Activity

The Cosmic Distance Ladder: Parallax

Determining the position of a star or other object in space is an important concept in astronomy. During this activity you will learn how the distances to nearby stars can be measured using the parallax effect, and put this method into practise to determine the distance to nearby stars.

Materials List

  • LCOGT Galaxy image
  • LCOGT Star image
  • Student worksheet
  • Chalk
  • Measuring wheel
  • Astrolabe

Background Science

When looking at the night sky, it’s fairly easy to measure how the stars differ in brightness, but discerning how distant the stars are is a more difficult task. One way to measure astronomical distances is called Parallax

Parallax is the apparent change in position of a nearby object caused by a change in the observer’s point of view.  This is demonstrated in the diagram below; as the observer moved between two positions, he would see the same tree but it would appear to move against the background.

To calculate distances in space, astronomers measure the parallax angle as demonstrated on the diagram below: 

Once the parallax angle has been measured twice, from opposite sides of Earth’s orbit around the Sun, we can calculate the distance to stars with the parallax method we use the following equation:

The parallax angle is measure in arcminutes (arcmin) and arcseconds (arcsec . Just as an hour is divided into 60 minutes and a minute into 60 seconds, a degree is divided into 60 arcminutes and an arcminute is divided into 60 arcseconds. 

1 degree = 1° = 1/360 of a circle
1 arcminute = 1' = 1/60 of a degree
1 arcsecond = 1" = 1/60 of an arcminute = 1/3600 of a degree

The parallax method can only be used to measure the distance to stars close enough to show a measurable parallax, this stretches to about 100 parsecs (or ~326 light years).

Instructions

In this activity you will investigate how the distance of an object is related to how far it appears to move when you view it from different perspectives and discuss how parallax can be used in astronomy to reveal the distance to nearby stars.

Preparation

Before the lesson, you will need to locate a suitable area to create your parallax diagram such as a playground that will not be disturbed. Ensure you have all the materials necessary (see Materials List) and follow the steps below:

1. Draw a straight line 24-metres long with chalk. Mark the centre of the line with a circle. This will represent the Sun. 

2. Measuring along the line to one side of the Sun, draw circles at 1-metre, 6-metres and 10-metres. These circles will represent Planet 1, Planet 3 and Planet 5 consecutively. 

3. To the other side of the Sun, draw circles at 5-metres, 8-metres and 12-metres. Label these as Planet 2, Planet 4 and Planet 6, consecutively.

4. Draw a new 30-metre line extending from the Sun at a perpendicular 90-degree angle.  Mark circles at the following points: 9-metres, 15-metres, 21-metres, 26-metres and 30-metres. These points will represent the distant stars, label them Star A-E.

Introduction

1. Ask the students if they know what the term “parallax” means. Explain that it is the apparent change in the position of a nearby object that is actually caused by the movement of the observer. This may seem like a foreign concept to some students, so explain that this is something that you use constantly, even though you might not notice it. 

2. Place a sticker somewhere in the room. Ask the students to extend their arm, close one eye and cover the object in their vision with their thumb. Then instruct the students to keep their hand still and switch eyes (close the other eye).  Ask them to switch back and forth several times. Discuss the following questions:

  • What did you notice?
  • Did your thumb physically move?
  • How did your thumb appear to move?
  • What could explain your thumb’s apparent movement?

3. Ask the students to move their thumb close to their face while switching eyes. Ask them the following questions:

  • How does the apparent motion of your thumb change?
  • What is it about your eyes that allows you to see this?
  • If you could make your arms longer, would you expect the apparent shift to increase or decrease? Why?

4. Explain to your students that the change in their thumb’s apparent position relative to the background object is due to a change in the viewer’s position. In this case, the “viewer” is your left or right eye.  The few centimetres of separation between your eyes means that their viewing positions are different. 

5. Now explain to your students that parallax allows us to find the position of distant objects such as stars, since just like our eyes, Earth changes position every 6 months as it orbits the Sun. 

