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.
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
or, with a little rearranging,
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.
For the following procedure, record your data on the worksheet.
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.
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 bx, the angle between the target and the reference object at position X. Then go to viewpoint Y and measure the angle by 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 - bx .
Since g + h = 180° (because they make up a straight line), we can substitute
h = a + bx .
But then, looking at the figure, and using what we know about parallel lines
h = by .
So then, substitute for h in the two equations above, and do a little rearranging, and we find
a = by - bx .
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.
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.
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).
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 bx is the difference between your two measurements. Record this number in the table. To increase your precision, measure bx 3 times by 3 different people, then average your readings to get your final value for bx. Make sure each person measures from the same antenna to the same position on the target.
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 https://dept.astro.lsa.umich.edu/ugactivities/Labs/Comp/docs.html
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