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Transcript of Intro to Astronautics
1
Welcome To..
Satellite Tool Kit
Astronautics Primerby
Jerry Jon Sellers
Based on
Understanding Space: AnIntroduction to Astronautics
Copyright 1996 McGrawHill Inc.
by
Jerry Jon Sellers
Wiley J. Larson (editor)
Your Understanding of Space Starts Here!
COPYRIGHT NOTICE
McGrawHill owns the copyright to the book Understanding Space: An Introduction toAstronautics. This primer was adapted from the book by Jerry Jon Sellers and Wiley J.
Larson. Analytical Graphics, Inc. retains copyright for this Primer. It is illegal toreproduce this material without permission.
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OBJECTIVE
The objective of this primer is to provide you with a fundamental reference for keyconcepts in astronautics, including:
♦ Space mission architecture
♦ History
♦ Dynamics
♦ Orbital mechanics
♦ Orbital elements
♦ Orbit propagation
♦ Ground tracks
♦ Satellite access
ABOUT THE AUTHOR
Jerry Jon Sellers was born in Harlan, Iowa. He has worked over 13 years at variousastronautics assignments including the NASA Johnson Space Center, where heworked in Space Shuttle Mission Control (guidance and navigation) and the U.S. AirForce Academy where he served on the faculty of the Department of Astronautics. Hewas a distinguished graduate from the U.S. Air Force Academy in 1984 and hasearned a Master’s Degree in Physical Science from the University of Houston, ClearLake, a Master’s Degree in Aeronautics and Astronautics from Stanford Universityand a Ph.D. in Satellite Engineering from the University of Surrey, UK. He currentlyworks as an international research and development liaison officer in London, UK,and continues to write and consult on space mission analysis and design.
CONTENTS
Stepping Into Space
The big picture of why space is important and how the pieces fit together
Exploring Space
Some early “explorers” who’ve shaped our current understanding of orbits
An Introduction to Orbit Motion
Key concepts necessary for understanding orbit motion
Describing Orbits
Understanding orbital elements (twoline element sets), ground tracks and how they relateto space missions
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Predicting Orbits
The “nuts and bolts” of predicting realworld satellite motion using orbit propagators
Satellite Access
The how, when and where of links between ground stations and satellites
Recommended Reading
Other great astronautics references
STEPPING INTO SPACE
Since the dawn of the Space Age only a few decades ago, we have come to rely moreand more on satellites for a variety of needs. Daily weather forecasts, instantaneousworldwide communication, and a constant ability to keep an eye on notsofriendlyneighbors are all examples of space technology that we’ve come to take for granted.
The purpose of this brief astronautics primer is to provide the reader with aconceptual overview of important topics in orbital mechanics. Understanding thesekey concepts will enhance your insight into the science behind Satellite Took Kit andbetter equip you to apply these concepts to practical problems in space. We’ll beginwith a brief overview of space, space missions and space history. Then we’ll get intothe details of orbital mechanics to see how you can use STK to plot your path to thestars.
Why is space so useful?
Getting into space is dangerous and expensive. So why bother? Space offers severalcompelling advantages for modern society
♦ A global perspective—the ultimate high ground
♦ A universal perspective—from space we have a clear view of the heavens,unobscured by the atmosphere
♦ A unique environment—freefall and abundant resources make space thetrue final frontier
Global Perspective
Space offers a global perspective. As you can see in Figure 1, the higher you are, themore you can see. For thousands of years, kings and rulers took advantage of thisfact by putting lookout posts atop the tallest mountains to survey more of their realmand fend off wouldbe attackers. Throughout history, many battles have been foughtto “take the high ground.” Space takes this quest for greater perspective to itsultimate end. From the vantage point of space, we can view large parts of the Earth’ssurface. Orbiting satellites can thus serve as “eyes in the sky” to provide a variety ofuseful services.
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Figure 1: Global perspective. From space, satellites can observe largescale featureson the Earth, track weather patterns, monitor the environment and view widely separatedpoints simultaneously, allowing them to communicate.
Universal Perspective
Space offers a clear view of the heavens. When we look at stars in the night sky, wesee their characteristic twinkle. This twinkle, caused by the blurring of “starlight” as itpasses through the atmosphere, is known as scintillation. Not only is the light blurred,but some of it is blocked or attenuated altogether. This attenuation is frustrating forastronomers who need access to all the regions of the spectrum to fully explore theuniverse. By placing observatories in space, we can sit above the atmosphere andgain an unobscured view of the universe, as depicted in Figure 2. The Hubble SpaceTelescope and the Gamma Ray Observatory are armed with sensors operating farbeyond the range of human senses. Already, results from these instruments arerevolutionizing our understanding of the cosmos.
Figure 2: Seeing beyond the clouds. Earthbased astronomy is obscured by theatmosphere. Astronomers don’t like the “twinkle” of star light. Some wavelengths are
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completely blocked. Spacebased astronomy opens the door to a whole new perspective onthe universe.
A Unique Environment
Space offers a unique environment with many advantages.
♦ Freefall environment—enables developing advanced materials
♦ Abundant resources—solar energy and minerals
Gravity makes some manufacturing processes difficult if not impossible. To formcertain new metal alloys, for example, we must blend two or more metals in just theright proportion. Unfortunately, gravity tends to pull the heavier metal to the bottom,making a uniform mixture difficult to obtain. But space offers the solution. Amanufacturing plant in orbit is literally falling toward Earth but never hitting it. Thisis a condition known as freefall (NOT zero gravity). In freefall there are no contactforces on an object, so it is said to be weightless. Unencumbered by the weight felt onthe Earth’s surface, factories in orbit can create exotic new metals for computers orother advanced technologies, as well as for promising new pharmaceutical products tobattle disease on Earth.
Figure 3: Early freefall manufacturing. In the 16th century, Italian weapons makersdeveloped a secret way of making lead shot for muskets. By dropping liquid lead from a“shot tower,” they found nearperfect spheres would form as the molten lead cooled andhardened in free fall.
Space also offers abundant resources. While some on Earth argue about how to carvethe pie of Earth’s resources into smaller and smaller pieces, others have argued that
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example, is known to be rich in oxygen and aluminum. The oxygen could be used asrocket propellant and for humans to breathe. Aluminum is an important metal forvarious industrial uses. These resources, coupled with the human drive to explore,mean the sky is truly the limit!
Space ApplicationsLet’s look at some important application of space that affect all of our lives today.
Communications Satellites
Science/Science Fiction writer Arthur C. Clarke first proposed putting satellites intoorbits with periods of 24 hours, 36,000 km above the equator, exactly matching therotation rate of the Earth. These geostationary orbits could serve as communicationhubs to link together remote parts of the planet. With the launch of the firstexperimental communications satellite, Echo I, into Earth orbit in 1960, Clarke’sfanciful idea showed promise of becoming reality. Although Echo I was little morethan a reflective balloon in lowEarth orbit, radio signals were bounced off it,demonstrating that space could be used to broaden our horizons of communication.An explosion of technology to exploit this idea quickly followed.
Satellites are now used for a large percentage of commercial and governmentcommunications and for most domestic cable television. Through satellite technology,relief workers can now stay in constant contact with their organizations, enablingthem to better distribute aid to refugees hungry for food. In addition, our modernmilitary now relies almost totally on satellites to communicate with forces deployedworldwide. Without satellites, global communication as we know it today would notbe possible.
Remote Sensing Missions
Satellites operating from the global perspective of space have also made possible thescience of remote sensing. Remote sensing is the act of observing Earth and otherobjects from space. For decades, military “spy satellites” have kept tabs on theactivities of potential adversaries using remotesensing technology. This sametechnology has been adapted for civilian uses such as
♦ monitoring Earth’s environment
♦ forecasting the weather
♦ managing resources
Satellites can now “spy” on crops, ocean currents, and natural resources to aidfarmers, resource managers, and planners on Earth. In countries where the failure ofa harvest may mean the difference between bounty and starvation, spacecraft havehelped planners manage scarce resources and head off potential disasters beforeinsects or other blights could wipe out an entire crop. Weather forecasting is a furtherapplication of remotesensing technology—one we’ve all come to rely on. Overall, we’vecome to rely more and more on the ability to monitor and map our entire planet. As
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Satellites can now “spy” on crops, ocean currents, and natural resources to aidfarmers, resource managers, and planners on Earth. In countries where the failure ofa harvest may mean the difference between bounty and starvation, spacecraft havehelped planners manage scarce resources and head off potential disasters beforeinsects or other blights could wipe out an entire crop. Weather forecasting is a furtherapplication of remotesensing technology—one we’ve all come to rely on. Overall, we’vecome to rely more and more on the ability to monitor and map our entire planet. Asthe pressure builds to better manage scarce resources and assess environmentaldamage, we’ll call upon remotesensing spacecraft to do even more.
Figure 4: Satellite remote sensing. From the vantage point of space, we can planurban development and plot the course of dangerous storms.
Spacebased Navigation
Early seafarers looked to the stars to guide their way. Modern seafarers look only asfar as satellites in Earth orbit. Systems such as the Global Positioning System (GPS),developed by the U.S. military, tell you where you are, in what direction you’reheading and how fast you’re going.
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Figure 5: Global Positioning System (GPS). GPS allows Earthbased users armedwith a simple, handheld receiver to triangulate from a constellation of 24 satellites. Theycan then determine their location to within a few meters and velocity, and a few m/secanywhere on Earth.
Exploration
Perhaps the most exciting missions are those which explore the unknown. Missionssuch as the Magellan spacecraft that orbited Venus with a powerful radar to peel backthe clouds of this once mysterious planet are a good example. A computerenhancedimage taken by Magellan is shown in Figure 6. These types of missions push back theboundaries of human knowledge, giving us new insight into planetary formation,weather and other important processes at work back here on Earth.
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Figure 6: Magellan at Venus. The powerful synthetic aperture radar on NASA’sMagellan spacecraft pierced the thick clouds of Venus, giving us the first details of theplanet’s surface. (Photo courtesy of NASA.)
Describing Space Missions
Space missions seem complex, and they are to a certain extent, but if you look atthem logically, you’ll see many similarities. Let’s begin with some key definitions:
♦ Mission Objective  Why we’re going to space and what we’re going to do oncewe get there.
♦ Users  The people or systems that use data or services provided by thesatellite or satellites.
♦ Operators The people who manage and run the mission from the ground.
♦ Mission Operations Concept How users, operators, ground and spaceelements all work together to make a mission successful.
All these come together to form the tangible elements of what is collectively called theSpace Mission Architecture. These elements are depicted in Figure 7; each one isdefined in the subsections following.
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The Space Mission
Space Operations
Spacecraft
Payload Bus
Payload
Space Transportation
Communications Network
Trajectories & Orbits
Figure 7: Space mission architecture. The key to understanding how missions arebuilt is to look at the space mission architecture that includes these critical elements.
Space Operations
The term space operations encompasses all activities needed to monitor and controlsatellites and the other elements that make up a space mission. Space operations areperformed by teams of people located at tracking sites and control centers around theworld.
Spacecraft Bus and Payload
A spacecraft has two basic parts, a payload (or payloads) and a bus. The payload includesspaceborne people and instruments that perform the primary mission. The spacecraft busprovides for the care and feeding of the payload—pointing, heating and cooling, structure,transportation and power. A simple analogy of a spacecraft bus and its payload is a goodoldfashioned school bus, as shown in Figure 8. It contains all of the same types ofsystems needed to support a spacecraft.
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Body & frame(Structures)Horn, radio & driver
(communications & data handling)
Steering (space vehicle control)
Battery & alternator (electrical power)
Radiator, air conditioning &heater (environment control &life support )
engine & drive train(space transportation)
Passengers(payload)
Figure 8: The spacecraft “bus.” A spacecraft has all the basic systems found in aregular school bus.
Trajectories and Orbits
A trajectory is any path an object follows through space. An orbit is a special type ofrepeating trajectory. The simplest way to imagine an orbit is to think of a “racetrack”around the Earth which satellites “drive” around, as shown in Figure 9.
Figure 9: Orbit racetrack. An orbit can be thought of as a fixed racetrack around aplanet, where the size of the racetrack depends on the velocity of the object in orbit.
Depending on the altitude of the orbit, a satellite has different perspectives on the Earth.The total fraction of the Earth a satellite can “see” using its onboard sensors is known asthe field of view. The projection of this field of view onto the Earth’s surface creates aswath width for the sensor as it sweeps around the Earth on its orbit. These twoparameters are illustrated in Figure 10.
