Mathematics Of Space - NASA
Educational ProductEducators Grades 5-12National Aeronautics andSpace AdministrationLiftoff to LearningMathematics of Space RendezvousA Videotape for MathematicsVideo Resource GuideEV-1998-02-014-HQMathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ1
Video SynopsisTitle: The Mathematics of Space RendezvousLength: 17 minutesSubjects: The mathematics of spacecraftorbital rendezvous.Description:This program addresses the basicmathematical operations of spacecraftrendezvous in Earth orbit. Middle schoolstudents in a mathematics class work to solvesome problems that permit the Space Shuttleto rendezvous and dock with the RussianSpace Station Mir. The video has stoppingpoints to permit viewers to work the problemsas well.Mathematics Standards:Mathematics as Problem SolvingMathematics as CommunicationMathematics as ReasoningComputation and tle astronauts have gained valuablepractice in the maneuvers they will need towork with the ISS. These maneuvers arecomplex and docking is a delicate operation.Each maneuver employs extensive use ofmathematics to achieve the objective.The objective of this video is to invitestudents to work out some of the fundamentalequations Shuttle crews use for rendezvous.The equations presented in the video arelisted in a following section of this guide.Some students may become confusedwith some of the mathematical operationsshown. The video is designed with stoppingplaces for you to work the mathematics withyour students.One rendezvous concept presented inthe video may need some additional explanation. By firing its rocket engines to increasevelocity, the Space Shuttle actually slowsdown. Conversely, firing engines to slowdown causes the vehicle to speed up. Theseparadoxical statements are easier to understand when you remember that movement inspace is a three-dimensional problem.Changes in velocity leads to changes inaltitude.On Earth, rendezvous between twoautomobiles is a relatively simple operation.Both drive at specific speeds to arrive at aspecific location. When the automobilesarrive at the right place, they stop. In space,the rendezvous location is over a specificplace on Earth but it is also at a specificaltitude. When the two spacecraft arrive, theycannot stop. Doing so will cause them to fallback to Earth. Instead, their rendezvous is ata specific location and altitude (three dimensions), and at a specific time. Time is important when you consider that the spacecraftwill be traveling at 5 or more kilometers persecond. A mere 5-second error will cause thespacecraft to miss each other by 25 kilometers.The nature of space rendezvous isalso complicated by some basic physicallaws. Altitude and velocity of a spacecraftIn a few years, the International Space Station (ISS) will be ready for full-time occupancyby crews of astronauts. From their vantagepoint in space, astronauts will study Earth'senvironment and conduct a variety of scientific and technological experiments that willultimately help to improve life on Earth.One of the critical tasks to the construction and use of a space station is theability to rendezvous in space. Consequently, the first phase of the InternationalSpace Station program is a series of SpaceShuttle mission rendezvous and dockingactivities with the Russian Space Station Mir.By bringing crew and equipment to Mir,Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ2
are related. Spacecraft in low orbits travelvery fast because the gravitational pull isstrong. In higher orbits, spacecraft travelslower because the force of gravity is less.The force of gravity between two objects(Earth and the Shuttle) is determined by thefollowing mathematical relationship that wasfirst formulated by Isaac Newton and latermodified by Henry Cavendish.f Gyour rocket engines to accelerate. The acceleration causes your spacecraft to climb higherabove Earth. As you climb higher, yourvelocity diminishes until you are traveling atthe right velocity for the higher orbit. It is aslower velocity than you were traveling beforethe firing of the engines. In other words, yousped up so that you could slow down in ahigher orbit.The reverse is true if you want to go toa lower orbit. To descend, you fire rocketengines in the opposite direction you aretraveling. This causes your spacecraft toslow. Earth's gravity pulls your spacecraftdownward, and as you fall, your speed increases until you are at the right speed for thenew altitude. Your speed is greater in thelower orbit. Thus, you slowed down to speedup.Although somewhat complicated, thisparadox helps to accomplish rendezvous.For example, when the two spacecraft are onopposite sides of Earth from each other,having one spacecraft in a lower orbit willenable it to close the distance. In the lowerorbit, the spacecraft will not only travel fasterthan the higher spacecraft, but the orbit has asmaller circumference as well. After closingthe distance, the lower spacecraft can beginmaneuvers to adjust to the right altitude forthe rendezvous. How long to fire rocketengines, when to do it, and in what directionsis determined with mathematics.m 1 m2r2G in the equation is the gravitational constant. The r in the equation is the distancebetween the center of Earth and the center ofthe Shuttle (not the altitude of the Shuttleover Earth's surface). As you can see, r hasan inverse square relationship in the equation. That means that the closer the centersof the two bodies are to each other, thegreater the force of attraction. It also meansthat increasing the distance between thecenters decreases the attraction by an inverse square.The difference in gravitational attraction with change in distance (orbital altitude)is where the speed-up/slow-down paradoxcomes in. You must travel faster in a lowerorbit than a higher one to stay in orbit. If youwant to go to a higher orbit, you must fireEquations Used In the ProgramDegrees longitude orbital ground track shiftseastward with each orbit.o360x 24 hrs (60 min/hr)92 minx 23oNumber of orbits so that Mir flies over Moscow.Mir - Moscow Longitude Distanceo105- 3768oo68o23 /orbit 2.9565 orbitsor 3 orbitsMathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ3
Time for Mir's orbit to cross Moscow.92 min/orbitx 3 orbits276 minor276 min 4.6 hr60 min/hrMir's orbital speed.Distance SpeedTimeDistance (circumference) Mir travels during oneorbit. (The altitude is the distance from Earth'scenter to Mir.)6,778 km x 2 π C Mir42.587 km C MirMir's orbital speed.42,587 km speed92 minor63 km/minx 60 min/hr27,780 km/hr463 km/min speedShuttle speed change needed to raise orbit 7kilometers. (It is stated in the video that a changein velocity of 0.4 meters per second raises theShuttle 1 kilometer.)target altitude 305 kmpresent altitude - 298 km7km/1km7 kmx 0.4 m/sec 2.8 m/sec1 km4Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ
Sine Curve Orbits?Classroom ActivitiesHow High?Materials:Earth globeMetric rulerObjective: To learn why it is necessary toexaggerate altitudes when orbits are shown inmodel form.Procedure:Measure the diameter of the globe you areusing for the model. Determine its scale. Todo this, you will need to know the actualdiameter of Earth (12,756 km). Using thesame scale, determine how high above theglobe's surface the Space Shuttle and Mirwould be (400 km).Discussion:Diagrams of planets and spacecraft orbitingthem are difficult to portray accurately. Thediagrams of Earth, the Space Shuttle, and MirSpace Station used in the video greatlyexaggerate the distance the orbitingspacecraft are above Earth. Without doingthis, the orbits would lie so closely to thesurface of Earth that the lines would beindistinguishable.Materials:Earth globePaperTapeMarkerScissorsObjective: To show why a sine curve orbitalplot is created when an orbit is portrayed on aflat map.Procedure:When shown on a flat map, Space Shuttleorbits resemble sine curves. To show whythis happens, wrap and tape a cylinder ofpaper around an Earth globe. Use a markerpen to draw an orbit around the cylinder.Start with an orbit inclined 28 degrees. Drawthe line around the cylinder so that it falls on aplane inclined to the globe's equator by 28degrees. Remove the cylinder and cut theExtension:Using the same scale, determine how far theMoon would be from Earth.Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQo28.55
paper along the line you drew. If you drewthe line carefully, the edge of the cut will fallon a plane. Unwrap the cylinder and look atthe shape of the orbit.Discussion:Orbital maps displayed in Mission Control atthe NASA Johnson Space Center show threeSpace Shuttle orbits at a time. A smallSpace Shuttle orbiter is displayed on one ofthe orbits over the geographic position theactual orbiter is flying. The curve of the orbitsresembles a sine curve. The steepness ofthe curve is determined by the angle in whichthe Space Shuttle was launched in respect toEarth's equator. Many Shuttle orbits areinclined at 28.5 degrees. This is thegeographic latitude of the Kennedy SpaceCenter. When a Shuttle is launched dueeast, its orbit is inclined 28.5 degrees. Thishappens because an orbit must be concentricwith the center of Earth. In geographic terms,the orbit must be a great circle.Extension:Create other cylinders for different orbits suchas 35 degrees and 51.6 degrees (orbit of theInternational Space Station). Compare thesteepness of the curves when the cylindersare flattened.Equatorsegree51.6 D28.5DegreesEquator6Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ
ReferencesNASA ON-LINE RESOURCES FOR EDUCATORSNASA On-line Resources for Educators providecurrent educational information and instructionalresource materials to teachers, faculty, and students.A wide range of information is available, includingscience, mathematics, engineering, and technologyeducation lesson plans, historical information relatedto the aeronautics and space program, current statusreports on NASA projects, news releases, informationon NASA educational programs, useful software andgraphics files. Educators and students can also useNASA resources as learning tools to explore theInternet, access information about educational grants,interact with other schools, and participate in on-lineinteractive projects with NASA scientists, engineers,and other team members to experience the excitementof real NASA projects.Access these resources through the NASA EducationHome Page:http://education.nasa.govOther web sites of 84 Crew BiographiesCommander: Charles J. Precourt (Col., USAF)Charles Precourt was born in