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Welcome To.. Satellite Tool Kit Astronautics Primer by Jerry Jon Sellers Based on Understanding Space: An Introduction to Astronautics Copyright 1996 McGraw-Hill Inc. by Jerry Jon Sellers Wiley J. Larson (editor) Your Understanding of Space Starts Here! COPYRIGHT NOTICE McGraw-Hill owns the copyright to the book Understanding Space: An Introduction to Astronautics. 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 to reproduce this material without permission. 1 STK Astronautics Primer OBJECTIVE The objective of this primer is to provide you with a fundamental reference for key concepts 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 various astronautics assignments including the NASA Johnson Space Center, where he worked in Space Shuttle Mission Control (guidance and navigation) and the U.S. Air Force Academy where he served on the faculty of the Department of Astronautics. He was a distinguished graduate from the U.S. Air Force Academy in 1984 and has earned a Master’s Degree in Physical Science from the University of Houston, Clear Lake, a Master’s Degree in Aeronautics and Astronautics from Stanford University and a Ph.D. in Satellite Engineering from the University of Surrey, UK. He currently works 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 (two-line element sets), ground tracks and how they relate to space missions 2 STK Astronautics Primer Predicting Orbits The “nuts and bolts” of predicting real-world 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 more and more on satellites for a variety of needs. Daily weather forecasts, instantaneous world-wide communication, and a constant ability to keep an eye on not-so-friendly neighbors 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 a conceptual overview of important topics in orbital mechanics. Understanding these key concepts will enhance your insight into the science behind Satellite Took Kit and better equip you to apply these concepts to practical problems in space. We’ll begin with a brief overview of space, space missions and space history. Then we’ll get into the details of orbital mechanics to see how you can use STK to plot your path to the stars. Why is space so useful? Getting into space is dangerous and expensive. So why bother? Space offers several compelling 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—free-fall and abundant resources make space the true final frontier Global Perspective Space offers a global perspective. As you can see in Figure 1, the higher you are, the more you can see. For thousands of years, kings and rulers took advantage of this fact by putting lookout posts atop the tallest mountains to survey more of their realm and fend off would-be attackers. Throughout history, many battles have been fought to “take the high ground.” Space takes this quest for greater perspective to its ultimate end. From the vantage point of space, we can view large parts of the Earth’s surface. Orbiting satellites can thus serve as “eyes in the sky” to provide a variety of useful services. 3 STK Astronautics Primer Figure 1: Global perspective. From space, satellites can observe large-scale features on the Earth, track weather patterns, monitor the environment and view widely separated points simultaneously, allowing them to communicate. Universal Perspective Space offers a clear view of the heavens. When we look at stars in the night sky, we see their characteristic twinkle. This twinkle, caused by the blurring of “starlight” as it passes 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 for astronomers who need access to all the regions of the spectrum to fully explore the universe. By placing observatories in space, we can sit above the atmosphere and gain an unobscured view of the universe, as depicted in Figure 2. The Hubble Space Telescope and the Gamma Ray Observatory are armed with sensors operating far beyond the range of human senses. Already, results from these instruments are revolutionizing our understanding of the cosmos. Figure 2: Seeing beyond the clouds. Earth-based astronomy is obscured by the atmosphere. Astronomers don’t like the “twinkle” of star light. Some wavelengths are 4 STK Astronautics Primer completely blocked. Space-based astronomy opens the door to a whole new perspective on the universe. A Unique Environment Space offers a unique environment with many advantages. ♦ Free-fall environment—enables developing advanced materials ♦ Abundant resources—solar energy and minerals Gravity makes some manufacturing processes difficult if not impossible. To form certain new metal alloys, for example, we must blend two or more metals in just the right proportion. Unfortunately, gravity tends to pull the heavier metal to the bottom, making a uniform mixture difficult to obtain. But space offers the solution. A manufacturing plant in orbit is literally falling toward Earth but never hitting it. This is a condition known as free-fall (NOT zero gravity). In free-fall there are no contact forces on an object, so it is said to be weightless. Unencumbered by the weight felt on the Earth’s surface, factories in orbit can create exotic new metals for computers or other advanced technologies, as well as for promising new pharmaceutical products to battle disease on Earth. Figure 3: Early free-fall manufacturing. In the 16th century, Italian weapons makers developed a secret way of making lead shot for muskets. By dropping liquid lead from a “shot tower,” they found near-perfect spheres would form as the molten lead cooled and hardened in free fall. Space also offers abundant resources. While some on Earth argue about how to carve the pie of Earth’s resources into smaller and smaller pieces, others have argued that 5 STK Astronautics Primer example, is known to be rich in oxygen and aluminum. The oxygen could be used as rocket propellant and for humans to breathe. Aluminum is an important metal for various industrial uses. These resources, coupled with the human drive to explore, mean the sky is truly the limit! Space Applications Let’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 into orbits with periods of 24 hours, 36,000 km above the equator, exactly matching the rotation rate of the Earth. These geostationary orbits could serve as communication hubs to link together remote parts of the planet. With the launch of the first experimental communications satellite, Echo I, into Earth orbit in 1960, Clarke’s fanciful idea showed promise of becoming reality. Although Echo I was little more than a reflective balloon in low-Earth 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 government communications and for most domestic cable television. Through satellite technology, relief workers can now stay in constant contact with their organizations, enabling them to better distribute aid to refugees hungry for food. In addition, our modern military now relies almost totally on satellites to communicate with forces deployed world-wide. Without satellites, global communication as we know it today would not be possible. Remote Sensing Missions Satellites operating from the global perspective of space have also made possible the science of remote sensing. Remote sensing is the act of observing Earth and other objects from space. For decades, military “spy satellites” have kept tabs on the activities of potential adversaries using remote-sensing technology. This same technology 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 aid farmers, resource managers, and planners on Earth. In countries where the failure of a harvest may mean the difference between bounty and starvation, spacecraft have helped planners manage scarce resources and head off potential disasters before insects or other blights could wipe out an entire crop. Weather forecasting is a further application of remote-sensing technology—one we’ve all come to rely on. Overall, we’ve come to rely more and more on the ability to monitor and map our entire planet. As 6 STK Astronautics Primer Satellites can now “spy” on crops, ocean currents, and natural resources to aid farmers, resource managers, and planners on Earth. In countries where the failure of a harvest may mean the difference between bounty and starvation, spacecraft have helped planners manage scarce resources and head off potential disasters before insects or other blights could wipe out an entire crop. Weather forecasting is a further application of remote-sensing technology—one we’ve all come to rely on. Overall, we’ve come to rely more and more on the ability to monitor and map our entire planet. As the pressure builds to better manage scarce resources and assess environmental damage, we’ll call upon remote-sensing spacecraft to do even more. Figure 4: Satellite remote sensing. From the vantage point of space, we can plan urban development and plot the course of dangerous storms. Space-based Navigation Early seafarers looked to the stars to guide their way. Modern seafarers look only as far 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’re heading and how fast you’re going. 7 STK Astronautics Primer Figure 5: Global Positioning System (GPS). GPS allows Earth-based users armed with a simple, hand-held receiver to triangulate from a constellation of 24 satellites. They can then determine their location to within a few meters and velocity, and a few m/sec anywhere on Earth. Exploration Perhaps the most exciting missions are those which explore the unknown. Missions such as the Magellan spacecraft that orbited Venus with a powerful radar to peel back the clouds of this once mysterious planet are a good example. A computer-enhanced image taken by Magellan is shown in Figure 6. These types of missions push back the boundaries of human knowledge, giving us new insight into planetary formation, weather and other important processes at work back here on Earth. 8 STK Astronautics Primer Figure 6: Magellan at Venus. The powerful synthetic aperture radar on NASA’s Magellan spacecraft pierced the thick clouds of Venus, giving us the first details of the planet’s surface. (Photo courtesy of NASA.) Describing Space Missions Space missions seem complex, and they are to a certain extent, but if you look at them 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 once we get there. ♦ Users - The people or systems that use data or services provided by the satellite or satellites. ♦ Operators -The people who manage and run the mission from the ground. ♦ Mission Operations Concept -How users, operators, ground and space elements all work together to make a mission successful. All these come together to form the tangible elements of what is collectively called the Space Mission Architecture. These elements are depicted in Figure 7; each one is defined in the subsections following. 9 STK Astronautics Primer Communications Network The Space Mission Trajectories & Orbits Space Transportation Payload Bus Space Operations Payload Spacecraft Figure 7: Space mission architecture. The key to understanding how missions are built 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 control satellites and the other elements that make up a space mission. Space operations are performed by teams of people located at tracking sites and control centers around the world. Spacecraft Bus and Payload A spacecraft has two basic parts, a payload (or payloads) and a bus. The payload includes space-borne people and instruments that perform the primary mission. The spacecraft bus provides 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 good old-fashioned school bus, as shown in Figure 8. It contains all of the same types of systems needed to support a spacecraft. 10 STK Astronautics Primer Horn, radio & driver (communications & data handling) Body & frame (Structures) Steering (space vehicle control) Radiator, air conditioning & heater (environment control & life support ) Battery & alternator (electrical power) engine & drive train (space transportation) Passengers (payload) Figure 8: The spacecraft “bus.” A spacecraft has all the basic systems found in a regular school bus. Trajectories and Orbits A trajectory is any path an object follows through space. An orbit is a special type of repeating 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 a planet, 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 as the field of view. The projection of this field of view onto the Earth’s surface creates a swath width for the sensor as it sweeps around the Earth on its orbit. These two parameters are illustrated in Figure 10. 