When you look at things from two different points of view, nearby objects appear to shift with respect to more distant ones. This is called parallax, and it's a basic tool for measuring astronomical distances. The same technique can be used to measure distances to objects on Earth.
Background Reading: Stars & Planets, p. 10 to 12 (Star distances).
Astronomers first used parallax to measure distances within the Solar System in 1672, but living organisms have been using parallax for several hundred million years - ever since the evolution of the first animal with two eyes in its head. Two eyes are better than one because they give you two slightly different views of the world; by combining these views, your brain can estimate 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 have no idea 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 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 more or less exactly on top of the coin. (If you consistently miss the coin, you may not be employing 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?
This diagram is an overhead view showing the geometry of a parallax measurement. Such a measurement requires 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 streetlight 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 between your target and the background landmark. The distance D to your target is
This formula is fairly easy to derive using simple geometry. We will cover the derivation in class. The angle should be measured in degrees. Note that 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. The first picture just shows the overall situation.
|Target and background for a parallax measurement. The target (top of pole, on right) and background landmark (transformer, on left) are marked by arrows.|
To make the first observation, I moved around to line up the target with background landmark, as shown below (on the right). 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 below (on the left). The distance to shift is arbitrary, as long as the target and background landmark now 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.
|Second observation: target is visibly shifted with respect to background.||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 between the target and the background object. This is a little tricky, since it's hard to keep the background, the target, and the ruler in focus at the same time; the picture below shows that my camera also had some trouble focusing. Nonetheless, even this fuzzy image is clear enough to show that the background landmark falls at the 16.0 cm mark on the ruler, while the target falls at about 16.8 cm. Thus the apparent separation between the target and the background is 16.8 cm - 16.0 cm = 0.8 cm. Since 1 cm on the ruler represents an angular separation of 1°, the angle between the target and the background landmark is about = 0.8°.
|Measurement of parallax angle. The dotted lines show where the background landmark (left) and target (right) fall along the cross-staff ruler.|
Using b = 45 inches and = 0.8° in the formula above, I get D = 3200 inches = 270 feet. These results are given to slightly better than one significant figure; the measurement of could 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 my use of a background landmark which is only a few times further away than the target. For example, if the background is about five times further away than the target, the resulting value of D will be about 20% too large.
In addition to the simple experiment on distance judging 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 should be made later, using a target and background that you chose. 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.
Do the experiments described in the sections on JUDGING DISTANCES and PARALLAX EXPERIMENTS, and write a report on your work. This report should include, in order,
In somewhat more detail, here are several things you should be sure to do in your lab report:
Note: keep in mind that binocular parallax is not the only way your brain judges distances. For example, if you move your head around, nearby objects appear to shift more than distant ones; this is another form of parallax which can help you judge distances using only one eye. The apparent sizes of objects, and the amounts of light they reflect, also provide some clues to their distances. All of these tricks, which are `built in' to our brains, are also used to measure distances in astronomy.
Last modified: January 21, 2003