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TE Activity: A Roundabout Way to Mars

Students explore orbit transfers and, specifically, Hohmann transfers. They investigate the orbits of Earth and Mars by using cardboard and string. Students learn about the planets' orbits around the sun, and about a transfer orbit from one planet to the other. After the activity, students will know exactly what is meant by a delta-v maneuver!

Aerospace engineers must be creative when planning the best routes and methods to send a spacecraft from Earth to another planet since space travel is never a direct linear path, but involves transfers between circular orbits. Engineers apply their understanding of math, especially geometry, to determining the most efficient interplanetary trajectories, and thus minimize costs by reducing the need for supplies and fuel as much as possible. They design techniques that take advantage of gravity and the behavior of forces in space.

Learning Objectives
Materials
Introduction/Motivation
Procedure
Safety Issues
Troubleshooting Tips
Assessment
Activity Scaling
References

Grade Level: 7 (6-8)	Group Size: 2
Time Required: 30 minutes	Activity Dependency : None
Expendable Cost Per Group : US$ 5 Under $5 total (minimal or none, depending on office-type supplies that are available).
Keywords: navigation, space, orbits, transfer orbit, trajectory

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Colorado Math
- 1. demonstrate meanings for integers, rational numbers, percents, exponents, square roots, and pi (π) use physical materials and technology in problem-solving situations; (Grades 5 - 8) [2005]
- 1. estimate, use, and describe measures of distance, perimeter, area, volume, capacity, weight, mass, and angle comparison; (Grades 5 - 8) [2005]
- 1. construct two- and three-dimensional models using a variety of materials and tools; (Grades 5 - 8) [2005]
Colorado Science
- Standard 1: Students understand the processes of scientific investigation and design, conduct, communicate about, and evaluate such investigations. (Grades 0 - 12) [1995]
- Standard 5: Students know and understand interrelationships among science, technology, and human activity and how they can affect the world. (Grades 0 - 12) [1995]
- 4.4 Students know the structure of the solar system, composition and interactions of objects in the universe, and how space is explored. (Grades 0 - 12) [1995]

Space travel is one of the most exciting, high tech, and challenging fields in engineering. In 1961, Soviet cosmonaut Yuri Gagarin was the first human in space. In 1969, American astronaut Neil Armstrong was the first to walk on the moon. Aerospace technology has developed enormously since then, and aerospace engineers continue to seek safe and efficient ways to travel in space.

When imagining space travel, one should think of it not as a direct linear path from point A to point B, but more like a transfer between circular orbits. It is possible to travel in a straight line from A to B; but in the vicinity of a planet or star, the gravity of the planet or star causes the natural motion of objects to follow an elliptical (or hyperbolic) path. A huge amount of fuel would be needed to overcome gravity and follow a straight path. Instead, orbital transfers are designed to take advantage of gravitational motion by scheduling velocity changes, known as delta-v maneuvers at precise times and places in the orbit.

Background

An Hohmann transfer is a fuel efficient way to transfer from one circular orbit to another circular orbit that is in the same plane (inclination), but a different altitude. This transfer occurs when the launch and arrival points are lined up on opposite sides of the Sun. With the planets in this position, the travel trajectory between them is an ellipse. If the trajectory was from Earth to Mars — like the one of this activity — the ellipse has its perihelion (closest point in to the sun) at the orbit of Earth and its aphelion (furthest point from the sun) at the orbit of Mars.

A diagram depicting the Hohmann transfer from Earth to Mars. The Sun is in the center of the image, Mars is lined up to the left and the Earth is lined up to the right of the Sun. A blue line indicates that upon leaving Earth, a craft traveling counter-clockwise would circle once around the sun, bypass Earth, and switch orbits to get to Mars.

Figure 1. Hohmann transfer from Earth to Mars.
click for copyright

Before the Activity

Gather all necessary materials.

A picture of a cardboard square, with blank paper placed on it with push pins in the upper corners. Sewing thread and a red, green and blue colored pencil are placed on top of the cardboard, indicating the supplies needed for this activity.

