Name: Partner(s): Day/Time:

Parallax

Objectives

• Learn about and do a little trigonometric parallax.
• Calculate the distance to a relatively nearby building using trigonometric parallax.
• Calculate the distance to the Moon by simulating two widely seperated observations.

Introduction

A most basic task for astronomers is to determine the distance to faraway objects, such as the Moon, the Sun, or the stars. This is more difficult than it might appear at first -- we can't simply use a giant tape measure. Neither can we make an estimate of the distance based on the apparent size of the (unfamiliar) object, since we could be looking at a large object very far away or a small object very nearby.

Luckily for the science of astronomy, there is a technique for measuring remote distances called parallax. In this lab you will become more familiar with parallax, and use it to measure the distance to a toy at the other end of your lab table, the distance to a building across campus, and finally the distance to the Moon.

The principle of parallax is illustrated below, in figure 1. The two ‘eyeballs' represent two viewpoints, separated by a distance (the baseline) labeled B. If you stood at the viewpoint labeled X, you would see the tree (at O) directly across the page (let's say that's due East). Now, if you stood at the other viewpoint, labeled Y, you see that the tree is no longer straight across from your position. You have to look at some angle, a, away from East to see the tree.

Figure 1.

Remembering a little geometry, notice that the angle XOY is equal to the angle, a. This is the key, since we now have a little triangle made up of the points XOY and can use the small angle formula (or "skinny triangle" method). This tells us that

Equation 1.

or, with a little rearranging,

Equation 2.

.

So, if we know the angle, a (called the parallax angle), and the distance, B, between the two viewing points (called the baseline), we can derive the distance, R, to the object, without ever having to leave our baseline (very handy when you could be looking at an object 30 light-years away!). Looking at both the figure and the formula, notice that the further away the object, O, is, the smaller the parallax angle we measure will be. Also, for a particular object, O, having a shorter baseline will result in a smaller parallax angle.

A part of you actually already understands parallax, although the parallax formula itself may be unfamiliar. Normal human vision uses parallax to estimate distances to things all the time. Look at a nearby object (at about arm's length) and alternatively blink each of your eyes: notice how it appears to move relative to objects in the background. This is parallax! In this case, the baseline, B, is just the distance between your two eyes. Your brain automatically measures the parallax angle a and gives you an intuitive guess for the distance of that nearby object. This is how depth-perception works.

The parallax method (sometimes called geometric or trigonometric parallax, or triangulation) is the principle, direct method of obtaining distances to objects outside the solar system. In this lab you will apply the parallax method to three different distances. Notice you won't know what the true distances are before you do the lab -- this is on purpose, so that you can compare the parallax distance you find to the true distance only after you do the experiment.

Part One: Tabletop Parallax

Your instructor has covered your lab table with some graph paper. Don't write on the graph paper, please! At the one end of the table is a toy, which will be the object O from Figure 1. Your job is to estimate the distance from the other end of the table to the toy, without leaving the end of the table. You will need a meter stick and a protractor.

1. The first step is to find which of the lines on the graph paper points straight toward the toy. Do this by putting your eye at the edge of the table and sighting down the graph paper toward the toy. Where the graph paper reaches the edge of the table (or the end of the graph paper) is point X.
   
   
2. Next, move your eye to point Y, which can be anywhere else along the graph paper line perpendicular to the one pointing to the toy. Choose a spot near to the corner of the table. Hey -- this graph paper line doesn't point at the toy! You want to measure the angle, a, between the graph paper line at Y and your line of sight (the direction you have to look) to the toy using the protractor. To make it easier (without drawing big lines all over the graph paper), you might want to place the protractor on the paper, line it up correctly, then just place your pencil point in the direction of the toy -- then you can get up and read off the angle a. This is your parallax angle; enter the number into your data table.
   
   
3. To determine the distance to the toy, you must know the length of your baseline, B. Use the meter stick to determine the distance between the two sighting points X and Y, and enter this into your data table.
   
