I’m trying to understand how to measure the distance to nearby stars using the parallax method.
My question is: how do you get the value of the p angle?
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$\begingroup$ It's half of the angle the star appears to move over the "fixed" stars in the background? .... Is this some sort of trick question, because that is very elementary geometry. It get a bit more complicated when the star in question is not exactly in the plane of Earth's orbit, but that's just one additional cosine to calculate. $\endgroup$– PcManJun 27, 2021 at 9:42
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$\begingroup$ The image is misleading. The distance to the nearest star compared to the Earths orbit is much larger than shown here. $\endgroup$– Thorbjørn Ravn AndersenJun 27, 2021 at 16:58
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$\begingroup$ There's some pretty good answers here now. It's of course up to you, but if you feel there needs to be something more before accepting one, perhaps consider leaving a comment? Thanks! Why don't question-askers accept answers? $\endgroup$– uhohJul 11, 2021 at 0:25
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2 Answers
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In actual fact its a bit more complicated, because
- the Earth's orbit is not perfectly circular,
- The star does not lie exactly on the Earth's orbital plane, so the observation is not a line of movement but an ellipse
- both the Sun and the Star are moving relative to everything, so the ellipse is not an ellipse but a helix
- Observational errors from various sources, such as atmospheric disturbances, so the helix is wobbly and fuzzy,
- The "fixed" stars not being at infinite distance, just very very far away (distant galaxies work better, simply because they are more distant)
- Etc.
The combination of these errors is what limits the use of parallax method for distance measurement to about 150 parsecs. For observations of more distant stars the errors overwhelm the observed motion and alternate measurement method are needed.
You might consider reading through https://en.wikipedia.org/wiki/Stellar_parallax , it explains a lot.
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$\begingroup$ PcMan, great comments. One other factor is that the star might not be perpendicular to the June-December axis as shown in the diagram. The maximum parallax might be at a different time of year, depending on the star. $\endgroup$– JimJun 27, 2021 at 13:02
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You measure it on the photographs.
If you take multiple images of the same star over the year you will find that it moves in a loop (compared to the very distant background stars) You measure the size of the loop. A larger loop is a larger angle. Because the angles are small, the size of the loop is in direct proportion to the angle. If you know the scale of the photograph, you can find the angle directly.
In practice, there are difficulties: The star is moving, so rather than a closed loop, it will trace a helical path in the sky. Your images may be distorted by the atmosphere, making it hard to exactly locate the star, and the star may not be visible for several months of the year when it is behind the sun (from our perspective). The difficulties can be overcome by taking a series of images over a longer period of time.
