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    Re: Star-star distances for arc error
    From: Frank Reed
    Date: 2009 Jun 19, 19:38 -0700

    Douglas you wrote:
    "I've done this. It provides a method of checking a sextant but is not easy 
    with accuracy unless you have some clamping system to hold the sextant still 
    and be able to rotate the sextant into alignment."
    With a little practice, it's not hard, but I agree that you can find yourself 
    in some awkward positions. It's good practice for lunars, too. Not that 
    lunars are for everybody, but some people get a kick out of shooting them.
    And then you wrote:
    "If you use horizontal angles with stars approximately of the same declination 
    the refraction element is negligable if they are above say 30 degrees."
    No, that just isn't right. First, you say 'same declination'. I assume that 
    this was just a "typo" and you meant altitudes. No problem. Even so, then 
    your claim would be that refraction is negligible for stars at the same 
    altitude (above 30 degrees). This is a "navigation urban legend". It isn't 
    true, but it has even been published in some recent editions of Bowditch and 
    other places where the authors really should know better. The angular 
    distances between stars at the same altitude are indeed affected by 
    Consider for example, the case of two stars at 45 degrees altitude on opposite 
    sides of the sky. Let's assume that the angle between them is exactly 90 
    degrees in the absence of refraction. Now, refraction lifts BOTH stars by 
    about 1'. They're pushed towards each other, so the observed angular distance 
    would be just about 89d 58' which is a difference that would be measurable 
    even with a fairly bad sextant.
    Now consider a case of two pairs of stars: two are at 45 degrees altitude 
    separated by 20 degrees more or less horizontally. Another pair is located 
    with a more or less vertical separation, one at 45 degrees altitude, the 
    other at 65 degrees altitude. What I was saying my previous message is that 
    you can calculate (clear) the effect of refraction in these cases, and indeed 
    in any case where both objects are above 45 degrees, simply by multiplying 
    the observed distance by 1.00034. So in these pairs, one where the stars are 
    horizontally placed and the other in which they're vertically placed, the 
    effect of refraction is the same. Horizontal relative orientation of the two 
    stars makes no difference.
    The idea that horizontal placement of the stars eliminates the effect of 
    reftraction is most likely due to a mistaken extrapolation from one extreme 
    case (which is never observable in practice but worth understanding in 
    principle). Imagine two stars right on the horizon (can't see stars that low, 
    but let's ignore that for the moment). The effect of refraction is to lift 
    the stars towards the zenith. There is no component in azimuth. Now near the 
    horizon the lines of constant azimuth show no convergence so if stars very 
    close to the horizon are shifted towards the zenith, the angle between them 
    will hardly change at all. So here we have a case where the distances between 
    stars at the same altitude (zero) are unaffected by refraction. It seems 
    likely that this case was mistakenly extrapolated decades ago leading to the 
    mistaken notion that the angles between stars at ANY altitude are unaffected 
    if the stars are at the same altitude. And again, this just isn't true.
    "Most sextants only read with accuracy to about a half of a minute of arc anyway."
    Accuracy in sextants depends on many factors, but I have seen plenty of folks 
    get significantly better accuracy than that with a decent metal sextant (so 
    long as it's equipped with a medium power telescope, e.g. 7x). And I myself 
    generally get 0.25' accuracy in lunars on individual sights or 0.1' when four 
    are averaged. But none of this will work unless you've checked for arc error 
    (and that's the motivation for measuring star-star distances for some 
    people). Arc error is generally a fixed error and therefore it is just as 
    easy to correct for it as it is to correct for index error.
    And you concluded:
    "Star distances can be calculated from declinations/ RA (or hour angles) with simple spherical trig."
    Of course. And I probably should have outlined the exact calculation for those 
    who like setting things up in spreadsheets and in their own software, so if 
    anybody is uncertain on those details, let me know. But neat little rules 
    like the one I described earlier have another interesting use. Knowing that 
    the distances between all stars change by the same percentage amount when 
    they're above 45 degrees gives us useful planning information. As long as 
    you're sure the altitudes above 45, you don't have to record any other 
    information. Local time doesn't matter. The actual altitudes of the stars 
    don't matter. You can just measure the distance at your leisure and the 
    clearing process will not care about the details. Just multiply by 1.00034 
    and you're done.
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