   
4. You can now calculate the distance R to the toy using the parallax formula in equation 2. Enter the answer into the table.
   
   
5. To check the answer, you can finally measure the distance to the toy with the meter stick. Enter the true distance R into the table.

Part Two: Parallax Across Town

Proceed to the roof, where you will use parallax on a slightly larger scale to determine the distance to a structure in downtown Ann Arbor. Since no one has covered Ann Arbor in graph paper, you do not have parallel lines to use for measuring your angles like we did on the table top. Instead, you must use a distant "reference object", as illustrated in Figure 2, where you see the target object O and a very distance reference object labeled V. Because V is very far away, the dotted lines connecting V to the viewing points X and Y are nearly parallel. (If you have a hard time accepting this concept, think of train tracks -- as they go off to the horizon they appear to come together into a point, but really the train tracks are parallel and as far apart as they always were). The further away V is, the closer to parallel the dotted lines will be. These dotted lines are the equivalent of the graph paper lines.

Unlike the graph paper on the table, the reference object will not line up exactly with the target object from viewpoint X. As a result, we must use a little geometry to find our angle a by measuring the angles between the reference object and the target object at both locations X and Y. Measure b_x, the angle between the target and the reference object at position X. Then go to viewpoint Y and measure the angle b_y between the target and the reference object at that position. Looking at Figure 2 and remembering that the angles in a triangle must add to 180°, we see that

g = 180° - a - b_x .

Since g + h = 180° (because they make up a straight line), we can substitute

h = a + b_x .

But then, looking at the figure, and using what we know about parallel lines

h = b_y .

So then, substitute for h in the two equations above, and do a little rearranging, and we find

a = b_y - b_x .

So the parallax angle a is just the difference between the two angles b that you measure between the target and the very distant reference object.

Figure 2.

For the purposes of our lab, the reference object can be any of the tall radio/TV antennas on the horizon as long as it's in the same direction as your object and it stays on the same side of the object as you move from viewpoint X to Y (i.e. if it appears on the right of the object at one end of the roof it should appear on the right side of the object at the other end of the roof, Y). The blinking lights on the antennas make them easy to spot if it is dark out. The target object will be pointed out by your lab instructor, but is usually a parking structure on the west side of the building -- if it is becoming dark out, make sure you pick a lighted feature on the parking lot.

1. Using one of the sighting telescopes, sight on the distant antenna as accurately as possible. Line up the top of the antenna with the center of the cross hairs, then have your partner read the angle indicated by the pointer on the base of the telescope. Estimate the angle to the nearest quarter of a degree (half the smallest division on the base).

2. Turn the telescope (not the base!) to align the target object with the crosshairs. It may help to leave both eyes open and move the telescope until you see the same thing with both eyes. Have your lab partner read the angle again. The angle b_x is the difference between your two measurements. Record this number in the table. To increase your precision, measure b_x 3 times by 3 different people, then average your readings to get your final value for b_x. Make sure each person measures from the same antenna to the same position on the target.

3. Now go to the other viewing station and measure b_y in the same way (also measuring three times, with three different people). Be sure to put the cross hairs on the same part of the building as before.
   
4. Calculate the parallax angle a by subtracting the averages of b_x from b_y and enter the value into your data table.
   
5. To derive the parallax distance to the parking structure, you still have to measure your baseline B. Do this by using meter sticks to measure the distance across the roof from viewpoint X to viewpoint Y, and enter the value into your table.
   
6. To get the true distance to the parking structure, use the attached map of Ann Arbor. The parking structure is on the corner of Fourth and William streets (unless your lab instructor told you to use a different building). Be sure to measure on the map between the center of your baseline (not X or Y itself) and the part of the building that you sighted.

