When you switch between two different points of view, nearby objects appear to shift with respect to more distant ones. This is called parallax, and it is a basic tool for measuring astronomical distances. The same technique is also used to measure distances to objects on Earth.
Reading: Stars & Planets, p. 12 — 14 (Star distances).
Astronomers first used parallax to measure distances to other planets in 1672, but living organisms have been using parallax for several hundred million years — ever since animals with two eyes (or more) first evolved. Two eyes are better than one because they give you two different views of the world; by combining these views, your brain can find distances to nearby objects. The parallax measurements we will make in this lab use a technique you have been practicing since infancy. In some sense, you are already an expert at using parallax to measure distances, but at the same time, you may not know how your brain accomplishes this very useful trick.
A simple experiment illustrates the role of binocular vision — that is, vision using two eyes — in judging distance. First, close both eyes and lift one hand over your head. Have your lab partner place a coin (or other small object) on the table within reach in front of you. Now open both eyes and quickly lower your hand so that the tip of your index finger lands on the middle of the coin. You should have no trouble doing this; try it a few times — with the coin in a different place each time — to convince yourself that you can always place your finger on top of the coin. (If you consistently miss the coin, you may not be using both eyes — get your vision checked!)
Now try the same thing again, but this time, open only one eye (no peeking — cover your other eye to make sure). You will probably have much more trouble putting your finger down on top of the coin. Again, try this a few times with the coin in a different place each time. About how often do you hit the coin? Do you tend to reach too far, or not far enough? Try using your other eye — is it any better?
Fig. 1 shows the geometry of a parallax measurement. Such a measurement involves observations from two different places separated by a known distance. This distance, the baseline, is represented by the symbol b. Pick a fairly nearby target which you can view in front of a background much further away (for example, you might use the pole of a street-light as your target, with the side of the valley as a background). For the first observation, line the target up with some definite landmark in the background (for example, a rock on the side of the valley). Now move to your second observation point, and use a cross-staff to measure the angle θ between your target and the background landmark.
|Fig. 1. Geometry of a parallax measurement. Observations of the target are made from the two positions on the left. From the first observation point, the target is aligned with a very distant landmark. From the second observation point, the angle between the target and the landmark is measured. Simple geometry gives the distance D to the target in terms of the distance b between the observation points.|
Once the baseline b and parallax angle θ have been measured, the distance D to the target is
This formula is easy to derive using simple geometry. It assumes that the angle θ is measured in degrees. It does not matter what units you use for b; you will automatically get D in the same units!
The pictures below show how to make a parallax measurement. For simplicity, I chose a fairly unexciting target — the top of an electricity pole near my home, which I can view in front of the side of a hill somewhat further away. As the background landmark, I used a transformer on another electricity pole on the distant hillside. Fig. 2 shows the overall situation. One important fact, which may not be obvious from this picture, is that the background landmark was much further away than the target.
|Fig. 2. Target and landmark for a parallax measurement. Arrows mark target (top of pole, on right) and landmark (transformer, on left).|
To make the first observation, I moved around to line up the target and background landmark, as shown in Fig. 3b. I used a pebble to mark the location of my first observation. I then shifted to my left until the target and the background were no longer lined up, as shown in Fig. 3a. The distance to shift depends on the situation, but the key thing is to make sure that the target and background landmark appear comfortably separated from each other. I used another pebble to mark the location of my second observation. The baseline distance between the two pebbles was b = 45 inches.
|Fig. 3a. Second observation: target is visibly shifted with respect to background.||Fig. 3b. First observation: target and background landmark are lined up with each other.|
Now, from my second observation point, I used a cross-staff to measure the angle θ between the target and the background object. This was a little tricky, since it's hard to keep the background, target, and ruler all in focus at the same time; Fig. 4 shows that my camera also had some trouble focusing. Nonetheless, even this fuzzy image is clear enough to show that the background landmark fell at the 16.0 cm mark on the ruler, while the target fell at about 16.8 cm. Thus the apparent separation between the target and the background was 16.8 cm - 16.0 cm = 0.8 cm. Since 1 cm on the ruler represents an angle of 1°, the angular separation between the target and the background landmark was about θ = 0.8°.
|Fig. 4. Measurement of parallax angle. Dotted lines show where the background landmark (left) and target (right) appear on the cross-staff ruler.|
Using b = 45 inches and θ = 0.8° in the parallax formula, I got D = 3200 inches = 270 feet. My results are given to slightly better than one significant figure; the measurement of θ could easily be off by ±0.1°, so there's no point in trying to claim any higher level of accuracy. The most serious source of error is probably my use of a background landmark which was only a few times further away than the target. For example, if the background was about five times further away than the target, the resulting value of D would be about 20% too large.
In addition to the simple experiment on judging distances described above, you will also make two measurements of distance using parallax. One measurement will be performed during the lab; we will set up a suitable target and coach everybody on the proper technique. The second measurement will be made after, using a target and background that you select. The key here is not just to make a measurement — you will also have to make some choices, and explain why you made those choices. At every step, your choices affect the accuracy of your result, so think carefully when choosing.
Joshua E. Barnes
(barnes at ifa.hawaii.edu)
19 October 2010