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    Re: The Navigation Lamp Post
    From: Frank Reed
    Date: 2013 Feb 16, 16:44 -0800

    Brad, you wrote:
    " That correction is far less than 1 arc second. This can be as small as zero when the object is at the zenith and as great as 1/10000 arcsecond when the altitude of the object is zero."

    Right. Nothing to worry about! Of course, we do have to worry about it with nearby objects. The parallax in altitude correction for the Sun is 9 seconds of arc. For any other object, it's inversely proportional to the distance in AUs. So for Saturn, since its distance from the Earth is between 8.5 (when it's best placed for observation) and 10.5 AUs (when it's impossible to see), the parallax in altitude is typically 1 second of arc. That's the maximum at the horizon. Multiply by the cosine of the altitude when it's higher in the sky. At the other extreme, our Moon is about 1/400 of an AU away from us, so its parallax in altitude is nearly 9*400 or 3600 seconds of arc, a full degree, at the horizon (its actual mean value is a few percent less than that). Even closer objects have still higher parallaxes. For example, if you wanted to observe the little asteroid "2012 DA14" yesterday, parallax was a huge issue. Even when it wasn't at its perigee, it was ten times closer than the Moon implying a parallax of as much as ten degrees depending on its altitude in the sky. I feel sorry for poor "DA14". Its fifteen minutes of fame were stolen by that nameless interloper in the skies over Russia. Oh, just a coincidence. A freaking AMAZING coincidence!

    You wrote:
    "In the parallax observation, I was attempting to illustrate how in the 1800's, the distance to stars was attempted by measuring the shift of foreground objects against the background objects. So Alpha Centauri, the angle to the star shifts by 0.88 arcseconds over a 6 month period. This is why the measurement was unsuccessful and we were forced to calculate the distances to stars by other means. "

    Hmmm. But those observations WERE successful. It was in 1838 (175 years ago, back when navigators were still shooting lunars on a regular basis on American vessels), that Bessel observed and published the first stellar parallax. It was for the double star "61 Cygni" easily observed in the northern hemisphere. He selected it because it has a high proper motion so it was likely that this was due to its proximity. Its annual parallax is around 0.3 seconds of arc.

    You wrote:
    "With todays orbital telescopes, the pointing precision is far greater than instruments of the 1800's, consequently, I believe we could measure the parallax to Alpha Centauri (Rigil Kentaurus)"

    Sure can. This used to be the work of astrometric observatories, and when I was a physics and astronomy student at Wesleyan University, I got to try my hand at measuring astrometric plates for parallax analysis (very dull). But that is now ancient history. Amazingly, we can make comparable observations today with backyard telescopes equipped with high-end digital cameras at a total cost of less than $2500. I remember being amazed ten years ago when I read an article, probably in Sky & Telescope, where the annual parallax wobble of Barnard's Star was plainly visible in data from a backyard setup. As for those orbital telescopes, the gold standard in recent astronomy was the ESA (European Space Agency) Hipparcos mission which, after the second round of analysis, managed to observe the parallaxes of a few hundred thousand stars to an accuracy of 1/4000 arc seconds (a quarter of a milli-arcsecond). But that ain't nothin! In October of this year, ESA's Gaia mission will launch and over some years, it will measure the parallaxes of a BILLION stars to an accuracy of about one micro-arcsecond. At no extra charge, it will also detect tens of thousands of Near Earth asteroids, and it would have been able to detect the very small asteroid that exploded over Russia yesterday. These automated telescopes, like Hipparcos and Gaia, and also the very successful NASA planet-hunting Kepler mission, are revolutionizing astronomy as they return vast quantities of data on stars and also minor Solar System bodies that are not even their primary targets. More variable star light curves have been returned by the Kepler mission in its short period of operation than in the entire previous history of astronomy!

    Back to navigation issues, there is an aspect of measuring parallax that is relevant to navigation. In celestial navigation, we have the very unusual circumstance that a 1 minute of arc error in ordinary altitude observations always yields a 1 nautical mile error in the line of position. Of course, the numerical value is due to the definition of the nautical mile. But the deeper point is that it depends on no other factors (until you get to exotic details like very low altitude refraction). For any observation at any altitude of any celestial body, 1' error in the observation shifts the LOP by 1 n.m. It doesn't have to be this way! If the Earth were much more oblate, then this relationship would vary with latitude. Or if we were measuring the altitudes of satellites, much closer to us, the impact of an error in angular observation would depend on the satellites altitudes in the sky. The error in the distances to stars that results from, say, a 1 milli-arcsecond error in angle is very similar to the distance from the "lamp post" that you mentioned in your first post. Suppose we try to get a circle of position from an actual lamp post or better yet for a nautical example, a lighthouse, of known height. If we make a 1 minute of arc error in measuring the apparent angular height of a lighthouse, how large is the error in the resulting circle of position? Clearly if the lighthouse is 30' in angular size, it's not much. But if it's 3' in angular height, then the difference is quite large. Notice that this is basically identical to the problem of determining the error in the distance to a star when there is a small error in its observed parallax.


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