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field of view
swath width
Figure 10: Field of view and swath width. The height of the orbit and the sensor fieldof view dictates the swath width that can be imaged on the ground.
There are a variety of different types of orbits that can be found in a typical space mission,including parking orbits, transfer orbits and final mission orbits. These are illustrated inFigure 11.
final orbitparking orbit
transfer orbit
Figure 11: Orbit types. Different types of orbits include the parking orbit, the transferorbit and the final or mission orbit. A satellite normally begins its life in a temporaryparking orbit. From there, an upper stage rocket is used to boost the satellite onto atransfer orbit. An additional boost places it into the final mission orbit.
Space Transportation
Space transportation includes all of the systems necessary to deliver our spacecraft to itsfinal mission orbit. Normally, this consists of a booster, such as the Space Shuttle or
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Ariane, an upper stage, such as the Inertial Upper Stage (IUS), and onboard thrusters forfinal maneuvers and station keeping. The Space Shuttle, shown in Figure 12, is one typeof complete space transportation system.
Figure 12: The Space Shuttle. Space transportation includes the systems that put thespacecraft in orbit, keep it there, and rotate and move it if necessary. Spacetransportation systems develop the velocity needed to obtain and stay in orbit. Spaceboosters are divided into stages that provide incremental changes in velocity and arethen discarded.
Communications Network
A space mission is more than just rockets and satellites. An entire system of ground andonorbit assets are needed to track, command and control all aspects of the mission. Thiscommunications network ties together various links needed to deliver bus telemetry andpayload data to operators and users, as shown in Figure 13.
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primary aircraft
Tracking & DataRelay Satellite(TDRS)
relay satellite
tracking site mission control center tracking site/users
Figure 13: Communications network. The communications network is the “glue”that holds the mission together. The network ties together space assets, groundcontrollers and users in a complex web of links that transfers data among the variousmission element nodes.
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EXPLORING SPACE
Long before rockets and interplanetary probes escaped the Earth’s atmosphere,people explored the heavens with just their eyes and imagination. Later, with the aidof telescopes and other instruments, humans continued their struggle to bring orderto the heavens. With order came some understanding and a concept of our place inthe universe. Thousands of years ago, priests of ancient Egypt and Babylon carefullyobserved the heavens to plan religious festivals, to control the planting and harvestingof various crops, and to understand at least partially the realm in which they believedmany of their gods lived. Later, philosophers such as Aristotle and Ptolemy developedcomplex theories to explain and predict the motions of the Sun, Moon, planets andstars.
The theories of Aristotle and Ptolemy dominated the world of astronomy and ourunderstanding of the heavens well into the 1600s. Combining ancient traditions withnew observations and insights, natural philosophers such as Copernicus and Kepleroffered rival explanations from the 1500s onward. Using their models and IsaacNewton’s new tools of physics, astronomers in the 1700s and 1800s made severalstartling discoveries, including two new planets—Uranus and Neptune. Let’s brieflyexplore some of these major contributors to our early understanding of space andorbits.
Copernicus
With the Renaissance and humanism came a new emphasis on the accessibility of theheavens to human thought. Nicolaus Copernicus (1473–1543), a Renaissancehumanist and Catholic clergyman, reordered the universe and enlarged man’shorizons. He placed the Sun at the center of the solar system, as shown in Figure 14,and had the Earth rotate on its axis once a day while revolving about the Sun once ayear.
Copernicus further observed that, with respect to a viewer located on the Earth, theplanets occasionally appear to back up in their orbits as they move against thebackground of the fixed stars. Ptolemy and others resorted to complex combinationsof circles to explain this backward motion of the planets, but Copernicus cleverlyexplained that this motion was simply the effect of the Earth overtaking, and beingovertaken by, the planets as they all revolved about the Sun.
However, Copernicus’ heliocentric system had its drawbacks. He couldn’t prove theEarth moved, and he couldn’t explain why the Earth rotated on its axis whilerevolving about the Sun. He also adhered to the Greek tradition that orbits followuniform circles, so his geometry was complex and somewhat erroneous. In addition,Copernicus wrestled with the problem of parallax—the apparent shift in the positionof bodies when viewed from different distances. If the Earth truly revolved about theSun, critics observed, a viewer stationed on the Earth should see an apparent shift in
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position of a closer star with respect to its more distant neighbors. Because no onesaw this shift, Copernicus’ Suncentered system was suspect. In response,Copernicus speculated that the stars must be at vast distances from the Earth, butsuch distances were far too great for most people to contemplate at the time, so thisidea was also widely rejected.
Figure 14: Copernicus redefines the center. Polish astronomer Copernicus reorderedour view of the universe. He promoted a heliocentric (Suncentered) universe, a simpler,more symmetric approach with all of the planets in circular orbits about the Sun.Unfortunately, these ideas were widely rejected because they disputed religiousteachings of his day.
Kepler
Johannes Kepler (1571  1630) revolutionized our understanding of orbits. In CosmicMystery, written before he was age 25, he calculated that the orbit of Mars was notcircular but elliptical. From this work, he developed three important laws of orbitmotion, described in Figures 15 through 17.
Figure 15: Kepler’s 1st law. The orbits of the planets are ellipses with the Sun at afocus.
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Planetarymotion over 30 days
Planetarymotion over30 days
Area 1 = Area 2
Area 1 Area 2
Figure 16: Kepler’s 2nd law. The orbits of the planets sweep out equal areas in equaltime.
Average distance
Figure 17: Kepler’s 3rd law. The square of the orbit period—the time it takes to goaround once—is proportional to the cube of the average distance to the Sun.
Galileo
In 1609, an innovative mathematician, Galileo Galilei (1564–1642), heard of a newoptical device that could magnify objects so they would appear to be closer andbrighter than when seen with the naked eye. Building a telescope that could magnifyan image 20 times, Galileo ushered in a new era of space exploration. He made somestartling telescopic observations of the Moon, the planets, and the stars, therebyattaining stardom in the eyes of his peers and potential patrons. Observing theplanets, Galileo noticed that Jupiter had four moons or satellites (a word coined byKepler in 1611) that moved about it. This disproved Aristotle’s claim that everythingrevolved about the Earth.
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Galileo also took on Aristotle’s physics. He rolled a sphere down a grooved ramp andused a water clock to measure the time it took to reach the bottom. He repeated theexperiment with heavier and lighter spheres, as well as steeper and shallower ramps,and cleverly extended his results to objects in freefall. Through these experiments,Galileo discovered, contrary to Aristotle, that all objects fall at the same rateregardless of their weight, as shown in Figure 18.
Galileo further contradicted Aristotle as to why objects, once in motion, tend to keepgoing. Aristotle held that objects in “violent” motion, such as arrows shot from bows,keep going only as long as something is physically in touch with them, pushing themonward. Once this push dies out, they resume their natural motion and drop straightto Earth. Galileo showed that objects in uniform motion keep going unless disturbedby some outside influence. He wrongly held that this uniform motion was circular,and he never used the term “inertia.” Nevertheless, we applaud Galileo today forgreatly refining the concept of inertia as we know it today.
Figure 18: Galileo on gravity. Through application of the scientific method, Galileo putAristotle’s ideas to the test and proved Aristotle wrong—all objects fall at the same rate.
Newton
To complete the astronomical revolution, which Copernicus had almost unwittinglystarted and which Kepler and Galileo had advanced, the terrestrial and heavenlyrealms had to be united under one set of natural laws. Isaac Newton (1642–1727)answered this challenge. Newton was a brilliant natural philosopher andmathematician who provided a majestic vision of nature’s unity and simplicity. 1665proved to be Newton’s “miracle year,” in which he significantly advanced the study ofcalculus, gravitation, and optics. Extending the groundbreaking work of Galileo indynamics, Newton published his three laws of motion and the law of universalgravitation in the Principia in 1687. With these laws, you could explain and predictmotion not only on Earth but also in tides, comets, moons, planets—in other words,motion everywhere. Newton’s laws are explained more thoroughly in the next section.
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INTRODUCTION TO ORBITAL MOTION
Orbits are one of the basic elements of any space mission. Understanding a satellitein motion may at first seem rather intimidating. After all, to fully describe orbitalmotion we need some basic physics along with a healthy dose of calculus andgeometry. However, as we’ll see, the complex trajectories of rockets flying into spacearen’t all that different from the paths of baseballs pitched across home plate. In fact,in most cases, both can be described in terms of the single force pinning you to yourchair right now—gravity.
Armed only with an understanding of this single pervasive force, we can predict,explain and understand the motion of nearly all objects in space, from baseballs toentire galaxies. Once we know an object’s position and velocity, as well as the natureof the local gravitational field, we can predict exactly where the object will be minutes,hours or even years from now.
Overview
Math for the Fainthearted
♦ What is a vector?
♦ What is a derivative?
♦ What is an integral good for?
Key Concepts in Dynamics
♦ What is the difference between mass, inertia and weight?
♦ What is momentum?
♦ What is energy?
♦ What are Newton’s Laws of Motion?
♦ What is gravity?
Orbits Made Simple
♦ OK, forget the math, what is an orbit, really?
♦ How are energy and momentum conserved in an orbit?
Math for the Fainthearted
Before delving too deeply into a discussion of dynamics, orbital mechanics andpropagators it’s useful to step back and briefly review a bit of math. BUT DON’TPANIC! This section is designed to simplify a few basic concepts and describe the
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notation used. You should walk away from this section with a “Math Survival Kit”that will make the rest of this Primer far more useful and enjoyable.
Vectors and Such
Scalar
A scalar is a quantity that has magnitude only. Speed, energy and temperatureare examples of scalars. None of these quantities has a unique meaning in anycertain direction. A single letter, such as E for Total Mechanical Energy, denotes ascalar quantity.
Vector
A vector is a quantity that has both magnitude and direction. For example, if I askyou where you drove in your car, you might answer: “I went south.” But thiswouldn’t tell me much. If I asked “How far?,” and you said “five miles,” I could puttogether a better picture. By knowing you drove five miles south, I have bothmagnitude and direction. A letter with an arrow over it, such as
vV for the velocity
vector is used to denote a vector quantity.
Unit Vector
A unit vector is a vector having a magnitude of one; it’s used to describe directiononly. For example, if I want to define where north is on a drawing, I could do sowith a unit vector indicating the direction. A letter with a caret or hat over it, such
as $I for the Idirection denotes a unit vector.
Example
We generally describe the velocity of an object in orbit in terms of three unit
vectors, $, $, $I J K . Thus, a typical velocity vector could be written as:
vV I J K= + −30 21 7 4. $ . $ . $ km / sec
This means the velocity is 3.0 km/sec in the Idirection, 2.1 km/sec in the Jdirection and 7.4 km/sec in the Kdirection.
Calculus—Just the important bits
Calculus was developed to analyze changing parameters. Let’s look at how thosechanges are described.
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Derivative
A derivative represents the rate of change of one parameter with respect toanother. For example, if you’re traveling north in your car, your position ischanging over time. The rate at which your position changes over time is yourvelocity. Thus, if you travel 30 miles north in 30 minutes, your velocity is:
Velocity VChange in
Change in = = = =
vv
position (R)time (t)
30 miles30 minutes
60 mph north
Several methods are commonly used to denote a vector. In this primer, we use twotypes of notation. The first is to represent a derivative as d. So the change inposition over time would be:
vv
VdRdt
=
We also use a “dot” over a symbol to represent the derivative with respect to time. For
example, v&R represents the derivative of the position vector or the rate of change of
position. v&&R is the second derivative of position, or the rate of change of the rate of change
of position (i.e. the acceleration).
Integral
An integral represents the cumulative effect of one parameter changing withrespect to another. If we were to graph both changing parameters, the integral isthe area under the curve. For example, if you drive north at 30 miles per hour for30 minutes, the integral of this velocity is your new position at the end of the time(e.g., 15 miles north of where you started). In other words, you add up all thechanges in position over time to get the total change. An integral is the reverse ofa derivative. Because acceleration is the derivative of velocity over time, theintegral of acceleration over time is velocity.
Key Concepts
Mass
Mass is a measure of how much matter or “stuff” an object possesses. For example, avolleyball and a cannon ball are about the same size, but the cannon ball has farmore mass because it is made of a more dense material.
Inertia
Inertia describes how hard it is to move an object. It is much easier to push a babycarriage than a bulldozer because the bulldozer, being more massive, has moreinertia.
