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Parallax and Stellar Distances
I. Introduction
Parallax is defined as the apparent motion of a nearby object relative to a more distant background due to
the observer changing position. There is a definite relationship between the amount of parallax observed
and the distance to the object: the greater the distance, the smaller the parallax. Also, there is a
relationship between the amount of movement of the observer and the amount of parallax observed: the
more the observer moves, the greater the parallax shift.
A person experiences parallax in a moving car when they observe the nearby road signs appearing to
move by much faster than a distant mountain range. Astronomers take advantage of the phenomenon of
parallax for determining the distances to stars. As the Earth moves around the Sun in its annual motion,
nearby stars appear to move relative to the background of more distant stars (see the figure below).
Earth
Nearby Star
Sun
This motion is very small - less than 1 arcsecond for the nearest star (Remember: the farther an object, the
smaller the parallax). By carefully measuring these shifts for nearby stars, astronomers have determined
the distances to several thousand stars using parallax. That may seem like a large number of stars, but
keep in mind that there are over 100 billion stars in just our galaxy!
Astronomers use a unit of distance called the parsec. The parsec is defined as the distance of an object
which produces a parallax of one arcsecond when the observer moves a distance of one astronomical
unit (AU). Referring to the figure on the previous page and remembering that 1 AU is the average
distance from the Earth to the Sun, the definition of a parsec starts to make sense. One parsec equals
approximately 3.26 Light Years. If distances to stars (d) are measured in parsecs and the parallax angles
(p) are in arcseconds, then the relationship between d and p is simply:
d= 1/p
where d is in parsecs and p is in arcseconds.
Remember, this relationship only works for the observer changing their perspective by 1 AU.
II. Measuring Angles
A. Construct a calibrated scale for measuring angles:
1. Take a ruler and put a piece of masking tape that covers over the ruler.
2. Tape a sheet of paper to a wall at approximately shoulder height.
3. Two marks on the paper which are separated by 0.087 times the distance away from the wall
will create an angle of 5o.
d x 0.087
5o
d
4. In the space below, calculate the separation of two marks (in centimeters) needed to create an
angle of 5o for a distance from the wall of 3.0 meters:
separation of marks = ______________ cm.
5. Make two marks with the required separation on your piece of paper. Now stand so that your
eye is 3.0 meters away and hold your ruler at arms length. Line up the ruler with the paper and
have your partner place a pencil mark on the masking tape on the ruler that lines up with the marks
on the paper.
6. Mark off 10o and 15o intervals on your ruler by measuring the length of the 5o markings.
Subdivide into 1o increments on your ruler.
Determining the height of your partner.
7. Have your partner stand 10 meters away and measure and record the angle for this distance.
8. Have your partner move 10 meters further away and record the angle for this distance. Repeat
this step 3 more times.
9. Graph your results on Logger Pro:
Distance (meters)
angle (o)
l
Size (meters)
a.
L
b.
l
c.
L
d.
L
e.
L
10. Using the relationship:
s = r/57,
where s is the height or size of an object, r is the distance to the object and  is the angle (in
degrees) made by the object at a distance r, calculate and record the height of the object
measured in step 6 for each measurement (show one sample calculation) and then record
the average here:
Object height (average) = ______________ meters.
III. Parallax and Distances
A. While outside, determine the distance to a lab partner standing some distance away:
1. Using the relationship from step 10 in the previous section, we can express the distance, r, as:
r = 57 s/
where the units of r and s are in meters and  is in degrees.
2. Using a distant reference point such as a feature on the McDowell Mountains, measure the
angle between your partner and the reference point from two different positions as shown below:
angle a
s

angle b
angle a = ________
angle b = __________
s= _________
3. It can be shown that  = a + b. Using this fact, calculate the distance to your lab partner:
distance to lab partner = r = _________________ meters
IV. Questions
1. From your graph made in part II, step 6, draw a smooth line through the data. Is the relationship linear
(that is, a straight line)?
2. Discuss 3 sources of error in your distance determination in part III.
3. The star Deneb is 1600 light years away. What parallax angle would be observed as the Earth orbited
the Sun? How likely is it that the distance to Deneb was determined with parallax?
4. The nearest star Alpha Centauri makes a parallax angle of 0.762 arcseconds. What is its distance in
parsecs? What is its distance in light years?