Parallax lab answers

Parallax Background

The Background information on this page will help you to understand the concept of parallax and how it is used to determine distances in astronomy. Once you understand these ideas, you may proceed to the lab activity by proceeding the the Lab Activity part of this module. (Click "Next" at the bottom of this page).

In the lab activity, you will calculate the height of an object using the same mathematical techniques involved in stellar parallax. To do this, you will construct a simple tool called an astrolabe.

Parallax Background Learning Objectives

By the end of this section, you will be able to:

· Understand the concept of triangulating distances to distant objects, including stars

· Explain why space-based satellites deliver more precise distances than ground-based methods

· Discuss astronomers’ efforts to study the stars closest to the Sun

It is an enormous step to go from the planets to the stars. For example, our Voyager 1 probe, which was launched in 1977, has now traveled farther from Earth than any other spacecraft. As this is written in 2016, Voyager 1 is 134 AU from the Sun.1 (Links to an external site.) The nearest star, however, is hundreds of thousands of AU from Earth. Even so, we can, in principle, survey distances to the stars using the same technique that a civil engineer employs to survey the distance to an inaccessible mountain or tree—the method of triangulation.

Triangulation in Space

A practical example of triangulation is your own depth perception. As you are pleased to discover every morning when you look in the mirror, your two eyes are located some distance apart. You therefore view the world from two different vantage points, and it is this dual perspective that allows you to get a general sense of how far away objects are.

To see what we mean, take a pen and hold it a few inches in front of your face. Look at it first with one eye (closing the other) and then switch eyes. Note how the pen seems to shift relative to objects across the room. Now hold the pen at arm’s length: the shift is less. If you play with moving the pen for a while, you will notice that the farther away you hold it, the less it seems to shift. Your brain automatically performs such comparisons and gives you a pretty good sense of how far away things in your immediate neighborhood are.

If your arms were made of rubber, you could stretch the pen far enough away from your eyes that the shift would become imperceptible. This is because our depth perception fails for objects more than a few tens of meters away. In order to see the shift of an object a city block or more from you, your eyes would need to be spread apart a lot farther.

Let’s see how surveyors take advantage of the same idea. Suppose you are trying to measure the distance to a tree across a deep river (Figure (Links to an external site.)). You set up two observing stations some distance apart. That distance (line AB in Figure (Links to an external site.)) is called the baseline. Now the direction to the tree (C in the figure) in relation to the baseline is observed from each station. Note that C appears in different directions from the two stations. This apparent change in direction of the remote object due to a change in vantage point of the observer is called parallax.

Triangulation.

Illustration of the Triangulation Method. In this illustration a surveyor’s transit is shown at two positions along a stream of water. Position “A” is at the center left of this image, and position “B” is just below the center of the illustration. They are separated by a distance labeled “Baseline,” with a black line drawn connecting the two. Both instruments are being used to measure the distance to a tree on the far side of the stream which is located at the upper right corner in the illustration. The tree is labeled “C.” Black lines are drawn from positions “A” and “B” to the tree at “C” to create the triangle ABC. A dashed line is drawn from the center of the baseline to point “C.” A curved arrow is drawn from the baseline to the line AC to represent the angle between the baseline and line AC.Triangulation allows us to measure distances to inaccessible objects. By getting the angle to a tree from two different vantage points, we can calculate the properties of the triangle they make and thus the distance to the tree.

The parallax is also the angle that lines AC and BC make—in mathematical terms, the angle subtended by the baseline. A knowledge of the angles at A and B and the length of the baseline, AB, allows the triangle ABC to be solved for any of its dimensions—say, the distance AC or BC. The solution could be reached by constructing a scale drawing or by using trigonometry to make a numerical calculation. If the tree were farther away, the whole triangle would be longer and skinnier, and the parallax angle would be smaller. Thus, we have the general rule that the smaller the parallax, the more distant the object we are measuring must be.

In practice, the kinds of baselines surveyors use for measuring distances on Earth are completely useless when we try to gauge distances in space. The farther away an astronomical object lies, the longer the baseline has to be to give us a reasonable chance of making a measurement. Unfortunately, nearly all astronomical objects are very far away. To measure their distances requires a very large baseline and highly precise angular measurements. The Moon is the only object near enough that its distance can be found fairly accurately with measurements made without a telescope. Ptolemy determined the distance to the Moon correctly to within a few percent. He used the turning Earth itself as a baseline, measuring the position of the Moon relative to the stars at two different times of night.

With the aid of telescopes, later astronomers were able to measure the distances to the nearer planets and asteroids using Earth’s diameter as a baseline. This is how the AU was first established. To reach for the stars, however, requires a much longer baseline for triangulation and extremely sensitive measurements. Such a baseline is provided by Earth’s annual trip around the Sun.

Distances to Stars

As Earth travels from one side of its orbit to the other, it graciously provides us with a baseline of 2 AU, or about 300 million kilometers. Although this is a much bigger baseline than the diameter of Earth, the stars are so far away that the resulting parallaxshift is still not visible to the naked eye—not even for the closest stars. This dilemma perplexed the ancient Greeks, some of whom had actually suggested that the Sun might be the center of the solar system, with Earth in motion around it. Aristotle and others argued, however, that Earth could not be revolving about the Sun. If it were, they said, we would surely observe the parallax of the nearer stars against the background of more distant objects as we viewed the sky from different parts of Earth’s orbit. Tycho Brahe (1546–1601) advanced the same faulty argument nearly 2000 years later, when his careful measurements of stellar positions with the unaided eye revealed no such shift.