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Weight
Weight actually describes the force produced by gravity acting on a mass. Your weightin various situations is illustrated in Figure 19.
Figure 19: Weight. Weight describes a contact force caused by the effect of gravity onmass. On Earth, your weight is one value. As you move further from the center of theEarth, say in a penthouse at the top of a 250milehigh skyscraper, your weight would beslightly less. In orbit at 250 miles altitude, the gravitational force is still the same;however, because you are in freefall and not in contact with the Earth, your weight iszero. In all cases, mass stays the same.
Momentum
Linear momentum describes the resistance a moving object has to changes in eitherdirection or speed. The more massive an object, or the faster it is moving, the harderit is to stop or change its direction of motion. As a result, linear momentum is theproduct of the mass and velocity of an object. Momentum for baby carriages andbulldozers is shown in Figure 20.
m = 25 kgv= 1000 m/smv = 25,000 kgm/s
m = 25,000 kgv = 1 m/smv= 25,000 kgm/s
Figure 20: Momentum, bulldozers and baby carriages. Linear momentum is theproduct of mass and velocity. For a baby carriage to have the same linear momentum asa bulldozer, it would have to be traveling at a much higher velocity.
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Angular momentum is a measure of the spinning properties of an object. As Figure 21illustrates, a nonspinning top immediately falls over. However, a spinning top hasangular momentum, which allows it to resist the force of gravity pulling it over (until itfinally slows down due to friction).
nonspinning top spinning top
angular momentum
Figure 21: Angular momentum. A spinning top has angular momentum which keeps itpointing upright even when pulled by outside forces such as gravity.
Energy
Potential energy is a function of an object’s position and mass. The greater the heightan object is raised to, the greater its potential energy. An object at the top of a deepwell, as shown in Figure 22, has more potential energy than an object at the bottom ofthe well.
PE = 0 at R = infinite
PE < 0 at R > 0
PE < 0 at R = 0
R
Figure 22: Potential energy. Because we normally define our coordinate systems aspositive outward from the center of the Earth, we measure potential energy “from thebottom up.” For example, at the top of a deep well we would say the potential energy iszero. As we get closer to the bottom of the well, the potential energy is less, or morenegative.
Kinetic energy is a function of an object’s mass and speed. Like momentum, the moremassive an object is or the faster it travels, the more kinetic energy it has. It is this
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energy that must be transformed in order to stop a speeding bulldozer, as shown inFigure 23. This is accomplished by applying the brakes, which turns kinetic energyinto heat (those brakes get hot!).
m = 25,000 kgv = 1 m/s1/2 mv2 = 12,500 kgm/s2
Figure 23: Newton’s 1st law. The kinetic energy of a bulldozer moving at only 1 m/s is12,500 kgm/s2.
Total energy is the sum of kinetic plus potential energy.
Total Energy = Kinetic Energy + Potential Energy
Newton’s Laws
Newton’s First Law
A body remains at rest or in constant motion unless acted upon by external forces.
In other words, if you were to pitch a baseball, it should continue on its path, in astraight line forever, unless disturbed by an outside force such as gravity or airresistance.
Newton’s Second Law
The time rate of change of an object’s momentum is equal to the applied force.
Change momentumChange time
( )
( )= Force Applied
Recall, momentum is the product of mass and velocity. Thus, as long as massstays constant (which it normally does as long as rockets aren’t firing) thisequation can be reduced to:
F ma=
or:
Force applied = mass (m ) times acceleration (a)
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The significance of this relationship can be felt every time you hit the brakes in you car.The more force you apply (the harder you hit the brakes) the faster you stop (the fasteryou decelerate). This principle is illustrated in Figure 24.
stops in 1second
25,000 N
1 m/s
1 m/s6.9 N
stops in 1 hour
Figure 24: Newton’s 2nd law. Force is proportional to acceleration (or deceleration). A25,000 N force is needed to stop a 1 m/s bulldozer in 1 second, while much smaller 6.9N force would take 1 hour to bring it to a stop.
Newton’s Third Law
For every action, there’s an equal and opposite reaction. This basic principle canbe illustrated by two rollerskating astronautics, as shown in Figure 25.
Figure 25: Newton’s 3rd law. If two people on roller skates push against each other,they both move backward. Their acceleration is proportional to their mass.
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Newton’s Law of Universal Gravitation
The force of gravity between two bodies is proportional to the product of themasses and inversely proportional to the square of the distance between them.This is illustrated in Figure 26.
R
FF
m
m
1
2g
g
Figure 26: Gravitational attraction. Two masses in space each exert a force on theother. The magnitude of this force depends on the product of their masses and thesquare of the distance between them.
Newton’s law can be summarized in equation form as follows:
FGM M
Rgravity = 1 2
2
where
Fgravity = Force of gravity (N)
G = Universal gravitational constant = 6.67 x 1011 (Nm/kg)
M1, M2 = Mass 1 and Mass 2 respectively (kg)
R = Distance between the two masses (m)
In other words, the more mass an object has, the more gravitational force itgenerates. Furthermore, the farther apart two objects are, the less the force is, infact, the force decreases with the square of the distance as illustrated in Figure 27.
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R
FF
m
m
1
2g
g
2R
m
m
1
2
F /4g
gF /4
Figure 27: Gravity and distance. The force of gravitational attraction decreases withthe square of the distance (e.g., if you double the distance, the force decreases by onefourth).
What exactly is gravity? The study of physics is still grappling to reconcile theforce of gravity with the other fundamental forces of nature. Already, extremelyweak “gravity waves” have been detected from distant galaxies. More sensitiveinstruments are being built to understand and quantify this mysterious force.
How strong is gravity? Let’s look at the EarthMoon system. The force of gravitybetween the Earth and Moon is 1.98 x 1020 Newtons! To put this into perspective,the Space Shuttle generates about 28 million Newtons thrust at liftoff. The EarthMoon gravity force is more than one trillion times as great as that of the Shuttle!
Regardless of what gravity really is, we know it’s a force that affects anything withmass (and that’s pretty much everything!). While Galileo right in that thegravitational force is greater on heavier objects than lighter objects, he was wrongin predicting the affect this would have on the rate at which they fall. Theacceleration of an object in a gravitational field is independent of its mass.
Figure 28: All things fall at the same rate. It is important to note that all thingsaccelerate at the same rate within a gravitational field. For example, if you drop a
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hammer and a feather, both objects impact the ground at the same time (neglecting airresistance). Of course, Galileo predicted this. Astronaut Dave Scott proved this with anexperiment on the Moon. He dropped a hammer and a feather at the same time. Both hitthe ground at the same instant (there is no air resistance on the Moon)!
Orbits Made Simple
What is an orbit? In the simplest sense, orbits are a type of “racetrack” in space that asatellite “drives” around.
Figure 29: Orbits as racetracks. The simplest way to think of orbits is as giant, fixed“racetracks” on which spacecraft “drive” around the Earth.
Baseballs and Satellites
But what makes these racetracks? Before diving into a complicated explanation, let’sbegin with a simple experiment that illustrates, conceptually, how orbits work. To dothis, we’ll arm ourselves with a bunch of baseballs and travel to the top of a tallmountain. Imagine you were standing on top of this mountain prepared to pitchbaseballs to the east. As the balls sail off the summit, what would you see? Thebaseballs would follow a curved path before hitting the ground. Why is this? The forceof your throw causes them to move outward, but the force of gravity pulls them down.Therefore, the “compromise” shape of the baseball’s path is a curve.
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Figure 30: Baseballs in motion. A baseballthrowing astronaut can be used to illustratethe simple motion of a satellite. Naturally, the harder the baseballs are thrown, thefurther they travel before hitting the ground.
As Figure 30 illustrates, the faster you throw the balls, the farther they travel beforehitting the ground. This could lead you to conclude that the faster you throw them,the longer it takes before they hit the ground. But is this really the case? Let’s tryanother experiment to see. As you watch, two astronauts, standing on flat ground,release baseballs. The first one simply drops a ball from a fixed height. At exactly thesame time, a second astronaut throws an identical ball horizontally as hard aspossible. What do you see? If the second astronaut throws a fast ball, it travels outabout 20 m (60 ft.) or so before it hits the ground. However, the ball dropped by thefirst astronaut hits the ground at exactly the same time as the pitched ball, as Figure31 shows!
Figure 31: Motion and gravity. Two astronauts each have a baseball held at the sameheight above the ground. If the first astronaut drops her baseball while the secondastronaut throws his, both baseballs hit the ground at exactly the same time. Gravity actson both baseballs in the same way, independent of their horizontal motion.
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How can this be? To understand this seeming paradox, we must recognize that, inthis case, the motion in one direction is independent of motion in another. Thus,while the second astronaut’s ball moves horizontally at 30 km/hr (20 m.p.h.) or so,it’s still falling at the same rate as the first ball. This rate is the constant gravitational
acceleration of all objects near the Earth’s surface: 9.798 m/s2. Thus, they hit theground at the same time. The only difference is that the pitched ball, because it hashorizontal velocity, manages to travel some distance before intercepting the ground.
Now let’s return to the top of our mountain and start throwing our baseballs fasterand faster to see what happens. No matter how fast we throw them, the balls still fallat the same rate. However, as we increase their horizontal velocity, they’re able totravel farther and farther before they hit the ground. Because the Earth is basicallyspherical in shape, something interesting happens. The Earth’s spherical shapecauses the surface to drop approximately 5 m for every 8 km we travel horizontallyacross it, as shown in Figure 32.
8 km
5 m
Figure 32: Our spherical Earth. We know the Earth is a nearly perfect sphere. Forevery 8 km of horizontal distance, the Earth curves down about 5 m. In other words, ifyou could lay an 8 km long board tangent to the Earth at one end, at the other end itwould be 5 m off the ground.
So, if we were able to throw a baseball at 8 km/s (assuming no air resistance), itspath would exactly match the rate of curvature of the Earth. That is, gravity wouldpull it down about 5 m for every 8 km it travels, and it would continue around theEarth at a constant height. If we don’t remember to duck, it will hit us in the back ofthe head about 85 minutes later. (Actually, because of the rotation of the Earth, itwould miss your head.) A ball thrown at a speed slower than 8 km/s falls faster thanthe Earth curves away beneath it.
If we analyze our various baseball trajectories, we see a range of different shapes.Only at exactly one particular velocity do we get a circular trajectory. Any slower thanthat and our trajectory impacts the Earth at some point. If we were to project thisshape into the Earth, we’d find the trajectory we see is really just a piece of an ellipse.If we throw the ball a bit harder than the circular velocity, we also obtain an ellipse.An object in orbit is literally falling around the Earth but, because of its horizontalvelocity, it never quite impacts the ground. If we throw the ball too hard, it leaves theEarth altogether on a parabolic or hyperbolic trajectory, never to return.
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circle
hyperbola
parabola
ellipse
Figure 33: Baseballs in orbit . If we throw a baseball fast, but not quite fast enough,eventually the ball will impact the ground (perhaps even on the other side of the Earth)like an ICBM (Intercontinental Ballistic Missile). If we throw it at just the right speed,gravity will cause the ball to fall 5 m for every 8 km of horizontal distance traveled. But,since the Earth also curves down 5 m for each 8 km of horizontal distance, the ball willstay at same the instantaneous height above the ground. We call this a circular orbit . Ifwe throw it faster than the circular orbit speed, the ball will be in an ellipticalshaped orbit.If we throw the ball faster yet, it will escape the Earth’s gravity altogether on a parabolic orhyperbolic trajectory.
Thus, it is important to note that no matter how hard we throw, our trajectoryresembles either a circle, ellipse, parabola or hyperbola. These four shapes are calledconic sections. Why conic sections? Because these are the shapes we get by slicingthrough a cone with a plane at different angles, as illustrated in Figure 34.
circle parabolahyperbolaellipse
Figure 34: Conic sections . The four basic conic sections: circle, ellipse, hyperbola andparabola.
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So how fast how fast do we have to throw our baseball to put it into a circular orbit?Let’s play with some math. The velocity of a satellite in a circular orbit can be foundusing:
VGM
RcircularEarth=
where
Vcircular = Satellite velocity in a circular orbit (km/sec)
G = Universal gravitational constant = 6.67 x 1011 km2/sec3
MEarth = mass of the Earth = 5.98 x 1015 kg
R = distance from the center of the Earth = 6378 km at the surface
Thus, at the surface of the Earth, the velocity in a circular orbit would be 7.9 km/sec(17,600 mph)! In other words, to move into a circular orbit that stays just above thesurface of the Earth (ignoring air drag) you’d have to throw the baseball at 17,600mph. Notice that the circular orbit speed depends on your distance from the center ofthe Earth. The lower you are, the faster you must travel to achieve a circular orbit.
