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Astronautics: The Physics of Space Flight
Astronautics: The Physics of Space Flight
Ulrich Walter
This introductory text covers all the key concepts, relationships, and ideas behind spaceflight and is the perfect companion for students pursuing courses on or related to astronautics. As a crew member of the STS55 Space Shuttle mission and a full professor of astronautics at the Technical University of Munich, Ulrich Walter is an acknowledged expert in the field. This book is based on his extensive teaching and work with students, and the text is backed up by numerous examples drawn from his own experience. With its endofchapter examples and problems, this work is suitable for graduate level or even undergraduate courses in spaceflight, as well as for professionals working in the space industry. This third edition includes substantial revisions of several sections to extend their coverage. These include both theoretical extensions such as the study of relative motion in nearcircular orbits, and more practical matters such as additional details about jetengine and general rocket performance. New sections address regularized equations of orbital motion and their algebraic solutions and also state vector propagation; two new chapters are devoted to orbit geometry and orbit determination and to thermal radiation physics and modelling.
Categories:
Physics\\Astronomy: Astrophysics
Year:
2018
Edition:
3
Publisher:
Springer Nature Switzerland AG
Language:
english
Pages:
849
ISBN 10:
3319743724
ISBN 13:
9783319743721
File:
PDF, 47.42 MB
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Most frequently terms
orbit^{1465}
fig^{798}
orbital^{731}
cos^{707}
orbits^{639}
equation^{636}
velocity^{532}
sect^{474}
thermal^{417}
rocket^{387}
gravitational^{378}
vector^{352}
axis^{332}
equations^{321}
trajectory^{263}
circular^{246}
reentry^{232}
momentum^{227}
see fig^{212}
radiation^{205}
angular^{203}
nozzle^{199}
maneuver^{195}
satellite^{195}
altitude^{193}
elliptic^{191}
spacecraft^{190}
propulsion^{179}
radial^{179}
external^{177}
rotation^{176}
perturbations^{175}
see sect^{173}
eqs^{173}
shuttle^{171}
derive^{168}
orbital elements^{159}
const^{156}
launch^{154}
flux^{153}
eccentricity^{149}
ascent^{149}
solar^{147}
atmospheric^{135}
hohmann^{133}
maneuvers^{131}
transfer orbit^{131}
inclination^{129}
dynamics^{129}
density^{129}
propellant^{128}
libration^{126}
flyby^{121}
periapsis^{121}
delta^{119}
iss^{118}
angular momentum^{117}
node^{117}
coefficient^{117}
rendezvous^{116}
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Ulrich Walter Astronautics The Physics of Space Flight Third Edition Astronautics Ulrich Walter Astronautics The Physics of Space Flight Third Edition 123 Ulrich Walter Institute of Astronautics Technical University of Munich Garching Germany ISBN 9783319743721 ISBN 9783319743738 (eBook) https://doi.org/10.1007/9783319743738 Library of Congress Control Number: 2017964237 1st and 2nd edition: © WileyVCH 2008, 2012 3rd edition: © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover: The Space Shuttle Atlantis launched on February 7, 2008, to ferry on its 29th flight the European science laboratory Columbus to the International Space Station. (Used with permission of NASA) This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland This book is dedicated to the astronauts and cosmonauts, who lost their lives in the pursuit of space exploration Preface to the Third Edition This textbook is about all basic physical aspects of spaceflight. Not all have been covered in the past editions. So, what is new in this third edition? First, there are new sections covering new topics, such as – Sections 1.2 and 1.3 dealing with the physics of a jet engine and general rocket performance have been widely extended to more sophisticated effects. – Sections 7.4.5 and 7.4.6 describe two general solutions to Newton’s gravita tional equation of motion. – Section 7.7 studies stellar orbits, which are not subject to the standard but more general types of gravitational potentials. – Hypersonic flow theory for reentry vehicles is expounded in Sect. 6.2 as a basis to understand how lift and drag come about and in particular how both depend on the angle of attack, the most important control parameter to guide a winged body through the flight corridor (see Fig. 10.22). – Accordingly, the reentry of a Space Shuttle, which in this book even more serves as a case study, is explained in Sect. 10.7 in greater detail and in terms of NASA terminology. – In Sect. 8.1, the different basic types of orbit maneuvers are discussed and exemplified. – A new form of solution of Lambert’s problem is derived in Sect. 8.2.3, which is visualized in Fig. 8.8. – Section 8.4.3 discusses modern supersynchronous transfer orbits to GEO. – Relative motion in nearcircular orbits is examined in Sect. 8.5.4. – The virial theorem for bounded and unbounded nbody systems is derived in Sect. 11.1.2 and used to discuss the stability of an nbody system. – Section 12.3 (Gravitational Perturbation Effects) has been revised and greatly extended including other and higher order perturbation terms. – Chapter 14 has been radically revised: There is a new Sect. 14.1 on orbit geometric issues (eclipse duration and access area) and a fully revised Sect. 14.2 on orbit determination. vii – There is a whole new Chap. 16 dedicated to thermal radiation physics and modeling. It serves the same purpose as Chap. 15 Spacecraft Attitude Dynamics, namely to provide insight into some basic and important physics of a spacecraft in space. Some sections have been substantially revised and there a hundreds more or less significant extensions of established topics of space fight as already covered in the 2nd version of this textbook. I put a lot of effort into introducing and using a proper terminology, or estab lishing one if not existent. An example of the former is the distinction between orbital velocity v, angular velocity x, angular frequency xi, and orbital frequency n, which are sometimes confused. Orbital velocity v is the speed of motion of a body on an orbit. Angular velocity x is the instantaneous speed of angular motion, while angular frequency xi is the number of revolutions in a given time. Finally, orbital frequency n (a.k.a. mean motion) is the time average of the angular velocity over one orbital period T (see Eq. (7.4.10)). Thus, n ¼ 2p=T; it therefore can be con sidered both as a mean angular velocity (i.e., mean angular motion) and as a frequency, the orbital frequency. Because proper terminology is essential, the conventional “symbols used” table on the following pages also serves the purpose of enabling one to look up the proper terminology for a physical quantity. Because physics is independent of the choice of the reference system, the third version consequently uses a reference systemfree vector notation (except auxiliary corotating reference systems in Sects. 6.3 and 7.3). All reference systems, the transformations between them, and the vector representations in the different common reference systems are summed up in Sect. 13.1. Finally, I feel the need to a very personal comment on textbooks in general. When I was a student, I bought some expensive but basic physics textbooks, which are still in my office shelf and serve as my reference books, because true physics is eternal. Compare buying a textbook with a marriage. You do not just buy it. It must have a kind of visual—a tactile sensuality: You open it with joyful anticipation. Your fingers glide over the pages, and they slowly turn one page after the other. You like the layout, the way the book talks to you, and how it explains the world from a point of view you have never considered before. You just love it, and thus it will become part of your daily scientific work. You may forget little physical details, but you will always remember that the one you are looking for is on top of the lefthand page somewhere in the middle of the book. You will never forget that visual detail, and therefore you will always find the answer to your question quite swiftly. I have about a handful of such key textbooks, which I would not sell in my lifetime. I sense that these books were written for guiding me through my scientific life. For me, writing this book was for giving back to other people what many scientists before had given to me. We all are standing on the shoulders of giants. May this textbook keep and pass the body of basic knowledge to you and future generations. Garching, Germany Ulrich Walter viii Preface to the Third Edition Preface to the Second Edition Textbooks are subject to continuous and critical scrutiny of students. So is this one. Having received many questions to the book in my lectures and by email, I constantly improved and updated the content such that already after three years it was time to have also the reader benefit from this. You will therefore find the textbook quite revised as for instance rocket staging (Chapter 3), engine design (Section 4.4), radial orbits (Section 7.5), or the circular restricted theebody prob lem (Section 11.4). But there are also new topics, namely Lambert transfer (Section 8.2), relative orbits (Section 8.5), and orbital rendezvous (Section 8.6), higher orbit perturbations including frozen orbits (Sections 12.3.6 and 12.3.7), resonant perturbations and resonant orbits (Section 12.4), and relativistic pertur bations (Section 12.6.2). Along with this also the structure of the content has changed slightly. Therefore the section and equation numbers are not always identical to the first edition. Nevertheless the overall structure still serves the same intention: It is set up for a two semester course on astronautics. Chapter 1–7 (except Sect. 1.4), Section 8.1, and Chapters 9–10 is the basic subject matter an aerospace student should know or have been exposed to at least once. The sequence of the chapters is first rocket basics (Chapter 1–5), thereafter a flight into space “once around”, starting with ascent flight (Chapter 6), then space orbits (Chapter 7) and basic orbital maneuvers (Section 8.1), interplanetary flight (Chapter 9), and finally reentry (Chapter 10). The second part of the textbook is more advanced material, which I lecture together with satellite technology in an advanced course for true rocket scientists and space engineers. The careful reader might have noticed that the book now comes with a subtitle: The Physics of Space Flight. This was decided to provide a quick comprehension of the nature of this textbook. In addition, because the Space Shuttle and the ISS are running examples in this textbook, a picture of the launching Space Shuttle Atlantis was chosen as a new frontispiece. Unfortunately, I couldn’t find an equally attractive picture of my Space Shuttle Columbia. ix Preface to the First Edition There is no substitute for true understanding Kai Lai Chung If you want to cope with science, you have to understand it – truly understand it. This holds in particular for astronautics. “To understand” means that you have a network of relationships in your mind, which permits you to deduce an unknown fact from wellknown facts. The evolution of a human being from birth to adult hood and beyond consists of building up a comprehensive knowledge network of the world, which makes it possible to cope with it. That you are intelligent just means that you are able to do that – sometimes you can do it better, and sometimes worse. True understanding is the basis of everything. There is nothing that would be able to substitute true understanding. Computers do not understand – they merely carry out programmed deterministic orders. They do not have any understanding of the world. This is why even a large language computer will always render a false translation of the phrase: “He fed her cat food.” Our world experience intuitively tells us that “He fed a woman’s cat some food.” But a computer does not have world experience, and thus does not generally know that cat food is