### Parallax and Stellar Aberration

August 29, 2012

I was a Boy Scout when I was in sixth grade. The Boy Scouts of America has recently been an object of criticism, but in those days it was all fun. I learned how to make fire using flint and steel, and I was awarded a few merit badges.

After a while, my interests switched to other areas; so, I dropped out of scouting, but not without learning that moss grows on the north side of a tree. Now, if only my house were always north of where I was. There is one person for whom that's possible (Hint: He wears a read suit, he has a white beard, and he husbands reindeer).

One amazing thing that I learned as a Boy Scout was how to measure great distances without pacing them out. The technique was especially useful for measuring things that couldn't be paced, such as the width of rivers. This was an application of geometry before the subject was introduced in elementary school.

Distance measurement of the width of a river using similar triangles.

This method should be readily apparent to all geometers, but it's explained in the text.

(Figure rendered by the author using Inkscape)

The figure above shows the method for finding the width of a river. All that's required is a reference object, such as a tree, on the opposite side of the river, and some handy rocks to mark some points on your side of the river. You just pace the lines, OA and AB, walk the line BP, and pace the line, PO. Simple algebra will give you the river width, OC, knowing that triangles ABC and OPC are similar.

Later, in high school physics, we did the same thing on the roof of our school. I doubt that today's students would be allowed to do the same thing, since there was no guard railing around the roof.

This geometrical method is good on Earth, but when you look upwards, towards the stars, you need to use trigonometry, instead. The ancient Greeks were able to measure the distance to the Moon by noting its shift in angular position, its parallax, when it was observed at one edge of the Earth, or the other. You don't need to be at two places at once. You just wait for the Earth's rotation to carry you around from Moon-rise to Moon-set, resulting in a diurnal parallax (see figure). The Moon does move over the course of those twelve hours, so a correction needs to be applied to get the proper distance. Of course, to the ancients, it wasn't the Earth that rotated; rather, the fixed stars revolved about the Earth.

Finding the distance to the Moon. In ancient astronomy, it was all about the "fixed stars," those stars that never changed position, since they were affixed to a distant celestial sphere. Figure rendered by the author using Inkscape)

Todd Timberlake of Berry College (Mount Berry, Georgia) has published a paper reviewing the history of parallax in astrometry on the arXiv Preprint Server.[1] Early astronomers weren't too concerned with the fact that parallax of the stars wasn't observed, since their geocentric model of the universe put the Earth at its center. The stars were just too distant for their parallax to be seen from opposite sides of the Earth.

All this changed when the Copernicus introduced his heliocentric system. It was now apparent that the parallax of stars could be measured across the much wider breadth of the Earth's orbit. Surprise! No parallax was found. Were the stars that distant; or, perhaps, was Copernicus wrong? When faced with a challenging measurement, scientists roll up their sleeves and get to work refining their instrumentation.

It took two centuries before any parallax was found. Friedrich Bessel, of the famous Bessel functions, detected the parallax of the star, 61 Cygni, in 1838. The star, 61 Cygni, is 11.41 light years from Earth, a small figure by cosmic standards, but this distance corresponds to a parallax of just 286 milliarcseconds, One interesting side-effect of this search was the earlier discovery of another astronomical phenomenon, stellar aberration.

The finite speed of light (299,792,458 m/sec), combined with the speed of the Earth along its orbit (on average, 29,780 m/s), leads to a shift in stellar position because the Earth could be moving traversely with respect to a star's light signal. The ratio of these speeds is just 0.01%, but the effect can be detected.

James Bradley (1693 - 1762, left) in a portrait by Thomas Hudson (1701-1779), from the National Portrait Gallery, London; and Friedrich Wilhelm Bessel (1784 - 1846) in a portrait by Christian Albrecht Jensen (1792–1870), from the Ny Carlsberg Glyptotek. Left image and Right image, via Wikimedia Commons)

This stellar aberration was discovered in 1725 by James Bradley, who was the third Astronomer Royal. Stellar aberration was detected long before stellar parallax, since its maximum value is about twenty arcseconds, or nearly a hundred times greater than the parallax of 61 Cygni.

I close with the colorful first sentence of "An Attempt to Prove the Motion of the Earth by Observations," by Robert Hooke.[2] Hooke attempted a parallax observation of gamma Draconis in 1669. The detection of this parallax was not possible, since gamma-Draconis is at 154.3 light year's distance, and it has a parallax of only 21 milliarcseconds.
"Wether the Earth move or stand still hath been a Problem, that since Copernicus revived it, hath much exercised the Wits our best modern Astronomers and Philosophers, amongst which notwithstanding there hath not been any one who hath found out a certain manifestation either of the one or the other Doctrine.[2]"

### References:

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