Parallax can be used to measure the distances to nearby stars. As the Earth orbits the sun, a nearby star will appear to move against the more distant background stars, in the same way the tree appears to move against the more distant mountains in the diagram above. 

Astronomers can measure a star's position twice in a year, with a 6-month gap between the first and second measurement, and calculate the apparent change in position. The star's apparent motion is called stellar parallax.

There is a simple relationship between a star's distance and its parallax angle:

d = 1/p

The distance d is measured in parsecs, the parallax angle p is measured in arcseconds. The radius of Earth’s orbit is 1 AU (149, 598, 000 km).

Outdoor Activity

1. Ask students to gather into groups (no more than 5 students per group) and lead them outside to the Parallax Diagram.  

2. Assign each team to one of the planets and provide them with an Astrolabe, Instruction Sheet and a Worksheet.

3. Ask the groups to use the Parallax Diagram to work through part one of the Student Worksheet

4. When they have finished part one of their worksheets, gather the groups and head back into the classroom. 

In the Classroom 

1. Open the Parallax table spreadsheet on your computer and use a projector to allow the class to see it for the class to see. 

2. Ask each group, one at a time, to provide the parallax angle for each star as seen from their planet. Leave this data on the screen for the class to see and ask them to complete part two of their activity worksheet.

3. Once the students have worked through the second part of their worksheet move on to the discussion section.

Discussion: 

1. Ask the students what they notice about the results – did each team end up the same distance? Is there a correlation between percent error and baseline distance (distance from the Sun?)

2. If each group measured the parallax angle with the same amount of care, there should be a notable trend showing teams assigned to planets further from the Sun getting more accurate results. Ask students why they think this is.  

The answer is that as the angle gets closer to 90-degrees, inaccuracies are more exaggerated by the tangent function. This means that a very long baseline is advantageous for parallax measurements. 

3. Ask students what is the longest possible baseline for Earth-based parallax measurements?  The answer is 2 AU, or twice the distance between the Earth and the Sun.

4. Do the students notice a correlation between the distance of a star and the percent error?  

For the reason stated previously, the further a star is the less accurate its distance calculations will be on average.  This is one of the problems with the parallax method for measuring distances in the cosmos. 

5. Ask the students to fill in part three of their worksheet before discussing the problems with parallax in further detail.

Conclusion

Since the parallax method cannot be used beyond 100 parsecs, have your students look up other possible techniques for measuring more distant astronomical objects. Examples include the use of Cepheid Variable stars and Type 1a supernovae for nearby galaxies and redshift for those further away.

Evaluation

It sounds fairly easy to measure the distance to another star; just make two measurements of its position six months apart. 

In practice, however, it is very difficult. The first successful measurement of stellar parallax came more than two hundred years after the invention of the telescope.

Ask your students to discuss some possible difficulties of measuring cosmic distances using parallax. Ensure they mention the following points:

Parallax shifts are always small. 

Parallax shift is even smaller than the apparent size of the star. In additional, starlight is refracted by Earth's atmosphere and causes the star to appear blurred. Determining the position of a star, plus that of several reference stars in the same field, to a very small fraction of this blurry dot is not an easy task. 

All stars in a field exhibit parallax. 

In practice, astronomers usually measure the shift of one star in an image relative to other stars in the same image. However, as the Earth moves from one side of the Sun to the other, we will see all the stars in the field shift, not only those of interest.
Astronomers must pick out a set of reference stars that happen to be much farther away than the target star. Distant stars will shift by a much smaller angle, so by measuring the position of the nearby target star relative to those distant ones, it is possible to detect its minute shift.

Limitations of Distance Measurement Using Stellar Parallax.

Parallax angles of less than 0.01 arcsec are very difficult to measure accurately from Earth because of the effects of the atmosphere. This limits Earth-based telescopes to measuring the distances to stars about 1/0.01 or 100 parsecs away. Space-based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method. However, most stars even in our own galaxy are much further away than 1000 parsecs, since the Milky Way is about 30,000 parsecs across. The next section describes how astronomers measure distances to more distant objects.