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    Re: Questions (about lunars)
    From: George Huxtable
    Date: 2002 Feb 28, 11:49 +0000

    May I welcome Arthur Pearson as a newcomer to this list, and clearly
    another searcher-after-truth about lunars with some technical expertise..
    
    He asks some interesting and well-defined questions about calculating
    lunars. Some are clearly for Bruce Stark to answer but I will have a shot
    at others, as follows-
    
    >1. Starting with the same Ma, Sa and Da, should I get the same cleared
    >distance from these calculations as from the tabular method in Bruce's
    >book? I seem to be close and I suspect the difference is from rounding
    >errors and/or differences in my formulas for refraction and
    >semi-diameter vs. his tabular values.
    
    I'll leave that one for Bruce.
    
    >2. Is there a good formula for deriving the augmented semi-diameter of
    >the moon using HP and Ma? Bruce has a table, but I don't have a formula,
    >so I am stuck with the value listed for that day in the Almanac which
    >does is not corrected for augmentation.
    
    To obtain augmented Moon semidiameter,multiply Moon semidiameter by a
    factor {1 + [(sin Moon alt)/60]}. This factor of 60 is a close-enough
    approximation to (Moon mean distance / Earth radius). This ought to give
    the same result as with the tables. Please say if it doesn't.
    
    >3. Bruce's has tables for determining GMT by comparing Do to the
    >calculated distances at the hours before and after the observation.
    >These tables appear to assume that Do changes at a constant rate during
    >any given hour. How robust is this assumption?
    
    Very robust! Maskelyne's original lunar distance tables, and those
    published for many years afterwards, required interpolation over 3-hour
    intervals, for which linear interpolation was perfectly satisfactory for
    mariners. One condition was that the observed body had to be reasonably
    close to the path of the Moon across the sky. Lunar distance measurements
    had to avoid the situation when a body was very close in the sky to the
    Moon, as the line joining it to the Moon might become quite skew to the
    Moon's path. Under those circumstances a significant non-linearity could
    result. Similar restrictions apply, but to a much lesser extent, even when
    interpolations are made over a 1-hour interval.
    
    >4. What other refinements could be made to the formulas provided in the
    >back of the Almanac to impart greater accuracy to clearing the distance?
    >I already have build in the refraction corrections for temperature and
    >pressure. Where is the greatest leverage in increasing accuracy?
    
    Arthur hasn't mentioned the "reduction of the Moon's horizontal parallax"
    to allow for the ellipsoidal shape of the Earth. This depends on the
    observer's (approximate) latitude, and is a correction which has its
    maximum value of 0.2 minutes when he reaches the poles. So if he is
    searching for the highest precision, he could take that into account by
    multiplying the Moon's HP by {1- [(sin lat) squared /300]}. The factor of
    1/300 is an approximation to the flattening of the Earth.
    
    There is indeed another effect of the Earth's ellipsoidal shape. I
    understand that it can cause a small shift in the apparent azimuth of the
    Moon, as well as the altitude effect of parallax referred to above. Because
    an azimuth shift has no effect on altitude, modern tables such as Norie's
    don't even mention it. I suspect that such an azimuthal shift might alter a
    lunar distance observation, because it is measured at a skewed angle to the
    vertical. W.M.Smart, in his "Textbook on Spherical Astronomy", touches on
    this matter, but to be honest I don't understand what he says well enough
    to make use of it. If any listmember does understand this matter, or knows
    of a fuller reference, I would like to hear about it.
    
    When adding semidiameters of Moon (and if necessary Sun) to the lunar
    distance between limbs, one assumes that their images are round. However,
    refraction veries with height, and this causes their images to appear to be
    slightly oval, less so as they get higher in the sky. When searching for
    ultimate accuracy, that may be a worthwhile correction to apply. However, I
    haven't come across a simple way of doing so.
    
    =====================
    
    Some additional comments from George.
    
    Clearly, Arthur is a dab-hand with his pocket computer, and I wonder if he
    has considered taking the next step of computing for himself the positions
    of Moon, Sun, planets, and stars, from the formulae that Meeus gives in
    "Astronomical Algorithms". As a by-product he will obtain semidiameters and
    horizontal parallaxes too. The predictions will remain valid for many years
    longer than he will, and he won't need to buy another Nautical Almanac ever
    again.
    
    I haven't yet found a copy of Bruce Stark's tables, so I would like to
    learn what accuracy the angles are tabulated to, and what accuracy is
    claimed for the whole process.
    
    I wonder if any list members have tried the simple "Letcher method" for
    correcting a lunar observation (as described in my "About Lunars-part 3")
    and if so, whether or not they have found it successful, easy, and
    accurate.. Feedback, please.
    
    George Huxtable.
    
    ------------------------------
    
    george---.u-net.com
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.
    ------------------------------
    
    
    

       
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