11 STK Astronautics Primer field of view swath width Figure 10: Field of view and swath width. The height of the orbit and the sensor field of 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 in Figure 11. fin al or bit parking orbit r orbit transfe Figure 11: Orbit types. Different types of orbits include the parking orbit, the transfer orbit and the final or mission orbit. A satellite normally begins its life in a temporary parking orbit. From there, an upper stage rocket is used to boost the satellite onto a transfer 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 its final mission orbit. Normally, this consists of a booster, such as the Space Shuttle or 12 STK Astronautics Primer Ariane, an upper stage, such as the Inertial Upper Stage (IUS), and onboard thrusters for final maneuvers and station keeping. The Space Shuttle, shown in Figure 12, is one type of complete space transportation system. Figure 12: The Space Shuttle. Space transportation includes the systems that put the spacecraft in orbit, keep it there, and rotate and move it if necessary. Space transportation systems develop the velocity needed to obtain and stay in orbit. Space boosters are divided into stages that provide incremental changes in velocity and are then discarded. Communications Network A space mission is more than just rockets and satellites. An entire system of ground and on-orbit assets are needed to track, command and control all aspects of the mission. This communications network ties together various links needed to deliver bus telemetry and payload data to operators and users, as shown in Figure 13. 13 STK Astronautics Primer Tracking & Data Relay Satellite (TDRS) relay satellite primary aircraft 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, ground controllers and users in a complex web of links that transfers data among the various mission element nodes. 14 STK Astronautics Primer 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 aid of telescopes and other instruments, humans continued their struggle to bring order to the heavens. With order came some understanding and a concept of our place in the universe. Thousands of years ago, priests of ancient Egypt and Babylon carefully observed the heavens to plan religious festivals, to control the planting and harvesting of various crops, and to understand at least partially the realm in which they believed many of their gods lived. Later, philosophers such as Aristotle and Ptolemy developed complex theories to explain and predict the motions of the Sun, Moon, planets and stars. The theories of Aristotle and Ptolemy dominated the world of astronomy and our understanding of the heavens well into the 1600s. Combining ancient traditions with new observations and insights, natural philosophers such as Copernicus and Kepler offered rival explanations from the 1500s onward. Using their models and Isaac Newton’s new tools of physics, astronomers in the 1700s and 1800s made several startling discoveries, including two new planets—Uranus and Neptune. Let’s briefly explore some of these major contributors to our early understanding of space and orbits. Copernicus With the Renaissance and humanism came a new emphasis on the accessibility of the heavens to human thought. Nicolaus Copernicus (1473–1543), a Renaissance humanist and Catholic clergyman, reordered the universe and enlarged man’s horizons. 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 a year. Copernicus further observed that, with respect to a viewer located on the Earth, the planets occasionally appear to back up in their orbits as they move against the background of the fixed stars. Ptolemy and others resorted to complex combinations of circles to explain this backward motion of the planets, but Copernicus cleverly explained that this motion was simply the effect of the Earth overtaking, and being overtaken by, the planets as they all revolved about the Sun. However, Copernicus’ heliocentric system had its drawbacks. He couldn’t prove the Earth moved, and he couldn’t explain why the Earth rotated on its axis while revolving about the Sun. He also adhered to the Greek tradition that orbits follow uniform circles, so his geometry was complex and somewhat erroneous. In addition, Copernicus wrestled with the problem of parallax—the apparent shift in the position of bodies when viewed from different distances. If the Earth truly revolved about the Sun, critics observed, a viewer stationed on the Earth should see an apparent shift in 15 STK Astronautics Primer position of a closer star with respect to its more distant neighbors. Because no one saw this shift, Copernicus’ Sun-centered system was suspect. In response, Copernicus speculated that the stars must be at vast distances from the Earth, but such distances were far too great for most people to contemplate at the time, so this idea was also widely rejected. Figure 14: Copernicus redefines the center. Polish astronomer Copernicus reordered our view of the universe. He promoted a heliocentric (Sun-centered) 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 religious teachings of his day. Kepler Johannes Kepler (1571 - 1630) revolutionized our understanding of orbits. In Cosmic Mystery, written before he was age 25, he calculated that the orbit of Mars was not circular but elliptical. From this work, he developed three important laws of orbit motion, described in Figures 15 through 17. st Figure 15: Kepler’s 1 law. The orbits of the planets are ellipses with the Sun at a focus. 16 STK Astronautics Primer Planetary motion over 30 days Area 1 Area 2 Planetary motion over 30 days Area 1 = Area 2 Figure 16: Kepler’s 2nd law. The orbits of the planets sweep out equal areas in equal time. Average distance Figure 17: Kepler’s 3rd law. The square of the orbit period—the time it takes to go around 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 new optical device that could magnify objects so they would appear to be closer and brighter than when seen with the naked eye. Building a telescope that could magnify an image 20 times, Galileo ushered in a new era of space exploration. He made some startling telescopic observations of the Moon, the planets, and the stars, thereby attaining stardom in the eyes of his peers and potential patrons. Observing the planets, Galileo noticed that Jupiter had four moons or satellites (a word coined by Kepler in 1611) that moved about it. This disproved Aristotle’s claim that everything revolved about the Earth. 17 STK Astronautics Primer Galileo also took on Aristotle’s physics. He rolled a sphere down a grooved ramp and used a water clock to measure the time it took to reach the bottom. He repeated the experiment with heavier and lighter spheres, as well as steeper and shallower ramps, and cleverly extended his results to objects in free-fall. Through these experiments, Galileo discovered, contrary to Aristotle, that all objects fall at the same rate regardless of their weight, as shown in Figure 18. Galileo further contradicted Aristotle as to why objects, once in motion, tend to keep going. 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 them onward. Once this push dies out, they resume their natural motion and drop straight to Earth. Galileo showed that objects in uniform motion keep going unless disturbed by some outside influence. He wrongly held that this uniform motion was circular, and he never used the term “inertia.” Nevertheless, we applaud Galileo today for greatly refining the concept of inertia as we know it today. Figure 18: Galileo on gravity. Through application of the scientific method, Galileo put Aristotle’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 unwittingly started and which Kepler and Galileo had advanced, the terrestrial and heavenly realms had to be united under one set of natural laws. Isaac Newton (1642–1727) answered this challenge. Newton was a brilliant natural philosopher and mathematician who provided a majestic vision of nature’s unity and simplicity. 1665 proved to be Newton’s “miracle year,” in which he significantly advanced the study of calculus, gravitation, and optics. Extending the groundbreaking work of Galileo in dynamics, Newton published his three laws of motion and the law of universal gravitation in the Principia in 1687. With these laws, you could explain and predict motion 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. 18 STK Astronautics Primer INTRODUCTION TO ORBITAL MOTION Orbits are one of the basic elements of any space mission. Understanding a satellite in motion may at first seem rather intimidating. After all, to fully describe orbital motion we need some basic physics along with a healthy dose of calculus and geometry. However, as we’ll see, the complex trajectories of rockets flying into space aren’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 your chair 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 to entire galaxies. Once we know an object’s position and velocity, as well as the nature of the local gravitational field, we can predict exactly where the object will be minutes, hours or even years from now. Overview Math for the Faint-hearted ♦ 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 Faint-hearted Before delving too deeply into a discussion of dynamics, orbital mechanics and propagators it’s useful to step back and briefly review a bit of math. BUT DON’T PANIC! This section is designed to simplify a few basic concepts and describe the 19 STK Astronautics Primer 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 temperature are examples of scalars. None of these quantities has a unique meaning in any certain direction. A single letter, such as E for Total Mechanical Energy, denotes a scalar quantity. Vector A vector is a quantity that has both magnitude and direction. For example, if I ask you where you drove in your car, you might answer: “I went south.” But this wouldn’t tell me much. If I asked “How far?,” and you said “five miles,” I could put together a better picture. By knowing you drove five miles south, I have both v magnitude and direction. A letter with an arrow over it, such as V 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 direction only. For example, if I want to define where north is on a drawing, I could do so with a unit vector indicating the direction. A letter with a caret or hat over it, such as I$ for the I-direction 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: v V = 30 . I$ + 21 . J$ − 7.4 K$ km / sec This means the velocity is 3.0 km/sec in the I-direction, 2.1 km/sec in the Jdirection and 7.4 km/sec in the K-direction. Calculus—Just the important bits Calculus was developed to analyze changing parameters. Let’s look at how those changes are described. 20 STK Astronautics Primer Derivative A derivative represents the rate of change of one parameter with respect to another. For example, if you’re traveling north in your car, your position is changing over time. The rate at which your position changes over time is your velocity. Thus, if you travel 30 miles north in 30 minutes, your velocity is: v v Change in position (R) 30 miles Velocity = V = = = 60 mph north Change in time (t) 30 minutes Several methods are commonly used to denote a vector. In this primer, we use two types of notation. The first is to represent a derivative as d. So the change in position over time would be: v v dR V = dt We also use a “dot” over a symbol to represent the derivative with respect to time. For v& &&v position. R is the second derivative of position, or the rate of change of the rate of change example, R represents the derivative of the position vector or the rate of change of of position (i.e. the acceleration). Integral An integral represents the cumulative effect of one parameter changing with respect to another. If we were to graph both changing parameters, the integral is the area under the curve. For example, if you drive north at 30 miles per hour for 30 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 the changes in position over time to get the total change. An integral is the reverse of a derivative. Because acceleration is the derivative of velocity over time, the integral of acceleration over time is velocity. Key Concepts Mass Mass is a measure of how much matter or “stuff” an object possesses. For example, a volleyball and a cannon ball are about the same size, but the cannon ball has far more 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 baby carriage than a bulldozer because the bulldozer, being more massive, has more inertia. 