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Mathematics of Space - Rendezvous A Videotape for

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EV-1998-02-014-HQ Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ 1 Video Synopsis Shuttle have gained valuable practice in the maneuvers they will need to work with the ISS. These maneuvers are Title: The Mathematics of Space - complex and docking is a delicate operation. Rendezvous Each maneuver employs extensive use of mathematics to achieve the objective. Length: 17 minutes The objective of this video is to invite students to work out some of the fundamental Subjects: The mathematics of Shuttle crews use for rendezvous. orbital rendezvous. The equations presented in the video are listed in a following section of this guide. Description: Some students may become confused This program addresses the basic with some of the mathematical operations mathematical operations of spacecraft shown. The video is designed with stopping rendezvous in Earth . Middle school places for you to work the mathematics with students in a mathematics work to solve your students. some problems that permit the One rendezvous concept presented in to rendezvous and dock with the Russian the video may need some additional explana- . The video has stopping tion. By firing its rocket engines to increase points to permit viewers to work the problems velocity, the Space Shuttle actually slows as well. down. Conversely, firing engines to slow down causes the vehicle to speed up. These Mathematics Standards: paradoxical statements are easier to under- stand when you remember that movement in Mathematics as Problem Solving space is a three-dimensional problem. Mathematics as Communication Changes in velocity leads to changes in Mathematics as Reasoning altitude. Computation and Estimation On Earth, rendezvous between two automobiles is a relatively simple operation. Geometry Both drive at specific speeds to arrive at a Measurement specific location. When the automobiles arrive at the right place, they stop. In space, Background the rendezvous location is over a specific In a few years, the International Space Sta- place on Earth but it is also at a specific tion (ISS) will be ready for full-time occupancy altitude. When the two spacecraft arrive, they by crews of astronauts. From their vantage cannot stop. Doing so will cause them to fall in space, astronauts will study Earth's back to Earth. Instead, their rendezvous is at environment and conduct a variety of scien- a specific location and altitude (three dimen- tific and technological experiments that will sions), and at a specific time. Time is impor- ultimately help to improve life on Earth. tant when you consider that the spacecraft One of the critical tasks to the con- will be traveling at 5 or more kilometers per struction and use of a space station is the second. A mere 5-second error will cause the ability to rendezvous in space. Conse- spacecraft to miss each other by 25 kilome- quently, the first phase of the International ters. Space Station program is a series of Space The nature of is Shuttle mission rendezvous and docking also complicated by some basic physical activities with the Russian Space Station Mir. laws. Altitude and velocity of a spacecraft By bringing crew and equipment to Mir, 2 Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ are related. Spacecraft in low travel your rocket engines to accelerate. The accel- very fast because the gravitational pull is eration causes your spacecraft to climb higher strong. In higher orbits, spacecraft travel above Earth. As you climb higher, your slower because the force of gravity is less. velocity diminishes until you are traveling at The force of gravity between two objects the right velocity for the higher orbit. It is a (Earth and the Shuttle) is determined by the slower velocity than you were traveling before following mathematical relationship that was the firing of the engines. In other words, you first formulated by Isaac Newton and later sped up so that you could slow down in a modified by Henry Cavendish. higher orbit. m m The reverse is true if you want to go to f =G12 a lower orbit. To descend, you fire rocket 2 engines in the opposite direction you are r traveling. This causes your spacecraft to G in the is the gravitational con- slow. Earth's gravity pulls your spacecraft stant. The r in the equation is the distance downward, and as you fall, your speed in- between the center of Earth and the center of creases until you are at the right speed for the the Shuttle (not the altitude of the Shuttle new altitude. Your speed is greater in the over Earth's surface). As you can see, r has lower orbit. Thus, you slowed down to speed an inverse square relationship in the equa- up. tion. That means that the closer the centers Although somewhat complicated, this of the two bodies are to each other, the paradox helps to accomplish rendezvous. greater the force of attraction. It also means For example, when the two spacecraft are on that increasing the distance between the opposite sides of Earth from each other, centers decreases the attraction by an in- having one spacecraft in a lower orbit will verse square. enable it to close the distance. In the lower The difference in gravitational attrac- orbit, the spacecraft will not only travel faster tion with change in distance (orbital altitude) than the higher spacecraft, but the orbit has a is where the speed-up/slow-down paradox smaller circumference as well. After closing comes in. You must travel faster in a lower the distance, the lower spacecraft can begin orbit than a higher one to stay in orbit. If you maneuvers to adjust to the right altitude for want to go to a higher orbit, you must fire the rendezvous. How long to fire rocket engines, when to do it, and in what directions is determined with mathematics. Equations Used In the Program

Degrees longitude orbital shifts eastward with each orbit. o x 360 o = x = 23 92 min 24 hrs (60 min/hr)

Number of orbits so that Mir flies over Moscow. Mir - Moscow = Longitude Distance o 68o 105 = o 2.9565 orbits - 37 23 /orbit o or 3 orbits 68 Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ 3 Time for Mir's orbit to cross Moscow.

