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    Re: Lunars
    From: Peter Smith
    Date: 1998 Jul 01, 2:07 PM

    On Wed, 1 Jul 1998 12:15:27 GMT, Rick Emerson  asked:
    > With the anniversary of Capt. Slocum's trip at hand and my mention of
    > older editions of Bowditch, the question of how "lunars" are done has
    > come around.  Aside from a vague notion that the system has something
    > to do with tables predicting the moon's position relative to key
    > stars, I'm clueless about this technique.  Does anyone have a modern
    > source explaining this system or, pushing my luck, a source for
    > tables, trusty alarm clocks[g], etc.?
    Since the Moon moves with respect to the Sun and stars, by measuring
    the distance between it and a body near its path, one can interpolate
    the time at which that distance was current from a table of lunar
    distances -- although such tables haven't been published since the
    beginning of this century (US Nautical Almanac dropped them in 1912).
    Bowditch editions printed before before 1914 give several methods of
    solving this.  Each is based on three simultaneous observations:
    lunar altitude, a second body's altitude, and the angular distance
    from the moon to the second body.  While the key to the solution is
    the moon-body distance, the altitudes are required to compute the
    refraction and parallax corrections necessary to correct the observed
    distance from the moon's limb to the second body's limb to an actual
    distance from the moon's center to the second body's center.
    When only a single observer or instrument is available, the navigator
    must approximate the altitudes at the time of the distance sight by
    taking timed altitudes of both bodies before and after the distance
    sight.  The altitudes at the time of the distance sight are then
    interpolated by proportional logs.  Clearly, this will only be accuate
    when both the moon and the second body are well off the meridian and
    their altitudes are changing fairly linearly.  However, since the
    corrections for refraction and parrallax do not change too rapidly at
    moderate altitudes, the errors introduced by imprecition here shouldn't
    be large.
    Once the true lunar distance is calculated, the navigator enters the
    almanac.  In Bowditch's day, the Nautical Almanac tabulated the true
    distance from the moon to the sun, four major planets, and nine stars
    for every three hours.  One would use the actual distance observed
    to interpolate between tabulations and approximate the Greenwich time
    of the observation.  Given the non-linearities in the motions of the
    two bodies, this was not precise. Ol' Nat has this to say for the
    expected accuracy:
            As the moon moves in her orbit about 1' in 2m of
            time, it follows that if her angular distance can be
            ascertained from the sun or star within 1', the time
            at Greenwich will be known within 2 minutes, and the
            longitude within 30 miles.
    As for the details of the calculation behind the now-extinct tables,
    I can send you (back-channel) a posting by once (and perhaps, still)
    listmember Jeff Gottfred, who's grasp of theory and wealth of
    experience far exceeds mine.
    Peter Smith -- psmith@wellspring.us.dg.com
    Data General Corp., Westboro, Massachusetts  (for whom I do not speak)
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