Materials

Meterstick

One of the most difficult problems in astronomy is determining the distances to objects in the sky. There are four basic methods of determining distances: radar, parallax, standard candles, and the Hubble Law. Each of these methods is most useful at certain distances, with radar being useful nearby (for example, the Moon), and the Hubble Law being useful at the most distant scales. In this exercise, we investigate the use of parallax to determine distances.

Even when observed with the largest telescopes, stars are still just points of light. Although we may be able to tell a lot about a star through its light, these observations do not give us a reference scale to use to measure their distances. We need to rely on a method with which you are actually already familiar: the parallax.

You can see the parallax effect by holding your thumb out at arm's length. View your thumb relative to a distant background while you alternate opening and closing each eye. Does your thumb seem to jump back and forth relative to this background? This is because the centers of your eyes are a few centimeters apart, so each eye has a different point of view.

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

   1. 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.

   2. Have your partner move the pen 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.).

   3. Now, have your lab partner move the pen 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.).

   The parallax of the pencil depends on the distance the pencil is from you -- the closer the object, the larger the parallax. Thus, although it may have been hard to tell precisely, when the pencil was half the original distance from you, it had twice the parallax; when it was double the original distance from you, it had one-half the parallax.

2. Here is a look at the approximate relationship between distance and parallax from a different viewpoint, from above the observer.

   • The distance d₂ is twice the distance d₁. Does it qualitatively appear that angle a₂ is one-half of a₁?

   • When the distance is large enough that the parallax angle is very small, the parallax angle is proportional to the inverse of the distance (1/d). Conversely, if we can measure the parallax angle, we know that the distance to the object is proportional to the inverse of that angle.

   
   

3. There is a limit at which parallax becomes ineffective. This occurs when the parallax angle is so small that you can't see a change from one eye to the other. This distance is effectively infinity. You and your partner can find your personal infinity by taking the pencil farther and farther away from the observer until the parallax becomes undetectable.

   1. One of you should take the role of "observer" while the other walks straight away, holding the pencil out at arm's length, stopping every meter or so. The observer should alternate opening and closing each eye as the partner stops. How far away are you from each other when the pencil stops moving relative to even more distant objects?

   2. What can you determine about the usefulness of parallax at different distances?

4. To measure a parallax, you use a baseline: the distance between observation points. In the exercise above, the "baseline" is equal to the distance between the center of your eyes. Measure this distance for you and your partner. Compare these two numbers to the distances found in Step 3a. Is there a relationship between the two sets of numbers?
5. In astronomy, we use a "baseline" of the diameter of the Earth's orbit. The infinite background is made of stars much farther away than the star or object in question. We can measure accurate parallaxes of thousands of stars; unfortunately, most stars are simply too far to get an accurate parallax, and we must resort to other methods to determine their distances. Label the following on the image on the worksheet: Reference star(s), target star, Earth, Sun, dates of observation (choose one, then find the other consistent with that choice).
6. Technology improves all the time, and our options for choosing a baseline may be expected to expand. Which of the following scenarios would give the longest baseline for measuring parallax (and therefore would be most desirable)? Which is most feasible with today's technology? Why?
   1. A satellite orbiting high above the Earth.
   2. A telescope on the Moon.
   3. A satellite orbiting Jupiter.
   4. A satellite orbiting Pluto.

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   © 1999 University of Washington
   Revised: May 1, 2000