21 STK Astronautics Primer Weight Weight actually describes the force produced by gravity acting on a mass. Your weight in various situations is illustrated in Figure 19. Figure 19: Weight. Weight describes a contact force caused by the effect of gravity on mass. On Earth, your weight is one value. As you move further from the center of the Earth, say in a penthouse at the top of a 250-mile-high skyscraper, your weight would be slightly less. In orbit at 250 miles altitude, the gravitational force is still the same; however, because you are in free-fall and not in contact with the Earth, your weight is zero. In all cases, mass stays the same. Momentum Linear momentum describes the resistance a moving object has to changes in either direction or speed. The more massive an object, or the faster it is moving, the harder it is to stop or change its direction of motion. As a result, linear momentum is the product of the mass and velocity of an object. Momentum for baby carriages and bulldozers is shown in Figure 20. m = 25 kg v= 1000 m/s mv = 25,000 kg-m/s m = 25,000 kg v = 1 m/s mv= 25,000 kg-m/s Figure 20: Momentum, bulldozers and baby carriages. Linear momentum is the product of mass and velocity. For a baby carriage to have the same linear momentum as a bulldozer, it would have to be traveling at a much higher velocity. 22 STK Astronautics Primer Angular momentum is a measure of the spinning properties of an object. As Figure 21 illustrates, a non-spinning top immediately falls over. However, a spinning top has angular momentum, which allows it to resist the force of gravity pulling it over (until it finally slows down due to friction). angular momentum non-spinning top spinning top Figure 21: Angular momentum. A spinning top has angular momentum which keeps it pointing 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 height an object is raised to, the greater its potential energy. An object at the top of a deep well, as shown in Figure 22, has more potential energy than an object at the bottom of the well. PE = 0 at R = infinite R PE < 0 at R > 0 PE < 0 at R = 0 Figure 22: Potential energy. Because we normally define our coordinate systems as positive outward from the center of the Earth, we measure potential energy “from the bottom up.” For example, at the top of a deep well we would say the potential energy is zero. As we get closer to the bottom of the well, the potential energy is less, or more negative. Kinetic energy is a function of an object’s mass and speed. Like momentum, the more massive an object is or the faster it travels, the more kinetic energy it has. It is this 23 STK Astronautics Primer energy that must be transformed in order to stop a speeding bulldozer, as shown in Figure 23. This is accomplished by applying the brakes, which turns kinetic energy into heat (those brakes get hot!). m = 25,000 kg v = 1 m/s 1/2 mv2 = 12,500 kg-m/s2 Figure 23: Newton’s 1st law. The kinetic energy of a bulldozer moving at only 1 m/s is 12,500 kg-m/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 a straight line forever, unless disturbed by an outside force such as gravity or air resistance. Newton’s Second Law The time rate of change of an object’s momentum is equal to the applied force. Change(momentum) = Force Applied Change(time) Recall, momentum is the product of mass and velocity. Thus, as long as mass stays constant (which it normally does as long as rockets aren’t firing) this equation can be reduced to: F = ma or: Force applied = mass (m ) times acceleration (a) 24 STK Astronautics Primer 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 faster you decelerate). This principle is illustrated in Figure 24. 25,000 N 1 m/s stops in 1 second 6.9 N 1 m/s stops in 1 hour Figure 24: Newton’s 2nd law. Force is proportional to acceleration (or deceleration). A 25,000 N force is needed to stop a 1 m/s bulldozer in 1 second, while much smaller 6.9 N 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 can be illustrated by two roller-skating 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. 25 STK Astronautics Primer Newton’s Law of Universal Gravitation The force of gravity between two bodies is proportional to the product of the masses and inversely proportional to the square of the distance between them. This is illustrated in Figure 26. R F g m F g m 2 1 Figure 26: Gravitational attraction. Two masses in space each exert a force on the other. The magnitude of this force depends on the product of their masses and the square of the distance between them. Newton’s law can be summarized in equation form as follows: Fgravity = GM1 M 2 R2 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 it generates. Furthermore, the farther apart two objects are, the less the force is, in fact, the force decreases with the square of the distance as illustrated in Figure 27. 26 STK Astronautics Primer R 2R F g F g m Fg /4 2 m 2 F /4 g m 1 m 1 Figure 27: Gravity and distance. The force of gravitational attraction decreases with the square of the distance (e.g., if you double the distance, the force decreases by one fourth). What exactly is gravity? The study of physics is still grappling to reconcile the force of gravity with the other fundamental forces of nature. Already, extremely weak “gravity waves” have been detected from distant galaxies. More sensitive instruments are being built to understand and quantify this mysterious force. How strong is gravity? Let’s look at the Earth-Moon system. The force of gravity between 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 lift-off. 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 with mass (and that’s pretty much everything!). While Galileo right in that the gravitational force is greater on heavier objects than lighter objects, he was wrong in predicting the affect this would have on the rate at which they fall. The acceleration 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 things accelerate at the same rate within a gravitational field. For example, if you drop a 27 STK Astronautics Primer hammer and a feather, both objects impact the ground at the same time (neglecting air resistance). Of course, Galileo predicted this. Astronaut Dave Scott proved this with an experiment on the Moon. He dropped a hammer and a feather at the same time. Both hit the 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 a satellite “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’s begin with a simple experiment that illustrates, conceptually, how orbits work. To do this, we’ll arm ourselves with a bunch of baseballs and travel to the top of a tall mountain. Imagine you were standing on top of this mountain prepared to pitch baseballs to the east. As the balls sail off the summit, what would you see? The baseballs would follow a curved path before hitting the ground. Why is this? The force of 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. 28 STK Astronautics Primer Figure 30: Baseballs in motion. A baseball-throwing astronaut can be used to illustrate the simple motion of a satellite. Naturally, the harder the baseballs are thrown, the further they travel before hitting the ground. As Figure 30 illustrates, the faster you throw the balls, the farther they travel before hitting 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 try another 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 the same time, a second astronaut throws an identical ball horizontally as hard as possible. What do you see? If the second astronaut throws a fast ball, it travels out about 20 m (60 ft.) or so before it hits the ground. However, the ball dropped by the first astronaut hits the ground at exactly the same time as the pitched ball, as Figure 31 shows! Figure 31: Motion and gravity. Two astronauts each have a baseball held at the same height above the ground. If the first astronaut drops her baseball while the second astronaut throws his, both baseballs hit the ground at exactly the same time. Gravity acts on both baseballs in the same way, independent of their horizontal motion. 29 STK Astronautics Primer How can this be? To understand this seeming paradox, we must recognize that, in this 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 the ground at the same time. The only difference is that the pitched ball, because it has horizontal 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 faster and faster to see what happens. No matter how fast we throw them, the balls still fall at the same rate. However, as we increase their horizontal velocity, they’re able to travel farther and farther before they hit the ground. Because the Earth is basically spherical in shape, something interesting happens. The Earth’s spherical shape causes the surface to drop approximately 5 m for every 8 km we travel horizontally across it, as shown in Figure 32. 8 km 5m Figure 32: Our spherical Earth. We know the Earth is a nearly perfect sphere. For every 8 km of horizontal distance, the Earth curves down about 5 m. In other words, if you could lay an 8 km long board tangent to the Earth at one end, at the other end it would be 5 m off the ground. So, if we were able to throw a baseball at 8 km/s (assuming no air resistance), its path would exactly match the rate of curvature of the Earth. That is, gravity would pull it down about 5 m for every 8 km it travels, and it would continue around the Earth at a constant height. If we don’t remember to duck, it will hit us in the back of the head about 85 minutes later. (Actually, because of the rotation of the Earth, it would miss your head.) A ball thrown at a speed slower than 8 km/s falls faster than the 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 than that and our trajectory impacts the Earth at some point. If we were to project this shape 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 horizontal velocity, it never quite impacts the ground. If we throw the ball too hard, it leaves the Earth altogether on a parabolic or hyperbolic trajectory, never to return. 30 STK Astronautics Primer hyperbola parabola circle 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 will stay at same the instantaneous height above the ground. We call this a circular orbit. If we throw it faster than the circular orbit speed, the ball will be in an elliptical-shaped orbit. If we throw the ball faster yet, it will escape the Earth’s gravity altogether on a parabolic or hyperbolic trajectory. Thus, it is important to note that no matter how hard we throw, our trajectory resembles either a circle, ellipse, parabola or hyperbola. These four shapes are called conic sections. Why conic sections? Because these are the shapes we get by slicing through a cone with a plane at different angles, as illustrated in Figure 34. circle ellipse hyperbola parabola Figure 34: Conic sections. The four basic conic sections: circle, ellipse, hyperbola and parabola. 31 STK Astronautics Primer 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 found using: Vcircular = GM Earth R where Vcircular = Satellite velocity in a circular orbit (km/sec) G = Universal gravitational constant = 6.67 x 10-11 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 the surface of the Earth (ignoring air drag) you’d have to throw the baseball at 17,600 mph. Notice that the circular orbit speed depends on your distance from the center of the 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 how conservation of energy and momentum affects the velocity of satellites. Gravity is a conservation force, which means that an object moving in a gravitational field doesn’t lose 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. 