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    Re: Star-star distances for arc error
    From: Frank Reed
    Date: 2009 Jun 26, 21:05 -0700

    George H, you wrote:
    "but one of its serious weaknesses is in that star-star distance table, appendix G."
    Well, correct me if I'm wrong, but couldn't those weaknesses in his appendix G 
    be addressed very simply? You're quite correct to point out the errors, of 
    course. But calling them "serious weaknesses" when there are simple solutions 
    strikes me as a bit of an exaggeration. The two issues are:
    Problem #1: "the star-star distances would only be correct during
    the rising part of path of star 1, to the Eastwards". 
    OK. So can't he just include a line in the instructions (and perhaps at the 
    bottom of each page) saying exactly that? 
    Problem #2: "No account has been taken of annual aberration, which varies 
    cyclically over the year, and in the worst case can amount to errors in 
    star-star angle amounting to 40 arc-seconds". 
    OK, but the error should be no more than 20 arcseconds if the mean distance 
    has been used (I haven't checked to see whether he did that or not). In any 
    case, to correct for that, all he needs is a short table, maybe on the right 
    margin, giving the change in each distance for each month of the year. 
    Something like: Jan -0.3, Feb -0.2, Mar -0.2, Apr -0.1, etc.
    Those are simple solutions to relatively minor problems. It's a shame that 
    they weren't caught in proofreading but it would take no more than an index 
    card sized "errata" sheet to cover those particular items (I do understand 
    that there are some other small errors, as in any book published in a finite 
    amount of time).
    And you wrote:
    "So, anyone wishing to make high-precision checks on a sextant using
    star-star distances would be well advised to calculate, and correct for, the
    refractions from first principles"
    Yes, of course, that's one solution. Or they could just write in by hand the 
    extremely brief monthly correction table (for each pair) as noted above for 
    Problem #2 and take note of the special instructions for dealing with Problem 
    And George, you concluded:
    "rather than follow Frank's recommendation of Table G in Karl's book. I wonder 
    if Frank has ever tried it?"
    I'm sorry my post confused you, George. If you re-read it, I did not issue a 
    "recommendation" to USE those tables. I merely pointed out there existence in 
    a popular new book. Douglas Denny had asserted repeatedly that using 
    star-star distances to measure arc error was not possible. You read those 
    posts, too. I suggested that Douglas might want to "take a peek" at those 
    tables in John Karl's book. My point was to persuade him, just maybe, that he 
    needed to re-think his certainty that such observations were worthless. Do 
    you understand now??
    So how should you clear star-star distances? One way would be to make those 
    small corrections to the tables in John Karl's book as noted above. But 
    probably the easiest way for a modern observer is with a spreadsheet or a 
    purpose-built piece of software. You calculate refraction for each star as 
    exactly as possible based on its altitude (since this is a "backyard" 
    observation the altitudes would presumably be calculated from the observer's 
    position and the GMT and date of the sight) and then you do the usual 
    spherical trig to get the correct apparent distance. Compare that with the 
    observed distance and you're done. No problem. BUT here's a fun trick (that 
    started this thread): for stars above 45 degrees altitude (about one-third of 
    the usable distances at any moment), the refraction in the star-star 
    distances does not change. The refraction is directly proportional to the 
    distance. All of the constellations above 45 degrees shrink proportionately 
    by a factor of just about 1.00034. Even if you don't do any calculations this 
    way, this refractional morsel is also great for planning a round of sights 
    because you know in advance that you don't have to record the time since the 
    refracted distances won't change by more than about 0.1 minutes of arc (until 
    one of the stars is below 45 degrees).
    Navigation List archive: www.fer3.com/arc
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