nasty for people. Most probably, and according to the syntax, it would translate it as: “He fed a woman some food that was intended for cats.”, what the Google translator actually does when translating this phase into other languages. No computer pro gram in the world is able to substitute understanding. You have to understand yourself. Only when you understand are you able to solve problems by designing excellent computer programs. Nowadays, real problems are only solved on com puters – written by bright engineers and scientists. The goal of this book is to build up a network of astronautic relationships in the mind of the reader. If you don’t understand something while reading this book, I made a mistake. The problem of a relational network, though, is that the underlying logic can be very complex, and sometimes it seems that our brains are not suitable for even the simplest logic. If I asked you, “You are not stupid, are you?”, you would normally answer, “No!” From a logical point of view, a double negation of xi an attribute is the attribute itself. So your “No!” means that you consider yourself stupid. You, and also we scientists and engineers, do not want this embarrassing mistake to happen time and time again, and so we use mathematics. Mathematical logic is the guardrail of human thinking. Physics, on the other hand, is the art of applying this logic consistently to nature in order to be able to understand how it works. So it comes as no surprise to find a huge amount of formulas and a lot of physics in this book. Some might think this is sheer horror. But now comes the good news. Most of the formulas are just intermediate steps of our elaborations. To understand astronautics, you only need to engage in the formulas shaded gray and to remember those bordered black. There you should pause and try to understand their meaning because they will tell you the essential story and lift the secrets of nature. Though you don’t need to remember all the other formulas, as a student you should be able to derive these stepping stones for yourself. Thereby you will always be able to link nodes in your relational network whenever you deem it necessary. To treat formulas requires knowing a lot of tricks. You will learn them only by watching others doing such “manipulation” and, most importantly, by doing it yourself. Sometimes you will see the word “exercise” in brackets. This indicates that the said calculation would be a good exercise for you to prove to yourself that you know the tricks. Sometimes it might denote that there is not the space to fully lay out the needed calculation because it is too lengthy or quite tricky. So, you have to guess for yourself whether or not you should do the exercise. Nonetheless, only very few of you will have to derive formulas professionally later. For the rest of you: just try to follow the story and understand how consistent and wonderful nature is. Those who succeed will understand the words of Richard Feynman, the great physicist, who once expressed his joy about this by saying: “The pleasure of finding things out.” Take the pleasure to find out about astronautics. xii Preface to the First Edition Acknowledgements Key parts of the new Chap. 16 are the two sections of thermal modeling. Thermal modeling of vehicles in space requires not only high skill, but also a lot of expert knowledge gathered in daily work. I am happy and very thankful to Philipp Hager (Thermal Engineer in the Thermal Control Section of the European Space Agency at ESTEC) and to Markus Czupalla (Full Professor at the Department of Aerospace Engineering, University of Applied Sciences Aachen, Germany) that they agreed to contribute these important sections. I am grateful to Olivier L. de Weck, Bernd Häusler, and HansJoachim Blome for carefully reading the manuscript and for many fruitful suggestions. My sincere thanks go to my research assistants Markus Wilde, who contributed Sects. 8.5 and 8.6 for the second edition, and equally also for this third edition; Winfried Hofstetter, who contributed Sect. 9.6 and the freereturn trajectories to Sect. 11.4.4; to my colleague Oskar Haidn for his expertise in Sect. 4.4; and to my master student Abhishek Chawan at Technical University, Munich, who provided numerical cal culations and figures to the subsection SuperSynchronous Transfer Orbits in Sect. 8.4.3. My special thanks go to Julia Bruder for her tedious work of translating the original German manuscript into English. Many expounding passages of this book would not be in place without the bright questions of my students, who reminded me of the fact that a lot of implicit meanings that scientists have become used to are not that trivial as they seem to be. Many figures in this book were drawn by the interactive plotting program gnuplot v4.0. My sincere thanks to its authors Thomas Williams, Colin Kelley, HansBernhard Bröker, and many others for establishing and maintaining this versatile and very useful tool for free public use. The author is grateful to the GeoForschungsZentrum Potsdam, Germany’s National Research Centre for Geosciences for providing the geoid views and the visualization of the spherical harmonics in the color tables on pages 566, 568, and 569. xiii Contents 1 Rocket Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Rocket Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Repulsion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Total Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Equation of Rocket Motion . . . . . . . . . . . . . . . . . . . . . 5 1.2 Jet Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Nozzle Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Pressure Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 Momentum versus Pressure Thrust . . . . . . . . . . . . . . . 12 1.3 Rocket Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Payload Considerations . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Rocket Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.3 Performance Parameters . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 Relativistic Rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.1 Space Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4.2 Relativistic Rocket Equation . . . . . . . . . . . . . . . . . . . . 28 1.4.3 Exhaust Considerations . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.4 External Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4.5 Space–Time Transformations . . . . . . . . . . . . . . . . . . . 32 1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 Rocket Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Rocket in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Rocket in a Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.1 Impulsive Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.2 Brief Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.3 Gravitational Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Deltav Budget and Fuel Demand . . . . . . . . . . . . . . . . . . . . . . 43 2.4.1 Deltav Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 xv 2.4.2 Fuel Demand—Star Trek Plugged . . . . . . . . . . . . . . . . 44 2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Rocket Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Serial Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.2 Rocket Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 SerialStage Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.1 Road to Stage Optimization . . . . . . . . . . . . . . . . . . . . 52 3.2.2 General Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 Uniform Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.2 Uniform Exhaust Velocities . . . . . . . . . . . . . . . . . . . . 60 3.3.3 Uneven Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Parallel Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Other Types of Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Thermal Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Engine Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.1 Physics of Propellant Gases . . . . . . . . . . . . . . . . . . . . 68 4.1.2 Flow Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.3 Flow at the Throat . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.4 Flow in the Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Ideally Adapted Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 IdealAdaptation Criterion . . . . . . . . . . . . . . . . . . . . . . 81 4.2.2 Ideal Nozzle Design . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.3 Shock Attenuation and Pogos . . . . . . . . . . . . . . . . . . . 85 4.2.4 Ideal Engine Performance . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Engine Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.1 Engine Performance Parameters . . . . . . . . . . . . . . . . . 89 4.3.2 Thrust Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3.3 Nozzle Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 Engine Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.1 Combustion Chamber . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4.2 Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.3 Design Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Electric Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 Ion Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 xvi Contents 5.2.1 Ion Acceleration and Flow . . . . . . . . . . . . . . . . . . . . . 107 5.2.2 Ideal Engine Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.3 Thruster Performance . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3 Electric Propulsion Optimization . . . . . . . . . . . . . . . . . . . . . . . 115 5.4 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6 Atmospheric and Ascent Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1 Earth’s Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1.1 Density Master Equation . . . . . . . . . . . . . . . . . . . . . . . 122 6.1.2 Atmospheric Structure . . . . . . . . . . . . . . . . . . . . . . . . 123 6.1.3 PiecewiseExponential Model . . . . . . . . . . . . . . . . . . . 129 6.2 Hypersonic Flow Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2.1 Free Molecular Flow . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2.2 Newtonian Flow Theory . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.3 Drag and Lift Coefficients . . . . . . . . . . . . . . . . . . . . . . 136 6.2.4 Drag in Free Molecular Flow . . . . . . . . . . . . . . . . . . . 137 6.2.5 Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.4 Ascent Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.4.1 Ascent Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4.2 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4.3 Gravity Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.4.4 Pitch Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.4.5 ConstantPitchRate Maneuver . . . . . . . . . . . . . . . . . . 157 6.4.6 Terminal State Control . . . . . . . . . . . . . . . . . . . . . . . . 160 6.4.7 Optimal Ascent Trajectory . . . . . . . . . . . . . . . . . . . . . 162 7 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1 Fundamental Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1.1 Gravitational Potential . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1.2 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.1.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.1.4 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . . 175 7.1.5 General TwoBody Problem . . . . . . . . . . . . . . . . . . . . 178 7.2 General Principles of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2.1 Vector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2.2 Motion in a Central Force Field . . . . . . . . . . . . . . . . . 182 7.2.3 VisViva Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.2.4 Effective Radial Motion . . . . . . . . . . . . . . . . . . . . . . . 188 7.3 Motion in a Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . 190 7.3.1 Orbit Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.3.2 Position on the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 194 Contents xvii 7.3.3 Orbital Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.3.4 Orbital Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.3.5 Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.4 Keplerian Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.4.1 Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.4.2 Elliptic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.4.3 Hyperbolic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.4.4 Parabolic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.4.5 eBased Transformation . . . . . . . . . . . . . . . . . . . . . . . 231 7.4.6 hBased Transformation . . . . . . . . . . . . . . . . . . . . . . . 235 7.4.7 Conventional State Vector Propagation . . . . . . . . . . . . 242 7.4.8 Universal Variable Formulation . . . . . . . . . . . . . . . . . . 244 7.5 Radial Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.5.1 Radial Elliptic Trajectory . . . . . . . . . . . . . . . . . . . . . . 248 7.5.2 Radial Hyperbolic Trajectory . . . . . . . . . . . . . . . . . . . 250 7.5.3 Radial Parabolic Trajectory . . . . . . . . . . . . . . . . . . . . . 251 7.5.4 Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.5.5 Bounded Vertical Motion . . . . . . . . . . . . . . . . . . . . . . 253 7.6 Life in Other Universes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.6.1 Equation of Motion in n Dimensions . . . . . . . . . . . . . . 256 7.6.2 4Dimensional Universe . . . . . . . . . . . . . . . . . . . . . . . 259 7.6.3 Universes with � 5 Dimensions . . . . . . . . . . . . . . . . . 260 7.6.4 Universes with � 2 Dimensions . . . . . . . . . . . . . . . . . 262 7.7 Stellar Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.7.1 Motion in General Gravitational Potentials . . . . . . . . . . 262 7.7.2 Stellar Motion in General Galaxies . . . . . . . . . . . . . . . 266 7.7.3 Stellar Orbits in Globular Cluster Galaxies . . . . . . . . . . 269 7.7.4 Stellar Motion in DiskShaped Galaxies . . . . . . . . . . . . 271 7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8 Orbital Maneuvering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.1 OneImpulse Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 8.1.1 Elementary Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . 281 8.1.2 Elementary Maneuvers in Circular Orbits . . . . . . . . . . 287 8.1.3 General Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.1.4 Tangent Plane Maneuvers . . . . . . . . . . . . . . . . . . . . . . 292 8.1.5 Genuine Plane Change Maneuvers . . . . . . . . . . . . . . . 293 8.1.6 Tangent Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.2 Lambert Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 8.2.1 Orbital Boundary Value Problem . . . . . . . . . . . . . . . . 298 8.2.2 Lambert Transfer Orbits . . . . . . . . . . . . . . . . . . . . . . . 301 8.2.3 Lambert’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 8.2.4 Minimum Effort Lambert Transfer . . . . . . . . . . . . . . . . 313 xviii Contents 8.3 Hohmann Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 8.3.1 The Minimum Principle . . . . . . . . . . . . . . . . . . . . . . . 314 8.3.2 Transfer Between Circular Orbits . . . . . . . . . . . . . . . . 317 8.3.3 Transfer Between NearCircular Orbits . . . . . . . . . . . . 321 8.3.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.4 Other Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 8.4.1 Parabolic Escape Transfer . . . . . . . . . . . . . . . . . . . . . . 325 8.4.2 Bielliptic Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.4.3 SuperSynchronous Transfer Orbits . . . . . . . . . . . . . . . 329 8.4.4 nImpulse Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . 334 8.4.5 Continuous Thrust Transfer . . . . . . . . . . . . . . . . . . . . . 334 8.5 Relative Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 8.5.1 General Equation of Motion . . . . . . . . . . . . . . . . . . . . 337 8.5.2 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8.5.3 Flyaround Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 345 8.5.4 NearCircular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.6 Orbital Rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 8.6.1 Launch Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 8.6.2 Phasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 8.6.3 Homing Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 8.6.4 Closing Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 8.6.5 Final Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 8.6.6 ShuttleISS Rendezvous . . . . . . . . . . . . . . . . . . . . . . . 375 8.6.7 Plume Impingement . . . . . . . . . . . . . . . . . . . . . . . . . . 378 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 9 Interplanetary Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 9.1 Patched Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 9.1.1 Sphere of Influence . . . . . . . . . . . . . . . . . . . . . . . . . . 386 9.1.2 Patched Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 9.2 Departure Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 9.3 Transfer Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 9.3.1 Hohmann Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . 393 9.3.2 NonHohmann Transfers . . . . . . . . . . . . . . . . . . . . . . . 396 9.4 Arrival Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 9.5 Flyby Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 9.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 9.5.2 Flyby Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 9.5.3 Planetocentric Flyby Analysis . . . . . . . . . . . . . . . . . . . 409 9.5.4 Heliocentric Flyby Analysis . . . . . . . . . . . . . . . . . . . . 415 9.5.5 Transition of Orbital Elements . . . . . . . . . . . . . . . . . . 418 9.6 Weak Stability Boundary Transfers . . . . . . . . . . . . . . . . . . . . . 422 9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Contents xix 10 Planetary Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 10.1.1 Aerothermodynamical Challenges . . . . . . . . . . . . . . . . 428 10.1.2 Entry Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 10.1.3 Deorbit Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 10.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 10.2.1 Normalized Equations of Motion . . . . . . . . . . . . . . . . . 437 10.2.2 Reduced Equations of Motion . . . . . . . . . . . . . . . . . . . 442 10.3 Elementary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 10.3.1 DragFree Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 10.3.2 Ballistic Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.3.3 Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 10.4 Reentry with Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 10.4.1 LiftOnly Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 10.4.2 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 10.4.3 NearBallistic Reentry . . . . . . . . . . . . . . . . . . . . . . . . . 459 10.5 Reflection and Skip Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . 466 10.5.1 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 10.5.2 Skip Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 10.5.3 Phugoid Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 10.6 Lifting Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 10.6.1 Reentry Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 10.6.2 Critical Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 479 10.6.3 Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 10.7 Space Shuttle Reentry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 10.7.1 Reentry Flight Design and Preentry Phase . . . . . . . . . 485 10.7.2 Constant Heat Rate Phase (Thermal Control Phase) . . . 487 10.7.3 Equilibrium Glide Phase . . . . . . . . . . . . . . . . . . . . . . . 488 10.7.4 ConstantDrag Phase . . . . . . . . . . . . . . . . . . . . . . . . . 489 10.7.5 Transition Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 10.7.6 TAEM Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 10.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 11 ThreeBody Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.1 The NBody Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.1.1 Integrals of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.1.2 Stability of an NBody System . . . . . . . . . . . . . . . . . . 494 11.1.3 NBody Choreographies . . . . . . . . . . . . . . . . . . . . . . . 498 11.2 Synchronous 3Body Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 11.2.1 Collinear Configuration . . . . . . . . . . . . . . . . . . . . . . . . 500 11.2.2 Equilateral Configuration . . . . . . . . . . . . . . . . . . . . . . 506 11.3 Restricted ThreeBody Problem . . . . . . . . . . . . . . . . . . . . . . . . 508 xx Contents 11.3.1 Collinear Libration Points . . . . . . . . . . . . . . . . . . . . . . 510 11.3.2 Equilateral Libration Points . . . . . . . . . . . . . . . . . . . . . 513 11.4 Circular Restricted ThreeBody Problem . . . . . . . . . . . . . . . . . 513 11.4.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 515 11.4.2 Jacobi’s Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 11.4.3 Stability at Libration Points . . . . . . . . . . . . . . . . . . . . . 520 11.4.4 General System Dynamics . . . . . . . . . . . . . . . . . . . . . 522 11.5 Dynamics About Libration Points . . . . . . . . . . . . . . . . . . . . . . 529 11.5.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 529 11.5.2 Collinear Libration Points . . . . . . . . . . . . . . . . . . . . . . 530 11.5.3 Equilateral Libration Points . . . . . . . . . . . . . . . . . . . . . 545 11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 12 Orbit Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 12.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 12.1.1 Origins of Perturbations . . . . . . . . . . . . . . . . . . . . . . . 555 12.1.2 Osculating Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 12.1.3 Gaussian Variational Equations . . . . . . . . . . . . . . . . . . 558 12.2 Gravitational Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 12.2.1 Geoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 12.2.2 Gravitational Potential . . . . . . . . . . . . . . . . . . . . . . . . 561 12.2.3 Lagrange’s Planetary Equations . . . . . . . . . . . . . . . . . . 570 12.2.4 Numerical Perturbation Methods . . . . . . . . . . . . . . . . . 570 12.3 Gravitational Perturbation Effects . . . . . . . . . . . . . . . . . . . . . . . 574 12.3.1 Classification of Effects . . . . . . . . . . . . . . . . . . . . . . . 574 12.3.2 Removing ShortPeriodic Effects . . . . . . . . . . . . . . . . . 576 12.3.3 Oblateness Perturbation . . . . . . . . . . . . . . . . . . . . . . . 578 12.3.4 HigherOrder Perturbations . . . . . . . . . . . . . . . . . . . . . 582 12.3.5 SunSynchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . 595 12.3.6 Frozen Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 12.3.7 Frozen SunSynchronous Orbits . . . . . . . . . . . . . . . . . 600 12.4 Resonant Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 12.4.1 Resonance Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 603 12.4.2 Resonance Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 605 12.4.3 Low Earth Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 12.4.4 GPS Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 12.4.5 Geostationary Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 615 12.5 Solar Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 12.5.1 Effects of Solar Radiation . . . . . . . . . . . . . . . . . . . . . . 622 12.5.2 Orbital Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 12.5.3 Correction Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . 629 12.6 Celestial Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 12.6.1 Lunisolar Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 632 Contents xxi 12.6.2 Relativistic Perturbations . . . . . . . . . . . . . . . . . . . . . . . 639 12.7 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 12.7.1 Drag Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 12.7.2 Orbit Circularization . . . . . . . . . . . . . . . . . . . . . . . . . . 