32 STK Astronautics Primer Maximum PE KE=0 Maximum PE KE=0 Max KE Min 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 is lowest. As you swing, you trade kinetic energy (speed) for potential energy (height) with the 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 for potential 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 kinetic energy apogee high potential energy high kinetic energy perigee low potential energy Figure 36: Energy conservation in orbit. As a satellite moves closer to the Earth in an orbit, it must speed up to conserve total energy. As it gets further away, the satellite trades kinetic energy for potential energy and slows down. Angular momentum is also always conserved. Ice skaters use this principle to speed up 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, 33 STK Astronautics Primer 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 with it) moves around the Sun, the orbit itself stays fixed in respect to a stationary reference. Figure 37: Conservation of angular momentum. Ice skaters use this principle to speed up or slow down as they spin. When their arms are extended, the moment of inertia is low, so they spin more slowly. As they draw their arms in, the moment of inertia decreases and the spin rate increases to keep total angular momentum constant. 34 STK Astronautics Primer DESCRIBING ORBITS Overview Understanding Coordinate Systems ♦ What is a coordinate system? The Geocentric-Equatorial 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 of definitions that allow us to describe positions and velocities in a more meaningful way. For example, if I simply told you a car is traveling south, you wouldn’t have very much information. But if I first tell you that our a reference frame is centered on Washington, DC, and then tell you that the car is 30 miles east of the city traveling south at 60 mph, you’d know something far more useful. In defining coordinate systems and describing position and velocities, we make extensive use of vectors. We use vectors because we want to keep track of the information contained in a particular parameter. Specifically, a vector is a parameters having both magnitude and direction. For example, 60 mph tells you speed (magnitude) without direction. But 60 mph south tells you both magnitude and direction. In defining coordinate systems, we are sometimes only interested in direction. In that case, we use unit vectors defined to have a specific direction and a magnitude of 1. Cartesian coordinate systems are laid out with three orthogonal unit vectors—vectors at right angles to each other. For example, if the origin of a coordinate system were chosen to be in one corner of a room, the floor would be the fundamental plane. We could then describe the position of every piece of furniture in the room with respect to this system. Such a collection of unit vectors allows us to establish the components of other vectors in 3-D space. 35 STK Astronautics Primer To create a coordinate system, we need to specify four pieces of information—an origin, a fundamental plane, a principle direction, and a third axis, as shown in Figure 38. The origin defines a physically identifiable starting point for the coordinate system. The other two parameters fix the orientation of the frame. The fundamental plane contains two axes of the system. Once we know the plane, we can define a direction perpendicular to that plane. The unit vector in this direction at the origin is one axis. Next, we need a principle direction within the plane. Again, we choose something 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 can find the third axis using the right-hand rule. The right-hand rule can be demonstrated by pointing the thumb of your right hand in the direction perpendicular to the fundamental plane. With you fingers pointing in the principle direction, curl your fingers 90° so that your thumb is pointing in the direction of the third axis. (2) pick fundamental plane & perpendicular to it (1) pick origin fundamental plane origin origin (3) pick principal direction (4) find 3rd axis fundamental plane fundamental plane origin origin principal direction 3rd axis, found using right-hand rule principal direction Figure 38: Defining a coordinate system. Remember—coordinate systems are defined to make our lives easier. If we choose the correct coordinate system, developing the equations of motion can be simple. If we choose the wrong system, it can be nearly impossible. The Geocentric-Equatorial Coordinate System For Earth-orbiting spacecraft, we’ll choose a tried-and-true system that we know makes solving the equations of motion relatively easy. We call this system the geocentric-equatorial coordinate system, shown in Figure 39. Here’s how it’s defined: 36 STK Astronautics Primer Origin The center of the Earth (hence the name geocentric). Fundamental plane Earth’s equator (hence geocentric-equatorial), where perpendicular to the fundamental plane is the direction of the north pole. Principle direction Vernal equinox direction, , or the vector pointing to the first point of Aries. The vernal equinox direction points at the zodiac constellation Aries and is found by drawing a line from the Earth to the Sun on the first day of spring. While this direction may not seem “convenient” to you, it’s significant to the astronomers who originally defined the system. Unfortunately, the vernal equinox direction is not perfectly constant. The Earth’s orbit precesses around the Sun and the Sun is moving through the galaxy. Therefore, exactly when this direction is defined is extremely important for the definition of the system. We can use two ways of defining these directions. The first is to use the mean or average direction at some point in time. The other is to use the true position at exactly one specific point in time. Various combinations of these definitions using different dates gives us several possibilities for coordinate systems used in STK: J2000 Defines the mean vernal equinox direction and mean Earth rotation axis on January 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 the orbit 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 the coordinate 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 exactly the orbit epoch time specified. 37 STK Astronautics Primer True of epoch Defines the true vernal equinox direction and true Earth rotation axis at the coordinate epoch time specified. B1950 Defines the mean vernal equinox direction and mean Earth rotation axis at the beginning of the Besselian year 1950. It corresponds to 31 December 1949 at 22:09:07.20 Greenwich Mean Time (GMT). Mean Equinox, True Equator Defines mean vernal equinox direction and true Earth rotation axis for the orbit epoch time specified. Third axis The third axis of the geocentric-equatorial coordinate system is found using the righthand rule. ^ K ^I ^ J Figure 39: The geocentric-equatorial 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’re r known as an object’s position vector, R . Three more pieces of information describe its r velocity vector, V . One additional item, time, tells us when the information provided is valid. These elements are know as Cartesian elements. While it is often convenient to describe orbit motion using simply position and velocity vectors in a Cartesian coordinate system, especially for computational work, these vectors provide us little insight into the orbit itself. For this reason, astronomers long ago developed orbital 38 STK Astronautics Primer elements. Orbital elements give us a short-hand way of expressing orbit size, shape and orientation, allowing us to tell at a glance the application for a given orbit. This section describes the Classical orbital elements, which are sometimes referred to as the Keplerian elements and are attributed originally to Kepler himself. Variations on these elements, the commonly used two-line element sets, are described in a later section. Orbit Size How big is an orbit? This depends on how fast we “throw” our satellite into orbit. The faster we throw, the more energy an orbit has, and the bigger it is. We express the size of an orbit in terms of its semimajor axis, a., as defined in Figure 40. perigee apogee 2a = major axis a = semimajor axis Figure 40: Semimajor axis. The major axis of an elliptical orbit is the distance between the point of closest approach (perigee) and furthest point (apogee). Semimajor axis is one-half this distance. We can express the semimajor axis in terms of the distance from the center of the Earth to apogee (Rapogee) and perigee (Rperigee). Perigee is the point in an orbit that is closest to the Earth. Apogee is the point where it is furthest away (apogee is undefined for a parabolic or hyperbolic trajectory). The semimajor axis can be found using: a= Rapogee + R perigee 2 where 39 a = semimajor axis (km) Rapogee = Distance from center of Earth to apogee (km) STK Astronautics Primer 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 one time), is proportional to the orbit size: P = 2π a3 GM Earth For example, a typical Space Shuttle orbit at an altitude of a few hundred kilometers has a period of about 90 minutes. It orbits the Earth about 16 times each day! For communications satellites in geosynchronous orbit at an altitude of 35,780 km, the period 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. e>1 e=1 hyperbola parabola ellipse circle 0<e<1 e=0 Figure 41: Eccentricity. A perfectly circular orbit has an eccentricity of 0. Elliptical orbits have an eccentricity of less than 1. A parabolic orbit has an eccentricity of exactly 1. Hyperbolic orbits (or trajectories) have eccentricities of greater than 1. In practice, a perfectly circular or parabolic orbit cannot be achieved. 40 STK Astronautics Primer Orbit Orientation How the orbit is tilted with respect to the equator is called its inclination, i. An orbit that stays directly over the equator has an inclination of 0° and is called an equatorial orbit. An orbit that goes directly over the north and south poles must have an inclination of exactly 90° and is called a polar orbit. Different classes of orbits have different inclinations, as shown in the table. Inclination Orbit Type 0° = i = 180° Equatorial i = 90° Polar 0° < i < 90° Direct or posigrade (moves in direction of Earth’s rotation) Diagram i=0o i=90o ascending node 90° < i <180° Indirect or retrograde (moves against the direction of Earth’s rotation) ascending node We measure how an orbit is twisted by locating its ascending node, the point where the satellite crosses the equator moving south to north. This point is referenced to the I-direction, 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. 41 STK Astronautics Primer ^ K equ a lp toria lane Ω ascending node ^ J ^ i Figure 42: Right Ascension of the ascending node (RAAN), Ω , is the angular distance from the vernal equinox direction to the ascending node. The ascending node of 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 ascending node. This angle is called the argument of perigee, ω; it is measured positive in the direction of satellite motion. 42 STK Astronautics Primer ^ K equ a lp toria lane ω ascending node ^ J ^ i Figure 43: Argument of perigee. The argument of perigee, ω, is the angular distance between the ascending node and perigee. Finally, we describe a satellite’s instantaneous position with respect to perigee using another angle, true anomaly, ν. It is the angle, measured positive in the direction of motion, between perigee and the satellite’s position. Of the six orbital elements, only true anomaly changes continually (ignoring perturbations). V R υ perigee Figure 44: True anomaly. The true anomaly, ν, is the angular distance from perigee to r the orbit position vector, R . 