann Kepler (1571–1630) developed the fundamental laws for planetary motion based on astronomical observations of the planet Mars compiled by the Danish nobleman Tycho Brahe (1546–1601). Kepler’s three laws are:

The orbit of a planet is an ellipse, with the sun located at a focus.

The radial line from the sun to the planet sweeps out equal areas during equal time intervals.

The square of a planet’s orbital period for one revolution is proportional to the cube of the planet’s “mean distance” from the sun.

The third law notes the planet’s “mean distance” from the sun. In Chapter 2 we will define this “mean distance” as one‐half of the length of the major axis of an ellipse. Kepler published his first two laws of planetary motion in 1609 and his third law in 1619. Kepler developed an expression for the time‐of‐flight between two points in an orbit; this expression is now known as Kepler’s equation.

Isaac Newton (1642–1727) was an English astronomer, mathematician, and physicist who developed calculus and formulated the laws of motion and universal gravitation. Newton’s three laws of motion are:

A body remains at rest or moves with a constant velocity unless acted upon by a force.

The vector sum of the forces acting on a body is equal to the mass of the body multiplied by its absolute acceleration vector (i.e.,

).

When a body exerts a force on a second body, the second body exerts an equal‐and‐opposite force on the first body.

The first and second laws hold relative to a fixed or inertial reference frame. Newton published the three laws of motion in Principia in 1687. Newton’s universal law of gravitation states that any two bodies attract one another with a force that is proportional to the product of their masses and inversely proportional to the square of their separation distance. Newton’s laws of motion and gravitation explain the planetary motion that Kepler described by geometrical means.

Leonhard Euler (1707–1783), a Swiss mathematician, made many mathematical and scientific contributions to the fields of calculus, mathematical analysis, analytical mechanics, fluid dynamics, and optics. Euler also developed equations that govern the motion of a rotating body; these equations serve as the foundation for analyzing the rotational motion of satellites in orbit. Johann Heinrich Lambert (1728–1777), also a Swiss mathematician, formulated and solved the problem of determining the orbit that passes through two known position vectors with a prescribed transit time. Known today as Lambert’s problem, its solution provides a method for the orbit‐determination process as well as planning orbital maneuvers. Joseph‐Louis Lagrange (1736–1813) was an Italian‐born mathematician who made significant contributions in analytical mechanics and celestial mechanics, including the determination of equilibrium orbits for a problem with three bodies and the formulation of Lagrange’s planetary equations for orbital motion. Pierre‐Simon Laplace (1749–1827) was a French mathematician who, among his many mathematical contributions, formulated the first orbit‐determination method based solely on angular measurements. Carl Friedrich Gauss (1777–1855), a German mathematician of great influence, made significant contributions to the field of orbit determination. In mid‐1801 he predicted the orbit of the dwarf planet Ceres using a limited amount of observational data taken before Ceres became obscured by the sun. In late 1801, astronomers rediscovered Ceres just as predicted by Gauss.

1.3 Early Twentieth Century

Let us next briefly describe the important figures in the early twentieth century. It is during this period when mathematical theories are augmented by experimentation, most notably in the field of rocket propulsion. It is interesting to note that the important figures of this period were inspired by the nineteenth century science fiction literature of H.G. Wells and Jules Verne and consequently were tantalized by the prospect of interplanetary space travel.

Konstantin Tsiolkovsky (1857–1935) was a Russian mathematician and village school teacher who worked in relative obscurity. He theorized the use of oxygen and hydrogen as the optimal combination for a liquid‐propellant rocket in 1903 (the same year as the Wright brothers’ first powered airplane flight). Tsiolkovsky also developed theories regarding rocket propulsion and a vehicle’s velocity change – the so‐called “rocket equation.”

Robert H. Goddard (1882–1945), a US physicist, greatly advanced rocket technology by combining theory and experimentation. On March 16, 1926, Goddard successfully launched the first liquid‐propellant rocket. In 1930, Goddard moved his laboratory to New Mexico and continued to develop larger and more powerful rocket engines.

Hermann J. Oberth (1894–1989) was born in Transylvania and later became a German citizen. A physicist by training, he independently developed theories regarding human space flight through rocket propulsion. Oberth was a key figure in the German Society for Space Travel, which was formed in 1927, and whose membership included the young student Wernher von Braun. Von Braun (1912–1977) led the Nazi rocket program at Peenemünde during World War II. Von Braun’s team developed the V‐2 rocket, the first long‐range rocket and the first vehicle to achieve space flight above the sensible atmosphere.