Step 3: Parallax Distance to the Moon

Now you will measure the distance to the Moon using parallax. The target object O is the Moon, and the distant reference object V will be a bright star near the Moon. If your baseline were Angell Hall, the parallax angle would be too small to measure. Instead, Angell Hall will be position X and Atlanta, Georgia will be position Y, giving you a baseline of 942 km. Since it’s too far to go to Atlanta during lab, you will simulate the observation on a computer. If the weather is poor, you will also simulate the Ann Arbor measurement on the computer.

 A quick guide to Starry Night is available at http://www.astro.lsa.umich.edu/undergrad/Labs/Comp/docs.html
You can also access the user’s guide from the Help menu.

1. Open Starry Night. 
2. Click the favorites tab.  If there is a favorite called “parallax” with location “Ann Arbor, United States”, click it to open it.  If not, you’ll have to set up Starry night using the following steps:
   1. Stop the time flow
   2. Set the date and time to August 26, 2010 at 1:32:55 AM.
   3. Click the options tab to open the options pane.  Under the Guides section, check the Meridian checkbox under “Alt-Az …ridian)” to turn on the local meridian.  Under the Solar System section, click on “Plan…ons” to pull up the planet and moon dialogue box.  In the “Other” section of the dialogue box, make sure  “Enlarge Moon size at large FOVs” is NOT checked, so the moon will be the correct relative size.  Click Ok to close the window.
   4. If the Moon isn’t in view, click the Find tab.  Click the arrow button next to The Moon and select “Centre” from the pop-up menu.  Zoom in until you can clearly see the Moon.  The Moon should be almost full and close to centered on the meridian.  If it isn’t, make sure you have the correct date and time set, or check with your GSI.
   5. Click the Find tab to collapse the pane.
3. Once you have everything set up, change the pointer to the angular separation tool.
4. Find a star that lies as close to the meridian as possible. If you hold the pointer over it, the HUD should pop up and tell you the star name, usually a catalogue number like ‘HIP 60236’ or “TYC 5242-26-1”.  Record the name of the star on the worksheet for reference.  Control + click on it and choose “Select <starname>” – it should be the very first item in the pop-up menu.  This should put a label on the star so you can find it again easily.  You can also use the Find tab to find it again.
5. Center the pointer on the star.  Click and drag to roughly the center of the Moon.  The information HUD should tell you about the Moon, and there should be an Angular Separation readout near the pointer.  Record the Angular separation in table 3 under the βx (Ann Arbor) column.
6. Click the location arrow to get the pop-up menu and choose “Other…”.  Click the Latitude/Longitude tab.  Enter 0 for the latitude, and click the “Set Location” button.  This will take you to a spot on the equator with the same longitude as Ann Arbor.  Since the Moon is roughly on the equator on this date, it should be straight overhead at the new location.  This also means that you’ll have a right triangle, like you did for the first part, which is much more accurate than the method used on the roof.
7. If the Moon isn’t in view, hit “z” on the keyboard to look towrd zenith.  The moon should be on the meridian right near the 90º mark.  Ctrl+click on the Moon and choose “centre” to center on the Moon.  You can also use the Find pane again, as in the set-up, above.
8. Adjust the zoom so it looks about like it did from Ann Arbor. 
9. Your star should be labeled.  If not, you can type its name into the find pane and click the checkbox to have the computer label it again. 
10. Measure the angular separation again, as you did in step 8.  Record it in table 3 under the βy(equator) column.
11. The parallax angle is the difference between the two measurements.  Calculate the difference and enter it in table 3 under the column labeled α.
12. The baseline from here to the equator is 4.268x10^3 km (note this is the straight line distance, not the distance along the surface).  Calculate the distance to the Moon.  Show your work and record this in table 3 under the “R (parallax)” column.
13. Hold the pointer over the Moon to get the information HUD.  Record the “Distance from observer:” in table 3 under the “R(SN)” column.
14. Caculate your % error: .  Show your work and record it in the space on the worksheet.