How do we find the distance to the Sun using the transit of Venus?

Introduction: Surveyors use a method called 'triangulation' to determine the distance to an object by using the two base angles and the distance between the surveyors, and solving for the height of the triangle whose vertex is at the distant object. Astronomers use a similar method which they call 'parallax' to gauge the same distance. If you hold a pencil in your hand at arms-length, and alternately open and close your eyes between right and left, you will see the pencil move relative to the background objects in the room. This angular shift, together with the distance between your eyes, can be used to determine the length of your arm.

The transit of Venus (see sketch above), when observed by astronomers at two stations far apart on the Earth, will see a slight shift in the location of Venus relative to the face of the sun across which it is moving as viewed from Earth. By knowing the distance between the observers on Earth, and the size of the 'parallax' angle, astronomers can determine how far Venus is from the Earth. This distance is known, precisely, in terms of the distance from Earth to Sun. If the Earth-Sun distance is called 1.0 'Astronomical Units', the distance to Venus will be 0.28 Astronomical Units. By using the parallax method to determine what the Earth-Venus distance is in kilometers, astronomers can then determine exactly what the Earth-Sun distance is in kilometers.

Objectives:

The students will apply the concepts of vertical angles and trigonometric ratios to calculate lengths and angles. The students will determine congruent angles.

Benchmark:

6-8 Technology is essential to science for such purposes as access to outer space, sample collection, measurement, storage and computation.

9-12 Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings.

9-12 Find answers to problems by substituting numerical values in simple algebraic formulas.

Materials:

Vocabulary: Vertical angles Trigonometric ratios Radius Diameter

Procedures:

Step 1: Teacher provides the students with the Student Worksheet. The students complete Problem 1 and Problem 2. Teacher and students discuss the answers to Problem 1 and Problem 2.

Step 2: Teacher and students discuss how Problem 3 is created from Problem 2. The students answer Problem 3 and the teacher and students discuss the answers.

Step 3: Teacher and students discuss how the drawing in Problem 3 was turned and changed to create the drawing in Problem 4. Teacher can guide the students through the questions for Problem 4 or provide time for the students to complete Problem 4. Discuss the answers to the questions.

Step 4: Teacher and students discuss and answer Problem 5.