Celestial MechanicsCelestial MechanicsGeometry in SpaceGeometry in Space

Stephanie BoydStephanie Boyd

EE33 Teacher Summer Research Program Teacher Summer Research Program

Department of Aerospace EngineeringDepartment of Aerospace Engineering

Texas A&M UniversityTexas A&M University

20032003

Stephanie BoydStephanie Boyd

Texas A&M University, Class of 2004Texas A&M University, Class of 2004

Candidate for B.A., Mathematics and Candidate for B.A., Mathematics and Secondary Teacher CertificationSecondary Teacher Certification

Program Goals: Program Goals: To increase personal knowledge and To increase personal knowledge and

awareness of engineeringawareness of engineering To learn from the experience of current math To learn from the experience of current math

and science teachersand science teachers

Unit GoalsUnit Goals

To provide high school geometry students with To provide high school geometry students with the answer to the question, “Why do I need to the answer to the question, “Why do I need to learn this?”learn this?”

To demonstrate relevant, real-world examples of To demonstrate relevant, real-world examples of geometry principlesgeometry principles

To connect each lesson plan to an application in To connect each lesson plan to an application in current aerospace engineering researchcurrent aerospace engineering research

To correlate each lesson plan with established To correlate each lesson plan with established standards (TEKS, NCTM) standards (TEKS, NCTM)

BackgroundBackground

What does “Celestial Mechanics” mean?What does “Celestial Mechanics” mean? ““Celestial” = spaceCelestial” = space ““Mechanics” = workings, machineryMechanics” = workings, machinery

What do aerospace engineers do?What do aerospace engineers do? Develop new ideas and technologyDevelop new ideas and technology Find ways to improve the quality, efficiency, and cost Find ways to improve the quality, efficiency, and cost

of machines in space (use of lighter materials, of machines in space (use of lighter materials, different structures, etc.)different structures, etc.)

Minimize the cost of putting the new ideas and Minimize the cost of putting the new ideas and technology into actiontechnology into action

Applicable ProblemsApplicable Problems

SatellitesSatellites CommunicationCommunication

Cell phones, Cell phones,

Satellite TV, etc. Satellite TV, etc. NavigationNavigation

GPS GPS

MilitaryMilitary

Emergency situationsEmergency situations Data-gatheringData-gathering

WeatherWeather

ObservationObservation

Geostationary Orbit Geostationary Orbit (GSO)(GSO)

Used for communication, Used for communication, navigation, and data-gathering navigation, and data-gathering satellitessatellites

Circular orbitCircular orbit Satellite Satellite rotates around a fixed point on rotates around a fixed point on Earth Earth

Contains hundreds of satellitesContains hundreds of satellites Launch costs $50-$400 millionLaunch costs $50-$400 million 35000 km from the surface of 35000 km from the surface of

the Earththe Earth 42000 km from the center of 42000 km from the center of

the Earththe Earth

SatellitesSatellites

SputnikSputnik GPS satelliteGPS satellite

First Satellite (1957)First Satellite (1957) Currently in GSO Currently in GSO

Geostationary Orbit (GSO)Geostationary Orbit (GSO)

Each of the little tick marks Each of the little tick marks represents one satellite in GSO!represents one satellite in GSO!

Unit ConceptsUnit Concepts

Lesson 1: Exploring CirclesLesson 1: Exploring Circles

Lesson 2: Angles and ArcsLesson 2: Angles and Arcs

Lesson 3: Equations of CirclesLesson 3: Equations of Circles

Resource: Glencoe: Geometry, 1998 ed. (Chapter 9)Resource: Glencoe: Geometry, 1998 ed. (Chapter 9)

Unit ObjectivesUnit Objectives

The student will: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 recognize major arcs, minor arcs, semicircles and

central angles.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 write and use the equation of a circle in the coordinate

plane.plane.

ExamplesExamples

Example 1: The radius of the Earth is Example 1: The radius of the Earth is 6378 km. What is the diameter and 6378 km. What is the diameter and circumference of the Earth in kilometers?circumference of the Earth in kilometers?

Solution:Solution: D = 2r = 2(6378) = 12756 kmD = 2r = 2(6378) = 12756 km C = 2C = 2ππr = 12756r = 12756ππ ≈ 40074 km ≈ 40074 km

ExamplesExamples

Example 2: Satellite S is in a Example 2: Satellite S is in a geostationary orbit around geostationary orbit around Earth. The major arc between Earth. The major arc between point A and point B measures point A and point B measures 240 degrees. If S begins its 240 degrees. If S begins its orbit at A and travels clockwise orbit at A and travels clockwise to B, how far has it traveled?to B, how far has it traveled?