Conservation of Energy & Momentum
Now that we’ve looked at the simple geometry of an orbit, we can consider howconservation of energy and momentum affects the velocity of satellites. Gravity is aconservation force, which means that an object moving in a gravitational field doesn’tlose any energy through friction or heat, etc. Additionally, total energy is constant, or:
Total Energy = Kinetic (KE) + Potential Energy (PE) = constant
This basic principle is illustrated by the swinging astronaut in Figure 35.
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Maximum PEKE=0
Maximum PEKE=0
Max KEMin PE
PE+KE=constant
Figure 35: Trading kinetic and potential energy. The conservation of energy(Potential Energy + Kinetic Energy = Constant) is illustrated by a simple swing.Neglecting losses from friction, the total energy of the astronaut on the swing is constant.At the low point in the swing, speed (kinetic energy) is greatest and potential energy islowest. As you swing, you trade kinetic energy (speed) for potential energy (height) withthe sum of the two constant.
When applied to an orbit, the same principle applies. Total energy must be conserved,thus the orbit speed varies throughout the orbit as kinetic energy is traded forpotential energy. A satellite travels fastest at perigee—the lowest point in the orbit—and slowest at apogee—the highest point in the orbit. This is shown in Figure 36.
low kineticenergy
apogee
high potentialenergy
low potentialenergy
high kineticenergy
perigee
Figure 36: Energy conservation in orbit. As a satellite moves closer to the Earth in anorbit, it must speed up to conserve total energy. As it gets further away, the satellitetrades kinetic energy for potential energy and slows down.
Angular momentum is also always conserved. Ice skaters use this principle to speedup or slow down as they spin, as shown in Figure 37. Just like a spinning ice skater,an orbit has angular momentum. Because angular momentum is a vector quantity,
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the direction as well as the magnitude of this momentum stays the same. As a result,even though the Earth rotates under the orbit and the Earth (and the orbit along withit) moves around the Sun, the orbit itself stays fixed in respect to a stationaryreference.
Figure 37: Conservation of angular momentum. Ice skaters use this principle tospeed up or slow down as they spin. When their arms are extended, the moment ofinertia is low, so they spin more slowly. As they draw their arms in, the moment of inertiadecreases and the spin rate increases to keep total angular momentum constant.
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DESCRIBING ORBITS
Overview
Understanding Coordinate Systems
♦ What is a coordinate system?
The GeocentricEquatorial Coordinate System
♦ What is the most common coordinate system used for satellites?
Classical Orbital Elements
♦ What do all those Greek letters tell me about an orbit?
Ground Tracks
♦ How do those squiggly lines on a map represent the path of a satellite?
Understanding Coordinate Systems
To be valid, Newton’s laws must be expressed in an inertial frame of reference,meaning a frame that is not accelerating. Any reference frame is just a collection ofdefinitions that allow us to describe positions and velocities in a more meaningfulway. For example, if I simply told you a car is traveling south, you wouldn’t have verymuch information. But if I first tell you that our a reference frame is centered onWashington, DC, and then tell you that the car is 30 miles east of the city travelingsouth at 60 mph, you’d know something far more useful.
In defining coordinate systems and describing position and velocities, we makeextensive use of vectors. We use vectors because we want to keep track of theinformation contained in a particular parameter. Specifically, a vector is a parametershaving both magnitude and direction. For example, 60 mph tells you speed(magnitude) without direction. But 60 mph south tells you both magnitude anddirection. In defining coordinate systems, we are sometimes only interested indirection. In that case, we use unit vectors defined to have a specific direction and amagnitude of 1.
Cartesian coordinate systems are laid out with three orthogonal unit vectors—vectorsat right angles to each other. For example, if the origin of a coordinate system werechosen to be in one corner of a room, the floor would be the fundamental plane. Wecould then describe the position of every piece of furniture in the room with respect tothis system. Such a collection of unit vectors allows us to establish the components ofother vectors in 3D space.
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To create a coordinate system, we need to specify four pieces of information—anorigin, a fundamental plane, a principle direction, and a third axis, as shown inFigure 38. The origin defines a physically identifiable starting point for the coordinatesystem. The other two parameters fix the orientation of the frame. The fundamentalplane contains two axes of the system. Once we know the plane, we can define adirection perpendicular to that plane. The unit vector in this direction at the origin isone axis. Next, we need a principle direction within the plane. Again, we choosesomething that is physically significant, like a star. Now that we have two directions,the principle direction and an axis perpendicular to the fundamental plane, we canfind the third axis using the righthand rule. The righthand rule can be demonstratedby pointing the thumb of your right hand in the direction perpendicular to thefundamental plane. With you fingers pointing in the principle direction, curl yourfingers 90° so that your thumb is pointing in the direction of the third axis.
(1) pick origin
origin
(2) pick fundamental plane& perpendicular to it
origin
fundamentalplane
(3) pick principal direction
fundamentalplane
fundamentalplane
originorigin
principal direction principal direction
3rd axis, foundusing righthandrule
(4) find 3rd axis
Figure 38: Defining a coordinate system.
Remember—coordinate systems are defined to make our lives easier. If we choose thecorrect coordinate system, developing the equations of motion can be simple. If wechoose the wrong system, it can be nearly impossible.
The GeocentricEquatorial Coordinate System
For Earthorbiting spacecraft, we’ll choose a triedandtrue system that we knowmakes solving the equations of motion relatively easy. We call this system thegeocentricequatorial coordinate system, shown in Figure 39. Here’s how it’s defined:
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Origin
The center of the Earth (hence the name geocentric).
Fundamental plane
Earth’s equator (hence geocentricequatorial), where perpendicular to the fundamentalplane is the direction of the north pole.
Principle direction
Vernal equinox direction, , or the vector pointing to the first point of Aries. Thevernal equinox direction points at the zodiac constellation Aries and is found bydrawing a line from the Earth to the Sun on the first day of spring. While thisdirection may not seem “convenient” to you, it’s significant to the astronomers whooriginally defined the system. Unfortunately, the vernal equinox direction is notperfectly constant. The Earth’s orbit precesses around the Sun and the Sun is movingthrough the galaxy. Therefore, exactly when this direction is defined is extremelyimportant for the definition of the system. We can use two ways of defining thesedirections. The first is to use the mean or average direction at some point in time. Theother is to use the true position at exactly one specific point in time. Variouscombinations of these definitions using different dates gives us several possibilities forcoordinate systems used in STK:
J2000
Defines the mean vernal equinox direction and mean Earth rotation axis onJanuary 1 of the year 2000 at approximately 12:00:00.00 GMT.
Mean of date
Defines the mean vernal equinox direction and mean Earth rotation axis at theorbit epoch time (the time for which the orbital elements being used is true).
Mean of epoch
Defines the mean vernal equinox direction and mean Earth rotation axis at thecoordinate epoch time (time at which the coordinate system being used is defined).
True of date
Defines the true vernal equinox direction and true Earth rotation axis at exactlythe orbit epoch time specified.
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True of epoch
Defines the true vernal equinox direction and true Earth rotation axis at thecoordinate epoch time specified.
B1950
Defines the mean vernal equinox direction and mean Earth rotation axis at thebeginning of the Besselian year 1950. It corresponds to 31 December 1949 at22:09:07.20 Greenwich Mean Time (GMT).
Mean Equinox, True Equator
Defines mean vernal equinox direction and true Earth rotation axis for the orbitepoch time specified.
Third axis
The third axis of the geocentricequatorial coordinate system is found using the righthand rule.
I
K
J
Figure 39: The geocentricequatorial coordinate system. The system is defined by:u Origin  Center of Earth; u Fundamental Plane  Equals equatorial plane;u Perpendicular to Plane  north pole; u Principle Direction  vernal equinox direction.
Classical Orbital Elements
Three pieces of information are needed to fix any point in space; collectively, they’reknown as an object’s position vector,
rR . Three more pieces of information describe its
velocity vector, r
V . One additional item, time, tells us when the information providedis valid. These elements are know as Cartesian elements. While it is often convenientto describe orbit motion using simply position and velocity vectors in a Cartesiancoordinate system, especially for computational work, these vectors provide us littleinsight into the orbit itself. For this reason, astronomers long ago developed orbital
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elements. Orbital elements give us a shorthand way of expressing orbit size, shapeand orientation, allowing us to tell at a glance the application for a given orbit. Thissection describes the Classical orbital elements, which are sometimes referred to asthe Keplerian elements and are attributed originally to Kepler himself. Variations onthese elements, the commonly used twoline element sets, are described in a latersection.
Orbit Size
How big is an orbit? This depends on how fast we “throw” our satellite into orbit. Thefaster we throw, the more energy an orbit has, and the bigger it is. We express the sizeof an orbit in terms of its semimajor axis, a., as defined in Figure 40.
apogee perigee
2a = major axis
a = semimajor axis
Figure 40: Semimajor axis. The major axis of an elliptical orbit is the distance betweenthe point of closest approach (perigee) and furthest point (apogee). Semimajor axis isonehalf this distance.
We can express the semimajor axis in terms of the distance from the center of theEarth to apogee (Rapogee) and perigee (Rperigee). Perigee is the point in an orbit that isclosest to the Earth. Apogee is the point where it is furthest away (apogee is undefinedfor a parabolic or hyperbolic trajectory). The semimajor axis can be found using:
aR Rapogee perigee
=+
2
where
a = semimajor axis (km)
Rapogee = Distance from center of Earth to apogee (km)
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Rperigee = Distance from center of Earth to perigee (km)
The orbit’s period, P (i.e., how long the satellite takes to travel around the orbit onetime), is proportional to the orbit size:
Pa
GM Earth= 2
3π
For example, a typical Space Shuttle orbit at an altitude of a few hundred kilometershas a period of about 90 minutes. It orbits the Earth about 16 times each day! Forcommunications satellites in geosynchronous orbit at an altitude of 35,780 km, theperiod is exactly 24 hours.
Orbit Shape
The less circular an orbit is, the more eccentric or “imperfect” it is. Eccentricity, e,describes the shape of orbit with respect to that of a circle.
0<e<1 e = 0
ellipse
parabola
hyperbolae = 1
circle
e>1
Figure 41: Eccentricity. A perfectly circular orbit has an eccentricity of 0. Ellipticalorbits have an eccentricity of less than 1. A parabolic orbit has an eccentricity of exactly1. Hyperbolic orbits (or trajectories) have eccentricities of greater than 1. In practice, aperfectly circular or parabolic orbit cannot be achieved.
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Orbit Orientation
How the orbit is tilted with respect to the equator is called its inclination, i. An orbitthat stays directly over the equator has an inclination of 0° and is called an equatorialorbit. An orbit that goes directly over the north and south poles must have aninclination of exactly 90° and is called a polar orbit.
Different classes of orbits have different inclinations, as shown in the table.
InclinationInclination Orbit TypeOrbit Type DiagramDiagram
0° = i = 180° Equatorial i=0o
i = 90° Polar i=90o
0° < i < 90° Direct or posigrade(moves in direction ofEarth’s rotation)
ascendingnode
90° < i <180° Indirect or retrograde(moves against thedirection of Earth’srotation) ascending
node
We measure how an orbit is twisted by locating its ascending node, the point wherethe satellite crosses the equator moving south to north. This point is referenced to theIdirection, which points in the Vernal Equinox direction. The angle between the Idirection and the ascending node is called the right ascension of the ascending node,RAAN, Ω.
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ascendingnode
equatorial plane
ΩΩ
i
J^
K
Figure 42: Right Ascension of the ascending node (RAAN), ΩΩ, is the angulardistance from the vernal equinox direction to the ascending node. The ascending nodeof an orbit is the point where it crosses the equatorial plane from south to north.
We describe the orbit’s orientation by locating perigee with respect to the ascendingnode. This angle is called the argument of perigee, ω; it is measured positive in thedirection of satellite motion.
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ascendingnode
equatorial plane
i
J^
K
ωω
Figure 43: Argument of perigee. The argument of perigee, ω, is the angular distancebetween the ascending node and perigee.
