«Lab 8 Parallax Measurements and Determining Distances 8.1 Introduction How do we know how far away stars and galaxies are from us? Determining the ...»
Parallax Measurements and
How do we know how far away stars and galaxies are from us? Determining the distances
to these distant objects is one of the most diﬃcult tasks that astronomers face. Since we
cannot simply pull out a very long ruler to make a few measurements, we have to use other
Inside the solar system, astronomers can bounce a radar signal oﬀ of a planet, asteroid or
comet to directly measure its distance. How does this work? A radar signal is an electromagnetic wave (a beam of light), so it always travels at the same speed, the speed of light.
Since we know how fast the signal travels, we just measure how long it takes to go out and to return to determine the object’s distance.
Some stars, however, are located hundreds, thousands or even tens of thousands of “lightyears” away from Earth. A light-year is the distance that light travels in a single year (about
9.5 trillion kilometers). To bounce a radar signal oﬀ of a star that is 100 light-years away, we would have to wait 200 years to get a signal back (remember the signal has to go out, bounce oﬀ the target, and come back). Obviously, radar is not a feasible method for determining how far away the stars are.
In fact, there is one, and only one, direct method to measure the distance to a star: the “parallax” method. Parallax is the angle by which something appears to move across the sky when an observer looking at that object changes position. By observing the size of this angle and knowing how far the observer has moved, one can determine the distance to the object. Today you will experiment with parallax, and develop an appreciation for the small angles that astronomers must measure to determine the distances to stars.
To get the basic idea, perform the following simple experiment. Hold your thumb out in front of you at arm’s length and look at it with your right eye closed and your left eye open.
Now close your left eye and open your right one. See how your thumb appeared to move to the left? Keep staring at your thumb, and change eyes several times. You should see your thumb appear to move back and forth, relative to the background. Of course, your thumb is not moving. Your vantage point is moving, and so your thumb appears to move. That’s the parallax eﬀect!
How does this work for stars? Instead of switching from eye to eye, we shift the position of our entire planet! We observe a star once, and then wait six months to observe it again. In six months, the Earth will have revolved half-way around the Sun. This shift of two A.U.
(twice the distance between the Earth and the Sun) is equivalent to the distance between your two eyes. Just as your thumb will appear to shift position relative to background objects when viewed from one eye and then the other, over six months a nearby star will appear to shift position in the sky relative to very distant stars.
8.1.1 Goals The primary goals of this laboratory exercise are to understand the theory and practice of using parallax to ﬁnd the distances to nearby stars, and to use it to measure the distance to objects for yourself.
8.1.2 Materials All online lab exercise components can be reached from the GEAS project lab URL, listed below.
You will need the following items to perform your parallax experiment:
• a protractor (provided on page 27)
• a 30-foot long tape measure, or a shorter tape measure (or a yardstick) and at least 15 feet of non-stretchy string (not yarn)
• a thin object and a tall object with vertical sides, such as a drinking straw and a full soft drink can, or a chopstick and a soup can
• a piece of cardboard, 30 inches by 6 inches
• a pair of scissors, and a roll of tape
• a needle, and 18 inches of brightly colored thread • 2 paper clips • 3 coins (use quarters, or even heavier objects, if windy)
• a pencil or pen
• a calculator • 2 large cardboard boxes, chairs, or stools (to create 2 ﬂat surfaces a couple of feet oﬀ the ground; recommended but not required)
• a clamp, to secure your measuring device to a ﬂat surface (helpful, but not required) You will also need a computer with an internet connection, to analyze the data you collect from your parallax experiment.
8.1.3 Primary Tasks This lab is built on two activities: 1) a parallax measurement experiment, to be performed in a safe, dry, well-lit space with a view of the horizon (or at least out to a distance of 200 feet), and 2) an application of the parallax technique to stars. Students will complete these two activities, answer a set of ﬁnal (Post-Lab) questions, and write a summary of the laboratory exercise.
8.1.4 Grading Scheme There are 100 points available for completing the exercise and submitting the lab report perfectly. They are allotted as shown below, with set numbers of points being awarded for individual questions and tasks within each section. Note that Section 8.6 (§8.6) contains 5 extra credit points.
8.1.5 Timeline Week 1: Read §8.1–§8.3, complete activities in §8.2 and §8.3, and begin ﬁnal (Post-Lab) questions in §8.4. Identify any issues that are not clear to you, so that you can receive feedback and assistance from your instructors before Week 2.
Week 2: Finish ﬁnal (Post-Lab) questions in §8.4, write lab summary, and submit completed lab report.
8.2 The Parallax Experiment In this experiment, you will develop a better understanding of parallax by measuring the apparent shift in position of a nearby object relative to the background as you view it from two locations. You will explore the eﬀects of changing the “object distance,” the distance between an object and an “observer” (you), and of changing the “vantage point separation” (the distance between your two viewing locations).
8.2.1 Setting up your experiment Our ﬁrst step is to create a measuring device for observing angles between various object and landmarks, as shown in Figure 8.5. The more carefully you build your device the more accurate your results will be, so take your time and work carefully. You can put the device together ahead of time, and then conduct your experiment later if this is more convenient than completing the entire experiment all at once.
Find the ﬁrst protractor provided in your lab manual at the end of this chapter (on page 27) and carefully cut it out. Make sure to follow the dotted lines, so that your edges are straight and form right angles (90◦ corners). Trim down a piece of cardboard (such as the side of a large box) to form a rectangle 30 inches wide and 6 inches tall. Cut the edges as perfectly straight as you can make them, again forming right angles.
Figure 8.1: The measuring device has a base formed from a piece of cardboard, 30 inches wide and 6 inches high.
