We have an idea of why we use angular measure. Now we will see how to use it.

Keep in mind that parallax measurements only work for the nearest stars. This forms the bottom rung of what is called "the distance ladder". Other distance measurement techniques build on the parallax method. That is why we are going to explore this method in detail. Let's start with the question: How do we measure the distance to something that is far away?

We need two things: A baseline and an angle. 

Most generally, when the angle is very small, and in astronomy we usually deal with very small angles, we can say that

 

θ (theta)   =    length of opposite side of triangle (Baseline)

                                      length of adjacent side (Distance)

For this to work θ must be in radians, and the two lengths in the same distance units for the equation to be used. This known as the Small Angle Approximation.

How do we use the Small Angle Approximation in astronomy? Two ways. First, if we observe an object from two different locations at the ends of a baseline, and measure the change in angle of the object in our field of view, we can get the distance to the object. This is Trigonometric Parallax. 

Terrestrial examples:

1). The figure below shows two surveyors separated by a baseline they know, and measuring the change in angle of a tree across the river to get the distance to the tree.

Distance = Baseline / θ

For astronomy, replace the surveyors with telescopes and the tree with a planet or star.

The second way to use the Small Angle Approximation is to measure the angle subtended by an object in space. We have to know how far away it is before we can work out the size of the object.

2). Now that the surveyors know the distance to the tree, one measures the angular size and uses the equation below to measure its height:



Size = θ x Distance

Again, for astronomy, replace the surveyor with a telescope and the tree with a planet, moon or comet, for example.

Astronomical example:

The Sun subtends (covers) about 0.5 (0.0085 radians) of the sky. We know the sun is 1.5e8 km from the Earth. Plugging these numbers into the above Small Angle Approximation equation we get:

Size = 0.0085 radians x 1.5e8 km = 1.3e6 km in diameter (note that Size and Distance are both in km)
The actual size is slightly different but once again this is an approximation and a pretty good one at that.


So you see that even in trigonometric parallax we can get around the trigonometry and just use plain division.

The two surveyors above are separated by a baseline of 3 meters.  The shift in angle of the tree across the river they measure is 10.  What do they find for the distance to the tree in meters?

2.

Now they know the distance to the tree, they can measure its height.  The surveyor in the second picture finds that the tree subtends an angle of 10.  How high is the tree in meters?

3.


Eclipses and Parallax

Let's consider one more example of parallax in astronomy. You may have heard that when a solar eclipse occurs the path of totality is very narrow. Why is that? Well, it's because of parallax. The picture below shows the predicted paths of future solar eclipses in North America. Those living along the path see the Moon aligned with the Sun. Those on either side of the path see it sufficiently offset from the Sun that the Moon does not cover all of the Sun. That is because of parallax. The Moon is so much closer to us than the Sun that viewing it from either side of the path introduces enough parallax that it appears to be displaced relative to the center of the Sun.

This problem deals with angular measurements. 

Using the small angle approximation calculate the distance of a ruler when it subtends 1 degree = .017 radians.  Just use the formula below to get your answer for the distance.   Don't forget the units on your answer.

    .017 = 1 foot / L

4. Our calculated L =

Now go out into the hall outside the lab. A ruler is taped up in the window down the hall from the lab. As you move toward the ruler hold your thumb out in front of you and close one eye. Find the place where your thumb entirely blocks your view of the ruler (remember your thumb at arms length subtends roughly a degree). This should be the distance that you calculated for L.  Each of the floor tiles is 1 foot across. Use these to make a measurement of the distance to the ruler.

5. Our measured L =

The difference between your measured value and the calculated value is the measurement error in this experiment. If the two values disagree, what do you think is the reason?

6.

Consider the following simple exercise. 

7.   Have your partner hold up a pencil about six inches from the wall.  Stand 3 feet from the pencil.  Alternately close your left and right eye noting how much the pencil shifts compared to a fixed location on the wall.  Now back away from the pencil and blink your eyes again.  Note again how much the pencil shifts compared to the fixed location on the wall.  At what distance can you no longer detect any shift of the pencil?  Is there any point in trying to make this measurement for even greater distances? 


8.  Stand at the last distance from the pencil you wrote down in the previous question.   Keep both of your eyes open and take three steps to the left, then back to your original spot, then three steps to the right.  Does the pencil appear to shift relative to the wall again?  How, then, does the length of the baseline affect your ability to measure parallax?