Celestial Mechanics Geometry in Space Stephanie Boyd E 3 Teacher Summer Research Program Department of Aerospace Engineering Texas A&M University 2003

Stephanie Boyd Texas A&M University, Class of 2004 Candidate for B.A., Mathematics and Secondary Teacher Certification Program Goals: To increase personal knowledge and awareness of engineering To increase personal knowledge and awareness of engineering To learn from the experience of current math and science teachers To learn from the experience of current math and science teachers

Unit Goals To provide high school geometry students with the answer to the question, “Why do I need to learn this?” To demonstrate relevant, real-world examples of geometry principles To connect each lesson plan to an application in current aerospace engineering research To correlate each lesson plan with established standards (TEKS, NCTM)

Background What does “Celestial Mechanics” mean? “Celestial” = space “Celestial” = space “Mechanics” = workings, machinery “Mechanics” = workings, machinery What do aerospace engineers do? Develop new ideas and technology Develop new ideas and technology Find ways to improve the quality, efficiency, and cost of machines in space (use of lighter materials, different structures, etc.) Find ways to improve the quality, efficiency, and cost of machines in space (use of lighter materials, different structures, etc.) Minimize the cost of putting the new ideas and technology into action Minimize the cost of putting the new ideas and technology into action

Applicable Problems Satellites Communication Communication Cell phones, Satellite TV, etc. Navigation NavigationGPSMilitary Emergency situations Data-gathering Data-gatheringWeatherObservation Geostationary Orbit (GSO) Used for communication, navigation, and data-gathering satellites Circular orbit  Satellite rotates around a fixed point on Earth Contains hundreds of satellites Launch costs $50-$400 million km from the surface of the Earth km from the center of the Earth

Satellites SputnikGPS satellite First Satellite (1957)Currently in GSO

Geostationary Orbit (GSO) Each of the little tick marks represents one satellite in GSO!

Unit Concepts Lesson 1: Exploring Circles Lesson 2: Angles and Arcs Lesson 3: Equations of Circles Resource: Glencoe: Geometry, 1998 ed. (Chapter 9)

Unit Objectives The student will: identify and use parts of circles. identify and use parts of circles. solve problems involving the circumference of a circle. solve problems involving the circumference of a circle. recognize major arcs, minor arcs, semicircles and central angles. recognize major arcs, minor arcs, semicircles and central angles. find measures of arcs and central angles. find measures of arcs and central angles. solve problems by making circle graphs. solve problems by making circle graphs. write and use the equation of a circle in the coordinate plane. write and use the equation of a circle in the coordinate plane.

Examples Example 1: The radius of the Earth is 6378 km. What is the diameter and circumference of the Earth in kilometers? Solution: D = 2r = 2(6378) = km D = 2r = 2(6378) = km C = 2πr = 12756π ≈ km C = 2πr = 12756π ≈ km

Examples Example 2: Satellite S is in a geostationary orbit around Earth. The major arc between point A and point B measures 240 degrees. If S begins its orbit at A and travels clockwise to B, how far has it traveled? Solution: Distance from S to center of Earth is km (GSO) Distance from S to center of Earth is km (GSO) C = 2 π r = 2(42000) π ≈ km C = 2 π r = 2(42000) π ≈ km (240/360) * C ≈ km (240/360) * C ≈ km A B Earth S 240

Examples Example 3: Since a satellite in GSO takes about 24 hours to make a complete revolution around Earth, how much time does it take S to travel from A to B? Solution: t = travel time from A to B t = travel time from A to B t hrs / 24 hrs ≈ km / km t hrs / 24 hrs ≈ km / km t ≈ 16 hrs t ≈ 16 hrs

Examples Example 4: Based on the previous example problems, how fast must a satellite travel to stay in GSO? Solution: v = velocity of a satellite in GSO v = velocity of a satellite in GSO v = km / 24 hrs ≈ km/hr v = km / 24 hrs ≈ km/hr

Independent Practice The discovery: NASA has just discovered a new planet in the universe, and it has been named Planet Aggie! They know a few facts about it, but your task is to help NASA plot a map of this new planet and its satellites! The data: If the Sun is located at the origin of the map (0,0), the new planet is located 5 units to the east and 12 units to the north. If the Sun is located at the origin of the map (0,0), the new planet is located 5 units to the east and 12 units to the north. There are 2 geostationary orbits around Planet Aggie! Each of these orbits contains 2 satellites. There are 2 geostationary orbits around Planet Aggie! Each of these orbits contains 2 satellites. Orbit “Whoop” has an altitude of 3 units. Satellite “Gig” is located at (5, 15) and Satellite “Em” is located at (8, 12). Orbit “Howdy” is given by the equation (x – 5) 2 + (y – 12) 2 = 25. Satellite “A” is located 12 units north of the Sun and Satellite “M” is located due south of “TX”.

Independent Practice Your task: On an 8x10 sheet of graph paper, draw a coordinate plane. Plot and label the Sun, Planet Aggie, the two orbits, and the four satellites. Then answer the following questions and show your work on a separate sheet of paper: Questions: How far is Planet Aggie from the Sun? How far is Planet Aggie from the Sun? Assume that Planet Aggie revolves around the Sun in a circular orbit. What is the equation of this orbit? Assume that Planet Aggie revolves around the Sun in a circular orbit. What is the equation of this orbit? How far does Planet Aggie travel during each revolution on its orbit around the Sun? How far does Planet Aggie travel during each revolution on its orbit around the Sun? What is the equation associated with “Whoop”? What is the equation associated with “Whoop”? What is the distance from “Gig” to “Em”? What is the distance from “Gig” to “Em”? What is the altitude (radius) of “Howdy”? What is the altitude (radius) of “Howdy”? What are the coordinates of “A”? What are the coordinates of “A”? What are the coordinates of “M”? What are the coordinates of “M”? What is the distance from “A” to “M”? What is the distance from “A” to “M”?

Independent Practice

Resources and Links – Comprehensive website with resources for teachers and students – Searchable directory of images, visualizations, and animations of Earth taken from satellites – Images of Earth taken from geostationary satellites Information about geostationary orbits Great resource for learning about satellites Glencoe: Geometry, 1998 edition (Chapter 9)

Any Questions? Thank you for this opportunity!