92 min/orbit 276 min x 3 orbits or = 4.6 hr 60 min/hr 276 min

Mir's . Distance = Speed Time

Distance (circumference) Mir travels during one orbit. (The altitude is the distance from Earth's center to Mir.) 6,778 km x 2 π = C Mir

42.587 km = C Mir

Mir's orbital speed.

42,587 km 63 km/min = speed 92 min or x 60 min/hr 27,780 km/hr 463 km/min = speed

Shuttle speed change needed to raise orbit 7 kilometers. (It is stated in the video that a change in velocity of 0.4 meters per second raises the Shuttle 1 kilometer.)

target altitude 305 km present altitude - 298 km 7km/1km

7 km x 0.4 m/sec = 2.8 m/sec 1 km

4 Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ Sine Curve Orbits? Classroom Activities Materials: Earth globe Paper How High? Tape Marker Materials: Earth globe Scissors Metric ruler Objective: To show why a sine curve orbital Objective: To learn why it is necessary to plot is created when an orbit is portrayed on a exaggerate altitudes when orbits are shown in flat map. model form. Procedure: Procedure: When shown on a flat map, Space Shuttle the diameter of the globe you are orbits resemble sine curves. To show why using for the model. Determine its scale. To this happens, wrap and tape a cylinder of do this, you will need to know the actual paper around an Earth globe. Use a marker diameter of Earth (12,756 km). Using the pen to draw an orbit around the cylinder. same scale, determine how high above the Start with an orbit inclined 28 degrees. Draw globe's surface the Space Shuttle and Mir the line around the cylinder so that it falls on a would be (400 km). plane inclined to the globe's equator by 28 degrees. Remove the cylinder and cut the Discussion: Diagrams of planets and spacecraft orbiting them are difficult to portray accurately. The diagrams of Earth, the Space Shuttle, and Mir Space Station used in the video greatly exaggerate the distance the orbiting spacecraft are above Earth. Without doing this, the orbits would lie so closely to the surface of Earth that the lines would be indistinguishable.

Extension: o Using the same scale, determine how far the 28.5 would be from Earth.

Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ 5 paper along the line you drew. If you drew Earth's equator. Many Shuttle orbits are the line carefully, the edge of the cut will fall inclined at 28.5 degrees. This is the on a plane. Unwrap the cylinder and look at geographic latitude of the Kennedy Space the shape of the orbit. Center. When a Shuttle is launched due east, its orbit is inclined 28.5 degrees. This Discussion: happens because an orbit must be concentric Orbital maps displayed in Mission Control at with the center of Earth. In geographic terms, the NASA show three the orbit must be a great circle. Space Shuttle orbits at a time. A small Space Shuttle orbiter is displayed on one of Extension: the orbits over the geographic position the Create other cylinders for different orbits such actual orbiter is flying. The curve of the orbits as 35 degrees and 51.6 degrees (orbit of the resembles a sine curve. The steepness of International Space Station). Compare the the curve is determined by the angle in which steepness of the curves when the cylinders the Space Shuttle was launched in respect to are flattened.

