Activity 10

 

Estimating Angular Size

 

 

Discussion. In Activity 8, you used your little finger at arm's length to estimate sizes and distances. This exercise will take you a step farther, to determining the apparent size of an object as an angle measured in degrees. You will also learn how the size/distance ratio for an object is related to its angular size.

 

Using the size/distance ratio from Activity 8, you can estimate the actual size of an object if you know its distance from you. For a celestial object, you do not know its size or its distance. The most obvious feature of an object is how large it appears to be - its apparent size. For apparent size, however, using units such as meters is not appropriate.

 

To describe the apparent size of an object, you might compare it to some object you know, such as a tennis ball. But how large a tennis ball appears to be depends on how far away it is from you: the greater its distance, the smaller it appears to be. To say that an object appears to be as large as a tennis ball is not enough; you must also indicate the distance from you to the tennis ball. A more appropriate description would be: "The object appears to be as large as a tennis ball looks 5 meters away." One problem arises here: just how large does a tennis ball appear to be when it is 5 meters away? Do you know? This description is not very helpful. There is, however, a convenient way to describe the apparent size of an object.

 

This method involves measuring the angular size of an object: how many degrees (or fractions of a degree) the object "covers." The angular size of an object is related to the object's size/distance ratio. For small angles only (when the object is far away), the angular size is directly proportional to the size/distance ratio.

 

An object with a size/distance ratio of 1/57 has an angular size of about 1 degree (1^o). Your little finger held at arm's length has a ratio of about 1/50 (see Activity 8). Thus, anything that you can just "cover up" with the width of your little finger has an angular size slightly greater than 1^o. As a convenient approximation, you may assume that the size/distance ratio for an angular size of 1^o is 1/60.

EXAMPLE 1

 

A ball has a diameter of 10 cm. How far away from you must it be to have an angular size of 1^o?

 

Since the size/distance ratio for 1^o is 1/60, the ball must be 60 times its diameter away from you:

 

10 cm X 60 = 600 cm = 6 m

 

The ball must be at a distance of 6 meters to have an angular size of 1^o.                                          

 

EXAMPLE 2

 

Suppose the 10-cm ball is 20 meters away. What is its angular size?

 

The size/distance ratio for the ball is 10 cm/20 m, or 10 cm/2000 cm, which simplifies to 1/200. The size/distance ratio for 1^o is 1/60.

 

1/200 ÷ 1/60 = 60/200 = 0.3

 

The angular size of the ball is 0.3^o (about one-third of a degree).

 

Procedure. Obtain two spheres of different sizes, perhaps a baseball and a volleyball or two StyrofoamÔ spheres of about these sizes.  First, measure the diameter of the smaller sphere in centimeters and record this measurement on your report sheet.

 

1. At what distance does the small sphere have to be from your eyes to have an angular size of 1^o? Show your calculations on your report sheet. (See Example 1.)

 

Place the small sphere at the distance you have just calculated. Place the large sphere next to the small sphere.

 

2. What is the size/distance ratio for the large sphere? Show your calculations. (See Example 2.)

 

3. What is the angular diameter of the large sphere? Show your calculations. (See Example 2.)

 

Close one eye and look at both spheres, with your eye at the same level as the spheres. The spheres do not appear to be the same size since they are both the same distance from your eye and their diameters are not equal.

 

4. How far away must the large sphere be to have an angular diameter of 1^o? Show your calculations.

           

Place the large sphere at the distance you have just calculated. Make sure that the small sphere is not moved from its previous position. Close one eye and look at both spheres, with your eye at the same level as the spheres.

 

5. How do the angular sizes of the spheres compare? Does one appear larger than the other? If so, which one? Why?

 

The moon is at a distance of about 110 of its diameters from Earth.

 

6. What is the angular size of the moon as seen from Earth? Show your calculations.

 

7. How far would a quarter (24 mm in diameter), observed face on, have to be from your eyes to have the same angular size as the moon? Show your calculations.

 

 

Activity 10 Report Sheet

 

Estimating Angular Size

 

 

 

 

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