How do we measure distances to other stars?
We're looking for a simple (algabraic, geometric) procedure to gauge the distance from the earth to the stars using triangulation. We would be very appreciative of any help you could offer.
What you're looking for is an experiment in measuring the "parallax" to nearby stars. (Parallax is what astronomers call the procedure you're interested in.) Parallax is the only direct way to measure distances to astronomical objects, and all other distance scales are predicated on parallax measurements. Unfortunately, parallaxes are extremely difficult to measure.
There's a very good geometrical treatment of parallax at this website. Usually, in introductory astronomy classes we don't bother with figuring out the direction of shift, but stop when we get to understanding its magnitude. Anyway, it's all there, and very clear if you speak geometry.
I wouldn't hold out any hope whatsoever for making parallax measurements on your own, however. Parallax angles are measured in fractions of seconds of arc, and there is no way to do the experiment to that accuracy without a major series of observations at a professional observatory. Even with an observatory, the procedure is extremely difficult. In fact, the inability to triangulate the distances to stars over the course of the year is one of the primary reasons the ancient Greek astronomer Ptolemy proposed the geocentric model for the solar system! He resoned that if he could not see the parallax motions of the stars, then the Earth could not be moving. He didn't fully comprehend the small radius of the Earth's orbit compared to the huge distances to the stars.
In fact, until the Hipparcos satellite, which measured parallaxes for a great many stars from space, we could only be sure of the parallax for a mere few hundred stars.
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