43 STK Astronautics Primer Summary of classical orbital elements Recall, the reason we wanted to develop the orbital elements in the first place was to give us a short-hand method of describing an orbit. We also wanted to use parameters that would have some physical meaning we could more easily visualize. The six classical orbital elements are summarized below. Name Symbol Describes Semimajor axis a Size (and energy) Eccentricity e Shape (e = 0 for circle, 0> e >1 for ellipse, e = 1 for parabola, e > 1 for hyperbola) Inclination i Tilt of orbit plane with respect to the equator Longitude of ascending node Ω Twist of orbit with respect to the ascending node location Argument of perigee ω Location of perigee with respect to the ascending node True anomaly ν Location of satellite with respect to perigee Satellite Missions As we already know, varying missions require different orbits, which can be described using Classical orbit elements. The table following shows various missions and their typical orbits. Technically speaking, a geostationary orbit is a circular orbit with a period of exactly 24 hours and an inclination of exactly 0°. A satellite in a geostationary orbit appears to be stationary to an Earth-based observer. Geosynchronous orbits are slightly inclined orbits with a period of 24 hours. In practice, it is almost impossible to achieve an orbit with exactly a 24-hour period and an inclination of 0°. Thus, the two terms are frequently used interchangeably. A semisynchronous orbit has a period of 12 hours. Sun-synchronous orbits are retrograde low-Earth orbits (LEO) inclined 95° to 105°; they are typically used in remote-sensing missions to observe Earth. A Molniya orbit is a semi-synchronous, eccentric orbit used for some communication missions. Super-synchronous orbits are usually circular orbits with periods longer than 24 hours. Mission 44 ♦ Communications ♦ Early Warning ♦ Nuclear detection ♦ Remote sensing Orbit Type Geostationary Semimajor Axis (Altitude) Period Inclinatio n 42,158 km 24 hours ~0° e ≅0 ~90 min ~95° e ≅0 (35,780 km) Sunsynchronous ~6500-7300 km (~150-900 km) Other STK Astronautics Primer Mission ♦ Navigation (GPS) Orbit Type Semimajor Axis (Altitude) Period Inclinatio n Semisynchronous 26,610 km 12 hours 55° e ≅0 (20,232 km) Other ♦ Space Shuttle Low-Earth obit ~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) 12 hours 63.4° ω = 270° e = 0.7 Ground Tracks Orbital elements allow us to visualize the shape of an orbit around the Earth. Because we use satellites for missions involving specific points on Earth—taking pictures, communications, navigation—we really would like to know what path the satellite traces over the Earth’s surface. A satellite ground track is the orbit path (usually for multiple orbits) projected onto a flat map of the Earth. These projections become complex because we must account for the satellite circumnavigating the entire Earth during 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’t rotate. Picture an orbit around this nonrotating Earth. Because the orbit plane must pass 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. For example, all lines of longitude are great circles. The equator is the only line of latitude that is a great circle—No other line of latitude “slices through” the center of the Earth. When the Earth is stretched out to a flat-map projection (called an equidistant cylindrical 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 the ground 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 a soda 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!) 45 STK Astronautics Primer X X Figure 45: Ground track for a nonrotating Earth. If the Earth didn’t rotate, the ground track would always look like a constant sine wave. As the figure shows, the ground track would always have the same relative orientation with respect to a stationary observer on the 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 stays the same with respect to a stationary observer. However, because the Earth rotates at 15° 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 from one orbit to the next, as shown in Figure 46. The amount it shifts depends on its period. The longer the period, the more time the Earth has to rotate between successive orbits. X X Figure 46: Satellite ground tracks. Satellite orbits are fixed in space with respect to a stationary observer. However, a stationary observer on the Earth is rotating to the east at 15° per hour. Thus, each successive orbit ground track shifts to the west. Let’s look at the ground track of some very different orbits. 46 STK Astronautics Primer E C B A D 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=24 hr. 47 STK Astronautics Primer 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 through space, we need to know where they are now and where they’ll be later so that we can predict sensor coverage and point our antennas at them to gather data. Although we can easily predict this motion when the orbit is a circle, the problem becomes more complicated 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 and velocity (or orbital elements) of an orbit given some initial conditions and assumptions. There are a wide variety of orbit propagation techniques available with widely different accuracy and applications. Knowing the assumptions built into different propagation schemes is key to knowing which one to use for a given application. Overview Understanding Propagators ♦ How do propagators work? The Two-Body Propagator ♦ What is meant by a “two-body” propagator? Orbit Perturbations ♦ What are “J2” and those other things that affect an orbit? Dealing with Perturbations ♦ How can I model orbit perturbations? Two-line 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 of our ball-throwing astronaut shown in Figure 48. What we’re after is a simple 48 STK Astronautics Primer mathematical algorithm that will allow us to predict the ball’s position and velocity at any point in time. Whether you’re analyzing the motion of baseballs or galaxies, the fundamental 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 baseball can 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 system with its origin at the point of release. The x-direction is defined to be positive to the right in the picture. The y-direction is positive down. Next, we need to make some assumptions to make our lives easier. The major assumption we’ll make is that Earth’s gravity is the only force acting on the ball, a force which is constant over the flight path we’re concerned with. That is, wind resistance and other forces (the gravitational pull of the Moon and stars, solar pressure, etc.) are negligible. This gravitational force can be expressed using Newton’s law, using vector notation, as: v Fgravity = mgy$ N Note y$ is a vector notation indicating the force of gravity acts only in the y-direction, i.e., down. See the section describing vectors for further explanation. 49 STK Astronautics Primer Finally, we need some initial conditions. Let’s pretend that the ball leaves the pitcher’s hand at a velocity of 10 m/s on a horizontal path (i.e., all motion in the x-direction). r r Symbolically we would say the initial velocity ( Vinitial ) and position ( Rinitial ) are: Vinitial = 10 x$ m / sec The ball starts out traveling 10 m/sec horizontal to the right r Rinitial = 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 only force on the ball is due to gravity, which acts to accelerate the ball in the positive ydirection. Thus, we have: r a = 0 x$ + gy$ m / sec 2 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 equation with respect to time. (Remember integrals from calculus? Basically, an integral is a mathematical means of adding together lots of small changes over time to develop the total change.) Thus, r r V (t ) = Vinitial + gty$ = 10 x$ + gty$ m / sec This equation tells us that the ball will keep its initial horizontal velocity constant but will speed up in the vertical direction due to gravity (which we already knew). To obtain the instantaneous position of the ball at any time (t), we must once again integrate this equation with respect to time so that: r r r 1 1 R( t ) = Rinitial + Vinitial t + gt 2 y$ = 10tx$ + gt 2 y$ 2 2 We can now use these relatively simple equations to propagate the motion of the baseball. Using a simple spreadsheet, we can determine the position and velocity of the baseball for each second for a total 10-second flight. These values and a graph depicting the trajectory are shown in Figure 49. Notice the trajectory we derived is exactly what we’d expect from experience. Often, people mistakenly refer to the 50 STK Astronautics Primer baseball trajectories as “parabolic;” however, as we know, this is actually a small section of an ellipse. Time (sec) X (m) 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 Y (m) Trajectory of the Baseball Y (m) 100 90 80 70 60 50 40 30 20 10 0.0 4.9 19.6 44.1 78.4 122.5 176.4 240.1 313.5 396.8 489.9 0 X (m) 0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 Figure 49: Results of baseball propagator: Using the simple equations of motion we derived for the baseball, we can use a spreadsheet to calculate the x and y positions at each point in time for a 10-second flight. Plotting these on a graph, we see the shape of the trajectory. Note that the technique we developed here was for analytic propagation. An analytic propagation technique has a close-form solution. In other words, given the initial conditions, we solve directly for the position and velocity at any future time using a straightforward “plug and chug” of the equations of motion. How accurate is this propagation technique? As we shall see, this (and any other method) is only as good as the assumptions we make. For example, we assumed no wind resistance, but we know from experience that a sudden gust of wind could make this trajectory change considerably. In the next section, we’ll apply this same basic technique to understand the slightly more complicated motion of a satellite in orbit. The Two-Body Propagator Now we’ll develop a simple method we can use to propagate the position and velocity of a satellite known at a given time to predict its position and velocity at some time in the future. By “simple method” we mean the restrictions placed on the complexity of the problem. In this case, one of the primary assumptions we will make is that there are only two bodies concerned—the Earth and the satellite. Thus, we arrive at the term used to describe this approach “the restricted two-body problem.” 51 STK Astronautics Primer Motion Analysis Process The approach we’ll take is exactly the same as the one we used to describe the motion of the baseball in the previous section. Using the same motion analysis process described, we can apply the three basic steps of our motion analysis model: define a coordinate system, make assumptions and identify equations of motion. Our coordinate system will be the geocentric-equatorial coordinate system described earlier. The assumptions we make will “restrict” our solution to cases in which these assumptions 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 drag force is small. ♦ the satellite won’t maneuver or change its path, so we can ignore the thrust force. ♦ we’re considering the motion of the satellite close to the Earth, so we can ignore the gravitational attraction of the Sun, the Moon or any third body. (That’s why we call this the two-body problem.) ♦ compared to Earth’s gravity, other forces such as those due to solar radiation, 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 be treated as a point mass. After all these assumptions, we’re left with gravity as the only force affecting the motion of a satellite for the restricted two-body problem. This can be expressed (using vector notation) as: v − GM Earth M satellite $ Fsatellite = R 2 Rsatellite The force due to gravity on a satellite depends on the mass of both the satellite and Earth and the distance to the Earth’s center. The direction is down, in the minus-R direction. For convenience, we often combine GMEarth to derive an expression µEarth , known as the Earth’s gravitational parameter. While Newton’s law of gravity describes the force on the satellite, we can use Newton’s 2nd law of motion to describe the effect of that force to develop our equations of motion. From Newton’s second law, the force on the satellite can be expressed as: 52 STK