643 12.7.3 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 12.7.4 Orbit Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 13 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 13.1 Space Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 13.1.1 Inertial Reference Frames . . . . . . . . . . . . . . . . . . . . . . 662 13.1.2 Heliocentric Reference Frames . . . . . . . . . . . . . . . . . . 664 13.1.3 Terrestrial Reference Frames . . . . . . . . . . . . . . . . . . . . 666 13.1.4 Orbital Reference Frames . . . . . . . . . . . . . . . . . . . . . . 667 13.1.5 Vector Representations . . . . . . . . . . . . . . . . . . . . . . . . 670 13.2 Time Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 14 Orbit Geometry and Determination . . . . . . . . . . . . . . . . . . . . . . . . 677 14.1 Orbit Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 14.1.1 Eclipse Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 14.1.2 Access Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 14.2 Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 14.2.1 Orbit Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 14.2.2 Generalized Orbit Determination Method . . . . . . . . . . . 686 14.2.3 GEO Orbit from AnglesOnly Data . . . . . . . . . . . . . . . 690 14.2.4 Simple Orbit Estimation . . . . . . . . . . . . . . . . . . . . . . . 692 14.2.5 Modified Battin’s Method . . . . . . . . . . . . . . . . . . . . . . 693 14.2.6 Advanced Orbit Determination . . . . . . . . . . . . . . . . . . 695 15 Spacecraft Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 15.1 Fundamentals of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 15.1.1 Elementary Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 700 15.1.2 Equations of Rotational Motion . . . . . . . . . . . . . . . . . . 706 15.1.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 708 15.1.4 RotationtoTranslation Equivalence . . . . . . . . . . . . . . 710 15.2 Attitude Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 15.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 15.2.2 Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 15.2.3 General TorqueFree Motion . . . . . . . . . . . . . . . . . . . . 717 15.3 Attitude Dynamics Under External Torque . . . . . . . . . . . . . . . . 719 15.3.1 External Torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 15.3.2 Road to Flat Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 xxii Contents 15.3.3 Flat Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 724 15.4 GravityGradient Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 726 15.4.1 GravityGradient Torque . . . . . . . . . . . . . . . . . . . . . . . 727 15.4.2 GravityGradient Dynamics . . . . . . . . . . . . . . . . . . . . . 729 16 Thermal Radiation Physics and Modeling . . . . . . . . . . . . . . . . . . . . 735 16.1 Radiation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 16.1.1 Radiometric Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 738 16.1.2 Diffuse Radiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 16.1.3 BlackBody Radiator . . . . . . . . . . . . . . . . . . . . . . . . . 743 16.1.4 Selective Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 16.1.5 Kirchhoff’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 16.2 Radiation Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 16.2.1 Transmitted and Absorbed Flux . . . . . . . . . . . . . . . . . 751 16.2.2 View Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 16.2.3 Point Radiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 16.2.4 Radiation Exchange Between Two Bodies . . . . . . . . . . 756 16.2.5 Spacecraft Thermal Balance . . . . . . . . . . . . . . . . . . . . 759 16.2.6 a/ɛ Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 16.3 Thermal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 16.3.1 Thermal Requirements and Boundary Conditions . . . . . 767 16.3.2 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768 16.3.3 Thermal Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . 770 16.3.4 Geometric Mathematical Model (GMM) . . . . . . . . . . . 774 16.3.5 Thermal Mathematical Model (TMM) . . . . . . . . . . . . . 780 16.3.6 Applied Thermal Design and Analysis . . . . . . . . . . . . . 784 16.3.7 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 16.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 Appendix A: Planetary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 Appendix B: Approximate Analytical Solution for Uneven Staging . . . . 801 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Contents xxiii Abbreviations AOA Angle of attack AU Astronomical unit CFPAR Constant flight path angle rate CM Center of mass CPR Constant pitch rate CR3BP Circular restricted threebody problem EGM96 Earth gravitational model 1996 EL1 SunEarth libration point L1 EoM Equation of motion ET External tank EQW Equinoctial coordinate system EQW (see Sect. 7.3.5) FPA Flight path angle GEO Geostationary orbit GEODSS Groundbased electrooptical deep space surveillance GG Gravity gradient GMT Greenwich mean time GMST Greenwich mean sidereal time GSO Geosynchronous orbit GTO Geostationary transfer orbit GVE Gaussian variational equation IAU International Astronomical Union ICRF International Celestial Reference Frame IJK Cartesian equatorial coordinate system (see Sect. 13.1.4) ITRF International Terrestrial Reference Frame ISS International Space Station JD Julian date LEO Low earth orbit, 100 km \ h \ 2000 km LL1 EarthMoon (Lunar) libration point L1 LPE Lagrange’s planetary equations LVLH Local vertical, local horizontal (reference frame) xxv MECO Main engines cutoff MEO Medium earth orbit, 2000 km \ h \ GEO MJD Modified julian date NTW Corotating Cartesian topocentric satellite coordinate system NTW (see Sect. 13.1.4) OMS Orbital maneuvering system PQW Cartesian geocentric perifocal coordinate system PQW (see Sect. 13.1.4) R&D Rendezvous and docking R3BP Restricted threebody problem RAAN Right ascension of ascending node RSW Corotating Cartesian topocentric satellite coordinate system RSW (see Sect. 13.1.4) RTG Radioisotope thermoelectric generator S/C Spacecraft SOI Sphere of influence SRB Solid rocket booster SSME Space shuttle main engine SSO Sunsynchronous orbit SSTO Supersynchronous transfer orbit TAEM Terminal area energy management TDRS Tracking and data relay satellite TDRSS Tracking and data relay satellite system TTPR Thrusttopower ratio UT Universal time VDF Velocity distribution function WSB Weak stability boundary xxvi Abbreviations Symbols Used and Terminology x; a; b :ð Þ; f �ð Þ Scalars/scalarvalued functions x; a; bð:Þ; f ð�Þ Vectors/vectorvalued functions X ; X; J �ð Þ;M �ð Þ Matrices/matrixvalued functions Superscripts T Transpose of a vector or matrix Subscripts 0 At the beginning (zero); or osculating (momentary) a With respect to the atmosphere air Atmosphere apo Apoapsis B Body system c Combustion; or commensurate CM Center of mass (a.k.a. barycenter) col Collision crit Critical (maximal deceleration) D Aerodynamic drag div (Jet) divergence e At exit, or ejection; or at entry interface eff Effective esc Escape (velocity) ex Exhaust ext External EQW Equinoctial coordinate system EQW (see Sect. 7.3.5) xxvii f Final (mass); or frozen orbit F Force G Gravitation GEO Geostationary GG Gravity gradient h Horizontal H Hohmann i Initial (mass) id Ideal engine I Inertial reference frame IJK Cartesian equatorial coordinate system (see Sect. 13.1.4) IR Infrared ion Ionic in Initial, at entry, incoming int Internal jet Propellant exhaust jet kin Kinetic (energy) L Aerodynamic lift; or payload; or libration point LVLH Local vertical, local horizontal (reference frame) max Maximum min Minimum micro Microscopic n Nozzle; or normal (vertically to …) NTW Corotating Cartesian topocentric satellite coordinate system NTW (see Sect. 13.1.4) opt Optimal (value) out Final, at exit, outgoing p Propellant; or planet; or perturbation; or periapsis (only in the case of epoch tp) P Principal axes system; or orbital period per Periapsis; or periodic PQW Cartesian geocentric perifocal coordinate system PQW (see Sect. 13.1.4) pot Potential (energy) xxviii Symbols Used and Terminology r Radial; or reflection; or radiation rms Rootmeansquare (a.k.a. quadratic mean) RSW Corotating Cartesian topocentric satellite coordinate system RSW (see Sect. 13.1.4) s Structural S/C Spacecraft sec Secular sk Station keeping sol Solar SOI Sphere of influence syn Synodic t Tangential; or throat (of thruster) T Transfer orbit tot Total trans Translation; or transition TTPR Thrusttopower ratio v Vertical VDF Velocity distribution function vib Vibration h Vertically to radial x Rotation (or centrifugal); or argument of periapsis (apsidal line) X Relating to the ascending node (draconitic) ∞ External, at infinity � Effective (thrust), total � Earth � Sun ∇ Spacecraft � Inner (orbit); or black body Outer (orbit); or in orbit plane £ Diameter; or cross section jj Parallel to … ? Vertical to … (A? ¼ effectively wetted surface area) At orbit crossing Symbols Used and Terminology xxix Latin Symbols a Semimajor axis (of a Keplerian orbit); or speed of sound; or acceleration A Area Ap Daily global index of geomagnetic activity, 0�Ap � 400 b Semiminor axis (of a Keplerian orbit); or b :¼ L � tan ce= 2Dð Þ B Ballistic coefficient (without index: for drag), (see Eq. (6.2.19)) c c :¼ L� cot ce= 2Dð Þ; or speed of light c� Characteristic velocity, c� :¼ p0At � mp cp Specific heat capacity at constant pressure cV Specific heat capacity at constant volume C Jacobi constant C3 Characteristic energy, C3 :¼ v21 C1 Infiniteexpansion coefficient CD Drag coefficient Cf Thrust coefficient; or skin friction drag coefficient CL Lift coefficient Cn Nozzle coefficient (a.k.a. nozzle efficiency) Cmn Multipole coefficient of the cosine term dx Variation (small changes) of x dvjj Differential increase in orbital velocity due to kickburn in flight direction dv?O Differential increase in orbital velocity due to kickburn vertical to flight direction, within orbital plane, outbound dv?? Differential increase in orbital velocity due to kickburn vertical to flight direction and vertical to orbital plane, parallel to angular momentum Dv Deltav budget D Drag force D Aerodynamic drag, D ¼ Dj j; or diameter diag(…) Diagonal matrix with elements (…) e Eccentricity; or electrical charge unit; or Eulerian number, e = 2.718281828 … E Energy; or (elliptic) eccentric anomaly Ei xð Þ Exponential integral (see Eq. (10.4.5)) f x f function (see definition Eq. (10.4.6)) F Force (without index: gravitational force); or hyperbolic anomaly xxx Symbols Used and Terminology F� Thrust force (total), (a.k.a. propellant force) F� Thrust (total) Fe Ejection thrust Fex Momentum thrust Fp Pressure thrust F10:7 Daily solar flux index at wavelength of 10.7 cm in units 10−22 W m2 Hz−1 (= 1 Jansky = 1 solar flux unit) g rð Þ (Earth’s) gravitational field g (Earth’s) mean gravitational acceleration, g ¼ GM� � r2 (see Sect. 7.1.2) g0 (Earth’s) mean gravitational acceleration at its surface, g0 :¼ g R�ð Þ ¼ GM� � R2� ¼ l� � R2� ¼ 9:7982876 m s�2 (see Sect. 7.1.2) G Gravitational constant, G = 6.67259 10−11 m3 kg−1 s−2; or generic anomaly h (Massspecific) angular momentum (i.e., per mass unit); or molar enthalpy; or height (above sea level); altitude H Enthalpy; or scale height i Inclination I Inertia tensor Ix; Iy; Iz Principal moments of inertia Isp (Weight)specific impulse j Charge flow density (a.k.a. charge flux) jn Reduced harmonic coefficient of order n Jnm Harmonic coefficient kB Boltzmann constant, kB ¼ 1:380650 1023 J K�1 L Lift force L Aerodynamic lift, L ¼ Lj j; or angular momentum L� Characteristic length of a combustion chamber m Body mass (without index: of a spacecraft) mi In the collinear configuration: masses ordered according to their index, m3\m2\m1 m0i In the collinear configuration: masses with m 0 2 located between m01 and m 0 3, and m 0 1 �m03 _m Mass flow rate (without index: of a spacecraft) M Central mass (central body); or total mass of a system of bodies; or mean anomaly; or molar mass Ma Mach number Ma :¼ v=a Symbols Used and Terminology xxxi n Rocket stage number; or mean motion (a.k.a orbital frequency); or mean number of excited degrees of freedom of gas molecules; or particle density N Particle number p Pressure; or propellant; or linear momentum p ¼ mv; or semilatus rectum, p :¼ h2�l ¼ a 1� e2ð Þ; or p :¼ H= eeRð Þ P Power Pn xð Þ Legendre polynomials of degree n Pmn xð Þ Unnormalized associated Legendre polynomials of degree n and order m q Electrical charge density; or q :¼ H cot2 ce � eeRð Þ _q Heat flux (a.k.a. heat flow density), _q :¼ _Q�A _Q Heat flow rate r Orbit radius; or ratio r Radial vector, a.k.a. position vector R Radius (of a celestial body, in particular Earth’s radius); or residual perturbational potential; or universal gas constant, R ¼ 8:314 JK�1 mol�1 R Rotation (matrix) R� Equatorial scale factor of the Earth gravitational model EGM96, R� ¼ 6378:1363 km, equaling roughly Earth’s mean equatorial radius Rs Specific gas constant of standard atmosphere, Rs ¼ 286:91 JK�1 kg�1 Re Reynolds number Smn Multipole coefficient of the sine term sgn xð Þ Sign function (sign of x): sgn xð Þ ¼ x= xj j St Stanton number, St � 0:1% t Time tp Time at passage through periapsis, a.k.a. epoch (see end of Sect. 7.3.1) T Temperature; or orbital period; or torque u Argument of latitude u Unit vector as a basis of a reference system (example: ur r̂ ¼ r=r) xxxii Symbols Used and Terminology U Internal energy of a gas; or total electrical voltage; or potential (for which holds F ¼ �m � dU=dr) (without index: gravitational potential) v Velocity (orbital v of the spacecraft, or drift v of propellant gas) v Velocity vector v. First cosmic velocity, v. ¼ ffiffiffiffiffiffiffiffiffiffiffi g0R� p ¼ 7:905 km s�1 (see Eq. (7.4.4)) v.. Second cosmic velocity, v.. ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2g0R� p ¼ 11:180 km s�1 (see Sect. 7.4.4) v� Effective exhaust velocity ve Ejection velocity �ve Mean ejection velocity, �ve :¼ hveil vex Exhaust velocity, vex ¼ gdiv�ve vh vh :¼ l=h V Volume; or electric potential yr Year(s) Greek Symbols a Thrust angle; or angle of attack (AOA); or proper acceleration; or massspecific power output of an electrical plant; or absorptivity (a.k.a. absorption coefficient) b b :¼ v=cor geocentric latitude; or (orbit) beta angle d Deflection angle d xð Þ Dirac delta function dnm Kronecker delta dx A finite (not differential) but small variation of x; or small variation of an orbital element over one orbital revolution; or small error of x dv An impulsive maneuver (kickburn) varying the orbital velocity by dv D Impact parameter Di Distance of the libration point i normalized to the distance between the two primaries in the R3BP (see Sect. 11.3.1) Symbols Used and Terminology xxxiii Dx The amount of change of parameter x e Structural ratio; or specific orbital energy (a.k.a. specific mechanical energy); or expansion ration; or e :¼ v2�v2c (see Eq. (10.2.11)); or emissivity (a.k.a. emission coefficient emittance) e0 Vacuum permittivity c Flight path angle; or c :¼ 1 . ffiffiffiffiffiffiffiffiffiffiffiffiffi 1� b2 p ¼1 . ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1� v2=c2 p j Heat capacity ratio; or reduced drag; jD, or lift, jL, coefficient k Payload ratio; or dimensionless altitude variable (see definition Eq. (10.2.12)); or geographic longitude knm Equilibrium longitude fd Discharge correction factor fv Velocity correction factor g Efficiency (a.k.a. loss factor), in particular thermal efficiency gdiv Nozzledivergence loss factor gec Energy conversion efficiency q (Atmospheric) mass density; or normalized position vector, q ¼ n; g; fð Þ ¼ r=d; or surface reflectivity; or inverse radius, q :¼ 1=r l Standard gravitational parameter: l :¼ GM. For Earth: l� ¼ GM� ¼ 3:9860044105 105 km3 s�2 (as of EGM96); or reduced mass, l :¼ m2= m1 þm2ð Þ with m2\m1; or mass ratio (see Eq. (1.3.1)); or mass flux (a.k.a. mass flow density), l :¼ _mp � A (see Sect. 4.1.3) l hð Þ Angular mass flow distribution function; or bank angle (roll angle) li Mass ratio of the ith partial rocket (see Eq. (3.1.7)) r Stefan–Boltzmann constant, r ¼ 5:6704 10�8 W m�2 K�4; or proper speed, r ¼ c � arctanhb s Proper time (a.k.a. eigentime); or dimensionless time h True anomaly, (a.k.a. orbit angle); or pitch angle, h :¼ aþ c v Collinear configuration parameter (see Sect. 11.2.2) v k; eeð Þ See definition Eq. (10.4.7) xxxiv Symbols Used and Terminology x Angular velocity vector x Angular velocity, x ¼ dh=dt ¼ xj j; or argument of periapsis xi Angular frequency (a.k.a. circular frequency) X Right ascension of ascending node (RAAN); or effective potential in the CR3BP; or solid angle Diacritics _x Firstorder time derivative of a quantity x, _x ¼ dx=dt €x Secondorder time derivative of a quantity x, €x ¼ d2x�dt2 x0 Firstorder derivative with respect to a specified variable x00 Secondorder derivative with respect to a specified variable �x Geometric mean expression The underlined letters of an arbitrary expression will be used as subscript for an upcoming variable to indicate its special meaning. Example: exhaust velocity vex r̂ Unit vector along direction r, r̂ ¼ r=r Others := or =: Definition equation. The symbol on the colon’s side is defined by the expression on the other side of the equation ¼ const The expression preceding the equation sign is constant (invariant) with respect to a given variable ) From this follows … @ The condition following this symbol applies to the equation preceding it OðenÞ Landau notation (a.k.a. Big O notation): OðenÞ is the magnitude (order) of the residual power (here en) of a power series expansion. OðenÞ means: The residual is of order en n! Factorial of the nonnegative integer n, n! ¼ Q n k¼1 k, 0! ¼ 1 ð2n� 1Þ!! Double factorial of the odd positive integer i ¼ 2k � 1, ð2n� 1Þ!! ¼ Q n k¼1 ð2k � 1Þ […] Square brakes denote the units of a given physical quantity [a, b] Closed interval between numbers a and b \ða; bÞ Angle between vector a and vector b yh ix Average of y with respect to x over interval [a, b], yh ix:¼ 1b�a Rb a y � dx yh i Time average, yh i yh it Symbols Used and Terminology xxxv � Inner orbit (relative to another given orbit) Outer orbit (relative to another given orbit) � What was to be shown (quod erat demonstrandum) ! First point of Aries, a.k.a. vernal point (see Sect. 13.1) " Increasing "" Strongly increasing # Decreasing ## Strongly decreasing xxxvi Symbols Used and Terminology Chapter 1 Rocket Fundamentals 1.1 Rocket Principles 1.1.1 Repulsion Principle Many people have had and still have misconceptions about the basic principle of rocket propulsion. Here is a comment of an unknown editorial writer of the renowned New York Times from January 13, 1920, about the pioneer of US astronautics, Robert Goddard, who at that time was carrying out the first experi ments with liquid propulsion engines: Professor Goddard … does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react – to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools. The publisher’s doubt whether rocket propulsion in the vacuum could work is based on our daily experience that you can only move forward by pushing backward against an object or medium. Rowing is based on the same principle. You use the blades of the oars to push against the water. But this example already shows that the medium you push against, which is water, does not have to be at rest, it may move backward. So basically it would suffice to fill a blade with water and push against it by very quickly guiding the water backward with the movement of the oars. Of course, the forward thrust of the boat gained hereby is much lower compared with rowing with the oars in the water, as the large displacement resistance in the water means that you push against a far bigger mass of water. But the principle is the same. Instead of pushing water backward with a blade, you could also use a pile of stones in the rear of your boat, and hurl them backward as fast as possible. With this, you would push ahead against the accelerating stone. And this is the basis of the propulsion principle of a rocket: it pushes against the gases it ejects backward with full brunt. So, with the propellant, the rocket carries the mass, against which it pushes to move forwards, and this is why it also works in vacuum. This repulsion principle is called the “rocket principle” in astronautics and is utilized in the classical rocket engine. © Springer Nature Switzerland AG 2018 U. Walter, Astronautics, https://doi.org/10.1007/9783319743738_1 1 The repulsion principle is based on the physical principle of conservation of momentum. It states that the total (linear) momentum of a closed system remains constant with time. So, if at initial time t0 the boat (rocket) with mass m1 plus stone (propellant) with mass m2 had velocity v0, implying that the initial total momentum was p t0ð Þ ¼ ðm1 þm2Þv0, this must remain the same at some time tþ [ t0 when the stone is hurled away with velocity v2, the boat has velocity v1 (neglecting water friction), and the total momentum is p tþð Þ ¼ m1v1 þm2v2. That is, p t0ð Þ ¼ p tþð Þ principle of the conservation of linearð Þmomentum from which follows m1 þm2ð Þ � v0 ¼ m1v1 þm2v2 Note The principle of the conservation of momentum is valid only for the vectorial form of the momentum equation, which is quite often ignored. A bomb that is ignited generates a huge amount of momentum out of nothing, which apparently would invalidate an absolute value form of the momentum equation. But if you add up the vectorial momentums of the bomb’s frag ments, it becomes obvious that the vectorial momentum has been conserved. Given m1, m2, v0 and velocity v2 of the stone (propellant) expelled, one is able to calculate from this equation the increased boat (rocket) velocity v1. Doing so, this equation affirms our daily experience that hurling the stone backward increases the speed of the boat, while doing it forward decreases its speed. With a rocket, the situation is a bit more complicated, as it does not eject one stone after the other, but it emits a continuous stream of tiny mass particles (typ ically molecules). In order to describe the gain of rocket speed by the continuous mass ejection stream adequately in mathematical and physical terms, we have to consider the ejected mass and time steps as infinitesimally small and in an external rest frame, a socalled inertial (unaccelerated, see Sect. 13.1) reference frame. This is depicted in Fig. 1.1 where in an inertial reference frame with its origin at the center of the Earth a rocket with mass m in space experiences no external forces. At a given time t, the rocket may have velocity v and momentum p tð Þ ¼ mv. By ejecting the propellant mass dmp [ 0 with exhaust velocity vex and hence with momentum pp tþ dtð Þ ¼ vþ vexð Þ � dmp, it will lose part of its mass dm ¼ �dmp\0 and hence gain rocket speed dv by acquiring momentum pr tþ dtð Þ ¼ mþ dmð Þ vþ dvð Þ. Note In literature, dm[ 0 often denotes the positive mass flow rate of the propellant, and m the mass of the rocket. This is inconsistent, and leads to an erroneous mathematical description of the relationships, because if m is the mass of the rocket, logically dm has to be the mass change of the rocket, and thus it has to be negative. This is why in this book, we will always discrim inate between rocket mass m and propulsion mass mp using the consistent description dm ¼ �dmp\0 implying _m ¼ � _mp\0 for their flows. 