Activity 9
Measuring Distances by Parallax
Discussion. Astronomers have a harder time of measuring distances than the rest of us
do. They cannot hop into a spaceship, fly to a star, and read the trip odometer. They have
to find ways to measure these extremely large distances without leaving Earth.
To measure the distance to a nearby star, astronomers use a geometric (trigonometric)
process of measurement by parallax. To get an idea of what is meant by parallax,
perform this short activity:
Choose an object (a picture on the wall, a piece of furniture, etc.) across the room.
With only one eye open, hold up your thumb at arm's length so that the thumb is
eclipsing the object. Now, holding your thumb still, close your open eye and
open your other eye. Does your thumb still eclipse the object?
Your thumb appears to "jump" to one side or the other of the object when you switch
eyes. This is because your eyes are set approximately 2½ to 3 inches apart. Each eye
sees your thumb against the background of the distant object at a slightly different angle
than the other eye.
Now imagine an astronomer trying to measure the distance to a relatively nearby star. On
the night of June 15, the astronomer takes a photograph of the star against the background
of other stars that lie in the same direction. Then, one-half year later on the night of
December 15 (or thereabouts), the astronomer takes another photograph of the star. By
measuring the very slight "shift" in the apparent position of the star against the
background of stars that are farther away, the astronomer can calculate the distance to the
star, assuming that the astronomer knows the straight-line distance between Earth's June
15 location and December 15 location. The drawing below (not to scale) illustrates this. Background
stars December 15 Sun Star June 15 43 The apparent "shift" of the target star against the background of farther stars can be
expressed as an angle . Using this angle, the known distance between the two positions
of Earth, and the appropriate geometric relationship, the astronomer can calculate the
distance to the target star.
This method does have its limitations. The farther away a star is, the smaller is its
apparent shift against the background stars. Ultimately the "shift" will be too small to
measure accurately or too small even to be seen.
In this activity, you will use a simple rangefinder to measure the distance between
yourself and some landmark fairly far away.
The rangefinder. The rangefinder you will use is a soft board or a rectangular piece of
Styrofoam approximately two feet long and six inches wide. Attach a thin nail at one
end of the board; the nail is fixed in its position. Attach a centimeter scale at the other
end of the board; the centimeter scale can be a strip of centimeter tape or a strip of paper
that has been carefully marked off in centimeters (to the nearest tenth of a centimeter).
Two moveable straight pins are also attached to the board, just in front of the centimeter
scale. Pin 1 should be the same distance from the long side of the board as the thin nail.
Pin 2 can initially be at any location in front of the centimeter scale.See the drawing
below.
Centimeter tape
Thin nail
1
Moveable pin 2 Cut a 10-meter length of clothesline or thick cord, being sure to measure the 10 meters
carefully. Make a mark on the line at each 1-meter interval. If you do not have access to
a meterstick or a metric tape measure, you may use a 10-yard length of line. If you use a
10-yard line, substitute the word "yards" for "meters" in any directions that follow.
For your measurement location, choose a place that is flat and level for at least 10 meters.
Select a fairly distant landmark that you can see from either end of the 10-meter line
when the line is laid on the ground. Select your initial viewing position and fasten or
weigh down one end of your 10-meter line. At a right angle to the line-of-sight between
your initial position and the distant landmark, stretch out the 10-meter line so that it is
perfectly straight and fasten or weigh down the other end of the line. 44 Stand at your initial position at one end of the 10-meter line so that the center of your
body is directly above the end of the line. Hold the rangefinder near one of your eyes and
resting against your cheek so that the rangefinder is pointed at the landmark. Move pin 1
until the thin nail, the pin, and the landmark are in a straight line (the pin should eclipse
the thin nail and at least some of the chosen landmark). Push the pin in the board at the
edge of the centimeter scale. Then walk to the other end of the 10-meter line.
Position yourself so that the center of your body is directly over the other end of the 10meter line and aim the rangefinder at a right angle to the 10-meter line (not toward the
landmark). Now move pin 2 until the thin nail, pin 2, and the landmark are in a straight
line. With a ruler or tape measure, measure the distance, in centimeters, between the nail
and the first pin. Use the centimeter scale on the board to determine the distance between
the two pins. Refer to the drawing below. A. Triangle on rangefinder Angle d'
x' B. Triangle on ground x = 10 m Landmark
d Angle Triangle A is the triangle formed on the rangefinder by the two pins and the thin nail.
Triangle B is the triangle formed on the ground by your two viewing positions and the
landmark. Although the triangle on the rangefinder is really much smaller than the
triangle on the ground, both triangles have the same set of three angles. Angle on
Triangle A is equal to angle on triangle B; angle on triangle A is equal to angle on
triangle B. Both triangles also have a right angle. 45 Two triangles that have equal corresponding angles are called similar triangles. In
similar triangles, the ratio of the lengths of corresponding sides is constant. In the
drawing, the distance d' in triangle A is the distance between the thin nail and the first pin
on the rangefinder, and distance d in triangle B is the distance between your initial
viewing position and the landmark. The distance x' on triangle A is the distance, in
centimeters, between the two pins on the rangefinder, while distance x on triangle B is the
10 meters between your two viewing positions. Since d/d' = x/x' = 10 m/x', and since you
have measured distances d' and x' on the rangefinder, you can solve the proportion for d.
Repeat the above procedure two more times, returning to your initial viewing position
each time. Record the calculated value of d for each trial on the report sheet. From the
three calculated values of d, determine the average value of d. Record the average value
of d on the report sheet. If your line is measured in yards, report the distance to the
landmark in yards.
Suggestions. If you do this activity at night, choose as your landmark something that can
be easily seen, such as a building with an exterior light or a lighted window, a tree
silhouetted against background light or the night sky, or a tower with a flashing anticollision light.
If the landmark you choose is not terribly distant, the viewing angle at your second
position may be too great to allow you to set pin 2 on the rangefinder. If this is the case,
either choose another, more distant, landmark or use an x distance shorter than 10 meters. Activity 9 Report Sheet
Measuring Distances by Parallax
Enter your data for each trial in the table below. If you used yards instead of
meters on the ground, change the units accordingly.
Trial
x’ (cm)
d’ (cm)
x (m)
d (m) 1 Average value of d: 2 3 m Show your calculations to determine d for Trial 1 only. 46 Questions
1. A student sighted on a distant lamppost, and then walked a distance of 15
meters perpendicular to the line-of-sight. After setting the pin 2, the student
measured the distance between the two pins and found it to be 2.8 cm. The
distance from the thin nail to the pin 1 was 45.0 cm. Calculate the distance
from the student's original location to the lamppost. 2. The initial distance between a student and a given water tower was 545
meters. The student sighted on the tower and pin 1. After walking 10 meters
perpendicular to the line-of-sight, the student sighted again and set the pin 2.
If the distance between the thin nail and pin 1 was 46.4 cm, how far apart are
the first and second pins? 47

 

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