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    Re: Star - Star Observations
    From: Frank Reed
    Date: 2010 Mar 14, 06:24 -0700

    To add one more option, here's another variation on a short formula for calculating star-star refraction:

    y = sin(h1)/sin(h2)
    x = (y - 1/y)/2
    r = 1.9*(x - cos(D))/sin(D)
    D' = D + r

    where D is the measured distance between the stars, D' is the corrected distance, and h1 and h2 are the measured or calculated altitudes of the stars. The result is the refraction in minutes of arc under the assumption that refraction is given by 0.95*tan(z) where z is zenith distance, which implies the altitudes should be above 15 degrees. This is the basis for the neat graphical solution in Letcher. It is also very easy to derive from the "corner cosines" approach (a.k.a. series expansion of the standard spherical triangle solution).

    The refraction can be corrected for temperature, pressure, and altitude above sea level before adding r to D just like any refraction value (details upon request). Various simple rules can be derived from this set of equations like the ones I've posted previously. For example, if h1=h2, then y=1 so x=0. Then r=1.9*tan(D). For D<=90, this is within a tenth of a minute of arc of the rule "a tenth of a minute or arc refraction for every five degrees of distance".

    After working this very short calculation, D' is compared with D0 which is the true distance between the stars on that given date. The calculation of D0 has to be done for every few weeks out of the year to account for aberration --or star pairs can be selected to minimize aberration. Also, unless stars with high proper motions are avoided, any tables of true distances should be updated every few years to deal with those fast-moving stars. One extreme case is Arcturus. Its proper motion is around 2.3 seconds of arc annually. Star-star distances involving Arcturus can change by 0.1 minutes of arc in just three years. None of this matters, of course, if you set it up in software (without printing/publishing tabulated results). Then you can just click a button and get the true distances for that moment of time to whatever accuracy your heart desires. In the software case, you can even include the planets, which, thanks to their distinct disks, are easier to align in a sextant.


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