2 1 Rocket Fundamentals For this line of events, we can apply the principle of conservation of momentum as follows pðtÞ ¼ p tþ dtð Þ ¼ pp tþ dtð Þþ pr tþ dtð Þ From this follows, mv ¼ �dm vþ vexð Þþ mþ dmð Þ vþ dvð Þ ¼ mv� dm � vex þm � dvþ dm � dv As the double differential dm � dv mathematically vanishes with respect to the single differentials dm and dv, we get with division by dt m _v ¼ _mvex According to Newton’s second law (Eq. (7.1.12)), F ¼ m _v, the term on the left side corresponds to a force, called momentum thrust force, due to the repulsion of the propellant, which we correspondingly indicate by Fex ¼ _mvex ð1:1:1Þ Fig. 1.1 A rocket in forcefree space before (above) and after (below) it ejected a mass dmp with exhaust velocity vex, thereby gaining speed dv. Velocities relative to the external inertial reference frame (Earth) are dashed and those with regard to the rocket are solid 1.1 Rocket Principles 3 Remark This equation can alternatively be derived from the fact that the momentum of the expelled propellant mass is dpp ¼ dmpvex. The equivalent force according to Newton’s second law (see Eq. (7.1.12)) is Fp ¼ dpp � dt ¼ _mpvex. This in turn causes a reaction force (Newton’s third law Eq. (7.1.11)) on the rocket of Fex ¼ �Fp ¼ � _mpvex ¼ _mvex. Although this derivation is more ele gant, we retain the conservation of linear momentum approach in the main text because it nicely expounds the physics behind the propulsion—the repulsion principle. This means that the thrust of a rocket is determined by the product of propellant mass flow rate and exhaust velocity. Observe that due to _m ¼ � _mp\0, Fex is exactly in opposite direction to vex (but depending on the steering angle of the engine, vex and hence Fex do not necessarily have to be in line with the flight direction v). Therefore, with regard to absolute values, we can write momentum thrust ð1:1:2Þ The term momentum thrust is well chosen, because if the expression _mpvex is integrated with regard to time, one obtains the momentum mpvex, which is merely the recoil momentum of the ejected propellant. 1.1.2 Total Thrust In the above, we have considered a simple propulsion mechanism, namely, repulsion from expelled mass. As we will see later there exist other physical effects, such as gas pressure for jet engines (Sect. 1.3) or relativistic effects close to the speed of light (Sect. 1.4), which contribute to the thrust. We take all these into account by a corresponding additional thrust term Fþ und thus obtaining the total thrust of a reaction engine F� :¼ Fex þFþ In reference to Eq. (1.1.1) we can formally write for the total thrust force ð1:1:3Þ and for its absolute values, i.e. for the total thrust ð1:1:4Þ By doing so we have defined the effective exhaust velocity ð1:1:5Þ 4 1 Rocket Fundamentals http://dx.doi.org/10.1007/9783319743738_7 We thus can interpret the total thrust F� as caused by expelling mass at rate _mp at an effective exhaust velocity v�. From this point of view, As we will see in Sect. 1.3.3, the effective exhaust velocity v� is identical to and therefore can also be understood as “the achievable total impulse of an engine with respect to a given exhausted propellant mass mp”, called the massspecific impulse: v� ¼ Isp. In essence, one can state that for each type of engine, one has to investigate what the thrustgenerating mechanisms are, how they act, and, by writing its total thrust in the form F� ¼ _mpv�, determine what the effective exhaust velocity of that engine is. Equations (1.1.3) or (1.1.4), respectively, is of vital importance for astronautics, as it describes basic physical facts, just like every other physical relationship, relating just three parameters, such as W ¼ F � s or U ¼ R � I. This is its statement: thrust is the product of effective exhaust velocity times mass flow rate. Only both properties together make up a powerful thruster. The crux of the propellant is not its “energy content” (actually, the energy to accelerate the propellant might be pro vided externally, which is the case with ion propulsions), but the fact that it pos sesses mass, which is ejected backward and thus accelerates the rocket forward by means of conservation of momentum. The higher the mass flow rate, the larger the thrust. If “a lot of thrust” is an issue, for instance, during launch, when the thrust has to overcome the pull of the Earth’s gravity, and since the exhaust speed of engines is limited, you need thrusters with a huge mass flow rate. The more the better. Each of the five first stage engines of a Saturn V rocket had a mass flow rate of about 2.5 metric tons per second, in total 12.5 tons per second, to achieve the required thrust of 33,000 kN (corresponds to 3400 tons of thrust). This tremendous mass flow rate is exactly why, for launch, chemical thrusters are matchless up to now, and they will certainly continue to be so for quite some time. 1.1.3 Equation of Rocket Motion Knowing the thrust of the rocket, we now wonder what the trajectory of a powered rocket looks like. To determine it, we have to account not only for the thrust but also for all possible external forces. They are typically summarized to one external force Fext: 1.1 Rocket Principles 5 Fext :¼ FG þFD þFL þ . . . ð1:1:6Þ with FG Gravitational force FD Aerodynamic drag FL Aerodynamic lift For each of these external forces a virtual point within the rocket can be assumed the external force effectively acts on (Fig. 1.2). This point has a unique location with regard to the geometry of the rocket, and it is in general different for every type of forces. For instance, the masses of the rocket can be treated as lumped together in the center of mass where the gravitational force applies; the aerodynamic drag and lift forces effectively impact the spacecraft at the socalled center of pressure; and possible magnetic fields have still another imaginary point of impact. If the latter do not coincide with the center of mass, which in general is the case, the distance in between results in torques due to the inertial forces acting effectively at the center of mass. In this textbook, we disregard the resulting complex rotational movements, and we just assume that all the points of impact coincide with the center of mass or, alternatively, that the torques are compensated by thrusters. Newton’s second law, Eq. (7.1.12), gives us an answer to the question of how the rocket will move under the influence of all noninertial forces Fi, including the rocket’s thrust F�: m _v ¼ X all i Fi We therefore find the following equation of motion for the rocket m _v ¼ F� þFext Fig. 1.2 External forces acting on a Space Shuttle upon reentry 6 1 Rocket Fundamentals and with Eq. (1.1.4), we finally obtain ð1:1:7Þ This is the key differential equation for themotion of the rocket. In principle, the speed can be obtained by a single integration step and its position by a double integration. Note that this equation not only applies to rockets but also to any type of spacecraft during ascent flight, reentry, or when flying in space with or without propulsion. 1.2 Jet Engine Any propulsion system that acts according to the repulsion principle provides thrust by expelling reaction mass is called a reaction engine. This definition includes not only the classical jet engines, such as the thermal jet engine (see Chap. 4), resistojets, or arcjets working with neutral gases, but also engines that work with “ion gases”, i.e., plasma, such as ion thrusters (see Chap. 5) or Hall effect thrusters, where the ions interact via the Coulomb interaction and therefore also create pressure. A rocket engine is a reaction engine that merely stores all the propellant in the rocket. A jet engine is a reaction engine that generates thrust by discharging a gas jet at high speed. Gas is a loose accumulation of molecules, which at high speed move around and collide with each other and with the volume boundary thus generating pressure. The gas pressure in the jet is highly specific for jet engines and determines their performance, because on one hand it creates an additional thrust component called pressure thrust, but on the other hand also causes thrust losses owing to nozzle divergence. Both effects are considered in this section. 1.2.1 Nozzle Divergence An exhaust jet that is everywhere parallel to the average thrust direction, as assumed in Sect. 1.1.1, is an ideal situation. In practice there is jet spraying, i.e., depending on the nozzle shape and on internal gas dynamics we have diverging components of the otherwise axisymmetric gas flow. We account for that by expressing the differential gas mass as a conical outflow shell with width dh as d _mp ¼ _mpl hð Þ sin h � dh where l hð Þ is the dimensionless and axisymmetric angular massflow distribution function, and h is the cone shell half angle, which is half of the aperture angle of the conical shell, as measured against its centerline. Any engine property x that depends on jet spraying then needs to be evaluated in terms of this mass flow angle distribution 1.1 Rocket Principles 7 xh il:¼ R p=2 0 x hð Þ � l hð Þ sin h � dhR p=2 0 l hð Þ sin h � dh ð1:2:1Þ Take the momentum thrust as an example. Let ve hð Þ be the ejection velocity (a.k.a. velocity distribution function, VDF) of the jet mass flow at ejection angle h. The contribution of this mass flow to the momentum thrust is the ejection thrust Fe hð Þ ¼ _mpve hð Þ projected onto the centerline, Fejj hð Þ ¼ _mpve hð Þ cos hð Þ. For the total momentum thrust it then follows from Eq. (1.2.1) ð1:2:2Þ with �ve :¼ veh il mean ejection velocity ð1:2:3Þ �Fe ¼ _mp�ve mean ejection thrust ð1:2:4Þ ð1:2:5Þ From this and Eq. (1.1.2), Fex ¼ _mpvex, follows that ð1:2:6Þ From this we see that the nozzledivergence loss factor is an important figure of merit of a jet engine because it affects the engine’s efficiency via a reduced exhaust velocity. Example What is the nozzledivergence loss factor for a common conical nozzle with cone half angle a and ve hð Þ ¼ const ¼ ve � �ve? Having l hð Þ ¼ 1 @ h\a 0 @ h� a � we find according to Eq. (1.2.1) ve hð Þ cos hð Þh il¼ ve R a 0 cos h sin h � dhR a 0 sin h � dh ¼ ve 2 1� cos2 a 1� cos a 8 1 Rocket Fundamentals We hence obtain with Eq. (1.2.5) and ve ¼ veh il ð1:2:7Þ Typical divergence cone half angles in use are a ¼ 12°–18°, with corresponding loss factors of gdiv ¼ 0.989–0.975 . So, these losses amount to 1.1–2.5%. In practice an ideal contoured Rao nozzle (see Sect. 4.4.2) is the most common. Its conical exit wall angle is typically h ¼ 7�–12�. The corresponding conical exhaust plume therefore causes a loss of gdiv ¼ 0:989–0:996. So, exhaust losses due to the Rao nozzle contour do not exceed 1% and are therefore a small contribution to the total thrust losses of typically 2–8% mainly due to shock formation in the nozzle and boundary layer losses owing to friction with the nozzle wall. 1.2.2 Pressure Thrust The working fluid of a jet engine is gas. While the jet engine has an internal gas pressure, there might also exist an external gas pressure from the gas molecules of the atmosphere surrounding a rocket during ascent. In order to understand the impact of the propellant gas pressure and external ambient pressure on the engine’s thrust, let us first have a look at the general pressure and flow conditions in a typical jet engine depicted in Fig. 1.3 by a thrust chamber of a rocket engine. Continuity Equation Let us have a look at the general propellant gas flow in a rocket engine. A propellant mass dmp perfuses a given engine cross section of area A with velocity v (see Fig. 1.4). During the time interval dt, it will have passed through the volume dV ¼ A � ds ¼ Av � dt. Therefore, dmp ¼ q � dV ¼ qAv � dt Fig. 1.3 Pressure and velocity conditions inside and outside a thrust chamber 1.2 Jet Engine 9 Fig. 1.4 The volume dV that a mass flow with velocity v passes in time dt where q is the mass density. As the number of molecules that enter and exit this volume, is preserved, we derive for the mass flow rate the equation ð1:2:8Þ The continuity equation is a direct outcome of the transport of mass particles as a conserved quantity, which expresses the fact that the number of mass particles cannot increase or decrease, but can only move from place to place. This is exactly what the word “continuity” means. Pressure Thrust Jet thrust is generated by gas pressure. To see how, we denote by p the varying engine pressure acting from inside on the wall and exerting the force dF ¼ p�dA on a wall segment dA, which points outward. In the area surrounding the chamber (here “chamber” is the abbreviation for thrust chamber, i.e. the engine’s casing including the nozzle, if existent) we assume a constant external pressure p1. Quite generally, the total propellant force F� generated by the chamber must be the sum of all effective forces acting on the entire chamber wall having surface SC: F� ¼ ZZ SC dF ¼ ZZ SC p� p1ð Þ � dA Of course, no force is acting at the imaginary exit surface. Therefore SC does not include the flat exit area Ae. By denoting the closed surface SO ¼ SC [Ae we can rewrite the above equation as F� ¼ ZZ � SO p� p1ð Þ � dA� pe � p1ð ÞAe ð1:2:9Þ The second term is called pressure thrust force Fp. Fp ¼ � pe � p1ð ÞAe pressure thrust force ð1:2:10Þ We will discuss its properties in a moment after having evaluated the first term in Eq. (1.2.9). 