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    Re: Parallax in azimuth
    From: Robin Stuart
    Date: 2020 May 28, 08:29 -0700

    My compliments to Dave Walden for posting a number of interesting and thought provoking navigational puzzles recently. With regard to parallax in azimuth; the topic of corrections to parallax in altitude due to the ellipsoidal figure of the Earth was discussed a while ago here and here.

    For the particular problem posed
    e2                                                                  0.00669454   
    Mean equatorial radius of the Earth, a          6378.14 km
    Moon's geocentric distance, R                  395191.48 km
    Topocentric altitude, h                                  32.05569°
    Latitude, L                                                    39°
    Topocentric azimuth,                                 88.83136°   

    The geocentric horizontal parallax is, sin HP = (a/R)

    The result expressed as a series expansion in e2 and (a/R) for the geocentric altitude h' is
    h' = h + (a/R) cos h - (a/2R) e2 sin2L cos h + (a/2R) e2 sin 2L cos Z sin h
    Assuming a mean HP of 57.7' and plugging in the numbers reproduces the formula found in the Nautical Almanac. At time I didn't derive the equivalent series result for the parallax in azimuth but, having set the problem up as described in the attached document here, it's a simple extension. The result for the geocentric azimuth, Z', is
    Z' = Z + (a/2R) e2 sin 2L sin Z / cos h
    I haven't found this formula in Chauvenet (although it might be there somewhere) and it seems more convenient to use than the ones he gives.
    Plugging the numbers gives  Z' = Z + 12.858" in close agreement with Kermit.

    Robin Stuart

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