How can the parallax method be used to determine the distance of a star?

[snafu]

Hey, high school junior in an AP-level Astrophysics class reporting in... I have a short question about the equation in the subject line.

This equation was in my textbook, but with almost no explanation whatsoever. Yes, it explained how to apply the equation, but not where it comes from, why it works, how it is derived, etc. So, my teacher wants us for homework to find out those things. I've googled and looked everywhere I have to look, but I can't find a proper explanation. I'm worried I'm going about this the wrong way... Any help would be appreciated, thank you.

 

[tony873004]

Normally, for distance you would use trig. The parallax is how many degrees an object seems to move against an infinite background while you the observer change your position between 2 points (called baseline).

For example, if you observed a tree against a distant mountain range and you found that the tree moved 1 degree when you moved 5 meters, you could approximate this with a right triangle and say that d = 5 / tan(1) or d = 5 / sin(1). You get the same answer (~286 meters is your distance to the tree) since the angle is small and your side adjacent to the angle is almost the same length as the hyponeneuse of your triangle.

When looking at stars the angle is also small. In fact, it is much smaller and rarely exceeds 1/3600 of a degree called an arcsecond.

The Earth moves in its orbit. The diameter of its orbit is 2 AU (1 AU = average distance of Earth to Sun). But we can not view most stars 6 months apart, when the Earth has moved 2 AU because the star is in the daytime sky during one of those periods. So we use 1 AU as the standard base line for computing distance.

That would make our formula distance = 1 AU / sin(parallax) or 1 AU / tan(parallax). But since the angles are extremely small when looking at stars,
we can simplify the math and get rid of the trig if we invent a new unit of distance called the parsec. Parsec stands for Parallax Arcsecond.

A parsec is defined as the distance at which the parallax equals 1 arcsecond.

So in distance = 1 AU / parallax (in arcseconds), and the distance comes out in our new unit: Parsecs.

 

[xanthym]

The following Web Sites supplement previous msg and summarize the parallax method. Web Site #1 diagrams the 6 month parallax technique and illustrates the difference in viewing angle ("parallax") over 6 months of the same nearby star against background of distant stars.

Web Site #2 provides mathematical details of the parallax method. It shows how d=(1/Parallax) is the star's Distance in PARSECS when "Parallax" is measured in Arc-Seconds, and is the star's Distance in AU ("Astronomical Units") when "Parallax" is measured in Radians. It also provides example calculations.

Web Site #3 summarizes parallax quantities. Finally, Web Site #4 is an animated, interactive demonstration of parallax. It requires time to load, and you should read instructions at bottom of page. (Star's distance from Earth can be changed using mouse. Hopefully it'll work on your browser.). Good Luck!

URL #1 ---> http://www.scri.fsu.edu/~capstick/AST1002/Chapter8/parallax.gif
URL #2 ---> http://www.rgcl3.com/a20p/parallax_2.htm
URL #3 ---> http://wwwhip.obspm.fr/hipparcos/SandT/images/parallax.jpg
URL #4 ---> http://instruct1.cit.cornell.edu/courses/astro101/java/parallax/parallax.html


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