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Help needed for Parallax in Astrometry and its early Application

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#1 DCWZ

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Posted 12 January 2021 - 10:08 AM

Hi all, sort of an unconventional question here. I hope it doesn't violate the forum rules, but if it does, do let me know and I'll retract it!

Can anyone who is well-versed in astrometry or the history of astronomy provide any sources (with mathematical workings) on how Friedrich Bessel managed to triangulate the position of 61 Cygni via the parallax angle of 0.314"? I've tried working it out myself, but cannot figure out the angle of deviation of 61 Cygni from the ecliptic given Earth's tilt and the star's Ra and declination.

 

Alternatively, likewise sources for how Thomas Henderson figured out that of Alpha Centauri or Friedrich Struve for Vega would work too. Or if any kind and mathematically inclined souls are able to help me on this, I would greatly appreciate it!



#2 555aaa

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Posted 12 January 2021 - 12:26 PM

Isn’t the parallax independent from the star’s position relative to the ecliptic? A star that’s one parsec away has a one arc second parallax anywhere on the sky. The proper motion of most nearby stars is typically larger than their parallax. If the sky positions are relative to distant background field stars then you don’t have to take into account the aberration from the earth moving towards that direction or away since that is the same for field stars. That aberration is many arc seconds if I recall. Isn’t Gauss’s method a least squares fit? So you have a wiggly line across the sky. Do a series of trial fits of a line with a sinusoid on it and adjust the parameters to find the best fit and the best fit gives you both parallax and proper motion.

#3 munkacsymj

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Posted 12 January 2021 - 10:05 PM

In the general case, the parallax motion of a star is in the shape of an ellipse (with proper motion added as an additional linear motion). The eccentricity of the ellipse is a function of the star's distance from the plane of the Earth's orbit (the ecliptic). As you approach the ecliptic, the ellipse collapses into a linear back-and-forth motion. For a star that is perpendicular to the Earth's orbital plane, the parallax motion is a circle.

 

Another way to think about it is to look at how the Earth's orbit would look as seen from the star. That motion of the Earth is the same shape as the parallax motion of the star as seen from the Earth. Thus, the apparent parallax motion is what you get when you project the shape of the Earth's orbit into a plane perpendicular to the vector from the sun to the star of interest. If you work out that projection, you'll get the shape of the ellipse.

 

As a special case, stars on the ecliptic have a simple back-and-forth linear motion in the plane of the ecliptic. As luck would have it, one of the closest stars (Wolf 359) sits almost exactly on the ecliptic. It's an excellent first star for anyone wanting to try to measure parallax. I mentored a local high school student who worked with me to measure Wolf 359's parallax and proper motion. We were within 2% on the proper motion and around 8% for the parallax angle (parallax angle is about 0.4 arcsec -- 1/4 of a pixel for our optical configuration). 

 

- Mark M



#4 StupendousMan

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Posted 19 January 2021 - 09:34 PM

You can read one of Bessel' original accounts from 1838 at the URL below, if you wish.

 

http://articles.adsa...MNRAS...4..152B


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#5 555aaa

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Posted 20 January 2021 - 09:29 AM

You can read one of Bessel' original accounts from 1838 at the URL below, if you wish.

http://articles.adsa...MNRAS...4..152B

I had forgotten that they did these observations visually with a filar micrometer eyepiece. I think that is what it is called. Thanks for the post.

Edited by 555aaa, 20 January 2021 - 09:32 AM.



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