At the end of World War II, von Braun and members of his team immigrated to the US and began a rocket program at the US Army’s Redstone Arsenal at Huntsville, Alabama. It was during this time that the US and the Soviet Union were rapidly developing long‐range intercontinental ballistic missiles (ICBMs) for delivering nuclear weapons.

1.4 Space Age

On October 4, 1957, the Soviet Union successfully launched the first artificial satellite (Sputnik 1) into an Earth orbit and thus ushered in the space age. Sputnik 1 was a polished 84 kg metal sphere and it completed an orbital revolution every 96 min. The US successfully launched its first satellite (Explorer 1) almost 4 months after Sputnik on January 31, 1958. Unlike Sputnik 1, Explorer 1 was a long, tube‐shaped satellite, and because of its shape, it unexpectedly entered into an end‐over‐end tumbling spin after achieving orbit.

Our abridged historical overview of the first half of the twentieth century illustrates the very rapid progress achieved in rocket propulsion and space flight. For example, in less than 20 years after Goddard’s 184 ft flight of the first liquid‐propellant rocket, Nazi Germany was bombarding London with long‐range V‐2 missiles. Twelve years after the end of World War II, the USSR successfully launched a satellite into orbit. Another point of interest is that in this short period, rocket propulsion and space flight transitioned from the realm of the singular individual figure to large team structures funded by governments. For example, the US established the National Aeronautics and Space Administration (NASA) on July 29, 1958.

The US and USSR space programs launched and operated many successful missions after the space age began in late 1957. Table 1.1 summarizes notable robotic space missions (i.e., no human crew). A complete list of successful space missions would be quite long; Table 1.1 is not an exhaustive list and instead presents a list of mission “firsts.” It is truly astounding that 15 months after Sputnik 1, the USSR sent a space probe (Luna 1) to the vicinity of the moon. Equally impressive is the first successful interplanetary mission (Mariner 2), which NASA launched less than 5 years after Explorer 1. Table 1.1 shows that spacecraft have visited all planets in our solar system and other celestial bodies such as comets and asteroids.

Table 1.1 Notable robotic space missions.

Mission

Date

Achievement

Country

Sputnik 1

October 4, 1957

First artificial satellite to achieve Earth orbit

USSR

Luna 1

January 2, 1959

First satellite to reach the vicinity of the moon

USSR

Mariner 2

December 14, 1962

First spacecraft to encounter (fly by) another planet (Venus)

US

Mariner 4

July 14, 1965

First spacecraft to fly by Mars

US

Luna 9

February 3, 1966

First spacecraft to land on another body (moon)

USSR

Luna 10

April 3, 1966

First spacecraft to orbit the moon

USSR

Venera 7

December 15, 1970

First spacecraft to land on another planet (Venus)

USSR

Mariner 9

November 14, 1971

First spacecraft to orbit another planet (Mars)

US

Pioneer 10

December 3, 1973

First spacecraft to fly by Jupiter

US

Mariner 10

March 29, 1974

First spacecraft to fly by Mercury

US

Viking 1

July 20, 1976

First spacecraft to land on Mars

US

Voyager 1

March 1979, November 1980

Fly by encounters with Jupiter, Saturn, and Saturn’s moon Titan

US

Voyager 2

January 1986, August 1989

First spacecraft to fly by Uranus and Neptune

US

Galileo

December 8, 1995

First spacecraft to orbit Jupiter

US

Mars Pathfinder

July 4, 1997

First rover on the planet Mars

US

NEAR Shoemaker

February 12, 2001

First spacecraft to land on an asteroid (433 Eros)

US

Cassini‐Huygens

July 2004, January 2005

First spacecraft to orbit Saturn (Cassini) and first spacecraft to land on the moon Titan (Huygens)

US and Europe

Stardust

January 16, 2006

First spacecraft to return samples from a comet

US

MESSENGER

March 18, 2011

First spacecraft to orbit Mercury

US

New Horizons

July 14, 2015

First spacecraft to fly by Pluto

US

On April 12, 1961, the USSR successfully sent the first human into space when Yuri Gagarin orbited the Earth in the Vostok 1 spacecraft. Less than 1 month later, the US launched its first human into space when Alan Shepard flew a suborbital mission in a Mercury spacecraft. Table 1.2 presents notable space missions with human crews (as with Table 1.1, Table 1.2 focuses on first‐time achievements). Tables 1.1 and 1.2 clearly illustrate the accelerated pace of accomplishments in space flight. Table 1.2 shows