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    Star-star distances for arc error
    From: Frank Reed
    Date: 2009 Jun 18, 22:21 -0700

    On a number of occasions, we've discussed using star-star distances as a means 
    of testing a sextant's arc error. The idea is that you measure the angle 
    between two stars and then compare it with known true angles. Any difference 
    is some combination of index error, arc error, shade error, personal error, 
    etc. 
    
    The true angles between stars are nearly fixed for many stars, especially 
    those at greater distances since they tend to have low proper motions, but 
    you do have to be careful to list those angles at monthly intervals since 
    aberration changes the angles during the year. A table of them can be 
    computed once and used for many years.
    
    The observed distance between stars has to be cleared or corrected for 
    refraction. There is a straight-forward mathematical procedure for this which 
    is basically the same as clearing a lunar distance without the large 
    corrections for parallax. There are also simple graphical means for doing 
    this. But recently it occurred to me that the correction for refraction can 
    be handled very nearly with a trivially simple calculation for all measured 
    distances when the stars are above 45 degrees altitude. 
    
    Refraction shrinks all of the constellations under all observing conditions by 
    lifting the stars towards the zenith. But for altitudes above 45 degrees, the 
    "shrinking" is nearly uniform. That is, it is nearly proportional to the 
    zenith distance (since the refraction is proportional to tan(z.d.)). And 
    therefore all of the apparent distances between all of the stars are reduced 
    by the same proportional amount. Thus to clear the distances between stars 
    (when both stars are above 45 degrees altitude), all you have to do is 
    multiply the measured observed distance by 1.00034. By my calculations, which 
    somebody should check, the results never differ by more than 7 seconds of arc 
    and usually less than 3 seconds of arc from the exact calculation which is 
    pretty good for such a simple trick.
    
    Another way of looking at this is to note that the apparent distances between 
    stars when they're both above 45 degrees does not change as they cross the 
    sky even though the refraction for each is changing continually. One could 
    publish a table listing a couple of dozen convenient pairs of bright stars at 
    various angular distances. The distances could be calculated with refraction 
    included for each month of the year (to deal with aberration), and there 
    would be no need for any further clearing. The observed distances could be 
    compared directly with the tabulated. For added utility, one could publish 
    three tables: one for standard pressure and temperature and one each for 
    pressure/temperature 10% above and 10% below the standard values (use factors 
    of 1+0.00034*P/T where P and T are given as ratios from their standard 
    values). That would cover nearly all observing conditions.
    
    I suppose I should point out that for both stars to be above 45 degrees, the 
    measured distance would necessarily be less than 90 degrees so this trick 
    would only work with distances below that.
    
    -FER
    
    
    
    
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