Parallax and Distance Measurement

What is Parallax?

Hold your finger about two inches in front of your nose. Look at it with just your left eye and then your right eye. Notice how far to the left and right it appears to shift against the background of your computer screen? Now move your finger closer to the screen and look at it again with each eye. The apparent left to right shift is much smaller, now, isn't it? That shift is an example of parallax. Looking with each eye shifted the point of view, which caused a large apparent shift when your finger was close to your eye, and a smaller apparent shift when it was farther away. The stars are so far away that we need a huge shift in our point of view before we can detect even a slight apparent shit in their position. The diagram shows how we can use Earth's orbit to get the largest actual shift in our point of view without leaving the planet.

Most stars are so very far away that they form a background against which we can measure the apparent shift of the stars that are nearest to Earth.

The two negatives (black stars on a clear piece of film) in the diagram are placed one on top of the other, so the apparent shift of the distant star can be measured very carefully. Knowing the power of the telescope used to make the two photographs, we can convert this shift into an angle of arch. Knowing the diameter of Earth's orbit, we can use trigonometry to calculate the distance to that star!

Calculating Sun-to-Star Distance:

The triangle formed by the Star and the two Earth positions is an isosceles triangle, which can be bisected (split down the middle) into two right triangles. The common leg between them represents the Sun-to-Star distance. One of these right triangles is shown in the diagram below.

This tiny shift angle, labeled a, is related to the Sun-to-Star distance. The stars that are farthest away from us (whose shift can be detected) have the smallest shifts. Remember from the experiment above that when your finger was furthee away from your nose, the apparent shift was smaller.

In trigonometry, we define the sine of angle a in the diagram as the opposite side (the radius of the Earth's orbit, r), divided by the hypotenuse (labeled h). The cosine of an angle is the adjacent side (the Sun-to-Star distance, d) divided by the hypotenuse. We can use algebra to get d, the Sun-to-Star distance in terms of these two trigonometric values.

When we are dealing with really small angles, the sine of an angle is said to be equal to that angle and the cosine is equal to 1.

We know the angle from the analysis of the photographs, and we know the radius of Earth's orbit. The Sun-to-Star distance is the quotient of these two known values. It should be noted that in practice, perhaps twenty or more photographs are made at various points along Earth's semiannual orbit. If a star is too far away to exhibit a shift, there are other methods Astronomers can use to estimate their distances. Enter "star distance measurement" into an Internet search engine for further details.

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