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    From: Arthur Pearson
    Date: 2002 Feb 27, 23:06 -0500

    I am also a newcomer to this mailing list which I found in my search for
    information about lunars. I have Bruce Stark's book and have used it to
    reduce a small number of lunar distances. I have also developed a
    spreadsheet that runs on my palm pilot that clears the distance and
    determines GMT from comparing distances calculated from Nautical Almanac
    data for the hours before and hour after the observation. I have some
    specific questions regarding the method I am using as they compare to
    the method in Bruce's book, and some questions seeking guidance on
    building more accurate formulas into my spreadsheet.
    I will briefly define my terms, then ask my questions. I will use labels
    similar to those in Bruce's book uses with some additions:
    Ds = Lunar distance directly off the sextant
    Da = Ds corrected for Index Error and augmented semi-diameter
    Do = Da corrected for parallax and refraction, which is the cleared
    lunar distance.
    Ms = Altitude of the moon off the sextant
    Ma = Ms corrected for IC, dip and semi-diameter
    Mo = Mo corrected for parallax and refraction
    Ss = Altitude of the comparing body
    Sa = Ss corrected for IC, dip and semi-diameter
    So = Sa corrected for parallax and refraction
    Z = Angle at the observer's zenith between the intersecting hour circles
    of the two observed bodies
    Previous emails on this site described the problem as solving two
    spherical triangles that had a common angle at Z. In the first triangle,
    we know the value of the three legs (Da, 90-Sa, and 90-Ma) and solve for
    the angle Z. Assuming Z is the same for the second triangle, we know the
    value of Z and of two legs (90-So and 90-Mo) and solve for the third leg
    Do. This clears the distance.
    The previous emails listed formulas using sines and cosines by which Z
    could be determined from Da, Sa and Ma and subsequently how Do could be
    determined from Z, Do and So. I have built these into the spreadsheet
    along with formulas from the back of the Nautical Almanac which I use to
    correct the altitudes for dip, semi-diameter, parallax and refraction in
    their turn. I use the tabular values in the Almanac as inputs for
    semi-diameter of the moon and sun, HP of the moon, and GHA and
    Declination of the bodies when calculating the comparing distances for
    the hours before and after observation. HP of the sun and Venus are also
    given in the back of the Almanac and can be built into a lookup table
    for spreadsheet calculations.
    My questions are 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.
    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.
    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?
    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?
    I would be delighted to hear from Bruce anyone else who understands his
    methodology. I should add a note of thanks to Bruce for the book that
    launched me into this realm, and to the past contributors of this
    discussion group for the emails and comments that helped me better
    understand these methods.
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Bruce Stark
    Sent: Wednesday, February 27, 2002 12:49 PM
    Subject: Re: Lunar by altitudes
    That's me, Dan. Thanks for the welcome.
    A couple of years ago the Navigation Foundation published a paper on the
    problems, ideas, and equations that led to my method. If you are a
    member of
    the Foundation you can get get back issues. It's in issue #60 of their
    Navigator's Newsletter.
    At present I've been using nearly all my free time to show how to work
    and Clark's astronomical observations. It's an opportunity to
    reintroduce the
    nautical astronomy Bowditch knew. But I'd be glad to answer questions
    the Tables, as long as it doesn't take too much time or mental effort.

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