Finally, we describe a satellite’s instantaneous position with respect to perigee usinganother angle, true anomaly, ν. It is the angle, measured positive in the direction ofmotion, between perigee and the satellite’s position. Of the six orbital elements, onlytrue anomaly changes continually (ignoring perturbations).
perigee
υυ
R
V
Figure 44: True anomaly. The true anomaly, ν, is the angular distance from perigee to
the orbit position vector, rR .
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Summary of classical orbital elements
Recall, the reason we wanted to develop the orbital elements in the first place was togive us a shorthand method of describing an orbit. We also wanted to use parametersthat would have some physical meaning we could more easily visualize. The sixclassical orbital elements are summarized below.
NameName SymbolSymbol DescribesDescribes
Semimajor axis a Size (and energy)
Eccentricity e Shape (e = 0 for circle, 0> e >1 forellipse, e = 1 for parabola, e > 1 forhyperbola)
Inclination i Tilt of orbit plane with respect to theequator
Longitude of ascending node Ω Twist of orbit with respect to theascending node location
Argument of perigee ω Location of perigee with respect to theascending node
True anomaly ν Location of satellite with respect toperigee
Satellite Missions
As we already know, varying missions require different orbits, which can be describedusing Classical orbit elements. The table following shows various missions and theirtypical orbits. Technically speaking, a geostationary orbit is a circular orbit with aperiod of exactly 24 hours and an inclination of exactly 0°. A satellite in ageostationary orbit appears to be stationary to an Earthbased observer.Geosynchronous orbits are slightly inclined orbits with a period of 24 hours. Inpractice, it is almost impossible to achieve an orbit with exactly a 24hour period andan inclination of 0°. Thus, the two terms are frequently used interchangeably. A semisynchronous orbit has a period of 12 hours. Sunsynchronous orbits are retrogradelowEarth orbits (LEO) inclined 95° to 105°; they are typically used in remotesensingmissions to observe Earth. A Molniya orbit is a semisynchronous, eccentric orbit usedfor some communication missions. Supersynchronous orbits are usually circularorbits with periods longer than 24 hours.
MissionMission Orbit TypeOrbit Type SemimajorSemimajorAxis (Altitude)Axis (Altitude)
PeriodPeriod InclinatioInclinationn
OtherOther
♦ Communications
♦ Early Warning
♦ Nuclear detection
Geostationary 42,158 km
(35,780 km)
24hours
~0° e ≅ 0
♦ Remote sensing Sunsynchronous
~65007300 km
(~150900 km)
~90 min ~95° e ≅ 0
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MissionMission Orbit TypeOrbit Type SemimajorSemimajorAxis (Altitude)Axis (Altitude)
PeriodPeriod InclinatioInclinationn
OtherOther
♦ Navigation (GPS) Semisynchronous
26,610 km
(20,232 km)
12hours
55° e ≅ 0
♦ Space Shuttle LowEarthobit
~6700 km(~300 km)
~90 min 28.5° or 57° e ≅ 0
♦ Communication/Intelligence
Molniya 26,571 km(RP = 7971 km);(RA = 45,170 km)
12hours
63.4° ω = 270°
e = 0.7
Ground Tracks
Orbital elements allow us to visualize the shape of an orbit around the Earth. Becausewe use satellites for missions involving specific points on Earth—taking pictures,communications, navigation—we really would like to know what path the satellitetraces over the Earth’s surface. A satellite ground track is the orbit path (usually formultiple orbits) projected onto a flat map of the Earth. These projections becomecomplex because we must account for the satellite circumnavigating the entire Earthduring each orbit while the Earth itself rotates at 1600 km/sec underneath it.
To visualize a satellite’s ground track, let’s begin by assuming the Earth doesn’trotate. Picture an orbit around this nonrotating Earth. Because the orbit plane mustpass through the Earth’s center, the ground track traces a great circle. By definition,a great circle is any circle on a sphere that can be projected through the center. Forexample, all lines of longitude are great circles. The equator is the only line of latitudethat is a great circle—No other line of latitude “slices through” the center of the Earth.
When the Earth is stretched out to a flatmap projection (called an equidistantcylindrical projection), things start to look different. Imagine yourself on the ground,watching the orbit pass by overhead. If the Earth didn’t rotate, the projection of theground track would always look the same—a sine wave over the surface of the Earth,.(If you have trouble picturing why it is a sine wave, roll a piece of paper around asoda can and draw an inclined circle around the can. When you unroll the paper,you’ll see a sine wave just like an orbit ground track!)
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XX XX
Figure 45: Ground track for a nonrotating Earth. If the Earth didn’t rotate, the groundtrack would always look like a constant sine wave. As the figure shows, the ground trackwould always have the same relative orientation with respect to a stationary observer onthe Earth (shown here as an X in the Pacific ocean).
Now let’s start the Earth rotating again. As you watch the orbit pass overhead,something happens from one orbit to the next—the ground track shifts to the west!What happened? The orbit plane is fixed in inertial space. This means the orbit staysthe same with respect to a stationary observer. However, because the Earth rotates at15° per hour, an observer on the Earth is not stationary. As the Earth (and an Earthfixed observer) rotates to the east, the satellite ground track shifts to the west fromone orbit to the next, as shown in Figure 46. The amount it shifts depends on itsperiod. The longer the period, the more time the Earth has to rotate betweensuccessive orbits.
XXXX
Figure 46: Satellite ground tracks. Satellite orbits are fixed in space with respect to astationary observer. However, a stationary observer on the Earth is rotating to the east at15° per hour. Thus, each successive orbit ground track shifts to the west.
Let’s look at the ground track of some very different orbits.
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E
D
C B A
Figure 47: Ground tracks for orbits with different periods. Orbit A: Period=2.67 hr,Orbit B: Period=8 hr, Orbit C: Period=18 hr, Orbit D: Period=24 hr, Orbit E: Period=24hr.
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PREDICTING ORBITS
One of the most important problems in mission planning and satellite command &control is being able to accurately predict orbital motion. To track satellites throughspace, we need to know where they are now and where they’ll be later so that we canpredict sensor coverage and point our antennas at them to gather data. Although wecan easily predict this motion when the orbit is a circle, the problem becomes morecomplicated when the orbit is an ellipse, and most orbits are at least slightly elliptical.
An orbit propagator is a mathematical algorithm for predicting the future position andvelocity (or orbital elements) of an orbit given some initial conditions andassumptions. There are a wide variety of orbit propagation techniques available withwidely different accuracy and applications. Knowing the assumptions built intodifferent propagation schemes is key to knowing which one to use for a givenapplication.
Overview
Understanding Propagators
♦ How do propagators work?
The TwoBody Propagator
♦ What is meant by a “twobody” propagator?
Orbit Perturbations
♦ What are “J2” and those other things that affect an orbit?
Dealing with Perturbations
♦ How can I model orbit perturbations?
Twoline Element (TLE) Sets
♦ What are TLE set?
STK Propagators
♦ What propagators are available in STK?
Understanding Propagators
To understand the basic problem of orbit propagators, let’s return to the example ofour ballthrowing astronaut shown in Figure 48. What we’re after is a simple
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mathematical algorithm that will allow us to predict the ball’s position and velocity atany point in time. Whether you’re analyzing the motion of baseballs or galaxies, thefundamental approach is the same. This motion analysis process has three steps:
♦ define a convenient coordinate system
♦ list simplifying assumptions
♦ define initial conditions
x
y
Figure 48: A baseball motion propagator. The simple example of a thrown baseballcan be used to describe the basic problem of orbit propagation.
We can now apply the motion analysis process to describe, and eventually propagate,the motion of the baseball. First, we select a simple, convenient coordinate systemwith its origin at the point of release. The xdirection is defined to be positive to theright in the picture. The ydirection is positive down. Next, we need to make someassumptions to make our lives easier. The major assumption we’ll make is thatEarth’s gravity is the only force acting on the ball, a force which is constant over theflight path we’re concerned with. That is, wind resistance and other forces (thegravitational pull of the Moon and stars, solar pressure, etc.) are negligible. Thisgravitational force can be expressed using Newton’s law, using vector notation, as:
vFgravity = mgy N$
Note $y is a vector notation indicating the force of gravity acts only in the ydirection, i.e.,down. See the section describing vectors for further explanation.
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Finally, we need some initial conditions. Let’s pretend that the ball leaves the pitcher’shand at a velocity of 10 m/s on a horizontal path (i.e., all motion in the xdirection).Symbolically we would say the initial velocity (
rVinitial ) and position (
rRinitial ) are:
V xinitial = 10 $ m / sec
The ball starts out traveling 10 m/sec horizontal to the right
rRinitial 0x 0y m= +$ $
The ball starts out at the origin, in the pitcher’s hand
To derive an expression for the velocity and position of the ball as a function of time,we begin by writing the acceleration as a function of time. Recall we assumed the onlyforce on the ball is due to gravity, which acts to accelerate the ball in the positive ydirection. Thus, we have:
ra x gy= +0 $ $ m / sec2
Acceleration is down at gravitational rate.
where g = gravitational acceleration at the Earth’s surface = 9.798 m/sec2
To obtain the instantaneous velocity at any time (t), we must integrate this equationwith respect to time. (Remember integrals from calculus? Basically, an integral is amathematical means of adding together lots of small changes over time to develop thetotal change.) Thus,
r rV t V gty x gtyinitial( ) $ $ $= + = +10 m / sec
This equation tells us that the ball will keep its initial horizontal velocity constant butwill speed up in the vertical direction due to gravity (which we already knew). Toobtain the instantaneous position of the ball at any time (t), we must once againintegrate this equation with respect to time so that:
r r rR t R V t gt y tx gt yinitial initial( ) $ $ $= + + = +1
210
1
22 2
We can now use these relatively simple equations to propagate the motion of thebaseball. Using a simple spreadsheet, we can determine the position and velocity ofthe baseball for each second for a total 10second flight. These values and a graphdepicting the trajectory are shown in Figure 49. Notice the trajectory we derived isexactly what we’d expect from experience. Often, people mistakenly refer to the
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baseball trajectories as “parabolic;” however, as we know, this is actually a smallsection of an ellipse.
TimeTime(sec)(sec)
X (m)X (m) Y (m)Y (m)
0 0 0.01 10 4.92 20 19.63 30 44.14 40 78.45 50 122.56 60 176.47 70 240.18 80 313.59 90 396.8
10 100 489.9
Figure 49: Results of baseball propagator: Using the simple equations of motion wederived for the baseball, we can use a spreadsheet to calculate the x and y positions ateach point in time for a 10second flight. Plotting these on a graph, we see the shape ofthe trajectory.
Note that the technique we developed here was for analytic propagation. An analyticpropagation technique has a closeform solution. In other words, given the initialconditions, we solve directly for the position and velocity at any future time using astraightforward “plug and chug” of the equations of motion. How accurate is thispropagation technique? As we shall see, this (and any other method) is only as goodas the assumptions we make. For example, we assumed no wind resistance, but weknow from experience that a sudden gust of wind could make this trajectory changeconsiderably. In the next section, we’ll apply this same basic technique to understandthe slightly more complicated motion of a satellite in orbit.
The TwoBody Propagator
Now we’ll develop a simple method we can use to propagate the position and velocityof a satellite known at a given time to predict its position and velocity at some time inthe future. By “simple method” we mean the restrictions placed on the complexity ofthe problem. In this case, one of the primary assumptions we will make is that thereare only two bodies concerned—the Earth and the satellite. Thus, we arrive at theterm used to describe this approach “the restricted twobody problem.”
Trajectory of the Baseball
0.050.0
100.0150.0200.0250.0
300.0350.0400.0450.0500.0
0 10 20 30 40 50 60 70 80 90 100
X (m)
Y (
m)
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Motion Analysis Process
The approach we’ll take is exactly the same as the one we used to describe the motionof the baseball in the previous section. Using the same motion analysis processdescribed, we can apply the three basic steps of our motion analysis model: define acoordinate system, make assumptions and identify equations of motion. Ourcoordinate system will be the geocentricequatorial coordinate system described earlier.The assumptions we make will “restrict” our solution to cases in which theseassumptions apply. Fortunately, this includes most of the situations we’ll encounter.We’ll assume that:
♦ satellites travel high enough above the Earth’s atmosphere so that the dragforce is small.
♦ the satellite won’t maneuver or change its path, so we can ignore the thrustforce.
♦ we’re considering the motion of the satellite close to the Earth, so we canignore the gravitational attraction of the Sun, the Moon or any third body.(That’s why we call this the twobody problem.)
