## Orbiting Satellites

- Lesson

In this lesson students explore the concept of orbits, focusing on altitude, velocity, and distance traveled. The lesson explores the connection of Earth-orbiting satellites to the study of the environment. Both geometric and algebraic concepts are presented to students in this application of science and mathematics to a real-world situation.

Using the background knowledge presented in this lesson, students will calculate two components of satellites, namely altitude and velocity. This lesson is comprised of two main problems. In each problem, students work in groups and use algebraic thinking to determine solutions to the problems posed. Suggested methods for solving these problems are described in detail. The teacher is encouraged to highlight alternative methods for solving the problems as discovered by students, when appropriate.

### Background Information

Weather satellites are important for collecting data about Earth. These unmanned spacecraft carry a variety of sensory equipment that scans Earth and electronically communicates the data back to scientists on Earth. Two types of environmental satellite systems named for their orbital characteristics, geostationary and polar-orbiting satellites, provide the data for scientists to study our environment.

http://spaceflight.nasa.gov/gallery/images/station/assembly/html/sts097-704
-071.html

Geostationary satellites orbit at a speed that enables them to remain constantly over the same area of Earth. This position provides constant "viewing" of a specific area of Earth. The Geostationary Operational Environmental Satellites are an important system for the United States. These satellites are maneuvered into orbits that ensure constant coverage of the Western Hemisphere. They continually produce infrared images, weather charts, ice charts, and other data important to the study of the environment.

Polar-orbiting satellites operate in orbits that cause them to pass almost over the poles of the Earth. Many of these satellites operate in Sun-synchronous orbits. The Sun-synchronous orbits are designed so that the satellite passes over the same terrain at the same time each day. This orbital feature is important for the collection of data about Earth.

To simplify the notion of orbits for middle school students, the orbit discussed in these activities are assumed to be circular. The orbital period is the time required to complete one orbit around Earth.

http://images.jsc.nasa.gov/images/pao/STS51I/10062241.jpg

### Getting Started

Engage the class in a large-group discussion about the satellite images they see on local television news broadcasts. In the discussion, help the students distinguish among animations of Shuttle missions, photographs from the Space Shuttle, pictures from remote robotic explorations such as Pathfinder on Mars, Hubble Space Telescope images, and satellite photographs of Earth. If possible, obtain a videotape of a television weather report of a fast-moving storm that has been tracked and shown in nearly real time. If the videotape cannot be obtained, ask students to recall watching such events on television. (Alternatively, students can use Internet resources to find video segments.)

As part of the discussion, ask students to speculate how the television station obtained these images. Some students may know that some TV weather graphics are made with satellite data.

This discussion can lead to an opportunity to present the notions of geostationary and polar-orbiting satellites. The notion of polar-orbiting satellites (those whose orbits take them above both poles) may be easier for students to understand. They seem to understand the notion of the ground track of polar-orbiting satellites. The reason may be that the movement of these satellites is similar to that of an airplane in the sky. Thus, presenting the polar-orbiting notion first may be desirable. The class discussion will determine if this seems appropriate.

*Note: For computations in this lesson, we use 3.14 as the value of pi.*

### Activity A

Pose the following question to the class: How do you determine the velocity of a polar-orbiting satellite when the altitude and the period of the orbit are known? For example, if a 750-mile high satellite orbits Earth every 3 hours, 45 minutes, how fast is it traveling? That is, what is the velocity of the satellite?

Allow the students to work in small groups to draw pictures of the situation and attempt to think through the solution to the problem without doing the actual computation. Encourage them to think about the information they already know, and what information they need to solve the problem. Students can record this information on their papers.

Once the group has a plan to compute the velocity of the satellite, have each student in the group determine the velocity.

*Sample method for solving the example*:

Step 1: Change 3 hours, 45 minutes to:

3 45/60 = 3¾ = 3.75 hours

Note that a fraction calculator can be used for this conversion.

The radius of an orbit is calculated from the center of the Earth. Consequently, the radius of an orbit equals the radius of Earth plus the altitude of the satellite above the Earth.

*Step 2*: According to the formula *C* = 2 × pi, the distance traveled is:

C= 2 × pi × (3960.5 + 750) ≈ 29,581 miles

Note: 3,960.5 miles is the radius of the Earth.

*Step 3*: The distance of the orbit (29,581 miles) is traveled each 3.75 hours. Therefore, the speed—or *velocity*—of the satellite is determined by dividing distance by time:

29,581 miles ÷ 3.75 hours ≈ 7,888 miles per hour

### Additional Exercises

When students are sharing solutions, emphasize solutions' methods. Talk about the different approaches (if any) the students used to reach the same solution.

### Activity B

Pose the following question to the class: How do you find the altitude of a geosynchronous satellite that is orbiting Earth above the equator? The velocity of the satellite must be known. Using a value for the velocity, we could ask, "If there is a satellite traveling at 6,900 miles per hour, how high must it be orbiting?"

This question is sometimes difficult for students because there is only one number in the problem. Putting students into groups to draw pictures and think about what is known about the problem will usually help them identify all the pertinent data. Drawing pictures will help their discussions.

*One possible solution method:*

*Step 1*: A geostationary satellite "orbits" the Earth once
every 24 hours. The time of the orbit is what makes it appear not to
move. It is moving in unison with Earth's rotation.

*Step 2*: If a satellite travels 6,900 miles in 1 hour, how far does it travel in the 1-day orbit (a period of one day)?

6,900 miles per hour × 24 hours = 165,600 miles

*Step 3*: 165,600 miles is the circumference (*C*) of a circle, and *C* = 2 × pi *r*. Therefore,

pi ×r= 165,600 ÷ 2 = 82,800 milesand

r= 82,800 ÷ pi ≈ 26,369 miles.

*Step 4*: 26,369 miles is the radius of a circle whose center
is the center of the Earth. To determine the satellite's altitude above
the surface of the Earth, the radius of Earth (3,960.5 miles) must be
subtracted from the radius of the circle. Therefore, the altitude above
the Earth is:

26,369 - 3,960.5 = 22,408.5 miles

Once again, it is important to have the students share their solutions to this problem. They may see different methods for solving this problem. When students share their thinking, some of their classmates will gain new or heightened understanding of the lesson topics.

### Reference

Adapted from Orbiting Satellites in Mission Mathematics, Linking Aerospace and the NCTM Standards, a NASA/NCTM project, NCTM 1997. Images from Web sites are credited by inclusion of the URL and/or direct links to their source.

- Calculators

### Learning Objectives

Students will:

- Use algebraic thinking to determine velocity and altitude of orbiting satellites.

### NCTM Standards and Expectations

- Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results.

- Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

- Understand and use the inverse relationships of addition and subtraction, multiplication and division.

- Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

### Common Core State Standards – Mathematics

Grade 7, Geometry

- CCSS.Math.Content.7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.