Astronautics Primer Fsatellite = M satellite a satellite Setting the two expressions for force on the satellite equal to each other, we develop an expression for the satellite’s acceleration: −µ v a satellite = 2 Earth R$ Rsatellite A satellite accelerates down [minus-R direction] due to gravity. The further away from the Earth’s center, the smaller the gravitational force and, therefore, the smaller the corresponding acceleration. This equation says the motion of a satellite depends only on the distance between the center of the Earth and the satellite. It is independent of satellite mass. Substituting the more common notation for acceleration, we get the two-body Equation of Motion. .. − m Earth R$ v R= 2 R 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$ r = unit vector in direction of R Two-Body Orbit Propagation What can the two-body equation of motion tell us about the movement of a satellite around the Earth? Unfortunately, in its present form—a second-order, non-linear, vector differential equation—it doesn’t help us visualize anything about this movement. So what good is it? To understand the significance of the two-body equation of motion, we must first “solve” it using rather complex mathematical slightof-hand. When the smoke clears, we’re left with an expression for the position of an object in space in terms of some variables we already know. R= 53 ( a 1 − e2 ) 1 + e cosν STK Astronautics Primer where r R = magnitude of R a = semimajor axis (km) e = eccentricity (dimensionless) ν = true anomaly (deg or radians) This equation represents the solution to the restricted two-body equation of motion and describes the location, R, of a satellite in terms of a few constants and some initial conditions. We can now make use of the solution to the two-body equation of motion 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 initial position to a future position is simple, because the satellite is moving at a constant speed. However, in an elliptical orbit this speed varies (recall a satellite travels fastest at perigee and slowest at apogee, keeping total energy constant). As a result, we don’t know how the true anomaly, ν, changes with time because it doesn’t change uniformly. Here’s where Johannes Kepler came to the rescue. He developed this technique 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: n= angle 2π = = time P µ a3 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 a future n, how to find out how long Mars would take to travel there. Kepler’s approach was purely geometrical—he related motion on a circle to motion on an ellipse. To do this, he had to invent a new angle called the mean anomaly, M, defined as: M = nT where M = mean anomaly (rad) 54 (1) STK Astronautics Primer 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 a picture. We’ll have to describe it mathematically. Expressing this equation in terms of two 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 called the eccentric anomaly, E, so that he could relate M to E and then E to n. With all of these things defined, Kepler was able to develop his now-famous equation, commonly called Kepler’s Equation. (For this equation to work, all angles must be in radians.) M = E − e sin E (3) where E = eccentric anomaly (rad) e = eccentricity Kepler then related E to n using: cos E = where ν = true anomaly (rad) And related ν to E through: 55 e + cos ν 1 + e cos E (4) STK Astronautics Primer cos ν = cos E − e 1 − e cos E (5) Finally, we have all the equations needed to build a complete two-body propagator. The first problem, and the easiest, is finding the time of flight between two points in an orbit. Given νinitial and νfuture, 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. This involves determining a satellite’s position at some future time, tfuture, as shown in Figure 50. Time of Flight υ Future υ Initial Figure 50: Time of flight on an elliptical orbit. The second problem Kepler tackled was 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 is at time tinitial, so we know νinitial. We start by finding Einitial, using Equation (4). Then we find Minitial using Kepler’s Equation (3). M initial = E initial − e sin Einitial Now, because we know tfuture, using Equation (2), we can find Mfuture: 56 STK Astronautics Primer ( ) M future − M initial = n t future − t initial − 2 kπ 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 future = M future + e sin E future (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’s Equation 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 the kids’ game Twenty Questions. In this game, your partner thinks of a person, place, or thing 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 right answer, a good player will systematically eliminate all other possibilities until only the correct answer remains. A mathematical application of this can be seen using another transcendental equation: y = cos(y) Because we can’t solve for y using algebra (we can’t get the y out of the cosine function to put all the ys on the left side), we must iterate. Begin by taking a guess at the value for y, and take the cosine to see how close you were. Then take this as the new value of y and use it for the next guess, and keep doing this iteration until the new 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/4 radians as your first guess for y. (Remember to set your calculator to use radians, not degrees.) Keep pressing the cosine function button and you’ll see the value slowly converges to 0.739085 radians (about 43°). Presto—you’ve now solved the transcendental 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 most eccentric orbits, so let’s use Mfuture for our first guess at Efuture. Here’s the algorithm: 57 ♦ Use Mfuture for the first E. ♦ Solve Equation (6) for a new Efuture. ♦ Use this new Efuture for the next guess for Equation (6). STK Astronautics Primer ♦ 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 better methods in use in most standard propagators. Let’s quickly summarize what we’ve learned. If we know where we are in an orbit and where we want to be, we can use Kepler’s Equation to solve for the time it takes to travel to the place we want to be. The solution is very straightforward. If, however, we know where we are and want to know where we’ll be at some future time, we can use Kepler’s Equation to find that location only by iterating a transcendental equation for eccentric anomaly. Orbit Perturbations In deriving the two-body 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 be treated as a point mass These assumptions led us to the restricted two-body equation of motion: &&v + µ R $ =0 R R2 (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 given orbit. Only the true anomaly, ν, varies with time as the satellite travels around its fixed orbit. What happens if we now change some of our original assumptions? Other classical orbital elements besides ν will begin to change as well. Any changes to these classical orbital elements due to other forces are called perturbations. To see which classical orbital elements will change and by how much, let’s look at our first assumption—gravity is the only force. 58 STK Astronautics Primer Atmospheric Drag Just as a sudden gust of wind changes the course of a football, atmospheric drag can affect satellites in low Earth orbit (below about 1000 km). Let’s look at how drag affects the orbital elements. Because drag is a nonconservative force, it takes energy away from the orbit in the form 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 why this is so. When a satellite in an elliptical orbit is at perigee, it has a greater speed than it would if the orbit were circular at that same altitude. The drag decreases the speed, making it closer to the circular orbit speed. That’s exactly what we see in Figure 51. It’s as if drag were giving the satellite a small negative velocity change, or delta (∆) V, (slowing it down) each time it passes perigee. successive orbits ∆V drag Earth’s atmosphere original orbit Figure 51: The effect of drag on an eccentric, low-Earth orbit. As a satellite passes through 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 upper atmosphere and the satellite’s attitude. The Earth’s day-night cycle, seasonal tilt, variable solar distance, the fluctuation of Earth’s magnetic field, the Sun’s 27-day rotation and the 11-year cycle for Sun spots make precise modeling nearly impossible. The force of drag also depends on the satellite’s coefficient of drag and frontal area, which can also vary widely, further complicating the modeling problem. The uncertainty in these variables is the main reason Skylab decayed and burned up in the atmosphere several years earlier than first predicted. For a given orbit, however, we can approximate how the semimajor axis and the eccentricity change 59 STK Astronautics Primer with time, at least for the short term. Different propagation techniques use different methods 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 two-body assumption. We call this squashed shape oblateness. What exactly does an oblate Earth look like? Imagine spinning a ball of jello around its axis and you can visualize how the middle (or equator) of the spinning jello would bulge out—the Earth is fatter at 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 constant describing the size of the bulge in the mathematical formulas used to model the oblate Earth. Why “J2?” This term arises from the mathematical short-hand used to describe Earth’s gravitational field. (Gravitational acceleration at any point on Earth is commonly expressed as a geopotential function expressed in terms of Legendre polynomials and dimensionless coefficients Jn—whew!). J2, J3 and J4 are the zonal coefficients that depend on latitude. Of these, J2 is by far the most important; it is roughly 1000 times greater than either J3 or J4. However, for more precise modeling of 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 the Earth into wedges that depend on longitude (sectoral terms) and slice it again into regions of longitude and latitude (tesseral terms). Let’s concentrate on the simplest and most profound case, J2. What effect does J2 have on the orbit? Let’s look at Figure 52. Here it’s shown exaggerated; actually the bulge is only about 22 km thick. That is, the Earth’s radius is about 22 km longer along the equator than through the poles. 60 STK Astronautics Primer F J2 F J2 Figure 52: Diagram of Earth oblateness. The Earth’s oblateness, shown here as a bulge at the equator (highly exaggerated to demonstrate the concept) causes a twisting force 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 force caused by the equatorial bulge is still gravity. Recall that gravity is a conservative force; therefore, the total mechanical energy in an orbit must be conserved. Total mechanical energy depends on the orbit’s semimajor axis. Thus, as long as energy remains constant (i.e., no drag or other forces adding or stealing energy), the semimajor axis also remains constant. It turns out that the eccentricity, e, also doesn’t change, although the explanation for this is beyond the scope of our discussion here. Although you might expect the inclination to change because the bulge pulls on our orbit, it doesn’t! However, it does affect the orbit by changing the right 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 spinning top. If you stand a nonspinning top on its point, gravity causes it to fall over. If you spin the top first, gravity still tries to make it fall but, because of its angular momentum, it begins to swivel—this motion is called precession. Let’s examine the effect 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 satellite because the force no longer originates from the center of the Earth. This causes the plane of the orbit to precess (like the spinning top), resulting in a movement of the ascending node, ∆Ω. This motion is westward for posigrade orbits (inclination <90°) and eastward for retrograde orbits (inclination > 90°). 