Waltham,Massachusetts, and grew up in Hudson,Massachusetts. He received a B.S. degreein aeronautical engineering from the USAF Academy,an M.S. degree in engineering management fromGolden Gate University, and an M.A. degree innational security affairs and strategic studies from theU.S. Naval War College. He also studied as anexchange student at the French Air Force Academy.Precourt flew the F-15 while based at Bitburg Air Basein Germany. As a test pilot at Edwards Air Force Base,California, Precourt flew the F-T5E, F-4, A-7, and A-37aircraft. His flight experience includes more than5,500 hours in over 50 types of civil and militaryaircraft. Precourt was selected as an astronaut in 1990and flew as a mission specialist on STS-55 in 1993and as a pilot aboard STS-71 (the first Space Shuttlemission to dock with the Russian Mir Space Stationand exchange crews) in 1995. He has spent nearly 20days in space. As commander of STS-84, Precourtwas in charge of the sixth Shuttle mission scheduled torendezvous and dock with Mir.Pilot: Eileen M, Collins (Lt. Col., USAF)Eileen Collins was born in Elmira, New York. Shereceived an A.A. degree in mathematics science fromCorning Community College, a B.S. degree inmathematics economics from Syracuse University, anM.S. degree in operations research from StanfordUniversity, and an M.A. degree in space systemsmanagement from Webster University. Collins was oneof the first women to graduate from pilot training atVance Air Force Base (AFB), Oklahoma, and laterbecame the first woman T-38 instructor pilot at thatbase. She served as a C-141 pilot, aircraftcommander, and instructor pilot at Travis AFB,California. In 1983, she flew in Operation Urgent Furyin Grenada. She was later assigned to the USAFAcademy where she was an assistant professor inmathematics and a T-41 instructor pilot. In 1990, shegraduated from the AF Test Pilot School at EdwardsAFB, California, where she was the class leader.Collins has logged over 4,000 hours in 30 differenttypes of aircraft and was selected as an astronaut in1990. Aboard STS-63, she spent more than 8 days inspace, and became the first woman pilot of a SpaceShuttle. STS-84 was her second Shuttle mission.Payload Commander, Mission Specialist:Jean-Francois Clervoy (Ingenieur en Chef de lArmement, ESA Astronaut)Jean-Francois Clervoy was born in Longeville-lesMetz, France. He graduated from EcolePolytechnique, Paris, in 1981; from Ecole Supérieurede l' Aéonautiqe et de l' Espace, Toulouse, in 1983;and from Ecole du Personnel Navigant d Essais et deReception,lstres, as a flight test engineer, in 1987.Seconded from the Délégation Générale pour L'Armement to the French Space Agency, Clervoyworked on various satellite projects when he wasselected in the second group of French astronauts. In1985, Clervoy was the Chief Test Director of theParabolic Flight Program and provided support to theEuropean Manned Space Programs Hermes andColumbus until he was selected as a European SpaceAgency (ESA) astronaut and reported to NASAJohnson Space Center for astronaut training in 1992.Clervoy has also received training on the RussianSpace Systems Soyuz and Mir at Star City, Moscow.He served as a mission specialist aboard STS-66during which he logged more than 10 days in space.STS-84 was Clervoy's second Space Shuttle mission.Mission Specialist: Edward T. Lu (Ph.D )Edward Lu was born in Springfield, Massachusetts,raised in Webster, New York and most recently,resided in Honolulu, Hawaii. He received a B.S. degreein electrical engineering from Cornell University and aPh.D. in applied physics from Stanford University.Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ7
Since receiving his Ph D., Dr. Lu has been a researchphysicist working in the fields of solar physics andastrophysics. He was a visiting scientist at the HighAltitude Observatory in Boulder, Colorado, and, at onepoint, simultaneously worked with the Joint Institute forLaboratory Astrophysics at the University of Colorado.He was a postdoctoral fellow at the Institute forAstronomy in Honolulu and has developed a numberof new theoretical advances which have provided, forthe first time a basic understanding of the underlyingphysics of solar flares. Dr. Lu has published articleson a wide range of topics including solar flares,cosmology, solar oscillations, statistical mechanics,and plasma physics. He has given numerous invitedlectures at various universities and internationalconferences. He also holds a commercial pilot certificate with instrument and multi-engine ratings. Dr. Luwas selected as a NASA astronaut in 1994 and hasworked in the Computer Support Branch of the Astronaut Office. STS-84 was Dr. Lu's first Space Shuttlemission.Mission Specialist: Carlos I. Noriega (Major,USMC) Carlos Noriega was born in Lima, Peru, andraised in Santa Clara, California. He received a B.S.degree in computer science from the University ofSouthern California. After flight school he flew CH-46Sea Knight helicopters. In addition to two 6-monthshipboard deployments which included operations insupport of the Multi-National Peacekeeping Force inBeirut, Lebanon, he served as an aviation safetyofficer and instructor pilot. He has logged approximately 2,000 flight hours in various fixed wing androtary wing aircraft. From