♦ compared to Earth’s gravity, other forces such as those due to solarradiation, electromagnetic fields, etc. are negligible.
♦ the mass of the Earth is much, much larger than the mass of the spacecraft.
♦ the Earth is spherically symmetrical with uniform density and can thus betreated as a point mass.
After all these assumptions, we’re left with gravity as the only force affecting themotion of a satellite for the restricted twobody problem. This can be expressed (usingvector notation) as:
vF
GM M
RRsatellite
Earth satellite
satellite=
−2
$
The force due to gravity on a satellite depends on the mass of both the satellite andEarth and the distance to the Earth’s center. The direction is down, in the minusRdirection.
For convenience, we often combine GMEarth to derive an expression µEarth , known as theEarth’s gravitational parameter.
While Newton’s law of gravity describes the force on the satellite, we can use Newton’s2nd law of motion to describe the effect of that force to develop our equations ofmotion. From Newton’s second law, the force on the satellite can be expressed as:
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F M asatellite satellite satellite=
Setting the two expressions for force on the satellite equal to each other, we developan expression for the satellite’s acceleration:
va
RRsatellite
Earth
satellite=
− µ2
$
A satellite accelerates down [minusR direction] due to gravity. The further away fromthe Earth’s center, the smaller the gravitational force and, therefore, the smaller thecorresponding acceleration.
This equation says the motion of a satellite depends only on the distance between thecenter of the Earth and the satellite. It is independent of satellite mass. Substitutingthe more common notation for acceleration, we get the twobody Equation of Motion.
..
RmEarth
R2Rv =
− $
where
R = distance from center of Earth (km) to satellite
v&&R = 2nd derivative of position = acceleration (km/sec2)
µEarth
= gravitational parameter (km3/sec2)
$R = unit vector in direction of rR
TwoBody Orbit Propagation
What can the twobody equation of motion tell us about the movement of a satellitearound the Earth? Unfortunately, in its present form—a secondorder, nonlinear,vector differential equation—it doesn’t help us visualize anything about thismovement. So what good is it? To understand the significance of the twobodyequation of motion, we must first “solve” it using rather complex mathematical slightofhand. When the smoke clears, we’re left with an expression for the position of anobject in space in terms of some variables we already know.
( )R
a e
e=
−
+
1
1
2
cosν
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where
R = magnitude of rR
a = semimajor axis (km)
e = eccentricity (dimensionless)
ν = true anomaly (deg or radians)
This equation represents the solution to the restricted twobody equation of motionand describes the location, R, of a satellite in terms of a few constants and someinitial conditions. We can now make use of the solution to the twobody equation ofmotion to propagate the position of a satellite to any point in time.
In nice circular orbits, determining how long a satellite takes to travel from an initialposition to a future position is simple, because the satellite is moving at a constantspeed. However, in an elliptical orbit this speed varies (recall a satellite travels fastestat perigee and slowest at apogee, keeping total energy constant). As a result, we don’tknow how the true anomaly, ν, changes with time because it doesn’t changeuniformly. Here’s where Johannes Kepler came to the rescue. He developed thistechnique to describe the orbit of Mars. To describe motion in an elliptical orbit,Kepler began by defining the mean motion, n, which tells us the mean, or average,speed in the orbit. The mean motion is defined as:
nangle
time P a= = =
23
π µ
where
n = mean motion (rad/sec)
P = period (sec)
µ = gravitational parameter (km3/sec2)
a = semimajor axis (km)
Kepler figured out how to move n to a time in the future and, conversely, given afuture n, how to find out how long Mars would take to travel there. Kepler’s approachwas purely geometrical—he related motion on a circle to motion on an ellipse. To dothis, he had to invent a new angle called the mean anomaly, M, defined as:
M = nT (1)
where
M = mean anomaly (rad)
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n = mean motion (rad/s)
T = the time since last perigee passage (sec)
Mean anomaly is an angle that has no physical meaning and can’t be drawn in apicture. We’ll have to describe it mathematically. Expressing this equation in terms oftwo points in the same orbit:
Mfuture  Minitial = n(tfuture tinitial)  2kπ (2)
where
Mfuture = mean anomaly when the satellite is in the future position (rad)
Minitial = mean anomaly when the satellite is in the initial position (rad)
tfuture – tinitial = time of flight (TOF)
tfuture = time when the satellite is in the final position (e.g., 3:47 a.m.)
tinitial = time when the satellite is in the initial position (e.g., 3:30 a.m.)
k = the number of times the satellite passes perigee
To relate elliptical motion to circular motion, Kepler defined another new angle calledthe eccentric anomaly, E, so that he could relate M to E and then E to n. With all ofthese things defined, Kepler was able to develop his nowfamous equation, commonlycalled Kepler’s Equation. (For this equation to work, all angles must be in radians.)
M E e E= − sin (3)
where
E = eccentric anomaly (rad)
e = eccentricity
Kepler then related E to n using:
coscos
cosE
ee E
=+
+ν
1(4)
where
ν = true anomaly (rad)
And related ν to E through:
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coscos
cosν =
−−
E ee E1
(5)
Finally, we have all the equations needed to build a complete twobody propagator.The first problem, and the easiest, is finding the time of flight between two points inan orbit. Given ν
initial and ν
future, we simply go through the following steps:
♦ Use Equation (4) to solve for Einitial and Efuture
♦ Use Equation (3) to solve for Minitial and Mfuture
♦ Use Equation (2) to solve for the time of flight (tfuture – tinitial)
Note that if n is between 0° and 180°, so are E and M.
The second problem we can solve using Kepler’s method is far more practical. Thisinvolves determining a satellite’s position at some future time, tfuture, as shown in
Figure 50.
υυ Future
υυ Initial
Time of Flight
Figure 50: Time of flight on an elliptical orbit. The second problem Kepler tackledwas predicting the future position of a satellite knowing only its initial position.
This second problem is much trickier. We assume that we know where the satellite isat time tinitial, so we know νinitial. We start by finding Einitial, using Equation (4). Then we
find Minitial using Kepler’s Equation (3).
M E e Einitial initial initial= − sin
Now, because we know tfuture, using Equation (2), we can find Mfuture:
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( )M M n t t kfuture initial future initial− = − − 2 π
Great. We’re now on our way to finding nfuture, which tells us where the satellite will
be. So we go to Kepler’s Equation again to find Efuture. Let’s rearrange this equation
and put E on the left side:
E M e Efuture future future= + sin (6)
OOPS! Efuture is on both sides of the equation. This is called a transcendental equation
and can’t be solved for Efuture directly. In fact, almost every notable mathematician
over the past 300 years has tried to find a direct solution to this form of Kepler’sEquation without success. So we must resort to “math tricks” to solve for Efuture. The
“math trick” we’ll use is called iteration. To see how iteration works, think about thekids’ game Twenty Questions. In this game, your partner thinks of a person, place, orthing and you must guess what he’s thinking of. You’re allowed 20 questions(guesses) to which your partner can answer only “yes” or “no.” In seeking the rightanswer, a good player will systematically eliminate all other possibilities until only thecorrect answer remains.
A mathematical application of this can be seen using another transcendentalequation:
y = cos(y)
Because we can’t solve for y using algebra (we can’t get the y out of the cosinefunction to put all the ys on the left side), we must iterate. Begin by taking a guess atthe value for y, and take the cosine to see how close you were. Then take this as thenew value of y and use it for the next guess, and keep doing this iteration until thenew y equals the old y (or is, say, within 0.000001 radians of the old value).
Let’s try it to see what the answer for y really is. Take out your calculator and use π/4radians as your first guess for y. (Remember to set your calculator to use radians, notdegrees.) Keep pressing the cosine function button and you’ll see the value slowlyconverges to 0.739085 radians (about 43°). Presto—you’ve now solved thetranscendental equation y = cos(y) using iteration!
We can use this same iterative technique to solve Equation (6) for Efuture. It turns out
that the values for M and E are always pretty close together, even for the mosteccentric orbits, so let’s use Mfuture for our first guess at Efuture. Here’s the algorithm:
♦ Use Mfuture for the first E.
♦ Solve Equation (6) for a new Efuture.
♦ Use this new Efuture for the next guess for Equation (6).
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♦ Keep doing the previous step until Efuture doesn’t change by much (less than
about 0.0001 rad). At this point, the solution is said to have converged.
This brute force iteration method will solve Equation (6), but there are much bettermethods in use in most standard propagators.
Let’s quickly summarize what we’ve learned. If we know where we are in an orbit andwhere we want to be, we can use Kepler’s Equation to solve for the time it takes totravel to the place we want to be. The solution is very straightforward. If, however, weknow where we are and want to know where we’ll be at some future time, we can useKepler’s Equation to find that location only by iterating a transcendental equation foreccentric anomaly.
Orbit Perturbations
In deriving the twobody equation of motion, we had to assume that:
♦ gravity was the only force
♦ the Earth’s mass was much greater than the satellite’s mass
♦ the Earth was spherically symmetric with uniform density, so it could betreated as a point mass
These assumptions led us to the restricted twobody equation of motion:
v&& $RR
R+ =µ
20 (7)
The solution to this equation gives us the six classical orbital elements:
a = semimajor axis
e = eccentricity
i = inclination
Ω = longitude of the ascending node
ω = argument of perigee
ν = true anomaly
Under our assumptions, the first five of these elements remain constant for a givenorbit. Only the true anomaly, ν, varies with time as the satellite travels around itsfixed orbit. What happens if we now change some of our original assumptions? Otherclassical orbital elements besides ν will begin to change as well. Any changes to theseclassical orbital elements due to other forces are called perturbations. To see whichclassical orbital elements will change and by how much, let’s look at our firstassumption—gravity is the only force.
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Atmospheric Drag
Just as a sudden gust of wind changes the course of a football, atmospheric drag canaffect satellites in low Earth orbit (below about 1000 km). Let’s look at how dragaffects the orbital elements.
Because drag is a nonconservative force, it takes energy away from the orbit in theform of friction on the satellite. Thus, we expect the semimajor axis, a, to decrease.The eccentricity also decreases, since the orbit becomes more circular. Let’s see whythis is so. When a satellite in an elliptical orbit is at perigee, it has a greater speedthan it would if the orbit were circular at that same altitude. The drag decreases thespeed, making it closer to the circular orbit speed. That’s exactly what we see inFigure 51. It’s as if drag were giving the satellite a small negative velocity change, ordelta (∆) V, (slowing it down) each time it passes perigee.
successive orbits
original orbit
Earth’s atmosphere
∆∆V drag
Figure 51: The effect of drag on an eccentric, lowEarth orbit. As a satellite passesthrough the upper atmosphere at perigee, drag acts to gradually slow it down,circularizing the orbit until it eventually decays.
Drag is very difficult to model because of the many factors affecting the Earth’s upperatmosphere and the satellite’s attitude. The Earth’s daynight cycle, seasonal tilt,variable solar distance, the fluctuation of Earth’s magnetic field, the Sun’s 27dayrotation and the 11year cycle for Sun spots make precise modeling nearlyimpossible. The force of drag also depends on the satellite’s coefficient of drag andfrontal area, which can also vary widely, further complicating the modeling problem.
The uncertainty in these variables is the main reason Skylab decayed and burned upin the atmosphere several years earlier than first predicted. For a given orbit,however, we can approximate how the semimajor axis and the eccentricity change
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with time, at least for the short term. Different propagation techniques use differentmethods of estimating drag, with widely varying accuracy.
Earth’s Oblateness—“J2”
Columbus was wrong! The Earth isn’t really round. From space, it looks like a big,blue spherical marble, but if you take a closer look, it’s really kind of squashed. Thus,it can’t most accurately be treated as a point mass, as it is treated in the twobodyassumption. We call this squashed shape oblateness. What exactly does an oblateEarth look like? Imagine spinning a ball of jello around its axis and you can visualizehow the middle (or equator) of the spinning jello would bulge out—the Earth is fatterat the equator than at the poles. This bulge can be modeled by complex mathematics(which we won’t do here) and is frequently referred to as the J2 effect. J2 is a constantdescribing the size of the bulge in the mathematical formulas used to model the oblateEarth. Why “J2?” This term arises from the mathematical shorthand used todescribe Earth’s gravitational field. (Gravitational acceleration at any point on Earth iscommonly expressed as a geopotential function expressed in terms of Legendrepolynomials and dimensionless coefficients Jn—whew!). J2, J3 and J4 are the zonalcoefficients that depend on latitude. Of these, J2 is by far the most important; it isroughly 1000 times greater than either J3 or J4. However, for more precise modelingof the Earth’s oblateness, all three of these must be taken into account. In addition,other, higher order terms can be included in the model. These terms serve to slice theEarth into wedges that depend on longitude (sectoral terms) and slice it again intoregions of longitude and latitude (tesseral terms).
