Measuring Distance using Parallax

Worth 20 points -- due Monday July 1st, 2002

This lab is best done after June 18th, when we will discuss the parallax of stars. However, it can be done before that as we will discuss the concept of angular size and distance earlier in the course.

This is a fun and easy lab designed by Prof. Gibor Basri of the UCB Astronomy Department. We will measure the angular size of the Moon, which will allow us to estimate the distance to the Moon. We will also measure the parallax of a pen held at arms length, which will allow us to estimate the length of your arm. The techniques in this lab are frequently used by professional astronomers.

Throughout this lab we will use the "small angle" formula. This is a formula that describes the relationship between size, angular size, and distance. Here we will use a variant of the formula which specifically uses degrees:

The exact formula involves trigonometry, but because we will be measuring small angles, we will manage just fine with this approximate formula. Each time we use this formula the meanings of size, angular size, and distance will change, but the underlying principle will be the same.

I. Building an Angletron

• Cut out a strip of paper or cardboard about 3 inches wide and 8.5 inches long.
• Hold the paper at arm's length in front of you and measure the distance from the paper to your eyes. You can measure the distance directly with a tape measure, or you can measure the distance with a string first and then measure the length of the string with a ruler. Be sure to be clear about the units of length you use (centimeters, inches, etc.). Get a friend to help you measure the distance, if possible. If you wish you can repeat the measurement multiple times and take an average, to achieve higher accuracy. Record the technique you used to measure the distance, and the answer you obtained.
• Now we want to make marks on the paper that will be 1° apart, when the paper is held at arm's length. We use the small-angle formula to calculate how far apart these marks should be. We have already measured the distance we need to use in the formula. We want the angular separation (i.e. angular size) of the marks to be 1°. The formula tells us how far apart our two marks need to be. The distance between the marks will be given by the "size" in the formula. Show your calculation of the spacing between tick marks.
• Now use a ruler to place big tick marks on the piece of paper we cut out earlier. The marks should be separated by the amount we just calculated with the small angle formula. Label the big tick marks 0°, 1°, 2°, etc..
• Divide the space between each big tick mark into 4 equally sized intervals by drawing 3 smaller tick marks between the major tick marks. These smaller tick marks will be 1/4 ° apart when the paper is held at arm's length. When you are done using your Angletron, attach it to your lab write-up.

II. Measuring the Moon

• Hold your Angletron at arm's length, so that it partially covers the Moon. Note how many small tick marks it takes to cross the diameter of the Moon and how many degrees this corresponds to. Estimate fractions of a small tick mark spacing in your measurements (i.e., you might get a measurement of 2.3 small tick marks). Again it may be wise to make the measurement several times and take an average. Record your answer.
• The actual angular size of the Moon is 0.5°. Compare this with your value. Offer some explanations for why they might be different.
• If your Angletron gave an angular size that was very far off, then the spacings of your tick marks may not be correct. Check your Angletron and make a new one if necessary. Also, be sure you are measuring the full diameter of the Moon.
• We have just measured the angular size of the Moon. The diameter (i.e., size) of the Moon is 3,476 km. Now use the small-angle formula to estimate the Moon's distance.
• Compare your calculated distance of the Moon with the actual average distance of the Moon, 3.84 x 10⁵ km. What is your percent error (the difference in the actual and measured value divided by the actual value)?

III. Measuring with Parallax

• Measure the distance between your eyes (center of one pupil to the center of the next pupil). Use a tape measure or a string. Record the technique you used and the answer you obtained.
• Now hold a pen at arm's length. Close one eye and note which background feature the pen is blocking. Then switch eyes and note which background feature the pen is now blocking. The background features should be as distant as possible.
• Use your Angletron to measure the angle between the two landmark features that were blocked by the pen. This angle is the same as the angle from one of your eyes to the pen to your other eye!
• We now know the separation of your eyes ("size") and the angular separation of your eyes, so we can use the small-angle formula to calculate the length ("distance") of your arm. Do this calculation now.
• Quantitatively compare this estimate with your more direct measurement in Part I.

This procedure is very similar to the way astronomers measure the distance of nearby stars (but astronomers' Angletrons are much more accurate!) The differences are:
- The distance to the pen becomes the distance to the nearby star.
- The angle between the background features becomes the angle between very distant stars or galaxies.
- The spacing between your eyes becomes the distance between Earth's position on opposite sides of its orbit around the Sun.

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