the Naval PostgraduateSchool, he received two masters degrees, one incomputer science and the other in space systemsoperations. Upon graduation, he was assigned to theUnited States Space Command in Colorado Springs.In addition to serving as a Space Surveillance CenterCommander, he was command representative for thedevelopment and integration of the major space andmissile warning computer system upgrades for Cheyenne Mountain Air Force Base. Noriega was selectedas an astronaut in 1994 and has worked technicalissues concerning extravehicular activity (spacewalks).He served as flight engineer (mission specialist) onSTS-84, his first Space Shuttle flight.Mission Specialist: Elena V. KondakovaElena Kondakova was born in Mitischi, MoscowRegion. She graduated from the Moscow BaumanTechnical Institute. Upon graduation, Kondakovastarted work in RSC-Energia, completing scienceprojects, experiments, and research work. In 1989,she was selected as a cosmonaut candidate by theRSC-Energia Main Design Bureau and sent to theGagarin Cosmonaut Training Center to start the8course of general space training. Upon coursecompletion, Kondakova was qualified as a flightengineer. During 1994 she was in training for theseventeenth main Mir mission known as the EuroMir94 flight. She completed her first flight on board thespacecraft Soyuz TM-17 and the orbital complex Miras a flight engineer in March 1995. Kondakova haslogged approximately 169 days in orbit. She servedas a mission specialist on exchange from the RussianSpace Agency during STS-84, her first Space Shuttlemission.Mission Specialist, NASA-Mir 4: Jerry M. Linenger(Capt., USN)Jerry Linenger was born and raised near Eastpointe,Michigan. He earned a B.S. degree from the U S.Naval Academy, an M.D. from Wayne State University, an M.S. in systems management from theUniversity of Southern California, and both an M.P.H.in health administration and a Ph.D in epidemiologyfrom the University of North Carolina. He completedsurgical internship, aerospace medicine, and preventive medicine programs. Linenger served as a navalflight surgeon at Cuba Point, Republic of the Philippines. He was then assigned as medical advisor tothe Commander, Naval Air Forces, U.S. Pacific Fleet,San Diego. He later became a research principalinvestigator at the Naval Health Research Center anda faculty member at the University of California-SanDiego School of Medicine in the Division of SportsMedicine. Dr. Linenger was selected to be an astronaut in 1992 and flew aboard STS-64, during whichhe spent 11 days in space. Following training in StarCity, he traveled to the Mir Space Station aboardSTS-81. Linenger returned with the STS-84 crewaboard the Space Shuttle after spending approximately 130 days aboard the Mir Space Station.Mission Specialist, NASA-Mir 5: C. Michael Foale(Ph.D.)Michael Foale was born in Louth and raised inCambridge, England. He attended the University ofCambridge Queens' College, where he earned a B.A.degree in physics, National Sciences Tripos, with firstclass honors. While at Queens' College he completeda Ph.D. in laboratory astrophysics. As a postgraduateat Cambridge University, Dr. Foale participated in theorganization and execution of scientific scuba divingprojects including surveying underwater antiquities inGreece. Dr. Foale joined NASA Johnson SpaceCenter in 1983 in the payload operations areas of theMission Operations Directorate. He was selected asan astronaut in 1987 and flew as a mission specialiston STS-45, the first of the ATLAS series of missionsto address the atmosphere and its interaction with thesun, and again as a mission specialist on STS-56,carrying ATLAS-2. He served as a mission specialistMathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ
on STS-63 (February 2-11, 1995), the first rendezvouswith the
4 Mathematics of Space - Rendezvous - Video Resource Guide - EV-1998-02-014-HQ Time for Mir's orbit to cross Moscow. Mir's orbital speed. Distance (circumference) Mir travels during one orbit. (The altitude is the distance from Earth's
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Jan 10, 2012 · The NASA STI program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA
The NASA STI program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server, thus providing one of the largest collections of aero-nautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA
The NASA STI program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server, thus providing one of the largest collections of aero-nautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes
The NASA STI program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server, thus providing one of the largest collections of aero-nautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes
provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes the following report types:Cited by: 11Page Count: 37File Size: 408KBAuthor: Corneli
The NASA STI program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for archiving, and disseminates NASA’s STI. The NASA STI program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical
NASA Aeronautics and Space Database and its public interface, the NASA Technical Reports Server, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, wh