Let’s concentrate on the simplest and most profound case, J2. What effect does J2have on the orbit? Let’s look at Figure 52. Here it’s shown exaggerated; actually thebulge is only about 22 km thick. That is, the Earth’s radius is about 22 km longeralong the equator than through the poles.
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FJ2
FJ2
Figure 52: Diagram of Earth oblateness. The Earth’s oblateness, shown here as abulge at the equator (highly exaggerated to demonstrate the concept) causes a twistingforce on satellite orbits that change various orbital elements over time.
Let’s see if we can reason out how this bulge will affect the orbital elements. The forcecaused by the equatorial bulge is still gravity. Recall that gravity is a conservativeforce; therefore, the total mechanical energy in an orbit must be conserved. Totalmechanical energy depends on the orbit’s semimajor axis. Thus, as long as energyremains constant (i.e., no drag or other forces adding or stealing energy), thesemimajor axis also remains constant. It turns out that the eccentricity, e, alsodoesn’t change, although the explanation for this is beyond the scope of ourdiscussion here. Although you might expect the inclination to change because thebulge pulls on our orbit, it doesn’t! However, it does affect the orbit by changing theright ascension of the ascending node, Ω, and moving the argument of perigee, ω,within the plane. That’s not very intuitive, but it’s like a force acting on a spinningtop. If you stand a nonspinning top on its point, gravity causes it to fall over. If youspin the top first, gravity still tries to make it fall but, because of its angularmomentum, it begins to swivel—this motion is called precession. Let’s examine theeffect of precession on the ascending node and the argument of perigee more closely.
How J2 Affects the Right Ascension of the Ascending Node, Ω
The gravitational effect of this equatorial bulge slightly perturbs the satellitebecause the force no longer originates from the center of the Earth. This causesthe plane of the orbit to precess (like the spinning top), resulting in a movement ofthe ascending node, ∆Ω. This motion is westward for posigrade orbits (inclination<90°) and eastward for retrograde orbits (inclination > 90°).
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Figure 54 shows this nodal regression rate, &Ω , as a function of inclination andorbital altitude. Let’s look more closely at this figure. It shows that the higher thesatellite is, the less effect the bulge has on the orbit. This makes sense becausegravity decreases with the inverse square of the distance (see Newton’s Law ofGravitation). It also says that if the satellite is in a polar orbit (center of the graph),the bulge has no effect. The greatest effect occurs at low altitudes with lowinclinations. This makes sense, too, because the satellite travels much closer tothe bulge during its orbit, and thus is pulled more by the bulge. For lowaltitudeand lowinclination orbits, the ascending node can move as much as 9° per day(lower left corner and upper right corner of Figure 53).
Nodal Regression Rate as a Function of Inclination and Eccentricity
10.0
5.0
0.0
5.0
10.0
0 20 40 60 80 100 120 140 160 180
Inclination (deg)
No
dal
Reg
ress
ion
Rat
e (d
eg/d
ay)
100km altitude circular orbit
4000km x 100km altitude elliptical orbit
Figure 53 Nodal Regression Rate, &Ω . The nodal regression rate caused by the Earth’sequatorial bulge. Positive numbers represent eastward movement; negative numbersrepresent westward movement. The less inclined an orbit is to the equator, the greaterthe effect of the bulge. The higher the orbit, the smaller the effect.
How J2 Affects the Argument of Perigee, ω
Figure 54 shows how perigee location rotates for an orbit with a perigee altitude of100 km depending on the inclination for various apogee altitudes. This perigeerotation rate, &ω , is difficult to explain physically, but it could be derivedmathematically from the equation for J2 effects on perigee location. With thisperturbation, the major axis, or line of nodes, rotates in the direction of satellitemotion if the inclination is less than 63.4° or greater than 116.6°. It rotatesopposite to satellite motion for inclinations between 63.4° and 116.6°.
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Perigee Rotation Rate as a Function of Inclination and Eccentricity
5.0
0.0
5.0
10.0
15.0
20.0
0 20 40 60 80 100 120 140 160 180
Inclination (deg)
Per
igee
Ro
tati
on
Rat
e (d
eg/d
ay)
100km altitude circular orbit
4000km x 100km altitude elliptical orbit
Figure 54: Affects of J2 on argument of perigee. The perigee rotation rate caused bythe Earth’s equatorial bulge depends on inclination and altitude at apogee.
SunSynchronous and Molniya Orbits
The effects of the Earth’s oblateness on the node and perigee positions give rise totwo unique orbits that have very practical applications. The first of these, the Sunsynchronous orbit, takes advantage of eastward nodal regression at inclinationsgreater than 90°. Looking at Figure 55, we see that the ascending node moveseastward about 1° per day at an inclination of about 98° (depending on thesatellite’s altitude).
Coincidentally, the Earth also moves around the Sun about 1° per day (360° in365 days), so at this Sunsynchronous inclination, the satellite’s orbital plane willalways maintain the same orientation to the Sun. This means the satellite can seethe same Sun angle when it passes over a particular point on the Earth’s surface.As a result, the Sun shadows cast by features on the Earth’s surface won’t changewhen pictures are taken days or even weeks apart. This is important for remotesensing missions such as reconnaissance, weather and monitoring of the Earth’sresources, because they use shadows to measure an object’s height. Bymaintaining the same Sun angle day after day, observers can better track changesin weather, terrain or manmade features.
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Sun lineSun line
Sun line
Sun line
Sun angle
Sun angle
Sun angle
Sun angle
orbit plane
Earth moves around theSun at ~1 deg/day
orbit planerotates at ~1 deg/day due to Earth’soblateness
Figure 55: SunSynchronous Orbit. Sunsynchronous orbits take advantage of the rateof change in right ascension of the ascending node caused by the Earth’s oblateness. Bycarefully selecting the proper inclination and altitude, we can match the rotation of Ω withthe movement of the Earth around the Sun. In this way, the same angle between theorbit plane and the Sun can be maintained without using rocket engines to change orbit.Such orbits are very useful for remote sensing missions that want to maintain the sameSun angle on targets on the Earth’s surface.
The second unique orbit is the Molniya orbit, named after the Russian word forlightning (as in “quickaslightning”). This is a 12hour orbit with high eccentricity(about e = 0.7) and a perigee location in the Southern Hemisphere. The inclinationis 63.4°—why? Because at this inclination, the perigee doesn’t rotate so thesatellite “hangs” over the Northern Hemisphere for nearly 11 hours of its 12hourperiod before it whips “quick as lightning” through perigee in the SouthernHemisphere. Figure 56 shows the orbit and ground tracks for a Molniya orbit. TheRussians used this orbit for their communications satellites because they didn’thave launch vehicles large enough to put them into geosynchronous orbits fromtheir far northern launch sites. Molniya orbits also offer better coverage oflatitudes above 80° north.
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Figure 56: Molniya orbit and ground tracks. Molniya orbits take advantage of the factthat ω, due to Earth’s oblateness, is zero at an inclination of 63.4°. Thus, apogee staysover the Northern Hemisphere, covering high latitudes for 11 hours of the 12hour orbitperiod.
Other Perturbations
Other perturbing forces can affect a satellite’s orbit and its orientation within thatorbit. These forces are usually much smaller than the J2 (oblate Earth) and dragforces but, depending on the required accuracy, satellite planners may need toanticipate their effects. These forces include:
♦ Solar radiation pressure, which can cause longterm orbit perturbations andunwanted satellite rotation.
♦ Thirdbody gravitational effects (Moon, Sun, planets, etc.), which can perturborbits at high altitudes and on interplanetary trajectories.
♦ Unexpected thrusting caused by either outgassing or malfunctioningthrusters, which can perturb the orbit and cause satellite rotation.
Dealing with Perturbations
Understanding and modeling orbit perturbations is one of the primary activities ofastrodynamics. Even very early space pioneers such as Kepler and Newton spentconsiderable effort grappling with the various forces that disturb a satellite from puretwobody motion. Let’s begin by classifying perturbations with respect to their relativeeffects on orbital elements. Perturbations can cause both secular and periodicchanges to orbital elements. Secular perturbations are those that cause elements tosteadily diverge over time. Periodic perturbations are those that impart a sinusoidalvariation in elements over time. Shortterm periodic perturbations are those with aperiod less than the orbit period. Longterm periodic perturbations are those with a
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period greater than one orbit period. We can now look at two techniques for modelingboth secular and periodic perturbations. The relative effects of these different typesare illustrated in Figure 57.
Long term effects on orbital elements for various types of perturbations
0
1
2
3
4
5
6
7
8
90 1 2 3 4 5 6 7 8
Orbit Periods
Orb
ital
ele
men
t va
riat
ion
(a
rbit
rary
un
its)
Secular
Short term Periodic
LongtermPeriodic
Figure 57: Types of orbit perturbations. Orbit perturbations are categorized based ontheir longterm effects on orbital elements.
General Perturbations Techniques
General perturbations techniques are those that generalize the effects on orbitalelements in order to develop analytic expressions allowing for direct computation. Inthe grossest sense, general perturbation techniques apply “fudge factors” to thesimple twobody solution to account for the effect of different perturbation sources.For example, returning to our baseballthrowing astronaut scenario, we could modelthe drag on the baseball using:
D V A=1
22ρ
where
D = Drag (N)
ρ = air density (kg/m3)
V = baseball velocity (m/sec)
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A = baseball crosssectional area (m2)
We could substitute this expression into our baseball equations to derive newequations of motion that would account for the general effects of drag. Even with thisadditional complexity, the equations could still be solved analytically.
One of the most widely used propagators was developed by the North AmericanAerospace Defense Command (NORAD) to track the 8000plus satellites and spacejunk in orbit around the Earth. Called Merged Simplified General Perturbations4,MSGP4, this technique uses the generalized approach to model orbit perturbations.
Special Perturbations Techniques
In contrast, special perturbation techniques are based on special case assumptionsabout the orbit scenario that allow for more detailed modeling of individualperturbation sources.
Twoline Element Sets
One of the most commonly used methods of communicating orbital parameters is the2line element sets generated by NORAD in Cheyenne Mountain, Colorado (literally, inCheyenne Mountain!). It is important to note that TLEs were developed specifically foruse with the MSGP4 propagator! Using TLEs with any other propagator mayinvalidate some of the builtin assumptions.
These elements contain many of the same elements as the classical orbital elements,along with some additional parameters for identification purposes and for use inmodeling perturbations in the MSGP4 propagator.
STK Propagators
In selecting the “best” propagator to use for a given application, it is important toconsider the assumptions on which they are based. The temptation is to use the most“accurate” propagation model available. However, this can lead to false accuracy,especially for very long term propagation over which time even the best models canbreak down. The relative accuracy among various propagators can vary widelydepending on the scenario. For example, a geostationary spacecraft is well above mostatmospheric drag and J2 perturbations. Therefore, a shortterm difference betweenthe twobody propagator and a more complex technique could be relatively small.However, for a spacecraft in lowEarth orbit, the shortterm differences between thetwo solutions could be significant. The objective is to choose the most appropriatepropagation scheme for a given application. Unfortunately, any propagation techniqueis simply an attempt to model events in the real world. Regardless of the techniquechosen, only frequent tracking of an orbit can guarantee that the predicted orbitalparameters will match the real world.
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TwoBody
The twobody propagator or Keplerian motion propagator uses the same basictechnique outlined in the twobody equation of motion development. This techniqueassumes the Earth is a perfect sphere and the only force acting on a satellite isgravity. This propagator doesn’t account for any perturbations.
J2
The J2 propagator accounts for the 1st order effects of J2 Earth oblateness. This effectcauses secular changes to the orbital elements over time.
J4
The J4 propagator accounts for 1st and 2nd order J2 effects as well as 1st order J4effects. J3, which causes longterm periodic effects, is not modeled. Because the 2nd
order J2 and 1st order J4 effects are very small, you’ll see very little differencesbetween the J2 and J4 propagators for most orbits considered.
MSGP4
MSGP4 stands for Merged Simplified General Perturbations4. It is one of the mostwidely used propagators in the industry. This technique uses the generalizedapproach to model orbit perturbations, including both secular and periodic variationssuch as Earth oblateness, solar and lunar gravitational effects and drag.
