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- Learn about and do a little trigonometric parallax.
- Calculate the distance to a relatively nearby building using trigonometric parallax.
- Calculate the distance to the Moon by simulating two widely separated observations.

A most basic task for astronomers is to determine the distance to faraway objects, such as the Moon, the Sun, or the stars. This is more difficult than it might appear at first -- we can't simply use a giant tape measure. Neither can we make an estimate of the distance based on the apparent size of the (unfamiliar) object, since we could be looking at a large object very far away or a small object very nearby.

Luckily for the science of astronomy, there is a technique for measuring
remote distances called **parallax**.

A part of you actually already understands parallax, although the parallax
formula itself may be unfamiliar. *Normal human vision* uses parallax to
estimate distances to things all the time. Look at your thumb held at about
arm's length and alternatively blink each of your eyes: notice how it appears
to move relative to objects in the background. This is parallax! In this case,
the baseline, B, is just the distance between your two eyes. Your brain
automatically measures the parallax angle a and gives you an
intuitive guess for the distance of that nearby object. This is how *depth-perception* works. You can also bring your thumb closer and see that it appears to move more - the angle is larger for closer objects.

The principle of parallax is illustrated below, in figure 1. The two ‘eyeballs' represent two viewpoints, separated by a distance (the baseline) labeled B. If you stood at the viewpoint labeled X, you would see the tree (at O) directly across the page (let's say that's due East). Now, if you stood at the other viewpoint, labeled Y, you see that the tree is no longer straight across from your position. You have to look at some angle, a, away from East to see the tree.

**Figure 1.**

Remembering a little geometry, notice that the angle XOY is equal to the angle, α. This is the key, since we now have a little triangle made up of the points XOY and can use the small angle formula (or "skinny triangle" method). For α measured in degrees, the small angle formula is

**Equation 1.**

Solving for the distance gives

**Equation 2.**

So, if we know the angle, α (called the
**parallax angle**), and the distance, B, between the two viewing
points (called the **baseline**), we can derive the distance, R, to
the object, without ever having to leave our baseline (very handy when you could
be looking at an object 30 light-years away!). Looking at both the figure and
the formula, notice that the further away the object, O, is, the
*smaller* the parallax angle we measure will be. Also, for a particular
object, O, having a shorter baseline will result in a smaller parallax
angle.

Unfortunately, we aren't always lucky enough to have an object align at a perfect right angle like the tree in figure 1. No problem, a little more geometry and an extra measurement and we can still find α. Figure 2 shows a more common situation.

Figure 2: A more common situation when trying to measure the distance to an object using parallax. The two diagonal lines indicate that a large section of the image has been left out to make the image fit on a piece of paper.

In this situation, we use a distant reference object, V and measure two angles, β_{x} and β_{y}. Because V is very far away, the dotted lines connecting V to the viewing
points X and Y are nearly parallel. (If you have a hard time accepting this
concept, think of train tracks -- as they go off to the horizon they appear to
come together into a point, but really the train tracks are parallel and as far
apart as they always were). The further away V is, the closer to parallel the
dotted lines will be.

Looking at Figure 2 and remembering that the angles in a triangle must add to 180°, we see that

g = 180° - α- β_{x} .

Since g + h = 180° (because they make up a straight line), we can substitute

h = α + β_{x} .

But then, looking at the figure, and using what we know about parallel lines

h = β_{y} .

So then, substitute for h in the two equations above, and do a little rearranging, and we find

α = β_{y} - β_{x} .

So the parallax angle a is just the difference between the two angles βy and β_{x} that
you measure between the target and the very distant reference object.

The parallax method (sometimes called **geometric** or
**trigonometric parallax**, or **triangulation**) is
the *principle*, *direct* method of obtaining distances to objects
outside the solar system. In this lab you will apply the parallax method to measure at least one distance. Notice you won't know what the true distances are
before you do the lab -- this is on purpose, so that you can compare the
parallax distance you find to the true distance only after you do the
experiment.

We are using the slightly simpler (and more common for terrestrial use) α for the parallax angle. Astronomers more often use half that angle, labeled *p,* for the parallax angle, because it allows them to calculate the distance in parsecs as d = 1/*p* as long as *p* is in arcseconds. You'll take a look at the parsec in the concluding questions. You should be aware of the difference between α and *p* if you look up parallax on the web or in an astronomy textbook.

To complete the activity, go to the worksheet.

Last updated: 3/8/10 by SAM based on the previous version.

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