10 1 Rocket Fundamentals Emergence of Momentum Thrust We now assume an axisymmetric thrust chamber, which generally is the case, with an according axially symmetric gas flow along the chamber axis. This implies that we can treat the hydrodynamics along the chamber axis, which we denote by ux and which points from the front of the chamber to the exit, as onedimensional. The propellant gas enters the chamber at the front side with velocity v0 0 and by arbitrary means is accelerated along the chamber axis to ve at the exit; therefore ux ¼ v̂ ¼ Âe ¼: ue. Overall, the gas is accelerated, although there may be times when it is decelerated along its path through the chamber. The axial symmetry reduces the first term in Eq. (1.2.9) to a force on the effective front side and a reversed force on the effective and imaginary closed rear side, i.e.,ZZ � SO p� p1ð Þ � dA ¼ � p0 � p1ð Þ � pe � p1ð Þ½ A/ux ¼ pe � p0ð ÞA/ux ð1:2:11Þ where A/ is the chamber crosssection. Note that at the front side where p ¼ p0 we have dÂ ¼ �ux, which causes the negative sign. According to hydrodynamics, an accelerated flow is intimately connected via the mass density q to a pressure gra dient as $p ¼ �q � dv=dt. This is the socalled Euler equation, which reads in our onedimensional case dp dx ¼ �q � dvx dt Euler equation So, the gas pressure decreases with increasing gas velocity. This seemingly para doxical effect is called the Bernoulli effect. After separating the variables dp and dx and then integrating this equation we have with the continuity equation _mp ¼ const (Eq. (1.2.8)) and because v0 0Zp p0 dp ¼ p� p0 ¼ � Zx 0 dmp A/dx dvx dt dx ¼ � 1 A/ Zv v0 dmp dt dvx ¼ � _mpA/ v Therefore, at any point along the chamber axis we have p� p0ð ÞA/ ¼ � _mpv Applying this result to the exit and inserting it into Eq. (1.2.11) and taking into account also nozzle divergence (see Sect. 1.2.1) we obtain the thrust component Fex ¼ ZZ � SO p� p1ð Þ � dA ¼ � _mpvex momentum thrust force ð1:2:12Þ which is the momentum thrust. It is remarkable that we have recovered Eq. (1.1.1) on hydrodynamic grounds rather than on first principles. 1.2 Jet Engine 11 Total Thrust Inserting the above results into Eq. (1.2.9) we obtain for the total thrust F� ¼ � _mpvex � pe � p1ð ÞAe � ue ð1:2:13Þ where ue is the unit vector of the exit surface in the direction of the exhaust jet. Because the thrust is antiparallel to the exit flow we finally have for the absolute value of the thrust ð1:2:14Þ Hence, the total thrust is the sum of the momentum thrust and the pressure thrust. The wording “pressure thrust”, on one hand, is conclusive because it originates from the very special fact that the rocket engine works with gases that produce pressure. On the other hand, and as according to Eq. (1.2.12), the exhaust and momentum thrust is also generated by a pressure on the chamber because of its internal pressure gradient. In the end, it is solely pressure that accelerates the gas engine, and with it the rocket. Effective Exhaust Velocity If we apply Eq. (1.2.14) to the definition of the effective exhaust velocity as given by Eq. (1.1.5) with Fþ ¼ Fp ¼ pe � p1ð ÞAe and keeping in mind Eq. (1.2.6), we get ð1:2:15Þ The expression “effective exhaust velocity” makes it clear that thrust is essentially caused by the exhaust velocity vex ¼ gdiv�ve modified by a pressurethrust equivalence exhaust velocity term. Indeed, as we will see from Eq. (1.2.18) the pressure thrust for a real engine chamber is only a small contribution. For an ideally adapted nozzle with pe ¼ p1 (see Sect. 4.2.1) it even vanishes. 1.2.3 Momentum versus Pressure Thrust Ultimately, if it is only pressure that drives a rocket engine, how does this fit together with the rocket principle discussed in Sect. 1.1.1, which was based on repulsion and not on pressure? And what is the physical meaning of pressure thrust? You often find the statement that pressure thrust occurs when the pressure at the exit (be it nozzle exit or combustion chamber exit) hits the external pressure. The pressure difference at this point times the surface is supposed to be the pressure thrust. Though the result is right, the explanation is not. First, the exit pressure does 12 1 Rocket Fundamentals not abruptly meet the external pressure. When the exhaust gas hits and merges with the ambient atmospheric gas there is rather a smooth pressure transition from the exit pressure to the external pressure covering in principle an infinite volume behind the engine. In the course of this process the pressure difference and with it the abstract force pe � p1ð ÞA/ is irreversibly lost. Second, even if such a pressure difference could be traced back mathematically to a specific surface, this would not cause a thrust, because, as we will see later, the gas in the nozzle expands backward with supersonic speed, and such a gas cannot have a causal effect on the engine to exert a thrust on it. Momentum Thrust For a true explanation, let us imagine for a moment and purely hypothetically, a fully closed and idealized rectangular thrust chamber (see Fig. 1.5) with the same pressure and flow conditions as in the real thrust chamber. The surface force on the front side would beFfront ¼ ðp0 � p1ÞA/, andFrear ¼ ðpe � p1ÞA/ on the rear side.Hence, the net forward thrust would be Fex ¼ Ffront � Frear ¼ p0 � peð Þ � A/. Owing to the Bernoulli effect, this translates into Fex ¼ _mpvex. Therefore, we can say the following, The momentum thrust can also be described in a different mathematical form. If we apply the continuity equation (1.2.8) to the exit of the engine we obtain _mp ¼ qe�veAe. Inserting this into Fex ¼ _mpvex yields ð1:2:16Þ This equation begs the question whether the momentum thrust of a rocket engine is linearly or quadratically dependent on �ve. The answer depends on the engine in question. Depending on the engine type (e.g., electric or chemical engine), a change of its design in general will vary all parameters ve and _mp; qe;Ae in a specific way. Fig. 1.5 Pressure conditions of the idealized rectangle thrust chamber if it would be, hypothetically, fully closed 1.2 Jet Engine 13 This is why the demanding goal of engine design is to tune all engine parameters, including �ve, such that the total thrust is maximized. Hence, it is not only �ve alone, which is decisive for the momentum thrust of a rocket engine but also it is necessary to adjust all relevant engine parameters in a coordinated way. Pressure Thrust and Its Significance In order to have the hypothetical gas flow indeed flowing, we need to make a hole with area Ae into the rear side of the hypothetically closed thrust chamber of Fig. 1.5. Once this is done, the counterthrust at the rear side decreases by DFrear ¼ �ðpe � p1ÞAe, which in turn increases the total thrust by the same amount. This contribution is the pressure thrust. Therefore, If the exit pressure happens to be equal to the external pressure, then the external pressure behaves like a wall, the pressure thrust vanishes, and we have an ideally adapted nozzle (see Sect. 4.2.1). Given Eq. (1.2.16) we are able to qualitatively derive the significance of pres sure thrust. We do so by rating it against the momentum thrust. Because Fp ¼ ðpe � p1ÞAe � peAe ¼ Fp �� 1 we have Fp Fex Fp Fe �Fp Fe ���� 1 ¼ peAe qeAe�v2e ¼ qeRTe � Mp qe�v2e ð1:2:17Þ For jet engines one can make use of various results of Sect. 4.1, namely Eq. (4.1.6); the ideal gas law Eq. (4.1.1) with the universal gas constant R, molar propellant mass Mp, average number of excited degrees of freedom of the gas molecules, n 8; and Eq. (4.1.3), which defines the thermal efficiency at the exit ge. This leads to Fp Fe � R 2cpMp 1� ge ge ¼ 1 nþ 2 1 ge � 1 � � With n 8 and ge ¼ 0:5� 1:0 for jet engines we finally have Fp Fex � 1 10 � 1 ge � 1 � � 0� 10% @ jet engines ð1:2:18Þ For a socalled ideally adapted nozzle where pe ¼ p1 (see Sect. 4.2.1) then of course Fp ¼ 0. Because the exhaust temperature Te generally decreases with increasing ve (cf. Eq. (4.1.6) for their general relation) we see that the pressure thrust becomes rapidly less important with increasing ejection velocity. This will be particularly important for ion thrusters with exhaust velocity 10 times larger than for thermal engines. 14 1 Rocket Fundamentals 1.3 Rocket Performance In this section, we define and examine those parameters that characterize a rocket, in particular its figures of merit. 1.3.1 Payload Considerations When looking at Eq. (2.2.3), at first glance one might think that the burnout final mass is identical to the payload mass, m ¼ mf ¼ mL. That would mean that if you only choose the launch mass big enough one would be able to get a payload of any size into space. However, this thought discards the structural mass ms of the rocket, which includes the mass of the outer and inner mechanical structure of the rocket in particular the tank mass, the mass of the propulsion engines including propellant supply (pumps), avionics incl. cable harness, energy support systems, emergency systems, and so on. Structural mass trades directly with payload mass, and hence mf ¼ ms þmL In practice, structural mass limits the payload mass to such a severe extent that later on we will have to look for alternative propulsion concepts, the socalled staging concepts, to reduce ms. For further considerations, in particular for the later stage optimization, we define the following mass ratios: l :¼ mf m0 ¼ ms þmL m0 mass ratio ð1:3:1Þ e :¼ ms m0 � mL ¼ ms ms þmp structural ratio ð1:3:2Þ k :¼ mL m0 � mL ¼ mL ms þmp payload ratio ð1:3:3Þ Observe that for the last two ratios the structural mass and the payload mass are not taken relative to the total mass, but to the total mass reduced by the payload mass. This is done in view of consistency with the equivalent, more general definitions for the upcoming rocket staging (see Eqs. (3.1.3)–(3.1.5)). From the above definitions it follows that l 1þ kð Þ ¼ ms þmL m0 ms þmp þmL ms þmp ¼ ms þmL ms þmp ¼ eþ k 1.3 Rocket Performance 15 hence l ¼ mf m0 ¼ eþ k 1þ k ð1:3:4Þ So the rocket Eq. (2.2.2) can be written as Dv v� ¼ � ln eþ k 1þ k or ð1:3:5Þ This equation is represented in Fig. 1.6. It directly relates the payload ratio to the achievable propulsion demand at a given structural ratio of the rocket and effective exhaust velocity of the engine. So, because the structural mass is not negligibly small, it is not possible to achieve any propulsion demands you like. In numbers this says that: Fig. 1.6 Obtainable payload ratios at a given propulsion demand for different structural ratios 16 1 Rocket Fundamentals The effective velocity of chemical rockets ascending through Earth’s atmosphere is limited to v� � 4 km/s, limiting the available propulsion demand to Dv < 10 km/s. If, for instance, the goal is to get in a single stage with e ¼ 0:1 into low Earth orbit (socalled SSTO) for which in practice Dv ¼ 9 km/s is required (see Sect. 6.4.7), then even with an optimal v� ¼ 4 km/s for a LOX/LH2 engine the achievable payload ratio is a mere k ¼ 0:6%. Even if the structural ratio would be a smashing e ¼ 0:075 we would arrive at only k ¼ 3:4%. So, in principle, a SSTO rocket is possible, but only at the expense of an unacceptable low payload mass. This is why there is no way around a staged rocket to which we come in Chap. 3 (cf. consid erations following Eq. (3.3.5)). 