The protractor is aﬃxed to the right side, and an “X” is marked exactly 24 inches to the left of its origin. This two-foot distance represents the two astronomical units between the Earth’s position around the Sun during January and July. Straightened paper clips are attached perpendicular to the surface at the protractor origin and the “X” mark, to mark sight-lines. A piece of thread is doubled and secured at the protractor origin, to create two threads that can be rotated around the protractor to mark various angles between 0 and 180 degrees.
Place the cardboard frame in front of you on a table, extending 15 inches on either side of you. Now place the protractor on the right end of the cardboard. Tape it securely to the cardboard, making sure that the dotted line forming the edge of the protractor paper is parallel to the long edge of the cardboard. Take your tape measure or yardstick, and measure a distance of two feet (24 inches) to the left of the origin of the protractor. (The origin is the dot in the center of the small circle, at the point where the dashed lines pointing to the numbers 10, 20, and so on up to 170, all meet.) Make sure to align your tape measure precisely with the line on the protractor extending from 180 to 0, and don’t let it tilt. Mark the point 24 inches to the left with a small “X”.
Next take a threaded needle, and thrust it through the origin of the protractor. Pull just enough thread through the hole that you can cut the needle free, and tape the two ends of thread securely to the back of the cardboard. You should have two pieces of thread coming up through the origin, each long enough to reach past the arc of numbers and trail oﬀ of the edge of the protractor paper.
Now take your two paper clips. Unfold one bend in each one, making the resultant long stick as straight as possible. This is very important, so spend a bit of time getting the bend out and straightening any kinks. The remaining portion will have two bends; force the ﬁrst into a perfect right angle and open up the last one as well. Your paper clip should now have the form of a small ﬂagpole, with a broad “V” shaped base. If you hold the base on a table, the ﬂagpole should point upward. Tape over the tips of the ﬂagpoles with small pieces of tape, so that they do not scratch anything or anyone.
Place the ﬁrst paper clip on top of the origin of the protractor, and tape it into place. Don’t secure it until you have checked that the ﬂagpole is centered directly over the center of the origin when viewed from all angles around it (you will probably have to touch up your paper clip angles a bit as you do this). This is another critical step, so take your time and do it well.
Place the second paper clip on top of the “X” mark 24 inches to the left of the protractor origin (at the left end of the cardboard). Again take your time and make sure that it is located directly above the mark, and points directly upward.
Our next step is to construct a support for the thin object which you will observe. If you are using a drinking straw and a soft drink can, tape the straw securely to the side of the can, pointing straight up, so that the top of the straw lies at least 6 inches above the table.
If you are using a thin chopstick and a soup can, tape the chopstick securely to the side of the can, pointing straight up, so that the top of the chopstick lies at least 6 inches above the table.
You will be lining up the position of the top of your thin object (the tip of the straw, or the thinner end of the chopstick) with your paper clip markers by eye, so it needs to be held straight, without drooping or falling.
8.2.2 Taking parallax measurements Complete the following, answering the 10 questions and completing Tables 8.2–8.5. Filling in the three tables correctly is worth 22 points, and the associated questions are worth 20 additional points (2 points per question).
You will need to ﬁnd a clear, ﬂat fairly-open space (ideally 30 feet long), with a view to the horizon (or to at least 200 feet away), as shown in Figure 8.2. An empty parking lot, a large room with an expansive view of the horizon (look for a ten-foot wide window on the ﬁrst few ﬂoors of a library or a community center, with a view of mountains or a radio tower), or Figure 8.2: Basic layout of the parallax experiment. Lay the coins in a fairly straight line, between 4 and 30 feet away from the measuring device. They are marking the position at which you will place objects for distance estimation, so if you are using existing objects such as street sign or lamp posts, chimneys, or trees, you will not need them.
a backyard with a low fence might all be possible places to work. Be sure that your location is safe, well-lit, and not too windy.
You will need a stable support for your measuring device, such as a wall, a table, a stool, or cardboard box. The support should raise it high enough oﬀ the ground that you can comfortably sight along both paper clip posts toward features on the horizon, and should keep it level. It is important that your measuring device does not shift position as you work, even slightly. If you have a clamp, use it to secure your measuring device in place; otherwise, tape may help. You are going to use the device to estimate the distance to several objects between you and the horizon (or a landmark at least 200 feet in front of you).
Once you have your measuring apparatus in place, select an initial location at which to place an object, some ﬁve feet or so in front of it. If there are existing objects with clearly-deﬁned vertical lines in front of you, such as a post supporting a street lamp, a chimney, or a radio antenna on a car, feel free to use them. Otherwise, place your foreground object (the can with an attached straw or chopstick) on top of a stool or cardboard box, and place your ﬁrst coin on the ground next to it at the same distance from the measuring device as the object.
Be sure to match the position of the coin with the position of the straw or chopstick, not with the supporting stool or box.
Before committing to a particular position, check that you can sight along both paper clip posts (ﬁrst one, and then the other) and the object, and have a clear view to the horizon or to a distant landmark. If you need to shift your object (or your measuring device) slightly to the left or right, that is ﬁne. (You can sight along the measuring device and direct your lab partner to shift the object for you if you are working in a team, to save time.) You are now going to estimate the distance to the object, directly and via the parallax method. First, measure directly by running the tape measure from the object to the center of the line between your two paper clip posts on your measuring device. (Your partner can help by holding one end of the tape measure for you.) This is the equivalent of measuring the distance between a nearby star and the Sun, if we had a really long, long tape measure.
Record this distance, in units of inches, in Table 8.2 under “Position 1.” (If you prefer to use the metric system, then give yourself a pat on the back and replace inches with centimeters throughout this exercise. Be sure to indicate the change of units clearly in Tables 8.2, 8.3, and 8.5, however.) We will use the parallax method to compute this distance independently without leaving our measuring device, and then we will compare our two distance measurements.