Equator

51.6 Degrees

28.5 Degrees

Equator

6 Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ References Pilot: Eileen M, Collins (Lt. Col., USAF) Eileen Collins was born in Elmira, New York. She NASA ON-LINE RESOURCES FOR EDUCATORS received an A.A. degree in mathematics science from Corning Community College, a B.S. degree in NASA On-line Resources for Educators provide mathematics economics from Syracuse University, an current educational information and instructional M.S. degree in operations research from Stanford resource materials to teachers, faculty, and students. University, and an M.A. degree in space systems A wide range of information is available, including management from Webster University. Collins was one science, mathematics, engineering, and technology of the first women to graduate from pilot training at education lesson plans, historical information related Vance Air Force Base (AFB), Oklahoma, and later to the aeronautics and space program, current status became the first woman T-38 instructor pilot at that reports on NASA projects, news releases, information base. She served as a C-141 pilot, aircraft on NASA educational programs, useful software and commander, and instructor pilot at Travis AFB, graphics files. Educators and students can also use California. In 1983, she flew in Operation Urgent Fury NASA resources as learning tools to explore the in Grenada. She was later assigned to the USAF Internet, access information about educational grants, Academy where she was an assistant professor in interact with other schools, and participate in on-line mathematics and a T-41 instructor pilot. In 1990, she interactive projects with NASA scientists, engineers, graduated from the AF Test Pilot School at Edwards and other team members to experience the excitement AFB, California, where she was the class leader. of real NASA projects. Collins has logged over 4,000 hours in 30 different types of aircraft and was selected as an in Access these resources through the NASA Education 1990. Aboard STS-63, she spent more than 8 days in Home Page: space, and became the first woman pilot of a Space http://education.nasa.gov Shuttle. STS-84 was her second Shuttle mission.