It is important to understand the purpose for which MSGP4 was developed. NORADwanted a simple propagator that would provide acceptable results for a wide variety oftracking tasks, from tracking highpriority military satellites to keeping tabs on spacejunk. Given the over 8000 objects NORAD must track, a technique was needed thatwould not be computationally intensive (is was first developed back when computerswere much slower than today). Furthermore, there are far fewer ground tracking sitesthan there are objects to track. Thus, it is important that the propagated solution begood enough to ensure the tracking radar can find a specific object the next time itgets around to tracking it (which, for some very low priority objects like pieces ofrocket boosters, may be days or even weeks).
Because MSGP4 is a generalized approach, it is specifically tailored to a given set ofinputs: the TLE sets that contain parameters that make the analytic calculationsvalid. For best results, MSGP4 should always be used with TLEs. Likewise, NORADgenerated TLEs should only be used in the MSGP4 propagator.
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HPOP
HPOP is the High Precision Orbit Propagator. As its name implies, it uses a powerfulpropagation technique to incorporate sophisticated orbit perturbation models. HPOPuses a variety of highfidelity models including:
♦ Joint Gravity Model (JGM) 2—a highly precise model of the Earth’soblateness.
♦ Lunar/solar gravitational effects—based on U.S. Naval Observatory data.Accurate to within 0.03 arc seconds.
♦ Atmospheric drag effects— using either the 1971 Jacchia or the HarrisPriester model, which takes into account daily variations in the height of theatmosphere due to solar heating among other parameters.
♦ Solar radiation pressure—yes, sunlight produces a small force on any exposedsurface. This force varies depending on how reflective the surface is—amirrored surface is more reflective than a black surface.
Depending on the application, HPOP can deliver accuracy on the order of 10 metersper orbit. But beware of false accuracy, always remember—“garbage in, garbage out.”To get this level of accuracy, your initial orbital elements must be at least thisaccurate to start with. Putting NORADgenerated TLEs into HPOP will not necessarilygive you a better solution. No propagator can create accuracy, at best it can onlyminimize the long term dispersions due to inherent limitations in our ability to modelthe effects of perturbations.
Great Arc
The Great Arc propagator allows the user the model the flight path of a vehicle flyingclose to Earth. By providing way points and speed, STK uses Great Arc to predictwhere and when it will be next. The propagation scheme is essentially the same as theTwoBody propagator, no perturbations are assumed.
Ballistic
The Ballistic propagator is a variation of the TwoBody propagator for use withballistic trajectories. These are the trajectories used by artillery shells, suborbitalsounding rockets and ballistic missiles, allowing the user to predict impact points ordetermined required velocity to reach a certain point. The propagation “engine” is thesame as the TwoBody propagator, no perturbations are modeled.
LOP
The Longterm Orbit Predictor (LOP) allows accurate prediction of a satellite’s orbitover many months or years. This is often used for long duration mission design, fuelbudget definition, and endoflife studies. For performance reasons, it is impractical
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to compute the longterm variation in a satellite’s orbit using high accuracy, smalltime step, propagators that compute a satellite’s position as it moves through itsorbit. LOP exploits a “variation of parameters” approach which integrates analyticallyderived equations of motion computing the average effects of perturbations over anorbit. This approach allows large multiorbit time steps and typically improvescomputational speed by several hundred times while still offering high fidelitycomputation of orbit parameters.
Lifetime
Lifetime estimates the amount of time a low Earth orbiting satellite can be expected toremain in orbit before the drag of the atmosphere causes reentry. While thecomputational algorithms are similar to those implemented in the Longterm OrbitPredictor, there are some important differences. First, a much more accurateatmospheric model is implemented to compute the drag effects. The gravitationalmodel for the Earth, however, is significantly simplified since the inclusion of thehigher order terms doesn’t impact orbit decay estimates.
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SATELLITE ACCESS
Overview
Line of sight
♦ Why is line of sight important for satellite viewing?
Communication Architecture
♦ What are the elements that make up a space mission communicationarchitecture?
Communication Links
♦ What are the communication paths used by satellites and groundstations?
Understanding Access
♦ What is meant by “satellite access?”
Describing Access
♦ How do I explain and quantify satellite access?
Line of Sight
Standing on a beach, looking out over the ocean on a clear day, you can seeright to the edge of the horizon, which is about 8 miles away. If you were towatch a ship sailing away from you, you would notice that the hull woulddisappear first, followed by the top of the mast The taller the mast, the furtherthe ship could be from the shore before disappearing completely from sight. Anobject is in your line of sight if you can draw a straight line between yourselfand the object without any interference, such as a mountain or a bend in aroad. An object beyond the horizon is below our line of sight and, therefore, canbe difficult to communicate with. Early methods of longdistancecommunications increased the effective line of sight by employing methodssuch as smoke signals or other means. Because the line of sight was raised sothat others could see, or receive the message being sent, communicationamong objects that didn’t really have a direct line of sight was achieved.
At the beginning of the 20th Century, radio engineers discovered that certainfrequencies could be bounced off the ionosphere, greatly extending the effectiveline of sight for communications and creating a new “radio horizon” far beyondthe more limited visual horizon. Today’s communications satellites take thisbasic principle to the extreme. Groundbased operators can “bounce” radio
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communications off satellites stationed in geosynchronous orbit, creating avirtual line of sight extending half a world away. Furthermore, by bouncingsignals between satellites, this virtual line of sight can be extended to cover theentire global community.
Satellite access is the problem of determining when, where and for how long asatellite (or any number of objects you may be interested in) is within line ofsight of other objects.
Communications Architecture
To understand the satellite access problem more clearly, let’s begin byreviewing the players in the access problem. Figure 58 illustrates the elementsthat make up a communications architecture.
crosslink
crosslink
return link
forward link
forward link
return linkdownlink
uplin
k
Figure 58 Communications Architecture. The communications architectureconsists of space and groundbased elements tied together by communicationspaths or links.
The communications architecture has four elements:
Spacecraft
The spaceborne elements of the system.
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Ground stations
The Earthbased antennas and receivers that talk to the spacecraft. These aretypically remote tracking sites or users of mission data.
Control Center
The command center that controls the spacecraft and all other elements of thesystem.
Relay satellites
Additional satellites that link the primary spacecraft with the ground stationsand control center.
Communications Links
Information moves among the elements of the communications architectureusing various communication paths or links.
Uplink
Data sent from a groundbased station to the primary satellite.
Downlink
Data sent from the primary satellite to a ground station.
Forward link
Data sent from a ground station to the primary satellite via a relay satellite.
Return link
Data sent from the primary satellite to a ground station via a relay satellite.
Crosslink
Data sent through either the forward or return link between the primary satelliteand a relay satellite.
Understanding Access
The simplest example of a satellite access problem is that between a satelliteTV dish and a directbroadcast geosynchronous satellite. As a user, you justwant to point your dish and start watching the big game. Thus, you’re onlyinterested in downlink.
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Figure 59 Downlinking. As a direct TV subscriber, you would only be interestedin downlinking.
For satellites in geostationary orbit, the geometry and dynamic nature of bothuplink and downlink is very stable. It is this stability that make geostationarysatellites so useful for pointtopoint message relaying. You can set up youdish, point it at a predetermined point in the sky and pretty much forget aboutit. Because the ground track of a geosynchronous orbit is at most a tiny figure8 centered on the equator, a line of sight between the dish and the satellite isalmost constant (at least constant enough to ensure uninterrupted broadcast ofthat big game!).
Describing Access
So how do you know where to point the satellite dish? This depends on twoimportant pieces of information:
♦ Your location (latitude and longitude)
♦ The satellite’s location (orbital elements)
From this information, STK can determine the azimuth and elevation settingsfor your dish. These two important parameters are defined below.
♦ Azimuth  The compass direction between the ground site and thesatellite direction, e.g., due south would be 180°.
♦ Elevation  Angle measured from the local horizontal to the satellitedirection, e.g., directly overhead would be 90°.
In addition, STK also computes range—the distance between the dish and thesatellite. While range is not so important to the average satellite TV user, itbecomes very important for communications engineers who must ensure thereis sufficient transmission power to effectively carry the signal across thisdistance.
These three parameters are fairly constant for a geosynchronous satellite;however, for any other satellite, we must include an additional parameter—
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time. Imagine it is exactly 13:20 (1:20 p.m.) local time and you want to point aradar antenna at an airplane flying over your position. The plane is initiallydue south of you, flying north but out of sight below the horizon. As the planefirst comes into view, azimuth will be 180° and the elevation 0°. As itcontinues to fly north, azimuth will stay constant (disregarding Earth rotation)and the elevation angle will increase. As it flies overhead, the azimuth anglewill switch around to 0° (it is now north of your position) and the elevationangle will gradually decrease until the plane once again drops from view belowthe horizon. If we kept track of the azimuth and elevation viewing angles to theplane at 10minute intervals we could build a simple table, or access report, asshown below.
Time Azimuth(deg)
Elevation(deg)
Comment
13:20 180 45 Below horizon (south of you)
13:30 180 0 Just coming into view
13:40 180 45 Well above horizon
13:50  90 Directly overhead (azimuth isundefined)
14:00 0 45 Azimuth has switched around, plane isnow north of your position.
14:10 0 0 Plane drops below horizon, out of view
The geometry with respect to a plane flying directly overhead is relatively easyto visualize. However, if you’re faced with the problem of a satellite in a highlyeccentric orbit flying over a position northwest of you on a descending node,things become much more complicated. Fortunately, STK works out thegeometry for you. Figure 60 shows a relatively simple satellite access geometry.
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Figure 60 Satellite Access. Satellite access refers to the problem ofdetermining the geometry and timing of lineofsight between various ground andspacebased objects.
Access reports can be easily generated using STK; the report provides azimuth,elevation and range (AER) data for specified time intervals between whateverobjects you choose to define. Cumulative access time or duration can also bereported.
Access information becomes even more complicated when multiple objectsmust be taken into account. For example, if you are relaying informationbetween various ground stations and relay satellites or using an entireconstellation of satellites, the geometry can become very complex. To handlethese tasks, STK allows you to link together various objects to create a “chain”for which access information can be determined. Figure 61 shows a morecomplex series of “chained” objects. If one of these “objects” is the Sun, orbitlighting—a critical parameter for power and thermal management—can becomputed.
Figure 61 Chained objects. A series of ground and spacebased objects canbe “chained” together, allowing STK to compute access information between allof them. In this picture, access among a facility, relay satellite and a secondfacility is shown.
RECOMMENDED READING
For a more detailed explanation of the topics in this primer as well as anintroduction to the space environment, spacecraft design, rockets and systems,we recommend:
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Understanding Space: An Introduction to Astronautics, Sellers, 1994,McGrawHill.
To purchase a copy of Understanding Space, please contact McGrawHill atwww.mcgrawhill.com, www.mhhe.com, or 8003383987 : The McGrawHillCompanies, >Order Services, PO Bos 545, Blacklick, Ohio. ISBN: 0070570275.
Understanding Space is part of the Space Technology Series, a cooperativeactivity of the United States Department of Defense and National Aeronauticsand Space Administration. Series editor is Dr. Wiley J. Larson. Other books inthe series include:
Fundamentals of Astrodynamics and Applications, Vallado, 1997, McGrawHill.
Space Mission Analysis and Design, 2nd edition, Larson & Wertz (ed.), 1996,Kluwer and Microcosm.
Space Propulsion Analysis and Design, Humble & Larson, 1995, McGrawHill.
Reducing Space Mission Cost, Larson & Wertz (ed), 1996. Kluwer andMicrocosm.
CostEffective Space Mission Operations, Boden and Larson, 1996, McGrawHill.
Spacecraft Structures and Mechanisms: From Concept to Launch, Sarafin andLarson, 1995, Kluwer and Microcosm.
For more information about these books, please contact Kluwer atwww.wkap.nl, McGrawHill at www.mcgrawhill.com or 8003383987, orMicrocosm at www.microcosm.com.
Future books in the series:
Modeling and Simulation: In Integrated Approach to Development anOperations, Cloud and Rainey.
Human Space Mission Analysis and Design, Connally, Giffen and Larson.
Other recommended reading:
Fundamentals of Astronautics, Bate, Mueller & White, 1971, Dover.
1997 Microcosm Directory of Space Technology Data Sources, 1997, Wertzand Dawson, Kluwer and Microcosm.