Figure 2. Activity materials.
click for copyright

With the Students

Open with a discussion question and a prediction to get students' minds on the activity. First, ask the students how they would go from the Earth to Mars. Brainstorm and encourage wild ideas. Write ideas on the board. Next, ask them if they know how far apart the Earth and Mars are from the sun. Encourage guesses. Tell them they will learn in this activity.
To draw orbits, we have to scale the actual orbital distance to the size of a paper. The average distance from the sun to the Earth is 149,600,000 km and from the sun to Mars is 227,940,000 km. Have the students select a scale (such as: 1 cm = 10,000,000 km) that allows both orbits to be drawn on the paper. Find the orbit radius for the Earth (RE) and Mars (RM) on your scale.
Ask for a vote from students on the following true or false question: the easiest way to travel between two planets is a direct line. Get a show of hands from students and write the numbers on the board. (Answer: False, it is possible to travel in a straight line from A to B; but in the vicinity of a planet or star, the gravity of the planet or star causes the natural motion of objects to follow an elliptical (or hyperbolic) path. A huge amount of fuel would be needed to overcome gravity and follow a straight path. Instead, orbital transfers are designed to take advantage of gravitational motion by scheduling velocity changes, known as delta-v maneuvers at precise times and places in the orbit. See lesson background information for more detail.)
Give the materials to the students.
Tell the students to place a piece of paper on the cardboard and use 4 tacks to hold down the corners. Put one pushpin in the center to represent the Sun. It can be labeled accordingly.

A picture of a student using a hand-made drawing compass to draw a red circle around an existing, smaller blue circle on the paper/cardboard square. Thread is looped around a push pin, which is placed in the center of the blue circle, at a point marked Sun. The thread loop is knotted together and a pencil is placed in the end enabling the student to draw a circle.

Figure 3. Drawing the Mars orbit.
click for copyright

Cut a piece of thread for the Earth orbit that is 2 times the radius (RE) plus a little extra. Tie the two ends together. The extra length of thread allows for the knot and the extra distance taken up by the pushpin. Do the same for Mars (2 x RM).
Using the two thread loops and two different colored pens or pencils, draw each orbit around the "Sun" (see figure 3). This is accomplished by putting the loop around the pushpin on one end and the pencil or pen on the other. Make sure the string is pulled tightly all the way around the orbit.
Now set up the transfer orbit from the Earth to Mars. To do this, you will need to put another pushpin at the focus of the elliptical transfer orbit (see Figure 4). This focus is located at a distance from the Sun equal to the difference between the Mars orbit radius and the Earth orbit radius (RM-RE)

A picture demonstrating how to draw a transfer orbit. First mark the focus of the elliptical transfer orbit by placing another pushpin to the right of the "Sun" and equidistant from the innermost circle and the Sun. Place a thread loop around both pushpins and hold a pencil near the Mars orbit, creating a triangle. Move the pencil to trace an ellipse — the transfer orbit.

Figure 4. Drawing the transfer orbit.
click for copyright

Use the Mars thread to trace the transfer orbit. Loop the thread around both tacks and the pencil so that the string is in the shape of a triangle as you draw the orbit. Use a different colored pencil to trace the ellipse created by the third string. The orbit should touch the Earth orbit on one side and the Mars orbit on the opposite side.
Run the post-assessment activity as directed in the Activity Assessment / Evaluation section below.

Pre-Activity Assessment

Prediction/Discussion: Ask the students to predict:

How they would go from the Earth to the Mars.
How far apart the Earth and Mars are from the sun. (Answer: The average distance from the sun to the Earth is 149,600,000 km and from the sun to Mars is 227,940,000 km.)

Activity Embedded Assessment

Voting: Ask for a vote from students on the following question. Get a show of hands from students and write the numbers on the board for each answer. Discuss the correct answer.

True or false. The easiest way to travel between two planets is a direct line. (Answer: False, it is possible to travel in a straight line from A to B; but in the vicinity of a planet or star, the gravity of the planet or star causes the natural motion of objects to follow an elliptical (or hyperbolic) path. A huge amount of fuel would be needed to overcome gravity and follow a straight path. Instead, orbital transfers are designed to take advantage of gravitational motion by scheduling velocity changes, known as delta-v maneuvers, at precise times and places in the orbit.)

Post-Activity Assessment

Inside/Outside Circle: Have the students stand in two circles such that each student has a partner. Three people may work together if necessary. The outside circle faces in and the inside circle faces out. Ask the students a question. Both members of each pair think about the question and discuss their answers. If they cannot agree on an answer, they can consult with another pair. Call for responses from the inside or outside circle or the class as a whole.

Define: Ask the students to define the following terms:

Orbit: The path of a celestial body or an artificial satellite as it revolves around another body.
Perihelion: The point nearest the sun in the orbit of a planet or other object.
Aphelion: The point on the orbit of an object that is farthest from the sun.

Integrated Teaching and Learning Program, College of Engineering, University of Colorado at Boulder Penny Axelrad, Janet Yowell, Malinda Schaefer Zarske © 2004 by Regents of the University of Colorado.
The contents of this digital library curriculum were developed under a grant from the Satellite Division of the Institute of Navigation (www.ion.org) and National Science Foundation GK-12 grant no. 0226322.

Last Modified: November 8, 2005