Other web sites of interest: Payload Commander, Mission Specialist: http://www.jsc.nasa.gov Jean-Francois Clervoy (Ingenieur en Chef de l http://station.nasa.gov/core.html Armement, ESA Astronaut) http://www.osf.hq.nasa.gov/heds Jean-Francois Clervoy was born in Longeville-les Metz, France. He graduated from Ecole Polytechnique, Paris, in 1981; from Ecole Supérieure STS-84 Crew Biographies de l' Aéonautiqe et de l' Espace, Toulouse, in 1983; and from Ecole du Personnel Navigant d Essais et de Commander: Charles J. Precourt (Col., USAF) Reception,lstres, as a flight test engineer, in 1987. Charles Precourt was born in Waltham, Seconded from the Délégation Générale pour L' Massachusetts, and grew up in Hudson, Armement to the French Space Agency, Clervoy Massachusetts. He received a B.S. degree worked on various satellite projects when he was in aeronautical engineering from the USAF Academy, selected in the second of French astronauts. In an M.S. degree in engineering management from 1985, Clervoy was the Chief Test Director of the Golden Gate University, and an M.A. degree in Parabolic Flight Program and provided support to the national security affairs and strategic studies from the European Manned Space Programs Hermes and U.S. Naval War College. He also studied as an Columbus until he was selected as a European Space exchange student at the French Air Force Academy. Agency (ESA) astronaut and reported to NASA Precourt flew the F-15 while based at Bitburg Air Base Johnson Space Center for in 1992. in Germany. As a test pilot at Edwards Air Force Base, Clervoy has also received training on the Russian California, Precourt flew the F-T5E, F-4, A-7, and A-37 Space Systems and Mir at Star City, Moscow. aircraft. His flight experience includes more than He served as a mission specialist aboard STS-66 5,500 hours in over 50 types of civil and military during which he logged more than 10 days in space. aircraft. Precourt was selected as an astronaut in 1990 STS-84 was Clervoy's second Space Shuttle mission. and flew as a mission specialist on STS-55 in 1993 and as a pilot aboard STS-71 (the first Space Shuttle Mission Specialist: Edward T. Lu (Ph.D ) mission to dock with the Russian Mir Space Station Edward Lu was born in Springfield, Massachusetts, and exchange crews) in 1995. He has spent nearly 20 raised in Webster, New York and most recently, days in space. As commander of STS-84, Precourt resided in Honolulu, Hawaii. He received a B.S. degree was in charge of the sixth Shuttle mission scheduled to in electrical engineering from Cornell University and a rendezvous and dock with Mir. Ph.D. in applied physics from Stanford University. Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ 7 Since receiving his Ph D., Dr. Lu has been a research course of general space training. Upon course physicist working in the fields of solar physics and completion, Kondakova was qualified as a flight astrophysics. He was a visiting scientist at the High engineer. During 1994 she was in training for the Altitude Observatory in Boulder, Colorado, and, at one seventeenth main Mir mission known as the EuroMir- point, simultaneously worked with the Joint Institute for 94 flight. She completed her first flight on board the Laboratory Astrophysics at the University of Colorado. spacecraft Soyuz TM-17 and the orbital complex Mir He was a postdoctoral fellow at the Institute for as a flight engineer in March 1995. Kondakova has Astronomy in Honolulu and has developed a number logged approximately 169 days in orbit. She served of new theoretical advances which have provided, for as a mission specialist on exchange from the Russian the first time a basic understanding of the underlying Space Agency during STS-84, her first Space Shuttle physics of solar flares. Dr. Lu has published articles mission. on a wide range of topics including solar flares, cosmology, solar oscillations, statistical mechanics, Mission Specialist, NASA-Mir 4: Jerry M. Linenger and plasma physics. He has given numerous invited (Capt., USN) lectures at various universities and international Jerry Linenger was born and raised near Eastpointe, conferences. He also holds a commercial pilot certifi- Michigan. He earned a B.S. degree from the U S. cate with instrument and multi-engine ratings. Dr. Lu Naval Academy, an M.D. from Wayne State Univer- was selected as a NASA astronaut in 1994 and has sity, an M.S. in systems management from the worked in the Computer Support Branch of the Astro- University of Southern California, and both an M.P.H. naut Office. STS-84 was Dr. Lu's first Space Shuttle in health administration and a Ph.D in epidemiology mission. from the University of North Carolina. He completed surgical internship, aerospace medicine, and preven- Mission Specialist: Carlos I. Noriega (Major, tive medicine programs. Linenger served as a naval USMC) Carlos Noriega was born in Lima, Peru, and flight surgeon at Cuba Point, Republic of the Philip- raised in Santa Clara, California. He received a B.S. pines. He was then assigned as medical advisor to degree in computer science from the University of the Commander, Naval Air Forces, U.S. Pacific Fleet, Southern California. After flight school he flew CH-46 San Diego. He later became a research principal Sea Knight helicopters. In addition to two 6-month investigator at the Naval Health Research Center and shipboard deployments which included operations in a faculty member at the University of California-San support of the Multi-National Peacekeeping Force in Diego School of Medicine in the Division of Sports Beirut, Lebanon, he served as an aviation safety Medicine. Dr. Linenger was selected to be an astro- officer and instructor pilot. He has logged approxi- naut in 1992 and flew aboard STS-64, during which mately 2,000 flight hours in various fixed wing and he spent 11 days in space. Following training in Star rotary wing aircraft. From the Naval Postgraduate City, he traveled to the Mir Space Station aboard School, he received two masters degrees, one in STS-81. Linenger returned with the STS-84 crew computer science and the other in space systems aboard the Space Shuttle after spending approxi- operations. Upon graduation, he was assigned to the mately 130 days aboard the Mir Space Station. United States Space Command in Colorado Springs. In addition to serving as a Space Surveillance Center Mission Specialist, NASA-Mir 5: C. Michael Foale Commander, he was command representative for the (Ph.D.) development and integration of the major space and Michael Foale was born in Louth and raised in missile warning computer system upgrades for Chey- Cambridge, England. He attended the University of enne Mountain Air Force Base. Noriega was selected Cambridge Queens' College, where he earned a B.A. as an astronaut in 1994 and has worked technical degree in physics, National Sciences Tripos, with first issues concerning extravehicular activity (spacewalks). class honors. While at Queens' College he completed He served as flight engineer (mission specialist) on a Ph.D. in laboratory astrophysics. As a postgraduate STS-84, his first Space Shuttle flight. at Cambridge University, Dr. Foale participated in the organization and execution of scientific scuba diving Mission Specialist: Elena V. Kondakova projects including surveying underwater antiquities in Elena Kondakova was born in Mitischi, Moscow Greece. Dr. Foale joined NASA Johnson Space Region. She graduated from the Moscow Bauman Center in 1983 in the payload operations areas of the Technical Institute. Upon graduation, Kondakova Mission Operations Directorate. He was selected as started work in RSC-Energia, completing science an astronaut in 1987 and flew as a mission specialist projects, experiments, and research work. In 1989, on STS-45, the first of the series of missions she was selected as a cosmonaut candidate by the to address the atmosphere and its interaction with the RSC-Energia Main Design Bureau and sent to the sun, and again as a mission specialist on STS-56, Gagarin Cosmonaut Training Center to start the carrying ATLAS-2. He served as a mission specialist

8 Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ on STS-63 (February 2-11, 1995), the first rendezvous with the Russian Space Station Mir. During the flight he made a spacewalk (extravehicular activity) for 4 hours, 39 minutes, evaluating the effects of extremely cold conditions on his spacesuit as well as moving the 2,800-pound Spartan satellite as part of a mass- handling experiment. He launched to Mir on STS-84. After spending 4 months aboard Mir as a part of the Mir-24 crew, he returned to Earth with the crew of STS-86.

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