61 STK Astronautics Primer Figure 54 shows this nodal regression rate, Ω& , as a function of inclination and orbital altitude. Let’s look more closely at this figure. It shows that the higher the satellite is, the less effect the bulge has on the orbit. This makes sense because gravity decreases with the inverse square of the distance (see Newton’s Law of Gravitation). 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 low inclinations. This makes sense, too, because the satellite travels much closer to the bulge during its orbit, and thus is pulled more by the bulge. For low-altitude and low-inclination 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 (deg/day) Nodal Regression Rate as a Function of Inclination and Eccentricity 10.0 4000km x 100km altitude elliptical orbit 5.0 0.0 100km altitude circular orbit -5.0 -10.0 0 20 40 60 80 100 120 140 160 180 Inclination (deg) & . The nodal regression rate caused by the Earth’s Figure 53 Nodal Regression Rate, Ω equatorial bulge. Positive numbers represent eastward movement; negative numbers represent westward movement. The less inclined an orbit is to the equator, the greater the 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 of 100 km depending on the inclination for various apogee altitudes. This perigee rotation rate, ω& , is difficult to explain physically, but it could be derived mathematically from the equation for J2 effects on perigee location. With this perturbation, the major axis, or line of nodes, rotates in the direction of satellite motion if the inclination is less than 63.4° or greater than 116.6°. It rotates opposite to satellite motion for inclinations between 63.4° and 116.6°. 62 STK Astronautics Primer Perigee Rotation Rate (deg/day) Perigee Rotation Rate as a Function of Inclination and Eccentricity 20.0 100km altitude circular orbit 15.0 10.0 4000km x 100km altitude elliptical orbit 5.0 0.0 -5.0 0 20 40 60 80 100 120 140 160 180 Inclination (deg) Figure 54: Affects of J2 on argument of perigee. The perigee rotation rate caused by the Earth’s equatorial bulge depends on inclination and altitude at apogee. Sun-Synchronous and Molniya Orbits The effects of the Earth’s oblateness on the node and perigee positions give rise to two unique orbits that have very practical applications. The first of these, the Sunsynchronous orbit, takes advantage of eastward nodal regression at inclinations greater than 90°. Looking at Figure 55, we see that the ascending node moves eastward about 1° per day at an inclination of about 98° (depending on the satellite’s altitude). Coincidentally, the Earth also moves around the Sun about 1° per day (360° in 365 days), so at this Sun-synchronous inclination, the satellite’s orbital plane will always maintain the same orientation to the Sun. This means the satellite can see the 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 change when pictures are taken days or even weeks apart. This is important for remotesensing missions such as reconnaissance, weather and monitoring of the Earth’s resources, because they use shadows to measure an object’s height. By maintaining the same Sun angle day after day, observers can better track changes in weather, terrain or man-made features. 63 STK Astronautics Primer Sun line Sun angle Sun angle Sun line orbit plane Sun line Sun angle Sun angle Sun line orbit plane rotates at ~1 deg/ day due to Earth’s oblateness Earth moves around the Sun at ~1 deg/day Figure 55: Sun-Synchronous Orbit. Sun-synchronous orbits take advantage of the rate of change in right ascension of the ascending node caused by the Earth’s oblateness. By carefully selecting the proper inclination and altitude, we can match the rotation of Ω with the movement of the Earth around the Sun. In this way, the same angle between the orbit 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 same Sun angle on targets on the Earth’s surface. The second unique orbit is the Molniya orbit, named after the Russian word for lightning (as in “quick-as-lightning”). This is a 12-hour orbit with high eccentricity (about e = 0.7) and a perigee location in the Southern Hemisphere. The inclination is 63.4°—why? Because at this inclination, the perigee doesn’t rotate so the satellite “hangs” over the Northern Hemisphere for nearly 11 hours of its 12-hour period before it whips “quick as lightning” through perigee in the Southern Hemisphere. Figure 56 shows the orbit and ground tracks for a Molniya orbit. The Russians used this orbit for their communications satellites because they didn’t have launch vehicles large enough to put them into geosynchronous orbits from their far northern launch sites. Molniya orbits also offer better coverage of latitudes above 80° north. 64 STK Astronautics Primer Figure 56: Molniya orbit and ground tracks. Molniya orbits take advantage of the fact that ω, due to Earth’s oblateness, is zero at an inclination of 63.4°. Thus, apogee stays over the Northern Hemisphere, covering high latitudes for 11 hours of the 12-hour orbit period. Other Perturbations Other perturbing forces can affect a satellite’s orbit and its orientation within that orbit. These forces are usually much smaller than the J2 (oblate Earth) and drag forces but, depending on the required accuracy, satellite planners may need to anticipate their effects. These forces include: ♦ Solar radiation pressure, which can cause long-term orbit perturbations and unwanted satellite rotation. ♦ Third-body gravitational effects (Moon, Sun, planets, etc.), which can perturb orbits at high altitudes and on interplanetary trajectories. ♦ Unexpected thrusting caused by either out-gassing or malfunctioning thrusters, which can perturb the orbit and cause satellite rotation. Dealing with Perturbations Understanding and modeling orbit perturbations is one of the primary activities of astrodynamics. Even very early space pioneers such as Kepler and Newton spent considerable effort grappling with the various forces that disturb a satellite from pure two-body motion. Let’s begin by classifying perturbations with respect to their relative effects on orbital elements. Perturbations can cause both secular and periodic changes to orbital elements. Secular perturbations are those that cause elements to steadily diverge over time. Periodic perturbations are those that impart a sinusoidal variation in elements over time. Short-term periodic perturbations are those with a period less than the orbit period. Long-term periodic perturbations are those with a 65 STK Astronautics Primer period greater than one orbit period. We can now look at two techniques for modeling both secular and periodic perturbations. The relative effects of these different types are illustrated in Figure 57. Long term effects on orbital elements for various types of perturbations Orbital element variation (arbitrary units) 9 Long-term Periodic 8 7 Secular 6 5 4 3 Short- term Periodic 2 1 8 7 6 5 4 3 2 1 0 0 Orbit Periods Figure 57: Types of orbit perturbations. Orbit perturbations are categorized based on their long-term effects on orbital elements. General Perturbations Techniques General perturbations techniques are those that generalize the effects on orbital elements in order to develop analytic expressions allowing for direct computation. In the grossest sense, general perturbation techniques apply “fudge factors” to the simple two-body solution to account for the effect of different perturbation sources. For example, returning to our baseball-throwing astronaut scenario, we could model the drag on the baseball using: D= where D = Drag (N) ρ = air density (kg/m3) V = baseball velocity (m/sec) 66 1 ρV 2 A 2 STK Astronautics Primer A = baseball cross-sectional area (m2) We could substitute this expression into our baseball equations to derive new equations of motion that would account for the general effects of drag. Even with this additional complexity, the equations could still be solved analytically. One of the most widely used propagators was developed by the North American Aerospace Defense Command (NORAD) to track the 8000-plus satellites and space junk in orbit around the Earth. Called Merged Simplified General Perturbations-4, MSGP-4, this technique uses the generalized approach to model orbit perturbations. Special Perturbations Techniques In contrast, special perturbation techniques are based on special case assumptions about the orbit scenario that allow for more detailed modeling of individual perturbation sources. Two-line Element Sets One of the most commonly used methods of communicating orbital parameters is the 2-line element sets generated by NORAD in Cheyenne Mountain, Colorado (literally, in Cheyenne Mountain!). It is important to note that TLEs were developed specifically for use with the MSGP-4 propagator! Using TLEs with any other propagator may invalidate some of the built-in 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 in modeling perturbations in the MSGP-4 propagator. STK Propagators In selecting the “best” propagator to use for a given application, it is important to consider 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 can break down. The relative accuracy among various propagators can vary widely depending on the scenario. For example, a geostationary spacecraft is well above most atmospheric drag and J2 perturbations. Therefore, a short-term difference between the two-body propagator and a more complex technique could be relatively small. However, for a spacecraft in low-Earth orbit, the short-term differences between the two solutions could be significant. The objective is to choose the most appropriate propagation scheme for a given application. Unfortunately, any propagation technique is simply an attempt to model events in the real world. Regardless of the technique chosen, only frequent tracking of an orbit can guarantee that the predicted orbital parameters will match the real world. 67 STK Astronautics Primer Two-Body The two-body propagator or Keplerian motion propagator uses the same basic technique outlined in the two-body equation of motion development. This technique assumes the Earth is a perfect sphere and the only force acting on a satellite is gravity. This propagator doesn’t account for any perturbations. J2 The J2 propagator accounts for the 1st order effects of J2 Earth oblateness. This effect causes 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 J4 effects. J3, which causes long-term periodic effects, is not modeled. Because the 2nd order J2 and 1st order J4 effects are very small, you’ll see very little differences between the J2 and J4 propagators for most orbits considered. MSGP-4 MSGP-4 stands for Merged Simplified General Perturbations-4. It is one of the most widely used propagators in the industry. This technique uses the generalized approach to model orbit perturbations, including both secular and periodic variations such as Earth oblateness, solar and lunar gravitational effects and drag. It is important to understand the purpose for which MSGP-4 was developed. NORAD wanted a simple propagator that would provide acceptable results for a wide variety of tracking tasks, from tracking high-priority military satellites to keeping tabs on space junk. Given the over 8000 objects NORAD must track, a technique was needed that would not be computationally intensive (is was first developed back when computers were much slower than today). Furthermore, there are far fewer ground tracking sites than there are objects to track. Thus, it is important that the propagated solution be good enough to ensure the tracking radar can find a specific object the next time it gets around to tracking it (which, for some very low priority objects like pieces of rocket boosters, may be days or even weeks). Because MSGP-4 is a generalized approach, it is specifically tailored to a given set of inputs: the TLE sets that contain parameters that make the analytic calculations valid. For best results, MSGP-4 should always be used with TLEs. Likewise, NORADgenerated TLEs should only be used in the MSGP-4 propagator. 