Solution:Solution: Distance from S to center of Distance from S to center of

Earth is 42000 km (GSO)Earth is 42000 km (GSO) C = 2C = 2ππr = r = 2(42000)2(42000)ππ ≈ 264000 km ≈ 264000 km (240/360) * C ≈ 176000 km(240/360) * C ≈ 176000 km

A

B

Earth

S

240

ExamplesExamples

Example 3: Since a satellite in GSO takes about Example 3: Since a satellite in GSO takes about 24 hours to make a complete revolution around 24 hours to make a complete revolution around Earth, how much time does it take S to travel Earth, how much time does it take S to travel from A to B?from A to B?

Solution: Solution: t = travel time from A to Bt = travel time from A to B t hrs / 24 hrs ≈ 176000 km / 264000 kmt hrs / 24 hrs ≈ 176000 km / 264000 km t ≈ 16 hrst ≈ 16 hrs

ExamplesExamples

Example 4: Based on the previous example Example 4: Based on the previous example problems, how fast must a satellite travel to stay problems, how fast must a satellite travel to stay in GSO?in GSO?

Solution:Solution: v = velocity of a satellite in GSOv = velocity of a satellite in GSO v = 264000 km / 24 hrs ≈ 11000 km/hrv = 264000 km / 24 hrs ≈ 11000 km/hr

Independent PracticeIndependent Practice

The discovery: The discovery: NASA has just discovered a new planet NASA has just discovered a new planet in the universe, and it has been named Planet Aggie! in the universe, and it has been named Planet Aggie! They know a few facts about it, but your task is to help They know a few facts about it, but your task is to help NASA plot a map of this new planet and its satellites! NASA plot a map of this new planet and its satellites!

The data:The data: If the Sun is located at the origin of the map (0,0), the new planet 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.is located 5 units to the east and 12 units to the north. There are 2 geostationary orbits around Planet Aggie! Each of There are 2 geostationary orbits around Planet Aggie! Each of

these orbits contains 2 satellites.these orbits contains 2 satellites.Orbit “Whoop” has an altitude of 3 units. Satellite “Gig” is located at Orbit “Whoop” has an altitude of 3 units. Satellite “Gig” is located at (5, 15) and Satellite “Em” is located at (8, 12). (5, 15) and Satellite “Em” is located at (8, 12). Orbit “Howdy” is given by the equation (x – 5)Orbit “Howdy” is given by the equation (x – 5)22 + (y – 12) + (y – 12)22 = 25. = 25. Satellite “A” is located 12 units north of the Sun and Satellite “M” is Satellite “A” is located 12 units north of the Sun and Satellite “M” is located due south of “TX”. located due south of “TX”.

Independent PracticeIndependent Practice

Your task: Your task: On an 8x10 sheet of graph paper, draw a coordinate On an 8x10 sheet of graph paper, draw a coordinate plane. Plot and label the Sun, Planet Aggie, the two orbits, and the plane. Plot and label the Sun, Planet Aggie, the two orbits, and the four satellites. Then answer the following questions and show your four satellites. Then answer the following questions and show your work on a separate sheet of paper:work on a separate sheet of paper:

QuestionsQuestions:: 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. Assume that Planet Aggie revolves around the Sun in a circular orbit.

What is the equation of this orbit?What is the equation of this orbit? How far does Planet Aggie travel during each revolution on its orbit How far does Planet Aggie travel during each revolution on its orbit

around the Sun? 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 PracticeIndependent Practice

Resources and LinksResources and Links

http://www.nasa.govhttp://www.nasa.gov – Comprehensive website with – Comprehensive website with resources for teachers and studentsresources for teachers and studentshttp://visibleearth.nasa.govhttp://visibleearth.nasa.gov – Searchable directory of – Searchable directory of images, visualizations, and animations of Earth taken images, visualizations, and animations of Earth taken from satellitesfrom satelliteshttp://www.goes.nasa.govhttp://www.goes.nasa.gov – Images of Earth taken from – Images of Earth taken from geostationary satellitesgeostationary satelliteshttp://www.geo-orbit.org/sizepgs/geodef.htmlhttp://www.geo-orbit.org/sizepgs/geodef.html - - Information about geostationary orbitsInformation about geostationary orbitswww.howstuffworks.com/satellite.htmwww.howstuffworks.com/satellite.htm - Great resource - Great resource for learning about satellitesfor learning about satellitesGlencoe: Geometry, 1998 edition (Chapter 9)Glencoe: Geometry, 1998 edition (Chapter 9)

Any Questions?Any Questions?

Thank you for this Thank you for this opportunity! opportunity!