1.3.2 Rocket Efficiency The principle of rocket propulsion is that a certain amount of energy is utilized to accelerate propulsion mass in order to gain rocket speed via repulsion and hence rocket kinetic energy. Of course, it is the goal to design a rocket that from a given amount of spent energy extracts as much kinetic energy as possible. The quantity to measure this is the total rocket efficiency gtot. It is defined as gtot :¼ gained rocket kinetic energy utilized energy ¼ Ekin v0 þDvð Þ � Ekin v0ð Þ E0 total rocket efficiency ð1:3:6Þ The utilized energy is converted into rocket kinetic energy in two steps. First the engine converts the utilized energy into thrust with internal efficiency gint gint :¼ generated thrust energy utilized energy ¼ 1 2 mpv 2 � E0 internal efficiency ð1:3:7Þ The internal efficiency (a.k.a. total engine efficiency gtot, see also Sect. 1.3.3) is independent from the motion state of the rocket. It is therefore characteristic for an engine and has to be evaluated separately for different kinds of engines (see for instance Eq. (4.2.7)). In a second (propulsion) step the thrust energy is converted into kinetic energy of the rocket based on the conservation of momentum. The efficiency of this second conversion step is called external efficiency—a.k.a. integral or mechanical efficiency—of a rocket. It is defined as gext :¼ gained rocket kinetic energy generated thrust energy ¼ 1 2mf v0 þDvð Þ2� 12m0v20 1 2mpv 2� external efficiency ð1:3:8Þ 1.3 Rocket Performance 17 In total, we have gtot ¼ gext � gint ð1:3:9Þ Let us have a closer look at the external efficiency. The key point is that velocity is a property relative to a reference frame. Velocity and hence kinetic energy changes when a different reference frame is assumed. Although never mentioned in literature explicitly, the reference frame assumed here is the one in which the rocket had zero velocity at the beginning of the propulsion phase, v0 ¼ 0. Applying this condition and Eq. (2.0.1) to Eq. (1.3.8) yields gext ¼ mfDv2 mpv2� ¼ mf m0 � mf Dv2 v2� ¼ Dv � v� � �2 m0 � mf � 1 With Eq. (2.2.3) we finally obtain ð1:3:10Þ This function is displayed in Fig. 1.7. It has a maximum at Dv=Dv ¼ 1:59362. . ., which, according to Eq. (2.2.3), corresponds to mf � m0 ¼ 0:203188. . .. From this it is sometimes inferred that the optimal operating point is around this maximum and an acceptable economic limit usually is reached at about Dv 3v�, when the payload portion is only 5.0%. It is therefore argued that a rocket can be operated efficiently only for Dv\3v�. Some words of caution are in place. The argument of external efficiency is pointless for practical considerations. First, because it depends on a reference frame that can be chosen arbitrarily. Second, the objective of a thrust maneuver is to achieve a given deltav. The kinetic energy gained by the maneuver is irrelevant in Fig. 1.7 External efficiency of a rocket as a function of the propulsion demand 18 1 Rocket Fundamentals contrast. The only thing that matters is this: How much propellant is to be expended to achieve a given Dv? The answer is provided by the rocket Eq. (2.2.2) or Eq. (2.2.4), respectively. Only the rocket equation is able to tell whether an impulsive maneuver is efficient or not—apart from the fact that efficiency is a discretionary notion (we could instead define a momentum efficiency). The external energy efficiency therefore is of no practical relevance, which is why it is rarely used. In contrast the internal efficiency is a valuable figure of merit of an engine because it determines v� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2E0gint � mp q : the better gint the higher v� (cf. Sect. 4.2.4) and all the more Dv is achieved. Transmitted Spacecraft Power The power transmitted to the spacecraft with velocity v is simply calculated according to classic physics with the product of force times velocity, i.e., PS=C ¼ F� � v transmitted spacecraft power ð1:3:11Þ Note that forces, such as F�, are independent of the chosen reference system. Therefore, the transmitted spacecraft power is valid both in the rocket system and the external inertial reference frame in which v is measured. However, observe that v depends on the chosen external reference frame. 1.3.3 Performance Parameters In this section we summarize all those engine figures, which characterize the per formance of any rocket engine. Total and Specific Impulse The socalled total impulse Itot of an engine is the integral product of total thrust and propulsion duration Itot :¼ Z t 0 F�dt ¼ v� Z t 0 _mp � dt ¼ mpv� @ v� ¼ const total impulse ð1:3:12Þ The latter is only valid as long as the effective exhaust velocity is constant. This is, in its strict sense, not the case during launch where the external pressure and hence the effective exhaust velocity varies due to the pressure thrust. The total impulse can be used to define the very important (weight)specific impulse, which is defined as “the achievable total impulse of an engine with respect to a given exhausted propellant weight mpg0”, i.e., with Eq. (1.3.12) ð1:3:13Þ By this definition the specific impulse has the curious, but simple, dimension “second”. Typical values are 300–400 s for chemical propulsion, 300–1500 s for 1.3 Rocket Performance 19 electrothermal propulsion (Resistojet, Arcjet), and approximately 2000–6000 s for electrostatic (ion thrusters) and electromagnetic engines (see Fig. 1.8). The specific impulse characterizes the general performance and besides the thrust is therefore a figure of merit of an engine. In Europe, in particular at ESA, the massspecific impulse with definition “Isp is the achievable total impulse of an engine with respect to a given exhausted pro pellant mass mp” is more common. This leads to the simple identity Isp ¼ v�. However the definition “Isp ¼ weightspecific impulse” is worldwide more estab lished, which is why we also will use it throughout this book. In either case you should keep in mind that quite generally: The specific impulse is an important figure of merit of an engine, and is in essence the effective exhaust velocity. Jet Power The mechanical power of an exhaust jet, the so called jet power, is defined as the change of the kinetic energy of the ejected gas (jet energy) per time unit. In other words, Pjet describes the time rate of expenditure of the jet energy. With this definition, and averaging over the massflow distribution (see Sect. 1.1.2), and from Eq. (1.2.2) we get Fig. 1.8 Specific impulse and specific thrust of different propulsion systems. Credit Sutton (2001) 20 1 Rocket Fundamentals Pjet :¼ dEjetdt � l ¼ d dt 1 2 mpv 2 e � � � l ¼ 1 2 _mp v 2 e � l¼ 1 2 �Fe�ve gVDF � 1 2 �Fe�ve jet power ð1:3:14Þ Here we have defined the socalled gVDF :¼ veh i2l v2e � l � 1 VDF loss factor ð1:3:15Þ where VDF is the velocity distribution function ve hð Þ as of Sect. 1.2.1. The VDF loss factor takes account of the power loss owing to the flow velocity in the nozzle to be weighted with the angular distribution of the mass flow. Note that, though the angular massflow distribution function l hð Þ determines the amount of loss, it is the VDF that causes the loss. Since if ve hð Þ ¼ const we have veh i2l¼ v2e � l. For the proof of gVDF � 1, and hence that gVDF is a true loss factor and not just a correction factor, see Problem 1.3. Note that this loss is due to comparing thrust with jet power. In the first case the flow velocity, in the second case the square of the flow velocity, has to be averaged. This has nothing to do with divergence losses, which are solely accounted for by gdiv. Note that forces, such as �Fe, are independent of the chosen reference system, whereas the velocity �ve is defined with respect to the rocket. So jet power is a property with respect to the rocket. In the case where we have uniform ejection velocities, then �ve ¼ ve and gVDF ¼ 1, and ð1:3:16Þ The latter holds because of Eq. (1.2.2) with �ve ¼ ve. Total Engine Efficiency With the jet power as defined in Eq. (1.3.14) we can express the dimensionless total engine efficiency (a.k.a. internal efficiency, see Eq. (1.3.7)) of a real rocket as gtot :¼ Ejet Ein ¼ Pjet Pin ¼ 1 2 _mp v2e � l Pin ¼ 1 2 �Fe�ve gVDFPin ð1:3:17Þ We now factorize the total efficiency by taking real versus ideal rocket engines (for the definition of an ideal rocket engine see box in Sect. 4.1.1) into consideration and accounting in the following equation for the losses consecutively from right to left starting from the power into the propellant to the ideal jet power 1.3 Rocket Performance 21 gtot ¼ �Fe�ve 2gVDFPin ¼ _mp veh i 2 l 2gVDFPin ¼ _mp veh i 2 l 2gVDFPjet Pjet Pjet;id Pjet;id Pin where Pjet ¼ 12 _mp v 2 e � l real jet power Pjet;id ¼ 12 _mp;id v 2 e;id D E l ideal jet power and _mp is the total exit massflow, while ve hð Þ is the exit flow velocity into the conical shell with cone halfangle h for any given real rocket engine. On the other hand _mp;id and ve;id hð Þ are those for an ideal rocket engine as expounded in Chap. 4 for thermal engines and Chap. 5 for ion engines. We hence obtain gtot ¼ 1 gVDF veh i2l v2e � l _mp _mp;id v2e � l v2e;id D E l Pjet;id Pin ¼ _mp _mp;id v2e � l v2e;id D E l Pjet;id Pin This factorization gives rise to the definition of the following three correction and efficiency factors fd :¼ _mp _mp;id ¼ _mpa0 ffiffiffi n p AtC1p0 ¼ 0:98� 1:15 discharge correction factor ð1:3:18Þ where the third term follows from Eq. (4.3.2) for a thermal engine. The discharge correction factor relates the total mass flow in the real rocket to that of the ideal rocket. In the real rocket the gas flow experiences friction with the chamber and nozzle walls, which decelerates the flow. On the other hand, chemical reactions in the flow leading to a higher molecular weight, liquid and solid particles in the combustion products, and a lower gas density owing to heat loss and hence cooling of the gas, all these three effects increase the mass flow. In total all these processes usually make up fd [ 1. fv :¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2e � l v2e;id D E l vuuut ¼ 0:85� 0:98 velocity correction factor ð1:3:19Þ This factor relates the real mean to the ideal mean ejection velocity. Decline in real velocity is due to the friction of the gas with the walls leading to a boundary layer with reduced velocity, which is transmitted to neighboring gas layers due to gas viscosity. But also velocity increases are possible, for instance if postcombustion occurs in the nozzle. 22 1 Rocket Fundamentals gec :¼ Pjet;id Pin ¼ Ejet;id Ein � 1 energy conversion efficiency ð1:3:20Þ gec accounts for combustion losses ( 1%) in the chamber (see Fig. 4.6), but also for heat losses to the chamber and nozzle walls ( 3%). A very huge contribution is the energy loss of about 25–30% into internal excitation of the gas molecules and gas liquid and solid particles. So, most of the combustion enthalpy as the total input power into a chemical engine is spent on this drain. From this it is evident that efficient combustion is at the heart of any thrusttopower optimization. In summary we get the following expression for the total engine efficiency ð1:3:21Þ As we will see in Sect. 4.2.4 we have gtot � 0:735 for today’s chemical thrusters. Thrust Correction Factor From Eq. (1.2.2) the important ratio of real total thrust of a rocket engine with an ideally adapted nozzle (see Sect. 4.2.1: Fp ¼ 0) to the ideal momentum thrust is derived to be F� �Fe;id ¼ gdiv �Fe �Fe;id ¼ gdivfd �ve �ve;id ¼ gdivfd ve � l ve;id D E l Since according to Eq. (1.3.15) veh il¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gVDF v2e � l q we obtain with Eq. (1.3.19) for pe ¼ p1 ð1:3:22Þ ThrusttoPower Ratio Finally, we define the important thrusttototalpower ratio, which describes the thrust received from the total electrical power. rTTPR :¼ F�Pin For a rocket engine with an ideally adapted nozzle (pe ¼ p1, see Sect. 4.2.1) we have F� ¼ Fex and therefore rTTPR ¼ FexPin @ pe ¼ p1 1.3