Parallax - Exercise
One of the arguments raised by Aristotle and Ptolemy against the heliocentric system was the lack of parallax seen in the positions of the "fixed" stars. If the earth were really in motion then the relative positions of the stars with respect to each other should change. There is nothing particular about the stars in this remark. We see parallax all the time in our daily lives. As we move about a room, for example, the relative positions of objects per our line of sight changes. For example, from one vantage point a chair and lamp may seem to be aligned, one behind the other. When we move a few steps the chair has moved to the left or right of the line of sight to the lamp.

Tycho Brahe's compulsive attention to the precision of his measurements was driven by his attempt to see a parallax effect in the stars. He never saw this with his instruments. In fact, it wasn't until the early 19th century, some 2200 years after Aristotle that parallax in the stars was definitely detected.

We still use the method of parallax in modern satellites to map out the distances to the stars within our galaxy.  The parallactic shift is the only absolute measurement we have of distances. All other techniques of distance measurement which enable us to determine distances outside our galaxy are eventually calibrated to the absolute technique of parallax shift.

Procedure:
We will go outside to Greenlee plaza, which is adjacent to and south of the Physical Sciences building. South of the campus on distant hills and west of Salazar hall you will see tall antennas and a microwave tower. On Greenlee plaza there are several lampposts. The distances to these lampposts will be determined by measuring their parallactic shift relative to the distant antennas. You will be doing this with a partner.

1) Choose a lamppost/antenna pair and position yourself so that the two line up. You should be about 20 to 40 paces away from the lamppost.

2) Line up the lamppost and antenna using only one eye, standing sideways. Your right toe should be facing the lamppost if you use your right eye. The same applies for the left foot/eye if you use the left eye. See figure 1 for steps 2, 3, 4.

3) Your partner should lay the meter stick on the ground perpendicular to the line of sight starting at your toe.

4) Step to the other end of the meter stick with your toe exactly one meter away from its former position. This distance is the baseline S. Sight on the lamppost again. Note whether the lamppost has shifted to the left or right of your line of sight to the antenna. Holding the small ruler at arm's length read off the distance on the ruler between the center of the lamppost and the antenna as seen against the ruler. This is the distance s . See figure 2.

5) Your partner should measure the distance between the ruler in your hand and  your eye using the meter stick. This is basically the length of your arm. This is the distance deye .

Calculation
6) Fill in the table with your numbers from part 4 and 5 above. Calculate the distance to the lamppost according to the formula given.

7) Using the meter stick, measure the distance to the lamppost. Calculate the percent difference between your distance as determined by parallax and the direct measure of the distance.
Geometrical derivation
 
 
s, cm deye , cm parallax angle 
( in radians) 
a = s/deye		 parallax angle 
(in degrees ) 
A = 57.29*a		 S, baseline 
in meters		 calculated 
distance 
Dcalc = S/a, 
in meters		 measured 
distance 
Dm , in meters		 percent difference 
(Dm - Dcalc)/Dm