These early observers did not realize how truly distant the stars were and how small the change in their positions therefore was, even with the entire orbit of Earth as a baseline. The problem was that they did not have tools to measure parallax shifts too small to be seen with the human eye. By the eighteenth century, when there was no longer serious doubt about Earth’s revolution, it became clear that the stars must be extremely distant. Astronomers equipped with telescopes began to devise instruments capable of measuring the tiny shifts of nearby stars relative to the background of more distant (and thus unshifting) celestial objects.

This was a significant technical challenge, since, even for the nearest stars, parallax angles are usually only a fraction of a second of arc. Recall that one second of arc (arcsec) is an angle of only 1/3600 of a degree. A coin the size of a US quarter would appear to have a diameter of 1 arcsecond if you were viewing it from a distance of about 5 kilometers (3 miles). Think about how small an angle that is. No wonder it took astronomers a long time before they could measure such tiny shifts.

The figure below shows how such measurements work. Seen from opposite sides of Earth’s orbit, a nearby star shifts position when compared to a pattern of more distant stars. Astronomers actually define parallax to be one-half the angle that a star shifts when seen from opposite sides of Earth’s orbit (the angle labeled P in the figure). The reason for this definition is just that they prefer to deal with a baseline of 1 AU instead of 2 AU.

Parallax.

Illustration of Parallax. The Sun is drawn as a yellow disk in the left hand portion of the diagram and is labeled “Sun.” A blue circle surrounds the Sun and is labeled “Earth’s orbit.” The Earth is shown at two positions on the blue circle. Position “A” at the bottom of the circle and “B” at the top. Above and to the right of the center, a nearby star is drawn as an unlabeled red dot. In the upper right is an unlabeled group of five stars that are more distant than the red star. A white line is drawn from position A through the red dot to the uppermost stars in the group. A white line is drawn from position B through the red dot to the middle star in the group. A dashed line is drawn from the Sun to the red dot. The parallax angle, “p,” is drawn between the dashed line and line B. To illustrate the effect of parallax, two insets are included near points A and B. The inset at point B is labeled “Sky as seen from B,” and shows the red dot near the middle star of the group of five stars that are illustrated in the upper right side of the figure. The inset at point A is labeled “Sky as seen from A,” and shows the red dot near the uppermost stars of the group of five stars that are illustrated in the upper right side of the figure.As Earth revolves around the Sun, the direction in which we see a nearby star varies with respect to distant stars. We define the parallax of the nearby star to be one half of the total change in direction, and we usually measure it in arcseconds.

Units of Stellar Distance

With a baseline of one AU, how far away would a star have to be to have a parallax of 1 arcsecond? The answer turns out to be 206,265 AU, or 3.26 light-years. This is equal to 3.1 × 1013 kilometers (in other words, 31 trillion kilometers). We give this unit a special name, the parsec (pc)—derived from “the distance at which we have a parallax of one second.” The distance (D) of a star in parsecs is just the reciprocal of its parallax (p) in arcseconds; that is,

D=1p

Thus, a star with a parallax of 0.1 arcsecond would be found at a distance of 10 parsecs, and one with a parallax of 0.05 arcsecond would be 20 parsecs away.

Back in the days when most of our distances came from parallax measurements, a parsec was a useful unit of distance, but it is not as intuitive as the light-year. One advantage of the light-year as a unit is that it emphasizes the fact that, as we look out into space, we are also looking back into time. The light that we see from a star 100 light-years away left that star 100 years ago. What we study is not the star as it is now, but rather as it was in the past. The light that reaches our telescopes today from distant galaxies left them before Earth even existed. To convert between the two distance units, just bear in mind: 1 parsec = 3.26 light-year, and 1 light-year = 0.31 parsec.

The Nearest Stars

No known star (other than the Sun) is within 1 light-year or even 1 parsec of Earth. The stellar neighbors nearest the Sun are three stars in the constellation of Centaurus. To the unaided eye, the brightest of these three stars is Alpha Centauri, which is only 30○from the south celestial pole and hence not visible from the mainland United States. Alpha Centauri itself is a binary star—two stars in mutual revolution—too close together to be distinguished without a telescope. These two stars are 4.4 light-years from us. Nearby is a third faint star, known as Proxima Centauri. Proxima, with a distance of 4.3 light-years, is slightly closer to us than the other two stars. If Proxima Centauri is part of a triple star system with the binary Alpha Centauri, as seems likely, then its orbital period may be longer than 500,000 years.

Proxima Centauri is an example of the most common type of star, and our most common type of stellar neighbor. Low-mass red M dwarfs make up about 70% of all stars and dominate the census of stars within 10 parsecs of the Sun. The latest survey of the solar neighborhood has counted 357 stars and brown dwarfs within 10 parsecs, and 248 of these are red dwarfs. Yet, if you wanted to see an M dwarf with your naked eye, you would be out of luck. These stars only produce a fraction of the Sun’s light, and nearly all of them require a telescope to be detected.

The nearest star visible without a telescope from most of the United States is the brightest appearing of all the stars, Sirius, which has a distance of a little more than 8 light-years. It too is a binary system, composed of a faint white dwarf orbiting a bluish-white, main-sequence star. It is an interesting coincidence of numbers that light reaches us from the Sun in about 8 minutes and from the next brightest star in the sky in about 8 years.

Measuring Parallaxes in Space

The measurements of stellar parallax were revolutionized by the launch of the spacecraft Hipparcos in 1989, which measured distances for thousands of stars out to about 300 light-years with an accuracy of 10 to 20% (see Figure (Links to an external site.) and the feature on Parallax and Space Astronomy (Links to an external site.)). However, even 300 light-years are less than 1% the size of our Galaxy’s main disk.