68 STK Astronautics Primer HPOP HPOP is the High Precision Orbit Propagator. As its name implies, it uses a powerful propagation technique to incorporate sophisticated orbit perturbation models. HPOP uses a variety of high-fidelity models including: ♦ Joint Gravity Model (JGM) 2—a highly precise model of the Earth’s oblateness. ♦ 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 Harris Priester model, which takes into account daily variations in the height of the atmosphere due to solar heating among other parameters. ♦ Solar radiation pressure—yes, sunlight produces a small force on any exposed surface. This force varies depending on how reflective the surface is—a mirrored surface is more reflective than a black surface. Depending on the application, HPOP can deliver accuracy on the order of 10 meters per 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 this accurate to start with. Putting NORAD-generated TLEs into HPOP will not necessarily give you a better solution. No propagator can create accuracy, at best it can only minimize the long term dispersions due to inherent limitations in our ability to model the effects of perturbations. Great Arc The Great Arc propagator allows the user the model the flight path of a vehicle flying close to Earth. By providing way points and speed, STK uses Great Arc to predict where and when it will be next. The propagation scheme is essentially the same as the Two-Body propagator, no perturbations are assumed. Ballistic The Ballistic propagator is a variation of the Two-Body propagator for use with ballistic trajectories. These are the trajectories used by artillery shells, suborbital sounding rockets and ballistic missiles, allowing the user to predict impact points or determined required velocity to reach a certain point. The propagation “engine” is the same as the Two-Body propagator, no perturbations are modeled. LOP The Long-term Orbit Predictor (LOP) allows accurate prediction of a satellite’s orbit over many months or years. This is often used for long duration mission design, fuel budget definition, and end-of-life studies. For performance reasons, it is impractical 69 STK Astronautics Primer to compute the long-term variation in a satellite’s orbit using high accuracy, small time step, propagators that compute a satellite’s position as it moves through its orbit. LOP exploits a “variation of parameters” approach which integrates analytically derived equations of motion computing the average effects of perturbations over an orbit. This approach allows large multi-orbit time steps and typically improves computational speed by several hundred times while still offering high fidelity computation of orbit parameters. Lifetime Lifetime estimates the amount of time a low Earth orbiting satellite can be expected to remain in orbit before the drag of the atmosphere causes reentry. While the computational algorithms are similar to those implemented in the Long-term Orbit Predictor, there are some important differences. First, a much more accurate atmospheric model is implemented to compute the drag effects. The gravitational model for the Earth, however, is significantly simplified since the inclusion of the higher order terms doesn’t impact orbit decay estimates. 70 STK Astronautics Primer 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 communication architecture? Communication Links ♦ What are the communication paths used by satellites and ground stations? 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 see right to the edge of the horizon, which is about 8 miles away. If you were to watch a ship sailing away from you, you would notice that the hull would disappear first, followed by the top of the mast The taller the mast, the further the ship could be from the shore before disappearing completely from sight. An object is in your line of sight if you can draw a straight line between yourself and the object without any interference, such as a mountain or a bend in a road. An object beyond the horizon is below our line of sight and, therefore, can be difficult to communicate with. Early methods of long-distance communications increased the effective line of sight by employing methods such as smoke signals or other means. Because the line of sight was raised so that others could see, or receive the message being sent, communication among objects that didn’t really have a direct line of sight was achieved. At the beginning of the 20th Century, radio engineers discovered that certain frequencies could be bounced off the ionosphere, greatly extending the effective line of sight for communications and creating a new “radio horizon” far beyond the more limited visual horizon. Today’s communications satellites take this basic principle to the extreme. Ground-based operators can “bounce” radio 70 STK Astronautics Primer communications off satellites stationed in geosynchronous orbit, creating a virtual line of sight extending half a world away. Furthermore, by bouncing signals between satellites, this virtual line of sight can be extended to cover the entire global community. Satellite access is the problem of determining when, where and for how long a satellite (or any number of objects you may be interested in) is within line of sight of other objects. Communications Architecture To understand the satellite access problem more clearly, let’s begin by reviewing the players in the access problem. Figure 58 illustrates the elements that make up a communications architecture. crosslink return link k lin upl ink k lin d ar rw fo do wn lin k rn tu re crosslink forward link Figure 58 Communications Architecture. The communications architecture consists of space and ground-based elements tied together by communications paths or links. The communications architecture has four elements: Spacecraft The spaceborne elements of the system. 71 STK Astronautics Primer Ground stations The Earth-based antennas and receivers that talk to the spacecraft. These are typically remote tracking sites or users of mission data. Control Center The command center that controls the spacecraft and all other elements of the system. Relay satellites Additional satellites that link the primary spacecraft with the ground stations and control center. Communications Links Information moves among the elements of the communications architecture using various communication paths or links. Uplink Data sent from a ground-based 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 satellite and a relay satellite. Understanding Access The simplest example of a satellite access problem is that between a satellite TV dish and a direct-broadcast geosynchronous satellite. As a user, you just want to point your dish and start watching the big game. Thus, you’re only interested in downlink. 72 STK Astronautics Primer Figure 59 Downlinking. As a direct TV subscriber, you would only be interested in downlinking. For satellites in geostationary orbit, the geometry and dynamic nature of both uplink and downlink is very stable. It is this stability that make geostationary satellites so useful for point-to-point message relaying. You can set up you dish, point it at a pre-determined point in the sky and pretty much forget about it. 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 is almost constant (at least constant enough to ensure uninterrupted broadcast of that big game!). Describing Access So how do you know where to point the satellite dish? This depends on two important pieces of information: ♦ Your location (latitude and longitude) ♦ The satellite’s location (orbital elements) From this information, STK can determine the azimuth and elevation settings for your dish. These two important parameters are defined below. ♦ Azimuth - The compass direction between the ground site and the satellite direction, e.g., due south would be 180°. ♦ Elevation - Angle measured from the local horizontal to the satellite direction, e.g., directly overhead would be 90°. In addition, STK also computes range—the distance between the dish and the satellite. While range is not so important to the average satellite TV user, it becomes very important for communications engineers who must ensure there is sufficient transmission power to effectively carry the signal across this distance. These three parameters are fairly constant for a geosynchronous satellite; however, for any other satellite, we must include an additional parameter— 73 STK Astronautics Primer time. Imagine it is exactly 13:20 (1:20 p.m.) local time and you want to point a radar antenna at an airplane flying over your position. The plane is initially due south of you, flying north but out of sight below the horizon. As the plane first comes into view, azimuth will be 180° and the elevation 0°. As it continues to fly north, azimuth will stay constant (disregarding Earth rotation) and the elevation angle will increase. As it flies overhead, the azimuth angle will switch around to 0° (it is now north of your position) and the elevation angle will gradually decrease until the plane once again drops from view below the horizon. If we kept track of the azimuth and elevation viewing angles to the plane at 10-minute intervals we could build a simple table, or access report, as shown 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 undefined) 14:00 0 45 Azimuth has switched around, plane is now north of your position. 14:10 0 0 Plane drops below horizon, out of view (azimuth is The geometry with respect to a plane flying directly overhead is relatively easy to visualize. However, if you’re faced with the problem of a satellite in a highly eccentric orbit flying over a position northwest of you on a descending node, things become much more complicated. Fortunately, STK works out the geometry for you. Figure 60 shows a relatively simple satellite access geometry. 74 STK Astronautics Primer Figure 60 Satellite Access. Satellite access refers to the problem of determining the geometry and timing of line-of-sight between various ground and space-based objects. Access reports can be easily generated using STK; the report provides azimuth, elevation and range (AER) data for specified time intervals between whatever objects you choose to define. Cumulative access time or duration can also be reported. Access information becomes even more complicated when multiple objects must be taken into account. For example, if you are relaying information between various ground stations and relay satellites or using an entire constellation of satellites, the geometry can become very complex. To handle these tasks, STK allows you to link together various objects to create a “chain” for which access information can be determined. Figure 61 shows a more complex series of “chained” objects. If one of these “objects” is the Sun, orbit lighting—a critical parameter for power and thermal management—can be computed. Figure 61 Chained objects. A series of ground and space-based objects can be “chained” together, allowing STK to compute access information between all of them. In this picture, access among a facility, relay satellite and a second facility is shown. RECOMMENDED READING For a more detailed explanation of the topics in this primer as well as an introduction to the space environment, spacecraft design, rockets and systems, we recommend: 75 STK Astronautics Primer Understanding Space: An Introduction to Astronautics, Sellers, 1994, McGraw-Hill. To purchase a copy of Understanding Space, please contact McGraw-Hill at www.mcgraw-hill.com, www.mhhe.com, or 800-338-3987 : The McGraw-Hill Companies, >Order Services, PO Bos 545, Blacklick, Ohio. ISBN: 0-07057027-5. Understanding Space is part of the Space Technology Series, a cooperative activity of the United States Department of Defense and National Aeronautics and Space Administration. Series editor is Dr. Wiley J. Larson. Other books in the 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 and Microcosm. Cost-Effective Space Mission Operations, Boden and Larson, 1996, McGrawHill. Spacecraft Structures and Mechanisms: From Concept to Launch, Sarafin and Larson, 1995, Kluwer and Microcosm. For more information about these books, please contact Kluwer at www.wkap.nl, McGraw-Hill at www.mcgraw-hill.com or 800-338-3987, or Microcosm at www.microcosm.com. Future books in the series: Modeling and Simulation: In Integrated Approach to Development an Operations, 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, Wertz and Dawson, Kluwer and Microcosm. 76