How Far



How Far

Part I

1. Look at the diagram to the right. Imagine that you are looking at the stars from Earth in January. Use a ruler to draw a straight line from Earth in January, through the nearby star (Star A), out to the distant stars. Which of the distant stars would appear closest to the nearby star in your night sky in January. Circle this star and label it “Jan”.

2. Repeat the process for July, circling and labeling the distant star “July”.

3. Describe the motion of the Star A as Earth orbits the Sun counterclockwise from January of one year, through July, to January of the following year.

Moves from right to left against the background stars.

The apparent motion of nearby objects relative to distant objects, which you just described, is called parallax.

4. Consider two stars (C and D) that both exhibit parallax. If star C appears to move back and forth by a greater amount than star D, which star is closer to you? Explain your reasoning.

Star C. Closer it is the greater the parallax angle.

5. Draw a straight line through the sun, the nearby star, to the distant stars at the top of the page.

6. Draw a straight line from the Sun to the Earth in January.

There is now a narrow triangle with the Earth-Sun distance as its base. The small angle, just below the nearby star, formed by the two longest sides of this triangle is called the parallax angle for the nearby star. Label this angle “Pa”.

Knowing a star’s parallax angle allows us to calculate the distance to the star. Since even the nearest stars are still very far away, parallax angles are extremely small. Parallax angles are measured in arcseconds, where an arcsecond is 1/3600 of 1 degree.

To describe the distances to stars, astronomers use a unit of length called the parsec. 1 parsec is defined as the distance to a star that has a parallax angle of exactly 1 arcsecond. The distance from the Sun to a star 1 parsec away is 206,265 times the Earth-Sun distance or 206,265 AU.

7. Using the ruler, measure the distance (in inches) from the Sun to the Earth in the diagram above. This is 1 AU. Knowing there are 12 inches in a foot and 5,280 feet in a mile, how many miles would the nearby star be away from Earth if the picture were drawn to scale and the parallax angle were 1 arcsecond?

0.75 inches = 1 au

0.75 inches x 206,265 AU = 154700 inches /12 in/ft = 12890 ft/5280 ft/mi = 2.44 miles

8. Consider the following discussion:

George: If the distance to the star is more than 1 parsec, then the parallax angle must be more than 1 arcsecond. Larger distance means larger angles.

Hillary: If we drew a diagram for a star that was much more than 1 parsec away from us, the triangle in the diagram would be pointier than the one we just drew in the diagram on the other page. That should make the parallax angle smaller for a star farther away.

Whose side are you on (on this astronomy question, that is.)

Hillary. The closer the star the greater the parallax angle.

9. Draw a second star that is closer to the Sun than Star A and label it Star B (very creative). Repeat the process you did before with Star A to determine its parallax angle. Label the parallax angel Pb.

Which star, Star A or B, has the larger parallax angle? Who’s right, George or Hillary?

Pb is a larger angle.

10. Let's test how the parallax of an object varies with distance.

a. One partner takes the meterstick and places the pencil at the 50 cm mark, centering the pencil on the meter stick. The other partner places the "zero" end of the meterstick against her/his chin, holding it out horizontally. This partner then alternates opening and closing each eye, noting how the pencil moves against specific background objects.

b. Have your partner move the pencil half of the original distance (to 25 cm). When you alternate opening and closing each eye does the pen appear to move more or less than before? Try to quantify how much more or less (twice as much? half as much? three times as much? etc.).

c. Now, have your lab partner move the pencil twice the original distance to you, to approximately the end of the meterstick. When you alternate opening and closing each eye does the pen appear to move more or less than before? Try to quantify how much more or less (twice as much? half as much? three times as much? etc.).

Describe how the pencil moved against background objects:

There is a greater parallax angle the closer the pencil is to the observer.

Part II

Imagine that you are standing in an open field. While facing south, you see a house in the distance. If you look to the east, you see a barn in the distance.

11. What is the angle between the house and the barn? 90 degrees

12. You see the Moon on the horizon just above the barn in the east, and also see a bright star directly overhead. What is the angle between the Moon and the overhead star? 90 degrees

13. Compare your answers for the barn-house angle and the Moon-star angle. Are they the same? Does this angle tell you anything about the actual distance between the barn and house or the Moon and star?

They are the same. It tells you nothing about

the actual distance between objects.

Consider the star field drawing in the diagram

to the right (Figure 1). In this drawing imagine

that the angle separating Stars A and B is just

½ of an arcsecond. Measure this distance in

centimeters.

6 centimeters = ½ arcsecond

Half of this distance (and angle) is equal to the parallax, so the parallax angle for stars A and B is 3 cm or ¼ arcsecond.

On the next page there are several pictures of this star field taken at different times during the year (Figure 2). One star in the field exhibits parallax as it moves back and forth across the star field with respect tot the other, more distant stars. Circle the star on each picture that exhibits parallax.

In Figure 1, draw a line that shows the range of motion for the star exhibiting parallax in the pictures from Figure 2. Label the endpoints of this line with the months when the star appears at those endpoints. Measure this distance in centimeters.

3 centimeters

14. How many times larger is the separation between stars A and B compared to the distance between the endpoints of the line showing the range of the motion for the star exhibiting parallax?

Twice as large.

15. The star’s parallax angle is half the angular separation between the endpoints (look at your drawing on the first page to verify that the narrow angle labeled Pa is the parallax angle). What is the parallax angle for the endpoints that you marked in Figure 1 for the nearby star exhibiting parallax?

½ as much, if Stars A & B are ½ arcsecond, thus the total parallax is ¼ arcsecond. Since the angle of parallax is one-half of this (see Pa & Pb on first page), thus the angle of parallax is 1/8 arcsecond.

We define 1 parsec as the distance to an object that has a parallax angle of 1 arcsecond. An object at 2 parsecs has a parallax angle of ½ arcsecond.

16. For a star with a parallax angle of ¼ arcsecond, what is its distance from us?

4 parsecs

17. From the pictures, what is the distance (in parsecs) from us to the nearby star exhibiting parallax?

8 parsecs

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Distant Stars

Nearby Star

Star A

Earth

(July)

Earth

(January)

Figure 1

Figure 2

July

Jan

Pa

Pb

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