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Re: Polaris - changes in SHA and Declination
From: Frank Reed
Date: 2019 Feb 22, 10:33 -0800

David Fleming, you wrote:
"Wikipedia has aberation at 20 arc sec for stars located perpendicular to plane of earth's orbit."

It's about 20 seconds of arc for any star located roughly 90° from the spot in the sky towards which the Earth is moving in its orbit around the Sun. This is a set of points that includes points perpendicular to the plane of the orbit and also right on the plane of the orbit.

You can find that spot toward which the Earth is travelling by looking along the ecliptic 90° "to the right" of the Sun (as seen in northern latitudes). So on March 21, the Sun is at the "First Point of Aries". If you go to the right 90°, you end up at the most southerly point of the ecliptic near the tale of Scorpius (23.5° S, 18h R.A.). The Earth is headed toward that spot on the celestial sphere on March 21. For lack of a better name, let's just call that point "X". There's also the point "anti-X" which is directly opposite X on the sky. It's the direction that the Earth is moving away from. Now to see the effect of aberration, imagine the whole sphere of stars as slightly elastic. Grab it by the waist, consisting of points that are halfway between X and anti-X --a sort of equatorial region around the celestial sphere with those two points X, anti-X acting as poles. Grab that middle region and push it slightly towards X. That's the effect of stellar aberration. It takes all of the stars in that waistband region --a zone 90° from X-- and shifts all of their positions towards X. Meanwhile stars that are closer to X or closer to anti-X are shifted by proportionately smaller amounts. The maximum shift is just about 20". Over the months during a year, that point X slides around the ecliptic, and six months later X and anti-X have swapped positions. So the net motion is +/-20 seconds of arc during the year.

"Polaris is 433 light-yr according to wiki.  that roughly implies parallax of 2/100 arc sec."

That's relevant for annual parallax which is distinctly different from annual aberration. Aberration does not depend on distance, and it's relatively large --measurable with a sextant. Annual parallax depends specifically on distance --it is, in fact, inversely proportional to distance-- and it is extremely small, completely unmeasurable with a sextant. Consider Kochab and Polaris. They're in the same part of the sky, and they both experience nearly identical +/-20" aberration (a range of nearly 0.7 minutes of arc). But Kochab is about a third of the distance of Polaris away from the Earth. The angular shift from (semi-)annual parallax is 3.26/D(ly) so given a distance of 433 ly for Polaris, its parallax is +/-0.0075" while that of Kochab is +/-0.025". The parallax shift of Kochab is about 800 times smaller than the aberration shift. The parallax shift for Polaris is about 2700 times smaller than the aberration.

In navigation we often deal with the parallax of the Moon, which is as much as a degree, and occasionally the parallaxes of the Sun, Venus, and Mars, which are all a fraction of a minute of arc. It's worth remembering that these are topocentric (earth-based) parallaxes, not annual parallaxes. For example, the 9" parallax of the Sun refers to the change in the apparent position of the Sun as an observer shifts from one side of the Earth to the other (e.g. on the equinoxes, if you moved from north to south pole, the position of the Sun would shift +/-9"). By contrast, the annual parallaxes of the stars, like that number +/-0.025" for Kochab above, refers to the annual parallax due to the Earth's motion around the Sun. It's a huge difference in scale. To convert the annual parallax of a star to a topocentric value that can be directly compared to the usual parallaxes we talk about in celestial navigation, you have to divide by about 23,500 (1AU/radiusOfEarth). This renders them all down to the micro-arcsecond range even for the closest of the navigational stars. A star that's 15 lightyears away (Altair is about that) has a topocentric parallax (a celestial navigator's HP) that's a million times smaller than the Sun's measly 9 seconds of arc. That's why stellar parallax is never mentioned in the computations of celestial navigation. Even if we get carried away with extra digits, we're a long way from worrying about stellar parallax.

Frank Reed

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