Stellar Parallax and Nearby Stars

Introduction

This lab takes you to the next step in the cosmic distance ladder: figuring out the distances to stars other than the Sun. Astronomers derive distances to the nearest stars by a form of triangulation called stellar parallax, a most simple and direct method that relies on no assumptions other than the geometry of the Earth’s orbit around the sun.

Part 1 Measuring Parallax

Hold out your thumb at arm’s length, close one of your eyes, and examine the relative position of your thumb against other distant (background) objects, such as a window, wall, a tree, etc. Do it!

• Now look at your thumb with your other eye. What do you notice? • Move your thumb closer to your face and repeat the experiment. What was different this

time?

This is a demonstration of the parallax effect: the apparent shift in position of a relatively nearby object against more distant ones when viewed from different vantage points. Parallaxes are usually measured as angles; your thumb should appear to move by about 3 degrees when your arm is fully extended.

Look at this pair of photographs from the 1995 passage of Comet Hyakutake. The pictures were taken at the same time by two amateur astronomers at different places: one in Portugal and the other in Denmark.

It is clear that the comet’s position appears shifted with respect to the reference star (SAO 101241). This is yet another example of a parallactic shift; the comet is much closer to the Earth than the star, so that its position in the sky depends on the observer’s location.

http://www.astro.washington.edu/labs/parallax/Hyak_geometry.html
http://www.astro.washington.edu/labs/parallax/Hyak_geometry.html
In the preceding demonstration you noticed that your thumb exhibits larger parallactic shifts as it gets closer to your face (this should make sense). It is easy to see, then, that the amount of shift is proportional to the ratio between the separation of the observation stations (the “baseline”) and the distance to the object. Or, in other words, if we know our baseline (say, the separation between our eyes), then we can determine the distance to any object by simply measuring its parallax.

The geometry of a stellar parallax is shown below. As the Earth orbits the Sun, nearby stars will appear to shift in position relative to other, more distant stars.

Stellar parallaxes were predicted by the ancient Greek philosophers as a consequence of heliocentricity, but were not detected until the nineteenth century. Aristotle (and centuries later, even the astronomer Tycho Brahe) actually used the lack of stellar parallax to argue the case in favor of a stationary Earth.

The angles involved are very small, typically less than 1 second or arc! (Remember that 1 arcsecond = 1/3600 of a degree). To determine the distance to a star we can approximate the equation given in the previous section with the small angle approximation:

d = r / p

where d is the distance to the star, p is the parallax angle expressed in radians (see diagram), and r is the baseline, in this case 1 Astronomical Unit (A.U.)

Since there are 206,265 arc-seconds per radian, the formula can be re-written as:

d (in AU) = 206,265 / p

with p measured in arc-seconds .

Or, if we define the distance of one parsec as 206,265 AU, we get:

d (in parsecs) = 1 / p

This is the distance unit astronomers use most frequently, and it is equivalent to 3.26 light-years.

Questions for Part 1

1. How long do you have to wait for a star to undergo its maximum parallax displacement?

2. How can the observation of stellar parallaxes in general be used as evidence against a geocentric view of the cosmos?

3. How far, in parsecs, is an object that has a parallax p of 0.010 arc-second? How far is it, in light-years?

4. How far, in parsecs, is an object that has a parallax p of 0.00023 arc-seconds? How far is it, in light-years?

Part 2: Measuring Parallax

The circle labeled E in the diagram at left represents the Earth in its orbit in January.

Use a ruler and draw a line from the Earth to the background stars going through star A. Mark the Earth-Star A-Sun angle. This is called the parallax angle.

5. Describe what you think will happen to that angle if we were to do what we did in step #1 again but for star B, which is farther away.

Now find where the Earth will be 6 months later. Repeat step #1 using star A at this new position in its orbit. Using the diagram below mark where Star A will appear to be in January and then 6 months later.

6. Extend our observations over a number of years. How will Star A appear to move against the background stars?

7. How about star B over the same number of years compared to the motion of star A?

From Earth we can see the apparent motion of a star against a background of distant stars over time. This is called stellar parallax.

When we see parallax with our eyes we close one and then the other as we did above. This is where depth perception comes from, which allows us to estimate distances.

8. Below are a set of parallax observations of the different stars. Rank them from nearest to farthest. Explain your logic. Nearest ______ ______ ______ Farthest. Explain your answer.

Castor Pollux Sirius

9. Currently satellites orbiting the Earth are doing all the measurements of stellar

parallax. Why are they more accurate than Earth-based telescopes.

10. Consider this conversation between two astronomy students:

Student 1: I think that if we measured the parallax of a star from Mars, the angle would be larger than if we measured it from Earth because Mars has a much larger orbit. This would cause the star to move an angle comparable to that of its orbit.

Student 2: If we measured the parallax from Mars, the angle would have to be smaller because Mars is farther from the Sun so the star would also have to be farther away from Mars.

With whom do you agree? Explain.

d = r / p
d (in AU) = 206,265 / p
d (in parsecs) = 1 / p
Part 2: Measuring Parallax